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Sage constants & tests codegen (#101)

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* Implement a Sage codegenerator for frobenius constants

* Sage codegen for pairings

* Autogen of endomorphism acceleration constants

* The autogen fixed a copy-paste bug in lattice decomposition. We can use conditional negation now and save an add+dbl in scalar mul

* small fixes

* sage code for square root bls12-377 is not old

* readme updates

* Provide test suggestions for derive_frobenius

* indentation + add equation form to sage

* Sage test vector generator

* Use the json vectors
- includes type system workaround: generic sandwich https://github.com/nim-lang/Nim/issues/11225
- converting NimNode to typedesc: https://github.com/nim-lang/Nim/issues/6785

* Delete old sage code

* Install nim-serialization and nim-json-serialization in CI

* CI nimble install force yes
This commit is contained in:
Mamy Ratsimbazafy 2020-10-10 16:19:23 +02:00 committed by GitHub
parent 71bb4c799a
commit a2f46f77b7
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56 changed files with 6077 additions and 5324 deletions
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2
.github/workflows/ci.yml vendored
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@ -201,7 +201,7 @@ jobs:
shell: bash
run: |
nimble refresh
nimble install -y gmp stew
nimble install -y gmp stew serialization json_serialization
- name: Run Constantine tests (with Assembler & with GMP)
if: (runner.os == 'Linux' || runner.os == 'macOS') && matrix.target.cpu == 'amd64'
shell: bash

2
.travis.yml
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@ -123,7 +123,7 @@ before_script:
- export PATH="$PWD/nim-${CHANNEL}/bin${PATH:+:$PATH}"
script:
- nimble refresh
- nimble install gmp stew
- nimble install -y gmp stew serialization json_serialization
# Installing Clang9.0 or later is a pain in Travis
# for inline assembly "flag output constraint"
# Also MacOS build is timing out with 2 series of tests.

13
README.md
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@ -1,4 +1,4 @@
# Constantine - Constant Time Pairing-Based & Elliptic Curve Cryptography
# Constantine - Fast, compact, hardened Pairing-Based Cryptography
[![License: Apache](https://img.shields.io/badge/License-Apache%202.0-blue.svg)](https://opensource.org/licenses/Apache-2.0)
[![License: MIT](https://img.shields.io/badge/License-MIT-blue.svg)](https://opensource.org/licenses/MIT)
@ -7,7 +7,10 @@
[![Build Status: Travis](https://img.shields.io/travis/com/mratsim/constantine/master?label=Travis%20%28Linux%20x86_64%2FARM64%2FPowerPC64,%20MacOS%20x86_64%29)](https://travis-ci.com/mratsim/constantine)\
[![Build Status: Azure](https://img.shields.io/azure-devops/build/numforge/07a2a7a5-995a-45d3-acd5-f5456fe7b04d/4?label=Azure%20%28Linux%2032%2F64-bit%2C%20Windows%2032%2F64-bit%2C%20MacOS%2064-bit%29)](https://dev.azure.com/numforge/Constantine/_build?definitionId=4&branchName=master)
This library provides [constant-time](https://en.wikipedia.org/wiki/Side-channel_attack) implementation of elliptic curve cryptography
> “A cryptographic system should be secure even if everything about the system, except the key, is public knowledge.”\
> — Auguste Kerckhoffs
This library provides [constant-time](https://en.wikipedia.org/wiki/Timing_attack) implementation of elliptic curve cryptography
with a particular focus on pairing-based cryptography.
The implementations are accompanied with SAGE code used as reference implementation and test vectors generators before writing highly optimized routines implemented in the [Nim language](https://nim-lang.org/)
@ -16,10 +19,10 @@ The implementations are accompanied with SAGE code used as reference implementat
## Target audience
The library aims to be a portable, compact and hardened library for elliptic curve cryptography needs, in particular for blockchain protocols and zero-knowledge proofs system.
The library aims to be a fast, compact and hardened library for elliptic curve cryptography needs, in particular for blockchain protocols and zero-knowledge proofs system.
The library focuses on following properties:
- constant-time (not leaking secret data via side-channels)
- constant-time (not leaking secret data via [side-channels](https://en.wikipedia.org/wiki/Side-channel_attack))
- performance
- generated code size, datatype size and stack usage
@ -51,7 +54,7 @@ The Nim language offers the following benefits for cryptography:
- Obscure embedded devices with proprietary C compilers can be targeted.
- WASM can be targeted.
- Performance reachable in C is reachable in Nim, easily.
- Rich type system: generics, dependent types, mutability-tracking and side-effect analysis, borrow-checking, distinct types (Miles != Meters, SecretBool != bool SecretWord != uint64).
- Rich type system: generics, dependent types, mutability-tracking and side-effect analysis, borrow-checking, compiler enforced distinct types (Miles != Meters, SecretBool != bool and SecretWord != uint64).
- Compile-time evaluation, including parsing hex string, converting them to BigInt or Finite Field elements and doing bigint operations.
- Assembly support either inline or ``__attribute__((naked))`` or a simple `{.compile: "myasm.S".}` away
- No GC if no GC-ed types are used (automatic memory management is set at the type level and optimized for latency/soft-realtime by default and can be totally deactivated).

4
azure-pipelines.yml
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@ -29,7 +29,7 @@ strategy:
VM: 'ubuntu-18.04'
UCPU: amd64
CHANNEL: devel
WEAVE_TEST_LANG: cpp
TEST_LANG: cpp
# Deactivated for now, this is cross-compilation and tricky to get right
# Linux_devel_32bit:
@ -219,7 +219,7 @@ steps:
- bash: |
echo "PATH=${PATH}"
nimble refresh
nimble install gmp stew
nimble install -y gmp stew serialization json_serialization
displayName: 'Installing package and testing dependencies'
- bash: |

8
benchmarks/bench_elliptic_template.nim
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@ -175,7 +175,7 @@ proc scalarMulGenericBench*(T: typedesc, window: static int, iters: int) =
let exponent = rng.random_unsafe(BigInt[bits])
bench("EC ScalarMul Generic " & G1_or_G2 & " (window = " & $window & ", scratchsize = " & $(1 shl window) & ')', T, iters):
bench("EC ScalarMul " & $bits & "-bit Generic " & G1_or_G2 & " (window = " & $window & ')', T, iters):
r = P
r.scalarMulGeneric(exponent, window)
@ -188,7 +188,7 @@ proc scalarMulEndo*(T: typedesc, iters: int) =
let exponent = rng.random_unsafe(BigInt[bits])
bench("EC ScalarMul " & G1_or_G2 & " (endomorphism accelerated)", T, iters):
bench("EC ScalarMul " & $bits & "-bit " & G1_or_G2 & " (endomorphism accelerated)", T, iters):
r = P
r.scalarMulEndo(exponent)
@ -201,7 +201,7 @@ proc scalarMulEndoWindow*(T: typedesc, iters: int) =
let exponent = rng.random_unsafe(BigInt[bits])
bench("EC ScalarMul Window-2 " & G1_or_G2 & " (endomorphism accelerated)", T, iters):
bench("EC ScalarMul " & $bits & "-bit " & G1_or_G2 & " (window-2, endomorphism accelerated)", T, iters):
r = P
when T.F is Fp:
r.scalarMulGLV_m2w2(exponent)
@ -217,6 +217,6 @@ proc scalarMulUnsafeDoubleAddBench*(T: typedesc, iters: int) =
let exponent = rng.random_unsafe(BigInt[bits])
bench("EC ScalarMul " & G1_or_G2 & " (unsafe reference DoubleAdd)", T, iters):
bench("EC ScalarMul " & $bits & "-bit " & G1_or_G2 & " (unsafe reference DoubleAdd)", T, iters):
r = P
r.unsafe_ECmul_double_add(exponent)

2
constantine.nimble
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@ -114,7 +114,7 @@ const testDesc: seq[tuple[path: string, useGMP: bool]] = @[
# Elliptic curve arithmetic vs Sagemath
("tests/t_ec_frobenius.nim", false),
("tests/t_ec_sage_bn254.nim", false),
("tests/t_ec_sage_bn254_snarks.nim", false),
("tests/t_ec_sage_bls12_377.nim", false),
("tests/t_ec_sage_bls12_381.nim", false),
# Edge cases highlighted by past bugs

64
constantine/curves/bls12_377_frobenius.nim
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@ -15,11 +15,11 @@ import
# -----------------------------------------------------------------
# c = (SNR^((p-1)/6)^coef).
# Then for frobenius(2): c * conjugate(c)
# And for frobenius(3): c² * conjugate(c)
# Then for frobenius(2): c * conjugate(c)
# And for frobenius(3): c² * conjugate(c)
const BLS12_377_FrobeniusMapCoefficients* = [
# frobenius(1)
[Fp2[BLS12_377].fromHex( # SNR^((p-1)/6)^0
# frobenius(1) -----------------------
[Fp2[BLS12_377].fromHex( # SNR^((p-1)/6)^0
"0x1",
"0x0"
),
@ -27,93 +27,89 @@ const BLS12_377_FrobeniusMapCoefficients* = [
"0x9a9975399c019633c1e30682567f915c8a45e0f94ebc8ec681bf34a3aa559db57668e558eb0188e938a9d1104f2031",
"0x0"
),
Fp2[BLS12_377].fromHex( # SNR^((p-1)/6)^2 = SNR^((p-1)/3)
Fp2[BLS12_377].fromHex( # SNR^((p-1)/6)^2
"0x9b3af05dd14f6ec619aaf7d34594aabc5ed1347970dec00452217cc900000008508c00000000002",
"0x0"
),
Fp2[BLS12_377].fromHex( # SNR^((p-1)/6)^3 = SNR^((p-1)/2)
Fp2[BLS12_377].fromHex( # SNR^((p-1)/6)^3
"0x1680a40796537cac0c534db1a79beb1400398f50ad1dec1bce649cf436b0f6299588459bff27d8e6e76d5ecf1391c63",
"0x0"
),
Fp2[BLS12_377].fromHex( # SNR^((p-1)/6)^4 = SNR^(2(p-1)/3)
Fp2[BLS12_377].fromHex( # SNR^((p-1)/6)^4
"0x9b3af05dd14f6ec619aaf7d34594aabc5ed1347970dec00452217cc900000008508c00000000001",
"0x0"
),
Fp2[BLS12_377].fromHex( # SNR^((p-1)/6)^5
Fp2[BLS12_377].fromHex( # SNR^((p-1)/6)^5
"0xcd70cb3fc936348d0351d498233f1fe379531411832232f6648a9a9fc0b9c4e3e21b7467077c05853e2c1be0e9fc32",
"0x0"
)],
# frobenius(2)
[Fp2[BLS12_377].fromHex( # norm(SNR)^((p-1)/6)^1
# frobenius(2) -----------------------
[Fp2[BLS12_377].fromHex( # norm(SNR)^((p-1)/6)^0
"0x1",
"0x0"
),
Fp2[BLS12_377].fromHex( # norm(SNR)^((p-1)/6)^2
Fp2[BLS12_377].fromHex( # norm(SNR)^((p-1)/6)^1
"0x9b3af05dd14f6ec619aaf7d34594aabc5ed1347970dec00452217cc900000008508c00000000002",
"0x0"
),
Fp2[BLS12_377].fromHex(
Fp2[BLS12_377].fromHex( # norm(SNR)^((p-1)/6)^2
"0x9b3af05dd14f6ec619aaf7d34594aabc5ed1347970dec00452217cc900000008508c00000000001",
"0x0"
),
Fp2[BLS12_377].fromHex(
Fp2[BLS12_377].fromHex( # norm(SNR)^((p-1)/6)^3
"0x1ae3a4617c510eac63b05c06ca1493b1a22d9f300f5138f1ef3622fba094800170b5d44300000008508c00000000000",
"0x0"
),
Fp2[BLS12_377].fromHex(
Fp2[BLS12_377].fromHex( # norm(SNR)^((p-1)/6)^4
"0x1ae3a4617c510eabc8756ba8f8c524eb8882a75cc9bc8e359064ee822fb5bffd1e945779fffffffffffffffffffffff",
"0x0"
),
Fp2[BLS12_377].fromHex(
Fp2[BLS12_377].fromHex( # norm(SNR)^((p-1)/6)^5
"0x1ae3a4617c510eabc8756ba8f8c524eb8882a75cc9bc8e359064ee822fb5bffd1e94577a00000000000000000000000",
"0x0"
)],
# frobenius(3)
[Fp2[BLS12_377].fromHex(
# frobenius(3) -----------------------
[Fp2[BLS12_377].fromHex( # (SNR²)^((p-1)/6)^0
"0x1",
"0x0"
),
Fp2[BLS12_377].fromHex(
Fp2[BLS12_377].fromHex( # (SNR²)^((p-1)/6)^1
"0x1680a40796537cac0c534db1a79beb1400398f50ad1dec1bce649cf436b0f6299588459bff27d8e6e76d5ecf1391c63",
"0x0"
),
Fp2[BLS12_377].fromHex(
Fp2[BLS12_377].fromHex( # (SNR²)^((p-1)/6)^2
"0x1ae3a4617c510eac63b05c06ca1493b1a22d9f300f5138f1ef3622fba094800170b5d44300000008508c00000000000",
"0x0"
),
Fp2[BLS12_377].fromHex(
Fp2[BLS12_377].fromHex( # (SNR²)^((p-1)/6)^3
"0x4630059e5fd9200575d0e552278a89da1f40fdf62334cd620d1860769e389d7db2d8ea700d82721691ea130ec6e39e",
"0x0"
),
Fp2[BLS12_377].fromHex(
Fp2[BLS12_377].fromHex( # (SNR²)^((p-1)/6)^4
"0x1",
"0x0"
),
Fp2[BLS12_377].fromHex(
Fp2[BLS12_377].fromHex( # (SNR²)^((p-1)/6)^5
"0x1680a40796537cac0c534db1a79beb1400398f50ad1dec1bce649cf436b0f6299588459bff27d8e6e76d5ecf1391c63",
"0x0"
)]]
# ψ (Psi) - Untwist-Frobenius-Twist Endomorphisms on twisted curves
# -----------------------------------------------------------------
# BLS12_377 is a D-Twist: psi1_coef1 = SNR^((p-1)/6)
# BLS12_377 is a D-Twist: SNR^((p-1)/6)
const BLS12_377_FrobeniusPsi_psi1_coef1* = Fp2[BLS12_377].fromHex(
"0x9a9975399c019633c1e30682567f915c8a45e0f94ebc8ec681bf34a3aa559db57668e558eb0188e938a9d1104f2031",
"0x0"
)
# SNR^((p-1)/3)
const BLS12_377_FrobeniusPsi_psi1_coef2* = Fp2[BLS12_377].fromHex(
# SNR^((p-1)/3)
const BLS12_377_FrobeniusPsi_psi1_coef2* = Fp2[BLS12_377].fromHex(
"0x9b3af05dd14f6ec619aaf7d34594aabc5ed1347970dec00452217cc900000008508c00000000002",
"0x0"
)
# SNR^((p-1)/2)
const BLS12_377_FrobeniusPsi_psi1_coef3* = Fp2[BLS12_377].fromHex(
# SNR^((p-1)/2)
const BLS12_377_FrobeniusPsi_psi1_coef3* = Fp2[BLS12_377].fromHex(
"0x1680a40796537cac0c534db1a79beb1400398f50ad1dec1bce649cf436b0f6299588459bff27d8e6e76d5ecf1391c63",
"0x0"
)
# norm(SNR)^((p-1)/3)
const BLS12_377_FrobeniusPsi_psi2_coef2* = Fp2[BLS12_377].fromHex(
# norm(SNR)^((p-1)/3)
const BLS12_377_FrobeniusPsi_psi2_coef2* = Fp2[BLS12_377].fromHex(
"0x9b3af05dd14f6ec619aaf7d34594aabc5ed1347970dec00452217cc900000008508c00000000001",
"0x0"
)
)

74
constantine/curves/bls12_377_glv.nim
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@ -10,60 +10,54 @@ import
../config/[curves, type_bigint, type_fp],
../io/[io_bigints, io_fields]
# BLS12-377 G1
# ----------------------------------------------------------------------------------------
# BLS12_377 G1
# ------------------------------------------------------------
const BLS12_377_cubicRootofUnity_mod_p* =
Fp[BLS12_377].fromHex"0x9b3af05dd14f6ec619aaf7d34594aabc5ed1347970dec00452217cc900000008508c00000000001"
const BLS12_377_cubicRootOfUnity_mod_p* =
Fp[BLS12_377].fromHex"0x1ae3a4617c510eabc8756ba8f8c524eb8882a75cc9bc8e359064ee822fb5bffd1e945779fffffffffffffffffffffff"
const BLS12_377_Lattice_G1* = (
# (BigInt, isNeg)
((BigInt[127].fromHex"0x452217cc900000010a11800000000000", false), # u² - 1
(BigInt[1].fromHex"0x1", true)), # -1
((BigInt[1].fromHex"0x1", false), # 1
(BigInt[127].fromHex"0x452217cc900000010a11800000000001", false)) # u²
((BigInt[127].fromHex"0x452217cc900000010a11800000000001", false),
(BigInt[1].fromHex"0x1", false)),
((BigInt[1].fromHex"0x1", false),
(BigInt[127].fromHex"0x452217cc900000010a11800000000000", true))
)
const BLS12_377_Babai_G1* = (
# Vector for Babai rounding
# (BigInt, isNeg)
(BigInt[130].fromHex"0x3b3f7aa969fd371607f72ed32af90182c", false),
(BigInt[130].fromHex"0x3b3f7aa969fd371607f72ed32af90181e", false),
(BigInt[4].fromHex"0xd", false)
)
# BLS12-377 G2
# ----------------------------------------------------------------------------------------
# BLS12_377 G2
# ------------------------------------------------------------
const BLS12_377_Lattice_G2* = (
# Curve of order 254 -> mini scalars of size 65
# x = -0xd201000000010000
# Value, isNeg
((BigInt[64].fromHex"0x8508c00000000001", true), # -x
(BigInt[1].fromHex"0x1", false), # 1
(BigInt[1].fromHex"0x0", false), # 0
(BigInt[1].fromHex"0x0", false)), # 0
((BigInt[1].fromHex"0x0", false), # 0
(BigInt[64].fromHex"0x8508c00000000001", true), # -x
(BigInt[1].fromHex"0x1", false), # 1
(BigInt[1].fromHex"0x0", false)), # 0
((BigInt[1].fromHex"0x0", false), # 0
(BigInt[1].fromHex"0x0", false), # 0
(BigInt[64].fromHex"0x8508c00000000001", true), # -x
(BigInt[1].fromHex"0x1", false)), # 1
((BigInt[1].fromHex"0x1", false), # 1
(BigInt[1].fromHex"0x0", false), # 0
(BigInt[1].fromHex"0x1", true), # -1
(BigInt[64].fromHex"0x8508c00000000001", true)) # -x
# (BigInt, isNeg)
((BigInt[64].fromHex"0x8508c00000000001", true),
(BigInt[1].fromHex"0x1", false),
(BigInt[1].fromHex"0x0", false),
(BigInt[1].fromHex"0x0", false)),
((BigInt[1].fromHex"0x0", false),
(BigInt[64].fromHex"0x8508c00000000001", true),
(BigInt[1].fromHex"0x1", false),
(BigInt[1].fromHex"0x0", false)),
((BigInt[1].fromHex"0x0", false),
(BigInt[1].fromHex"0x0", false),
(BigInt[64].fromHex"0x8508c00000000001", true),
(BigInt[1].fromHex"0x1", false)),
((BigInt[1].fromHex"0x1", false),
(BigInt[1].fromHex"0x0", false),
(BigInt[1].fromHex"0x1", true),
(BigInt[64].fromHex"0x8508c00000000001", false))
)
const BLS12_377_Babai_G2* = (
# Vector for Babai rounding
# Value, isNeg
(BigInt[193].fromHex"0x1eca0125755aed064f63abaff9084ce152979759b442f60d1", true),
(BigInt[130].fromHex"0x3b3f7aa969fd371607f72ed32af90181f", true),
(BigInt[67].fromhex"0x72030ba8ee9c06415", true),
(BigInt[1].fromhex"0x0", false)
# (BigInt, isNeg)
(BigInt[193].fromHex"0x1eca0125755aed064f63abaff9084ce152979759b442f60d0", true),
(BigInt[130].fromHex"0x3b3f7aa969fd371607f72ed32af90181e", true),
(BigInt[67].fromHex"0x72030ba8ee9c06422", true),
(BigInt[4].fromHex"0xd", false)
)

16
constantine/curves/bls12_377_pairing.nim
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@ -8,24 +8,28 @@
import
../config/[curves, type_bigint],
../towers,
../io/io_bigints,
../towers,
../pairing/cyclotomic_fp12
# Slow generic implementation
# ------------------------------------------------------------
# The bit count must be exact for the Miller loop
const BLS12_377_pairing_ate_param* = block:
# BLS Miller loop is parametrized by u
# +1 so that we can take *3 and NAF encode it
BigInt[64+1].fromHex("0x8508c00000000001")
# BLS12 Miller loop is parametrized by u
# +1 to bitlength so that we can mul by 3 for NAF encoding
BigInt[64+1].fromHex"0x8508c00000000001"
const BLS12_377_pairing_ate_param_isNeg* = false
const BLS12_377_pairing_finalexponent* = block:
# (p^12 - 1) / r
# BigInt[4269].fromHex"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"
# (p^12 - 1) / r * 3
BigInt[4271].fromHex"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"
# Addition chain
# ------------------------------------------------------------
func pow_x*(r: var Fp12[BLS12_377], a: Fp12[BLS12_377], invert = BLS12_377_pairing_ate_param_isNeg) =
## f^x with x the curve parameter
## For BLS12_377 f^-0x8508c00000000001

64
constantine/curves/bls12_381_frobenius.nim
View File

@ -15,11 +15,11 @@ import
# -----------------------------------------------------------------
# c = (SNR^((p-1)/6)^coef).
# Then for frobenius(2): c * conjugate(c)
# And for frobenius(3): c² * conjugate(c)
# Then for frobenius(2): c * conjugate(c)
# And for frobenius(3): c² * conjugate(c)
const BLS12_381_FrobeniusMapCoefficients* = [
# frobenius(1)
[Fp2[BLS12_381].fromHex( # SNR^((p-1)/6)^0
# frobenius(1) -----------------------
[Fp2[BLS12_381].fromHex( # SNR^((p-1)/6)^0
"0x1",
"0x0"
),
@ -27,93 +27,89 @@ const BLS12_381_FrobeniusMapCoefficients* = [
"0x1904d3bf02bb0667c231beb4202c0d1f0fd603fd3cbd5f4f7b2443d784bab9c4f67ea53d63e7813d8d0775ed92235fb8",
"0xfc3e2b36c4e03288e9e902231f9fb854a14787b6c7b36fec0c8ec971f63c5f282d5ac14d6c7ec22cf78a126ddc4af3"
),
Fp2[BLS12_381].fromHex( # SNR^((p-1)/6)^2 = SNR^((p-1)/3)
Fp2[BLS12_381].fromHex( # SNR^((p-1)/6)^2
"0x0",
"0x1a0111ea397fe699ec02408663d4de85aa0d857d89759ad4897d29650fb85f9b409427eb4f49fffd8bfd00000000aaac"
),
Fp2[BLS12_381].fromHex( # SNR^((p-1)/6)^3 = SNR^((p-1)/2)
Fp2[BLS12_381].fromHex( # SNR^((p-1)/6)^3
"0x6af0e0437ff400b6831e36d6bd17ffe48395dabc2d3435e77f76e17009241c5ee67992f72ec05f4c81084fbede3cc09",
"0x6af0e0437ff400b6831e36d6bd17ffe48395dabc2d3435e77f76e17009241c5ee67992f72ec05f4c81084fbede3cc09"
),
Fp2[BLS12_381].fromHex( # SNR^((p-1)/6)^4 = SNR^(2(p-1)/3)
Fp2[BLS12_381].fromHex( # SNR^((p-1)/6)^4
"0x1a0111ea397fe699ec02408663d4de85aa0d857d89759ad4897d29650fb85f9b409427eb4f49fffd8bfd00000000aaad",
"0x0"
),
Fp2[BLS12_381].fromHex( # SNR^((p-1)/6)^5
Fp2[BLS12_381].fromHex( # SNR^((p-1)/6)^5
"0x5b2cfd9013a5fd8df47fa6b48b1e045f39816240c0b8fee8beadf4d8e9c0566c63a3e6e257f87329b18fae980078116",
"0x144e4211384586c16bd3ad4afa99cc9170df3560e77982d0db45f3536814f0bd5871c1908bd478cd1ee605167ff82995"
)],
# frobenius(2)
[Fp2[BLS12_381].fromHex( # norm(SNR)^((p-1)/6)^1
# frobenius(2) -----------------------
[Fp2[BLS12_381].fromHex( # norm(SNR)^((p-1)/6)^0
"0x1",
"0x0"
),
Fp2[BLS12_381].fromHex( # norm(SNR)^((p-1)/6)^2
Fp2[BLS12_381].fromHex( # norm(SNR)^((p-1)/6)^1
"0x5f19672fdf76ce51ba69c6076a0f77eaddb3a93be6f89688de17d813620a00022e01fffffffeffff",
"0x0"
),
Fp2[BLS12_381].fromHex(
Fp2[BLS12_381].fromHex( # norm(SNR)^((p-1)/6)^2
"0x5f19672fdf76ce51ba69c6076a0f77eaddb3a93be6f89688de17d813620a00022e01fffffffefffe",
"0x0"
),
Fp2[BLS12_381].fromHex(
Fp2[BLS12_381].fromHex( # norm(SNR)^((p-1)/6)^3
"0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2a0f6b0f6241eabfffeb153ffffb9feffffffffaaaa",
"0x0"
),
Fp2[BLS12_381].fromHex(
Fp2[BLS12_381].fromHex( # norm(SNR)^((p-1)/6)^4
"0x1a0111ea397fe699ec02408663d4de85aa0d857d89759ad4897d29650fb85f9b409427eb4f49fffd8bfd00000000aaac",
"0x0"
),
Fp2[BLS12_381].fromHex(
Fp2[BLS12_381].fromHex( # norm(SNR)^((p-1)/6)^5
"0x1a0111ea397fe699ec02408663d4de85aa0d857d89759ad4897d29650fb85f9b409427eb4f49fffd8bfd00000000aaad",
"0x0"
)],
# frobenius(3)
[Fp2[BLS12_381].fromHex(
# frobenius(3) -----------------------
[Fp2[BLS12_381].fromHex( # (SNR²)^((p-1)/6)^0
"0x1",
"0x0"
),
Fp2[BLS12_381].fromHex(
Fp2[BLS12_381].fromHex( # (SNR²)^((p-1)/6)^1
"0x135203e60180a68ee2e9c448d77a2cd91c3dedd930b1cf60ef396489f61eb45e304466cf3e67fa0af1ee7b04121bdea2",
"0x6af0e0437ff400b6831e36d6bd17ffe48395dabc2d3435e77f76e17009241c5ee67992f72ec05f4c81084fbede3cc09"
),
Fp2[BLS12_381].fromHex(
Fp2[BLS12_381].fromHex( # (SNR²)^((p-1)/6)^2
"0x0",
"0x1"
),
Fp2[BLS12_381].fromHex(
Fp2[BLS12_381].fromHex( # (SNR²)^((p-1)/6)^3
"0x135203e60180a68ee2e9c448d77a2cd91c3dedd930b1cf60ef396489f61eb45e304466cf3e67fa0af1ee7b04121bdea2",
"0x135203e60180a68ee2e9c448d77a2cd91c3dedd930b1cf60ef396489f61eb45e304466cf3e67fa0af1ee7b04121bdea2"
),
Fp2[BLS12_381].fromHex(
Fp2[BLS12_381].fromHex( # (SNR²)^((p-1)/6)^4
"0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2a0f6b0f6241eabfffeb153ffffb9feffffffffaaaa",
"0x0"
),
Fp2[BLS12_381].fromHex(
Fp2[BLS12_381].fromHex( # (SNR²)^((p-1)/6)^5
"0x6af0e0437ff400b6831e36d6bd17ffe48395dabc2d3435e77f76e17009241c5ee67992f72ec05f4c81084fbede3cc09",
"0x135203e60180a68ee2e9c448d77a2cd91c3dedd930b1cf60ef396489f61eb45e304466cf3e67fa0af1ee7b04121bdea2"
)]]
# ψ (Psi) - Untwist-Frobenius-Twist Endomorphisms on twisted curves
# -----------------------------------------------------------------
# BLS12_381 is a M-Twist: psi1_coef1 = (1/SNR)^((p-1)/6)
# BLS12_381 is a M-twist: (1/SNR)^((p-1)/6)
const BLS12_381_FrobeniusPsi_psi1_coef1* = Fp2[BLS12_381].fromHex(
"0x5b2cfd9013a5fd8df47fa6b48b1e045f39816240c0b8fee8beadf4d8e9c0566c63a3e6e257f87329b18fae980078116",
"0x5b2cfd9013a5fd8df47fa6b48b1e045f39816240c0b8fee8beadf4d8e9c0566c63a3e6e257f87329b18fae980078116"
)
# (1/SNR)^((p-1)/3)
const BLS12_381_FrobeniusPsi_psi1_coef2* = Fp2[BLS12_381].fromHex(
# (1/SNR)^((p-1)/3)
const BLS12_381_FrobeniusPsi_psi1_coef2* = Fp2[BLS12_381].fromHex(
"0x0",
"0x1a0111ea397fe699ec02408663d4de85aa0d857d89759ad4897d29650fb85f9b409427eb4f49fffd8bfd00000000aaad"
)
# (1/SNR)^((p-1)/2)
const BLS12_381_FrobeniusPsi_psi1_coef3* = Fp2[BLS12_381].fromHex(
# (1/SNR)^((p-1)/2)
const BLS12_381_FrobeniusPsi_psi1_coef3* = Fp2[BLS12_381].fromHex(
"0x135203e60180a68ee2e9c448d77a2cd91c3dedd930b1cf60ef396489f61eb45e304466cf3e67fa0af1ee7b04121bdea2",
"0x6af0e0437ff400b6831e36d6bd17ffe48395dabc2d3435e77f76e17009241c5ee67992f72ec05f4c81084fbede3cc09"
)
# norm(SNR)^((p-1)/3)
const BLS12_381_FrobeniusPsi_psi2_coef2* = Fp2[BLS12_381].fromHex(
# norm((1/SNR))^((p-1)/3)
const BLS12_381_FrobeniusPsi_psi2_coef2* = Fp2[BLS12_381].fromHex(
"0x1a0111ea397fe699ec02408663d4de85aa0d857d89759ad4897d29650fb85f9b409427eb4f49fffd8bfd00000000aaac",
"0x0"
)
)

70
constantine/curves/bls12_381_glv.nim
View File

@ -10,60 +10,54 @@ import
../config/[curves, type_bigint, type_fp],
../io/[io_bigints, io_fields]
# BLS12-381 G1
# ----------------------------------------------------------------------------------------
# BLS12_381 G1
# ------------------------------------------------------------
const BLS12_381_cubicRootOfUnity_mod_p* =
Fp[BLS12_381].fromHex"0x1a0111ea397fe699ec02408663d4de85aa0d857d89759ad4897d29650fb85f9b409427eb4f49fffd8bfd00000000aaac"
Fp[BLS12_381].fromHex"0x5f19672fdf76ce51ba69c6076a0f77eaddb3a93be6f89688de17d813620a00022e01fffffffefffe"
const BLS12_381_Lattice_G1* = (
# (BigInt, isNeg)
((BigInt[128].fromHex"0xac45a4010001a40200000000ffffffff", false), # u² - 1
(BigInt[1].fromHex"0x1", true)), # -1
((BigInt[1].fromHex"0x1", false), # 1
(BigInt[128].fromHex"0xac45a4010001a4020000000100000000", false)) # u²
((BigInt[128].fromHex"0xac45a4010001a4020000000100000000", false),
(BigInt[1].fromHex"0x1", false)),
((BigInt[1].fromHex"0x1", false),
(BigInt[128].fromHex"0xac45a4010001a40200000000ffffffff", true))
)
const BLS12_381_Babai_G1* = (
# Vector for Babai rounding
# (BigInt, isNeg)
(BigInt[129].fromHex"0x17c6becf1e01faadd63f6e522f6cfee30", false),
(BigInt[129].fromHex"0x17c6becf1e01faadd63f6e522f6cfee2e", false),
(BigInt[2].fromHex"0x2", false)
)
# BLS12-381 G2
# ----------------------------------------------------------------------------------------
# BLS12_381 G2
# ------------------------------------------------------------
const BLS12_381_Lattice_G2* = (
# Curve of order 254 -> mini scalars of size 65
# x = -0xd201000000010000
# Value, isNeg
((BigInt[64].fromHex"0xd201000000010000", false), # -x
(BigInt[1].fromHex"0x1", false), # 1
(BigInt[1].fromHex"0x0", false), # 0
(BigInt[1].fromHex"0x0", false)), # 0
((BigInt[1].fromHex"0x0", false), # 0
(BigInt[64].fromHex"0xd201000000010000", false), # -x
(BigInt[1].fromHex"0x1", false), # 1
(BigInt[1].fromHex"0x0", false)), # 0
((BigInt[1].fromHex"0x0", false), # 0
(BigInt[1].fromHex"0x0", false), # 0
(BigInt[64].fromHex"0xd201000000010000", false), # -x
(BigInt[1].fromHex"0x1", false)), # 1
((BigInt[1].fromHex"0x1", false), # 1
(BigInt[1].fromHex"0x0", false), # 0
(BigInt[1].fromHex"0x1", true), # -1
(BigInt[64].fromHex"0xd201000000010000", false)) # -x
# (BigInt, isNeg)
((BigInt[64].fromHex"0xd201000000010000", false),
(BigInt[1].fromHex"0x1", false),
(BigInt[1].fromHex"0x0", false),
(BigInt[1].fromHex"0x0", false)),
((BigInt[1].fromHex"0x0", false),
(BigInt[64].fromHex"0xd201000000010000", false),
(BigInt[1].fromHex"0x1", false),
(BigInt[1].fromHex"0x0", false)),
((BigInt[1].fromHex"0x0", false),
(BigInt[1].fromHex"0x0", false),
(BigInt[64].fromHex"0xd201000000010000", false),
(BigInt[1].fromHex"0x1", false)),
((BigInt[1].fromHex"0x1", false),
(BigInt[1].fromHex"0x0", false),
(BigInt[1].fromHex"0x1", true),
(BigInt[64].fromHex"0xd201000000010000", true))
)
const BLS12_381_Babai_G2* = (
# Vector for Babai rounding
# Value, isNeg
# (BigInt, isNeg)
(BigInt[193].fromHex"0x1381204ca56cd56b533cfcc0d3e76ec2892078a5e8573b29c", false),
(BigInt[129].fromHex"0x17c6becf1e01faadd63f6e522f6cfee2f", true),
(BigInt[65].fromhex"0x1cfbe4f7bd0027db0", false),
(BigInt[1].fromhex"0x0", false)
(BigInt[129].fromHex"0x17c6becf1e01faadd63f6e522f6cfee2e", true),
(BigInt[65].fromHex"0x1cfbe4f7bd0027db2", false),
(BigInt[2].fromHex"0x2", false)
)

15
constantine/curves/bls12_381_pairing.nim
View File

@ -8,23 +8,28 @@
import
../config/[curves, type_bigint],
../towers,
../io/io_bigints,
../towers,
../pairing/cyclotomic_fp12
# Slow generic implementation
# ------------------------------------------------------------
# The bit count must be exact for the Miller loop
const BLS12_381_pairing_ate_param* = block:
# BLS Miller loop is parametrized by u
BigInt[64+2].fromHex("0xd201000000010000") # +2 so that we can take *3 and NAF encode it
# BLS12 Miller loop is parametrized by u
# +2 to bitlength so that we can mul by 3 for NAF encoding
BigInt[64+2].fromHex"0xd201000000010000"
const BLS12_381_pairing_ate_param_isNeg* = true
const BLS12_381_pairing_finalexponent* = block:
# (p^12 - 1) / r
# BigInt[4314].fromHex"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"
# (p^12 - 1) / r * 3
BigInt[4316].fromHex"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"
# Addition chain
# ------------------------------------------------------------
func pow_xdiv2*(r: var Fp12[BLS12_381], a: Fp12[BLS12_381], invert = BLS12_381_pairing_ate_param_isNeg) =
## f^(x/2) with x the curve parameter
## For BLS12_381 f^-0xd201000000010000

64
constantine/curves/bn254_nogami_frobenius.nim
View File

@ -15,11 +15,11 @@ import
# -----------------------------------------------------------------
# c = (SNR^((p-1)/6)^coef).
# Then for frobenius(2): c * conjugate(c)
# And for frobenius(3): c² * conjugate(c)
# Then for frobenius(2): c * conjugate(c)
# And for frobenius(3): c² * conjugate(c)
const BN254_Nogami_FrobeniusMapCoefficients* = [
# frobenius(1)
[Fp2[BN254_Nogami].fromHex( # SNR^((p-1)/6)^0
# frobenius(1) -----------------------
[Fp2[BN254_Nogami].fromHex( # SNR^((p-1)/6)^0
"0x1",
"0x0"
),
@ -27,93 +27,89 @@ const BN254_Nogami_FrobeniusMapCoefficients* = [
"0x1b377619212e7c8cb6499b50a846953f850974924d3f77c2e17de6c06f2a6de9",
"0x9ebee691ed1837503eab22f57b96ac8dc178b6db2c08850c582193f90d5922a"
),
Fp2[BN254_Nogami].fromHex( # SNR^((p-1)/6)^2 = SNR^((p-1)/3)
Fp2[BN254_Nogami].fromHex( # SNR^((p-1)/6)^2
"0x0",
"0x25236482400000017080eb4000000006181800000000000cd98000000000000b"
),
Fp2[BN254_Nogami].fromHex( # SNR^((p-1)/6)^3 = SNR^((p-1)/2)
Fp2[BN254_Nogami].fromHex( # SNR^((p-1)/6)^3
"0x23dfc9d1a39f4db8c69b87a8848aa075a7333a0e62d78cbf4b1b8eeae58b81c5",
"0x23dfc9d1a39f4db8c69b87a8848aa075a7333a0e62d78cbf4b1b8eeae58b81c5"
),
Fp2[BN254_Nogami].fromHex( # SNR^((p-1)/6)^4 = SNR^(2(p-1)/3)
Fp2[BN254_Nogami].fromHex( # SNR^((p-1)/6)^4
"0x25236482400000017080eb4000000006181800000000000cd98000000000000c",
"0x0"
),
Fp2[BN254_Nogami].fromHex( # SNR^((p-1)/6)^5
Fp2[BN254_Nogami].fromHex( # SNR^((p-1)/6)^5
"0x19f3db6884cdca43c2b0d5792cd135accb1baea0b017046e859975ab54b5ef9b",
"0xb2f8919bb3235bdf7837806d32eca5b9605515f4fe8fba521668a54ab4a1078"
)],
# frobenius(2)
[Fp2[BN254_Nogami].fromHex( # norm(SNR)^((p-1)/6)^1
# frobenius(2) -----------------------
[Fp2[BN254_Nogami].fromHex( # norm(SNR)^((p-1)/6)^0
"0x1",
"0x0"
),
Fp2[BN254_Nogami].fromHex( # norm(SNR)^((p-1)/6)^2
Fp2[BN254_Nogami].fromHex( # norm(SNR)^((p-1)/6)^1
"0x49b36240000000024909000000000006cd80000000000008",
"0x0"
),
Fp2[BN254_Nogami].fromHex(
Fp2[BN254_Nogami].fromHex( # norm(SNR)^((p-1)/6)^2
"0x49b36240000000024909000000000006cd80000000000007",
"0x0"
),
Fp2[BN254_Nogami].fromHex(
Fp2[BN254_Nogami].fromHex( # norm(SNR)^((p-1)/6)^3
"0x2523648240000001ba344d80000000086121000000000013a700000000000012",
"0x0"
),
Fp2[BN254_Nogami].fromHex(
Fp2[BN254_Nogami].fromHex( # norm(SNR)^((p-1)/6)^4
"0x25236482400000017080eb4000000006181800000000000cd98000000000000b",
"0x0"
),
Fp2[BN254_Nogami].fromHex(
Fp2[BN254_Nogami].fromHex( # norm(SNR)^((p-1)/6)^5
"0x25236482400000017080eb4000000006181800000000000cd98000000000000c",
"0x0"
)],
# frobenius(3)
[Fp2[BN254_Nogami].fromHex(
# frobenius(3) -----------------------
[Fp2[BN254_Nogami].fromHex( # (SNR²)^((p-1)/6)^0
"0x1",
"0x0"
),
Fp2[BN254_Nogami].fromHex(
Fp2[BN254_Nogami].fromHex( # (SNR²)^((p-1)/6)^1
"0x1439ab09c60b248f398c5d77b755f92b9edc5f19d2873545be471151a747e4e",
"0x23dfc9d1a39f4db8c69b87a8848aa075a7333a0e62d78cbf4b1b8eeae58b81c5"
),
Fp2[BN254_Nogami].fromHex(
Fp2[BN254_Nogami].fromHex( # (SNR²)^((p-1)/6)^2
"0x0",
"0x1"
),
Fp2[BN254_Nogami].fromHex(
Fp2[BN254_Nogami].fromHex( # (SNR²)^((p-1)/6)^3
"0x1439ab09c60b248f398c5d77b755f92b9edc5f19d2873545be471151a747e4e",
"0x1439ab09c60b248f398c5d77b755f92b9edc5f19d2873545be471151a747e4e"
),
Fp2[BN254_Nogami].fromHex(
Fp2[BN254_Nogami].fromHex( # (SNR²)^((p-1)/6)^4
"0x2523648240000001ba344d80000000086121000000000013a700000000000012",
"0x0"
),
Fp2[BN254_Nogami].fromHex(
Fp2[BN254_Nogami].fromHex( # (SNR²)^((p-1)/6)^5
"0x23dfc9d1a39f4db8c69b87a8848aa075a7333a0e62d78cbf4b1b8eeae58b81c5",
"0x1439ab09c60b248f398c5d77b755f92b9edc5f19d2873545be471151a747e4e"
)]]
# ψ (Psi) - Untwist-Frobenius-Twist Endomorphisms on twisted curves
# -----------------------------------------------------------------
# BN254_Nogami is a D-Twist: psi1_coef1 = SNR^((p-1)/6)
# BN254_Snarks is a D-Twist: SNR^((p-1)/6)
const BN254_Nogami_FrobeniusPsi_psi1_coef1* = Fp2[BN254_Nogami].fromHex(
"0x1b377619212e7c8cb6499b50a846953f850974924d3f77c2e17de6c06f2a6de9",
"0x9ebee691ed1837503eab22f57b96ac8dc178b6db2c08850c582193f90d5922a"
)
# SNR^((p-1)/3)
const BN254_Nogami_FrobeniusPsi_psi1_coef2* = Fp2[BN254_Nogami].fromHex(
# SNR^((p-1)/3)
const BN254_Nogami_FrobeniusPsi_psi1_coef2* = Fp2[BN254_Nogami].fromHex(
"0x0",
"0x25236482400000017080eb4000000006181800000000000cd98000000000000b"
)
# SNR^((p-1)/2)
const BN254_Nogami_FrobeniusPsi_psi1_coef3* = Fp2[BN254_Nogami].fromHex(
# SNR^((p-1)/2)
const BN254_Nogami_FrobeniusPsi_psi1_coef3* = Fp2[BN254_Nogami].fromHex(
"0x23dfc9d1a39f4db8c69b87a8848aa075a7333a0e62d78cbf4b1b8eeae58b81c5",
"0x23dfc9d1a39f4db8c69b87a8848aa075a7333a0e62d78cbf4b1b8eeae58b81c5"
)
# norm(SNR)^((p-1)/3)
const BN254_Nogami_FrobeniusPsi_psi2_coef2* = Fp2[BN254_Nogami].fromHex(
# norm(SNR)^((p-1)/3)
const BN254_Nogami_FrobeniusPsi_psi2_coef2* = Fp2[BN254_Nogami].fromHex(
"0x49b36240000000024909000000000006cd80000000000007",
"0x0"
)
)

10
constantine/curves/bn254_nogami_pairing.nim
View File

@ -8,14 +8,17 @@
import
../config/[curves, type_bigint],
../towers,
../io/io_bigints,
../towers,
../pairing/cyclotomic_fp12
# Slow generic implementation
# ------------------------------------------------------------
# The bit count must be exact for the Miller loop
const BN254_Nogami_pairing_ate_param* = block:
# BN Miller loop is parametrized by 6u+2
# 65+2 bit for NAF x3 encoding
# +2 to bitlength so that we can mul by 3 for NAF encoding
BigInt[65+2].fromHex"0x18300000000000004"
const BN254_Nogami_pairing_ate_param_isNeg* = true
@ -24,6 +27,9 @@ const BN254_Nogami_pairing_finalexponent* = block:
# (p^12 - 1) / r
BigInt[2786].fromHex"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"
# Addition chain
# ------------------------------------------------------------
func pow_u*(r: var Fp12[BN254_Nogami], a: Fp12[BN254_Nogami], invert = BN254_Nogami_pairing_ate_param_isNeg) =
## f^u with u the curve parameter
## For BN254_Nogami f^-0x4080000000000001

64
constantine/curves/bn254_snarks_frobenius.nim
View File

@ -15,11 +15,11 @@ import
# -----------------------------------------------------------------
# c = (SNR^((p-1)/6)^coef).
# Then for frobenius(2): c * conjugate(c)
# And for frobenius(3): c² * conjugate(c)
# Then for frobenius(2): c * conjugate(c)
# And for frobenius(3): c² * conjugate(c)
const BN254_Snarks_FrobeniusMapCoefficients* = [
# frobenius(1)
[Fp2[BN254_Snarks].fromHex( # SNR^((p-1)/6)^0
# frobenius(1) -----------------------
[Fp2[BN254_Snarks].fromHex( # SNR^((p-1)/6)^0
"0x1",
"0x0"
),
@ -27,93 +27,89 @@ const BN254_Snarks_FrobeniusMapCoefficients* = [
"0x1284b71c2865a7dfe8b99fdd76e68b605c521e08292f2176d60b35dadcc9e470",
"0x246996f3b4fae7e6a6327cfe12150b8e747992778eeec7e5ca5cf05f80f362ac"
),
Fp2[BN254_Snarks].fromHex( # SNR^((p-1)/6)^2 = SNR^((p-1)/3)
Fp2[BN254_Snarks].fromHex( # SNR^((p-1)/6)^2
"0x2fb347984f7911f74c0bec3cf559b143b78cc310c2c3330c99e39557176f553d",
"0x16c9e55061ebae204ba4cc8bd75a079432ae2a1d0b7c9dce1665d51c640fcba2"
),
Fp2[BN254_Snarks].fromHex( # SNR^((p-1)/6)^3 = SNR^((p-1)/2)
Fp2[BN254_Snarks].fromHex( # SNR^((p-1)/6)^3
"0x63cf305489af5dcdc5ec698b6e2f9b9dbaae0eda9c95998dc54014671a0135a",
"0x7c03cbcac41049a0704b5a7ec796f2b21807dc98fa25bd282d37f632623b0e3"
),
Fp2[BN254_Snarks].fromHex( # SNR^((p-1)/6)^4 = SNR^(2(p-1)/3)
Fp2[BN254_Snarks].fromHex( # SNR^((p-1)/6)^4
"0x5b54f5e64eea80180f3c0b75a181e84d33365f7be94ec72848a1f55921ea762",
"0x2c145edbe7fd8aee9f3a80b03b0b1c923685d2ea1bdec763c13b4711cd2b8126"
),
Fp2[BN254_Snarks].fromHex( # SNR^((p-1)/6)^5
Fp2[BN254_Snarks].fromHex( # SNR^((p-1)/6)^5
"0x183c1e74f798649e93a3661a4353ff4425c459b55aa1bd32ea2c810eab7692f",
"0x12acf2ca76fd0675a27fb246c7729f7db080cb99678e2ac024c6b8ee6e0c2c4b"
)],
# frobenius(2)
[Fp2[BN254_Snarks].fromHex( # norm(SNR)^((p-1)/6)^1
# frobenius(2) -----------------------
[Fp2[BN254_Snarks].fromHex( # norm(SNR)^((p-1)/6)^0
"0x1",
"0x0"
),
Fp2[BN254_Snarks].fromHex( # norm(SNR)^((p-1)/6)^2
Fp2[BN254_Snarks].fromHex( # norm(SNR)^((p-1)/6)^1
"0x30644e72e131a0295e6dd9e7e0acccb0c28f069fbb966e3de4bd44e5607cfd49",
"0x0"
),
Fp2[BN254_Snarks].fromHex(
Fp2[BN254_Snarks].fromHex( # norm(SNR)^((p-1)/6)^2
"0x30644e72e131a0295e6dd9e7e0acccb0c28f069fbb966e3de4bd44e5607cfd48",
"0x0"
),
Fp2[BN254_Snarks].fromHex(
Fp2[BN254_Snarks].fromHex( # norm(SNR)^((p-1)/6)^3
"0x30644e72e131a029b85045b68181585d97816a916871ca8d3c208c16d87cfd46",
"0x0"
),
Fp2[BN254_Snarks].fromHex(
Fp2[BN254_Snarks].fromHex( # norm(SNR)^((p-1)/6)^4
"0x59e26bcea0d48bacd4f263f1acdb5c4f5763473177fffffe",
"0x0"
),
Fp2[BN254_Snarks].fromHex(
Fp2[BN254_Snarks].fromHex( # norm(SNR)^((p-1)/6)^5
"0x59e26bcea0d48bacd4f263f1acdb5c4f5763473177ffffff",
"0x0"
)],
# frobenius(3)
[Fp2[BN254_Snarks].fromHex(
# frobenius(3) -----------------------
[Fp2[BN254_Snarks].fromHex( # (SNR²)^((p-1)/6)^0
"0x1",
"0x0"
),
Fp2[BN254_Snarks].fromHex(
Fp2[BN254_Snarks].fromHex( # (SNR²)^((p-1)/6)^1
"0x19dc81cfcc82e4bbefe9608cd0acaa90894cb38dbe55d24ae86f7d391ed4a67f",
"0xabf8b60be77d7306cbeee33576139d7f03a5e397d439ec7694aa2bf4c0c101"
),
Fp2[BN254_Snarks].fromHex(
Fp2[BN254_Snarks].fromHex( # (SNR²)^((p-1)/6)^2
"0x856e078b755ef0abaff1c77959f25ac805ffd3d5d6942d37b746ee87bdcfb6d",
"0x4f1de41b3d1766fa9f30e6dec26094f0fdf31bf98ff2631380cab2baaa586de"
),
Fp2[BN254_Snarks].fromHex(
Fp2[BN254_Snarks].fromHex( # (SNR²)^((p-1)/6)^3
"0x2a275b6d9896aa4cdbf17f1dca9e5ea3bbd689a3bea870f45fcc8ad066dce9ed",
"0x28a411b634f09b8fb14b900e9507e9327600ecc7d8cf6ebab94d0cb3b2594c64"
),
Fp2[BN254_Snarks].fromHex(
Fp2[BN254_Snarks].fromHex( # (SNR²)^((p-1)/6)^4
"0xbc58c6611c08dab19bee0f7b5b2444ee633094575b06bcb0e1a92bc3ccbf066",
"0x23d5e999e1910a12feb0f6ef0cd21d04a44a9e08737f96e55fe3ed9d730c239f"
),
Fp2[BN254_Snarks].fromHex(
Fp2[BN254_Snarks].fromHex( # (SNR²)^((p-1)/6)^5
"0x13c49044952c0905711699fa3b4d3f692ed68098967c84a5ebde847076261b43",
"0x16db366a59b1dd0b9fb1b2282a48633d3e2ddaea200280211f25041384282499"
)]]
# ψ (Psi) - Untwist-Frobenius-Twist Endomorphisms on twisted curves
# -----------------------------------------------------------------
# BN254_Snarks is a D-Twist: psi1_coef1 = SNR^((p-1)/6)
# BN254_Snarks is a D-Twist: SNR^((p-1)/6)
const BN254_Snarks_FrobeniusPsi_psi1_coef1* = Fp2[BN254_Snarks].fromHex(
"0x1284b71c2865a7dfe8b99fdd76e68b605c521e08292f2176d60b35dadcc9e470",
"0x246996f3b4fae7e6a6327cfe12150b8e747992778eeec7e5ca5cf05f80f362ac"
)
# SNR^((p-1)/3)
const BN254_Snarks_FrobeniusPsi_psi1_coef2* = Fp2[BN254_Snarks].fromHex(
# SNR^((p-1)/3)
const BN254_Snarks_FrobeniusPsi_psi1_coef2* = Fp2[BN254_Snarks].fromHex(
"0x2fb347984f7911f74c0bec3cf559b143b78cc310c2c3330c99e39557176f553d",
"0x16c9e55061ebae204ba4cc8bd75a079432ae2a1d0b7c9dce1665d51c640fcba2"
)
# SNR^((p-1)/2)
const BN254_Snarks_FrobeniusPsi_psi1_coef3* = Fp2[BN254_Snarks].fromHex(
# SNR^((p-1)/2)
const BN254_Snarks_FrobeniusPsi_psi1_coef3* = Fp2[BN254_Snarks].fromHex(
"0x63cf305489af5dcdc5ec698b6e2f9b9dbaae0eda9c95998dc54014671a0135a",
"0x7c03cbcac41049a0704b5a7ec796f2b21807dc98fa25bd282d37f632623b0e3"
)
# norm(SNR)^((p-1)/3)
const BN254_Snarks_FrobeniusPsi_psi2_coef2* = Fp2[BN254_Snarks].fromHex(
# norm(SNR)^((p-1)/3)
const BN254_Snarks_FrobeniusPsi_psi2_coef2* = Fp2[BN254_Snarks].fromHex(
"0x30644e72e131a0295e6dd9e7e0acccb0c28f069fbb966e3de4bd44e5607cfd48",
"0x0"
)
)

79
constantine/curves/bn254_snarks_glv.nim
View File

@ -10,63 +10,54 @@ import
../config/[curves, type_bigint, type_fp],
../io/[io_bigints, io_fields]
# BN254 Snarks G1
# ----------------------------------------------------------------------------------------
# BN254_Snarks G1
# ------------------------------------------------------------
const BN254_Snarks_cubicRootofUnity_mod_p* =
const BN254_Snarks_cubicRootOfUnity_mod_p* =
Fp[BN254_Snarks].fromHex"0x30644e72e131a0295e6dd9e7e0acccb0c28f069fbb966e3de4bd44e5607cfd48"
# Chapter 6.3.1 - Guide to Pairing-based Cryptography
const BN254_Snarks_Lattice_G1* = (
# Curve of order 254 -> mini scalars of size 127
# u = 0x44E992B44A6909F1
# (BigInt, isNeg)
((BigInt[64].fromHex"0x89d3256894d213e3", false), # 2u + 1
(BigInt[127].fromHex"0x6f4d8248eeb859fd0be4e1541221250b", false)), # 6u² + 4u + 1
((BigInt[127].fromHex"0x6f4d8248eeb859fc8211bbeb7d4f1128", false), # 6u² + 2u
(BigInt[64].fromHex"0x89d3256894d213e3", true)) # -2u - 1
((BigInt[127].fromHex"0x6f4d8248eeb859fc8211bbeb7d4f1128", false),
(BigInt[64].fromHex"0x89d3256894d213e3", true)),
((BigInt[64].fromHex"0x89d3256894d213e3", true),
(BigInt[127].fromHex"0x6f4d8248eeb859fd0be4e1541221250b", true))
)
const BN254_Snarks_Babai_G1* = (
# Vector for Babai rounding
# (BigInt, isNeg)
(BigInt[66].fromHex"0x2d91d232ec7e0b3d7", false), # (2u + 1) << 2^256 // r
(BigInt[130].fromHex"0x24ccef014a773d2d25398fd0300ff6565", false) # (6u² + 4u + 1) << 2^256 // r
(BigInt[130].fromHex"0x24ccef014a773d2d25398fd0300ff6565", false),
(BigInt[66].fromHex"0x2d91d232ec7e0b3d7", true)
)
# BN254 Snarks G2
# ----------------------------------------------------------------------------------------
# BN254_Snarks G2
# ------------------------------------------------------------
const BN254_Snarks_Lattice_G2* = (
# Curve of order 254 -> mini scalars of size 65
# x = 0x44E992B44A6909F1
# Value, isNeg
((BigInt[63].fromHex"0x44e992b44a6909f2", false), # x+1
(BigInt[63].fromHex"0x44e992b44a6909f1", false), # x
(BigInt[63].fromHex"0x44e992b44a6909f1", false), # x
(BigInt[64].fromHex"0x89d3256894d213e2", true)), # -2x
((BigInt[64].fromHex"0x89d3256894d213e3", false), # 2x+1
(BigInt[63].fromHex"0x44e992b44a6909f1", true), # -x
(BigInt[63].fromHex"0x44e992b44a6909f2", true), # -x-1
(BigInt[63].fromHex"0x44e992b44a6909f1", true)), # -x
((BigInt[64].fromHex"0x89d3256894d213e2", false), # 2x
(BigInt[64].fromHex"0x89d3256894d213e3", false), # 2x+1
(BigInt[64].fromHex"0x89d3256894d213e3", false), # 2x+1
(BigInt[64].fromHex"0x89d3256894d213e3", false)), # 2x+1
((BigInt[63].fromHex"0x44e992b44a6909f0", false), # x-1
(BigInt[65].fromHex"0x113a64ad129a427c6", false), # 4x+2
(BigInt[64].fromHex"0x89d3256894d213e1", true), # -2x+1
(BigInt[63].fromHex"0x44e992b44a6909f0", false)), # x-1
)
# (BigInt, isNeg)
((BigInt[64].fromHex"0x89d3256894d213e2", false),
(BigInt[63].fromHex"0x44e992b44a6909f2", false),
(BigInt[63].fromHex"0x44e992b44a6909f1", true),
(BigInt[63].fromHex"0x44e992b44a6909f1", false)),
((BigInt[63].fromHex"0x44e992b44a6909f1", true),
(BigInt[63].fromHex"0x44e992b44a6909f1", false),
(BigInt[63].fromHex"0x44e992b44a6909f1", true),
(BigInt[64].fromHex"0x89d3256894d213e3", true)),
((BigInt[63].fromHex"0x44e992b44a6909f2", false),
(BigInt[63].fromHex"0x44e992b44a6909f1", false),
(BigInt[63].fromHex"0x44e992b44a6909f1", false),
(BigInt[64].fromHex"0x89d3256894d213e2", true)),
((BigInt[64].fromHex"0x89d3256894d213e3", false),
(BigInt[63].fromHex"0x44e992b44a6909f1", true),
(BigInt[63].fromHex"0x44e992b44a6909f2", true),
(BigInt[63].fromHex"0x44e992b44a6909f1", true))
)
const BN254_Snarks_Babai_G2* = (
# Vector for Babai rounding
# Value, isNeg
(BigInt[128].fromHex"0xc444fab18d269b9dd0cb46fd51906254", false), # 2x²+3x+1 << 2^256 // r
(BigInt[193].fromHex"0x13d00631561b2572922df9f942d7d77c7001378f5ee78976d", false), # 3x³+8x²+x << 2^256 // r
(BigInt[192].fromhex"0x9e80318ab0d92b94916fcfca16bebbe436510546a93478ab", false), # 6x³+4x²+x << 2^256 // r
(BigInt[128].fromhex"0xc444fab18d269b9af7ae23ce89afae7d", true) # -2x²-x << 2^256 // r
# (BigInt, isNeg)
(BigInt[192].fromHex"0x9e80318ab0d92b9308e5da66fc7184ae46f4bda995d51bb1", false),
(BigInt[192].fromHex"0x9e80318ab0d92b9555b4ca7ba3e5577f2dff291532e42728", true),
(BigInt[192].fromHex"0x9e80318ab0d92b9555b4ca7ba3e55782071c4c43fac4daff", false),
(BigInt[192].fromHex"0x9e80318ab0d92b9555b4ca7ba3e5577dc170977dcef3cd3f", false)
)

9
constantine/curves/bn254_snarks_pairing.nim
View File

@ -8,13 +8,17 @@
import
../config/[curves, type_bigint],
../towers,
../io/io_bigints,
../towers,
../pairing/cyclotomic_fp12
# Slow generic implementation
# ------------------------------------------------------------
# The bit count must be exact for the Miller loop
const BN254_Snarks_pairing_ate_param* = block:
# BN Miller loop is parametrized by 6u+2
# +2 to bitlength so that we can mul by 3 for NAF encoding
BigInt[65+2].fromHex"0x19d797039be763ba8"
const BN254_Snarks_pairing_ate_param_isNeg* = false
@ -23,6 +27,9 @@ const BN254_Snarks_pairing_finalexponent* = block:
# (p^12 - 1) / r
BigInt[2790].fromHex"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"
# Addition chain
# ------------------------------------------------------------
func pow_u*(r: var Fp12[BN254_Snarks], a: Fp12[BN254_Snarks], invert = BN254_Snarks_pairing_ate_param_isNeg) =
## f^u with u the curve parameter
## For BN254_Snarks f^0x44e992b44a6909f1

36
constantine/elliptic/ec_endomorphism_accel.nim
View File

@ -39,6 +39,7 @@ type
func decomposeEndo*[M, scalBits, L: static int](
miniScalars: var MultiScalar[M, L],
negatePoints: var array[M, SecretBool],
scalar: BigInt[scalBits],
F: typedesc[Fp or Fp2]
) =
@ -109,6 +110,9 @@ func decomposeEndo*[M, scalBits, L: static int](
else:
k[miniScalarIdx] -= alphaB
let isNeg = k[miniScalarIdx].isMsbSet()
negatePoints[miniScalarIdx] = isNeg
k[miniScalarIdx].cneg(isNeg)
miniScalars[miniScalarIdx].copyTruncatedFrom(k[miniScalarIdx])
# Secret scalar + dynamic point
@ -304,9 +308,10 @@ func scalarMulEndo*[scalBits; EC](
{.error: "Unconfigured".}
# 2. Decompose scalar into mini-scalars
const L = (scalBits + M - 1) div M + 1 + 1 # A "+1" to handle negative
const L = (scalBits + M - 1) div M + 1 # Alternatively, negative can be handled with an extra "+1"
var miniScalars {.noInit.}: array[M, BigInt[L]]
miniScalars.decomposeEndo(scalar, P.F)
var negatePoints {.noInit.}: array[M, SecretBool]
miniScalars.decomposeEndo(negatePoints, scalar, P.F)
# 3. Handle negative mini-scalars
# A scalar decomposition might lead to negative miniscalar.
@ -319,17 +324,10 @@ func scalarMulEndo*[scalBits; EC](
# - Conditional negate is about 10 cycles per Fp, on G2 projective we have 3 (coords) * 2 (Fp2) * 10 (cycles) ~= 60 cycles
# We need to test the mini scalar, which is 65 bits so 2 Fp so about 2 cycles
# and negate it as well.
#
# However solution 1 seems to cause issues (TODO)
# with some of the BLS12-381 G2 test cases (6 and 9) as one of the miniscalars is 65 bits
# instead of the expected maximum of 64.
# - 0x5668a2332db27199dcfb7cbdfca6317c2ff128db26d7df68483e0a095ec8e88f
# - 0x644dc62869683f0c93f38eaef2ba6912569dc91ec2806e46b4a3dd6a4421dad1
#
# Furthermore solution 2 isn't enough on BLS12-377 G2 as test fails when miniScalars[0] is negative
let isNeg0 = miniscalars[0].isMsbSet()
miniscalars[0].cneg(isNeg0)
P.cneg(isNeg0)
block:
P.cneg(negatePoints[0])
staticFor i, 1, M:
endomorphisms[i-1].cneg(negatePoints[i])
# 4. Precompute lookup table
var lut {.noInit.}: array[1 shl (M-1), typeof(P)]
@ -458,8 +456,8 @@ func w2TableIndex(glv: GLV_SAC, bit2: int, isNeg: var SecretBool): SecretWord {.
func computeRecodedLength(bitWidth, window: int): int =
# Strangely in the paper this doesn't depend
# "m", the GLV decomposition dimension.
# lw = ⌈log2 r/w⌉+1+1 (a "+1" to handle negative mini scalars)
let lw = (bitWidth + window - 1) div window + 1 + 1
# lw = ⌈log2 r/w⌉+1 (optionally a second "+1" to handle negative mini scalars)
let lw = (bitWidth + window - 1) div window + 1
result = (lw mod window) + lw
func scalarMulGLV_m2w2*[scalBits; EC](
@ -488,12 +486,16 @@ func scalarMulGLV_m2w2*[scalBits; EC](
# 2. Decompose scalar into mini-scalars
const L = computeRecodedLength(C.getCurveOrderBitwidth(), 2)
var miniScalars {.noInit.}: array[2, BigInt[L]]
miniScalars.decomposeEndo(scalar, P0.F)
var negatePoints {.noInit.}: array[2, SecretBool]
miniScalars.decomposeEndo(negatePoints, scalar, P0.F)
# 3. TODO: handle negative mini-scalars
# 3. Handle negative mini-scalars
# Either negate the associated base and the scalar (in the `endomorphisms` array)
# Or use Algorithm 3 from Faz et al which can encode the sign
# in the GLV representation at the low low price of 1 bit
block:
P0.cneg(negatePoints[0])
P1.cneg(negatePoints[1])
# 4. Precompute lookup table
var lut {.noInit.}: array[8, EC]

2
constantine/elliptic/ec_shortweierstrass_jacobian.nim
View File

@ -383,7 +383,7 @@ func affineFromJacobian*[F; Tw](
aff.y.prod(jac.y, invZ)
aff.y.prod(jac.y, invZ2)
func projectiveFromJacobian*[F; Tw](
func jacobianFromAffine*[F; Tw](
jac: var ECP_ShortW_Jac[F, Tw],
aff: ECP_ShortW_Aff[F, Tw]) {.inline.} =
jac.x = aff.x

20
constantine/io/io_bigints.nim
View File

@ -407,7 +407,7 @@ func nativeEndianToHex(bytes: openarray[byte], order: static[Endianness]): strin
#
# ############################################################
func fromHex*(T: type BigInt, s: string): T {.noInit.} =
func fromHex*(a: var BigInt, s: string) =
## Convert a hex string to BigInt that can hold
## the specified number of bits
##
@ -421,12 +421,26 @@ func fromHex*(T: type BigInt, s: string): T {.noInit.} =
## Can work at compile-time to declare curve moduli from their hex strings
# 1. Convert to canonical uint
const canonLen = (T.bits + 8 - 1) div 8
const canonLen = (BigInt.bits + 8 - 1) div 8
var bytes: array[canonLen, byte]
hexToPaddedByteArray(s, bytes, bigEndian)
# 2. Convert canonical uint to Big Int
result.fromRawUint(bytes, bigEndian)
a.fromRawUint(bytes, bigEndian)
func fromHex*(T: type BigInt, s: string): T {.noInit.} =
## Convert a hex string to BigInt that can hold
## the specified number of bits
##
## For example `fromHex(BigInt[256], "0x123456")`
##
## Hex string is assumed big-endian
##
## This API is intended for configuration and debugging purposes
## Do not pass secret or private data to it.
##
## Can work at compile-time to declare curve moduli from their hex strings
result.fromHex(s)
func appendHex*(dst: var string, big: BigInt, order: static Endianness = bigEndian) =
## Append the BigInt hex into an accumulator

1
constantine/io/io_ec.nim
View File

@ -8,7 +8,6 @@
import
./io_bigints, ./io_fields, ./io_towers,
../config/curves,
../arithmetic,
../towers,
../elliptic/[

9
sage/README.md
View File

@ -1,7 +1,10 @@
# Sage scripts
This folder holds sage scripts:
- either for automating curve configuration
for example for computing the prime and order of BN or BLS curve families,
for irreducible polynomials for extension fields.
- for automating curve configuration
- Cofactor determination
- Frobenius precomputations
- Scalar docomposition for endomorphism acceleration
- Pairing parameters for naive implementation (ate and final exponent)
- Square-root parameters for naive implementation.
- for test vectors against a reference implementation.

106
sage/curve_family_bls12.sage
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@ -1,106 +0,0 @@
# Constantine
# Copyright (c) 2018-2019 Status Research & Development GmbH
# Copyright (c) 2020-Present Mamy André-Ratsimbazafy
# Licensed and distributed under either of
# * MIT license (license terms in the root directory or at http://opensource.org/licenses/MIT).
# * Apache v2 license (license terms in the root directory or at http://www.apache.org/licenses/LICENSE-2.0).
# at your option. This file may not be copied, modified, or distributed except according to those terms.
# ############################################################
#
# BLS12 Curves parameters
# (Barreto-Lynn-Scott with embedding degree of 12)
#
# ############################################################
#
# This module derives a BLS12 curve parameters from
# its base parameter x
def compute_curve_characteristic(x_str):
x = sage_eval(x_str)
p = (x - 1)^2 * (x^4 - x^2 + 1)//3 + x
r = x^4 - x^2 + 1
t = x + 1
print(f'BLS12 family - {p.nbits()} bits')
print(' Prime modulus p: 0x' + p.hex())
print(' Curve order r: 0x' + r.hex())
print(' trace t: 0x' + t.hex())
print(' Parameter x: ' + x_str)
if x < 0:
print(' Parameter x (hex): -0x' + (-x).hex())
else:
print(' Parameter x (hex): 0x' + x.hex())
print()
print(f' p mod 3: ' + str(p % 3))
print(f' p mod 4: ' + str(p % 4))
print(f' p mod 8: ' + str(p % 8))
print(f' p mod 12: ' + str(p % 12))
print(f' p mod 16: ' + str(p % 16))
print()
print(f' p^2 mod 3: ' + str(p^2 % 3))
print(f' p^2 mod 4: ' + str(p^2 % 4))
print(f' p^2 mod 8: ' + str(p^2 % 8))
print(f' p^2 mod 12: ' + str(p^2 % 12))
print(f' p^2 mod 16: ' + str(p^2 % 16))
print()
print(f' Endomorphism-based acceleration when p mod 3 == 1')
print(f' Endomorphism can be field multiplication by one of the non-trivial cube root of unity 𝜑')
print(f' Rationale:')
print(f' curve equation is y² = x³ + b, and y² = (x𝜑)³ + b <=> y² = x³ + b (with 𝜑³ == 1) so we are still on the curve')
print(f' this means that multiplying by 𝜑 the x-coordinate is equivalent to a scalar multiplication by some λᵩ')
print(f' with λᵩ² + λᵩ + 1 ≡ 0 (mod CurveOrder), see below. Hence we have a 2 dimensional decomposition of the scalar multiplication')
print(f' i.e. For any [s]P, we can find a corresponding [k1]P + [k2][λᵩ]P with [λᵩ]P being a simple field multiplication by 𝜑')
print(f' Finding cube roots:')
print(f' x³−1=0 <=> (x−1)(x²+x+1) = 0, if x != 1, x solves (x²+x+1) = 0 <=> x = (-1±√3)/2')
print(f' cube roots of unity: ' + str(['0x' + Integer(root).hex() for root in GF(p)(1).nth_root(3, all=True)]))
print(f' GLV-2 decomposition of s into (k1, k2) on G1')
print(f' (k1, k2) = (s, 0) - 𝛼1 b1 - 𝛼2 b2')
print(f' 𝛼i = 𝛼\u0302i * s / r')
print(f' Lattice b1: ' + str(['0x' + b.hex() for b in [x^2-1, -1]]))
print(f' Lattice b2: ' + str(['0x' + b.hex() for b in [1, x^2]]))
# Babai rounding
ahat1 = x^2
ahat2 = 1
# We want a1 = ahat1 * s/r with m = 2 (for a 2-dim decomposition) and r the curve order
# To handle rounding errors we instead multiply by
# 𝜈 = (2^WordBitWidth)^w (i.e. the same as the R magic constant for Montgomery arithmetic)
# with 𝜈 > r and w minimal so that 𝜈 > r
# a1 = ahat1*𝜈/r * s/𝜈
v = int(r).bit_length()
print(f' r.bit_length(): {v}')
v = int(((v + 64 - 1) // 64) * 64) # round to next multiple of 64
print(f' 𝜈 > r, 𝜈: 2^{v}')
print(f' Babai roundings')
print(f' 𝛼\u03021: ' + '0x' + ahat1.hex())
print(f' 𝛼\u03022: ' + '0x' + ahat2.hex())
print(f' Handle rounding errors')
print(f' 𝛼\u03051 = 𝛼\u03021 * s / r with 𝛼1 = (𝛼\u03021 * 𝜈/r) * s/𝜈')
print(f' 𝛼\u03052 = 𝛼\u03022 * s / r with 𝛼2 = (𝛼\u03022 * 𝜈/r) * s/𝜈')
print(f' -----------------------------------------------------')
l1 = Integer(ahat1 << v) // r
l2 = Integer(ahat2 << v) // r
print(f' 𝛼1 = (0x{l1.hex()} * s) >> {v}')
print(f' 𝛼2 = (0x{l2.hex()} * s) >> {v}')
if __name__ == "__main__":
# Usage
# BLS12-381
# sage sage/curve_family_bls12.sage '-(2^63 + 2^62 + 2^60 + 2^57 + 2^48 + 2^16)'
# BLS12-377
# sage sage/curve_family_bls12.sage '3 * 2^46 * (7 * 13 * 499) + 1'
from argparse import ArgumentParser
parser = ArgumentParser()
parser.add_argument("curve_param",nargs="+")
args = parser.parse_args()
compute_curve_characteristic(args.curve_param[0])

104
sage/curve_family_bn.sage
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@ -1,104 +0,0 @@
# Constantine
# Copyright (c) 2018-2019 Status Research & Development GmbH
# Copyright (c) 2020-Present Mamy André-Ratsimbazafy
# Licensed and distributed under either of
# * MIT license (license terms in the root directory or at http://opensource.org/licenses/MIT).
# * Apache v2 license (license terms in the root directory or at http://www.apache.org/licenses/LICENSE-2.0).
# at your option. This file may not be copied, modified, or distributed except according to those terms.
# ############################################################
#
# BN Curves parameters
# (Barreto-Naehrig curves)
#
# ############################################################
#
# This module derives a BN curve parameters from
# its base parameterx
def compute_curve_characteristic(x_str):
x = sage_eval(x_str)
p = 36*x^4 + 36*x^3 + 24*x^2 + 6*x + 1
r = 36*x^4 + 36*x^3 + 18*x^2 + 6*x + 1
t = 6*x^2 + 1
print(f'BN family - {p.nbits()} bits')
print(' Prime modulus p: 0x' + p.hex())
print(' Curve order r: 0x' + r.hex())
print(' trace t: 0x' + t.hex())
print(' Parameterx: ' + x_str)
if x < 0:
print(' Parameterx (hex): -0x' + (-x).hex())
else:
print(' Parameterx (hex): 0x' +x.hex())
print(f' p mod 3: ' + str(p % 3))
print(f' p mod 4: ' + str(p % 4))
print(f' p mod 8: ' + str(p % 8))
print(f' p mod 12: ' + str(p % 12))
print(f' p mod 16: ' + str(p % 16))
print()
print(f' p^2 mod 3: ' + str(p^2 % 3))
print(f' p^2 mod 4: ' + str(p^2 % 4))
print(f' p^2 mod 8: ' + str(p^2 % 8))
print(f' p^2 mod 12: ' + str(p^2 % 12))
print(f' p^2 mod 16: ' + str(p^2 % 16))
print()
print(f' Endomorphism-based acceleration when p mod 3 == 1')
print(f' Endomorphism can be field multiplication by one of the non-trivial cube root of unity 𝜑')
print(f' Rationale:')
print(f' curve equation is y² = x³ + b, and y² = (x𝜑)³ + b <=> y² = x³ + b (with 𝜑³ == 1) so we are still on the curve')
print(f' this means that multiplying by 𝜑 the x-coordinate is equivalent to a scalar multiplication by some λᵩ')
print(f' with λᵩ² + λᵩ + 1 ≡ 0 (mod r) and 𝜑² + 𝜑 + 1 ≡ 0 (mod p), see below.')
print(f' Hence we have a 2 dimensional decomposition of the scalar multiplication')
print(f' i.e. For any [s]P, we can find a corresponding [k1]P + [k2][λᵩ]P with [λᵩ]P being a simple field multiplication by 𝜑')
print(f' Finding cube roots:')
print(f' x³−1=0 <=> (x−1)(x²+x+1) = 0, if x != 1, x solves (x²+x+1) = 0 <=> x = (-1±√3)/2')
print(f' cube roots of unity 𝜑 (mod p): ' + str(['0x' + Integer(root).hex() for root in GF(p)(1).nth_root(3, all=True)]))
print(f' cube roots of unity λᵩ (mod r): ' + str(['0x' + Integer(root).hex() for root in GF(r)(1).nth_root(3, all=True)]))
print(f' GLV-2 decomposition of s into (k1, k2) on G1')
print(f' (k1, k2) = (s, 0) - 𝛼1 b1 - 𝛼2 b2')
print(f' 𝛼i = 𝛼\u0302i * s / r')
print(f' Lattice b1: ' + str(['0x' + b.hex() for b in [2*x+1, 6*x^2+4*x+1]]))
print(f' Lattice b2: ' + str(['0x' + b.hex() for b in [6*x^2+2*x, -2*x-1]]))
# Babai rounding
ahat1 = 2*x+1
ahat2 = 6*x^2+4*x+1
# We want a1 = ahat1 * s/r with m = 2 (for a 2-dim decomposition) and r the curve order
# To handle rounding errors we instead multiply by
# 𝜈 = (2^WordBitWidth)^w (i.e. the same as the R magic constant for Montgomery arithmetic)
# with 𝜈 > r and w minimal so that 𝜈 > r
# a1 = ahat1*𝜈/r * s/𝜈
v = int(r).bit_length()
print(f' r.bit_length(): {v}')
v = int(((v + 64 - 1) // 64) * 64) # round to next multiple of 64
print(f' 𝜈 > r, 𝜈: 2^{v}')
print(f' Babai roundings')
print(f' 𝛼\u03021: ' + '0x' + ahat1.hex())
print(f' 𝛼\u03022: ' + '0x' + ahat2.hex())
print(f' Handle rounding errors')
print(f' 𝛼1 = 𝛼\u03021 * s / r with 𝛼1 = (𝛼\u03021 * 𝜈/r) * s/𝜈')
print(f' 𝛼2 = 𝛼\u03022 * s / r with 𝛼2 = (𝛼\u03022 * 𝜈/r) * s/𝜈')
print(f' -----------------------------------------------------')
l1 = Integer(ahat1 << v) // r
l2 = Integer(ahat2 << v) // r
print(f' 𝛼1 = (0x{l1.hex()} * s) >> {v}')
print(f' 𝛼2 = (0x{l2.hex()} * s) >> {v}')
if __name__ == "__main__":
# Usage
# sage sage/curve_family_bn.sage '-(2^62 + 2^55 + 1)'
# sage sage/curve_family_bn.sage 4965661367192848881
from argparse import ArgumentParser
parser = ArgumentParser()
parser.add_argument("curve_param",nargs="+")
args = parser.parse_args()
compute_curve_characteristic(args.curve_param[0])

117
sage/curves.sage Normal file
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@ -0,0 +1,117 @@
#!/usr/bin/sage
# vim: syntax=python
# vim: set ts=2 sw=2 et:
# Constantine
# Copyright (c) 2018-2019 Status Research & Development GmbH
# Copyright (c) 2020-Present Mamy André-Ratsimbazafy
# Licensed and distributed under either of
# * MIT license (license terms in the root directory or at http://opensource.org/licenses/MIT).
# * Apache v2 license (license terms in the root directory or at http://www.apache.org/licenses/LICENSE-2.0).
# at your option. This file may not be copied, modified, or distributed except according to those terms.
# ############################################################
#
# Curves configuration
#
# ############################################################
import inspect
# Accelerate arithmetic by accepting probabilistic proofs
from sage.structure.proof.all import arithmetic
arithmetic(False)
def derive_BN_field(x):
params = {
'param': x,
'modulus': 36*x^4 + 36*x^3 + 24*x^2 + 6*x + 1,
'order': 36*x^4 + 36*x^3 + 18*x^2 + 6*x + 1,
'trace': 6*x^2 + 1,
'family': 'BN'
}
return params
def derive_BLS12_field(x):
params = {
'param': x,
'modulus': (x - 1)^2 * (x^4 - x^2 + 1)//3 + x,
'order': x^4 - x^2 + 1,
'trace': x + 1,
'family': 'BLS12'
}
return params
def copyright():
return inspect.cleandoc("""
# Constantine
# Copyright (c) 2018-2019 Status Research & Development GmbH
# Copyright (c) 2020-Present Mamy André-Ratsimbazafy
# Licensed and distributed under either of
# * MIT license (license terms in the root directory or at http://opensource.org/licenses/MIT).
# * Apache v2 license (license terms in the root directory or at http://www.apache.org/licenses/LICENSE-2.0).
# at your option. This file may not be copied, modified, or distributed except according to those terms.
""")
Curves = {
'BN254_Nogami': {
'field': derive_BN_field(-(2^62 + 2^55 + 1)),
'curve': {
'form': 'short_weierstrass',
'a': 0,
'b': 2
},
'tower': {
'embedding_degree': 12,
'twist_degree': 6,
'QNR_Fp': -1,
'SNR_Fp2': [1, 1],
'twist': 'D_Twist'
}
},
'BN254_Snarks': {
'field': derive_BN_field(Integer('0x44e992b44a6909f1')),
'curve': {
'form': 'short_weierstrass',
'a': 0,
'b': 3
},
'tower': {
'embedding_degree': 12,
'twist_degree': 6,
'QNR_Fp': -1,
'SNR_Fp2': [9, 1],
'twist': 'D_Twist'
}
},
'BLS12_377': {
'field': derive_BLS12_field(3 * 2^46 * (7 * 13 * 499) + 1),
'curve': {
'form': 'short_weierstrass',
'a': 0,
'b': 1
},
'tower': {
'embedding_degree': 12,
'twist_degree': 6,
'QNR_Fp': -5,
'SNR_Fp2': [0, 1],
'twist': 'D_Twist'
}
},
'BLS12_381': {
'field': derive_BLS12_field(-(2^63 + 2^62 + 2^60 + 2^57 + 2^48 + 2^16)),
'curve': {
'form': 'short_weierstrass',
'a': 0,
'b': 4
},
'tower': {
'embedding_degree': 12,
'twist_degree': 6,
'QNR_Fp': -1,
'SNR_Fp2': [1, 1],
'twist': 'M_Twist'
}
}
}

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#!/usr/bin/sage
# vim: syntax=python
# vim: set ts=2 sw=2 et:
# Constantine
# Copyright (c) 2018-2019 Status Research & Development GmbH
# Copyright (c) 2020-Present Mamy André-Ratsimbazafy
# Licensed and distributed under either of
# * MIT license (license terms in the root directory or at http://opensource.org/licenses/MIT).
# * Apache v2 license (license terms in the root directory or at http://www.apache.org/licenses/LICENSE-2.0).
# at your option. This file may not be copied, modified, or distributed except according to those terms.
# ############################################################
#
# Endomorphism acceleration constants
#
# ############################################################
# Imports
# ---------------------------------------------------------
import os
import inspect, textwrap
# Working directory
# ---------------------------------------------------------
os.chdir(os.path.dirname(__file__))
# Sage imports
# ---------------------------------------------------------
# Accelerate arithmetic by accepting probabilistic proofs
from sage.structure.proof.all import arithmetic
arithmetic(False)
load('curves.sage')
# Utilities
# ---------------------------------------------------------
def fp2_to_hex(a):
v = vector(a)
return '0x' + Integer(v[0]).hex() + ' + β * ' + '0x' + Integer(v[1]).hex()
def pretty_print_lattice(Lat):
print('Lattice:')
latHex = [['0x' + x.hex() if x >= 0 else '-0x' + (-x).hex() for x in vec] for vec in Lat]
maxlen = max([len(cell) for row in latHex for cell in row])
for row in latHex:
row = ' '.join(cell.rjust(maxlen + 2) for cell in row)
print(row)
def pretty_print_babai(Basis):
print('Babai:')
for i, v in enumerate(Basis):
if v < 0:
print(f' 𝛼\u0305{i}: -0x{Integer(int(-v)).hex()}')
else:
print(f' 𝛼\u0305{i}: 0x{Integer(int(v)).hex()}')
def derive_lattice(r, lambdaR, m):
lat = Matrix(matrix.identity(m))
lat[0, 0] = r
for i in range(1, m):
lat[i, 0] = -lambdaR^i
return lat.LLL()
def derive_babai(r, lattice, m):
basis = m * [0]
basis[0] = r
ahat = vector(basis) * lattice.inverse()
v = int(r).bit_length()
v = int(((v + 64 - 1) // 64) * 64)
return [(a << v) // r for a in ahat]
# TODO: maximum infinity norm
# G1 Endomorphism
# ---------------------------------------------------------
def check_cubic_root_endo(G1, Fp, r, cofactor, lambdaR, phiP):
## Check the Endomorphism for p mod 3 == 1
## Endomorphism can be field multiplication by one of the non-trivial cube root of unity 𝜑
## Rationale:
## curve equation is y² = x³ + b, and y² = (x𝜑)³ + b <=> y² = x³ + b (with 𝜑³ == 1) so we are still on the curve
## this means that multiplying by 𝜑 the x-coordinate is equivalent to a scalar multiplication by some λᵩ
## with λᵩ² + λᵩ + 1 ≡ 0 (mod r) and 𝜑² + 𝜑 + 1 ≡ 0 (mod p), see below.
## Hence we have a 2 dimensional decomposition of the scalar multiplication
## i.e. For any [s]P, we can find a corresponding [k1]P + [k2][λᵩ]P with [λᵩ]P being a simple field multiplication by 𝜑
## Finding cube roots:
## x³−1=0 <=> (x−1)(x²+x+1) = 0, if x != 1, x solves (x²+x+1) = 0 <=> x = (-1±√3)/2
assert phiP^3 == Fp(1)
assert lambdaR^3 % r == 1
Prand = G1.random_point()
P = Prand * cofactor
assert P != G1([0, 1, 0])
(Px, Py, Pz) = P
Qendo = G1([Px*phiP, Py, Pz])
Qlambda = lambdaR * P
assert P != Qendo
assert P != Qlambda
assert Qendo == Qlambda
print('Endomorphism OK')
def genCubicRootEndo(curve_name, curve_config):
p = curve_config[curve_name]['field']['modulus']
r = curve_config[curve_name]['field']['order']
b = curve_config[curve_name]['curve']['b']
Fp = GF(p)
G1 = EllipticCurve(Fp, [0, b])
cofactor = G1.order() // r
(phi1, phi2) = (Fp(root) for root in Fp(1).nth_root(3, all=True) if root != 1)
(lambda1, lambda2) = (GF(r)(root) for root in GF(r)(1).nth_root(3, all=True) if root != 1)
print('𝜑1 (mod p): 0x' + Integer(phi1).hex())
print('λᵩ1 (mod r): 0x' + Integer(lambda1).hex())
print('𝜑2 (mod p): 0x' + Integer(phi2).hex())
print('λᵩ2 (mod r): 0x' + Integer(lambda2).hex())
# TODO: is there a better way than spray-and-pray?
# TODO: Should we maximize or minimize lambda
# to maximize/minimize the scalar norm?
# TODO: Or is there a way to ensure
# that the Babai basis is mostly positive?
if lambda1 < lambda2:
lambda1, lambda2 = lambda2, lambda1
try:
check_cubic_root_endo(G1, Fp, r, cofactor, int(lambda1), phi1)
except:
print('Failure with:')
print(' 𝜑 (mod p): 0x' + Integer(phi1).hex())
print(' λᵩ (mod r): 0x' + Integer(lambda1).hex())
phi1, phi2 = phi2, phi1
check_cubic_root_endo(G1, Fp, r, cofactor, int(lambda1), phi1)
finally:
print('Success with:')
print(' 𝜑 (mod p): 0x' + Integer(phi1).hex())
print(' λᵩ (mod r): 0x' + Integer(lambda1).hex())
lattice = derive_lattice(r, lambda1, 2)
pretty_print_lattice(lattice)
babai = derive_babai(r, lattice, 2)
pretty_print_babai(babai)
return phi1, lattice, babai
# G2 Endomorphism
# ---------------------------------------------------------
def genPsiEndo(curve_name, curve_config):
t = curve_config[curve_name]['field']['trace']
r = curve_config[curve_name]['field']['order']
k = curve_config[curve_name]['tower']['embedding_degree']
# Decomposition factor depends on the embedding degree
m = CyclotomicField(k).degree()
# λψ is the trace of Frobenius - 1
lambda_psi = t - 1
lattice = derive_lattice(r, lambda_psi, m)
pretty_print_lattice(lattice)
babai = derive_babai(r, lattice, m)
pretty_print_babai(babai)
return lattice, babai
# Dump
# ---------------------------------------------------------
def dumpLattice(lattice):
result = ' # (BigInt, isNeg)\n'
lastRow = lattice.nrows() - 1
lastCol = lattice.ncols() - 1
for rowID, row in enumerate(lattice):
for colID, val in enumerate(row):
result += ' '
result += '(' if colID == 0 else ' '
result += f'(BigInt[{max(1, int(abs(val)).bit_length())}].fromHex"0x{Integer(int(abs(val))).hex()}", '
result += ('false' if val >= 0 else 'true') + ')'
result += ')' if colID == lastCol else ''
result += ',\n' if (rowID != lastRow or colID != lastCol) else '\n'
return result
def dumpBabai(vec):
result = ' # (BigInt, isNeg)\n'
lastRow = len(vec) - 1
for rowID, val in enumerate(vec):
result += ' '
result += f'(BigInt[{max(1, int(abs(val)).bit_length())}].fromHex"0x{Integer(int(abs(val))).hex()}", '
result += ('false' if val >= 0 else 'true') + ')'
result += ',\n' if rowID != lastRow else '\n'
return result
def dumpConst(name, inner):
result = f'const {name}* = (\n'
result += inner
result += ')\n'
return result
# CLI
# ---------------------------------------------------------
if __name__ == "__main__":
# Usage
# BLS12-381
# sage sage/derive_pairing.sage BLS12_381
from argparse import ArgumentParser
parser = ArgumentParser()
parser.add_argument("curve",nargs="+")
args = parser.parse_args()
curve = args.curve[0]
if curve not in Curves:
raise ValueError(
curve +
' is not one of the available curves: ' +
str(Curves.keys())
)
else:
print('\nPrecomputing G1 - 𝜑 (phi) cubic root endomorphism')
print('----------------------------------------------------\n')
cubeRootModP, g1lat, g1babai = genCubicRootEndo(curve, Curves)
print('\n\nPrecomputing G2 - ψ (Psi) - untwist-Frobenius-twist endomorphism')
print('----------------------------------------------------\n')
g2lat, g2babai = genPsiEndo(curve, Curves)
with open(f'{curve.lower()}_glv.nim', 'w') as f:
f.write(copyright())
f.write('\n\n')
f.write(inspect.cleandoc(f"""
import
../config/[curves, type_bigint, type_fp],
../io/[io_bigints, io_fields]
# {curve} G1
# ------------------------------------------------------------
"""))
f.write('\n\n')
f.write(inspect.cleandoc(f"""
const {curve}_cubicRootOfUnity_mod_p* =
Fp[{curve}].fromHex"0x{Integer(cubeRootModP).hex()}"
"""))
f.write('\n\n')
f.write(dumpConst(
f'{curve}_Lattice_G1',
dumpLattice(g1lat)
))
f.write('\n')
f.write(dumpConst(
f'{curve}_Babai_G1',
dumpBabai(g1babai)
))
f.write('\n\n')
f.write(inspect.cleandoc(f"""
# {curve} G2
# ------------------------------------------------------------
"""))
f.write('\n\n')
f.write(dumpConst(
f'{curve}_Lattice_G2',
dumpLattice(g2lat)
))
f.write('\n')
f.write(dumpConst(
f'{curve}_Babai_G2',
dumpBabai(g2babai)
))

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#!/usr/bin/sage
# vim: syntax=python
# vim: set ts=2 sw=2 et:
# Constantine
# Copyright (c) 2018-2019 Status Research & Development GmbH
# Copyright (c) 2020-Present Mamy André-Ratsimbazafy
# Licensed and distributed under either of
# * MIT license (license terms in the root directory or at http://opensource.org/licenses/MIT).
# * Apache v2 license (license terms in the root directory or at http://www.apache.org/licenses/LICENSE-2.0).
# at your option. This file may not be copied, modified, or distributed except according to those terms.
# ############################################################
#
# Frobenius constants
#
# ############################################################
# Imports
# ---------------------------------------------------------
import os
import inspect, textwrap
# Working directory
# ---------------------------------------------------------
os.chdir(os.path.dirname(__file__))
# Sage imports
# ---------------------------------------------------------
# Accelerate arithmetic by accepting probabilistic proofs
from sage.structure.proof.all import arithmetic
arithmetic(False)
load('curves.sage')
# Utilities
# ---------------------------------------------------------
def fp2_to_hex(a):
v = vector(a)
return '0x' + Integer(v[0]).hex() + ' + β * ' + '0x' + Integer(v[1]).hex()
def field_to_nim(value, field, curve, prefix = "", comment_above = "", comment_right = ""):
if field == 'Fp2':
v = vector(value)
result = '# ' + comment_above + '\n' if comment_above else ''
comment_right = ' # ' + comment_right if comment_right else ''
result += inspect.cleandoc(f"""
{prefix}Fp2[{curve}].fromHex( {comment_right}
"0x{Integer(v[0]).hex()}",
"0x{Integer(v[1]).hex()}"
)""")
return result
else:
raise newException(NotImplementedError)
# Code generators
# ---------------------------------------------------------
def genFrobeniusMapConstants(curve_name, curve_config):
embdeg = curve_config[curve_name]['tower']['embedding_degree']
twdeg = curve_config[curve_name]['tower']['twist_degree']
g2field = f'Fp{embdeg//twdeg}' if (embdeg//twdeg) > 1 else 'Fp'
p = curve_config[curve_name]['field']['modulus']
Fp = GF(p)
K.<u> = PolynomialRing(Fp)
if g2field == 'Fp2':
QNR_Fp = curve_config[curve_name]['tower']['QNR_Fp']
Fp2.<beta> = Fp.extension(u^2 - QNR_Fp)
SNR = curve_config[curve_name]['tower']['SNR_Fp2']
if g2field == 'Fp2':
cur = Fp2([1, 0])
SNR = Fp2(SNR)
else:
cur = Fp(1)
SNR = Fp(SNR)
print('\n----> Frobenius extension field constants <----\n')
buf = inspect.cleandoc(f"""
# Frobenius map - on extension fields
# -----------------------------------------------------------------
# c = (SNR^((p-1)/{twdeg})^coef).
# Then for frobenius(2): c * conjugate(c)
# And for frobenius(3): c² * conjugate(c)
const {curve_name}_FrobeniusMapCoefficients* = [
""")
FrobConst_map = SNR^((p-1)/6)
FrobConst_map_list = []
arr = ""
for i in range(twdeg):
if i == 0:
arr += '\n# frobenius(1) -----------------------\n'
arr += '['
arr += field_to_nim(cur, g2field, curve_name, comment_right = f'SNR^((p-1)/{twdeg})^{i}')
FrobConst_map_list.append(cur)
cur *= FrobConst_map
if i == twdeg - 1:
arr += ']'
arr += ',\n'
for i in range(twdeg):
if i == 0:
arr += '# frobenius(2) -----------------------\n'
arr += '['
val = FrobConst_map_list[i]*conjugate(FrobConst_map_list[i])
arr += field_to_nim(val, g2field, curve_name, comment_right = f'norm(SNR)^((p-1)/{twdeg})^{i}')
if i == twdeg - 1:
arr += ']'
arr += ',\n'
for i in range(twdeg):
if i == 0:
arr += '# frobenius(3) -----------------------\n'
arr += '['
val = FrobConst_map_list[i]^2 * conjugate(FrobConst_map_list[i])
arr += field_to_nim(val, g2field, curve_name, comment_right = f'(SNR²)^((p-1)/{twdeg})^{i}')
if i == twdeg - 1:
arr += ']]'
else:
arr += ',\n'
buf += textwrap.indent(arr, ' ')
return buf
def genFrobeniusPsiConstants(curve_name, curve_config):
embdeg = curve_config[curve_name]['tower']['embedding_degree']
twdeg = curve_config[curve_name]['tower']['twist_degree']
twkind = curve_config[curve_name]['tower']['twist']
g2field = f'Fp{embdeg//twdeg}' if (embdeg//twdeg) > 1 else 'Fp'
p = curve_config[curve_name]['field']['modulus']
Fp = GF(p)
K.<u> = PolynomialRing(Fp)
if g2field == 'Fp2':
QNR_Fp = curve_config[curve_name]['tower']['QNR_Fp']
Fp2.<beta> = Fp.extension(u^2 - QNR_Fp)
SNR = curve_config[curve_name]['tower']['SNR_Fp2']
if g2field == 'Fp2':
cur = Fp2([1, 0])
SNR = Fp2(SNR)
else:
cur = Fp(1)
SNR = Fp(SNR)
print('\n----> ψ (Psi) - Untwist-Frobenius-Twist Endomorphism constants <----\n')
buf = inspect.cleandoc(f"""
# ψ (Psi) - Untwist-Frobenius-Twist Endomorphisms on twisted curves
# -----------------------------------------------------------------
""")
buf += '\n'
if twkind == 'D_Twist':
buf += f'# {curve_name} is a D-Twist: psi1_coef1 = SNR^((p-1)/{twdeg})\n\n'
FrobConst_psi = SNR^((p-1)/twdeg)
snrUsed = 'SNR'
else:
buf += f'# {curve_name} is a M-Twist: psi1_coef1 = (1/SNR)^((p-1)/{twdeg})\n\n'
FrobConst_psi = (1/SNR)^((p-1)/twdeg)
snrUsed = '(1/SNR)'
FrobConst_psi1_coef2 = FrobConst_psi^2
FrobConst_psi1_coef3 = FrobConst_psi1_coef2 * FrobConst_psi
buf += field_to_nim(
FrobConst_psi1_coef2, g2field, curve_name,
prefix = f'const {curve_name}_FrobeniusPsi_psi1_coef2* = ',
comment_above = f'{snrUsed}^((p-1)/{twdeg//2})'
) + '\n'
buf += field_to_nim(
FrobConst_psi1_coef3, g2field, curve_name,
prefix = f'const {curve_name}_FrobeniusPsi_psi1_coef3* = ',
comment_above = f'{snrUsed}^((p-1)/{twdeg//3})'
) + '\n'
FrobConst_psi2_coef2 = FrobConst_psi1_coef2 * FrobConst_psi1_coef2**p
buf += field_to_nim(
FrobConst_psi2_coef2, g2field, curve_name,
prefix = f'const {curve_name}_FrobeniusPsi_psi2_coef2* = ',
comment_above = f'norm({snrUsed})^((p-1)/{twdeg//2})'
)
# psi2_coef3 is always -1 (mod p^m) with m = embdeg/twdeg
# Recap, with ξ (xi) the sextic non-residue
# psi_2 = ((1/ξ)^((p-1)/6))^2 = (1/ξ)^((p-1)/3)
# psi_3 = psi_2 * (1/ξ)^((p-1)/6) = (1/ξ)^((p-1)/3) * (1/ξ)^((p-1)/6) = (1/ξ)^((p-1)/2)
#
# Reminder, in 𝔽p2, frobenius(a) = a^p = conj(a)
# psi2_2 = psi_2 * psi_2^p = (1/ξ)^((p-1)/3) * (1/ξ)^((p-1)/3)^p = (1/ξ)^((p-1)/3) * frobenius((1/ξ))^((p-1)/3)
# = norm(1/ξ)^((p-1)/3)
# psi2_3 = psi_3 * psi_3^p = (1/ξ)^((p-1)/2) * (1/ξ)^((p-1)/2)^p = (1/ξ)^((p-1)/2) * frobenius((1/ξ))^((p-1)/2)
# = norm(1/ξ)^((p-1)/2)
#
# In Fp²:
# - quadratic non-residues respect the equation a^((p²-1)/2) ≡ -1 (mod p²) by the Legendre symbol
# - sextic non-residues are also quadratic non-residues so ξ^((p²-1)/2) ≡ -1 (mod p²)
# - QRT(1/a) = QRT(a) with QRT the quadratic residuosity test
#
# We have norm(ξ)^((p-1)/2) = (ξ*frobenius(ξ))^((p-1)/2) = (ξ*(ξ^p))^((p-1)/2) = ξ^(p+1)^(p-1)/2
# = ξ^((p²-1)/2)
# And ξ^((p²-1)/2) ≡ -1 (mod p²)
# So psi2_3 ≡ -1 (mod p²)
return buf
# CLI
# ---------------------------------------------------------
if __name__ == "__main__":
# Usage
# BLS12-381
# sage sage/derive_frobenius.sage BLS12_381
from argparse import ArgumentParser
parser = ArgumentParser()
parser.add_argument("curve",nargs="+")
args = parser.parse_args()
curve = args.curve[0]
if curve not in Curves:
raise ValueError(
curve +
' is not one of the available curves: ' +
str(Curves.keys())
)
else:
FrobMap = genFrobeniusMapConstants(curve, Curves)
FrobPsi = genFrobeniusPsiConstants(curve, Curves)
with open(f'{curve.lower()}_frobenius.nim', 'w') as f:
f.write(copyright())
f.write('\n\n')
f.write(inspect.cleandoc("""
import
../config/curves,
../towers,
../io/io_towers
"""))
f.write('\n\n')
f.write(FrobMap)
f.write('\n\n')
f.write(FrobPsi)
print(f'Successfully created {curve}_frobenius.nim')
print(inspect.cleandoc("""\n
For testing you can verify the following invariants:
Galbraith-Lin-Scott, 2008, Theorem 1
Fuentes-Castaneda et al, 2011, Equation (2)
ψ(ψ(P)) - t*ψ(P) + p*P == Infinity
Galbraith-Scott, 2008, Lemma 1
The cyclotomic polynomial with GΦ(ψ(P)) == Infinity
Hence for embedding degree k=12
ψ⁴(P) - ψ²(P) + P == Infinity
for embedding degree k=6
ψ²(P) - ψ(P) + P == Infinity
"""))

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#!/usr/bin/sage
# vim: syntax=python
# vim: set ts=2 sw=2 et:
# Constantine
# Copyright (c) 2018-2019 Status Research & Development GmbH
# Copyright (c) 2020-Present Mamy André-Ratsimbazafy
# Licensed and distributed under either of
# * MIT license (license terms in the root directory or at http://opensource.org/licenses/MIT).
# * Apache v2 license (license terms in the root directory or at http://www.apache.org/licenses/LICENSE-2.0).
# at your option. This file may not be copied, modified, or distributed except according to those terms.
# ############################################################
#
# Pairing constants
#
# ############################################################
# Imports
# ---------------------------------------------------------
import os
import inspect, textwrap
# Working directory
# ---------------------------------------------------------
os.chdir(os.path.dirname(__file__))
# Sage imports
# ---------------------------------------------------------
# Accelerate arithmetic by accepting probabilistic proofs
from sage.structure.proof.all import arithmetic
arithmetic(False)
load('curves.sage')
# Utilities
# ---------------------------------------------------------
# Code generators
# ---------------------------------------------------------
def genAteParam(curve_name, curve_config):
u = curve_config[curve_name]['field']['param']
family = curve_config[curve_name]['field']['family']
if family == 'BLS12':
ate_param = u
ate_comment = ' # BLS12 Miller loop is parametrized by u\n'
elif family == 'BN':
ate_param = 6*u+2
ate_comment = ' # BN Miller loop is parametrized by 6u+2\n'
else:
raise ValueError(f'family: {family} is not implemented')
buf = '# The bit count must be exact for the Miller loop\n'
buf += f'const {curve_name}_pairing_ate_param* = block:\n'
buf += ate_comment
ate_bits = int(ate_param).bit_length()
naf_bits = int(3*ate_param).bit_length() - ate_bits
buf += f' # +{naf_bits} to bitlength so that we can mul by 3 for NAF encoding\n'
buf += f' BigInt[{ate_bits}+{naf_bits}].fromHex"0x{Integer(abs(ate_param)).hex()}"\n\n'
buf += f'const {curve_name}_pairing_ate_param_isNeg* = {"true" if ate_param < 0 else "false"}'
return buf
def genFinalExp(curve_name, curve_config):
p = curve_config[curve_name]['field']['modulus']
r = curve_config[curve_name]['field']['order']
k = curve_config[curve_name]['tower']['embedding_degree']
family = curve_config[curve_name]['field']['family']
fexp = (p^k - 1)//r
if family == 'BLS12':
fexp *= 3
buf = f'const {curve_name}_pairing_finalexponent* = block:\n'
buf += f' # (p^{k} - 1) / r' + (' * 3' if family == 'BLS12' else '')
buf += '\n'
buf += f' BigInt[{int(fexp).bit_length()}].fromHex"0x{Integer(fexp).hex()}"'
return buf
# CLI
# ---------------------------------------------------------
if __name__ == "__main__":
# Usage
# BLS12-381
# sage sage/derive_pairing.sage BLS12_381
from argparse import ArgumentParser
parser = ArgumentParser()
parser.add_argument("curve",nargs="+")
args = parser.parse_args()
curve = args.curve[0]
if curve not in Curves:
raise ValueError(
curve +
' is not one of the available curves: ' +
str(Curves.keys())
)
else:
ate = genAteParam(curve, Curves)
fexp = genFinalExp(curve, Curves)
with open(f'{curve.lower()}_pairing.nim', 'w') as f:
f.write(copyright())
f.write('\n\n')
f.write(inspect.cleandoc("""
import
../config/[curves, type_bigint],
../io/io_bigints
# Slow generic implementation
# ------------------------------------------------------------
"""))
f.write('\n\n')
f.write(ate)
f.write('\n\n')
f.write(fexp)
f.write('\n\n')
f.write(inspect.cleandoc("""
# Addition chain
# ------------------------------------------------------------
"""))
print(f'Successfully created {curve}_pairing.nim')

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sage/frobenius_bls12_377.sage
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# Constantine
# Copyright (c) 2018-2019 Status Research & Development GmbH
# Copyright (c) 2020-Present Mamy André-Ratsimbazafy
# Licensed and distributed under either of
# * MIT license (license terms in the root directory or at http://opensource.org/licenses/MIT).
# * Apache v2 license (license terms in the root directory or at http://www.apache.org/licenses/LICENSE-2.0).
# at your option. This file may not be copied, modified, or distributed except according to those terms.
# ############################################################
#
# BLS12-377
# Frobenius Endomorphism
# Untwist-Frobenius-Twist isogeny
#
# ############################################################
# Parameters
x = 3 * 2^46 * (7 * 13 * 499) + 1
p = (x - 1)^2 * (x^4 - x^2 + 1)//3 + x
r = x^4 - x^2 + 1
t = x + 1
print('p : ' + p.hex())
print('r : ' + r.hex())
print('t : ' + t.hex())
# Finite fields
Fp = GF(p)
K2.<u> = PolynomialRing(Fp)
Fp2.<beta> = Fp.extension(u^2+5) # √-5 quadratic non-residue
# K6.<v> = PolynomialRing(F2)
# Fp6.<eta> = Fp2.extension(v^3-Fp2([0, 1])
# K12.<w> = PolynomialRing(Fp6)
# Fp12.<gamma> = Fp6.extension(w^2-eta)
# Curves
b = 1
SNR = Fp2([0, 1]) # √-5 sextic non-residue
G1 = EllipticCurve(Fp, [0, b])
G2 = EllipticCurve(Fp2, [0, b/SNR])
# Utilities
def fp2_to_hex(a):
v = vector(a)
return '0x' + Integer(v[0]).hex() + ' + β * ' + '0x' + Integer(v[1]).hex()
# Frobenius map constants
print('\nFrobenius extension field constants')
FrobConst_map = SNR^((p-1)/6)
FrobConst_map_list = []
cur = Fp2([1, 0])
for i in range(6):
FrobConst_map_list.append(cur)
print(f'FrobConst_map_{i} : {fp2_to_hex(cur)}')
cur *= FrobConst_map
print('')
for i in range(6):
print(f'FrobConst_map_{i}_pow2 : {fp2_to_hex(FrobConst_map_list[i]*conjugate(FrobConst_map_list[i]))}')
print('')
for i in range(6):
print(f'FrobConst_map_{i}_pow3 : {fp2_to_hex(FrobConst_map_list[i]**2 * conjugate(FrobConst_map_list[i]))}')
# Frobenius psi constants (D type: use SNR, M type use 1/SNR)
print('\nψ (Psi) - Untwist-Frobenius-Twist constants')
FrobConst_psi = SNR^((p-1)/6)
FrobConst_psi_2 = FrobConst_psi * FrobConst_psi
FrobConst_psi_3 = FrobConst_psi_2 * FrobConst_psi
print('FrobConst_psi : ' + fp2_to_hex(FrobConst_psi))
print('FrobConst_psi_2 : ' + fp2_to_hex(FrobConst_psi_2))
print('FrobConst_psi_3 : ' + fp2_to_hex(FrobConst_psi_3))
print('')
FrobConst_psi2_2 = FrobConst_psi_2 * FrobConst_psi_2**p
FrobConst_psi2_3 = FrobConst_psi_3 * FrobConst_psi_3**p
print('FrobConst_psi2_2 : ' + fp2_to_hex(FrobConst_psi2_2))
print('FrobConst_psi2_3 : ' + fp2_to_hex(FrobConst_psi2_3))
print('')
FrobConst_psi3_2 = FrobConst_psi_2 * FrobConst_psi2_2**p
FrobConst_psi3_3 = FrobConst_psi_3 * FrobConst_psi2_3**p
print('FrobConst_psi3_2 : ' + fp2_to_hex(FrobConst_psi3_2))
print('FrobConst_psi3_3 : ' + fp2_to_hex(FrobConst_psi3_3))
# Recap, with ξ (xi) the sextic non-residue
# psi_2 = (ξ^((p-1)/6))^2 = ξ^((p-1)/3)
# psi_3 = psi_2 * ξ^((p-1)/6) = ξ^((p-1)/3) * ξ^((p-1)/6) = ξ^((p-1)/2)
#
# Reminder, in 𝔽p2, frobenius(a) = a^p = conj(a)
# psi2_2 = psi_2 * psi_2^p = ξ^((p-1)/3) * ξ^((p-1)/3)^p = ξ^((p-1)/3) * frobenius(ξ)^((p-1)/3)
# = norm(ξ)^((p-1)/3)
# psi2_3 = psi_3 * psi_3^p = ξ^((p-1)/2) * ξ^((p-1)/2)^p = ξ^((p-1)/2) * frobenius(ξ)^((p-1)/2)
# = norm(ξ)^((p-1)/2)
#
# In Fp²:
# - quadratic non-residues respect the equation a^((p²-1)/2) ≡ -1 (mod p²) by the Legendre symbol
# - sextic non-residues are also quadratic non-residues so ξ^((p²-1)/2) ≡ -1 (mod p²)
#
# We have norm(ξ)^((p-1)/2) = (ξ*frobenius(ξ))^((p-1)/2) = (ξ*(ξ^p))^((p-1)/2) = ξ^(p+1)^(p-1)/2
# = ξ^((p²-1)/2)
# And ξ^((p²-1)/2) ≡ -1 (mod p²)
# So psi2_3 ≡ -1 (mod p²)
#
# TODO: explain why psi3_2 = [0, -1]
# Frobenius Fp2
A = Fp2([5, 7])
Aconj = Fp2([5, -7])
AF = A.frobenius(1) # or pth_power(1)
AF2 = A.frobenius(2)
AF3 = A.frobenius(3)
print('')
print('A : ' + fp2_to_hex(A))
print('A conjugate: ' + fp2_to_hex(Aconj))
print('')
print('AF1 : ' + fp2_to_hex(AF))
print('AF2 : ' + fp2_to_hex(AF2))
print('AF3 : ' + fp2_to_hex(AF3))
def psi(P):
(Px, Py, Pz) = P
return G2([
FrobConst_psi_2 * Px.frobenius(),
FrobConst_psi_3 * Py.frobenius()
# Pz.frobenius() - Always 1 after extract
])
def psi2(P):
(Px, Py, Pz) = P
return G2([
FrobConst_psi2_2 * Px.frobenius(2),
FrobConst_psi2_3 * Py.frobenius(2)
# Pz - Always 1 after extract
])
# Test generator
set_random_seed(1337)
# Vectors
print('\nTest vectors:')
for i in range(4):
P = G2.random_point()
(Px, Py, Pz) = P
vPx = vector(Px)
vPy = vector(Py)
# Pz = vector(Pz)
print(f'\nTest {i}')
print(' Px: ' + Integer(vPx[0]).hex() + ' + β * ' + Integer(vPx[1]).hex())
print(' Py: ' + Integer(vPy[0]).hex() + ' + β * ' + Integer(vPy[1]).hex())
# Galbraith-Lin-Scott, 2008, Theorem 1
# Fuentes-Castaneda et al, 2011, Equation (2)
assert psi(psi(P)) - t*psi(P) + p*P == G2([0, 1, 0])
# Galbraith-Scott, 2008, Lemma 1
# k-th cyclotomic polynomial with k = 12
assert psi2(psi2(P)) - psi2(P) + P == G2([0, 1, 0])
assert psi(psi(P)) == psi2(P)
(Qx, Qy, Qz) = psi(P)
vQx = vector(Qx)
vQy = vector(Qy)
print(' Qx: ' + Integer(vQx[0]).hex() + ' + β * ' + Integer(vQx[1]).hex())
print(' Qy: ' + Integer(vQy[0]).hex() + ' + β * ' + Integer(vQy[1]).hex())

180
sage/frobenius_bls12_381.sage
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# Constantine
# Copyright (c) 2018-2019 Status Research & Development GmbH
# Copyright (c) 2020-Present Mamy André-Ratsimbazafy
# Licensed and distributed under either of
# * MIT license (license terms in the root directory or at http://opensource.org/licenses/MIT).
# * Apache v2 license (license terms in the root directory or at http://www.apache.org/licenses/LICENSE-2.0).
# at your option. This file may not be copied, modified, or distributed except according to those terms.
# ############################################################
#
# BLS12-381
# Frobenius Endomorphism
# Untwist-Frobenius-Twist isogeny
#
# ############################################################
# Parameters
x = -(2^63 + 2^62 + 2^60 + 2^57 + 2^48 + 2^16)
p = (x - 1)^2 * (x^4 - x^2 + 1)//3 + x
r = x^4 - x^2 + 1
t = x + 1
print('p : ' + p.hex())
print('r : ' + r.hex())
print('t : ' + t.hex())
print('p (mod r) == t-1 (mod r) == 0x' + (p % r).hex())
# Finite fields
Fp = GF(p)
K2.<u> = PolynomialRing(Fp)
Fp2.<beta> = Fp.extension(u^2+1)
# Curves
b = 4
SNR = Fp2([1, 1])
G1 = EllipticCurve(Fp, [0, b])
G2 = EllipticCurve(Fp2, [0, b*SNR])
# https://crypto.stackexchange.com/questions/64064/order-of-twisted-curve-in-pairings
# https://math.stackexchange.com/questions/144194/how-to-find-the-order-of-elliptic-curve-over-finite-field-extension
cofactorG1 = G1.order() // r
cofactorG2 = G2.order() // r
print('')
print('cofactor G1: ' + cofactorG1.hex())
print('cofactor G2: ' + cofactorG2.hex())
print('')
# Utilities
def fp2_to_hex(a):
v = vector(a)
return '0x' + Integer(v[0]).hex() + ' + β * ' + '0x' + Integer(v[1]).hex()
# Frobenius map constants
print('\nFrobenius extension field constants')
FrobConst_map = SNR^((p-1)/6)
FrobConst_map_list = []
cur = Fp2([1, 0])
for i in range(6):
FrobConst_map_list.append(cur)
print(f'FrobConst_map_{i} : {fp2_to_hex(cur)}')
cur *= FrobConst_map
print('')
for i in range(6):
print(f'FrobConst_map_{i}_pow2 : {fp2_to_hex(FrobConst_map_list[i]*conjugate(FrobConst_map_list[i]))}')
print('')
for i in range(6):
print(f'FrobConst_map_{i}_pow3 : {fp2_to_hex(FrobConst_map_list[i]**2 * conjugate(FrobConst_map_list[i]))}')
# Frobenius psi constants (D type: use SNR, M type use 1/SNR)
print('\nψ (Psi) - Untwist-Frobenius-Twist constants')
print('1/sextic_non_residue: ' + fp2_to_hex(1/SNR))
FrobConst_psi = (1/SNR)^((p-1)/6)
FrobConst_psi_2 = FrobConst_psi * FrobConst_psi
FrobConst_psi_3 = FrobConst_psi_2 * FrobConst_psi
print('FrobConst_psi : ' + fp2_to_hex(FrobConst_psi))
print('FrobConst_psi_2 : ' + fp2_to_hex(FrobConst_psi_2))
print('FrobConst_psi_3 : ' + fp2_to_hex(FrobConst_psi_3))
print('')
FrobConst_psi2_2 = FrobConst_psi_2 * FrobConst_psi_2**p
FrobConst_psi2_3 = FrobConst_psi_3 * FrobConst_psi_3**p
print('FrobConst_psi2_2 : ' + fp2_to_hex(FrobConst_psi2_2))
print('FrobConst_psi2_3 : ' + fp2_to_hex(FrobConst_psi2_3))
print('')
FrobConst_psi3_2 = FrobConst_psi_2 * FrobConst_psi2_2**p
FrobConst_psi3_3 = FrobConst_psi_3 * FrobConst_psi2_3**p
print('FrobConst_psi3_2 : ' + fp2_to_hex(FrobConst_psi3_2))
print('FrobConst_psi3_3 : ' + fp2_to_hex(FrobConst_psi3_3))
# Recap, with ξ (xi) the sextic non-residue
# psi_2 = ((1/ξ)^((p-1)/6))^2 = (1/ξ)^((p-1)/3)
# psi_3 = psi_2 * (1/ξ)^((p-1)/6) = (1/ξ)^((p-1)/3) * (1/ξ)^((p-1)/6) = (1/ξ)^((p-1)/2)
#
# Reminder, in 𝔽p2, frobenius(a) = a^p = conj(a)
# psi2_2 = psi_2 * psi_2^p = (1/ξ)^((p-1)/3) * (1/ξ)^((p-1)/3)^p = (1/ξ)^((p-1)/3) * frobenius((1/ξ))^((p-1)/3)
# = norm(1/ξ)^((p-1)/3)
# psi2_3 = psi_3 * psi_3^p = (1/ξ)^((p-1)/2) * (1/ξ)^((p-1)/2)^p = (1/ξ)^((p-1)/2) * frobenius((1/ξ))^((p-1)/2)
# = norm(1/ξ)^((p-1)/2)
#
# In Fp²:
# - quadratic non-residues respect the equation a^((p²-1)/2) ≡ -1 (mod p²) by the Legendre symbol
# - sextic non-residues are also quadratic non-residues so ξ^((p²-1)/2) ≡ -1 (mod p²)
# - QRT(1/a) = QRT(a) with QRT the quadratic residuosity test
#
# We have norm(ξ)^((p-1)/2) = (ξ*frobenius(ξ))^((p-1)/2) = (ξ*(ξ^p))^((p-1)/2) = ξ^(p+1)^(p-1)/2
# = ξ^((p²-1)/2)
# And ξ^((p²-1)/2) ≡ -1 (mod p²)
# So psi2_3 ≡ -1 (mod p²)
#
# TODO: explain why psi3_2 = [0, -1]
# Frobenius Fp2
A = Fp2([5, 7])
Aconj = Fp2([5, -7])
AF = A.frobenius(1) # or pth_power(1)
AF2 = A.frobenius(2)
AF3 = A.frobenius(3)
print('')
print('A : ' + fp2_to_hex(A))
print('A conjugate: ' + fp2_to_hex(Aconj))
print('')
print('AF1 : ' + fp2_to_hex(AF))
print('AF2 : ' + fp2_to_hex(AF2))
print('AF3 : ' + fp2_to_hex(AF3))
def psi(P):
(Px, Py, Pz) = P
return G2([
FrobConst_psi_2 * Px.frobenius(),
FrobConst_psi_3 * Py.frobenius()
# Pz.frobenius() - Always 1 after extract
])
def psi2(P):
(Px, Py, Pz) = P
return G2([
FrobConst_psi2_2 * Px.frobenius(2),
FrobConst_psi2_3 * Py.frobenius(2)
# Pz - Always 1 after extract
])
def clearCofactorG2(P):
return cofactorG2 * P
# Test generator
set_random_seed(1337)
# Vectors
print('\nTest vectors:')
for i in range(4):
P = G2.random_point()
P = clearCofactorG2(P)
(Px, Py, Pz) = P
vPx = vector(Px)
vPy = vector(Py)
# Pz = vector(Pz)
print(f'\nTest {i}')
print(' Px: ' + Integer(vPx[0]).hex() + ' + β * ' + Integer(vPx[1]).hex())
print(' Py: ' + Integer(vPy[0]).hex() + ' + β * ' + Integer(vPy[1]).hex())
# Galbraith-Lin-Scott, 2008, Theorem 1
# Fuentes-Castaneda et al, 2011, Equation (2)
assert psi(psi(P)) - t*psi(P) + p*P == G2([0, 1, 0]), "Always true"
# Galbraith-Scott, 2008, Lemma 1
# k-th cyclotomic polynomial with k = 12
assert psi2(psi2(P)) - psi2(P) + P == G2([0, 1, 0]), "Always true"
assert psi(psi(P)) == psi2(P), "Always true"
(Qx, Qy, Qz) = psi(P)
vQx = vector(Qx)
vQy = vector(Qy)
print(' Qx: ' + Integer(vQx[0]).hex() + ' + β * ' + Integer(vQx[1]).hex())
print(' Qy: ' + Integer(vQy[0]).hex() + ' + β * ' + Integer(vQy[1]).hex())
assert psi(P) == (p % r) * P, "Can be false if the cofactor was not cleared"

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sage/frobenius_bn254_nogami.sage
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# Constantine
# Copyright (c) 2018-2019 Status Research & Development GmbH
# Copyright (c) 2020-Present Mamy André-Ratsimbazafy
# Licensed and distributed under either of
# * MIT license (license terms in the root directory or at http://opensource.org/licenses/MIT).
# * Apache v2 license (license terms in the root directory or at http://www.apache.org/licenses/LICENSE-2.0).
# at your option. This file may not be copied, modified, or distributed except according to those terms.
# ############################################################
#
# BN254-Nogami
# Frobenius Endomorphism
# Untwist-Frobenius-Twist isogeny
#
# ############################################################
# Parameters
x = -(2^62 + 2^55 + 1)
p = 36*x^4 + 36*x^3 + 24*x^2 + 6*x + 1
r = 36*x^4 + 36*x^3 + 18*x^2 + 6*x + 1
t = 6*x^2 + 1
print('p : ' + p.hex())
print('r : ' + r.hex())
print('t : ' + t.hex())
print('p (mod r) == t-1 (mod r) == 0x' + (p % r).hex())
# Finite fields
Fp = GF(p)
K2.<u> = PolynomialRing(Fp)
Fp2.<beta> = Fp.extension(u^2+1)
# Curves
b = 2
SNR = Fp2([1, 1])
G1 = EllipticCurve(Fp, [0, b])
G2 = EllipticCurve(Fp2, [0, b/SNR])
# https://crypto.stackexchange.com/questions/64064/order-of-twisted-curve-in-pairings
# https://math.stackexchange.com/questions/144194/how-to-find-the-order-of-elliptic-curve-over-finite-field-extension
cofactorG1 = G1.order() // r
cofactorG2 = G2.order() // r
print('')
print('cofactor G1: ' + cofactorG1.hex())
print('cofactor G2: ' + cofactorG2.hex())
print('')
# Utilities
def fp2_to_hex(a):
v = vector(a)
return '0x' + Integer(v[0]).hex() + ' + β * ' + '0x' + Integer(v[1]).hex()
# Frobenius map constants
print('\nFrobenius extension field constants')
FrobConst_map = SNR^((p-1)/6)
FrobConst_map_list = []
cur = Fp2([1, 0])
for i in range(6):
FrobConst_map_list.append(cur)
print(f'FrobConst_map_{i} : {fp2_to_hex(cur)}')
cur *= FrobConst_map
print('')
for i in range(6):
print(f'FrobConst_map_{i}_pow2 : {fp2_to_hex(FrobConst_map_list[i]*conjugate(FrobConst_map_list[i]))}')
print('')
for i in range(6):
print(f'FrobConst_map_{i}_pow3 : {fp2_to_hex(FrobConst_map_list[i]**2 * conjugate(FrobConst_map_list[i]))}')
# Frobenius psi constants (D type: use SNR, M type use 1/SNR)
print('\nψ (Psi) - Untwist-Frobenius-Twist constants')
FrobConst_psi = SNR^((p-1)/6)
FrobConst_psi_2 = FrobConst_psi * FrobConst_psi
FrobConst_psi_3 = FrobConst_psi_2 * FrobConst_psi
print('FrobConst_psi : ' + fp2_to_hex(FrobConst_psi))
print('FrobConst_psi_2 : ' + fp2_to_hex(FrobConst_psi_2))
print('FrobConst_psi_3 : ' + fp2_to_hex(FrobConst_psi_3))
print('')
FrobConst_psi2_2 = FrobConst_psi_2 * FrobConst_psi_2^p
FrobConst_psi2_3 = FrobConst_psi_3 * FrobConst_psi_3^p
print('FrobConst_psi2_2 : ' + fp2_to_hex(FrobConst_psi2_2))
print('FrobConst_psi2_3 : ' + fp2_to_hex(FrobConst_psi2_3))
print('')
FrobConst_psi3_2 = FrobConst_psi_2 * FrobConst_psi2_2^p
FrobConst_psi3_3 = FrobConst_psi_3 * FrobConst_psi2_3^p
print('FrobConst_psi3_2 : ' + fp2_to_hex(FrobConst_psi3_2))
print('FrobConst_psi3_3 : ' + fp2_to_hex(FrobConst_psi3_3))
# Recap, with ξ (xi) the sextic non-residue
# psi_2 = (ξ^((p-1)/6))^2 = ξ^((p-1)/3)
# psi_3 = psi_2 * ξ^((p-1)/6) = ξ^((p-1)/3) * ξ^((p-1)/6) = ξ^((p-1)/2)
#
# Reminder, in 𝔽p2, frobenius(a) = a^p = conj(a)
# psi2_2 = psi_2 * psi_2^p = ξ^((p-1)/3) * ξ^((p-1)/3)^p = ξ^((p-1)/3) * frobenius(ξ)^((p-1)/3)
# = norm(ξ)^((p-1)/3)
# psi2_3 = psi_3 * psi_3^p = ξ^((p-1)/2) * ξ^((p-1)/2)^p = ξ^((p-1)/2) * frobenius(ξ)^((p-1)/2)
# = norm(ξ)^((p-1)/2)
#
# In Fp²:
# - quadratic non-residues respect the equation a^((p²-1)/2) ≡ -1 (mod p²) by the Legendre symbol
# - sextic non-residues are also quadratic non-residues so ξ^((p²-1)/2) ≡ -1 (mod p²)
#
# We have norm(ξ)^((p-1)/2) = (ξ*frobenius(ξ))^((p-1)/2) = (ξ*(ξ^p))^((p-1)/2) = ξ^(p+1)^(p-1)/2
# = ξ^((p²-1)/2)
# And ξ^((p²-1)/2) ≡ -1 (mod p²)
# So psi2_3 ≡ -1 (mod p²)
# Frobenius Fp2
A = Fp2([5, 7])
Aconj = Fp2([5, -7])
AF = A.frobenius(1) # or pth_power(1)
AF2 = A.frobenius(2)
AF3 = A.frobenius(3)
print('')
print('A : ' + fp2_to_hex(A))
print('A conjugate: ' + fp2_to_hex(Aconj))
print('')
print('AF1 : ' + fp2_to_hex(AF))
print('AF2 : ' + fp2_to_hex(AF2))
print('AF3 : ' + fp2_to_hex(AF3))
def psi(P):
(Px, Py, Pz) = P
return G2([
FrobConst_psi_2 * Px.frobenius(1),
FrobConst_psi_3 * Py.frobenius(1)
# Pz.frobenius() - Always 1 after extract
])
def psi2(P):
(Px, Py, Pz) = P
return G2([
FrobConst_psi2_2 * Px.frobenius(2),
FrobConst_psi2_3 * Py.frobenius(2)
# Pz - Always 1 after extract
])
def clearCofactorG2(P):
return cofactorG2 * P
# Test generator
set_random_seed(1337)
# Vectors
print('\nTest vectors:')
for i in range(4):
P = G2.random_point()
P = clearCofactorG2(P)
(Px, Py, Pz) = P
vPx = vector(Px)
vPy = vector(Py)
# vPz = vector(Pz)
print(f'\nTest {i}')
print(' Px: ' + Integer(vPx[0]).hex() + ' + β * ' + Integer(vPx[1]).hex())
print(' Py: ' + Integer(vPy[0]).hex() + ' + β * ' + Integer(vPy[1]).hex())
# print(' Pz: ' + Integer(vPz[0]).hex() + ' + β * ' + Integer(vPz[1]).hex())
# Galbraith-Lin-Scott, 2008, Theorem 1
# Fuentes-Castaneda et al, 2011, Equation (2)
assert psi(psi(P)) - t*psi(P) + p*P == G2([0, 1, 0])
# Galbraith-Scott, 2008, Lemma 1
# k-th cyclotomic polynomial with k = 12
assert psi2(psi2(P)) - psi2(P) + P == G2([0, 1, 0])
assert psi(psi(P)) == psi2(P)
(Qx, Qy, Qz) = psi(P)
vQx = vector(Qx)
vQy = vector(Qy)
print(' Qx: ' + Integer(vQx[0]).hex() + ' + β * ' + Integer(vQx[1]).hex())
print(' Qy: ' + Integer(vQy[0]).hex() + ' + β * ' + Integer(vQy[1]).hex())
(Rx, Ry, Rz) = (p % r) * P
vRx = vector(Rx)
vRy = vector(Ry)
print(' Rx: ' + Integer(vRx[0]).hex() + ' + β * ' + Integer(vRx[1]).hex())
print(' Ry: ' + Integer(vRy[0]).hex() + ' + β * ' + Integer(vRy[1]).hex())
assert psi(P) == (p % r) * P, "Can be false if the cofactor was not cleared"

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sage/frobenius_bn254_snarks.sage
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# Constantine
# Copyright (c) 2018-2019 Status Research & Development GmbH
# Copyright (c) 2020-Present Mamy André-Ratsimbazafy
# Licensed and distributed under either of
# * MIT license (license terms in the root directory or at http://opensource.org/licenses/MIT).
# * Apache v2 license (license terms in the root directory or at http://www.apache.org/licenses/LICENSE-2.0).
# at your option. This file may not be copied, modified, or distributed except according to those terms.
# ############################################################
#
# BN254-Snarks
# Frobenius Endomorphism
# Untwist-Frobenius-Twist isogeny
#
# ############################################################
# Parameters
x = Integer('0x44E992B44A6909F1')
p = 36*x^4 + 36*x^3 + 24*x^2 + 6*x + 1
r = 36*x^4 + 36*x^3 + 18*x^2 + 6*x + 1
t = 6*x^2 + 1
cofactor = 1
print('p : ' + p.hex())
print('r : ' + r.hex())
print('t : ' + t.hex())
print('p (mod r) == t-1 (mod r) == 0x' + (p % r).hex())
# Finite fields
Fp = GF(p)
K2.<u> = PolynomialRing(Fp)
Fp2.<beta> = Fp.extension(u^2+1)
# Curves
b = 3
SNR = Fp2([9, 1])
G1 = EllipticCurve(Fp, [0, b])
G2 = EllipticCurve(Fp2, [0, b/SNR])
# https://crypto.stackexchange.com/questions/64064/order-of-twisted-curve-in-pairings
# https://math.stackexchange.com/questions/144194/how-to-find-the-order-of-elliptic-curve-over-finite-field-extension
cofactorG1 = G1.order() // r
cofactorG2 = G2.order() // r
print('')
print('cofactor G1: ' + cofactorG1.hex())
print('cofactor G2: ' + cofactorG2.hex())
print('')
# Utilities
def fp2_to_hex(a):
v = vector(a)
return '0x' + Integer(v[0]).hex() + ' + β * ' + '0x' + Integer(v[1]).hex()
# Frobenius map constants
print('\nFrobenius extension field constants')
FrobConst_map = SNR^((p-1)/6)
FrobConst_map_list = []
cur = Fp2([1, 0])
for i in range(6):
FrobConst_map_list.append(cur)
print(f'FrobConst_map_{i} : {fp2_to_hex(cur)}')
cur *= FrobConst_map
print('')
for i in range(6):
print(f'FrobConst_map_{i}_pow2 : {fp2_to_hex(FrobConst_map_list[i]*conjugate(FrobConst_map_list[i]))}')
print('')
for i in range(6):
print(f'FrobConst_map_{i}_pow3 : {fp2_to_hex(FrobConst_map_list[i]**2 * conjugate(FrobConst_map_list[i]))}')
# Frobenius psi constants (D type: use SNR, M type use 1/SNR)
print('\nψ (Psi) - Untwist-Frobenius-Twist constants')
FrobConst_psi = SNR^((p-1)/6)
FrobConst_psi_2 = FrobConst_psi * FrobConst_psi
FrobConst_psi_3 = FrobConst_psi_2 * FrobConst_psi
print('FrobConst_psi : ' + fp2_to_hex(FrobConst_psi))
print('FrobConst_psi_2 : ' + fp2_to_hex(FrobConst_psi_2))
print('FrobConst_psi_3 : ' + fp2_to_hex(FrobConst_psi_3))
print('')
FrobConst_psi2_2 = FrobConst_psi_2 * FrobConst_psi_2^p
FrobConst_psi2_3 = FrobConst_psi_3 * FrobConst_psi_3^p
print('FrobConst_psi2_2 : ' + fp2_to_hex(FrobConst_psi2_2))
print('FrobConst_psi2_3 : ' + fp2_to_hex(FrobConst_psi2_3))
print('')
FrobConst_psi3_2 = FrobConst_psi_2 * FrobConst_psi2_2^p
FrobConst_psi3_3 = FrobConst_psi_3 * FrobConst_psi2_3^p
print('FrobConst_psi3_2 : ' + fp2_to_hex(FrobConst_psi3_2))
print('FrobConst_psi3_3 : ' + fp2_to_hex(FrobConst_psi3_3))
# Recap, with ξ (xi) the sextic non-residue
# psi_2 = (ξ^((p-1)/6))^2 = ξ^((p-1)/3)
# psi_3 = psi_2 * ξ^((p-1)/6) = ξ^((p-1)/3) * ξ^((p-1)/6) = ξ^((p-1)/2)
#
# Reminder, in 𝔽p2, frobenius(a) = a^p = conj(a)
# psi2_2 = psi_2 * psi_2^p = ξ^((p-1)/3) * ξ^((p-1)/3)^p = ξ^((p-1)/3) * frobenius(ξ)^((p-1)/3)
# = norm(ξ)^((p-1)/3)
# psi2_3 = psi_3 * psi_3^p = ξ^((p-1)/2) * ξ^((p-1)/2)^p = ξ^((p-1)/2) * frobenius(ξ)^((p-1)/2)
# = norm(ξ)^((p-1)/2)
#
# In Fp²:
# - quadratic non-residues respect the equation a^((p²-1)/2) ≡ -1 (mod p²) by the Legendre symbol
# - sextic non-residues are also quadratic non-residues so ξ^((p²-1)/2) ≡ -1 (mod p²)
#
# We have norm(ξ)^((p-1)/2) = (ξ*frobenius(ξ))^((p-1)/2) = (ξ*(ξ^p))^((p-1)/2) = ξ^(p+1)^(p-1)/2
# = ξ^((p²-1)/2)
# And ξ^((p²-1)/2) ≡ -1 (mod p²)
# So psi2_3 ≡ -1 (mod p²)
# Frobenius Fp2
A = Fp2([5, 7])
Aconj = Fp2([5, -7])
AF = A.frobenius(1) # or pth_power(1)
AF2 = A.frobenius(2)
AF3 = A.frobenius(3)
print('')
print('A : ' + fp2_to_hex(A))
print('A conjugate: ' + fp2_to_hex(Aconj))
print('')
print('AF1 : ' + fp2_to_hex(AF))
print('AF2 : ' + fp2_to_hex(AF2))
print('AF3 : ' + fp2_to_hex(AF3))
def psi(P):
(Px, Py, Pz) = P
return G2([
FrobConst_psi_2 * Px.frobenius(1),
FrobConst_psi_3 * Py.frobenius(1)
# Pz.frobenius() - Always 1 after extract
])
def psi2(P):
(Px, Py, Pz) = P
return G2([
FrobConst_psi2_2 * Px.frobenius(2),
FrobConst_psi2_3 * Py.frobenius(2)
# Pz - Always 1 after extract
])
def clearCofactorG2(P):
return cofactorG2 * P
# Test generator
set_random_seed(1337)
# Vectors
print('\nTest vectors:')
for i in range(4):
P = G2.random_point()
P = clearCofactorG2(P)
(Px, Py, Pz) = P
vPx = vector(Px)
vPy = vector(Py)
# vPz = vector(Pz)
print(f'\nTest {i}')
print(' Px: ' + Integer(vPx[0]).hex() + ' + β * ' + Integer(vPx[1]).hex())
print(' Py: ' + Integer(vPy[0]).hex() + ' + β * ' + Integer(vPy[1]).hex())
# print(' Pz: ' + Integer(vPz[0]).hex() + ' + β * ' + Integer(vPz[1]).hex())
# Galbraith-Lin-Scott, 2008, Theorem 1
# Fuentes-Castaneda et al, 2011, Equation (2)
assert psi(psi(P)) - t*psi(P) + p*P == G2([0, 1, 0])
# Galbraith-Scott, 2008, Lemma 1
# k-th cyclotomic polynomial with k = 12
assert psi2(psi2(P)) - psi2(P) + P == G2([0, 1, 0])
assert psi(psi(P)) == psi2(P)
(Qx, Qy, Qz) = psi(P)
vQx = vector(Qx)
vQy = vector(Qy)
print(' Qx: ' + Integer(vQx[0]).hex() + ' + β * ' + Integer(vQx[1]).hex())
print(' Qy: ' + Integer(vQy[0]).hex() + ' + β * ' + Integer(vQy[1]).hex())
assert psi(P) == (p % r) * P, "Can be false if the cofactor was not cleared"

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sage/lattice_decomposition_bls12_377_g1.sage
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# Constantine
# Copyright (c) 2018-2019 Status Research & Development GmbH
# Copyright (c) 2020-Present Mamy André-Ratsimbazafy
# Licensed and distributed under either of
# * MIT license (license terms in the root directory or at http://opensource.org/licenses/MIT).
# * Apache v2 license (license terms in the root directory or at http://www.apache.org/licenses/LICENSE-2.0).
# at your option. This file may not be copied, modified, or distributed except according to those terms.
# ############################################################
#
# BLS12-381 GLS Endomorphism
# Lattice Decomposition
#
# ############################################################
# Parameters
x = 3 * 2^46 * (7 * 13 * 499) + 1
p = (x - 1)^2 * (x^4 - x^2 + 1)//3 + x
r = x^4 - x^2 + 1
print('p : ' + p.hex())
print('r : ' + r.hex())
# Cube root of unity (mod r) formula for any BLS12 curves
lambda1_r = x^2 - 1
assert lambda1_r^3 % r == 1
print('λᵩ1 : ' + lambda1_r.hex())
print('λᵩ1+r: ' + (lambda1_r+r).hex())
lambda2_r = x^4
assert lambda2_r^3 % r == 1
print('λᵩ2 : ' + lambda2_r.hex())
# Finite fields
F = GF(p)
# Curves
b = 1
G1 = EllipticCurve(F, [0, b])
cofactorG1 = G1.order() // r
print('')
print('cofactor G1: ' + cofactorG1.hex())
print('')
(phi1, phi2) = (root for root in GF(p)(1).nth_root(3, all=True) if root != 1)
print('𝜑1 :' + Integer(phi1).hex())
print('𝜑2 :' + Integer(phi2).hex())
assert phi1^3 % p == 1
assert phi2^3 % p == 1
def clearCofactorG1(P):
return cofactorG1 * P
# Test generator
set_random_seed(1337)
# Check
def checkEndo():
Prand = G1.random_point()
P = clearCofactorG1(Prand)
assert P != G1([0, 1, 0]) # Infinity
(Px, Py, Pz) = P
Qendo1 = G1([Px*phi1 % p, Py, Pz])
Qendo2 = G1([Px*phi2 % p, Py, Pz])
Q1 = lambda1_r * P
Q2 = lambda2_r * P
assert P != Q1
assert P != Q2
assert (F(Px)*F(phi1))^3 == F(Px)^3
assert (F(Px)*F(phi2))^3 == F(Px)^3
assert Q1 == Qendo1
assert Q2 == Qendo1
print('Endomorphism OK with 𝜑1')
checkEndo()
# Decomposition generated by LLL-algorithm and Babai rounding
# to solve the Shortest (Basis) Vector Problem
# Lattice from Guide to Pairing-Based Cryptography
Lat = [
[x^2-1, -1],
[1, x^2]
]
ahat = [x^2, 1]
n = int(r).bit_length()
n = int(((n + 64 - 1) // 64) * 64) # round to next multiple of 64
v = [Integer(a << n) // r for a in ahat]
def pretty_print_lattice(Lat):
latHex = [['0x' + x.hex() if x >= 0 else '-0x' + (-x).hex() for x in vec] for vec in Lat]
maxlen = max([len(cell) for row in latHex for cell in row])
for row in latHex:
row = ' '.join(cell.rjust(maxlen + 2) for cell in row)
print(row)
print('\nLattice')
pretty_print_lattice(Lat)
print('\nbasis:')
print(' 𝛼\u03050: 0x' + v[0].hex())
print(' 𝛼\u03051: 0x' + v[1].hex())
print('')
maxInfNorm = abs(x^2 + 1)
print('\nmax infinity norm:')
print(' ||(a0 , a1)||∞ ≤ 0x' + str(maxInfNorm.hex()))
print(' infinity norm bitlength: ' + str(int(maxInfNorm).bit_length()))
# Contrary to Faz2013 paper, we use the max infinity norm
# to properly dimension our recoding instead of ⌈log2 r/m⌉ + 1
# which fails for some inputs
#
# +1 for signed column
# Optional +1 for handling negative miniscalars
L = int(maxInfNorm).bit_length() + 1
L += 1
def getGLV1_decomp(scalar):
maxLen = (int(r).bit_length() + 1) // 2 + 1
a0 = (v[0] * scalar) >> n
a1 = (v[1] * scalar) >> n
k0 = scalar - a0 * Lat[0][0] - a1 * Lat[1][0]
k1 = 0 - a0 * Lat[0][1] - a1 * Lat[1][1]
assert int(k0).bit_length() <= maxLen
assert int(k1).bit_length() <= maxLen
assert scalar == (k0 + k1 * (lambda1_r % r)) % r
assert scalar == (k0 + k1 * (lambda2_r % r)) % r
return k0, k1
def recodeScalars(k):
m = 2
b = [[0] * L, [0] * L]
b[0][L-1] = 0
for i in range(0, L-1): # l-2 inclusive
b[0][i] = 1 - ((k[0] >> (i+1)) & 1)
for j in range(1, m):
for i in range(0, L):
b[j][i] = k[j] & 1
k[j] = k[j]//2 + (b[j][i] & b[0][i])
return b
def buildLut(P0, P1):
m = 2
lut = [0] * (1 << (m-1))
lut[0] = P0
lut[1] = P0 + P1
return lut
def pointToString(P):
(Px, Py, Pz) = P
return '(x: ' + Integer(Px).hex() + ', y: ' + Integer(Py).hex() + ', z: ' + Integer(Pz).hex() + ')'
def scalarMulEndo(scalar, P0):
m = 2
print('L: ' + str(L))
print('scalar: ' + Integer(scalar).hex())
k0, k1 = getGLV1_decomp(scalar)
print('k0: ' + k0.hex())
print('k1: ' + k1.hex())
P1 = (lambda1_r % r) * P0
(Px, Py, Pz) = P0
P1_endo = G1([Px*phi1 % p, Py, Pz])
assert P1 == P1_endo
expected = scalar * P0
decomp = k0*P0 + k1*P1
assert expected == decomp
print('------ recode scalar -----------')
even = k0 & 1 == 0
if even:
k0 += 1
b = recodeScalars([k0, k1])
print('b0: ' + str(list(reversed(b[0]))))
print('b1: ' + str(list(reversed(b[1]))))
print('------------ lut ---------------')
lut = buildLut(P0, P1)
print('------------ mul ---------------')
# b[0][L-1] is always 0
Q = lut[b[1][L-1]]
for i in range(L-2, -1, -1):
Q *= 2
Q += (1 - 2 * b[0][i]) * lut[b[1][i]]
if even:
Q -= P0
print('final Q: ' + pointToString(Q))
print('expected: ' + pointToString(expected))
assert Q == expected
# Test generator
set_random_seed(1337)
for i in range(1):
print('---------------------------------------')
scalar = randrange(r) # Pick an integer below curve order
P = G1.random_point()
P = clearCofactorG1(P)
scalarMulEndo(scalar, P)

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sage/lattice_decomposition_bls12_377_g2.sage
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# Constantine
# Copyright (c) 2018-2019 Status Research & Development GmbH
# Copyright (c) 2020-Present Mamy André-Ratsimbazafy
# Licensed and distributed under either of
# * MIT license (license terms in the root directory or at http://opensource.org/licenses/MIT).
# * Apache v2 license (license terms in the root directory or at http://www.apache.org/licenses/LICENSE-2.0).
# at your option. This file may not be copied, modified, or distributed except according to those terms.
# ############################################################
#
# BLS12-381 GLS Endomorphism
# Lattice Decomposition
#
# ############################################################
# Parameters
x = 3 * 2^46 * (7 * 13 * 499) + 1
p = (x - 1)^2 * (x^4 - x^2 + 1)//3 + x
r = x^4 - x^2 + 1
t = x+1
print(' Prime modulus p: 0x' + p.hex())
print(' Curve order r: 0x' + r.hex())
print(' trace t: 0x' + t.hex())
# Finite fields
Fp = GF(p)
K2.<u> = PolynomialRing(Fp)
Fp2.<beta> = Fp.extension(u^2+5)
SNR = Fp2([0, 1]) # Sextic Non-Residue for Sextic Twist
# Curves
b = 1
G1 = EllipticCurve(Fp, [0, b])
G2 = EllipticCurve(Fp2, [0, b/SNR])
# https://crypto.stackexchange.com/questions/64064/order-of-twisted-curve-in-pairings
# https://math.stackexchange.com/questions/144194/how-to-find-the-order-of-elliptic-curve-over-finite-field-extension
cofactorG1 = G1.order() // r
cofactorG2 = G2.order() // r
print('')
print('cofactor G1: ' + cofactorG1.hex())
print('cofactor G2: ' + cofactorG2.hex())
print('')
# Frobenius constants (D type: use SNR, M type use 1/SNR)
FrobConst_psi = SNR^((p-1)/6)
FrobConst_psi_2 = FrobConst_psi * FrobConst_psi
FrobConst_psi_3 = FrobConst_psi_2 * FrobConst_psi
FrobConst_psi2_2 = FrobConst_psi_2 * FrobConst_psi_2^p
FrobConst_psi2_3 = FrobConst_psi_3 * FrobConst_psi_3^p
def psi(P):
(Px, Py, Pz) = P
return G2([
FrobConst_psi_2 * Px.frobenius(1),
FrobConst_psi_3 * Py.frobenius(1)
# Pz.frobenius() - Always 1 after extract
])
def psi2(P):
(Px, Py, Pz) = P
return G2([
FrobConst_psi2_2 * Px.frobenius(2),
FrobConst_psi2_3 * Py.frobenius(2)
# Pz - Always 1 after extract
])
def clearCofactorG2(P):
return cofactorG2 * P
# Test generator
set_random_seed(1337)
# Check
def checkEndo():
P = G2.random_point()
P = clearCofactorG2(P)
(Px, Py, Pz) = P
# Galbraith-Lin-Scott, 2008, Theorem 1
assert psi(psi(P)) - t*psi(P) + p*P == G2([0, 1, 0])
# Galbraith-Scott, 2008, Lemma 1
# k-th cyclotomic polynomial with k = 12
assert psi2(psi2(P)) - psi2(P) + P == G2([0, 1, 0])
assert p % r == (t-1) % r
# assert (p^4 - p^2 + 1) % r == 0
assert ((t-1)^4 - (t-1)^2 + 1) % r == 0
assert (t-1)*P == (p % r)*P
assert (t-1)*P == psi(P)
print('Endomorphism OK')
checkEndo()
def subgroup_check(P):
ppP = psi2(P)
assert x * psi(ppP) - ppP + P == G2([0,1,0])
# Decomposition generated by LLL-algorithm and Babai rounding
# to solve the Shortest (Basis) Vector Problem
#
# TODO: This lattice is generating wrong result
# Lattice from Guide to Pairing-Based Cryptography
# Lat = [
# [ x, 1, 0, 0],
# [ 0, x, 1, 0],
# [ 0, 0, x, 1],
# [ 1, 0,-1, x]
# ]
# ahat = [x*(x^2+1), -(x^2+1), x, -1]
# Lattice from my own LLL+Babai rounding routines
Lat = Matrix([
[-x, 1, 0, 0],
[ 0,-x, 1, 0],
[ 0, 0,-x, 1],
[ 1, 0,-1,-x]
])
# print('Lat: ' + str(Lat))
ahat = vector([r, 0, 0, 0]) * Lat.inverse()
# print('ahat: ' + str(ahat))
n = int(r).bit_length()
n = int(((n + 64 - 1) // 64) * 64) # round to next multiple of 64
v = [Integer(int(a) << n) // r for a in ahat]
def pretty_print_lattice(Lat):
latHex = [['0x' + x.hex() if x >= 0 else '-0x' + (-x).hex() for x in vec] for vec in Lat]
maxlen = max([len(cell) for row in latHex for cell in row])
for row in latHex:
row = ' '.join(cell.rjust(maxlen + 2) for cell in row)
print(row)
print('\nLattice')
pretty_print_lattice(Lat)
print('\nbasis:')
print(' 𝛼\u03050: 0x' + v[0].hex())
print(' 𝛼\u03051: 0x' + v[1].hex())
print(' 𝛼\u03052: 0x' + v[2].hex())
print(' 𝛼\u03053: 0x' + v[3].hex())
print('')
maxInfNorm = abs(x + 2)
print('\nmax infinity norm:')
print(' ||(a0, a1, a2, a3)||∞ ≤ 0x' + str(maxInfNorm.hex()))
print(' infinity norm bitlength: ' + str(int(maxInfNorm).bit_length()))
# Contrary to Faz2013 paper, we use the max infinity norm
# to properly dimension our recoding instead of ⌈log2 r/m⌉ + 1
# which fails for some inputs
# +1 for signed column
# Optional +1 for handling negative miniscalars
L = int(maxInfNorm).bit_length() + 1
L += 1
lambda1 = (t-1) % r
lambda2 = lambda1^2 % r
lambda3 = lambda1^3 % r
def getGLV2_decomp(scalar):
maxLen = (int(r).bit_length() + 3) // 4 + 1
maxLen += 1 # Deal with negative miniscalars
a0 = (v[0] * scalar) >> n
a1 = (v[1] * scalar) >> n
a2 = (v[2] * scalar) >> n
a3 = (v[3] * scalar) >> n
k0 = scalar - a0 * Lat[0][0] - a1 * Lat[1][0] - a2 * Lat[2][0] - a3 * Lat[3][0]
k1 = 0 - a0 * Lat[0][1] - a1 * Lat[1][1] - a2 * Lat[2][1] - a3 * Lat[3][1]
k2 = 0 - a0 * Lat[0][2] - a1 * Lat[1][2] - a2 * Lat[2][2] - a3 * Lat[3][2]
k3 = 0 - a0 * Lat[0][3] - a1 * Lat[1][3] - a2 * Lat[2][3] - a3 * Lat[3][3]
print("k0.bitlength(): " + str(int(k0).bit_length()))
print("k1.bitlength(): " + str(int(k1).bit_length()))
print("k2.bitlength(): " + str(int(k2).bit_length()))
print("k3.bitlength(): " + str(int(k3).bit_length()))
print('k0: ' + k0.hex())
print('k1: ' + k1.hex())
print('k2: ' + k2.hex())
print('k3: ' + k3.hex())
assert scalar == (k0 + k1*lambda1 + k2*lambda2 + k3*lambda3) % r
assert int(k0).bit_length() <= maxLen
assert int(k1).bit_length() <= maxLen
assert int(k2).bit_length() <= maxLen
assert int(k3).bit_length() <= maxLen
return k0, k1, k2, k3
def recodeScalars(k):
m = 4
b = [[0] * L, [0] * L, [0] * L, [0] * L]
b[0][L-1] = 0
for i in range(0, L-1): # l-2 inclusive
b[0][i] = 1 - ((k[0] >> (i+1)) & 1)
for j in range(1, m):
for i in range(0, L):
b[j][i] = k[j] & 1
k[j] = k[j]//2 + (b[j][i] & b[0][i])
return b
def clearBit(v, bit):
return v & ~int(1 << bit)
def buildLut(P0, P_endos):
m = 4
assert len(P_endos) == m-1
lut = [0] * (1 << (m-1))
lut[0] = P0
lutS = [''] * (1 << (m-1))
lutS[0] = 'P0'
endoS = ['P1', 'P2', 'P3']
for u in range(1, 1 << (m-1)):
msb = u.bit_length() - 1
idx = clearBit(u, msb)
lut[u] = lut[clearBit(u, msb)] + P_endos[msb]
lutS[u] = lutS[clearBit(u, msb)] + ' + ' + endoS[msb]
print('LUT: ' + str(lutS))
return lut
def pointToString(P):
(Px, Py, Pz) = P
vPx = vector(Px)
vPy = vector(Py)
result = 'Point(\n'
result += ' Px: ' + Integer(vPx[0]).hex() + ' + β * ' + Integer(vPx[1]).hex() + '\n'
result += ' Py: ' + Integer(vPy[0]).hex() + ' + β * ' + Integer(vPy[1]).hex() + '\n'
result += ')'
return result
def getIndex(glvRecoding, bit):
m = 4
index = 0
for k in range(1, m):
index |= ((glvRecoding[k][bit] & 1) << (k-1))
return index
def updateFactors(factors, recoded, bit):
index = getIndex(recoded, bit)
if recoded[0][bit] == 0: # Positive
factors[0] += 1
factors[1] += (index >> 0) & 1
factors[2] += (index >> 1) & 1
factors[3] += (index >> 2) & 1
else:
factors[0] -= 1
factors[1] -= (index >> 0) & 1
factors[2] -= (index >> 1) & 1
factors[3] -= (index >> 2) & 1
def doubleFactors(factors):
for i in range(len(factors)):
factors[i] *= 2
def printFactors(factors):
print('Multiplication done: ')
for i in range(len(factors)):
print(f' f{i}: {factors[i].hex()}')
def scalarMulEndo(scalar, P0):
m = 4
print('L: ' + str(L))
print('scalar: ' + Integer(scalar).hex())
k0, k1, k2, k3 = getGLV2_decomp(scalar)
P1 = psi(P0)
P2 = psi2(P0)
P3 = psi(P2)
expected = scalar * P0
decomp = k0*P0 + k1*P1 + k2*P2 + k3*P3
print('expected: ' + pointToString(expected))
print('decomp: ' + pointToString(decomp))
assert expected == decomp
# Alternative to adding an extra bit
# to deal with miniscalars
# if k0 < 0: k0 = -k0; P0 = -P0
# if k1 < 0: k1 = -k1; P1 = -P1
# if k2 < 0: k2 = -k2; P2 = -P2
# if k3 < 0: k3 = -k3; P3 = -P3
# assert expected == k0*P0 + k1*P1 + k2*P2 + k3*P3
# Somehow the recoding doesn't cope with first scalar being negative
if k0 < 0:
k0 = -k0
P0 = -P0
print('------ recode scalar -----------')
even = k0 & 1 == 0
print('was even: ' + str(even))
if even:
k0 += 1
b = recodeScalars([k0, k1, k2, k3])
print('b0: ' + str(list(reversed(b[0]))))
print('b1: ' + str(list(reversed(b[1]))))
print('b2: ' + str(list(reversed(b[2]))))
print('b3: ' + str(list(reversed(b[3]))))
print('------------ lut ---------------')
lut = buildLut(P0, [P1, P2, P3])
print('------------ mul ---------------')
# b[0][L-1] is always 0
print(f'L-1: {getIndex(b, L-1)}')
print(f'L-2: {getIndex(b, L-2)}')
print(f'L-3: {getIndex(b, L-3)}')
print(f'L-4: {getIndex(b, L-4)}')
print(f'L-5: {getIndex(b, L-5)}')
print(f'L-6: {getIndex(b, L-6)}')
factors = [0, 0, 0, 0] # Track the decomposed scalar applied (debugging)
updateFactors(factors, b, L-1)
Q = lut[getIndex(b, L-1)]
for bit in range(L-2, -1, -1):
Q *= 2
Q += (1 - 2 * b[0][bit]) * lut[getIndex(b, bit)]
doubleFactors(factors)
updateFactors(factors, b, bit)
if even:
Q -= P0
print('----')
print('final Q: ' + pointToString(Q))
print('expected: ' + pointToString(expected))
print('----')
printFactors(factors)
print('Mul expected:')
print(' k0: ' + k0.hex())
print(' k1: ' + k1.hex())
print(' k2: ' + k2.hex())
print(' k3: ' + k3.hex())
assert Q == expected
# Test generator
set_random_seed(1337)
for i in range(1):
print('---------------------------------------')
scalar = randrange(r) # Pick an integer below curve order
# P = G2.random_point()
# P = clearCofactorG2(P)
scalar = Integer('0x9d432eb58ec68bbc09d10961451d99c7796fb2f795eca603d6feaf3e2a1634b')
P = G2([
Fp2([Integer('0x267401f3ef554fe74ae131d56a10edf14ae40192654901b4618d2bf7af22e77c2a9b79e407348dbd4aad13ca73b33a'),
Integer('0x12dcca838f46a3e0418e5dd8b978362757a16bfd78f0b77f4a1916ace353938389ae3ea228d0eb5020a0aaa58884aec')]),
Fp2([Integer('0x11799118d2e054aabd9f74c0843fecbdc1c0d56f61c61c5854c2507ae2416e48a6b2cd3bc8bf7495a4d3d8270eafe2b'),
Integer('0x823b9f8fb9f8297734a14359fa2c2a0de275e7e638197eaaaa7cff28f9cb3101bdabb570016672455f1ecae625e294')])
])
subgroup_check(P)
scalarMulEndo(scalar, P)

227
sage/lattice_decomposition_bls12_381_g1.sage
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@ -1,227 +0,0 @@
# Constantine
# Copyright (c) 2018-2019 Status Research & Development GmbH
# Copyright (c) 2020-Present Mamy André-Ratsimbazafy
# Licensed and distributed under either of
# * MIT license (license terms in the root directory or at http://opensource.org/licenses/MIT).
# * Apache v2 license (license terms in the root directory or at http://www.apache.org/licenses/LICENSE-2.0).
# at your option. This file may not be copied, modified, or distributed except according to those terms.
# ############################################################
#
# BLS12-381 GLS Endomorphism
# Lattice Decomposition
#
# ############################################################
# Parameters
x = -(2^63 + 2^62 + 2^60 + 2^57 + 2^48 + 2^16)
p = (x - 1)^2 * (x^4 - x^2 + 1)//3 + x
r = x^4 - x^2 + 1
print('p : ' + p.hex())
print('r : ' + r.hex())
# Cube root of unity (mod r) formula for any BLS12 curves
lambda1_r = x^2 - 1
assert lambda1_r^3 % r == 1
print('λᵩ1 : ' + lambda1_r.hex())
print('λᵩ1+r: ' + (lambda1_r+r).hex())
lambda2_r = x^4
assert lambda2_r^3 % r == 1
print('λᵩ2 : ' + lambda2_r.hex())
# Finite fields
F = GF(p)
# Curves
b = 4
G1 = EllipticCurve(F, [0, b])
cofactorG1 = G1.order() // r
print('')
print('cofactor G1: ' + cofactorG1.hex())
print('')
(phi1, phi2) = (root for root in GF(p)(1).nth_root(3, all=True) if root != 1)
print('𝜑1 :' + Integer(phi1).hex())
print('𝜑2 :' + Integer(phi2).hex())
assert phi1^3 % p == 1
assert phi2^3 % p == 1
def clearCofactorG1(P):
return cofactorG1 * P
# Test generator
set_random_seed(1337)
# Check
def checkEndo():
Prand = G1.random_point()
P = clearCofactorG1(Prand)
assert P != G1([0, 1, 0]) # Infinity
(Px, Py, Pz) = P
Qendo1 = G1([Px*phi1 % p, Py, Pz])
Qendo2 = G1([Px*phi2 % p, Py, Pz])
Q1 = lambda1_r * P
Q2 = lambda2_r * P
assert P != Q1
assert P != Q2
assert (F(Px)*F(phi1))^3 == F(Px)^3
assert (F(Px)*F(phi2))^3 == F(Px)^3
assert Q1 == Qendo2
assert Q2 == Qendo2
print('Endomorphism OK with 𝜑2')
checkEndo()
# Decomposition generated by LLL-algorithm and Babai rounding
# to solve the Shortest (Basis) Vector Problem
# Lattice from Guide to Pairing-Based Cryptography
Lat = [
[x^2-1, -1],
[1, x^2]
]
ahat = [x^2, 1]
n = int(r).bit_length()
n = int(((n + 64 - 1) // 64) * 64) # round to next multiple of 64
v = [Integer(a << n) // r for a in ahat]
def pretty_print_lattice(Lat):
latHex = [['0x' + x.hex() if x >= 0 else '-0x' + (-x).hex() for x in vec] for vec in Lat]
maxlen = max([len(cell) for row in latHex for cell in row])
for row in latHex:
row = ' '.join(cell.rjust(maxlen + 2) for cell in row)
print(row)
print('\nLattice')
pretty_print_lattice(Lat)
print('\nbasis:')
print(' 𝛼\u03050: 0x' + v[0].hex())
print(' 𝛼\u03051: 0x' + v[1].hex())
print('')
maxInfNorm = abs(x + 2)
print('\nmax infinity norm:')
print(' ||(a0, a1)||∞ ≤ 0x' + str(maxInfNorm.hex()))
print(' infinity norm bitlength: ' + str(int(maxInfNorm).bit_length()))
# Contrary to Faz2013 paper, we use the max infinity norm
# to properly dimension our recoding instead of ⌈log2 r/m⌉ + 1
# which fails for some inputs
# +1 for signed column
# Optional +1 for handling negative miniscalars
L = int(maxInfNorm).bit_length() + 1
L += 1
def getGLV1_decomp(scalar):
maxLen = (int(r).bit_length() + 1) // 2 + 1
a0 = (v[0] * scalar) >> n
a1 = (v[1] * scalar) >> n
k0 = scalar - a0 * Lat[0][0] - a1 * Lat[1][0]
k1 = 0 - a0 * Lat[0][1] - a1 * Lat[1][1]
assert int(k0).bit_length() <= maxLen
assert int(k1).bit_length() <= maxLen
assert scalar == (k0 + k1 * (lambda1_r % r)) % r
assert scalar == (k0 + k1 * (lambda2_r % r)) % r
return k0, k1
def recodeScalars(k):
m = 2
b = [[0] * L, [0] * L]
b[0][L-1] = 0
for i in range(0, L-1): # l-2 inclusive
b[0][i] = 1 - ((k[0] >> (i+1)) & 1)
for j in range(1, m):
for i in range(0, L):
b[j][i] = k[j] & 1
k[j] = k[j]//2 + (b[j][i] & b[0][i])
return b
def buildLut(P0, P1):
m = 2
lut = [0] * (1 << (m-1))
lut[0] = P0
lut[1] = P0 + P1
return lut
def pointToString(P):
(Px, Py, Pz) = P
return '(x: ' + Integer(Px).hex() + ', y: ' + Integer(Py).hex() + ', z: ' + Integer(Pz).hex() + ')'
def scalarMulEndo(scalar, P0):
m = 2
print('L: ' + str(L))
print('scalar: ' + Integer(scalar).hex())
k0, k1 = getGLV1_decomp(scalar)
print('k0: ' + k0.hex())
print('k1: ' + k1.hex())
P1 = (lambda1_r % r) * P0
(Px, Py, Pz) = P0
P1_endo = G1([Px*phi2 % p, Py, Pz])
assert P1 == P1_endo
expected = scalar * P0
decomp = k0*P0 + k1*P1
assert expected == decomp
print('------ recode scalar -----------')
even = k0 & 1 == 0
if even:
k0 += 1
b = recodeScalars([k0, k1])
print('b0: ' + str(list(reversed(b[0]))))
print('b1: ' + str(list(reversed(b[1]))))
print('------------ lut ---------------')
lut = buildLut(P0, P1)
print('------------ mul ---------------')
# b[0][L-1] is always 0
Q = lut[b[1][L-1]]
for i in range(L-2, -1, -1):
Q *= 2
Q += (1 - 2 * b[0][i]) * lut[b[1][i]]
if even:
Q -= P0
print('final Q: ' + pointToString(Q))
print('expected: ' + pointToString(expected))
assert Q == expected
# Test generator
set_random_seed(1337)
for i in range(1):
print('---------------------------------------')
# scalar = randrange(r) # Pick an integer below curve order
# P = G1.random_point()
# P = clearCofactorG1(P)
scalar = Integer('0xf7e60a832eb77ac47374bc93251360d6c81c21add62767ff816caf11a20d8db')
P = G1([
Integer('0xf9679bb02ee7f352fff6a6467a5e563ec8dd38c86a48abd9e8f7f241f1cdd29d54bc3ddea3a33b62e0d7ce22f3d244a'),
Integer('0x50189b992cf856846b30e52205ff9ef72dc081e9680726586231cbc29a81a162120082585f401e00382d5c86fb1083f'),
Integer(1)
])
scalarMulEndo(scalar, P)

415
sage/lattice_decomposition_bls12_381_g2.sage
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@ -1,415 +0,0 @@
# Constantine
# Copyright (c) 2018-2019 Status Research & Development GmbH
# Copyright (c) 2020-Present Mamy André-Ratsimbazafy
# Licensed and distributed under either of
# * MIT license (license terms in the root directory or at http://opensource.org/licenses/MIT).
# * Apache v2 license (license terms in the root directory or at http://www.apache.org/licenses/LICENSE-2.0).
# at your option. This file may not be copied, modified, or distributed except according to those terms.
# ############################################################
#
# BLS12-381 GLS Endomorphism
# Lattice Decomposition
#
# ############################################################
# Parameters
x = -(2^63 + 2^62 + 2^60 + 2^57 + 2^48 + 2^16)
p = (x - 1)^2 * (x^4 - x^2 + 1)//3 + x
r = x^4 - x^2 + 1
t = x+1
print(' Prime modulus p: 0x' + p.hex())
print(' Curve order r: 0x' + r.hex())
print(' trace t: 0x' + t.hex())
# Finite fields
Fp = GF(p)
K2.<u> = PolynomialRing(Fp)
Fp2.<beta> = Fp.extension(u^2+1)
SNR = Fp2([1, 1]) # Sextic Non-Residue for Sextic Twist
# Curves
b = 4
G1 = EllipticCurve(Fp, [0, b])
G2 = EllipticCurve(Fp2, [0, b*SNR])
# https://crypto.stackexchange.com/questions/64064/order-of-twisted-curve-in-pairings
# https://math.stackexchange.com/questions/144194/how-to-find-the-order-of-elliptic-curve-over-finite-field-extension
cofactorG1 = G1.order() // r
cofactorG2 = G2.order() // r
print('')
print('cofactor G1: ' + cofactorG1.hex())
print('cofactor G2: ' + cofactorG2.hex())
print('')
# Frobenius constants (D type: use SNR, M type use 1/SNR)
FrobConst_psi = (1/SNR)^((p-1)/6)
FrobConst_psi_2 = FrobConst_psi * FrobConst_psi
FrobConst_psi_3 = FrobConst_psi_2 * FrobConst_psi
FrobConst_psi2_2 = FrobConst_psi_2 * FrobConst_psi_2^p
FrobConst_psi2_3 = FrobConst_psi_3 * FrobConst_psi_3^p
def psi(P):
(Px, Py, Pz) = P
return G2([
FrobConst_psi_2 * Px.frobenius(1),
FrobConst_psi_3 * Py.frobenius(1)
# Pz.frobenius() - Always 1 after extract
])
def psi2(P):
(Px, Py, Pz) = P
return G2([
FrobConst_psi2_2 * Px.frobenius(2),
FrobConst_psi2_3 * Py.frobenius(2)
# Pz - Always 1 after extract
])
def clearCofactorG2(P):
return cofactorG2 * P
# Test generator
set_random_seed(1337)
# Check
def checkEndo():
P = G2.random_point()
P = clearCofactorG2(P)
(Px, Py, Pz) = P
# Galbraith-Lin-Scott, 2008, Theorem 1
assert psi(psi(P)) - t*psi(P) + p*P == G2([0, 1, 0])
# Galbraith-Scott, 2008, Lemma 1
# k-th cyclotomic polynomial with k = 12
assert psi2(psi2(P)) - psi2(P) + P == G2([0, 1, 0])
assert p % r == (t-1) % r
# assert (p^4 - p^2 + 1) % r == 0
assert ((t-1)^4 - (t-1)^2 + 1) % r == 0
assert (t-1)*P == (p % r)*P
assert (t-1)*P == psi(P)
print('Endomorphism OK')
checkEndo()
def subgroup_check(P):
ppP = psi2(P)
assert x * psi(ppP) - ppP + P == G2([0,1,0])
# Decomposition generated by LLL-algorithm and Babai rounding
# to solve the Shortest (Basis) Vector Problem
#
# TODO: This lattice is generating wrong result
# Lattice from Guide to Pairing-Based Cryptography
# Lat = [
# [ x, 1, 0, 0],
# [ 0, x, 1, 0],
# [ 0, 0, x, 1],
# [ 1, 0,-1, x]
# ]
# ahat = [x*(x^2+1), -(x^2+1), x, -1]
# Lattice from my own LLL+Babai rounding routines
Lat = Matrix([
[-x, 1, 0, 0],
[ 0,-x, 1, 0],
[ 0, 0,-x, 1],
[ 1, 0,-1,-x]
])
# print('Lat: ' + str(Lat))
ahat = vector([r, 0, 0, 0]) * Lat.inverse()
# print('ahat: ' + str(ahat))
n = int(r).bit_length()
n = int(((n + 64 - 1) // 64) * 64) # round to next multiple of 64
v = [Integer(int(a) << n) // r for a in ahat]
def pretty_print_lattice(Lat):
latHex = [['0x' + x.hex() if x >= 0 else '-0x' + (-x).hex() for x in vec] for vec in Lat]
maxlen = max([len(cell) for row in latHex for cell in row])
for row in latHex:
row = ' '.join(cell.rjust(maxlen + 2) for cell in row)
print(row)
print('\nLattice')
pretty_print_lattice(Lat)
print('\nbasis:')
print(' 𝛼\u03050: 0x' + v[0].hex())
print(' 𝛼\u03051: 0x' + v[1].hex())
print(' 𝛼\u03052: 0x' + v[2].hex())
print(' 𝛼\u03053: 0x' + v[3].hex())
print('')
maxInfNorm = abs(x + 2)
print('\nmax infinity norm:')
print(' ||(a0 , a1 , a2 , a3)||∞ ≤ 0x' + str(maxInfNorm.hex()))
print(' infinity norm bitlength: ' + str(int(maxInfNorm).bit_length()))
# Contrary to Faz2013 paper, we use the max infinity norm
# to properly dimension our recoding instead of ⌈log2 r/m⌉ + 1
# which fails for some inputs
# +1 for signed column
# Optional +1 for handling negative miniscalars
L = int(maxInfNorm).bit_length() + 1
L += 1
lambda1 = (t-1) % r
lambda2 = lambda1^2 % r
lambda3 = lambda1^3 % r
def getGLV2_decomp(scalar):
maxLen = (int(r).bit_length() + 3) // 4 + 1
maxLen += 1 # Deal with negative miniscalars
a0 = (v[0] * scalar) >> n
a1 = (v[1] * scalar) >> n
a2 = (v[2] * scalar) >> n
a3 = (v[3] * scalar) >> n
k0 = scalar - a0 * Lat[0][0] - a1 * Lat[1][0] - a2 * Lat[2][0] - a3 * Lat[3][0]
k1 = 0 - a0 * Lat[0][1] - a1 * Lat[1][1] - a2 * Lat[2][1] - a3 * Lat[3][1]
k2 = 0 - a0 * Lat[0][2] - a1 * Lat[1][2] - a2 * Lat[2][2] - a3 * Lat[3][2]
k3 = 0 - a0 * Lat[0][3] - a1 * Lat[1][3] - a2 * Lat[2][3] - a3 * Lat[3][3]
print("k0.bitlength(): " + str(int(k0).bit_length()))
print("k1.bitlength(): " + str(int(k1).bit_length()))
print("k2.bitlength(): " + str(int(k2).bit_length()))
print("k3.bitlength(): " + str(int(k3).bit_length()))
print('k0: ' + k0.hex())
print('k1: ' + k1.hex())
print('k2: ' + k2.hex())
print('k3: ' + k3.hex())
assert scalar == (k0 + k1*lambda1 + k2*lambda2 + k3*lambda3) % r
assert int(k0).bit_length() <= maxLen
assert int(k1).bit_length() <= maxLen
assert int(k2).bit_length() <= maxLen
assert int(k3).bit_length() <= maxLen
return k0, k1, k2, k3
def recodeScalars(k):
m = 4
b = [[0] * L, [0] * L, [0] * L, [0] * L]
b[0][L-1] = 0
for i in range(0, L-1): # l-2 inclusive
b[0][i] = 1 - ((k[0] >> (i+1)) & 1)
for j in range(1, m):
for i in range(0, L):
b[j][i] = k[j] & 1
k[j] = k[j]//2 + (b[j][i] & b[0][i])
return b
def clearBit(v, bit):
return v & ~int(1 << bit)
def buildLut(P0, P_endos):
m = 4
assert len(P_endos) == m-1
lut = [0] * (1 << (m-1))
lut[0] = P0
lutS = [''] * (1 << (m-1))
lutS[0] = 'P0'
endoS = ['P1', 'P2', 'P3']
for u in range(1, 1 << (m-1)):
msb = u.bit_length() - 1
idx = clearBit(u, msb)
lut[u] = lut[clearBit(u, msb)] + P_endos[msb]
lutS[u] = lutS[clearBit(u, msb)] + ' + ' + endoS[msb]
print('LUT: ' + str(lutS))
return lut
def pointToString(P):
(Px, Py, Pz) = P
vPx = vector(Px)
vPy = vector(Py)
result = 'Point(\n'
result += ' Px: ' + Integer(vPx[0]).hex() + ' + β * ' + Integer(vPx[1]).hex() + '\n'
result += ' Py: ' + Integer(vPy[0]).hex() + ' + β * ' + Integer(vPy[1]).hex() + '\n'
result += ')'
return result
def getIndex(glvRecoding, bit):
m = 4
index = 0
for k in range(1, m):
index |= ((glvRecoding[k][bit] & 1) << (k-1))
return index
def updateFactors(factors, recoded, bit):
index = getIndex(recoded, bit)
if recoded[0][bit] == 0: # Positive
factors[0] += 1
factors[1] += (index >> 0) & 1
factors[2] += (index >> 1) & 1
factors[3] += (index >> 2) & 1
else:
factors[0] -= 1
factors[1] -= (index >> 0) & 1
factors[2] -= (index >> 1) & 1
factors[3] -= (index >> 2) & 1
def doubleFactors(factors):
for i in range(len(factors)):
factors[i] *= 2
def printFactors(factors):
print('Multiplication done: ')
for i in range(len(factors)):
print(f' f{i}: {factors[i].hex()}')
def scalarMulEndo(scalar, P0):
m = 4
print('r bits: ' + str(int(r).bit_length()))
print('L: ' + str(L))
print('scalar: ' + Integer(scalar).hex())
k0, k1, k2, k3 = getGLV2_decomp(scalar)
P1 = psi(P0)
P2 = psi2(P0)
P3 = psi(P2)
expected = scalar * P0
decomp = k0*P0 + k1*P1 + k2*P2 + k3*P3
print('expected: ' + pointToString(expected))
print('decomp: ' + pointToString(decomp))
assert expected == decomp
# Alternative to adding an extra bit
# to deal with miniscalars, unfortunately broken
# for some input
# for example 0x5668a2332db27199dcfb7cbdfca6317c2ff128db26d7df68483e0a095ec8e88f
# which is missing bits for b[2]
# if k0 < 0: k0 = -k0; P0 = -P0
# if k1 < 0: k1 = -k1; P1 = -P1
# if k2 < 0: k2 = -k2; P2 = -P2
# if k3 < 0: k3 = -k3; P3 = -P3
print('------ recode scalar -----------')
even = k0 & 1 == 0
print('was even: ' + str(even))
if even:
k0 += 1
b = recodeScalars([k0, k1, k2, k3])
print('b0: ' + str(list(reversed(b[0]))))
print('b1: ' + str(list(reversed(b[1]))))
print('b2: ' + str(list(reversed(b[2]))))
print('b3: ' + str(list(reversed(b[3]))))
print('------------ lut ---------------')
lut = buildLut(P0, [P1, P2, P3])
print('------------ mul ---------------')
# b[0][L-1] is always 0
print(f'L-1: {getIndex(b, L-1)}')
print(f'L-2: {getIndex(b, L-2)}')
print(f'L-3: {getIndex(b, L-3)}')
print(f'L-4: {getIndex(b, L-4)}')
print(f'L-5: {getIndex(b, L-5)}')
print(f'L-6: {getIndex(b, L-6)}')
factors = [0, 0, 0, 0] # Track the decomposed scalar applied (debugging)
updateFactors(factors, b, L-1)
Q = lut[getIndex(b, L-1)]
for bit in range(L-2, -1, -1):
Q *= 2
Q += (1 - 2 * b[0][bit]) * lut[getIndex(b, bit)]
doubleFactors(factors)
updateFactors(factors, b, bit)
if even:
Q -= P0
print('----')
print('final Q: ' + pointToString(Q))
print('expected: ' + pointToString(expected))
print('----')
printFactors(factors)
print('Mul expected:')
print(' k0: ' + k0.hex())
print(' k1: ' + k1.hex())
print(' k2: ' + k2.hex())
print(' k3: ' + k3.hex())
assert Q == expected
# Test generator
set_random_seed(1337)
for i in range(1):
print('---------------------------------------')
# scalar = randrange(r) # Pick an integer below curve order
# P = G2.random_point()
# P = clearCofactorG2(P)
# scalar = Integer('0x1f7bef2a74f3bf8ac0225a9edfa514bb5666b15e7be3e929059f2ef75f0035a6')
# P = G2([
# Fp2([Integer('0x989f16bcb9da60ef72383e6134ba194f57e30109806304336c0c995e2857ed20bf5b6e03d6fe1424332e9c666cbd10a'),
# Integer('0x16692643cb5e7466e3730d3ea775c7741ac34d670b3be685761a7d6ab722a2673ce374ddab87b7c4d2675ba2199f9121')]),
# Fp2([Integer('0x931e416488bef7cb4a053e4bd86ef44818bc03a5be5b04606b2a4dc1d139a3a452f5f7172f24eeaad84702b73b155bb'),
# Integer('0x192c3e2a6619473216b7bb2447448cdbeb9f7e3c9486b0a05aadf6dcd91d7cb275a5d84c1a362628efffbc8711a62a67')])
# ])
# This integer leads to negative miniscalar for proper handling it requires either:
# 1. Negating it and then negating the corresponding curve point P
# 2. Adding an extra bit to the recoding, which will do the right thing™
#
# For implementation solution 1 is faster:
# - Double + Add is about 5000~8000 cycles on 6 64-bits limbs (BLS12-381)
# - Conditional negate is about 10 cycles per Fp, on G2 projective we have 3 (coords) * 2 (Fp2) * 10 (cycles) ~= 60 cycles
# We need to test the mini scalar, which is 65 bits so 2 Fp so about 2 cycles
# and negate it as well.
# scalar = Integer('0x6448f296d9b1a8d81319a0b789df04c587c6165776ccf39f50a354204aabe0da')
# P = G2([
# Fp2([Integer('0x5adc112fb04bf4ca642d5a7d7343ccd6b93546442d2fff5b9d32c15e456d54884cba49dd7f94ce4ddaad4018e55d0f2'),
# Integer('0x5d1c5bbf5d7a833dc76ba206bfa99c281fc37941be050e18f8c6d267b2376b3634d8ad6eb951e52a6d096315abd17d6')]),
# Fp2([Integer('0x15a959e54981fab9ac3c6f5bfd6fb60a50a916bd43d96a09922a54309b84812736581bfa728670cba864b08b9e391bb9'),
# Integer('0xf5d6d74f1dd3d9c07451340b8f6990fe93a28fe5e176564eb920bf17eb02df8b6f1e626eda5542ff415f89d51943001')])
# ])
# The following input fails in Constantine when negating the base point
# but not when adding an extra bit
# scalar = Integer('0x5668a2332db27199dcfb7cbdfca6317c2ff128db26d7df68483e0a095ec8e88f')
# P = G2([
# Fp2([Integer('0xa8c5649d2df1bae84fd9e8bfcde5113937b3acea22d67ddfedaf1fb8de8c1ef4c70591cf505c24c31e54020c2c510c3'),
# Integer('0xa0553f98229a6a067489c3ee204161c11e96f421b3e9c145dc3865b03e9d4ff6cab14c5b5308ecd31173f954463690c')]),
# Fp2([Integer('0xb29d8dfe18dc41b4826c3a102c1bf8f306cb42433cc36ee38080f47a324c02a678f9daed0a2bc577c18b9865de029f0'),
# Integer('0x558cdabf11e37c5c5e8abd668bbdd71bb3f07f320948ccaac8a207359fffe38424bfd9b1ef1d24b28b2fbb9f76faff1')])
# ])
# The following fails when we have both extra bit and negation of the first
# scalar if it is negative.
# it also uses 65 bits instead of teh expected max of 64
# And triggers an off by 1 when negating
scalar = Integer('0x6448f296d9b1a8d81319a0b789df04c587c6165776ccf39f50a354204aabe0da')
P = G2([
Fp2([Integer('0x5adc112fb04bf4ca642d5a7d7343ccd6b93546442d2fff5b9d32c15e456d54884cba49dd7f94ce4ddaad4018e55d0f2'),
Integer('0x5d1c5bbf5d7a833dc76ba206bfa99c281fc37941be050e18f8c6d267b2376b3634d8ad6eb951e52a6d096315abd17d6')]),
Fp2([Integer('0x15a959e54981fab9ac3c6f5bfd6fb60a50a916bd43d96a09922a54309b84812736581bfa728670cba864b08b9e391bb9'),
Integer('0xf5d6d74f1dd3d9c07451340b8f6990fe93a28fe5e176564eb920bf17eb02df8b6f1e626eda5542ff415f89d51943001')])
])
subgroup_check(P)
scalarMulEndo(scalar, P)

219
sage/lattice_decomposition_bn254_snarks_g1.sage
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@ -1,219 +0,0 @@
# Constantine
# Copyright (c) 2018-2019 Status Research & Development GmbH
# Copyright (c) 2020-Present Mamy André-Ratsimbazafy
# Licensed and distributed under either of
# * MIT license (license terms in the root directory or at http://opensource.org/licenses/MIT).
# * Apache v2 license (license terms in the root directory or at http://www.apache.org/licenses/LICENSE-2.0).
# at your option. This file may not be copied, modified, or distributed except according to those terms.
# ############################################################
#
# BN254-Snarks GLV Endomorphism
# Lattice Decomposition
#
# ############################################################
# Parameters
x = Integer('0x44E992B44A6909F1')
p = 36*x^4 + 36*x^3 + 24*x^2 + 6*x + 1
r = 36*x^4 + 36*x^3 + 18*x^2 + 6*x + 1
# Cube root of unity (mod r) formula for any BN curves
lambda1_r = (-(36*x^3+18*x^2+6*x+2))
assert lambda1_r^3 % r == 1
print('λᵩ1 : ' + lambda1_r.hex())
print('λᵩ1+r: ' + (lambda1_r+r).hex())
lambda2_r = (36*x^4-1)
assert lambda2_r^3 % r == 1
print('λᵩ2 : ' + lambda2_r.hex())
# Finite fields
F = GF(p)
# Curves
b = 3
G1 = EllipticCurve(F, [0, b])
cofactorG1 = G1.order() // r
assert cofactorG1 == 1, "BN curve have a prime order"
print('')
print('cofactor G1: ' + cofactorG1.hex())
print('')
(phi1, phi2) = (root for root in GF(p)(1).nth_root(3, all=True) if root != 1)
print('𝜑1 :' + Integer(phi1).hex())
print('𝜑2 :' + Integer(phi2).hex())
# Test generator
set_random_seed(1337)
# Check
def checkEndo():
P = G1.random_point()
assert P != G1([0, 1, 0]) # Infinity
(Px, Py, Pz) = P
Qendo1 = G1([Px*phi1 % p, Py, Pz])
Qendo2 = G1([Px*phi2 % p, Py, Pz])
Q1 = lambda1_r * P
Q2 = lambda2_r * P
assert P != Q1
assert P != Q2
assert (F(Px)*F(phi1))^3 == F(Px)^3
assert (F(Px)*F(phi2))^3 == F(Px)^3
assert Q1 == Qendo1
assert Q2 == Qendo1
print('Endomorphism OK with 𝜑1')
checkEndo()
# Decomposition generated by LLL-algorithm and Babai rounding
# to solve the Shortest (Basis) Vector Problem
# Lattice from Guide to Pairing-Based Cryptography
Lat = [
[2*x+1, 6*x^2+4*x+1],
[6*x^2+2*x, -2*x-1]
]
ahat = [2*x+1, 6*x^2+4*x+1]
n = int(r).bit_length()
n = int(((n + 64 - 1) // 64) * 64) # round to next multiple of 64
v = [Integer(a << n) // r for a in ahat]
def pretty_print_lattice(Lat):
latHex = [['0x' + x.hex() if x >= 0 else '-0x' + (-x).hex() for x in vec] for vec in Lat]
maxlen = max([len(cell) for row in latHex for cell in row])
for row in latHex:
row = ' '.join(cell.rjust(maxlen + 2) for cell in row)
print(row)
print('\nLattice')
pretty_print_lattice(Lat)
print('\nbasis:')
print(' 𝛼\u03050: 0x' + v[0].hex())
print(' 𝛼\u03051: 0x' + v[1].hex())
print('')
maxInfNorm = abs(6*x^2+6*x+2)
print('\nmax infinity norm:')
print(' ||(a0, a1)||∞ ≤ 0x' + str(maxInfNorm.hex()))
print(' infinity norm bitlength: ' + str(int(maxInfNorm).bit_length()))
# Contrary to Faz2013 paper, we use the max infinity norm
# to properly dimension our recoding instead of ⌈log2 r/m⌉ + 1
# which fails for some inputs
# +1 for signed column
# Optional +1 for handling negative miniscalars
L = int(maxInfNorm).bit_length() + 1
L += 1
def getGLV2_decomp(scalar):
maxLen = (int(r).bit_length() + 1) // 2 + 1
a0 = (v[0] * scalar) >> n
a1 = (v[1] * scalar) >> n
k0 = scalar - a0 * Lat[0][0] - a1 * Lat[1][0]
k1 = 0 - a0 * Lat[0][1] - a1 * Lat[1][1]
assert int(k0).bit_length() <= maxLen
assert int(k1).bit_length() <= maxLen
assert scalar == (k0 + k1 * (lambda1_r % r)) % r
assert scalar == (k0 + k1 * (lambda2_r % r)) % r
return k0, k1
def recodeScalars(k):
m = 2
b = [[0] * L, [0] * L]
b[0][L-1] = 0
for i in range(0, L-1): # l-2 inclusive
b[0][i] = 1 - ((k[0] >> (i+1)) & 1)
for j in range(1, m):
for i in range(0, L):
b[j][i] = k[j] & 1
k[j] = k[j]//2 + (b[j][i] & b[0][i])
return b
def buildLut(P0, P1):
m = 2
lut = [0] * (1 << (m-1))
lut[0] = P0
lut[1] = P0 + P1
return lut
def pointToString(P):
(Px, Py, Pz) = P
return '(x: ' + Integer(Px).hex() + ', y: ' + Integer(Py).hex() + ', z: ' + Integer(Pz).hex() + ')'
def scalarMulEndo(scalar, P0):
m = 2
print('L: ' + str(L))
print('scalar: ' + Integer(scalar).hex())
k0, k1 = getGLV2_decomp(scalar)
print('k0: ' + k0.hex())
print('k1: ' + k1.hex())
P1 = (lambda1_r % r) * P0
(Px, Py, Pz) = P0
P1_endo = G1([Px*phi1 % p, Py, Pz])
assert P1 == P1_endo
expected = scalar * P0
decomp = k0*P0 + k1*P1
assert expected == decomp
print('------ recode scalar -----------')
even = k0 & 1 == 0
if even:
k0 += 1
b = recodeScalars([k0, k1])
print('b0: ' + str(list(reversed(b[0]))))
print('b1: ' + str(list(reversed(b[1]))))
print('------------ lut ---------------')
lut = buildLut(P0, P1)
print('------------ mul ---------------')
# b[0][L-1] is always 0
Q = lut[b[1][L-1]]
for i in range(L-2, -1, -1):
Q *= 2
Q += (1 - 2 * b[0][i]) * lut[b[1][i]]
if even:
Q -= P0
print('final Q: ' + pointToString(Q))
print('expected: ' + pointToString(expected))
assert Q == expected
# Test generator
set_random_seed(1337)
for i in range(1):
print('---------------------------------------')
# scalar = randrange(r) # Pick an integer below curve order
# P = G1.random_point()
scalar = Integer('0x0e08a292f940cfb361cc82bc24ca564f51453708c9745a9cf8707b11c84bc448')
P = G1([
Integer('0x22d3af0f3ee310df7fc1a2a204369ac13eb4a48d969a27fcd2861506b2dc0cd7'),
Integer('0x1c994169687886ccd28dd587c29c307fb3cab55d796d73a5be0bbf9aab69912e'),
Integer(1)
])
scalarMulEndo(scalar, P)

385
sage/lattice_decomposition_bn254_snarks_g2.sage
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@ -1,385 +0,0 @@
# Constantine
# Copyright (c) 2018-2019 Status Research & Development GmbH
# Copyright (c) 2020-Present Mamy André-Ratsimbazafy
# Licensed and distributed under either of
# * MIT license (license terms in the root directory or at http://opensource.org/licenses/MIT).
# * Apache v2 license (license terms in the root directory or at http://www.apache.org/licenses/LICENSE-2.0).
# at your option. This file may not be copied, modified, or distributed except according to those terms.
# ############################################################
#
# BN254-Snarks GLS Endomorphism
# Lattice Decomposition
#
# ############################################################
# Parameters
x = Integer('0x44E992B44A6909F1')
p = 36*x^4 + 36*x^3 + 24*x^2 + 6*x + 1
r = 36*x^4 + 36*x^3 + 18*x^2 + 6*x + 1
t = 6*x^2 + 1
print(' Prime modulus p: 0x' + p.hex())
print(' Curve order r: 0x' + r.hex())
print(' trace t: 0x' + t.hex())
# Finite fields
Fp = GF(p)
K2.<u> = PolynomialRing(Fp)
Fp2.<beta> = Fp.extension(u^2+1)
SNR = Fp2([9, 1]) # Sextic Non-Residue for Sextic Twist
# Curves
b = 3
G1 = EllipticCurve(Fp, [0, b])
G2 = EllipticCurve(Fp2, [0, b/SNR])
# https://crypto.stackexchange.com/questions/64064/order-of-twisted-curve-in-pairings
# https://math.stackexchange.com/questions/144194/how-to-find-the-order-of-elliptic-curve-over-finite-field-extension
cofactorG1 = G1.order() // r
cofactorG2 = G2.order() // r
print('')
print('cofactor G1: ' + cofactorG1.hex())
print('cofactor G2: ' + cofactorG2.hex())
print('')
# Frobenius constants (D type: use SNR, M type use 1/SNR)
FrobConst_psi = SNR^((p-1)/6)
FrobConst_psi_2 = FrobConst_psi * FrobConst_psi
FrobConst_psi_3 = FrobConst_psi_2 * FrobConst_psi
FrobConst_psi2_2 = FrobConst_psi_2 * FrobConst_psi_2^p
FrobConst_psi2_3 = FrobConst_psi_3 * FrobConst_psi_3^p
def psi(P):
(Px, Py, Pz) = P
return G2([
FrobConst_psi_2 * Px.frobenius(1),
FrobConst_psi_3 * Py.frobenius(1)
# Pz.frobenius() - Always 1 after extract
])
def psi2(P):
(Px, Py, Pz) = P
return G2([
FrobConst_psi2_2 * Px.frobenius(2),
FrobConst_psi2_3 * Py.frobenius(2)
# Pz - Always 1 after extract
])
def clearCofactorG2(P):
return cofactorG2 * P
# Test generator
set_random_seed(1337)
# Check
def checkEndo():
P = G2.random_point()
P = clearCofactorG2(P)
(Px, Py, Pz) = P
# Galbraith-Lin-Scott, 2008, Theorem 1
assert psi(psi(P)) - t*psi(P) + p*P == G2([0, 1, 0])
# Galbraith-Scott, 2008, Lemma 1
# k-th cyclotomic polynomial with k = 12
assert psi2(psi2(P)) - psi2(P) + P == G2([0, 1, 0])
assert p % r == (t-1) % r
# assert (p^4 - p^2 + 1) % r == 0
assert ((t-1)^4 - (t-1)^2 + 1) % r == 0
assert (t-1)*P == (p % r)*P
assert (t-1)*P == psi(P)
print('Endomorphism OK')
checkEndo()
# Decomposition generated by LLL-algorithm and Babai rounding
# to solve the Shortest (Basis) Vector Problem
# TODO: This Lattice from Guide to Pairing-Based Cryptography
# gives miniscalars bigger than (r/4)+1 = 65 bits
# for unknown reason
# The linear combination is correct though.
# Lattice from Guide to Pairing-Based Cryptography
# Lat = [
# [ - x , 2*x , 2*x+1, - x ],
# [ -2*x-1, x , x+1, x ],
# [ 2*x+1, 0, 2*x , 1],
# [ -1, 2*x+1, 1, 2*x ]
# ]
# ahat = [2*x+1, -(12*x^3+6*x^2+2*x+1), 2*x*(3*x^2+3*x+1), 6*x^2-1]
# Lattice from Galbraith-Scott 2008
Lat = Matrix([
[ x+1, x , x , -2*x ],
[ 2*x+1, -x , -x-1, -x ],
[ 2*x , 2*x+1, 2*x+1, 2*x+1],
[ x-1, 4*x+2, -2*x+1, x-1]
])
ahat = [2*x^2+3*x+1, 12*x^3+8*x^2+x, 6*x^3+4*x^2+x, -2*x^2-x]
n = int(r).bit_length()
n = int(((n + 64 - 1) // 64) * 64) # round to next multiple of 64
v = [Integer(a << n) // r for a in ahat]
def pretty_print_lattice(Lat):
latHex = [['0x' + x.hex() if x >= 0 else '-0x' + (-x).hex() for x in vec] for vec in Lat]
maxlen = max([len(cell) for row in latHex for cell in row])
for row in latHex:
row = ' '.join(cell.rjust(maxlen + 2) for cell in row)
print(row)
print('\nLattice')
pretty_print_lattice(Lat)
print('\nbasis:')
print(' 𝛼\u03050: 0x' + v[0].hex())
print(' 𝛼\u03051: 0x' + v[1].hex())
print(' 𝛼\u03052: 0x' + v[2].hex())
print(' 𝛼\u03053: 0x' + v[3].hex())
print('')
maxInfNorm = abs(5*x + 3)
print('\nmax infinity norm:')
print(' ||(a0 , a1 , a2 , a3)||∞ ≤ 0x' + str(maxInfNorm.hex()))
print(' infinity norm bitlength: ' + str(int(maxInfNorm).bit_length()))
# Contrary to Faz2013 paper, we use the max infinity norm
# to properly dimension our recoding instead of ⌈log2 r/m⌉ + 1
# which fails for some inputs
# +1 for signed column
# Optional +1 for handling negative miniscalars
L = int(maxInfNorm).bit_length() + 1
L += 1
lambda1 = (t-1) % r
lambda2 = lambda1^2 % r
lambda3 = lambda1^3 % r
def getGLV2_decomp(scalar):
maxLen = (int(r).bit_length() + 3) // 4 + 1
a0 = (v[0] * scalar) >> n
a1 = (v[1] * scalar) >> n
a2 = (v[2] * scalar) >> n
a3 = (v[3] * scalar) >> n
print('𝛼0: ' + a0.hex())
print('𝛼1: ' + a1.hex())
print('𝛼2: ' + a2.hex())
print('𝛼3: ' + a3.hex())
print('𝛼3 unred: ' + (v[3] * scalar).hex())
print('')
print('Lat[3][0]: ' + Lat[3][0].hex())
print('a3 * Lat[3][0]: ' + (a3 * Lat[3][0]).hex())
print('')
k0 = scalar - a0 * Lat[0][0] - a1 * Lat[1][0] - a2 * Lat[2][0] - a3 * Lat[3][0]
k1 = 0 - a0 * Lat[0][1] - a1 * Lat[1][1] - a2 * Lat[2][1] - a3 * Lat[3][1]
k2 = 0 - a0 * Lat[0][2] - a1 * Lat[1][2] - a2 * Lat[2][2] - a3 * Lat[3][2]
k3 = 0 - a0 * Lat[0][3] - a1 * Lat[1][3] - a2 * Lat[2][3] - a3 * Lat[3][3]
k = [scalar, 0, 0, 0]
a = [a0, a1, a2, a3]
for i in range(4):
for j in range(4):
elem = a[j] * Lat[j][i]
print(f'a{j} * Lat[{j}][{i}] = {elem.hex()}')
k[i] -= elem
print(f' k{i} = {k[i].hex()}')
print('k: ' + str([ki.hex() for ki in k]))
print("k0.bitlength(): " + str(int(k0).bit_length()))
print("k1.bitlength(): " + str(int(k1).bit_length()))
print("k2.bitlength(): " + str(int(k2).bit_length()))
print("k3.bitlength(): " + str(int(k3).bit_length()))
print('k0: ' + k0.hex())
print('k1: ' + k1.hex())
print('k2: ' + k2.hex())
print('k3: ' + k3.hex())
assert scalar == (k0 + k1*lambda1 + k2*lambda2 + k3*lambda3) % r
assert int(k0).bit_length() <= maxLen
assert int(k1).bit_length() <= maxLen
assert int(k2).bit_length() <= maxLen
assert int(k3).bit_length() <= maxLen
return k0, k1, k2, k3
def recodeScalars(k):
m = 4
b = [[0] * L, [0] * L, [0] * L, [0] * L]
b[0][L-1] = 0
for i in range(0, L-1): # l-2 inclusive
b[0][i] = 1 - ((k[0] >> (i+1)) & 1)
for j in range(1, m):
for i in range(0, L):
b[j][i] = k[j] & 1
k[j] = k[j]//2 + (b[j][i] & b[0][i])
return b
def clearBit(v, bit):
return v & ~int(1 << bit)
def buildLut(P0, P_endos):
m = 4
assert len(P_endos) == m-1
lut = [0] * (1 << (m-1))
lut[0] = P0
lutS = [''] * (1 << (m-1))
lutS[0] = 'P0'
endoS = ['P1', 'P2', 'P3']
for u in range(1, 1 << (m-1)):
msb = u.bit_length() - 1
idx = clearBit(u, msb)
lut[u] = lut[clearBit(u, msb)] + P_endos[msb]
lutS[u] = lutS[clearBit(u, msb)] + ' + ' + endoS[msb]
print('LUT: ' + str(lutS))
return lut
def pointToString(P):
(Px, Py, Pz) = P
vPx = vector(Px)
vPy = vector(Py)
result = 'Point(\n'
result += ' Px: ' + Integer(vPx[0]).hex() + ' + β * ' + Integer(vPx[1]).hex() + '\n'
result += ' Py: ' + Integer(vPy[0]).hex() + ' + β * ' + Integer(vPy[1]).hex() + '\n'
result += ')'
return result
def getIndex(glvRecoding, bit):
m = 4
index = 0
for k in range(1, m):
index |= ((glvRecoding[k][bit] & 1) << (k-1))
return index
def updateFactors(factors, recoded, bit):
index = getIndex(recoded, bit)
if recoded[0][bit] == 0: # Positive
factors[0] += 1
factors[1] += (index >> 0) & 1
factors[2] += (index >> 1) & 1
factors[3] += (index >> 2) & 1
else:
factors[0] -= 1
factors[1] -= (index >> 0) & 1
factors[2] -= (index >> 1) & 1
factors[3] -= (index >> 2) & 1
def doubleFactors(factors):
for i in range(len(factors)):
factors[i] *= 2
def printFactors(factors):
for i in range(len(factors)):
print(f'f{i}: {factors[i].hex()}')
def scalarMulEndo(scalar, P0):
m = 4
print('L: ' + str(L))
print('scalar: ' + Integer(scalar).hex())
k0, k1, k2, k3 = getGLV2_decomp(scalar)
P1 = psi(P0)
P2 = psi2(P0)
P3 = psi(P2)
expected = scalar * P0
decomp = k0*P0 + k1*P1 + k2*P2 + k3*P3
print('expected: ' + pointToString(expected))
print('decomp: ' + pointToString(decomp))
assert expected == decomp
print('------ recode scalar -----------')
even = k0 & 1 == 0
print('was even: ' + str(even))
if even:
k0 += 1
b = recodeScalars([k0, k1, k2, k3])
print('b0: ' + str(list(reversed(b[0]))))
print('b1: ' + str(list(reversed(b[1]))))
print('b2: ' + str(list(reversed(b[2]))))
print('b3: ' + str(list(reversed(b[3]))))
print('------------ lut ---------------')
lut = buildLut(P0, [P1, P2, P3])
print('------------ mul ---------------')
# b[0][L-1] is always 0
print(f'L-1: {getIndex(b, L-1)}')
print(f'L-2: {getIndex(b, L-2)}')
print(f'L-3: {getIndex(b, L-3)}')
print(f'L-4: {getIndex(b, L-4)}')
print(f'L-5: {getIndex(b, L-5)}')
print(f'L-6: {getIndex(b, L-6)}')
factors = [0, 0, 0, 0] # Track the decomposed scalar applied (debugging)
updateFactors(factors, b, L-1)
Q = lut[getIndex(b, L-1)]
for bit in range(L-2, -1, -1):
Q *= 2
Q += (1 - 2 * b[0][bit]) * lut[getIndex(b, bit)]
doubleFactors(factors)
updateFactors(factors, b, bit)
if even:
Q -= P0
print('----')
print('final Q: ' + pointToString(Q))
print('expected: ' + pointToString(expected))
print('----')
printFactors(factors)
print('Mul expected:')
print(' k0: ' + k0.hex())
print(' k1: ' + k1.hex())
print(' k2: ' + k2.hex())
print(' k3: ' + k3.hex())
assert Q == expected
# Test generator
set_random_seed(1337)
for i in range(1):
print('---------------------------------------')
# scalar = randrange(r) # Pick an integer below curve order
# P = G2.random_point()
# P = clearCofactorG2(P)
# scalar = Integer('0x2c02275a71bb41c911faf48cab4f7ac7fc6672a5c15586185c8cff3203181da0')
# P = G2([
# Fp2([Integer('0x2a028c1328bb0abf252edfbf7133b84eef2a5f20163fe61685b4b54229ca585d'),
# Integer('0x8f80ad79e8e7e79bbdc645d9f5b339c52dd99a901b90de2494492656f11a9d5')]),
# Fp2([Integer('0x1f04320578e31ffa2e2b59ad8fcb1aba622b5f307ac540cf2ccdab07dec56503'),
# Integer('0x2973900c0fdf651b64f5b1a990baec7c582e0743d501bdb991374776d6c73b28')])
# ])
scalar = Integer('0x24c5b2ce21615dca82231f5fb0fc8d05aa07c6df4bb5aa7c2381ac7b61a6290c')
P = G2([
Fp2([Integer('0x1132e63c444e1abce6fc4c39bdf5be5caad586837cbf5ca9d3891482bdefe77'),
Integer('0x22b71f598dab789f055fc9669ddf66f0d75f581af0e9e8863d7f95a51ef34862')]),
Fp2([Integer('0x58e39050f64c9948d7238b99ecaee947cb934688a6e9f483c5c36b6e07aa31b'),
Integer('0x2e64b920f498e12992f2a4ae3f9ced43f3f64705c9008169f3b930a760d055fb')])
])
scalarMulEndo(scalar, P)

38
sage/lattice_decomposition_finder.sage
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# Constantine
# Copyright (c) 2018-2019 Status Research & Development GmbH
# Copyright (c) 2020-Present Mamy André-Ratsimbazafy
# Licensed and distributed under either of
# * MIT license (license terms in the root directory or at http://opensource.org/licenses/MIT).
# * Apache v2 license (license terms in the root directory or at http://www.apache.org/licenses/LICENSE-2.0).
# at your option. This file may not be copied, modified, or distributed except according to those terms.
# ############################################################
#
# Lattice decomposition finder
#
# ############################################################
# Example of BLS12-381 with the ψ (Psi) - Untwist-Frobenius-Twist endomorphism
x = -(2^63 + 2^62 + 2^60 + 2^57 + 2^48 + 2^16)
p = (x - 1)^2 * (x^4 - x^2 + 1)//3 + x
r = x^4 - x^2 + 1
t = x + 1 # Trace of Frobenius
lambda_psi = t - 1
Lpsi = Matrix([
[ r, 0, 0, 0],
[-lambda_psi, 1, 0, 0],
[-lambda_psi^2, 0, 1, 0],
[-lambda_psi^3, 0, 0, 1],
])
Lpsi = Lpsi.LLL()
print(Lpsi)
ahat = vector([r, 0, 0, 0]) * Lpsi.inverse()
print('ahat: ' + str(ahat))
v = int(r).bit_length()
v = int(((v + 64 - 1) // 64) * 64)
print([(a << v) // r for a in ahat])

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sage/non_residues.sage
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# Constantine
# Copyright (c) 2018-2019 Status Research & Development GmbH
# Copyright (c) 2020-Present Mamy André-Ratsimbazafy
# Licensed and distributed under either of
# * MIT license (license terms in the root directory or at http://opensource.org/licenses/MIT).
# * Apache v2 license (license terms in the root directory or at http://www.apache.org/licenses/LICENSE-2.0).
# at your option. This file may not be copied, modified, or distributed except according to those terms.
# This script checks polynomial irreducibility
#
# Constructing Tower Extensions for the implementation of Pairing-Based Cryptography
# Naomi Benger and Michael Scott, 2009
# https://eprint.iacr.org/2009/556
# Note: Some of the curves here are not pairing friendly and never used in an extension field.
# We still check them to potentially add them as additional test vectors in
# 𝔽p2, 𝔽p6, 𝔽p12, ... since as they are most 0xFF bytes they
# trigger "carry" code-paths that are not triggered by pairing-friendly moduli.
Curves = {
'P224': Integer('0xffffffffffffffffffffffffffffffff000000000000000000000001'),
'BN254_Nogami': Integer('0x2523648240000001ba344d80000000086121000000000013a700000000000013'),
'BN254_Snarks': Integer('0x30644e72e131a029b85045b68181585d97816a916871ca8d3c208c16d87cfd47'),
'Curve25519': Integer('0x7fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffed'),
'P256': Integer('0xffffffff00000001000000000000000000000000ffffffffffffffffffffffff'),
'Secp256k1': Integer('0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F'),
'BLS12_377': Integer('0x01ae3a4617c510eac63b05c06ca1493b1a22d9f300f5138f1ef3622fba094800170b5d44300000008508c00000000001'),
'BLS12_381': Integer('0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2a0f6b0f6241eabfffeb153ffffb9feffffffffaaab'),
'BN446': Integer('0x2400000000000000002400000002d00000000d800000021c0000001800000000870000000b0400000057c00000015c000000132000000067'),
'FKM12_447': Integer('0x4ce300001338c00001c08180000f20cfffffe5a8bffffd08a000000f228000007e8ffffffaddfffffffdc00000009efffffffca000000007'),
'BLS12_461': Integer('0x15555545554d5a555a55d69414935fbd6f1e32d8bacca47b14848b42a8dffa5c1cc00f26aa91557f00400020000555554aaaaaac0000aaaaaaab'),
'BN462': Integer('0x240480360120023ffffffffff6ff0cf6b7d9bfca0000000000d812908f41c8020ffffffffff6ff66fc6ff687f640000000002401b00840138013')
}
def find_quadratic_non_residues(A, B, Field, modulus):
result = false
for a in A:
for b in B:
residue = Fp(a^2 + b^2).residue_symbol(Fp.ideal(modulus),2)
if residue < 0:
print(f' 𝔽p4 = 𝔽p2[v] / v² - ({a} ± {b}𝑖) is an irreducible polynomial')
result = true
return result
def find_cubic_non_residues_pmod3eq1(A, B, modulus):
assert modulus % 3 == 1
result = false
for a in A:
for b in B:
# The following `residue_symbol` is not satisfactory for cubic root
# It just returns exceptions for all values
#
#
# residue = Fp(a^2 + b^2).residue_symbol(Fp.ideal(modulus),3)
# if residue < 0:
# print(f' 𝔽p2[v] / v³ - ({a} ± {b}𝑖) is an irreducible polynomial')
# for p ≡ 1 (mod 3)
# we have ``a`` a cubic residue iff a^((p-1)/3) ≡ 1 (mod p)
residue = pow(a^2 + b^2, (modulus-1)//3, modulus)
if residue != 1:
print(f' 𝔽p6 = 𝔽p2[v] / v³ - ({a} ± {b}𝑖) is a possible extension')
result = true
return result
for curve, modulus in Curves.items():
print(f'Curve {curve}:')
print(f' Modulus 0x{modulus.hex()}:')
pMod4 = modulus % 4
print(f' p mod 4: {pMod4}')
if pMod4 == 3:
# This is actually the hard case, but given that most pairing friendly curves somehow end up in that case
# this is the one we will focus on.
print(f' ^ suggested irreducible polynomial for 𝔽p2: u² + 1 (𝔽p2 complex)')
else:
print(f' ⚠️ p mod 4 != 3: to be reviewed manually. See Theorem 1 of Scott 2009 Constructing Tower Extensions for the implementation of Pairing-Based Cryptography')
print(f' p mod 8: {modulus % 8}')
print(f' p mod 12: {modulus % 12}')
if pMod4 != 3:
print(f' p mod 4 != 3 => find a square/cubic root and then successively adjoin roots of the roots to build the tower.')
print(f' Skipping to next curve.')
continue
Fp.<p> = NumberField(x - 1)
print('')
print(' Searching for valid irreducible polynomials ...')
# Constructing 𝔽p4
print(' 𝔽p4 = 𝔽p2[v] / v² - (a ± 𝑖 b))')
found = find_quadratic_non_residues([0, 1, 2], [1, 2], Fp, modulus)
if not found:
found = find_quadratic_non_residues(range(5), range(1, 5), Fp, modulus)
assert found
found = false
# Constructing 𝔽p6
print(' 𝔽p6 = 𝔽p2[v] / v³ - (a ± 𝑖 b))')
pMod3 = modulus % 3
print(f' p mod 3: {pMod3}')
if pMod3 != 1:
# A remark on the computation of cube roots in finite fields
# https://eprint.iacr.org/2009/457.pdf
print(f' p mod 3 != 1 => to be reviewed manually')
print(f' Skipping to next curve.')
continue
if not found:
found = find_cubic_non_residues_pmod3eq1([0, 1, 2], [1, 2], modulus)
if not found:
found = find_cubic_non_residues_pmod3eq1(range(5), range(1, 5), modulus)
if not found:
found = find_cubic_non_residues_pmod3eq1(range(17), range(1, 17), modulus)
assert found
# ############################################################
#
# Failed experiments of actually instantiating
# the tower of extension fields
#
# ############################################################
# ############################################################
# 1st try
# # Create the field of x ∈ [0, p-1]
# K.<p> = NumberField(x - 1)
#
# # Tower Fp² with Fp[u] / (u² + 1) <=> u = 𝑖
# L.<im> = K.extension(x^2 + 1)
#
# TODO how to make the following work?
# # Tower Fp^6 with Fp²[v] / (v³ - (u + 1))
# M.<xi> = L.extension(x^3 - (im + 1))
# ############################################################
# 2nd try
# # Create the field of u ∈ [0, p-1]
# p = Integer('0x30644e72e131a029b85045b68181585d97816a916871ca8d3c208c16d87cfd47')
# Fp = GF(p)
# Elem.<u> = Fp[]
# print("p mod 4 = ", p % 4)
#
# # Tower Fp² with Fp[u] / (u² + 1) <=> u = 𝑖
# Fp2.<im> = Fp.extension(u^2 + 1)
# Elem2.<v> = Fp2[]
#
# # Tower Fp^6 with Fp²[v] / (v³ - (u + 1))
# Fp6.<xi> = Fp.extension(v^3 - (im + 1))
# Elem6.<w> = Fp6[]
# ############################################################
# 3rd try
# K.<xi, im, p> = NumberField([x^3 - I - 1, x^2 + 1, x - 1])
# ############################################################
# 4th try, just trying to verify Fp6
# print('Verifying non-residues')
# modulus = Integer('0x30644e72e131a029b85045b68181585d97816a916871ca8d3c208c16d87cfd47')
# Fp.<p> = NumberField(x - 1)
# r1 = Fp(-1).residue_symbol(Fp.ideal(modulus),2)
# print('Fp² = Fp[sqrt(-1)]: ' + str(r1))
# Fp2.<im> = Fp.extension(x^2 + 1)
# xi = Fp2(1+im)
# r2 = xi.residue_symbol(Fp2.ideal(modulus),3)
# # ValueError: The residue symbol to that power is not defined for the number field
# # ^ AFAIK that means that Fp2 doesn't contain the 3rd root of unity
# # so we are clear
# print('Fp6 = Fp²[cubicRoot(1+I)]: ' + str(r2))

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sage/testgen_bls12_377.sage
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# Constantine
# Copyright (c) 2018-2019 Status Research & Development GmbH
# Copyright (c) 2020-Present Mamy André-Ratsimbazafy
# Licensed and distributed under either of
# * MIT license (license terms in the root directory or at http://opensource.org/licenses/MIT).
# * Apache v2 license (license terms in the root directory or at http://www.apache.org/licenses/LICENSE-2.0).
# at your option. This file may not be copied, modified, or distributed except according to those terms.
# ############################################################
#
# BN254 test generator
#
# ############################################################
# Parameters
x = 3 * 2^46 * (7 * 13 * 499) + 1
p = (x - 1)^2 * (x^4 - x^2 + 1)//3 + x
r = x^4 - x^2 + 1
# Finite fields
Fp = GF(p)
K2.<u> = PolynomialRing(Fp)
Fp2.<beta> = Fp.extension(u^2+5)
# Curves
b = 1
SNR = Fp2([0, 1])
G1 = EllipticCurve(Fp, [0, b])
G2 = EllipticCurve(Fp2, [0, b/SNR])
# https://crypto.stackexchange.com/questions/64064/order-of-twisted-curve-in-pairings
# https://math.stackexchange.com/questions/144194/how-to-find-the-order-of-elliptic-curve-over-finite-field-extension
cofactorG1 = G1.order() // r
cofactorG2 = G2.order() // r
print('')
print('cofactor G1: ' + cofactorG1.hex())
print('cofactor G2: ' + cofactorG2.hex())
print('')
def clearCofactorG1(P):
return cofactorG1 * P
def clearCofactorG2(P):
return cofactorG2 * P
# Test generator
set_random_seed(1337)
print('=========================================')
print('G1 vectors: ')
for i in range(10):
print(f'--- test {i} ------------------------------')
Prand = G1.random_point()
P = clearCofactorG1(Prand)
(Px, Py, Pz) = P
print('Px: ' + Integer(Px).hex())
print('Py: ' + Integer(Py).hex())
# print('Pz: ' + Integer(Pz).hex())
exponent = randrange(r) # Pick an integer below curve order
print('scalar: ' + Integer(exponent).hex())
Q = exponent * P
(Qx, Qy, Qz) = Q
print('Qx: ' + Integer(Qx).hex())
print('Qy: ' + Integer(Qy).hex())
# print('Qz: ' + Integer(Qz).hex())
print('=========================================')
print('G2 vectors: ')
for i in range(10):
print(f'--- test {i} ------------------------------')
Prand = G2.random_point()
P = clearCofactorG2(Prand)
(Px, Py, Pz) = P
vPx = vector(Px)
vPy = vector(Py)
# Pz = vector(Pz)
print('Px: ' + Integer(vPx[0]).hex() + ' + β * ' + Integer(vPx[1]).hex())
print('Py: ' + Integer(vPy[0]).hex() + ' + β * ' + Integer(vPy[1]).hex())
exponent = randrange(r) # Pick an integer below curve order
print('scalar: ' + Integer(exponent).hex())
Q = exponent * P
(Qx, Qy, Qz) = Q
Qx = vector(Qx)
Qy = vector(Qy)
print('Qx: ' + Integer(Qx[0]).hex() + ' + β * ' + Integer(Qx[1]).hex())
print('Qy: ' + Integer(Qy[0]).hex() + ' + β * ' + Integer(Qy[1]).hex())
print('=========================================')
# CurveOrder sanity check
#
# P = G1.random_point()
# (Px, Py, Pz) = P
# print('Px: ' + Integer(Px).hex())
# print('Py: ' + Integer(Py).hex())
# print('Pz: ' + Integer(Pz).hex())
#
# print('order: ' + Integer(r).hex())
#
# Q = (r * cofactor) * P
# (Qx, Qy, Qz) = Q
# print('Qx: ' + Integer(Qx).hex())
# print('Qy: ' + Integer(Qy).hex())
# print('Qz: ' + Integer(Qz).hex())

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sage/testgen_bls12_381.sage
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# Constantine
# Copyright (c) 2018-2019 Status Research & Development GmbH
# Copyright (c) 2020-Present Mamy André-Ratsimbazafy
# Licensed and distributed under either of
# * MIT license (license terms in the root directory or at http://opensource.org/licenses/MIT).
# * Apache v2 license (license terms in the root directory or at http://www.apache.org/licenses/LICENSE-2.0).
# at your option. This file may not be copied, modified, or distributed except according to those terms.
# ############################################################
#
# BN254 test generator
#
# ############################################################
# Parameters
x = -(2^63 + 2^62 + 2^60 + 2^57 + 2^48 + 2^16)
p = (x - 1)^2 * (x^4 - x^2 + 1)//3 + x
r = x^4 - x^2 + 1
# Finite fields
Fp = GF(p)
K2.<u> = PolynomialRing(Fp)
Fp2.<beta> = Fp.extension(u^2+1)
# K6.<v> = PolynomialRing(Fp2)
# Fp6.<eta> = Fp2.extension(v^3-Fp2([1, 1])
# K12.<w> = PolynomialRing(Fp6)
# K12.<gamma> = F6.extension(w^2-eta)
# Curves
b = 4
SNR = Fp2([1, 1])
G1 = EllipticCurve(Fp, [0, b])
G2 = EllipticCurve(Fp2, [0, b*SNR])
# https://crypto.stackexchange.com/questions/64064/order-of-twisted-curve-in-pairings
# https://math.stackexchange.com/questions/144194/how-to-find-the-order-of-elliptic-curve-over-finite-field-extension
cofactorG1 = G1.order() // r
cofactorG2 = G2.order() // r
print('')
print('cofactor G1: ' + cofactorG1.hex())
print('cofactor G2: ' + cofactorG2.hex())
print('')
def clearCofactorG1(P):
return cofactorG1 * P
def clearCofactorG2(P):
return cofactorG2 * P
# Test generator
set_random_seed(1337)
print('=========================================')
print('G1 vectors: ')
for i in range(10):
print(f'--- test {i} ------------------------------')
Prand = G1.random_point()
P = clearCofactorG1(Prand)
(Px, Py, Pz) = P
print('Px: ' + Integer(Px).hex())
print('Py: ' + Integer(Py).hex())
# print('Pz: ' + Integer(Pz).hex())
exponent = randrange(r) # Pick an integer below curve order
print('scalar: ' + Integer(exponent).hex())
Q = exponent * P
(Qx, Qy, Qz) = Q
print('Qx: ' + Integer(Qx).hex())
print('Qy: ' + Integer(Qy).hex())
# print('Qz: ' + Integer(Qz).hex())
print('=========================================')
print('G2 vectors: ')
for i in range(10):
print(f'--- test {i} ------------------------------')
Prand = G2.random_point()
P = clearCofactorG2(Prand)
(Px, Py, Pz) = P
vPx = vector(Px)
vPy = vector(Py)
# Pz = vector(Pz)
print('Px: ' + Integer(vPx[0]).hex() + ' + β * ' + Integer(vPx[1]).hex())
print('Py: ' + Integer(vPy[0]).hex() + ' + β * ' + Integer(vPy[1]).hex())
exponent = randrange(r) # Pick an integer below curve order
print('scalar: ' + Integer(exponent).hex())
Q = exponent * P
(Qx, Qy, Qz) = Q
Qx = vector(Qx)
Qy = vector(Qy)
print('Qx: ' + Integer(Qx[0]).hex() + ' + β * ' + Integer(Qx[1]).hex())
print('Qy: ' + Integer(Qy[0]).hex() + ' + β * ' + Integer(Qy[1]).hex())
print('=========================================')
# CurveOrder sanity check
#
# P = G1.random_point()
# (Px, Py, Pz) = P
# print('Px: ' + Integer(Px).hex())
# print('Py: ' + Integer(Py).hex())
# print('Pz: ' + Integer(Pz).hex())
#
# print('order: ' + Integer(r).hex())
#
# Q = (r * cofactor) * P
# (Qx, Qy, Qz) = Q
# print('Qx: ' + Integer(Qx).hex())
# print('Qy: ' + Integer(Qy).hex())
# print('Qz: ' + Integer(Qz).hex())

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sage/testgen_bn254_snarks.sage
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# Constantine
# Copyright (c) 2018-2019 Status Research & Development GmbH
# Copyright (c) 2020-Present Mamy André-Ratsimbazafy
# Licensed and distributed under either of
# * MIT license (license terms in the root directory or at http://opensource.org/licenses/MIT).
# * Apache v2 license (license terms in the root directory or at http://www.apache.org/licenses/LICENSE-2.0).
# at your option. This file may not be copied, modified, or distributed except according to those terms.
# ############################################################
#
# BN254 test generator
#
# ############################################################
# Parameters
x = Integer('0x44E992B44A6909F1')
p = 36*x^4 + 36*x^3 + 24*x^2 + 6*x + 1
r = 36*x^4 + 36*x^3 + 18*x^2 + 6*x + 1
# Finite fields
Fp = GF(p)
K2.<u> = PolynomialRing(Fp)
Fp2.<beta> = Fp.extension(u^2+1)
# K6.<v> = PolynomialRing(Fp2)
# Fp6.<eta> = Fp2.extension(v^3-Fp2([9, 1]))
# K12.<w> = PolynomialRing(F6)
# K12.<gamma> = Fp6.extension(w^2-eta)
# Curves
b = 3
SNR = Fp2([9, 1])
G1 = EllipticCurve(Fp, [0, b])
G2 = EllipticCurve(Fp2, [0, b/SNR])
# https://crypto.stackexchange.com/questions/64064/order-of-twisted-curve-in-pairings
# https://math.stackexchange.com/questions/144194/how-to-find-the-order-of-elliptic-curve-over-finite-field-extension
cofactorG1 = G1.order() // r
cofactorG2 = G2.order() // r
print('')
print('cofactor G1: ' + cofactorG1.hex())
print('cofactor G2: ' + cofactorG2.hex())
print('')
def clearCofactorG1(P):
return cofactorG1 * P
def clearCofactorG2(P):
return cofactorG2 * P
# Test generator
set_random_seed(1337)
print('=========================================')
print('G1 vectors: ')
for i in range(10):
print(f'--- test {i} ------------------------------')
P = G1.random_point()
P = clearCofactorG1(P)
(Px, Py, Pz) = P
print('Px: ' + Integer(Px).hex())
print('Py: ' + Integer(Py).hex())
# print('Pz: ' + Integer(Pz).hex())
exponent = randrange(r) # Pick an integer below curve order
print('scalar: ' + Integer(exponent).hex())
Q = exponent * P
(Qx, Qy, Qz) = Q
print('Qx: ' + Integer(Qx).hex())
print('Qy: ' + Integer(Qy).hex())
# print('Qz: ' + Integer(Qz).hex())
print('---------------------------------------')
print('=========================================')
print('G2 vectors: ')
for i in range(10):
print(f'--- test {i} ------------------------------')
P = G2.random_point()
P = clearCofactorG2(P)
(Px, Py, Pz) = P
vPx = vector(Px)
vPy = vector(Py)
# Pz = vector(Pz)
print('Px: ' + Integer(vPx[0]).hex() + ' + β * ' + Integer(vPx[1]).hex())
print('Py: ' + Integer(vPy[0]).hex() + ' + β * ' + Integer(vPy[1]).hex())
exponent = randrange(r) # Pick an integer below curve order
print('scalar: ' + Integer(exponent).hex())
assert exponent < r
Q = exponent * P
(Qx, Qy, Qz) = Q
Qx = vector(Qx)
Qy = vector(Qy)
print('Qx: ' + Integer(Qx[0]).hex() + ' + β * ' + Integer(Qx[1]).hex())
print('Qy: ' + Integer(Qy[0]).hex() + ' + β * ' + Integer(Qy[1]).hex())
print('---------------------------------------')
print('=========================================')
# CurveOrder sanity check
#
# P = G1.random_point()
# (Px, Py, Pz) = P
# print('Px: ' + Integer(Px).hex())
# print('Py: ' + Integer(Py).hex())
# print('Pz: ' + Integer(Pz).hex())
#
# print('order: ' + Integer(r).hex())
#
# Q = (r * cofactor) * P
# (Qx, Qy, Qz) = Q
# print('Qx: ' + Integer(Qx).hex())
# print('Qy: ' + Integer(Qy).hex())
# print('Qz: ' + Integer(Qz).hex())

246
sage/testgen_scalar_mul.sage Normal file
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@ -0,0 +1,246 @@
#!/usr/bin/sage
# vim: syntax=python
# vim: set ts=2 sw=2 et:
# Constantine
# Copyright (c) 2018-2019 Status Research & Development GmbH
# Copyright (c) 2020-Present Mamy André-Ratsimbazafy
# Licensed and distributed under either of
# * MIT license (license terms in the root directory or at http://opensource.org/licenses/MIT).
# * Apache v2 license (license terms in the root directory or at http://www.apache.org/licenses/LICENSE-2.0).
# at your option. This file may not be copied, modified, or distributed except according to those terms.
# ############################################################
#
# Scalar multiplication test generator
#
# ############################################################
# Imports
# ---------------------------------------------------------
import os, json
import inspect, textwrap
# Working directory
# ---------------------------------------------------------
os.chdir(os.path.dirname(__file__))
# Sage imports
# ---------------------------------------------------------
# Accelerate arithmetic by accepting probabilistic proofs
from sage.structure.proof.all import arithmetic
arithmetic(False)
load('curves.sage')
# Utilities
# ---------------------------------------------------------
def progressbar(it, prefix="", size=60, file=sys.stdout):
count = len(it)
def show(j):
x = int(size*j/count)
file.write("%s[%s%s] %i/%i\r" % (prefix, "#"*x, "."*(size-x), j, count))
file.flush()
show(0)
for i, item in enumerate(it):
yield item
show(i+1)
file.write("\n")
file.flush()
def serialize_bigint(x):
return '0x' + Integer(x).hex()
def serialize_G1(P):
(Px, Py, Pz) = P
coords = {
'x': serialize_bigint(Px),
'y': serialize_bigint(Py)
}
return coords
def serialize_G2(P):
(Px, Py, Pz) = P
Px = vector(Px)
Py = vector(Py)
coords = {
'x': {
'c0': serialize_bigint(Px[0]),
'c1': serialize_bigint(Px[1])
},
'y': {
'c0': serialize_bigint(Py[0]),
'c1': serialize_bigint(Py[1])
}
}
return coords
# Generator
# ---------------------------------------------------------
def genScalarMulG1(curve_name, curve_config, count, seed):
p = curve_config[curve_name]['field']['modulus']
r = curve_config[curve_name]['field']['order']
form = curve_config[curve_name]['curve']['form']
a = curve_config[curve_name]['curve']['a']
b = curve_config[curve_name]['curve']['b']
Fp = GF(p)
G1 = EllipticCurve(Fp, [0, b])
cofactor = G1.order() // r
out = {
'curve': curve_name,
'group': 'G1',
'modulus': serialize_bigint(p),
'order': serialize_bigint(r),
'cofactor': serialize_bigint(cofactor),
'form': form
}
if form == 'short_weierstrass':
out['a'] = serialize_bigint(a)
out['b'] = serialize_bigint(b)
vectors = []
set_random_seed(seed)
for i in progressbar(range(count)):
v = {}
P = G1.random_point()
scalar = randrange(r)
P *= cofactor # clear cofactor
v['id'] = i
v['P'] = serialize_G1(P)
v['scalar'] = serialize_bigint(scalar)
Q = scalar * P
v['Q'] = serialize_G1(Q)
vectors.append(v)
out['vectors'] = vectors
return out
def genScalarMulG2(curve_name, curve_config, count, seed):
p = curve_config[curve_name]['field']['modulus']
r = curve_config[curve_name]['field']['order']
form = curve_config[curve_name]['curve']['form']
a = curve_config[curve_name]['curve']['a']
b = curve_config[curve_name]['curve']['b']
embedding_degree = curve_config[curve_name]['tower']['embedding_degree']
twist_degree = curve_config[curve_name]['tower']['twist_degree']
twist = curve_config[curve_name]['tower']['twist']
G2_field_degree = embedding_degree // twist_degree
G2_field = f'Fp{G2_field_degree}' if G2_field_degree > 1 else 'Fp'
if G2_field_degree == 2:
non_residue_fp = curve_config[curve_name]['tower']['QNR_Fp']
else:
raise NotImplementedError()
Fp = GF(p)
K.<u> = PolynomialRing(Fp)
if G2_field == 'Fp2':
Fp2.<beta> = Fp.extension(u^2 - non_residue_fp)
G2F = Fp2
if twist_degree == 6:
non_residue_twist = curve_config[curve_name]['tower']['SNR_Fp2']
else:
raise NotImplementedError()
else:
G2F = Fp
raise NotImplementedError()
if twist == 'D_Twist':
G2 = EllipticCurve(G2F, [0, b/G2F(non_residue_twist)])
elif twist == 'M_Twist':
G2 = EllipticCurve(G2F, [0, b*G2F(non_residue_twist)])
else:
raise ValueError('G2 must be a D_Twist or M_Twist but found ' + twist)
cofactor = G2.order() // r
out = {
'curve': curve_name,
'group': 'G2',
'modulus': serialize_bigint(p),
'order': serialize_bigint(r),
'cofactor': serialize_bigint(cofactor),
'form': form,
'twist_degree': int(twist_degree),
'twist': twist,
'non_residue_fp': int(non_residue_fp),
'G2_field': G2_field,
'non_residue_twist': [int(coord) for coord in non_residue_twist] if isinstance(non_residue_twist, list) else int(non_residue_twist)
}
if form == 'short_weierstrass':
out['a'] = serialize_bigint(a)
out['b'] = serialize_bigint(b)
vectors = []
set_random_seed(seed)
for i in progressbar(range(count)):
v = {}
P = G2.random_point()
scalar = randrange(r)
P *= cofactor # clear cofactor
v['id'] = i
v['P'] = serialize_G2(P)
v['scalar'] = serialize_bigint(scalar)
Q = scalar * P
v['Q'] = serialize_G2(Q)
vectors.append(v)
out['vectors'] = vectors
return out
# CLI
# ---------------------------------------------------------
if __name__ == "__main__":
# Usage
# BLS12-381
# sage sage/derive_pairing.sage BLS12_381 G1
from argparse import ArgumentParser
parser = ArgumentParser()
parser.add_argument("curve",nargs="+")
args = parser.parse_args()
curve = args.curve[0]
group = args.curve[1]
if curve not in Curves:
raise ValueError(
curve +
' is not one of the available curves: ' +
str(Curves.keys())
)
elif group not in ['G1', 'G2']:
raise ValueError(
group +
' is not a valid group, expected G1 or G2 instead'
)
else:
print(f'\nGenerating test vectors tv_{curve}_scalar_mul_{group}.json')
print('----------------------------------------------------\n')
count = 40
seed = 1337
if group == 'G1':
out = genScalarMulG1(curve, Curves, count, seed)
elif group == 'G2':
out = genScalarMulG2(curve, Curves, count, seed)
with open(f'tv_{curve}_scalar_mul_{group}.json', 'w') as f:
json.dump(out, f, indent=2)

1
tests/support/ec_reference_scalar_mult.nim
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@ -8,7 +8,6 @@
import
# Internals
../../constantine/config/[common, curves],
../../constantine/arithmetic,
../../constantine/io/io_bigints

582
tests/t_ec_sage_bls12_377.nim
View File

@ -7,572 +7,30 @@
# at your option. This file may not be copied, modified, or distributed except according to those terms.
import
# Standard library
std/[unittest, times],
# Internals
../constantine/config/[common, curves],
../constantine/arithmetic,
../constantine/config/[type_fp, curves],
../constantine/towers,
../constantine/io/[io_bigints, io_ec],
../constantine/elliptic/[
ec_shortweierstrass_projective,
ec_shortweierstrass_jacobian,
ec_scalar_mul, ec_endomorphism_accel],
../constantine/elliptic/ec_shortweierstrass_jacobian,
../constantine/elliptic/ec_shortweierstrass_projective,
# Test utilities
./support/ec_reference_scalar_mult
./t_ec_sage_template
echo "\n------------------------------------------------------\n"
run_scalar_mul_test_vs_sage(
ECP_ShortW_Proj[Fp[BLS12_377], NotOnTwist],
"t_ec_sage_bls12_377_g1_projective"
)
proc test(
id: int,
EC: typedesc[ECP_ShortW_Proj or ECP_ShortW_Jac],
Px, Py: string,
scalar: string,
Qx, Qy: string
) =
run_scalar_mul_test_vs_sage(
ECP_ShortW_Jac[Fp[BLS12_377], NotOnTwist],
"t_ec_sage_bls12_377_g1_jacobian"
)
test "test " & $id & " - " & $EC:
var P: EC
let pOK = P.fromHex(Px, Py)
doAssert pOK
run_scalar_mul_test_vs_sage(
ECP_ShortW_Proj[Fp2[BLS12_377], OnTwist],
"t_ec_sage_bls12_377_g2_projective"
)
var Q: EC
let qOK = Q.fromHex(Qx, Qy)
let exponent = BigInt[EC.F.C.getCurveOrderBitwidth()].fromHex(scalar)
var
impl = P
reference = P
endo = P
endoW = P
impl.scalarMulGeneric(exponent)
reference.unsafe_ECmul_double_add(exponent)
endo.scalarMulEndo(exponent)
endoW.scalarMulGLV_m2w2(exponent)
doAssert: bool(Q == reference)
doAssert: bool(Q == impl)
doAssert: bool(Q == endo)
doAssert: bool(Q == endoW)
suite "Scalar Multiplication (cofactor cleared): BLS12_377 implementation vs SageMath" & " [" & $WordBitwidth & "-bit mode]":
# Generated via sage sage/testgen_bls12_377.sage
test(
id = 0,
EC = ECP_ShortW_Proj[Fp[BLS12_377], NotOnTwist],
Px = "4e7e6dfa01ed0ceb6e66708b07cb5c6cd30a42eeb13d7b76f8d103a0a4d491450be6f526fc12f15209b792220c041e",
Py = "c782515159b7e7b9371e0e0caa387951317e993b1625d91869d4346621058a0960ef1b8b6eabb33cd5719694908a05",
scalar = "cf815cb4d44d3d691b7c82a40b4b70caa9b0e8fe9586648abf3f1e2e639ca1b",
Qx = "4a1203db4af8f0efc18c7ceb24999eb6e0dbdfc8f44a9edd5ba2f9eced38e81ecae287ab1c184eea8e8753d1178604",
Qy = "f7509cdce5de473e37f36c69d93ff1b2ab04c3ae25a3b5e41f2c138cdcc69ed5eaf4ac95a7a857eef336ebac19bcd3"
)
test(
id = 1,
EC = ECP_ShortW_Proj[Fp[BLS12_377], NotOnTwist],
Px = "13d735b28405253dcc0bc60bcdc13633475ffc187d38a9b97655b0d0fa1d56c4548f11ea0a795391ee85c953aaf9b83",
Py = "1693101123fd13a20f9c0569c52c29507ba1c8b6dd412660bc82e7974022f1a10f9137b4ba59d3f0aab67027cefec19",
scalar = "913aa7b9fa2f940b70b6dcf538cc08da1a369809ab86a8ee49cead0ed6bfef6",
Qx = "10e0e8582ec3456f7569473892c23997d004f2542d914fa75db8f1798ed8ce505836e8b7af5cf1503e14d85fadd65ee",
Qy = "d5151ae60e43100e20aad7eaf8659e690e8034910d7717078031520fcbf6f9a00b22c6a9894aec88c9182f13335639"
)
test(
id = 2,
EC = ECP_ShortW_Proj[Fp[BLS12_377], NotOnTwist],
Px = "6dd2211113cada09d3b96848cbf21ed14f6fc238b581c0afd49aa776980101e4ee279a256ce8f1428d3ed3f70afd85",
Py = "3b406b4433a3f44f8e196012f50c520e876412fbcae2651916f133c1fd3899c79f676e1abba01d84bab7ad100c9295",
scalar = "4cf47669aeb0f30b6c6c5aa02808a87dc787fba22da32875e614a54a50b6a0c",
Qx = "3caf55e1f82d746df2bf47defbee127a3e6f7e79a9575929704b25489ffb801dbce07999acbddfd79352b0633a2708",
Qy = "3fc275c136b07456f0b6ee814508876037add5e95357fae6b4195744ba5ccccaf93581e34feb2babe652a9a704731b"
)
test(
id = 3,
EC = ECP_ShortW_Proj[Fp[BLS12_377], NotOnTwist],
Px = "18648abbe3261f93cbf679777eac66e419035041a967d1e6a0f0afdb92810d7122e0984f1d6efc8fe518500464ee803",
Py = "7b6f518d4d06309aad4d60d29118310b6c8b17c7bf5db2251f4701b13b89a8c0f04bb5d0386785e55ffbbd7ecc445e",
scalar = "69498486a06c18f836a8e9ed507bbb563d6d03545e03e08f628e8fbd2e5d098",
Qx = "6a871d872673879fd23ec8c150f8d63e8130dc60343fc0c2ec9ff1b02e769e20eeec0288102ccec6ff2f6ac6973b4",
Qy = "cd95a31afa4f0bbcc063a1585fec4a6c51812b2baaee42a04525fd55b3a9db3bdb7cbc77f7cd32f42d2e0eb26ddfa2"
)
test(
id = 4,
EC = ECP_ShortW_Proj[Fp[BLS12_377], NotOnTwist],
Px = "745e9944549522486f3446676cc62fb666682e10c661a13b8110b42cb9a37676b6f33a46f9495f4fafb342d5809db5",
Py = "bc595854bc42ccec60dd9ec573608d736aa59996cef1c2e8f6c5d424f525a6f3e3d4beeedfac6b959dbd71ced95b13",
scalar = "6e08d8714102a5aa3e9f46e33c70a759c27253c7b0196c3e46c7cb42671197e",
Qx = "18a967a80785de8ec6ac9d98cffa06a8c633b5fa0f36431a32f7bd955946edc3d55f79bfdf5335db405560a6cbe2415",
Qy = "4552ff2eb6ade0c6b33c3603460d9d62099201d842c9883b33b7ed1147cb17268338a77a9417776ddbd774e91a8d2"
)
test(
id = 5,
EC = ECP_ShortW_Proj[Fp[BLS12_377], NotOnTwist],
Px = "f7b56cf212a8906ed77a758164c0bd05ce1fbd3ee3c4357e7a09b3aedc748a29ace254f1f6df35b8cb74361060337f",
Py = "2640ef641d20fea19b28947833e53faceeafa57a8761b807049f3d707d70c01f1c69a57edd993d301a64517bf47f77",
scalar = "a5c46d7a52938fed7f3093d5867f65361dc8b48c83bd7db490c26736196e20e",
Qx = "434f024c4afd7b9a44375011d663af0ae0fe79442e9caf36518e053bb13d49998ec1d2da4bc1c4a812812119b3221f",
Qy = "32c959e2a678cf93f77e621fdf887454a5f9cb7a67f29065669f48234c25e321ceb5758dfba987431a0c2caec94616"
)
test(
id = 6,
EC = ECP_ShortW_Proj[Fp[BLS12_377], NotOnTwist],
Px = "43b387c12e4cc91fb1a5593b7671356dceb0fe6e6ac666d5bac94f2a44c8db54976b649a678aae038b21144de04e23",
Py = "1a6366a50f1f9ba64eef73d82bec86177bf184be048a9d66326ccb0122203569ddcb8cf74445cadaff7f47a66d1b1a2",
scalar = "bddd07231bc7fe89ee4a859a00ea1f9d236be9e7fd561303d566904c1b0a07c",
Qx = "8ddb7a959483b51a1471de988146b7d5b166f660734b4d55166c5a23d781e923261927b1012dce73b822bb6e56bfd2",
Qy = "dc7178bf5597ef31209a5b0409fc42c64b81260a0046a562ca08c7a9dc67b444c25e94fb97b9a6bb4c137cadd81021"
)
test(
id = 7,
EC = ECP_ShortW_Proj[Fp[BLS12_377], NotOnTwist],
Px = "6984c9385a67081f97b3d33444077466cca1d0442a4da8c083a0957578e0b21011435b126a5ab143da9da1cf5b216f",
Py = "18a87c7f5f6c5a8101773e63956b9addd4becf5177acc560d548e5331638121934842fdc9f654b3f456a7df5a2e471a",
scalar = "b72c41a6ffaff0aacb5d62e3dcb16acfec66b6a9639e19d3128fd43c18e7dbe",
Qx = "19279201c3c7a9d50b546aa99d3e1a6625fe2a7bc64a09625b683534638b9e87b9102d4dba6684956b6be7668a658c6",
Qy = "195f15ad1edc05f8b289c7eee7bd8f78116a2d5ba8b83643d9e7cb2cdc6550bcf8c2145008e900ca9cba4d5040e9f4"
)
test(
id = 8,
EC = ECP_ShortW_Proj[Fp[BLS12_377], NotOnTwist],
Px = "efc34595823e7333616f7768bc82f407268df6f029bf4c94d15f1785dc7ccc08f22e7301dfc48dadc0ea383c8bb3e",
Py = "2459bd9f71977ef122d2102e8bfd07a5737066075058cfa8bcaa9f9690ed065919c844363ceaea6f9bb650906a535f",
scalar = "8f90f6ab0ffa6e4acc601b44c062745f2935b3dc153d0da07977470080d5c18",
Qx = "35d08b33e02579581905941975c8b1cc5be1c9670a7f7ef390daa363b0abd3571a802d8c27f156fba40573094f6c7a",
Qy = "111ba3bdfa4260dc2b636479edb1fcf2fc9478aa722da0118908e1db1551cf7e131c8521b2f3708ef670684dbe8d181"
)
test(
id = 9,
EC = ECP_ShortW_Proj[Fp[BLS12_377], NotOnTwist],
Px = "3eec93c2a9c5fd03f0de5ede2fdac9e361090fbaea38e4a0f1828745f1d14a057d9fd7c46b9168bd95a45a182a3a62",
Py = "e912dc7e95f90d91e3274ec5639edacb88be1b092c47c13d31a29ecd579885cc09f197f8207d23b2260ab10c94d5f5",
scalar = "203300e949aff816d084a388f07c74b9152bb11b523543afd65c805a389980",
Qx = "8c23ae9c51e7c92c6a59f0ed07a59f148b4d79394fc026931c264612041eedd3782e4f249bbfad1799212788c1a00d",
Qy = "191628b702ebef397bb7c52102ee4522f864ff594b24dc385ece081e8066bcde20aa5c7dfd00fb1f3cea221b3ad4ea2"
)
# ----------------------------------------
test(
id = 0,
EC = ECP_ShortW_Jac[Fp[BLS12_377], NotOnTwist],
Px = "4e7e6dfa01ed0ceb6e66708b07cb5c6cd30a42eeb13d7b76f8d103a0a4d491450be6f526fc12f15209b792220c041e",
Py = "c782515159b7e7b9371e0e0caa387951317e993b1625d91869d4346621058a0960ef1b8b6eabb33cd5719694908a05",
scalar = "cf815cb4d44d3d691b7c82a40b4b70caa9b0e8fe9586648abf3f1e2e639ca1b",
Qx = "4a1203db4af8f0efc18c7ceb24999eb6e0dbdfc8f44a9edd5ba2f9eced38e81ecae287ab1c184eea8e8753d1178604",
Qy = "f7509cdce5de473e37f36c69d93ff1b2ab04c3ae25a3b5e41f2c138cdcc69ed5eaf4ac95a7a857eef336ebac19bcd3"
)
test(
id = 1,
EC = ECP_ShortW_Jac[Fp[BLS12_377], NotOnTwist],
Px = "13d735b28405253dcc0bc60bcdc13633475ffc187d38a9b97655b0d0fa1d56c4548f11ea0a795391ee85c953aaf9b83",
Py = "1693101123fd13a20f9c0569c52c29507ba1c8b6dd412660bc82e7974022f1a10f9137b4ba59d3f0aab67027cefec19",
scalar = "913aa7b9fa2f940b70b6dcf538cc08da1a369809ab86a8ee49cead0ed6bfef6",
Qx = "10e0e8582ec3456f7569473892c23997d004f2542d914fa75db8f1798ed8ce505836e8b7af5cf1503e14d85fadd65ee",
Qy = "d5151ae60e43100e20aad7eaf8659e690e8034910d7717078031520fcbf6f9a00b22c6a9894aec88c9182f13335639"
)
test(
id = 2,
EC = ECP_ShortW_Jac[Fp[BLS12_377], NotOnTwist],
Px = "6dd2211113cada09d3b96848cbf21ed14f6fc238b581c0afd49aa776980101e4ee279a256ce8f1428d3ed3f70afd85",
Py = "3b406b4433a3f44f8e196012f50c520e876412fbcae2651916f133c1fd3899c79f676e1abba01d84bab7ad100c9295",
scalar = "4cf47669aeb0f30b6c6c5aa02808a87dc787fba22da32875e614a54a50b6a0c",
Qx = "3caf55e1f82d746df2bf47defbee127a3e6f7e79a9575929704b25489ffb801dbce07999acbddfd79352b0633a2708",
Qy = "3fc275c136b07456f0b6ee814508876037add5e95357fae6b4195744ba5ccccaf93581e34feb2babe652a9a704731b"
)
test(
id = 3,
EC = ECP_ShortW_Jac[Fp[BLS12_377], NotOnTwist],
Px = "18648abbe3261f93cbf679777eac66e419035041a967d1e6a0f0afdb92810d7122e0984f1d6efc8fe518500464ee803",
Py = "7b6f518d4d06309aad4d60d29118310b6c8b17c7bf5db2251f4701b13b89a8c0f04bb5d0386785e55ffbbd7ecc445e",
scalar = "69498486a06c18f836a8e9ed507bbb563d6d03545e03e08f628e8fbd2e5d098",
Qx = "6a871d872673879fd23ec8c150f8d63e8130dc60343fc0c2ec9ff1b02e769e20eeec0288102ccec6ff2f6ac6973b4",
Qy = "cd95a31afa4f0bbcc063a1585fec4a6c51812b2baaee42a04525fd55b3a9db3bdb7cbc77f7cd32f42d2e0eb26ddfa2"
)
test(
id = 4,
EC = ECP_ShortW_Jac[Fp[BLS12_377], NotOnTwist],
Px = "745e9944549522486f3446676cc62fb666682e10c661a13b8110b42cb9a37676b6f33a46f9495f4fafb342d5809db5",
Py = "bc595854bc42ccec60dd9ec573608d736aa59996cef1c2e8f6c5d424f525a6f3e3d4beeedfac6b959dbd71ced95b13",
scalar = "6e08d8714102a5aa3e9f46e33c70a759c27253c7b0196c3e46c7cb42671197e",
Qx = "18a967a80785de8ec6ac9d98cffa06a8c633b5fa0f36431a32f7bd955946edc3d55f79bfdf5335db405560a6cbe2415",
Qy = "4552ff2eb6ade0c6b33c3603460d9d62099201d842c9883b33b7ed1147cb17268338a77a9417776ddbd774e91a8d2"
)
test(
id = 5,
EC = ECP_ShortW_Jac[Fp[BLS12_377], NotOnTwist],
Px = "f7b56cf212a8906ed77a758164c0bd05ce1fbd3ee3c4357e7a09b3aedc748a29ace254f1f6df35b8cb74361060337f",
Py = "2640ef641d20fea19b28947833e53faceeafa57a8761b807049f3d707d70c01f1c69a57edd993d301a64517bf47f77",
scalar = "a5c46d7a52938fed7f3093d5867f65361dc8b48c83bd7db490c26736196e20e",
Qx = "434f024c4afd7b9a44375011d663af0ae0fe79442e9caf36518e053bb13d49998ec1d2da4bc1c4a812812119b3221f",
Qy = "32c959e2a678cf93f77e621fdf887454a5f9cb7a67f29065669f48234c25e321ceb5758dfba987431a0c2caec94616"
)
test(
id = 6,
EC = ECP_ShortW_Jac[Fp[BLS12_377], NotOnTwist],
Px = "43b387c12e4cc91fb1a5593b7671356dceb0fe6e6ac666d5bac94f2a44c8db54976b649a678aae038b21144de04e23",
Py = "1a6366a50f1f9ba64eef73d82bec86177bf184be048a9d66326ccb0122203569ddcb8cf74445cadaff7f47a66d1b1a2",
scalar = "bddd07231bc7fe89ee4a859a00ea1f9d236be9e7fd561303d566904c1b0a07c",
Qx = "8ddb7a959483b51a1471de988146b7d5b166f660734b4d55166c5a23d781e923261927b1012dce73b822bb6e56bfd2",
Qy = "dc7178bf5597ef31209a5b0409fc42c64b81260a0046a562ca08c7a9dc67b444c25e94fb97b9a6bb4c137cadd81021"
)
test(
id = 7,
EC = ECP_ShortW_Jac[Fp[BLS12_377], NotOnTwist],
Px = "6984c9385a67081f97b3d33444077466cca1d0442a4da8c083a0957578e0b21011435b126a5ab143da9da1cf5b216f",
Py = "18a87c7f5f6c5a8101773e63956b9addd4becf5177acc560d548e5331638121934842fdc9f654b3f456a7df5a2e471a",
scalar = "b72c41a6ffaff0aacb5d62e3dcb16acfec66b6a9639e19d3128fd43c18e7dbe",
Qx = "19279201c3c7a9d50b546aa99d3e1a6625fe2a7bc64a09625b683534638b9e87b9102d4dba6684956b6be7668a658c6",
Qy = "195f15ad1edc05f8b289c7eee7bd8f78116a2d5ba8b83643d9e7cb2cdc6550bcf8c2145008e900ca9cba4d5040e9f4"
)
test(
id = 8,
EC = ECP_ShortW_Jac[Fp[BLS12_377], NotOnTwist],
Px = "efc34595823e7333616f7768bc82f407268df6f029bf4c94d15f1785dc7ccc08f22e7301dfc48dadc0ea383c8bb3e",
Py = "2459bd9f71977ef122d2102e8bfd07a5737066075058cfa8bcaa9f9690ed065919c844363ceaea6f9bb650906a535f",
scalar = "8f90f6ab0ffa6e4acc601b44c062745f2935b3dc153d0da07977470080d5c18",
Qx = "35d08b33e02579581905941975c8b1cc5be1c9670a7f7ef390daa363b0abd3571a802d8c27f156fba40573094f6c7a",
Qy = "111ba3bdfa4260dc2b636479edb1fcf2fc9478aa722da0118908e1db1551cf7e131c8521b2f3708ef670684dbe8d181"
)
test(
id = 9,
EC = ECP_ShortW_Jac[Fp[BLS12_377], NotOnTwist],
Px = "3eec93c2a9c5fd03f0de5ede2fdac9e361090fbaea38e4a0f1828745f1d14a057d9fd7c46b9168bd95a45a182a3a62",
Py = "e912dc7e95f90d91e3274ec5639edacb88be1b092c47c13d31a29ecd579885cc09f197f8207d23b2260ab10c94d5f5",
scalar = "203300e949aff816d084a388f07c74b9152bb11b523543afd65c805a389980",
Qx = "8c23ae9c51e7c92c6a59f0ed07a59f148b4d79394fc026931c264612041eedd3782e4f249bbfad1799212788c1a00d",
Qy = "191628b702ebef397bb7c52102ee4522f864ff594b24dc385ece081e8066bcde20aa5c7dfd00fb1f3cea221b3ad4ea2"
)
proc test(
id: int,
EC: typedesc[ECP_ShortW_Proj or ECP_ShortW_Jac],
Px0, Px1, Py0, Py1: string,
scalar: string,
Qx0, Qx1, Qy0, Qy1: string
) =
test "test " & $id & " - " & $EC:
var P: EC
let pOK = P.fromHex(Px0, Px1, Py0, Py1)
doAssert pOK
var Q: EC
let qOK = Q.fromHex(Qx0, Qx1, Qy0, Qy1)
let exponent = BigInt[EC.F.C.getCurveOrderBitwidth()].fromHex(scalar)
var
impl = P
reference = P
endo = P
impl.scalarMulGeneric(exponent)
reference.unsafe_ECmul_double_add(exponent)
endo.scalarMulEndo(exponent)
doAssert: bool(Q == reference)
doAssert: bool(Q == impl)
doAssert: bool(Q == endo)
suite "Scalar Multiplication G2: BLS12-381 implementation vs SageMath" & " [" & $WordBitwidth & "-bit mode]":
# Generated via sage sage/testgen_bls12_377.sage
test(
id = 0,
EC = ECP_ShortW_Proj[Fp2[BLS12_377], OnTwist],
Px0 = "267401f3ef554fe74ae131d56a10edf14ae40192654901b4618d2bf7af22e77c2a9b79e407348dbd4aad13ca73b33a",
Px1 = "12dcca838f46a3e0418e5dd8b978362757a16bfd78f0b77f4a1916ace353938389ae3ea228d0eb5020a0aaa58884aec",
Py0 = "11799118d2e054aabd9f74c0843fecbdc1c0d56f61c61c5854c2507ae2416e48a6b2cd3bc8bf7495a4d3d8270eafe2b",
Py1 = "823b9f8fb9f8297734a14359fa2c2a0de275e7e638197eaaaa7cff28f9cb3101bdabb570016672455f1ecae625e294",
scalar = "9d432eb58ec68bbc09d10961451d99c7796fb2f795eca603d6feaf3e2a1634b",
Qx0 = "12cfcb50345d43271d2a20e8208789c8ca82f2b8732fae4e7cccd87eb0883741d5e77166971c38c54170bf7635ca2f3",
Qx1 = "17467c786368f2f6eabd78f1c24aa668eae9cd00f6045bfd86b1b3c40966f023a71927026ae5a1281b432ac980d2b6b",
Qy0 = "6f5f1068b6e6a8fafd96894d8b0a8548acee24319d8e2d8b6e6982b1ced8970d9fe33155e74b33d6c2a7835196ca9",
Qy1 = "dc3c7c91c712f9a2631d2b5e49497d8cdf2ea4b30859b43edd716e84cd6a8b61e63cf708d263a7845913cdfb9c7f3b"
)
test(
id = 1,
EC = ECP_ShortW_Proj[Fp2[BLS12_377], OnTwist],
Px0 = "3a3055d6a46901c1b2227a0e334ffa9f654e62a6d3608f3a672e5816f9e9b04c0e668c3e9f8c807b269422afdc7de9",
Px1 = "25803bd55f37b254865d5fc7ac9843fb306c2eb09d34ee0c4ecb705b5e10f6911f07fd707a2e28681a421f45b1a4d",
Py0 = "15a0594a3c9dddc535472c4827aa443774a06f77bec2d20837c6574aa5fac35a279bac756531fa75f979a7a97f297d6",
Py1 = "15e8c3e013cbc64110f075295f39a4f85c9591a52b8e4903047b1f4b44bb5216b6339788c82fd90e82b1027756e7987",
scalar = "a0067f5b4294fbacd24730fed25c936a08b5f1a77824149ad6c2ce476518d17",
Qx0 = "6280dbf019f8d94acac4a39213e03f221be732849b45437edef7617460e83f972883c20791a86670a8a4c6c0125629",
Qx1 = "ab3a185addbd6c7db907170d75bd2f1997666e93d8cedbef9e10a103110d4ea09d9a889f505d7026e5c498b0355c1",
Qy0 = "1a6dd15b4c2bd6bd97858e5a75ac4365f8becaa885c4e3267d0ae256655cd057061d61b2b9acbd10bdf678a9c1e7f8c",
Qy1 = "16204b5825eec903cb7798cb62ccadb9004032d72bb815958fcd2f613c77e147656449fdaa210994338978676ad6b0e"
)
test(
id = 2,
EC = ECP_ShortW_Proj[Fp2[BLS12_377], OnTwist],
Px0 = "18ebf549cd24a64badff4ec177205be121757b68589020434aba816756d2e6fa95fdb389c8639d4bf77575122e1c04e",
Px1 = "11cd0f0976658a6f2ec113034428ef1605befdaa5642944f5c4e571b24fc166c368c30473e25ab148209be4c0b4e37",
Py0 = "1190d818317495201732feb2cfc8f507adac6273debff46bb6aea4a3f3e3fe7d28d893c90b21a6f28d2fbc72d9fc528",
Py1 = "df5fcb2daa302a5c64aeef96835e0a6b39f5d7bf0e70cc10401f966745a6b3fa682b7e5b45d9295e744e1dd7855fd4",
scalar = "7a49802ba58c87c30b631b2f90a3b876c7143e09b542c9c14706bddf9bd4117",
Qx0 = "6cdff80576d8695a646f915caba5bc6748eee1707bb0d4ebcabaf8d236b780aef6a953a07f48ef4696f02db4906c71",
Qx1 = "6b64ca5f0d46702e7e9e7beb1993b698c9b3e9991f545241d5fd44a27dc692d5f5a4c2fe6871af0653128f960307d4",
Qy0 = "cfad02b664495d42d1c6b598306229c67bcf76cc923abb13ae038c90e959a7611161a98d607729577d55ec18bf1e74",
Qy1 = "e433d9dd7d47d94e9f61a5bc27a688c63efa05408d6fd40a5c4e2711a37d011dc80f5dbaaafd939a07235c770feead"
)
test(
id = 3,
EC = ECP_ShortW_Proj[Fp2[BLS12_377], OnTwist],
Px0 = "52b54c30c1dcadbcb698d0cb7fd65de1eb8f7590c7afe46e019bdc9e0ef8bc4c060339220d3615e4b1f2b12ffa6d83",
Px1 = "dd53b483c2ab1aaa7ed22ef619b5e979237ae95476436f2c51c8b70da39a4e54a989f10f6d12ee098154911aa052f6",
Py0 = "2a8c96662a7c76cb7d8ca6571a5b99abfd2d7343dd668425e5fa4a8c880b313b70f09a15a825fac63ae7065c0c51d",
Py1 = "7bd93bb9d5bfdd15636a34a13385e9abbc1088ecc02c86cbc733122ccbbb657ec8d75e47102ce46d35a9612c01fc3d",
scalar = "e0c564e69ad68343fd4ec3d4dafdf6a92f44c13fc70a9aad95d10b2c96ee747",
Qx0 = "126f6039b7031b73a7f926bd443ace416f62e56f03864d660bbe1f1a2a994672d408f9936293f367d3fb5d8e275ad40",
Qx1 = "1654a2c26479ace1eb68e279eceac6aa10680549b09f42d288767f6296c98ba4299fa6df37a946b88823a5deebc231c",
Qy0 = "23026839059191f71e2abcf582234aa2ce4b4225c3d503e8fc6c119df1168192e894f33ed48bc571e5527365dd92b5",
Qy1 = "11f8fcce53f5e3211290e6c72736eb1c9d2e159dd243f3e493c430f0411864e109af09a3b9379c4b332815e012caa80"
)
test(
id = 4,
EC = ECP_ShortW_Proj[Fp2[BLS12_377], OnTwist],
Px0 = "fb6dd822e327a60c9e55d760c4f5d26783617a06f868835c2f46e902108eaca5d60fd64c0a0c8a6dc12f9352dfef33",
Px1 = "85e97eef3b4e42a93b8421644e9334f4543a25a36ccd41d448c385146baf3b1efbaeac49c202b04cdbf8b50f3fd962",
Py0 = "8db78b315a524f17ac3d69333604e6bc8aa0b2f138f9adf7edb19f49c847eda6b64df9fe2576b7687e7b55cb5f0bd0",
Py1 = "d4a0c7e5e7aaacab0e0e60c6c49624971a161de4f0daa7968f80998fb7b4761b1196964b26fefa9337fd133784e3d9",
scalar = "1abce9e9c079c686864304d479d68087db33f9edb3f7b2e4655fe0c1da4993f",
Qx0 = "1238814ea497bc1b65ba1c0fbfc394427b10b2d8783bf6a04e01505bc165b577cb2366c19c24c555235ff9b16e7233a",
Qx1 = "b156205cad5d2d97d24c45fe218cd412f47248f26d579fb368af5ba2d80b8c95e2a68f4c84540a8b78b2f0a33a48a4",
Qy0 = "12154ce4a569b39ccfe464a7559b8baf03d3fb6462e67dc0c4ae79a3c33b7756e0c558eff64f9558ebd6500086684f8",
Qy1 = "74571a18fc091f4d77cc7d363bf7a34ab4d0dca86c160b261b383e416f68fa553d7b8b8f6317c8e340ce454daffb5f"
)
test(
id = 5,
EC = ECP_ShortW_Proj[Fp2[BLS12_377], OnTwist],
Px0 = "b56e23f2558c3c124f024952e1858067223541f51a8885d161133beb7bf8d64f22163769e604afbf7fbfb02e1fcb43",
Px1 = "e10c4366d667b40488a9ae32daf3e2a5cc8ddf25ed407afe1a38b855f3ac4f7ea20455924e71369eed07114613b633",
Py0 = "f42d3ab4716d10384bcdfec38cba997c2bafb8d8de32a47225a3e2d2835be1ad02a63323d3dd4145db21e3cbf5cc7a",
Py1 = "15f48a901f3dfd3b812a455f1297e8311c4608869ea02d3d16e9aafdf610d7e72f34b02830de8cc0d0f4e909af5827d",
scalar = "401e70df8ccaa504da72b1d649c067f3b752156dcea5b48dcb601710ab0baa3",
Qx0 = "949099ea1ece1be6fa3f3537e8ba39587b40003fb409a4beb831bb1d9d999b7c621b9a1ced3223f710e0bd05f4a018",
Qx1 = "cdfc0d47788bca3672126e0facc05c19a4fb24a43eee32c8e08a6e5152f6af1d7c2efa48907046f90a721ed28fcf7",
Qy0 = "11d608a01cfe1a40806b9951f85a7bb1b6df6e4c4e2c7c50c17e8fad6ed397a430aff9b4742408367d6536fe503692a",
Qy1 = "19c25d8a782b9d8d9f067d345fd03afe50439c82f99a06abca790f2fae944677d8095d2e276fcb7a75cfb2cbccf89d7"
)
test(
id = 6,
EC = ECP_ShortW_Proj[Fp2[BLS12_377], OnTwist],
Px0 = "137001541ab8a479362ee39f818fc38788231e5288ceea5ebe0c08d0bbca3be9519aa415bde98428df26d361b7311ea",
Px1 = "411b1a4f0abb9fd255a7ae26ac39c1f2c88a48f82c7623b6f225aec9755206e27084b23cbc98f31399405a6599dc54",
Py0 = "833ef097986116cab669c4fddff9b831535d100644f732fb0da0a2dce17d69beaeed67230dd66392e840679afbae1e",
Py1 = "f63e809bd55617c31570d686dbc9b94b3cb96c154f9983181cb913a0671c0c16143332087e44a0283ebf01eea6d73b",
scalar = "10697220c755c800861d377c55b22ae48c25dc144e1fa7a1e5bbdf82993aa33c",
Qx0 = "10fc7772bb2bc946e3ff659c01a0452ebf5dd1472bd1aa71e198597fd05361bf0d173817f91fcb8a1b96b46c35d2675",
Qx1 = "d7b00ddf32f9d8e549c05dd605931ef737d8a4bc62795f73044482957b8093b1cdad0f55c7d4edf6f59594a13f9b67",
Qy0 = "148756a592dec0bf2fda41ac219c4a891ef427665ba1cf47b58bed4996cf8937fb07929c041f10e8a55b2238944fd30",
Qy1 = "e4a240f5c32dab76575ace4d035cbd1e2b87d7585fee7bfde9c88e918a7ad2cd8eb4982acfacfac58c33c8a54ddac7"
)
test(
id = 7,
EC = ECP_ShortW_Proj[Fp2[BLS12_377], OnTwist],
Px0 = "1580ffccc21b057dfe4831f950cbb8f1436f999df657404ecec20225a929a5d56920e8662abc426de23402643087308",
Px1 = "8bf8ff20713a9054aa1cce9635eb3d6ac8371fc052b747a8414595708ff9462d64a0a11ff2c1c5121f4ecc5f22df5e",
Py0 = "e7a5f7df0cec03de25da415fdda485ecc1443b95352e47b23e5506b820c9bc9ea9c2d9e99dd48e1f74e1befe58ce80",
Py1 = "12b5638a0f7e508016fddd2dae2d84736efa59c53f2a5d979b9a29fc7c9c406407bb5f5788bcea3529ac1a763c72e8b",
scalar = "bd936c7066316d3f86b077419a0c0fb9867d4612208241fc004b548f1f54fad",
Qx0 = "69e3411725062858ca7e5fc4d648037baf10e34ba6830ff0090e7188932112b9458c4f4b82f2f0f06c6501fcdaa86c",
Qx1 = "127b75c55c90a872c1d749d05609d2499065ac3c9eb96bd6a9816bd7a2315795ddf40944baef1907676395df306860e",
Qy0 = "13fd0e314159909d3091e382017b35ec21ee2004e9082cd24338817046ed2202ffed52e7714cf436b953f95f49f9856",
Qy1 = "5a8ccdf4a1600fafcc86c646d83199c41cbfebf7a1fa2b1cb9c408352be46a33d0f411b175348f42382d170e17ec2b"
)
test(
id = 8,
EC = ECP_ShortW_Proj[Fp2[BLS12_377], OnTwist],
Px0 = "10898a074b6931cc4ada152b64dd1c6b2bb47912e6502b9e88f638e489c144f8aa9b9f915bb428e082ec84676607f40",
Px1 = "18484823648fe1699468e2d265cd2f2e381a0e67f35f8d192259e2a14573692b4e1fbdeed639e9b6eb0731be820b166",
Py0 = "1840bc6fcb224efe00e32f827fa4f9694cd4186493089c66a936e912b50346b75542b6edbe51ba95c88d3b0fcac34ed",
Py1 = "18cb4201510933776a28ff1c44d356ceab065880f5242d8cc7cdf86874568df679b20f34b6a216d0047d2e1c1a16b85",
scalar = "bdb46c0eece9c8bac8a382def90b522b1d5197f09cf9f7cd98b710845ddb7b4",
Qx0 = "102779766b0068b176e118348e4163e89e3171e6eaad0e47f00cad626f8db025ec3ea990363813882c3dcfcbb642240",
Qx1 = "131eb563d2b0145eee52ee7586acd0254f103f02a20de3bf0202d1cffec2c3628501ec20f9f2ae5f461b59604afc04c",
Qy0 = "1463527db149d2c76f30c082786edb3fe1a9c7f4e04f9496e7adb40aab89340d9792f7d75cf3f5f81b0a28f09625175",
Qy1 = "f3190261db082c207ccd88a3ba762d2c0ba330548444afca03c8535ff11c7c90659a867dda3712a83f02ad184eb24a"
)
test(
id = 9,
EC = ECP_ShortW_Proj[Fp2[BLS12_377], OnTwist],
Px0 = "13c3a392307124afb5f219ba0f8062fa9b75654d3fff12bc924592d284af550f039b6ac58880d2c6fea146b3982f03c",
Px1 = "188169bc937fcc20cc9c289adef30580188f64ecb126faadb5b888f31b813727ff7046d1a19b81abeea6609b8b208c6",
Py0 = "2f9bde8fdd43c5f4de30f335a480dca3bf0858464d8368984406f10ddc1ecabb15fcfd11cebd4fef426e7ca9411221",
Py1 = "25505653e199c354f2da8a13ed9de47f9b51a06ad2fa8826c4b8a9c61c6e75268807d7053e06bfc2899d1a3d6deeb4",
scalar = "91c6249ee16ef94d0b905575982419a7cf31d125775e7ede5c0ab4f86defd33",
Qx0 = "1967547943b197311a7d6723dcc37c3e649f173cec1c88ffca27df86518b3392ae0ee13dbca375c928b94d129226852",
Qx1 = "18b793916195f2d7d2a47c3f8243e5bdc239e78a3eb26971dac01b5f4ce14081a06529ab66cda8e8d5537808223fd00",
Qy0 = "a784902a14ad39adcfdc52bc30d8b8711b93dd869af2375a1f408ba8610b5c558bbebfe9f8f9875954780f4b4262c0",
Qy1 = "fb1c766957c89e4d67747549f650070983e6c19d0208b7e65478e4dd4f72fc54fca450d96329182e3f00d16ae0f3a"
)
# -----------------
test(
id = 0,
EC = ECP_ShortW_Jac[Fp2[BLS12_377], OnTwist],
Px0 = "267401f3ef554fe74ae131d56a10edf14ae40192654901b4618d2bf7af22e77c2a9b79e407348dbd4aad13ca73b33a",
Px1 = "12dcca838f46a3e0418e5dd8b978362757a16bfd78f0b77f4a1916ace353938389ae3ea228d0eb5020a0aaa58884aec",
Py0 = "11799118d2e054aabd9f74c0843fecbdc1c0d56f61c61c5854c2507ae2416e48a6b2cd3bc8bf7495a4d3d8270eafe2b",
Py1 = "823b9f8fb9f8297734a14359fa2c2a0de275e7e638197eaaaa7cff28f9cb3101bdabb570016672455f1ecae625e294",
scalar = "9d432eb58ec68bbc09d10961451d99c7796fb2f795eca603d6feaf3e2a1634b",
Qx0 = "12cfcb50345d43271d2a20e8208789c8ca82f2b8732fae4e7cccd87eb0883741d5e77166971c38c54170bf7635ca2f3",
Qx1 = "17467c786368f2f6eabd78f1c24aa668eae9cd00f6045bfd86b1b3c40966f023a71927026ae5a1281b432ac980d2b6b",
Qy0 = "6f5f1068b6e6a8fafd96894d8b0a8548acee24319d8e2d8b6e6982b1ced8970d9fe33155e74b33d6c2a7835196ca9",
Qy1 = "dc3c7c91c712f9a2631d2b5e49497d8cdf2ea4b30859b43edd716e84cd6a8b61e63cf708d263a7845913cdfb9c7f3b"
)
test(
id = 1,
EC = ECP_ShortW_Jac[Fp2[BLS12_377], OnTwist],
Px0 = "3a3055d6a46901c1b2227a0e334ffa9f654e62a6d3608f3a672e5816f9e9b04c0e668c3e9f8c807b269422afdc7de9",
Px1 = "25803bd55f37b254865d5fc7ac9843fb306c2eb09d34ee0c4ecb705b5e10f6911f07fd707a2e28681a421f45b1a4d",
Py0 = "15a0594a3c9dddc535472c4827aa443774a06f77bec2d20837c6574aa5fac35a279bac756531fa75f979a7a97f297d6",
Py1 = "15e8c3e013cbc64110f075295f39a4f85c9591a52b8e4903047b1f4b44bb5216b6339788c82fd90e82b1027756e7987",
scalar = "a0067f5b4294fbacd24730fed25c936a08b5f1a77824149ad6c2ce476518d17",
Qx0 = "6280dbf019f8d94acac4a39213e03f221be732849b45437edef7617460e83f972883c20791a86670a8a4c6c0125629",
Qx1 = "ab3a185addbd6c7db907170d75bd2f1997666e93d8cedbef9e10a103110d4ea09d9a889f505d7026e5c498b0355c1",
Qy0 = "1a6dd15b4c2bd6bd97858e5a75ac4365f8becaa885c4e3267d0ae256655cd057061d61b2b9acbd10bdf678a9c1e7f8c",
Qy1 = "16204b5825eec903cb7798cb62ccadb9004032d72bb815958fcd2f613c77e147656449fdaa210994338978676ad6b0e"
)
test(
id = 2,
EC = ECP_ShortW_Jac[Fp2[BLS12_377], OnTwist],
Px0 = "18ebf549cd24a64badff4ec177205be121757b68589020434aba816756d2e6fa95fdb389c8639d4bf77575122e1c04e",
Px1 = "11cd0f0976658a6f2ec113034428ef1605befdaa5642944f5c4e571b24fc166c368c30473e25ab148209be4c0b4e37",
Py0 = "1190d818317495201732feb2cfc8f507adac6273debff46bb6aea4a3f3e3fe7d28d893c90b21a6f28d2fbc72d9fc528",
Py1 = "df5fcb2daa302a5c64aeef96835e0a6b39f5d7bf0e70cc10401f966745a6b3fa682b7e5b45d9295e744e1dd7855fd4",
scalar = "7a49802ba58c87c30b631b2f90a3b876c7143e09b542c9c14706bddf9bd4117",
Qx0 = "6cdff80576d8695a646f915caba5bc6748eee1707bb0d4ebcabaf8d236b780aef6a953a07f48ef4696f02db4906c71",
Qx1 = "6b64ca5f0d46702e7e9e7beb1993b698c9b3e9991f545241d5fd44a27dc692d5f5a4c2fe6871af0653128f960307d4",
Qy0 = "cfad02b664495d42d1c6b598306229c67bcf76cc923abb13ae038c90e959a7611161a98d607729577d55ec18bf1e74",
Qy1 = "e433d9dd7d47d94e9f61a5bc27a688c63efa05408d6fd40a5c4e2711a37d011dc80f5dbaaafd939a07235c770feead"
)
test(
id = 3,
EC = ECP_ShortW_Jac[Fp2[BLS12_377], OnTwist],
Px0 = "52b54c30c1dcadbcb698d0cb7fd65de1eb8f7590c7afe46e019bdc9e0ef8bc4c060339220d3615e4b1f2b12ffa6d83",
Px1 = "dd53b483c2ab1aaa7ed22ef619b5e979237ae95476436f2c51c8b70da39a4e54a989f10f6d12ee098154911aa052f6",
Py0 = "2a8c96662a7c76cb7d8ca6571a5b99abfd2d7343dd668425e5fa4a8c880b313b70f09a15a825fac63ae7065c0c51d",
Py1 = "7bd93bb9d5bfdd15636a34a13385e9abbc1088ecc02c86cbc733122ccbbb657ec8d75e47102ce46d35a9612c01fc3d",
scalar = "e0c564e69ad68343fd4ec3d4dafdf6a92f44c13fc70a9aad95d10b2c96ee747",
Qx0 = "126f6039b7031b73a7f926bd443ace416f62e56f03864d660bbe1f1a2a994672d408f9936293f367d3fb5d8e275ad40",
Qx1 = "1654a2c26479ace1eb68e279eceac6aa10680549b09f42d288767f6296c98ba4299fa6df37a946b88823a5deebc231c",
Qy0 = "23026839059191f71e2abcf582234aa2ce4b4225c3d503e8fc6c119df1168192e894f33ed48bc571e5527365dd92b5",
Qy1 = "11f8fcce53f5e3211290e6c72736eb1c9d2e159dd243f3e493c430f0411864e109af09a3b9379c4b332815e012caa80"
)
test(
id = 4,
EC = ECP_ShortW_Jac[Fp2[BLS12_377], OnTwist],
Px0 = "fb6dd822e327a60c9e55d760c4f5d26783617a06f868835c2f46e902108eaca5d60fd64c0a0c8a6dc12f9352dfef33",
Px1 = "85e97eef3b4e42a93b8421644e9334f4543a25a36ccd41d448c385146baf3b1efbaeac49c202b04cdbf8b50f3fd962",
Py0 = "8db78b315a524f17ac3d69333604e6bc8aa0b2f138f9adf7edb19f49c847eda6b64df9fe2576b7687e7b55cb5f0bd0",
Py1 = "d4a0c7e5e7aaacab0e0e60c6c49624971a161de4f0daa7968f80998fb7b4761b1196964b26fefa9337fd133784e3d9",
scalar = "1abce9e9c079c686864304d479d68087db33f9edb3f7b2e4655fe0c1da4993f",
Qx0 = "1238814ea497bc1b65ba1c0fbfc394427b10b2d8783bf6a04e01505bc165b577cb2366c19c24c555235ff9b16e7233a",
Qx1 = "b156205cad5d2d97d24c45fe218cd412f47248f26d579fb368af5ba2d80b8c95e2a68f4c84540a8b78b2f0a33a48a4",
Qy0 = "12154ce4a569b39ccfe464a7559b8baf03d3fb6462e67dc0c4ae79a3c33b7756e0c558eff64f9558ebd6500086684f8",
Qy1 = "74571a18fc091f4d77cc7d363bf7a34ab4d0dca86c160b261b383e416f68fa553d7b8b8f6317c8e340ce454daffb5f"
)
test(
id = 5,
EC = ECP_ShortW_Jac[Fp2[BLS12_377], OnTwist],
Px0 = "b56e23f2558c3c124f024952e1858067223541f51a8885d161133beb7bf8d64f22163769e604afbf7fbfb02e1fcb43",
Px1 = "e10c4366d667b40488a9ae32daf3e2a5cc8ddf25ed407afe1a38b855f3ac4f7ea20455924e71369eed07114613b633",
Py0 = "f42d3ab4716d10384bcdfec38cba997c2bafb8d8de32a47225a3e2d2835be1ad02a63323d3dd4145db21e3cbf5cc7a",
Py1 = "15f48a901f3dfd3b812a455f1297e8311c4608869ea02d3d16e9aafdf610d7e72f34b02830de8cc0d0f4e909af5827d",
scalar = "401e70df8ccaa504da72b1d649c067f3b752156dcea5b48dcb601710ab0baa3",
Qx0 = "949099ea1ece1be6fa3f3537e8ba39587b40003fb409a4beb831bb1d9d999b7c621b9a1ced3223f710e0bd05f4a018",
Qx1 = "cdfc0d47788bca3672126e0facc05c19a4fb24a43eee32c8e08a6e5152f6af1d7c2efa48907046f90a721ed28fcf7",
Qy0 = "11d608a01cfe1a40806b9951f85a7bb1b6df6e4c4e2c7c50c17e8fad6ed397a430aff9b4742408367d6536fe503692a",
Qy1 = "19c25d8a782b9d8d9f067d345fd03afe50439c82f99a06abca790f2fae944677d8095d2e276fcb7a75cfb2cbccf89d7"
)
test(
id = 6,
EC = ECP_ShortW_Jac[Fp2[BLS12_377], OnTwist],
Px0 = "137001541ab8a479362ee39f818fc38788231e5288ceea5ebe0c08d0bbca3be9519aa415bde98428df26d361b7311ea",
Px1 = "411b1a4f0abb9fd255a7ae26ac39c1f2c88a48f82c7623b6f225aec9755206e27084b23cbc98f31399405a6599dc54",
Py0 = "833ef097986116cab669c4fddff9b831535d100644f732fb0da0a2dce17d69beaeed67230dd66392e840679afbae1e",
Py1 = "f63e809bd55617c31570d686dbc9b94b3cb96c154f9983181cb913a0671c0c16143332087e44a0283ebf01eea6d73b",
scalar = "10697220c755c800861d377c55b22ae48c25dc144e1fa7a1e5bbdf82993aa33c",
Qx0 = "10fc7772bb2bc946e3ff659c01a0452ebf5dd1472bd1aa71e198597fd05361bf0d173817f91fcb8a1b96b46c35d2675",
Qx1 = "d7b00ddf32f9d8e549c05dd605931ef737d8a4bc62795f73044482957b8093b1cdad0f55c7d4edf6f59594a13f9b67",
Qy0 = "148756a592dec0bf2fda41ac219c4a891ef427665ba1cf47b58bed4996cf8937fb07929c041f10e8a55b2238944fd30",
Qy1 = "e4a240f5c32dab76575ace4d035cbd1e2b87d7585fee7bfde9c88e918a7ad2cd8eb4982acfacfac58c33c8a54ddac7"
)
test(
id = 7,
EC = ECP_ShortW_Jac[Fp2[BLS12_377], OnTwist],
Px0 = "1580ffccc21b057dfe4831f950cbb8f1436f999df657404ecec20225a929a5d56920e8662abc426de23402643087308",
Px1 = "8bf8ff20713a9054aa1cce9635eb3d6ac8371fc052b747a8414595708ff9462d64a0a11ff2c1c5121f4ecc5f22df5e",
Py0 = "e7a5f7df0cec03de25da415fdda485ecc1443b95352e47b23e5506b820c9bc9ea9c2d9e99dd48e1f74e1befe58ce80",
Py1 = "12b5638a0f7e508016fddd2dae2d84736efa59c53f2a5d979b9a29fc7c9c406407bb5f5788bcea3529ac1a763c72e8b",
scalar = "bd936c7066316d3f86b077419a0c0fb9867d4612208241fc004b548f1f54fad",
Qx0 = "69e3411725062858ca7e5fc4d648037baf10e34ba6830ff0090e7188932112b9458c4f4b82f2f0f06c6501fcdaa86c",
Qx1 = "127b75c55c90a872c1d749d05609d2499065ac3c9eb96bd6a9816bd7a2315795ddf40944baef1907676395df306860e",
Qy0 = "13fd0e314159909d3091e382017b35ec21ee2004e9082cd24338817046ed2202ffed52e7714cf436b953f95f49f9856",
Qy1 = "5a8ccdf4a1600fafcc86c646d83199c41cbfebf7a1fa2b1cb9c408352be46a33d0f411b175348f42382d170e17ec2b"
)
test(
id = 8,
EC = ECP_ShortW_Jac[Fp2[BLS12_377], OnTwist],
Px0 = "10898a074b6931cc4ada152b64dd1c6b2bb47912e6502b9e88f638e489c144f8aa9b9f915bb428e082ec84676607f40",
Px1 = "18484823648fe1699468e2d265cd2f2e381a0e67f35f8d192259e2a14573692b4e1fbdeed639e9b6eb0731be820b166",
Py0 = "1840bc6fcb224efe00e32f827fa4f9694cd4186493089c66a936e912b50346b75542b6edbe51ba95c88d3b0fcac34ed",
Py1 = "18cb4201510933776a28ff1c44d356ceab065880f5242d8cc7cdf86874568df679b20f34b6a216d0047d2e1c1a16b85",
scalar = "bdb46c0eece9c8bac8a382def90b522b1d5197f09cf9f7cd98b710845ddb7b4",
Qx0 = "102779766b0068b176e118348e4163e89e3171e6eaad0e47f00cad626f8db025ec3ea990363813882c3dcfcbb642240",
Qx1 = "131eb563d2b0145eee52ee7586acd0254f103f02a20de3bf0202d1cffec2c3628501ec20f9f2ae5f461b59604afc04c",
Qy0 = "1463527db149d2c76f30c082786edb3fe1a9c7f4e04f9496e7adb40aab89340d9792f7d75cf3f5f81b0a28f09625175",
Qy1 = "f3190261db082c207ccd88a3ba762d2c0ba330548444afca03c8535ff11c7c90659a867dda3712a83f02ad184eb24a"
)
test(
id = 9,
EC = ECP_ShortW_Jac[Fp2[BLS12_377], OnTwist],
Px0 = "13c3a392307124afb5f219ba0f8062fa9b75654d3fff12bc924592d284af550f039b6ac58880d2c6fea146b3982f03c",
Px1 = "188169bc937fcc20cc9c289adef30580188f64ecb126faadb5b888f31b813727ff7046d1a19b81abeea6609b8b208c6",
Py0 = "2f9bde8fdd43c5f4de30f335a480dca3bf0858464d8368984406f10ddc1ecabb15fcfd11cebd4fef426e7ca9411221",
Py1 = "25505653e199c354f2da8a13ed9de47f9b51a06ad2fa8826c4b8a9c61c6e75268807d7053e06bfc2899d1a3d6deeb4",
scalar = "91c6249ee16ef94d0b905575982419a7cf31d125775e7ede5c0ab4f86defd33",
Qx0 = "1967547943b197311a7d6723dcc37c3e649f173cec1c88ffca27df86518b3392ae0ee13dbca375c928b94d129226852",
Qx1 = "18b793916195f2d7d2a47c3f8243e5bdc239e78a3eb26971dac01b5f4ce14081a06529ab66cda8e8d5537808223fd00",
Qy0 = "a784902a14ad39adcfdc52bc30d8b8711b93dd869af2375a1f408ba8610b5c558bbebfe9f8f9875954780f4b4262c0",
Qy1 = "fb1c766957c89e4d67747549f650070983e6c19d0208b7e65478e4dd4f72fc54fca450d96329182e3f00d16ae0f3a"
)
run_scalar_mul_test_vs_sage(
ECP_ShortW_Jac[Fp2[BLS12_377], OnTwist],
"t_ec_sage_bls12_377_g2_jacobian"
)

583
tests/t_ec_sage_bls12_381.nim
View File

@ -7,573 +7,30 @@
# at your option. This file may not be copied, modified, or distributed except according to those terms.
import
# Standard library
std/[unittest, times],
# Internals
../constantine/config/[common, curves],
../constantine/arithmetic,
../constantine/config/[type_fp, curves],
../constantine/towers,
../constantine/io/[io_bigints, io_ec],
../constantine/elliptic/[
ec_shortweierstrass_projective,
ec_shortweierstrass_jacobian,
ec_scalar_mul,
ec_endomorphism_accel],
../constantine/elliptic/ec_shortweierstrass_jacobian,
../constantine/elliptic/ec_shortweierstrass_projective,
# Test utilities
./support/ec_reference_scalar_mult
./t_ec_sage_template
echo "\n------------------------------------------------------\n"
run_scalar_mul_test_vs_sage(
ECP_ShortW_Proj[Fp[BLS12_381], NotOnTwist],
"t_ec_sage_bls12_381_g1_projective"
)
proc test(
id: int,
EC: typedesc[ECP_ShortW_Proj or ECP_ShortW_Jac],
Px, Py: string,
scalar: string,
Qx, Qy: string
) =
run_scalar_mul_test_vs_sage(
ECP_ShortW_Jac[Fp[BLS12_381], NotOnTwist],
"t_ec_sage_bls12_381_g1_jacobian"
)
test "test " & $id & " - " & $EC:
var P: EC
let pOK = P.fromHex(Px, Py)
doAssert pOK
run_scalar_mul_test_vs_sage(
ECP_ShortW_Proj[Fp2[BLS12_381], OnTwist],
"t_ec_sage_bls12_381_g2_projective"
)
var Q: EC
let qOK = Q.fromHex(Qx, Qy)
let exponent = BigInt[EC.F.C.getCurveOrderBitwidth()].fromHex(scalar)
var
impl = P
reference = P
endo = P
endoW = P
impl.scalarMulGeneric(exponent)
reference.unsafe_ECmul_double_add(exponent)
endo.scalarMulEndo(exponent)
endoW.scalarMulGLV_m2w2(exponent)
doAssert: bool(Q == reference)
doAssert: bool(Q == impl)
doAssert: bool(Q == endo)
doAssert: bool(Q == endoW)
suite "Scalar Multiplication (cofactor cleared): BLS12_381 implementation vs SageMath" & " [" & $WordBitwidth & "-bit mode]":
# Generated via sage sage/testgen_bls12_381.sage
test(
id = 0,
EC = ECP_ShortW_Proj[Fp[BLS12_381], NotOnTwist],
Px = "f9679bb02ee7f352fff6a6467a5e563ec8dd38c86a48abd9e8f7f241f1cdd29d54bc3ddea3a33b62e0d7ce22f3d244a",
Py = "50189b992cf856846b30e52205ff9ef72dc081e9680726586231cbc29a81a162120082585f401e00382d5c86fb1083f",
scalar = "f7e60a832eb77ac47374bc93251360d6c81c21add62767ff816caf11a20d8db",
Qx = "c344f3bcc86df380186311fa502b7943a436a629380f8ee1960515522eedc58fe67ddd47615487668bcf12842c524d8",
Qy = "189e0c154f2631ad26e24ca73d84fb60a21d385fe205df04cf9f2f6fc0c3aa72afe9fbea71a930fa71d9bbfddb2fa571"
)
test(
id = 1,
EC = ECP_ShortW_Proj[Fp[BLS12_381], NotOnTwist],
Px = "17d71835ff84f150fabf5c77ac90bf7f6249143abd1f5d8a46a76f243d424d82e1e258fc7983ba8af97a2462adebe090",
Py = "d3e108ee1332067cbe4f4193eae10381acb69f493b40e53d9dee59506b49c6564c9056494a7f987982eb4069512c1c6",
scalar = "5f10367bdae7aa872d90b5ac209321ce5a15181ce22848d032a8d452055cbfd0",
Qx = "21073bee733a07b15d83afcd4e6ee11b01e6137fd5ad4589c5045e12d79a9a9490a3ebc59f30633a60fc3635a3c1e51",
Qy = "eb7a97a9d3dfff1667b8fa559bdcdf37c7767e6afb8ca93ad9dd44feb93761e10aa2c4c1a79728a21cd4a6f705398b5"
)
test(
id = 2,
EC = ECP_ShortW_Proj[Fp[BLS12_381], NotOnTwist],
Px = "f92c9572692e8f3d450483a7a9bb4694e3b54c9cd09441a4dd7f579b0a6984e47f8090c31c172b33d87f3de186d6b58",
Py = "286ede4cb2ae19ead4932d5550c5d3ec8ce3a3ada5e1ed6d202e93dd1b16d3513f0f9b62adc6323f18e272a426ee955",
scalar = "4c321d72220c098fc0fd52306de98f8be9446bf854cf1e4d8dbae62375d18faf",
Qx = "4bb385e937582ae32aa7ba89632fcef2eace3f7b57309d979cf35298a430de9ef4d9ac5ba2335c1a4b6e7e5c38d0036",
Qy = "1801154d3a7b0daea772345b7f72a4c88c9677743f267da63490dad4dece2ecc9ec02d4d4d063086ee5d356aa2db914e"
)
test(
id = 3,
EC = ECP_ShortW_Proj[Fp[BLS12_381], NotOnTwist],
Px = "ec23ff3435b8ebd5e8e0a879d432e11eb974161664b1341fd28f1ffc4c228bf6ada2ae4a565f18c9b66f67a7573502d",
Py = "10c4b647be08db0b49b75320ae891f9f9c5d7bb7c798947e800d681d205d1b24b12e4dfa993d1bd16851b00356627cc1",
scalar = "1738857afb76c55f615c2a20b44ca90dcb3267d804ec23fddea431dbee4eb37f",
Qx = "dc7ae7801152918ee3c13590407b4242a80d0b855a0bf585d3dc30719601d2d5d9e01e99ae735003ecb7c20ef48265",
Qy = "142c01a6aa390426a4ce2f36df43f86442732c35d4e05e5b67f3623832944f0ea5a29138624cb939330652a3cfb282b5"
)
test(
id = 4,
EC = ECP_ShortW_Proj[Fp[BLS12_381], NotOnTwist],
Px = "df127083c2a5ef2388b02af913c0e4002a52a82db9e5ecbf23ee4f557d3b61c91ebcfe9d4973070b46bc5ea6897bca1",
Py = "318960aeea262ec23ffdd42ec1ba72ae6fa2186a1e2a0fc2659073fb7b5adfb50d581a4d998a94d1accf78b1b3a0163",
scalar = "19c47811813444020c999a2b263940b5054cf45bb8ad8e086ff126bfcd5507e1",
Qx = "5f93c42fd76a29063efa2ee92607e0b3ae7edc4e419b3914661e5162d6beaeb96a34d2007ff817bc102651f61dca8d1",
Qy = "18dde8666bb1d0a379719d7d1b1512de809b70e49d9553303274ea872e56f7f39da551d6bcb7c57ae88ec7dc1fb354a4"
)
test(
id = 5,
EC = ECP_ShortW_Proj[Fp[BLS12_381], NotOnTwist],
Px = "101123de23c0f240c583c2368c4118dc942db219c55f58cf54acd500c1fcfa06f651ad75319ebf840cbdb6bddea7fde4",
Py = "5268587d4b844b0708e0336d1bbf48da185aaf5b948eccc3b565d00a856dd55882b9bb31c52af0e275b168cb35eb7b0",
scalar = "43ffcda71e45a3e90b7502d92b30a0b06c54c95a91aa21e0438677b1c2714ecb",
Qx = "f9871b682c1c76c7f4f0a7ca57ad876c10dc108b65b76987264873278d9f54db95101c173aed06d07062efc7d47ca0c",
Qy = "20d9628d611e72a4251a1f2357d4f53e68e4915383b6a0d126273d216b1a8c5e2cb7b2688ad702ef1682f4c5228fcd9"
)
test(
id = 6,
EC = ECP_ShortW_Proj[Fp[BLS12_381], NotOnTwist],
Px = "1457ba1bae6eb3afae3261941c65c93e3ae7d784907d15b8d559100da5e13fd29e4a4d6e3103b781a95237b7b2d80a8e",
Py = "6a869a47cb48d01e7d29660932afd7617720262b55de5f430b8aa3d74f9fd2b9d3a07ce192425da58014764fc9532cd",
scalar = "64ad0d6c36dba5368e71f0010aebf860288f54611e5aaf18082bae7a404ebfd8",
Qx = "93e540e26190e161038d985d40f2ab897cbc2346be7d8f2b201a689b59d4020a8740e252606f2f79ba0e121ccc9976d",
Qy = "10568d68f1b993aa1eded3869eda14e509f1cb4d8553bdf97feee175467cea4c0c1316fdb4e5a68440ad04b96b2d3bfc"
)
test(
id = 7,
EC = ECP_ShortW_Proj[Fp[BLS12_381], NotOnTwist],
Px = "2615f843e8fe68d4c337bcf83b2cf13cbae638edd0740f1eac520dc2146afa3b8d36c540878c1d207ef913634b1e593",
Py = "1787d6eeeceb6e7793073f0bbe7bae522529c126b650c43d5d41e732c581a57df1bfb818061b7b4e6c9145da5df2c43e",
scalar = "b0ac3d0e685583075aa46c03a00859dfbec24ccb36e2cae3806d82275adcc03",
Qx = "d95ed29c2e15fd2205d83a71478341d6022deb93af4d49f704437678a72ce141d2f6043aa0e34e26f60d17e16b97053",
Qy = "b37cbded112c84116b74ff311b10d148f3e203cb88d4a011b096c74cd2bfdb27255727de4aa8299ae10b32d661d48a7"
)
test(
id = 8,
EC = ECP_ShortW_Proj[Fp[BLS12_381], NotOnTwist],
Px = "10bc0c4e1ed87246a9d4d7d38546369f275a245f6e1d3b882e8c9a7f05bc6ee8ff97a96a54084c2bef15ed8bfefb1465",
Py = "1782377e5f588576b5ab42fea224e88873dda957202f0c6d72ce8728c2d58dc654be77226fbda385d5f269354e4a176a",
scalar = "23941bb3c3659423d6fdafb7cff52e0e02de0ac91e64c537c6203d64905b63d0",
Qx = "83f1e7e8bd963c1ccd837dae7bc9336531aaf0aee717537a9a7e2712e220f74cdb73a99f331c0eb6b377be3dafc211f",
Qy = "cd87773d072b1305dfc85c2983aecae2ab316e5e8f31306c32d58d6ce2e431b12685d18c58b6a35ad2113c5b689eeb"
)
test(
id = 9,
EC = ECP_ShortW_Proj[Fp[BLS12_381], NotOnTwist],
Px = "be4f9f721d98a761a5562bd80ea06f369e9cbb7d33bbb2f0191d4b77d0fd2a10c4083b54157b525f36c522ca3a6ca09",
Py = "166c315ecdd20acb3c5efcc7e038b17d0b37a06ffbf77873f15fc0cd091a1e4102a8b8bf5507919453759e744391b04d",
scalar = "4203156dcf70582ea8cbd0388104f47fd5a18ae336b2fed8458e1e4e74d7baf5",
Qx = "c72bc7087cd22993b7f6d2e49026abfde678a384073ed373b95df722b1ab658eb5ae42211e5528af606e38b59511bc6",
Qy = "96d80593b42fe44e64793e490b1257af0aa26b36773aac93c3686fdb14975917cf60a1a19e32623218d0722dbb88a85"
)
# -----
test(
id = 0,
EC = ECP_ShortW_Jac[Fp[BLS12_381], NotOnTwist],
Px = "f9679bb02ee7f352fff6a6467a5e563ec8dd38c86a48abd9e8f7f241f1cdd29d54bc3ddea3a33b62e0d7ce22f3d244a",
Py = "50189b992cf856846b30e52205ff9ef72dc081e9680726586231cbc29a81a162120082585f401e00382d5c86fb1083f",
scalar = "f7e60a832eb77ac47374bc93251360d6c81c21add62767ff816caf11a20d8db",
Qx = "c344f3bcc86df380186311fa502b7943a436a629380f8ee1960515522eedc58fe67ddd47615487668bcf12842c524d8",
Qy = "189e0c154f2631ad26e24ca73d84fb60a21d385fe205df04cf9f2f6fc0c3aa72afe9fbea71a930fa71d9bbfddb2fa571"
)
test(
id = 1,
EC = ECP_ShortW_Jac[Fp[BLS12_381], NotOnTwist],
Px = "17d71835ff84f150fabf5c77ac90bf7f6249143abd1f5d8a46a76f243d424d82e1e258fc7983ba8af97a2462adebe090",
Py = "d3e108ee1332067cbe4f4193eae10381acb69f493b40e53d9dee59506b49c6564c9056494a7f987982eb4069512c1c6",
scalar = "5f10367bdae7aa872d90b5ac209321ce5a15181ce22848d032a8d452055cbfd0",
Qx = "21073bee733a07b15d83afcd4e6ee11b01e6137fd5ad4589c5045e12d79a9a9490a3ebc59f30633a60fc3635a3c1e51",
Qy = "eb7a97a9d3dfff1667b8fa559bdcdf37c7767e6afb8ca93ad9dd44feb93761e10aa2c4c1a79728a21cd4a6f705398b5"
)
test(
id = 2,
EC = ECP_ShortW_Jac[Fp[BLS12_381], NotOnTwist],
Px = "f92c9572692e8f3d450483a7a9bb4694e3b54c9cd09441a4dd7f579b0a6984e47f8090c31c172b33d87f3de186d6b58",
Py = "286ede4cb2ae19ead4932d5550c5d3ec8ce3a3ada5e1ed6d202e93dd1b16d3513f0f9b62adc6323f18e272a426ee955",
scalar = "4c321d72220c098fc0fd52306de98f8be9446bf854cf1e4d8dbae62375d18faf",
Qx = "4bb385e937582ae32aa7ba89632fcef2eace3f7b57309d979cf35298a430de9ef4d9ac5ba2335c1a4b6e7e5c38d0036",
Qy = "1801154d3a7b0daea772345b7f72a4c88c9677743f267da63490dad4dece2ecc9ec02d4d4d063086ee5d356aa2db914e"
)
test(
id = 3,
EC = ECP_ShortW_Jac[Fp[BLS12_381], NotOnTwist],
Px = "ec23ff3435b8ebd5e8e0a879d432e11eb974161664b1341fd28f1ffc4c228bf6ada2ae4a565f18c9b66f67a7573502d",
Py = "10c4b647be08db0b49b75320ae891f9f9c5d7bb7c798947e800d681d205d1b24b12e4dfa993d1bd16851b00356627cc1",
scalar = "1738857afb76c55f615c2a20b44ca90dcb3267d804ec23fddea431dbee4eb37f",
Qx = "dc7ae7801152918ee3c13590407b4242a80d0b855a0bf585d3dc30719601d2d5d9e01e99ae735003ecb7c20ef48265",
Qy = "142c01a6aa390426a4ce2f36df43f86442732c35d4e05e5b67f3623832944f0ea5a29138624cb939330652a3cfb282b5"
)
test(
id = 4,
EC = ECP_ShortW_Jac[Fp[BLS12_381], NotOnTwist],
Px = "df127083c2a5ef2388b02af913c0e4002a52a82db9e5ecbf23ee4f557d3b61c91ebcfe9d4973070b46bc5ea6897bca1",
Py = "318960aeea262ec23ffdd42ec1ba72ae6fa2186a1e2a0fc2659073fb7b5adfb50d581a4d998a94d1accf78b1b3a0163",
scalar = "19c47811813444020c999a2b263940b5054cf45bb8ad8e086ff126bfcd5507e1",
Qx = "5f93c42fd76a29063efa2ee92607e0b3ae7edc4e419b3914661e5162d6beaeb96a34d2007ff817bc102651f61dca8d1",
Qy = "18dde8666bb1d0a379719d7d1b1512de809b70e49d9553303274ea872e56f7f39da551d6bcb7c57ae88ec7dc1fb354a4"
)
test(
id = 5,
EC = ECP_ShortW_Jac[Fp[BLS12_381], NotOnTwist],
Px = "101123de23c0f240c583c2368c4118dc942db219c55f58cf54acd500c1fcfa06f651ad75319ebf840cbdb6bddea7fde4",
Py = "5268587d4b844b0708e0336d1bbf48da185aaf5b948eccc3b565d00a856dd55882b9bb31c52af0e275b168cb35eb7b0",
scalar = "43ffcda71e45a3e90b7502d92b30a0b06c54c95a91aa21e0438677b1c2714ecb",
Qx = "f9871b682c1c76c7f4f0a7ca57ad876c10dc108b65b76987264873278d9f54db95101c173aed06d07062efc7d47ca0c",
Qy = "20d9628d611e72a4251a1f2357d4f53e68e4915383b6a0d126273d216b1a8c5e2cb7b2688ad702ef1682f4c5228fcd9"
)
test(
id = 6,
EC = ECP_ShortW_Jac[Fp[BLS12_381], NotOnTwist],
Px = "1457ba1bae6eb3afae3261941c65c93e3ae7d784907d15b8d559100da5e13fd29e4a4d6e3103b781a95237b7b2d80a8e",
Py = "6a869a47cb48d01e7d29660932afd7617720262b55de5f430b8aa3d74f9fd2b9d3a07ce192425da58014764fc9532cd",
scalar = "64ad0d6c36dba5368e71f0010aebf860288f54611e5aaf18082bae7a404ebfd8",
Qx = "93e540e26190e161038d985d40f2ab897cbc2346be7d8f2b201a689b59d4020a8740e252606f2f79ba0e121ccc9976d",
Qy = "10568d68f1b993aa1eded3869eda14e509f1cb4d8553bdf97feee175467cea4c0c1316fdb4e5a68440ad04b96b2d3bfc"
)
test(
id = 7,
EC = ECP_ShortW_Jac[Fp[BLS12_381], NotOnTwist],
Px = "2615f843e8fe68d4c337bcf83b2cf13cbae638edd0740f1eac520dc2146afa3b8d36c540878c1d207ef913634b1e593",
Py = "1787d6eeeceb6e7793073f0bbe7bae522529c126b650c43d5d41e732c581a57df1bfb818061b7b4e6c9145da5df2c43e",
scalar = "b0ac3d0e685583075aa46c03a00859dfbec24ccb36e2cae3806d82275adcc03",
Qx = "d95ed29c2e15fd2205d83a71478341d6022deb93af4d49f704437678a72ce141d2f6043aa0e34e26f60d17e16b97053",
Qy = "b37cbded112c84116b74ff311b10d148f3e203cb88d4a011b096c74cd2bfdb27255727de4aa8299ae10b32d661d48a7"
)
test(
id = 8,
EC = ECP_ShortW_Jac[Fp[BLS12_381], NotOnTwist],
Px = "10bc0c4e1ed87246a9d4d7d38546369f275a245f6e1d3b882e8c9a7f05bc6ee8ff97a96a54084c2bef15ed8bfefb1465",
Py = "1782377e5f588576b5ab42fea224e88873dda957202f0c6d72ce8728c2d58dc654be77226fbda385d5f269354e4a176a",
scalar = "23941bb3c3659423d6fdafb7cff52e0e02de0ac91e64c537c6203d64905b63d0",
Qx = "83f1e7e8bd963c1ccd837dae7bc9336531aaf0aee717537a9a7e2712e220f74cdb73a99f331c0eb6b377be3dafc211f",
Qy = "cd87773d072b1305dfc85c2983aecae2ab316e5e8f31306c32d58d6ce2e431b12685d18c58b6a35ad2113c5b689eeb"
)
test(
id = 9,
EC = ECP_ShortW_Jac[Fp[BLS12_381], NotOnTwist],
Px = "be4f9f721d98a761a5562bd80ea06f369e9cbb7d33bbb2f0191d4b77d0fd2a10c4083b54157b525f36c522ca3a6ca09",
Py = "166c315ecdd20acb3c5efcc7e038b17d0b37a06ffbf77873f15fc0cd091a1e4102a8b8bf5507919453759e744391b04d",
scalar = "4203156dcf70582ea8cbd0388104f47fd5a18ae336b2fed8458e1e4e74d7baf5",
Qx = "c72bc7087cd22993b7f6d2e49026abfde678a384073ed373b95df722b1ab658eb5ae42211e5528af606e38b59511bc6",
Qy = "96d80593b42fe44e64793e490b1257af0aa26b36773aac93c3686fdb14975917cf60a1a19e32623218d0722dbb88a85"
)
proc test(
id: int,
EC: typedesc[ECP_ShortW_Proj or ECP_ShortW_Jac],
Px0, Px1, Py0, Py1: string,
scalar: string,
Qx0, Qx1, Qy0, Qy1: string
) =
test "test " & $id & " - " & $EC:
var P: EC
let pOK = P.fromHex(Px0, Px1, Py0, Py1)
doAssert pOK
var Q: EC
let qOK = Q.fromHex(Qx0, Qx1, Qy0, Qy1)
let exponent = BigInt[EC.F.C.getCurveOrderBitwidth()].fromHex(scalar)
var
impl = P
reference = P
endo = P
impl.scalarMulGeneric(exponent)
reference.unsafe_ECmul_double_add(exponent)
endo.scalarMulEndo(exponent)
doAssert: bool(Q == reference)
doAssert: bool(Q == impl)
doAssert: bool(Q == endo)
suite "Scalar Multiplication G2: BLS12-381 implementation vs SageMath" & " [" & $WordBitwidth & "-bit mode]":
# Generated via sage sage/testgen_bls12_381.sage
test(
id = 0,
EC = ECP_ShortW_Proj[Fp2[BLS12_381], OnTwist],
Px0 = "10fbddd49246ac4b0faa489e3474507ebc96a5da194b2f7a706fad6bf8435021e1598700088abfe0ae7343296c1b7f52",
Px1 = "324102fa5bd71d9048c3c6a6c62d1f35195d7067bf00dc5eaedd14eecc688383446aba4e8fda059d3f619f00be7890",
Py0 = "f3e974aafa7a3fb3a1209f3af4492c9d9c52f1ae738e1e08309dd0f438f131f8ddd8b934eb8ff2cb078b8c524c11fab",
Py1 = "15e75704edffe7b975cf1d3f27f1dceb89d02e5660650195e0288e5d26c5e9087c241d1bd3263c991d10e2695d1611f1",
scalar = "1f7bef2a74f3bf8ac0225a9edfa514bb5666b15e7be3e929059f2ef75f0035a6",
Qx0 = "1328e4ac12458f9e0c22e925d2fc5593eaf0c126154b484de986895030f830262f54493edcc26aa39c5ab8e714b784b3",
Qx1 = "14d375c704f9187984fe2d64d5517ce7c6eb09981cee68cd48370df21d7f4d0b19431347be3b9eae9d46605cb229e293",
Qy0 = "1580cb4fceea4e71e222945825d4352c97a02d3118ffbd0e006e467b1ff6d4207acd2bb58a197894e9870cbd1bfb369c",
Qy1 = "5df16d4223c040dffcd1912359cfae1f3a99e7f519b2aedaaeb6d77115c63acf309f9effc69d4d9a0d7de420dbf9f1a"
)
test(
id = 1,
EC = ECP_ShortW_Proj[Fp2[BLS12_381], OnTwist],
Px0 = "4b472ab5d0995a22c95f0805eb459b147d2033b8c13c1c07c857a97e66952d4df3e3b5f346997cc7bd19886492fae83",
Px1 = "12228345592511c7327176258097c70dffad1ff53b37163cbd4747d0085ed0bcfe90b9150d2f7e49580a42110b1d9c6b",
Py0 = "19220ed4e423d3274a8e9a58624b8762d7831d6f65fcaf6718b933bf77a0c41d3bb713a2224dbc448cfc735101a5bb1e",
Py1 = "1d4c7565e4831130ae0ddd95aaad4033e57daab2518b3f0d934c7abc7db8614adbf39beaef5ebb85d34f5add8a6d341",
scalar = "b500f1fa8ffa8d1c0aa7d65054a9aaa0d9ed2fff83b40516def10b03cc80026",
Qx0 = "aaf3fc4e3c92df5b76863d64d19d26213757173c1a8ef4beddbee7c22f5a6a00418527f83c9ab5b17797cd2212d3d73",
Qx1 = "1643ce8e3a0f0997e949c04cb28aa77b7184033dd0056caee2488311e0a3567a0cd9d051ee531d6a83be8ac516c3f075",
Qy0 = "12923eb157d8ee0a940e100408250beded20e4185cbec484431f705ab605c452122c66d6f08e1b06305677dc872a024f",
Qy1 = "460b65406a6e3935dfd75977b12c02971d2673b9c23b032494e6684e08727f35da0407be6708e98cd309ff0c09be00d"
)
test(
id = 2,
EC = ECP_ShortW_Proj[Fp2[BLS12_381], OnTwist],
Px0 = "11c9f03cd130f7b4d6675b902d9b4dddfa41577b7673c31a508760675ca083abedfe3f6c1c69eb46737d4877adb527c6",
Px1 = "c64be8d22d478378784c4f38e386635a8ab606d2b35101ebecfe97b3bb5132d26e9a7ea9690d07a78a22f458045a8c5",
Py0 = "6253c05b48fde95024644efd87cdf0cf15414c36c35625e383ea7b5ab839eaa783563918cd9e5e391ef1512a6ac28e0",
Py1 = "2d214172d2a0326ed45b60945c424ac30f416fa8c6e11f243034a9de26f4aaa69c0d4cc8405227f26c6ee4085ea5bd4",
scalar = "3638a1f09b542c9c14706bddf9bd411747489f3d398a5c286d28f3a950e33406",
Qx0 = "a6e22837968f191d848297f60b511acf4cc375e53161e7869b8d98455375d8ee69367513b3439b6c4ee66f9232badbb",
Qx1 = "10f95cc0c70943519c30c04a8625bb56c1656da634018b1e6e35786b610785f41978983c61e78d4be003074d7fd76660",
Qy0 = "bae732b1dd39cb84c7e0649ed641f9d275bfc45721e3ac35d6cd35faa356235ce0b69fa0f2882d6d2762ed8846368d6",
Qy1 = "bcab3bee59706e0f8c381d2a15a55ce6b2c38d1639b43852af2a62f1366d1cdb8433240cbb237750f2f3e0435b23141"
)
test(
id = 3,
EC = ECP_ShortW_Proj[Fp2[BLS12_381], OnTwist],
Px0 = "5adc112fb04bf4ca642d5a7d7343ccd6b93546442d2fff5b9d32c15e456d54884cba49dd7f94ce4ddaad4018e55d0f2",
Px1 = "5d1c5bbf5d7a833dc76ba206bfa99c281fc37941be050e18f8c6d267b2376b3634d8ad6eb951e52a6d096315abd17d6",
Py0 = "15a959e54981fab9ac3c6f5bfd6fb60a50a916bd43d96a09922a54309b84812736581bfa728670cba864b08b9e391bb9",
Py1 = "f5d6d74f1dd3d9c07451340b8f6990fe93a28fe5e176564eb920bf17eb02df8b6f1e626eda5542ff415f89d51943001",
scalar = "6448f296d9b1a8d81319a0b789df04c587c6165776ccf39f50a354204aabe0da",
Qx0 = "689e9532c287233d12c3d7196361a06479ecb762a8542f6ba4003a5863d5731f3a0cd3fe4e1405aa3c6f9bd87c08b26",
Qx1 = "b74b46e87a05b7413282d63d63a2b1eeb23e8c482fa8097623888d4be2b90c7cdb49fcb93c74f2b98da15aa9382e285",
Qy0 = "44250df4c1e38bfd7fef6c61649afe7ab5d5252b98589d71f1a64efac5ae8ecf0ccafa173acaafc30a0b2a1616b08e4",
Qy1 = "173b324bb2e0596d0ab560c2a447a1867fea87c5712bd06f2503385b0e145509036c119440973396fc263a02838c8433"
)
test(
id = 4,
EC = ECP_ShortW_Proj[Fp2[BLS12_381], OnTwist],
Px0 = "99f8b62e82892d323847f64ab2b422478d207b780fbb097cdca6a1a89e70ff09213dee8534eaf63dac9f7f7feff2548",
Px1 = "12e7b53d802fa9cd897b5470614d57b1620bfe53f36158466f83e7cc6a6cccb1ac7557a8d5a2208c7c1366835c2cba59",
Py0 = "115d6b7a8bc5628690ec750207b3252d4121c20c2106d0277cd41dee7b1d4ed1ff856883719bccb545054b9a745a53e2",
Py1 = "4b55174d273b358a72687d52956af3e94d97db8d2cc508b2a4ec5b0c0b4073b8fcc52eadaea35e3eae9a441b3f86cbc",
scalar = "150caebc321c53c0658c5cecb45e564620b57bfbad0f5d5a277be71a184937b7",
Qx0 = "3f474d7fa1ff31949c7b61f1ff71a7ccd282201ef88ef12fcd1fd3fef6a3ca18d8bb31fa7e9f4abc713f37e02abef3",
Qx1 = "cb94daed6079cbe5a628ae4c27e8ad31a17f14e68050e8ce1b03d5a3c0e6a6cc5a34b3115034349b2ebe8de6a2c441e",
Qy0 = "15b09aac63ac527ad719ee2bcebda6bb646044d9060c4f72280d41186798912b6e29b2f7782744b8cdce927a9c1b9340",
Qy1 = "5204820d6336aade860fb1cb983bcc66e10cfb352ea645b4cdf74e643e9fdf545609bd7181a67daac891f551a6ce566"
)
test(
id = 5,
EC = ECP_ShortW_Proj[Fp2[BLS12_381], OnTwist],
Px0 = "7f56aa6111f341d6381090f058835b3d60200032b382108194188c40afe4225eb6fecaba734084283771923e004b5ca",
Px1 = "18abca4d9eca6d3ef3c720ca6b27f824fdd12dcac72c167f0212f707fa22752f291c9c20a4b92417d05c64207b8e6da6",
Py0 = "10e08fc323d2ef92c5fd9a0ba38e32e16068ac5a4a0f95b0390c2e8ad6caa446adebe16bbf628a0c2de007bfa1218317",
Py1 = "6394379cc76d50b41865b619007de5a7cda3bb7ae6fc696bf2f83e2de562039dab890a9e2b4d61045bac606f224ba42",
scalar = "138ecc47a9d5b6cf2a052731b8f016734614949862a9f2be703935a5e0cd43bd",
Qx0 = "19ec61da69ffdbf79411f041276d4e9fe02df31caa79406135a7522a12965364bb3549b7310393208082465feca7c9eb",
Qx1 = "457ddcc53f5338d34713a91ca4298896bfa7ede3b939cd1c8b320a4c9ba4c1038ac09740e5b428f5c15bdf18ca6af0b",
Qy0 = "9e334e01a81dc6da2004d2f13eec7c15ba0ab8fb8a9628ce24a0de073b251f955ee758f5b0a298c4fadd641610c7fe0",
Qy1 = "935bf9c37ed2ce17b9e5f8014816d8a1a3d8debfc72e39ea9121018f3a990561cfbc64c25621d16a921f458d7f40344"
)
test(
id = 6,
EC = ECP_ShortW_Proj[Fp2[BLS12_381], OnTwist],
Px0 = "a8c5649d2df1bae84fd9e8bfcde5113937b3acea22d67ddfedaf1fb8de8c1ef4c70591cf505c24c31e54020c2c510c3",
Px1 = "a0553f98229a6a067489c3ee204161c11e96f421b3e9c145dc3865b03e9d4ff6cab14c5b5308ecd31173f954463690c",
Py0 = "b29d8dfe18dc41b4826c3a102c1bf8f306cb42433cc36ee38080f47a324c02a678f9daed0a2bc577c18b9865de029f0",
Py1 = "558cdabf11e37c5c5e8abd668bbdd71bb3f07f320948ccaac8a207359fffe38424bfd9b1ef1d24b28b2fbb9f76faff1",
scalar = "5668a2332db27199dcfb7cbdfca6317c2ff128db26d7df68483e0a095ec8e88f",
Qx0 = "490e99b9a27cb49b9446a7bedb8c22ef802cfdc3609cbecd8de4e227fdb72aeafb27c53d74361008ce9ea806e25dc85",
Qx1 = "1624f7ed9ca1fcfda7651608be2acb1d76cb37ab989c1aecf06a6401ee66afdddf283039496c2320dca4d720e8b8a337",
Qy0 = "c6fef37a864fa1602824a2128d4a62e3221413e1ded862f11347576f43c27d69b1957385ad6ba7a9168c05e6fc2ab85",
Qy1 = "11d4a696f8e366929fe0b7eae94702a89aef43725218f95e0bfe7d7abd726e884604838e4d7d670c9579431f9d8012e0"
)
test(
id = 7,
EC = ECP_ShortW_Proj[Fp2[BLS12_381], OnTwist],
Px0 = "eb79d9e425feb105ec09ce60949721b12dac5e6721c8a6b505aa2d83270d2a3e6c6bcce16a1b510f6822504c5e86416",
Px1 = "9f5d2dc403a2e96c3f59c7bb98a36cc8be68500510fd88b09f55938efd192d9653f4bcfd1451518c535e9d1996a924",
Py0 = "114825899129828ee0b946811ff98a79af1b53c4511bc45a8e41a07a7d9600c824ed7c5cd608781d0a98a13e69b0c002",
Py1 = "57600cfce779277faf31d1b18a39c752c179a76b301cbdc317263c7e8770df0d5896c9dec84958bf558c16b5ec5869c",
scalar = "45b4bca2f783bba42b2d41a8d00f4c5f18a78738a5678fc3707523e7c62dafcb",
Qx0 = "71bee2c58caace914434607aff9c1730af2da8bd78d1a44215d9c7cc422623e9b65ba1fc7207de9d0af96f8211eda2b",
Qx1 = "80ea7a46f9352599f67bbbed1253b142422cb79af32b1a544e77ff02c25057396569182341e5492c4a520a756568a41",
Qy0 = "73470439d1202c87e54d9de94121f5e36371fe527eca699f48caba867111707a250b2b0357e79a8263babb907c9f43a",
Qy1 = "cf2f2cca752462c5fc2827b7871d517e50d7ecd819e0b9419f9b1e357874acca15c8f0a1f611aa243b9d21ce1962d3a"
)
test(
id = 8,
EC = ECP_ShortW_Proj[Fp2[BLS12_381], OnTwist],
Px0 = "17ce8a849a475599245ad6b84cf5284cd15d6346ae52a833954ec50d5cf7f0be1cd8473fdc9dfd500d7f1d80bf6fa6ca",
Px1 = "15d8128bc60c8e83846bf6748982a7188df6393a9379b2959fa7e1cb72f1c1da066fe3a6d927f97ecec3725fac65eb10",
Py0 = "a05421595b36750e134b91500962201e9f57ac068732b9fb34ec50ff22f274d395d34d133e131e6dc7bc42d66149767",
Py1 = "178cb2541ce0c60b8860d59762daa6a5b55a0ca034aa18b1fcc6deb23aaa093fb2a6129db0d58de044c4bbb23f1fa298",
scalar = "6083671fcc66dc084ad73eba100830555fcfcc5eccaa6acb27cda0d3fa8d6f64",
Qx0 = "175a067a5cca1e18c2c60c1ed84a2d0fdad20ad8e2b9b67a13d8a173d6c097bd2bb8fa6d83ff5ab3668d56d39f34cbe0",
Qx1 = "14bdf8d62a088a6a71e5ea236bd3caa07934da726fc1f0b7ae462272c0778615734fac562a0293ae5da759dcfc78480c",
Qy0 = "59ead353a2a4189dd9dc0d93ac86d0f3f8fd60bf3db10fd77e4295b1dfaeccb28aa481b1efce7198093a7f2e8ba3617",
Qy1 = "a4661ce0260e251df9d2b01b5af29262e019b8ea12b3a0a2c4c125c09cadd906dcd8a25b7876e76892acfd74dfab8c5"
)
test(
id = 9,
EC = ECP_ShortW_Proj[Fp2[BLS12_381], OnTwist],
Px0 = "13eb9e9906446be49345c08de406cd104d6bb9901ee12af0b80a8351027152d6f6d158d5a906e4a58c5602e97347cfd5",
Px1 = "1218df3f2a9cd7685325a4a7bb6a3636a458a52ea7f1e1d73c2429acb74a2a9beb838c109541120b095118c90868eb0f",
Py0 = "3ac16edac6898f11ff8ddb48fad6f59f4842cd427d72fa964171801be172b8ecd2fdffb4882d4aa6f1e730f6e53f8c5",
Py1 = "b2d251295859b1be1854b1db06eae2ff8e3879d8ba5d9f86f4bb805c65696b48d0771cf10150983e322bdf9eb659af1",
scalar = "644dc62869683f0c93f38eaef2ba6912569dc91ec2806e46b4a3dd6a4421dad1",
Qx0 = "1888f703e7525e4ac29788eb6e3afde14e4c8b36f74a3058ab7e991630cdb8332fb164766db6d14186ac7fa2e593f513",
Qx1 = "1496076ec35db3760a3cfe22a1759b01ac89b7ae21ccb9c4d7553a7881f4a610ef88a56b2e5a027ab49507fd710dd9ea",
Qy0 = "1357e8db4a105bf81a94e9b9130a892b1ec78d564f77b2717451cce777c16a409cd19f450247c75882f1b84678d7c46d",
Qy1 = "14dd91161426cd5d831914706ee9f427512a789f4953f82538f3fb17553840eae31c992de2ed91a6695d291b5ef4c204"
)
# -----
test(
id = 0,
EC = ECP_ShortW_Jac[Fp2[BLS12_381], OnTwist],
Px0 = "10fbddd49246ac4b0faa489e3474507ebc96a5da194b2f7a706fad6bf8435021e1598700088abfe0ae7343296c1b7f52",
Px1 = "324102fa5bd71d9048c3c6a6c62d1f35195d7067bf00dc5eaedd14eecc688383446aba4e8fda059d3f619f00be7890",
Py0 = "f3e974aafa7a3fb3a1209f3af4492c9d9c52f1ae738e1e08309dd0f438f131f8ddd8b934eb8ff2cb078b8c524c11fab",
Py1 = "15e75704edffe7b975cf1d3f27f1dceb89d02e5660650195e0288e5d26c5e9087c241d1bd3263c991d10e2695d1611f1",
scalar = "1f7bef2a74f3bf8ac0225a9edfa514bb5666b15e7be3e929059f2ef75f0035a6",
Qx0 = "1328e4ac12458f9e0c22e925d2fc5593eaf0c126154b484de986895030f830262f54493edcc26aa39c5ab8e714b784b3",
Qx1 = "14d375c704f9187984fe2d64d5517ce7c6eb09981cee68cd48370df21d7f4d0b19431347be3b9eae9d46605cb229e293",
Qy0 = "1580cb4fceea4e71e222945825d4352c97a02d3118ffbd0e006e467b1ff6d4207acd2bb58a197894e9870cbd1bfb369c",
Qy1 = "5df16d4223c040dffcd1912359cfae1f3a99e7f519b2aedaaeb6d77115c63acf309f9effc69d4d9a0d7de420dbf9f1a"
)
test(
id = 1,
EC = ECP_ShortW_Jac[Fp2[BLS12_381], OnTwist],
Px0 = "4b472ab5d0995a22c95f0805eb459b147d2033b8c13c1c07c857a97e66952d4df3e3b5f346997cc7bd19886492fae83",
Px1 = "12228345592511c7327176258097c70dffad1ff53b37163cbd4747d0085ed0bcfe90b9150d2f7e49580a42110b1d9c6b",
Py0 = "19220ed4e423d3274a8e9a58624b8762d7831d6f65fcaf6718b933bf77a0c41d3bb713a2224dbc448cfc735101a5bb1e",
Py1 = "1d4c7565e4831130ae0ddd95aaad4033e57daab2518b3f0d934c7abc7db8614adbf39beaef5ebb85d34f5add8a6d341",
scalar = "b500f1fa8ffa8d1c0aa7d65054a9aaa0d9ed2fff83b40516def10b03cc80026",
Qx0 = "aaf3fc4e3c92df5b76863d64d19d26213757173c1a8ef4beddbee7c22f5a6a00418527f83c9ab5b17797cd2212d3d73",
Qx1 = "1643ce8e3a0f0997e949c04cb28aa77b7184033dd0056caee2488311e0a3567a0cd9d051ee531d6a83be8ac516c3f075",
Qy0 = "12923eb157d8ee0a940e100408250beded20e4185cbec484431f705ab605c452122c66d6f08e1b06305677dc872a024f",
Qy1 = "460b65406a6e3935dfd75977b12c02971d2673b9c23b032494e6684e08727f35da0407be6708e98cd309ff0c09be00d"
)
test(
id = 2,
EC = ECP_ShortW_Jac[Fp2[BLS12_381], OnTwist],
Px0 = "11c9f03cd130f7b4d6675b902d9b4dddfa41577b7673c31a508760675ca083abedfe3f6c1c69eb46737d4877adb527c6",
Px1 = "c64be8d22d478378784c4f38e386635a8ab606d2b35101ebecfe97b3bb5132d26e9a7ea9690d07a78a22f458045a8c5",
Py0 = "6253c05b48fde95024644efd87cdf0cf15414c36c35625e383ea7b5ab839eaa783563918cd9e5e391ef1512a6ac28e0",
Py1 = "2d214172d2a0326ed45b60945c424ac30f416fa8c6e11f243034a9de26f4aaa69c0d4cc8405227f26c6ee4085ea5bd4",
scalar = "3638a1f09b542c9c14706bddf9bd411747489f3d398a5c286d28f3a950e33406",
Qx0 = "a6e22837968f191d848297f60b511acf4cc375e53161e7869b8d98455375d8ee69367513b3439b6c4ee66f9232badbb",
Qx1 = "10f95cc0c70943519c30c04a8625bb56c1656da634018b1e6e35786b610785f41978983c61e78d4be003074d7fd76660",
Qy0 = "bae732b1dd39cb84c7e0649ed641f9d275bfc45721e3ac35d6cd35faa356235ce0b69fa0f2882d6d2762ed8846368d6",
Qy1 = "bcab3bee59706e0f8c381d2a15a55ce6b2c38d1639b43852af2a62f1366d1cdb8433240cbb237750f2f3e0435b23141"
)
test(
id = 3,
EC = ECP_ShortW_Jac[Fp2[BLS12_381], OnTwist],
Px0 = "5adc112fb04bf4ca642d5a7d7343ccd6b93546442d2fff5b9d32c15e456d54884cba49dd7f94ce4ddaad4018e55d0f2",
Px1 = "5d1c5bbf5d7a833dc76ba206bfa99c281fc37941be050e18f8c6d267b2376b3634d8ad6eb951e52a6d096315abd17d6",
Py0 = "15a959e54981fab9ac3c6f5bfd6fb60a50a916bd43d96a09922a54309b84812736581bfa728670cba864b08b9e391bb9",
Py1 = "f5d6d74f1dd3d9c07451340b8f6990fe93a28fe5e176564eb920bf17eb02df8b6f1e626eda5542ff415f89d51943001",
scalar = "6448f296d9b1a8d81319a0b789df04c587c6165776ccf39f50a354204aabe0da",
Qx0 = "689e9532c287233d12c3d7196361a06479ecb762a8542f6ba4003a5863d5731f3a0cd3fe4e1405aa3c6f9bd87c08b26",
Qx1 = "b74b46e87a05b7413282d63d63a2b1eeb23e8c482fa8097623888d4be2b90c7cdb49fcb93c74f2b98da15aa9382e285",
Qy0 = "44250df4c1e38bfd7fef6c61649afe7ab5d5252b98589d71f1a64efac5ae8ecf0ccafa173acaafc30a0b2a1616b08e4",
Qy1 = "173b324bb2e0596d0ab560c2a447a1867fea87c5712bd06f2503385b0e145509036c119440973396fc263a02838c8433"
)
test(
id = 4,
EC = ECP_ShortW_Jac[Fp2[BLS12_381], OnTwist],
Px0 = "99f8b62e82892d323847f64ab2b422478d207b780fbb097cdca6a1a89e70ff09213dee8534eaf63dac9f7f7feff2548",
Px1 = "12e7b53d802fa9cd897b5470614d57b1620bfe53f36158466f83e7cc6a6cccb1ac7557a8d5a2208c7c1366835c2cba59",
Py0 = "115d6b7a8bc5628690ec750207b3252d4121c20c2106d0277cd41dee7b1d4ed1ff856883719bccb545054b9a745a53e2",
Py1 = "4b55174d273b358a72687d52956af3e94d97db8d2cc508b2a4ec5b0c0b4073b8fcc52eadaea35e3eae9a441b3f86cbc",
scalar = "150caebc321c53c0658c5cecb45e564620b57bfbad0f5d5a277be71a184937b7",
Qx0 = "3f474d7fa1ff31949c7b61f1ff71a7ccd282201ef88ef12fcd1fd3fef6a3ca18d8bb31fa7e9f4abc713f37e02abef3",
Qx1 = "cb94daed6079cbe5a628ae4c27e8ad31a17f14e68050e8ce1b03d5a3c0e6a6cc5a34b3115034349b2ebe8de6a2c441e",
Qy0 = "15b09aac63ac527ad719ee2bcebda6bb646044d9060c4f72280d41186798912b6e29b2f7782744b8cdce927a9c1b9340",
Qy1 = "5204820d6336aade860fb1cb983bcc66e10cfb352ea645b4cdf74e643e9fdf545609bd7181a67daac891f551a6ce566"
)
test(
id = 5,
EC = ECP_ShortW_Jac[Fp2[BLS12_381], OnTwist],
Px0 = "7f56aa6111f341d6381090f058835b3d60200032b382108194188c40afe4225eb6fecaba734084283771923e004b5ca",
Px1 = "18abca4d9eca6d3ef3c720ca6b27f824fdd12dcac72c167f0212f707fa22752f291c9c20a4b92417d05c64207b8e6da6",
Py0 = "10e08fc323d2ef92c5fd9a0ba38e32e16068ac5a4a0f95b0390c2e8ad6caa446adebe16bbf628a0c2de007bfa1218317",
Py1 = "6394379cc76d50b41865b619007de5a7cda3bb7ae6fc696bf2f83e2de562039dab890a9e2b4d61045bac606f224ba42",
scalar = "138ecc47a9d5b6cf2a052731b8f016734614949862a9f2be703935a5e0cd43bd",
Qx0 = "19ec61da69ffdbf79411f041276d4e9fe02df31caa79406135a7522a12965364bb3549b7310393208082465feca7c9eb",
Qx1 = "457ddcc53f5338d34713a91ca4298896bfa7ede3b939cd1c8b320a4c9ba4c1038ac09740e5b428f5c15bdf18ca6af0b",
Qy0 = "9e334e01a81dc6da2004d2f13eec7c15ba0ab8fb8a9628ce24a0de073b251f955ee758f5b0a298c4fadd641610c7fe0",
Qy1 = "935bf9c37ed2ce17b9e5f8014816d8a1a3d8debfc72e39ea9121018f3a990561cfbc64c25621d16a921f458d7f40344"
)
test(
id = 6,
EC = ECP_ShortW_Jac[Fp2[BLS12_381], OnTwist],
Px0 = "a8c5649d2df1bae84fd9e8bfcde5113937b3acea22d67ddfedaf1fb8de8c1ef4c70591cf505c24c31e54020c2c510c3",
Px1 = "a0553f98229a6a067489c3ee204161c11e96f421b3e9c145dc3865b03e9d4ff6cab14c5b5308ecd31173f954463690c",
Py0 = "b29d8dfe18dc41b4826c3a102c1bf8f306cb42433cc36ee38080f47a324c02a678f9daed0a2bc577c18b9865de029f0",
Py1 = "558cdabf11e37c5c5e8abd668bbdd71bb3f07f320948ccaac8a207359fffe38424bfd9b1ef1d24b28b2fbb9f76faff1",
scalar = "5668a2332db27199dcfb7cbdfca6317c2ff128db26d7df68483e0a095ec8e88f",
Qx0 = "490e99b9a27cb49b9446a7bedb8c22ef802cfdc3609cbecd8de4e227fdb72aeafb27c53d74361008ce9ea806e25dc85",
Qx1 = "1624f7ed9ca1fcfda7651608be2acb1d76cb37ab989c1aecf06a6401ee66afdddf283039496c2320dca4d720e8b8a337",
Qy0 = "c6fef37a864fa1602824a2128d4a62e3221413e1ded862f11347576f43c27d69b1957385ad6ba7a9168c05e6fc2ab85",
Qy1 = "11d4a696f8e366929fe0b7eae94702a89aef43725218f95e0bfe7d7abd726e884604838e4d7d670c9579431f9d8012e0"
)
test(
id = 7,
EC = ECP_ShortW_Jac[Fp2[BLS12_381], OnTwist],
Px0 = "eb79d9e425feb105ec09ce60949721b12dac5e6721c8a6b505aa2d83270d2a3e6c6bcce16a1b510f6822504c5e86416",
Px1 = "9f5d2dc403a2e96c3f59c7bb98a36cc8be68500510fd88b09f55938efd192d9653f4bcfd1451518c535e9d1996a924",
Py0 = "114825899129828ee0b946811ff98a79af1b53c4511bc45a8e41a07a7d9600c824ed7c5cd608781d0a98a13e69b0c002",
Py1 = "57600cfce779277faf31d1b18a39c752c179a76b301cbdc317263c7e8770df0d5896c9dec84958bf558c16b5ec5869c",
scalar = "45b4bca2f783bba42b2d41a8d00f4c5f18a78738a5678fc3707523e7c62dafcb",
Qx0 = "71bee2c58caace914434607aff9c1730af2da8bd78d1a44215d9c7cc422623e9b65ba1fc7207de9d0af96f8211eda2b",
Qx1 = "80ea7a46f9352599f67bbbed1253b142422cb79af32b1a544e77ff02c25057396569182341e5492c4a520a756568a41",
Qy0 = "73470439d1202c87e54d9de94121f5e36371fe527eca699f48caba867111707a250b2b0357e79a8263babb907c9f43a",
Qy1 = "cf2f2cca752462c5fc2827b7871d517e50d7ecd819e0b9419f9b1e357874acca15c8f0a1f611aa243b9d21ce1962d3a"
)
test(
id = 8,
EC = ECP_ShortW_Jac[Fp2[BLS12_381], OnTwist],
Px0 = "17ce8a849a475599245ad6b84cf5284cd15d6346ae52a833954ec50d5cf7f0be1cd8473fdc9dfd500d7f1d80bf6fa6ca",
Px1 = "15d8128bc60c8e83846bf6748982a7188df6393a9379b2959fa7e1cb72f1c1da066fe3a6d927f97ecec3725fac65eb10",
Py0 = "a05421595b36750e134b91500962201e9f57ac068732b9fb34ec50ff22f274d395d34d133e131e6dc7bc42d66149767",
Py1 = "178cb2541ce0c60b8860d59762daa6a5b55a0ca034aa18b1fcc6deb23aaa093fb2a6129db0d58de044c4bbb23f1fa298",
scalar = "6083671fcc66dc084ad73eba100830555fcfcc5eccaa6acb27cda0d3fa8d6f64",
Qx0 = "175a067a5cca1e18c2c60c1ed84a2d0fdad20ad8e2b9b67a13d8a173d6c097bd2bb8fa6d83ff5ab3668d56d39f34cbe0",
Qx1 = "14bdf8d62a088a6a71e5ea236bd3caa07934da726fc1f0b7ae462272c0778615734fac562a0293ae5da759dcfc78480c",
Qy0 = "59ead353a2a4189dd9dc0d93ac86d0f3f8fd60bf3db10fd77e4295b1dfaeccb28aa481b1efce7198093a7f2e8ba3617",
Qy1 = "a4661ce0260e251df9d2b01b5af29262e019b8ea12b3a0a2c4c125c09cadd906dcd8a25b7876e76892acfd74dfab8c5"
)
test(
id = 9,
EC = ECP_ShortW_Jac[Fp2[BLS12_381], OnTwist],
Px0 = "13eb9e9906446be49345c08de406cd104d6bb9901ee12af0b80a8351027152d6f6d158d5a906e4a58c5602e97347cfd5",
Px1 = "1218df3f2a9cd7685325a4a7bb6a3636a458a52ea7f1e1d73c2429acb74a2a9beb838c109541120b095118c90868eb0f",
Py0 = "3ac16edac6898f11ff8ddb48fad6f59f4842cd427d72fa964171801be172b8ecd2fdffb4882d4aa6f1e730f6e53f8c5",
Py1 = "b2d251295859b1be1854b1db06eae2ff8e3879d8ba5d9f86f4bb805c65696b48d0771cf10150983e322bdf9eb659af1",
scalar = "644dc62869683f0c93f38eaef2ba6912569dc91ec2806e46b4a3dd6a4421dad1",
Qx0 = "1888f703e7525e4ac29788eb6e3afde14e4c8b36f74a3058ab7e991630cdb8332fb164766db6d14186ac7fa2e593f513",
Qx1 = "1496076ec35db3760a3cfe22a1759b01ac89b7ae21ccb9c4d7553a7881f4a610ef88a56b2e5a027ab49507fd710dd9ea",
Qy0 = "1357e8db4a105bf81a94e9b9130a892b1ec78d564f77b2717451cce777c16a409cd19f450247c75882f1b84678d7c46d",
Qy1 = "14dd91161426cd5d831914706ee9f427512a789f4953f82538f3fb17553840eae31c992de2ed91a6695d291b5ef4c204"
)
run_scalar_mul_test_vs_sage(
ECP_ShortW_Jac[Fp2[BLS12_381], OnTwist],
"t_ec_sage_bls12_381_g2_jacobian"
)

579
tests/t_ec_sage_bn254.nim
View File

@ -1,579 +0,0 @@
# Constantine
# Copyright (c) 2018-2019 Status Research & Development GmbH
# Copyright (c) 2020-Present Mamy André-Ratsimbazafy
# Licensed and distributed under either of
# * MIT license (license terms in the root directory or at http://opensource.org/licenses/MIT).
# * Apache v2 license (license terms in the root directory or at http://www.apache.org/licenses/LICENSE-2.0).
# at your option. This file may not be copied, modified, or distributed except according to those terms.
import
# Standard library
std/[unittest, times],
# Internals
../constantine/config/[common, curves],
../constantine/arithmetic,
../constantine/towers,
../constantine/io/[io_bigints, io_ec],
../constantine/elliptic/[
ec_shortweierstrass_projective,
ec_shortweierstrass_jacobian,
ec_scalar_mul,
ec_endomorphism_accel],
# Test utilities
./support/ec_reference_scalar_mult
echo "\n------------------------------------------------------\n"
proc test(
id: int,
EC: typedesc[ECP_ShortW_Proj or ECP_ShortW_Jac],
Px, Py: string,
scalar: string,
Qx, Qy: string
) =
test "test " & $id & " - " & $EC:
var P: EC
let pOK = P.fromHex(Px, Py)
doAssert pOK
var Q: EC
let qOK = Q.fromHex(Qx, Qy)
let exponent = BigInt[EC.F.C.getCurveOrderBitwidth()].fromHex(scalar)
var
impl = P
reference = P
endo = P
endoW = P
impl.scalarMulGeneric(exponent)
reference.unsafe_ECmul_double_add(exponent)
endo.scalarMulEndo(exponent)
endoW.scalarMulGLV_m2w2(exponent)
doAssert: bool(Q == reference)
doAssert: bool(Q == impl)
doAssert: bool(Q == endo)
doAssert: bool(Q == endoW)
suite "Scalar Multiplication G1: BN254 implementation vs SageMath" & " [" & $WordBitwidth & "-bit mode]":
# Generated via sage sage/testgen_bn254_snarks.sage
test(
id = 0,
EC = ECP_ShortW_Proj[Fp[BN254_Snarks], NotOnTwist],
Px = "22d3af0f3ee310df7fc1a2a204369ac13eb4a48d969a27fcd2861506b2dc0cd7",
Py = "1c994169687886ccd28dd587c29c307fb3cab55d796d73a5be0bbf9aab69912e",
scalar = "e08a292f940cfb361cc82bc24ca564f51453708c9745a9cf8707b11c84bc448",
Qx = "267c05cd49d681c5857124876748365313b9c285e783206f48513ce06d3df931",
Qy = "2fa00719ce37465dbe7037f723ed5df08c76b9a27a4dd80d86c0ee5157349b96"
)
test(
id = 1,
EC = ECP_ShortW_Proj[Fp[BN254_Snarks], NotOnTwist],
Px = "2724750abe620fce759b6f18729e40f891a514160d477811a44b222372cc4ea3",
Py = "105cdcbe363921790a56bf2696e73642447c60b814827ca4dba86c814912c98a",
scalar = "2f5c2960850eabadab1e5595ff0bf841206885653e7f2024248b281a86744790",
Qx = "57d2dcbc665fb93fd5119bb982c29700d025423d60a42b5fe17210fd5a868fd",
Qy = "2abad564ff78fbc266dfb77bdd110b22271136b33ce5049fb3ca05107787abc"
)
test(
id = 2,
EC = ECP_ShortW_Proj[Fp[BN254_Snarks], NotOnTwist],
Px = "39bc19c41835082f86ca046b71875b051575072e4d6a4aeedac31eee34b07df",
Py = "1fdbf42fc20421e1e775fd93ed1888d614f7e39067e7443f21b6a4817481c346",
scalar = "29e140c33f706c0111443699b0b8396d8ead339a3d6f3c212b08749cf2a16f6b",
Qx = "83895d1c7a2b15a5dfe9371983196591415182978e8ff0e83262e32d768c712",
Qy = "2ed8b88e1cd08814ce1d1929d0e4bba6fb5897f915b3525cf12349256da95499"
)
test(
id = 3,
EC = ECP_ShortW_Proj[Fp[BN254_Snarks], NotOnTwist],
Px = "157a3e1ff9dabccced9746e19855a9438098be6d734f07d1c069aa1bd05b8d87",
Py = "1c96bf3e48bc1a6635d93d4f1302a0eba39bd907c5d861f2a9d0c714ee60f04d",
scalar = "29b05bd55963e262e0fa458c76297fb5be3ec1421fdb1354789f68fdce81dc2c",
Qx = "196aeca74447934eeaba0f2263177fcb7eb239985814f8ef2d7bf08677108c9",
Qy = "1f5aa4c7df4a9855113c63d8fd55c512c7e919b8ae0352e280bdb1009299c3b2"
)
test(
id = 4,
EC = ECP_ShortW_Proj[Fp[BN254_Snarks], NotOnTwist],
Px = "2f260967d4cd5d15f98c0a0a9d5abaae0c70d3b8d83e1e884586cd6ece395fe7",
Py = "2a102c7aebdfaa999d5a99984148ada142f72f5d4158c10368a2e13dded886f6",
scalar = "1796de74c1edac90d102e7c33f3fad94304eaff4a67a018cae678774d377f6cd",
Qx = "28c73e276807863ecf4ae60b1353790f10f176ca8c55b3db774e33c569ef39d5",
Qy = "c386e24828cead255ec7657698559b23a26fc9bd5db70a1fe20b48ecfbd6db9"
)
test(
id = 5,
EC = ECP_ShortW_Proj[Fp[BN254_Snarks], NotOnTwist],
Px = "1b4ccef57f4411360a02b8228e4251896c9492ff93a69ba3720da0cd46a04e83",
Py = "1fabcb215bd7c06ead2e6b0167497efc2cdd3dbacf69bcb0244142fd63c1e405",
scalar = "116741cd19dac61c5e77877fc6fef40f363b164b501dfbdbc09e17ea51d6beb0",
Qx = "192ca2e120b0f5296baf7cc47bfebbbc74748c8847bbdbe485bcb796de2622aa",
Qy = "8bc6b1aa4532c727be8fd21a8176d55bc721c727af327f601f7a8dff655b0b9"
)
test(
id = 6,
EC = ECP_ShortW_Proj[Fp[BN254_Snarks], NotOnTwist],
Px = "2807c88d6759280d6bd83a54d349a533d1a66dc32f72cab8114ab707f10e829b",
Py = "dbf0d486aeed3d303880f324faa2605aa0219e35661bc88150470c7df1c0b61",
scalar = "2a5976268563870739ced3e6efd8cf53887e8e4426803377095708509dd156ca",
Qx = "2841f67de361436f64e582a134fe36ab7196334c758a07e732e1cf1ccb35a476",
Qy = "21fb9b8311e53832044be5ff024f737aee474bc504c7c158fe760cc999da8612"
)
test(
id = 7,
EC = ECP_ShortW_Proj[Fp[BN254_Snarks], NotOnTwist],
Px = "2754a174a33a55f2a31573767e9bf5381b47dca1cbebc8b68dd4df58b3f1cc2",
Py = "f222f59c8893ad87c581dacb3f8b6e7c20e7a13bc5fb6e24262a3436d663b1",
scalar = "25d596bf6caf4565fbfd22d81f9cef40c8f89b1e5939f20caa1b28056e0e4f58",
Qx = "2b48dd3ace8e403c2905f00cdf13814f0dbecb0c0465e6455fe390cc9730f5a",
Qy = "fe65f0cd4ae0d2e459daa4163f32deed1250b5c384eb5aeb933162a41793d25"
)
test(
id = 8,
EC = ECP_ShortW_Proj[Fp[BN254_Snarks], NotOnTwist],
Px = "273bf6c679d8e880034590d16c007bbabc6c65ed870a263b5d1ce7375c18fd7",
Py = "2904086cb9e33657999229b082558a74c19b2b619a0499afb2e21d804d8598ee",
scalar = "67a499a389129f3902ba6140660c431a56811b53de01d043e924711bd341e53",
Qx = "1d827e4569f17f068457ffc52f1c6ed7e2ec89b8b520efae48eff41827f79128",
Qy = "be8c488bb9587bcb0faba916277974afe12511e54fbd749e27d3d7efd998713"
)
test(
id = 9,
EC = ECP_ShortW_Proj[Fp[BN254_Snarks], NotOnTwist],
Px = "ec892c09a5f1c68c1bfec7780a1ebd279739383f2698eeefbba745b3e717fd5",
Py = "23d273a1b9750fe1d4ebd4b7c25f4a8d7d94f6662c436305cca8ff2cdbd3f736",
scalar = "d2f09ceaa2638b7ac3d7d4aa9eff7a12e93dc85db0f9676e5f19fb86d6273e9",
Qx = "305d7692b141962a4a92038adfacc0d2691e5589ed097a1c661cc48c84e2b64e",
Qy = "bafa230a0f5cc2fa3cf07fa46312cb724fc944b097890fa60f2cf42a1be7963"
)
# --------------------------------------------------------------------------
test(
id = 0,
EC = ECP_ShortW_Jac[Fp[BN254_Snarks], NotOnTwist],
Px = "22d3af0f3ee310df7fc1a2a204369ac13eb4a48d969a27fcd2861506b2dc0cd7",
Py = "1c994169687886ccd28dd587c29c307fb3cab55d796d73a5be0bbf9aab69912e",
scalar = "e08a292f940cfb361cc82bc24ca564f51453708c9745a9cf8707b11c84bc448",
Qx = "267c05cd49d681c5857124876748365313b9c285e783206f48513ce06d3df931",
Qy = "2fa00719ce37465dbe7037f723ed5df08c76b9a27a4dd80d86c0ee5157349b96"
)
test(
id = 1,
EC = ECP_ShortW_Jac[Fp[BN254_Snarks], NotOnTwist],
Px = "2724750abe620fce759b6f18729e40f891a514160d477811a44b222372cc4ea3",
Py = "105cdcbe363921790a56bf2696e73642447c60b814827ca4dba86c814912c98a",
scalar = "2f5c2960850eabadab1e5595ff0bf841206885653e7f2024248b281a86744790",
Qx = "57d2dcbc665fb93fd5119bb982c29700d025423d60a42b5fe17210fd5a868fd",
Qy = "2abad564ff78fbc266dfb77bdd110b22271136b33ce5049fb3ca05107787abc"
)
test(
id = 2,
EC = ECP_ShortW_Jac[Fp[BN254_Snarks], NotOnTwist],
Px = "39bc19c41835082f86ca046b71875b051575072e4d6a4aeedac31eee34b07df",
Py = "1fdbf42fc20421e1e775fd93ed1888d614f7e39067e7443f21b6a4817481c346",
scalar = "29e140c33f706c0111443699b0b8396d8ead339a3d6f3c212b08749cf2a16f6b",
Qx = "83895d1c7a2b15a5dfe9371983196591415182978e8ff0e83262e32d768c712",
Qy = "2ed8b88e1cd08814ce1d1929d0e4bba6fb5897f915b3525cf12349256da95499"
)
test(
id = 3,
EC = ECP_ShortW_Jac[Fp[BN254_Snarks], NotOnTwist],
Px = "157a3e1ff9dabccced9746e19855a9438098be6d734f07d1c069aa1bd05b8d87",
Py = "1c96bf3e48bc1a6635d93d4f1302a0eba39bd907c5d861f2a9d0c714ee60f04d",
scalar = "29b05bd55963e262e0fa458c76297fb5be3ec1421fdb1354789f68fdce81dc2c",
Qx = "196aeca74447934eeaba0f2263177fcb7eb239985814f8ef2d7bf08677108c9",
Qy = "1f5aa4c7df4a9855113c63d8fd55c512c7e919b8ae0352e280bdb1009299c3b2"
)
test(
id = 4,
EC = ECP_ShortW_Jac[Fp[BN254_Snarks], NotOnTwist],
Px = "2f260967d4cd5d15f98c0a0a9d5abaae0c70d3b8d83e1e884586cd6ece395fe7",
Py = "2a102c7aebdfaa999d5a99984148ada142f72f5d4158c10368a2e13dded886f6",
scalar = "1796de74c1edac90d102e7c33f3fad94304eaff4a67a018cae678774d377f6cd",
Qx = "28c73e276807863ecf4ae60b1353790f10f176ca8c55b3db774e33c569ef39d5",
Qy = "c386e24828cead255ec7657698559b23a26fc9bd5db70a1fe20b48ecfbd6db9"
)
test(
id = 5,
EC = ECP_ShortW_Jac[Fp[BN254_Snarks], NotOnTwist],
Px = "1b4ccef57f4411360a02b8228e4251896c9492ff93a69ba3720da0cd46a04e83",
Py = "1fabcb215bd7c06ead2e6b0167497efc2cdd3dbacf69bcb0244142fd63c1e405",
scalar = "116741cd19dac61c5e77877fc6fef40f363b164b501dfbdbc09e17ea51d6beb0",
Qx = "192ca2e120b0f5296baf7cc47bfebbbc74748c8847bbdbe485bcb796de2622aa",
Qy = "8bc6b1aa4532c727be8fd21a8176d55bc721c727af327f601f7a8dff655b0b9"
)
test(
id = 6,
EC = ECP_ShortW_Jac[Fp[BN254_Snarks], NotOnTwist],
Px = "2807c88d6759280d6bd83a54d349a533d1a66dc32f72cab8114ab707f10e829b",
Py = "dbf0d486aeed3d303880f324faa2605aa0219e35661bc88150470c7df1c0b61",
scalar = "2a5976268563870739ced3e6efd8cf53887e8e4426803377095708509dd156ca",
Qx = "2841f67de361436f64e582a134fe36ab7196334c758a07e732e1cf1ccb35a476",
Qy = "21fb9b8311e53832044be5ff024f737aee474bc504c7c158fe760cc999da8612"
)
test(
id = 7,
EC = ECP_ShortW_Jac[Fp[BN254_Snarks], NotOnTwist],
Px = "2754a174a33a55f2a31573767e9bf5381b47dca1cbebc8b68dd4df58b3f1cc2",
Py = "f222f59c8893ad87c581dacb3f8b6e7c20e7a13bc5fb6e24262a3436d663b1",
scalar = "25d596bf6caf4565fbfd22d81f9cef40c8f89b1e5939f20caa1b28056e0e4f58",
Qx = "2b48dd3ace8e403c2905f00cdf13814f0dbecb0c0465e6455fe390cc9730f5a",
Qy = "fe65f0cd4ae0d2e459daa4163f32deed1250b5c384eb5aeb933162a41793d25"
)
test(
id = 8,
EC = ECP_ShortW_Jac[Fp[BN254_Snarks], NotOnTwist],
Px = "273bf6c679d8e880034590d16c007bbabc6c65ed870a263b5d1ce7375c18fd7",
Py = "2904086cb9e33657999229b082558a74c19b2b619a0499afb2e21d804d8598ee",
scalar = "67a499a389129f3902ba6140660c431a56811b53de01d043e924711bd341e53",
Qx = "1d827e4569f17f068457ffc52f1c6ed7e2ec89b8b520efae48eff41827f79128",
Qy = "be8c488bb9587bcb0faba916277974afe12511e54fbd749e27d3d7efd998713"
)
test(
id = 9,
EC = ECP_ShortW_Jac[Fp[BN254_Snarks], NotOnTwist],
Px = "ec892c09a5f1c68c1bfec7780a1ebd279739383f2698eeefbba745b3e717fd5",
Py = "23d273a1b9750fe1d4ebd4b7c25f4a8d7d94f6662c436305cca8ff2cdbd3f736",
scalar = "d2f09ceaa2638b7ac3d7d4aa9eff7a12e93dc85db0f9676e5f19fb86d6273e9",
Qx = "305d7692b141962a4a92038adfacc0d2691e5589ed097a1c661cc48c84e2b64e",
Qy = "bafa230a0f5cc2fa3cf07fa46312cb724fc944b097890fa60f2cf42a1be7963"
)
proc test(
id: int,
EC: typedesc[ECP_ShortW_Proj or ECP_ShortW_Jac],
Px0, Px1, Py0, Py1: string,
scalar: string,
Qx0, Qx1, Qy0, Qy1: string
) =
test "test " & $id & " - " & $EC:
var P: EC
let pOK = P.fromHex(Px0, Px1, Py0, Py1)
doAssert pOK
var Q: EC
let qOK = Q.fromHex(Qx0, Qx1, Qy0, Qy1)
let exponent = BigInt[EC.F.C.getCurveOrderBitwidth()].fromHex(scalar)
var
impl = P
reference = P
endo = P
impl.scalarMulGeneric(exponent)
reference.unsafe_ECmul_double_add(exponent)
endo.scalarMulEndo(exponent)
doAssert: bool(Q == reference)
doAssert: bool(Q == impl)
doAssert: bool(Q == endo)
suite "Scalar Multiplication G2: BN254 implementation vs SageMath" & " [" & $WordBitwidth & "-bit mode]":
# Generated via sage sage/testgen_bn254_snarks.sage
test(
id = 0,
EC = ECP_ShortW_Proj[Fp2[BN254_Snarks], OnTwist],
Px0 = "1dcee2242ae85da43d02d38032b85836660f9a0a8777ab66c84ffbbde3ac3b25",
Px1 = "1e2eb4c305e3b6c36a4081888b7a953eb44804b8b5120306331f8c89a3bb950",
Py0 = "1db75f495edd522cae161ceeb86ca466ca2efd80ef979028d7aa39679de675fd",
Py1 = "b1b6edeb6a7689595098a58a916657dcc09f53f5fc1a1a64b34a2b80447692e",
scalar = "3075e23caee5579e5c96f1ca7b206862c2cf3ce21d79182d58b140074b7bd34",
Qx0 = "8d63bb4368f94f1629f33ef0c970b3a6fcec6979e423f54ce657f0493c08fde",
Qx1 = "29d1af77bb890dcb27e685198f23dffd5ae5733bd6dd55757dcb44ce8c396742",
Qy0 = "13c24efab7e517c2337ba9cbb8cfb2166666a44550fc4d314f4c81a4812fb8a",
Qy1 = "c3ad1f7a175fa21e1d18595f8fc793688e1a33feda902805a52569ea7a787bb"
)
test(
id = 1,
EC = ECP_ShortW_Proj[Fp2[BN254_Snarks], OnTwist],
Px0 = "5ed8c937273562944e0f1ebfb40e6511202188c1cabf588ed38735016d37b32",
Px1 = "23f75e8322c4b540cd5b8fd144a89ab5206a040f498b7b59385770bc841cf012",
Py0 = "2150beef17f5c22a65a4129390f47eece8f0c7d0c516790ea2632e7fd594ed8",
Py1 = "78281fb396e2b923453fae943af96b57e8b283fc40b0be4be3a1750d0daf121",
scalar = "1eac341ad699cba0cb13ae35b8215bfe0f34e931f8e51e33bf90d9849767bb",
Qx0 = "1bb6e8c1be4d9da9ef294ab066c82bb6dd805efa0c73f289e25f5cc6fc4f12e4",
Qx1 = "8ca44ff91e6484ecadc2a866ec64031e71c4a9d7a902f4280bef3db4dbf1bc9",
Qy0 = "39d151a22c49d4c71c8258e9664ead46ddccd49c596056509d9f9e6055def62",
Qy1 = "b37a08b453b96880f435d9fb2555960571a76e72a9d0d0ec622b055b7c97cdb"
)
test(
id = 2,
EC = ECP_ShortW_Proj[Fp2[BN254_Snarks], OnTwist],
Px0 = "2ac4b3d0a70d6686c63d9d63cce1370eafd4dc67f616c13582c6c64f6670513e",
Px1 = "f1daeb6a2581ba6c8027a645ab5c10e303db4aee85c70c8fe11f4c1adcc7029",
Py0 = "25807ff21967759cab64844741d006e2aa0221d9836613b1239da1a167d15131",
Py1 = "148dae725e508dbb43616972e2945f4868b2a193d828ed2efcf9c8a37b6b83f5",
scalar = "b29535765163b5d5f2d01c8fcde010d11f3f0f250a2cc84b8bc13dd9083356c",
Qx0 = "5f4b3a8a5fe74bf3f2ddc0bc2024f18c7958b2846dab1c602b8256a6035361a",
Qx1 = "ba3dad609b1ba8c78cbbfb7fae2d2f9398ef02265e3b4c0f3c8c18d8d0e59d6",
Qy0 = "2c226aee4621895d63df069c4b6951e2201ee1508d5d54e6ee860533b73b534a",
Qy1 = "2aa5384592339bff0a6e4664c931c0ec9f5a3d2fb2fff87a52245c0d95d3d130"
)
test(
id = 3,
EC = ECP_ShortW_Proj[Fp2[BN254_Snarks], OnTwist],
Px0 = "2a028c1328bb0abf252edfbf7133b84eef2a5f20163fe61685b4b54229ca585d",
Px1 = "8f80ad79e8e7e79bbdc645d9f5b339c52dd99a901b90de2494492656f11a9d5",
Py0 = "1f04320578e31ffa2e2b59ad8fcb1aba622b5f307ac540cf2ccdab07dec56503",
Py1 = "2973900c0fdf651b64f5b1a990baec7c582e0743d501bdb991374776d6c73b28",
scalar = "2c02275a71bb41c911faf48cab4f7ac7fc6672a5c15586185c8cff3203181da0",
Qx0 = "2f39a0772a0bd75db3e6ec0745b18824118e534fdefec811d0a4ca2ca3ce7606",
Qx1 = "23e1601a4338502dbc619a8bde251114272ca87662a851a56035c81d46877d81",
Qy0 = "1f0ee85e7c590124f89319f2450f7b8421d9f6d6414fd3b5cc18d781b69b30c9",
Qy1 = "29e4ff75afecaf732419409a5e0d8e94df6acec29fb5b298c7aad5ceef63e5f9"
)
test(
id = 4,
EC = ECP_ShortW_Proj[Fp2[BN254_Snarks], OnTwist],
Px0 = "1132e63c444e1abce6fc4c39bdf5be5caad586837cbf5ca9d3891482bdefe77",
Px1 = "22b71f598dab789f055fc9669ddf66f0d75f581af0e9e8863d7f95a51ef34862",
Py0 = "58e39050f64c9948d7238b99ecaee947cb934688a6e9f483c5c36b6e07aa31b",
Py1 = "2e64b920f498e12992f2a4ae3f9ced43f3f64705c9008169f3b930a760d055fb",
scalar = "24c5b2ce21615dca82231f5fb0fc8d05aa07c6df4bb5aa7c2381ac7b61a6290c",
Qx0 = "25a0637163a40813529e8a22a3e8be6db96f6dc1cdb8e1844729cad6be11668e",
Qx1 = "16de42461c4db7f9f72eb28bb16da850a42fc153dec64baf021f67f5564f36d8",
Qy0 = "27f2d743d3ce0c1f92c51110a6b9ca93a95693161f1d1cd85a0cf5a2492b4901",
Qy1 = "2c5a8df4fe93e31e374595c0605b1a8b93bb429232cf40f45c847739790c191e"
)
test(
id = 5,
EC = ECP_ShortW_Proj[Fp2[BN254_Snarks], OnTwist],
Px0 = "6a20c456e80e2bfe37d8631d41ffed31cba5d1c411e816d64014d0088d16554",
Px1 = "9d1555c77222abd79e17e2806386c53aba9609375276e817f52f03dc3f75676",
Py0 = "127e76f384726e56dfaa46e6fde6dc644f5fd494d056191059e2bebc525ce835",
Py1 = "2d80f2570da62adc61d794ac17c251f9f9f3de2b45f39c8ede5a9e215e60363",
scalar = "263e44e282fe22216cc054a39e40d4e38e71780bdc84563c340bdaaf4562534b",
Qx0 = "6d7e15b98352e445d1bdacb48826e5a7e8cf854fb9bc18839b30a164a2a9c53",
Qx1 = "12aa3294f6a9bb17f91d5875a5a1aa559c29b06134c6d165d83c8e9e02947568",
Qy0 = "271b8b552e52bdd310c46e07327a18861c91a739da178be909f0a4fe53ae0d05",
Qy1 = "1f4f200de96541e826f0bd536b1401e05e2a7c5a96c567b6dff21a21119bbf7"
)
test(
id = 6,
EC = ECP_ShortW_Proj[Fp2[BN254_Snarks], OnTwist],
Px0 = "4c591d080257375d6189340185c6fe4c1fa796722d07e1bec0b17080b6b1154",
Px1 = "241e2f2eb2a58afc2b5410d4ccbf75b53744ca1ac0bb28d7c60449f8d06204a4",
Py0 = "eaddea52f2e884a5e2635965ca4146b12127fe8a8883e07def8e8720f410781",
Py1 = "cc60dec6ad90e38c851924cf39ddd11b70eeb3fac95731eafd984e9aba2cae",
scalar = "1471f378000d557f0c33c1f8c8313edd53422479440cbd0fdc4cc55f1163deec",
Qx0 = "2a86b4867d7f63afdc09048210a3ef6d363c7896ccc1bb248f3aad4174e1f8fa",
Qx1 = "84c200018461c84ef9ce6de2c691b95cc2c41edc87107331f107ac49de76656",
Qy0 = "2ea1b6d71adb183d9a8dd319a21a679cb0b4e06bc96583d3a786f82b88b5e3ba",
Qy1 = "834e2ff738dcb5e8db7e4dae9336dede51524313b476019ea29ebadbb4ba12d"
)
test(
id = 7,
EC = ECP_ShortW_Proj[Fp2[BN254_Snarks], OnTwist],
Px0 = "115e772198e3f0003e31c0a1c48b6ac870d96020a4d634b7f14c15422d001cfe",
Px1 = "1913447ff41836e0b6a3b01be670a2457e6119e02ae35903fb71be4369e269f7",
Py0 = "14cb779c640aad2731b93b07c623c621a5585d0374f0394e5332f20ac28ca49d",
Py1 = "13a4c4b67f5976431158be8137b5017127fdbc312dc49825dae218a9a7878817",
scalar = "411315458c0c0cb27a058faf15dcbf2e467128f9c958d92de37902e1912f81",
Qx0 = "243a8808a891428d01ef28a77d0766488a98272a5dd394b2992599ff522f264",
Qx1 = "1baebf873402812e4345a8b1783fd25272c64d6194bd9f50b32b8e67ee737dc7",
Qy0 = "1f1001ba8b8b27159568f72e80662e352adfc00c9341d8b4fb8ef6f75ff628d2",
Qy1 = "169af215aa2456c6a65e13ac4df1ba1c982ca791058612679ef26dcb8fb0a989"
)
test(
id = 8,
EC = ECP_ShortW_Proj[Fp2[BN254_Snarks], OnTwist],
Px0 = "13faa1f28e6bfe89765c284e67face6ce0a29006ebc1551d4243e754c59f88ad",
Px1 = "640cebb80938dfcb998d84a8e5aafd47ffbcba0aa2f8c9b1c585baf119a8942",
Py0 = "1de793a9ef8f4dea5dad12fb09ddefa07ce197d4d7389a29ad3d8c6484582afe",
Py1 = "fc6e1f8bf75d1b7e48fdb6b2869c2de553663742c151b390cede6712da5a084",
scalar = "111e6f761ce48e22970752bd56cab93d47caa252d52948367b21163591f7b7b1",
Qx0 = "1c25702bf3b6f5fb453c6473b6dc2d67cd3cc21c65b589df1bfde254d50cffdd",
Qx1 = "14d03eb2075d6b5995240cc54a01ebe50b43933863f0316760226fbfa3a0280",
Qy0 = "1a075c9f2f6afa6e07e54b143a33c17e81d0ac8b91e0c8e0fdd5082bd6b9a72d",
Qy1 = "8e5ef57bb0f95fb6538dfaeac677977e3f3d6f49142c09584d8ec5c2ccd9b2d"
)
test(
id = 9,
EC = ECP_ShortW_Proj[Fp2[BN254_Snarks], OnTwist],
Px0 = "2fc3da947b78ac524a57670cef36ca89f1dad71b337bc3c18305c582a59648ad",
Px1 = "2f7cc845d8c1ef0613f919d8c47f3c62f83608a45b1e186748ac5dcccd4c6baf",
Py0 = "18ddc4718a4161f72f8d188fc61a609a3d592e186a65f4158483b719ffb05b8f",
Py1 = "45b9c3629ed709784db63ff090e2e891c5b5c6b2222cb6afc56638d7389d689",
scalar = "6223903d4bc2adea7b0a0db92822b6c2638691e4388df93f567e11edd6f23",
Qx0 = "9ec612ab0cf4a48e1c15d22284bce8e34619bfb9afb688a9a7930afcc1bd0f3",
Qx1 = "d796e5f5ae1a15622d2284ada34166b9e7c717bd2ff9b2cf2c6e48c33db5ff2",
Qy0 = "2a8ecb09a01cd2f89b316e7569331e9da3bfbd8a40114913b3e5477442c0e4ef",
Qy1 = "282b14bc00df2dd1733e3187a9845ef3a123c17ce4f6154e5cad26c3b48d1b98"
)
# --------------------------------------------------------------------------
test(
id = 0,
EC = ECP_ShortW_Jac[Fp2[BN254_Snarks], OnTwist],
Px0 = "1dcee2242ae85da43d02d38032b85836660f9a0a8777ab66c84ffbbde3ac3b25",
Px1 = "1e2eb4c305e3b6c36a4081888b7a953eb44804b8b5120306331f8c89a3bb950",
Py0 = "1db75f495edd522cae161ceeb86ca466ca2efd80ef979028d7aa39679de675fd",
Py1 = "b1b6edeb6a7689595098a58a916657dcc09f53f5fc1a1a64b34a2b80447692e",
scalar = "3075e23caee5579e5c96f1ca7b206862c2cf3ce21d79182d58b140074b7bd34",
Qx0 = "8d63bb4368f94f1629f33ef0c970b3a6fcec6979e423f54ce657f0493c08fde",
Qx1 = "29d1af77bb890dcb27e685198f23dffd5ae5733bd6dd55757dcb44ce8c396742",
Qy0 = "13c24efab7e517c2337ba9cbb8cfb2166666a44550fc4d314f4c81a4812fb8a",
Qy1 = "c3ad1f7a175fa21e1d18595f8fc793688e1a33feda902805a52569ea7a787bb"
)
test(
id = 1,
EC = ECP_ShortW_Jac[Fp2[BN254_Snarks], OnTwist],
Px0 = "5ed8c937273562944e0f1ebfb40e6511202188c1cabf588ed38735016d37b32",
Px1 = "23f75e8322c4b540cd5b8fd144a89ab5206a040f498b7b59385770bc841cf012",
Py0 = "2150beef17f5c22a65a4129390f47eece8f0c7d0c516790ea2632e7fd594ed8",
Py1 = "78281fb396e2b923453fae943af96b57e8b283fc40b0be4be3a1750d0daf121",
scalar = "1eac341ad699cba0cb13ae35b8215bfe0f34e931f8e51e33bf90d9849767bb",
Qx0 = "1bb6e8c1be4d9da9ef294ab066c82bb6dd805efa0c73f289e25f5cc6fc4f12e4",
Qx1 = "8ca44ff91e6484ecadc2a866ec64031e71c4a9d7a902f4280bef3db4dbf1bc9",
Qy0 = "39d151a22c49d4c71c8258e9664ead46ddccd49c596056509d9f9e6055def62",
Qy1 = "b37a08b453b96880f435d9fb2555960571a76e72a9d0d0ec622b055b7c97cdb"
)
test(
id = 2,
EC = ECP_ShortW_Jac[Fp2[BN254_Snarks], OnTwist],
Px0 = "2ac4b3d0a70d6686c63d9d63cce1370eafd4dc67f616c13582c6c64f6670513e",
Px1 = "f1daeb6a2581ba6c8027a645ab5c10e303db4aee85c70c8fe11f4c1adcc7029",
Py0 = "25807ff21967759cab64844741d006e2aa0221d9836613b1239da1a167d15131",
Py1 = "148dae725e508dbb43616972e2945f4868b2a193d828ed2efcf9c8a37b6b83f5",
scalar = "b29535765163b5d5f2d01c8fcde010d11f3f0f250a2cc84b8bc13dd9083356c",
Qx0 = "5f4b3a8a5fe74bf3f2ddc0bc2024f18c7958b2846dab1c602b8256a6035361a",
Qx1 = "ba3dad609b1ba8c78cbbfb7fae2d2f9398ef02265e3b4c0f3c8c18d8d0e59d6",
Qy0 = "2c226aee4621895d63df069c4b6951e2201ee1508d5d54e6ee860533b73b534a",
Qy1 = "2aa5384592339bff0a6e4664c931c0ec9f5a3d2fb2fff87a52245c0d95d3d130"
)
test(
id = 3,
EC = ECP_ShortW_Jac[Fp2[BN254_Snarks], OnTwist],
Px0 = "2a028c1328bb0abf252edfbf7133b84eef2a5f20163fe61685b4b54229ca585d",
Px1 = "8f80ad79e8e7e79bbdc645d9f5b339c52dd99a901b90de2494492656f11a9d5",
Py0 = "1f04320578e31ffa2e2b59ad8fcb1aba622b5f307ac540cf2ccdab07dec56503",
Py1 = "2973900c0fdf651b64f5b1a990baec7c582e0743d501bdb991374776d6c73b28",
scalar = "2c02275a71bb41c911faf48cab4f7ac7fc6672a5c15586185c8cff3203181da0",
Qx0 = "2f39a0772a0bd75db3e6ec0745b18824118e534fdefec811d0a4ca2ca3ce7606",
Qx1 = "23e1601a4338502dbc619a8bde251114272ca87662a851a56035c81d46877d81",
Qy0 = "1f0ee85e7c590124f89319f2450f7b8421d9f6d6414fd3b5cc18d781b69b30c9",
Qy1 = "29e4ff75afecaf732419409a5e0d8e94df6acec29fb5b298c7aad5ceef63e5f9"
)
test(
id = 4,
EC = ECP_ShortW_Jac[Fp2[BN254_Snarks], OnTwist],
Px0 = "1132e63c444e1abce6fc4c39bdf5be5caad586837cbf5ca9d3891482bdefe77",
Px1 = "22b71f598dab789f055fc9669ddf66f0d75f581af0e9e8863d7f95a51ef34862",
Py0 = "58e39050f64c9948d7238b99ecaee947cb934688a6e9f483c5c36b6e07aa31b",
Py1 = "2e64b920f498e12992f2a4ae3f9ced43f3f64705c9008169f3b930a760d055fb",
scalar = "24c5b2ce21615dca82231f5fb0fc8d05aa07c6df4bb5aa7c2381ac7b61a6290c",
Qx0 = "25a0637163a40813529e8a22a3e8be6db96f6dc1cdb8e1844729cad6be11668e",
Qx1 = "16de42461c4db7f9f72eb28bb16da850a42fc153dec64baf021f67f5564f36d8",
Qy0 = "27f2d743d3ce0c1f92c51110a6b9ca93a95693161f1d1cd85a0cf5a2492b4901",
Qy1 = "2c5a8df4fe93e31e374595c0605b1a8b93bb429232cf40f45c847739790c191e"
)
test(
id = 5,
EC = ECP_ShortW_Jac[Fp2[BN254_Snarks], OnTwist],
Px0 = "6a20c456e80e2bfe37d8631d41ffed31cba5d1c411e816d64014d0088d16554",
Px1 = "9d1555c77222abd79e17e2806386c53aba9609375276e817f52f03dc3f75676",
Py0 = "127e76f384726e56dfaa46e6fde6dc644f5fd494d056191059e2bebc525ce835",
Py1 = "2d80f2570da62adc61d794ac17c251f9f9f3de2b45f39c8ede5a9e215e60363",
scalar = "263e44e282fe22216cc054a39e40d4e38e71780bdc84563c340bdaaf4562534b",
Qx0 = "6d7e15b98352e445d1bdacb48826e5a7e8cf854fb9bc18839b30a164a2a9c53",
Qx1 = "12aa3294f6a9bb17f91d5875a5a1aa559c29b06134c6d165d83c8e9e02947568",
Qy0 = "271b8b552e52bdd310c46e07327a18861c91a739da178be909f0a4fe53ae0d05",
Qy1 = "1f4f200de96541e826f0bd536b1401e05e2a7c5a96c567b6dff21a21119bbf7"
)
test(
id = 6,
EC = ECP_ShortW_Jac[Fp2[BN254_Snarks], OnTwist],
Px0 = "4c591d080257375d6189340185c6fe4c1fa796722d07e1bec0b17080b6b1154",
Px1 = "241e2f2eb2a58afc2b5410d4ccbf75b53744ca1ac0bb28d7c60449f8d06204a4",
Py0 = "eaddea52f2e884a5e2635965ca4146b12127fe8a8883e07def8e8720f410781",
Py1 = "cc60dec6ad90e38c851924cf39ddd11b70eeb3fac95731eafd984e9aba2cae",
scalar = "1471f378000d557f0c33c1f8c8313edd53422479440cbd0fdc4cc55f1163deec",
Qx0 = "2a86b4867d7f63afdc09048210a3ef6d363c7896ccc1bb248f3aad4174e1f8fa",
Qx1 = "84c200018461c84ef9ce6de2c691b95cc2c41edc87107331f107ac49de76656",
Qy0 = "2ea1b6d71adb183d9a8dd319a21a679cb0b4e06bc96583d3a786f82b88b5e3ba",
Qy1 = "834e2ff738dcb5e8db7e4dae9336dede51524313b476019ea29ebadbb4ba12d"
)
test(
id = 7,
EC = ECP_ShortW_Jac[Fp2[BN254_Snarks], OnTwist],
Px0 = "115e772198e3f0003e31c0a1c48b6ac870d96020a4d634b7f14c15422d001cfe",
Px1 = "1913447ff41836e0b6a3b01be670a2457e6119e02ae35903fb71be4369e269f7",
Py0 = "14cb779c640aad2731b93b07c623c621a5585d0374f0394e5332f20ac28ca49d",
Py1 = "13a4c4b67f5976431158be8137b5017127fdbc312dc49825dae218a9a7878817",
scalar = "411315458c0c0cb27a058faf15dcbf2e467128f9c958d92de37902e1912f81",
Qx0 = "243a8808a891428d01ef28a77d0766488a98272a5dd394b2992599ff522f264",
Qx1 = "1baebf873402812e4345a8b1783fd25272c64d6194bd9f50b32b8e67ee737dc7",
Qy0 = "1f1001ba8b8b27159568f72e80662e352adfc00c9341d8b4fb8ef6f75ff628d2",
Qy1 = "169af215aa2456c6a65e13ac4df1ba1c982ca791058612679ef26dcb8fb0a989"
)
test(
id = 8,
EC = ECP_ShortW_Jac[Fp2[BN254_Snarks], OnTwist],
Px0 = "13faa1f28e6bfe89765c284e67face6ce0a29006ebc1551d4243e754c59f88ad",
Px1 = "640cebb80938dfcb998d84a8e5aafd47ffbcba0aa2f8c9b1c585baf119a8942",
Py0 = "1de793a9ef8f4dea5dad12fb09ddefa07ce197d4d7389a29ad3d8c6484582afe",
Py1 = "fc6e1f8bf75d1b7e48fdb6b2869c2de553663742c151b390cede6712da5a084",
scalar = "111e6f761ce48e22970752bd56cab93d47caa252d52948367b21163591f7b7b1",
Qx0 = "1c25702bf3b6f5fb453c6473b6dc2d67cd3cc21c65b589df1bfde254d50cffdd",
Qx1 = "14d03eb2075d6b5995240cc54a01ebe50b43933863f0316760226fbfa3a0280",
Qy0 = "1a075c9f2f6afa6e07e54b143a33c17e81d0ac8b91e0c8e0fdd5082bd6b9a72d",
Qy1 = "8e5ef57bb0f95fb6538dfaeac677977e3f3d6f49142c09584d8ec5c2ccd9b2d"
)
test(
id = 9,
EC = ECP_ShortW_Jac[Fp2[BN254_Snarks], OnTwist],
Px0 = "2fc3da947b78ac524a57670cef36ca89f1dad71b337bc3c18305c582a59648ad",
Px1 = "2f7cc845d8c1ef0613f919d8c47f3c62f83608a45b1e186748ac5dcccd4c6baf",
Py0 = "18ddc4718a4161f72f8d188fc61a609a3d592e186a65f4158483b719ffb05b8f",
Py1 = "45b9c3629ed709784db63ff090e2e891c5b5c6b2222cb6afc56638d7389d689",
scalar = "6223903d4bc2adea7b0a0db92822b6c2638691e4388df93f567e11edd6f23",
Qx0 = "9ec612ab0cf4a48e1c15d22284bce8e34619bfb9afb688a9a7930afcc1bd0f3",
Qx1 = "d796e5f5ae1a15622d2284ada34166b9e7c717bd2ff9b2cf2c6e48c33db5ff2",
Qy0 = "2a8ecb09a01cd2f89b316e7569331e9da3bfbd8a40114913b3e5477442c0e4ef",
Qy1 = "282b14bc00df2dd1733e3187a9845ef3a123c17ce4f6154e5cad26c3b48d1b98"
)

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# Constantine
# Copyright (c) 2018-2019 Status Research & Development GmbH
# Copyright (c) 2020-Present Mamy André-Ratsimbazafy
# Licensed and distributed under either of
# * MIT license (license terms in the root directory or at http://opensource.org/licenses/MIT).
# * Apache v2 license (license terms in the root directory or at http://www.apache.org/licenses/LICENSE-2.0).
# at your option. This file may not be copied, modified, or distributed except according to those terms.
import
# Internals
../constantine/config/[type_fp, curves],
../constantine/towers,
../constantine/elliptic/ec_shortweierstrass_jacobian,
../constantine/elliptic/ec_shortweierstrass_projective,
# Test utilities
./t_ec_sage_template
run_scalar_mul_test_vs_sage(
ECP_ShortW_Proj[Fp[BN254_Snarks], NotOnTwist],
"t_ec_sage_bn254_snarks_g1_projective"
)
run_scalar_mul_test_vs_sage(
ECP_ShortW_Jac[Fp[BN254_Snarks], NotOnTwist],
"t_ec_sage_bn254_snarks_g1_jacobian"
)
run_scalar_mul_test_vs_sage(
ECP_ShortW_Proj[Fp2[BN254_Snarks], OnTwist],
"t_ec_sage_bn254_snarks_g2_projective"
)
run_scalar_mul_test_vs_sage(
ECP_ShortW_Jac[Fp2[BN254_Snarks], OnTwist],
"t_ec_sage_bn254_snarks_g2_jacobian"
)

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# Constantine
# Copyright (c) 2018-2019 Status Research & Development GmbH
# Copyright (c) 2020-Present Mamy André-Ratsimbazafy
# Licensed and distributed under either of
# * MIT license (license terms in the root directory or at http://opensource.org/licenses/MIT).
# * Apache v2 license (license terms in the root directory or at http://www.apache.org/licenses/LICENSE-2.0).
# at your option. This file may not be copied, modified, or distributed except according to those terms.
import
# Standard library
std/[unittest, times, os, strutils, macros],
# 3rd party
serialization, json_serialization,
# Internals
../constantine/config/[common, curves, type_bigint, type_fp],
../constantine/towers,
../constantine/io/[io_bigints, io_ec],
../constantine/elliptic/[
ec_shortweierstrass_affine,
ec_shortweierstrass_projective,
ec_shortweierstrass_jacobian,
ec_scalar_mul,
ec_endomorphism_accel],
# Test utilities
./support/ec_reference_scalar_mult
# Workaround
# --------------------------------------------------------------------------
# Generic sandwich - https://github.com/nim-lang/Nim/issues/11225
export serialization, json_serialization
# When run_scalar_mul_test_vs_sage is not instantiated from this exact file
# "nim-serialization" somehow tries to serialize SecretWord
# json_serialization/reader.nim(522, 12) Error: Failed to convert to JSON an unsupported type: SecretWord
#
# This obscure error actually requires exporting the `readValue` proc
# Serialization
# --------------------------------------------------------------------------
macro matchingScalar*(EC: type ECP_ShortW_Aff): untyped =
## Workaround the annoying type system
## 1. Higher-kinded type
## 2. Computation in type section needs template or macro indirection
## 3. Converting NimNode to typedesc
## https://github.com/nim-lang/Nim/issues/6785
# BigInt[EC.F.C.getCurveOrderBitwidth()]
let ec = EC.getTypeImpl()
# echo ec.treerepr
# BracketExpr
# Sym "typeDesc"
# BracketExpr
# Sym "ECP_ShortW_Aff"
# BracketExpr
# Sym "Fp"
# IntLit 12
# IntLit 0
doAssert ec[0].eqIdent"typedesc"
doAssert ec[1][0].eqIdent"ECP_ShortW_Aff"
ec[1][1].expectkind(nnkBracketExpr)
doAssert ($ec[1][1][0]).startsWith"Fp"
let curve = Curve(ec[1][1][1].intVal)
let bitwidth = getAST(getCurveOrderBitwidth(curve))
result = nnkBracketExpr.newTree(
bindSym"BigInt",
bitwidth
)
type
TestVector*[EC: ECP_ShortW_Aff] = object
id: int
P: EC
scalar: matchingScalar(EC)
Q: EC
EC_G1_hex = object
x: string
y: string
Fp2_hex = object
c0: string
c1: string
EC_G2_hex = object
x: Fp2_hex
y: Fp2_hex
ScalarMulTestG1[EC: ECP_ShortW_Aff] = object
curve: string
group: string
modulus: string
order: string
cofactor: string
form: string
a: string
b: string
# vectors ------------------
vectors: seq[TestVector[EC]]
ScalarMulTestG2[EC: ECP_ShortW_Aff] = object
curve: string
group: string
modulus: string
order: string
cofactor: string
form: string
a: string
b: string
# G2 -----------------------
twist_degree: int
twist: string
non_residue_fp: int
G2_field: string
non_residue_twist: array[2, int]
# vectors ------------------
vectors: seq[TestVector[EC]]
const
TestVectorsDir* =
currentSourcePath.rsplit(DirSep, 1)[0] / "vectors"
proc readValue*(reader: var JsonReader, value: var BigInt) =
value.fromHex(reader.readValue(string))
proc readValue*(reader: var JsonReader, value: var ECP_ShortW_Aff) =
when ECP_ShortW_Aff.F is Fp:
let P = reader.readValue(EC_G1_hex)
let ok = value.fromHex(P.x, P.y)
doAssert ok, "\nDeserialization error on G1 for\n" &
" P.x: " & P.x & "\n" &
" P.y: " & P.x & "\n"
elif ECP_ShortW_Aff.F is Fp2:
let P = reader.readValue(EC_G2_hex)
let ok = value.fromHex(P.x.c0, P.x.c1, P.y.c0, P.y.c1)
doAssert ok, "\nDeserialization error on G2 for\n" &
" P.x0: " & P.x.c0 & "\n" &
" P.x1: " & P.x.c1 & "\n" &
" P.y0: " & P.y.c0 & "\n" &
" P.y1: " & P.y.c1 & "\n"
else:
{.error: "Not Implemented".}
proc loadVectors(TestType: typedesc): TestType =
const group = when TestType.EC.Tw == NotOnTwist: "G1"
else: "G2"
const filename = "tv_" & $TestType.EC.F.C & "_scalar_mul_" & group & ".json"
result = Json.loadFile(TestVectorsDir/filename, TestType)
# Testing
# ------------------------------------------------------------------------
proc run_scalar_mul_test_vs_sage*(
EC: typedesc,
moduleName: string
) =
echo "\n------------------------------------------------------\n"
echo moduleName & '\n'
when EC.Tw == NotOnTwist:
const G1_or_G2 = "G1"
let vec = loadVectors(ScalarMulTestG1[ECP_ShortW_Aff[EC.F, EC.Tw]])
else:
const G1_or_G2 = "G2"
let vec = loadVectors(ScalarMulTestG2[ECP_ShortW_Aff[EC.F, EC.Tw]])
const coord = when EC is ECP_ShortW_Proj: " Projective coordinates "
elif EC is ECP_ShortW_Jac: " Jacobian coordinates "
const testSuiteDesc = "Scalar Multiplication " & $EC.F.C & " " & G1_or_G2 & " vs SageMath"
suite testSuiteDesc & " [" & $WordBitwidth & "-bit mode]":
for i in 0 ..< vec.vectors.len:
test "test " & $vec.vectors[i].id & " - " & $EC:
var
P{.noInit.}: EC
Q {.noInit.}: EC
impl {.noInit.}: EC
reference {.noInit.}: EC
endo {.noInit.}: EC
when EC is ECP_ShortW_Proj:
P.projectiveFromAffine(vec.vectors[i].P)
Q.projectiveFromAffine(vec.vectors[i].Q)
else:
P.jacobianFromAffine(vec.vectors[i].P)
Q.jacobianFromAffine(vec.vectors[i].Q)
impl = P
reference = P
endo = P
impl.scalarMulGeneric(vec.vectors[i].scalar)
reference.unsafe_ECmul_double_add(vec.vectors[i].scalar)
endo.scalarMulEndo(vec.vectors[i].scalar)
doAssert: bool(Q == reference)
doAssert: bool(Q == impl)
doAssert: bool(Q == endo)
when EC.F is Fp: # Test windowed endomorphism acceleration
var endoW = P
endoW.scalarMulGLV_m2w2(vec.vectors[i].scalar)
doAssert: bool(Q == endoW)

492
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{
"curve": "BLS12_377",
"group": "G1",
"modulus": "0x1ae3a4617c510eac63b05c06ca1493b1a22d9f300f5138f1ef3622fba094800170b5d44300000008508c00000000001",
"order": "0x12ab655e9a2ca55660b44d1e5c37b00159aa76fed00000010a11800000000001",
"cofactor": "0x170b5d44300000000000000000000000",
"form": "short_weierstrass",
"a": "0x0",
"b": "0x1",
"vectors": [
{
"id": 0,
"P": {
"x": "0x4e7e6dfa01ed0ceb6e66708b07cb5c6cd30a42eeb13d7b76f8d103a0a4d491450be6f526fc12f15209b792220c041e",
"y": "0xc782515159b7e7b9371e0e0caa387951317e993b1625d91869d4346621058a0960ef1b8b6eabb33cd5719694908a05"
},
"scalar": "0xcf815cb4d44d3d691b7c82a40b4b70caa9b0e8fe9586648abf3f1e2e639ca1b",
"Q": {
"x": "0x4a1203db4af8f0efc18c7ceb24999eb6e0dbdfc8f44a9edd5ba2f9eced38e81ecae287ab1c184eea8e8753d1178604",
"y": "0xf7509cdce5de473e37f36c69d93ff1b2ab04c3ae25a3b5e41f2c138cdcc69ed5eaf4ac95a7a857eef336ebac19bcd3"
}
},
{
"id": 1,
"P": {
"x": "0x13d735b28405253dcc0bc60bcdc13633475ffc187d38a9b97655b0d0fa1d56c4548f11ea0a795391ee85c953aaf9b83",
"y": "0x1693101123fd13a20f9c0569c52c29507ba1c8b6dd412660bc82e7974022f1a10f9137b4ba59d3f0aab67027cefec19"
},
"scalar": "0x913aa7b9fa2f940b70b6dcf538cc08da1a369809ab86a8ee49cead0ed6bfef6",
"Q": {
"x": "0x10e0e8582ec3456f7569473892c23997d004f2542d914fa75db8f1798ed8ce505836e8b7af5cf1503e14d85fadd65ee",
"y": "0xd5151ae60e43100e20aad7eaf8659e690e8034910d7717078031520fcbf6f9a00b22c6a9894aec88c9182f13335639"
}
},
{
"id": 2,
"P": {
"x": "0x6dd2211113cada09d3b96848cbf21ed14f6fc238b581c0afd49aa776980101e4ee279a256ce8f1428d3ed3f70afd85",
"y": "0x3b406b4433a3f44f8e196012f50c520e876412fbcae2651916f133c1fd3899c79f676e1abba01d84bab7ad100c9295"
},
"scalar": "0x4cf47669aeb0f30b6c6c5aa02808a87dc787fba22da32875e614a54a50b6a0c",
"Q": {
"x": "0x3caf55e1f82d746df2bf47defbee127a3e6f7e79a9575929704b25489ffb801dbce07999acbddfd79352b0633a2708",
"y": "0x3fc275c136b07456f0b6ee814508876037add5e95357fae6b4195744ba5ccccaf93581e34feb2babe652a9a704731b"
}
},
{
"id": 3,
"P": {
"x": "0x18648abbe3261f93cbf679777eac66e419035041a967d1e6a0f0afdb92810d7122e0984f1d6efc8fe518500464ee803",
"y": "0x7b6f518d4d06309aad4d60d29118310b6c8b17c7bf5db2251f4701b13b89a8c0f04bb5d0386785e55ffbbd7ecc445e"
},
"scalar": "0x69498486a06c18f836a8e9ed507bbb563d6d03545e03e08f628e8fbd2e5d098",
"Q": {
"x": "0x6a871d872673879fd23ec8c150f8d63e8130dc60343fc0c2ec9ff1b02e769e20eeec0288102ccec6ff2f6ac6973b4",
"y": "0xcd95a31afa4f0bbcc063a1585fec4a6c51812b2baaee42a04525fd55b3a9db3bdb7cbc77f7cd32f42d2e0eb26ddfa2"
}
},
{
"id": 4,
"P": {
"x": "0x745e9944549522486f3446676cc62fb666682e10c661a13b8110b42cb9a37676b6f33a46f9495f4fafb342d5809db5",
"y": "0xbc595854bc42ccec60dd9ec573608d736aa59996cef1c2e8f6c5d424f525a6f3e3d4beeedfac6b959dbd71ced95b13"
},
"scalar": "0x6e08d8714102a5aa3e9f46e33c70a759c27253c7b0196c3e46c7cb42671197e",
"Q": {
"x": "0x18a967a80785de8ec6ac9d98cffa06a8c633b5fa0f36431a32f7bd955946edc3d55f79bfdf5335db405560a6cbe2415",
"y": "0x4552ff2eb6ade0c6b33c3603460d9d62099201d842c9883b33b7ed1147cb17268338a77a9417776ddbd774e91a8d2"
}
},
{
"id": 5,
"P": {
"x": "0xf7b56cf212a8906ed77a758164c0bd05ce1fbd3ee3c4357e7a09b3aedc748a29ace254f1f6df35b8cb74361060337f",
"y": "0x2640ef641d20fea19b28947833e53faceeafa57a8761b807049f3d707d70c01f1c69a57edd993d301a64517bf47f77"
},
"scalar": "0xa5c46d7a52938fed7f3093d5867f65361dc8b48c83bd7db490c26736196e20e",
"Q": {
"x": "0x434f024c4afd7b9a44375011d663af0ae0fe79442e9caf36518e053bb13d49998ec1d2da4bc1c4a812812119b3221f",
"y": "0x32c959e2a678cf93f77e621fdf887454a5f9cb7a67f29065669f48234c25e321ceb5758dfba987431a0c2caec94616"
}
},
{
"id": 6,
"P": {
"x": "0x43b387c12e4cc91fb1a5593b7671356dceb0fe6e6ac666d5bac94f2a44c8db54976b649a678aae038b21144de04e23",
"y": "0x1a6366a50f1f9ba64eef73d82bec86177bf184be048a9d66326ccb0122203569ddcb8cf74445cadaff7f47a66d1b1a2"
},
"scalar": "0xbddd07231bc7fe89ee4a859a00ea1f9d236be9e7fd561303d566904c1b0a07c",
"Q": {
"x": "0x8ddb7a959483b51a1471de988146b7d5b166f660734b4d55166c5a23d781e923261927b1012dce73b822bb6e56bfd2",
"y": "0xdc7178bf5597ef31209a5b0409fc42c64b81260a0046a562ca08c7a9dc67b444c25e94fb97b9a6bb4c137cadd81021"
}
},
{
"id": 7,
"P": {
"x": "0x6984c9385a67081f97b3d33444077466cca1d0442a4da8c083a0957578e0b21011435b126a5ab143da9da1cf5b216f",
"y": "0x18a87c7f5f6c5a8101773e63956b9addd4becf5177acc560d548e5331638121934842fdc9f654b3f456a7df5a2e471a"
},
"scalar": "0xb72c41a6ffaff0aacb5d62e3dcb16acfec66b6a9639e19d3128fd43c18e7dbe",
"Q": {
"x": "0x19279201c3c7a9d50b546aa99d3e1a6625fe2a7bc64a09625b683534638b9e87b9102d4dba6684956b6be7668a658c6",
"y": "0x195f15ad1edc05f8b289c7eee7bd8f78116a2d5ba8b83643d9e7cb2cdc6550bcf8c2145008e900ca9cba4d5040e9f4"
}
},
{
"id": 8,
"P": {
"x": "0xefc34595823e7333616f7768bc82f407268df6f029bf4c94d15f1785dc7ccc08f22e7301dfc48dadc0ea383c8bb3e",
"y": "0x2459bd9f71977ef122d2102e8bfd07a5737066075058cfa8bcaa9f9690ed065919c844363ceaea6f9bb650906a535f"
},
"scalar": "0x8f90f6ab0ffa6e4acc601b44c062745f2935b3dc153d0da07977470080d5c18",
"Q": {
"x": "0x35d08b33e02579581905941975c8b1cc5be1c9670a7f7ef390daa363b0abd3571a802d8c27f156fba40573094f6c7a",
"y": "0x111ba3bdfa4260dc2b636479edb1fcf2fc9478aa722da0118908e1db1551cf7e131c8521b2f3708ef670684dbe8d181"
}
},
{
"id": 9,
"P": {
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}

980
tests/vectors/tv_BLS12_377_scalar_mul_G2.json Normal file
View File

@ -0,0 +1,980 @@
{
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"non_residue_twist": [
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]
}

492
tests/vectors/tv_BLS12_381_scalar_mul_G1.json Normal file
View File

@ -0,0 +1,492 @@
{
"curve": "BLS12_381",
"group": "G1",
"modulus": "0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2a0f6b0f6241eabfffeb153ffffb9feffffffffaaab",
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"form": "short_weierstrass",
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{
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}

980
tests/vectors/tv_BLS12_381_scalar_mul_G2.json Normal file
View File

@ -0,0 +1,980 @@
{
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]
}

492
tests/vectors/tv_BN254_Snarks_scalar_mul_G1.json Normal file
View File

@ -0,0 +1,492 @@
{
"curve": "BN254_Snarks",
"group": "G1",
"modulus": "0x30644e72e131a029b85045b68181585d97816a916871ca8d3c208c16d87cfd47",
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"cofactor": "0x1",
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}

980
tests/vectors/tv_BN254_Snarks_scalar_mul_G2.json Normal file
View File

@ -0,0 +1,980 @@
{
"curve": "BN254_Snarks",
"group": "G2",
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"twist": "D_Twist",
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"G2_field": "Fp2",
"non_residue_twist": [
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"a": "0x0",
"b": "0x3",
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