210 lines
5.5 KiB
Python
210 lines
5.5 KiB
Python
# 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('')
|
||
|
||
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
|
||
L = ((int(r).bit_length() + m-1) // m) + 1 # l = ⌈log2 r/m⌉ + 1
|
||
|
||
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
|
||
L = ((int(r).bit_length() + m-1) // m) + 1 # l = ⌈log2 r/m⌉ + 1
|
||
|
||
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)
|