nim-dagger/dagger/storageproofs/bls.nim

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## Nim-POS
## Copyright (c) 2021 Status Research & Development GmbH
## Licensed under either of
## * Apache License, version 2.0, ([LICENSE-APACHE](LICENSE-APACHE))
## * MIT license ([LICENSE-MIT](LICENSE-MIT))
## at your option.
## This file may not be copied, modified, or distributed except according to
## those terms.
# Implementation of the BLS-based public PoS scheme from
# Shacham H., Waters B., "Compact Proofs of Retrievability"
# using pairing over BLS12-381 ECC
#
# Notation from the paper
# In Z:
# - n: number of blocks
# - s: number of sectors per block
#
# In Z_p: modulo curve order
# - m_{ij}: sectors of the file i:0..n-1 j:0..s-1
# - α: PoS secret key
# - name: random string
# - μ_j: part of proof, j:0..s-1
#
# In G_1: multiplicative cyclic group
# - H: {0,1} →G_1 : hash function
# - u_1,…,u_s ←R G_1 : random coefficients
# - σ_i: authenticators
# - σ: part of proof
#
# In G_2: multiplicative cyclic group
# - g: generator of G_2
# - v ← g^α: PoS public key
#
# In G_T:
# - used only to calculate the two pairings during validation
#
# Implementation:
# Our implementation uses additive cyclic groups instead of the multiplicative
# cyclic group in the paper, thus changing the name of the group operation as in
# blscurve and blst. Thus, point multiplication becomes point addition, and scalar
# exponentiation becomes scalar multiplicaiton.
#
# Number of operations:
# The following table summarizes the number of operations in different phases
# using the following notation:
# - f: file size expressed in units of 31 bytes
# - n: number of blocks
# - s: number of sectors per block
# - q: number of query items
#
# Since f = n * s and s is a parameter of the scheme, it is better to express
# the cost as a function of f and s. This only matters for Setup, all other
# phases are independent of the file size assuming a given q.
#
# | | Setup | Challenge | Proof | Verify |
# |----------------|-----------|---------------|-----------|-----------|-----------|
# | G1 random | s = s | q | | |
# | G1 scalar mult | n * (s+1) = f * (1 + 1/s) | | q | q + s |
# | G1 add | n * s = f | | q-1 | q-1 + s-1 |
# | Hash to G1 | n = f / s | | | q |
# | Z_p mult | = | | s * q | |
# | Z_p add | = | | s * (q-1) | |
# | pairing | = | | | 2 |
#
#
# Storage and communication cost:
# The storage overhead for a file of f_b bytes is given by the n authenticators
# calculated in the setup phase.
# f_b = f * 31 = n * s * 31
# Each authenticator is a point on G_1, which occupies 48 bytes in compressed form.
# Thus, the overall sorage size in bytes is:
# f_pos = fb + n * 48 = fb * (1 + (48/31) * (1/s))
#
# Communicaiton cost in the Setup phase is simply related to the storage cost.
# The size of the challenge is
# q * (8 + 48) bytes
# The size of the proof is instead
# s * 32 + 48 bytes
import blscurve
import blscurve/blst/blst_abi
import ../rng
import endians
# sector size in bytes. Must be smaller than the subgroup order r
# which is 255 bits long for BLS12-381
const bytespersector = 31
# length in bytes of the unique (random) name
const namelen = 512
type
# a single sector
ZChar = array[bytespersector, byte]
# secret key combining the metadata signing key and the POR generation key
SecretKey = object
signkey: blscurve.SecretKey
key: blst_scalar
# public key combining the metadata signing key and the POR validation key
PublicKey = object
signkey: blscurve.PublicKey
key: blst_p2
# POR metadata (called "file tag t_0" in the original paper)
TauZero = object
name: array[namelen, byte]
n: int64
u: seq[blst_p1]
# signed POR metadata (called "signed file tag t" in the original paper)
Tau = object
t: TauZero
signature: array[96, byte]
# PoR query element
QElement = object
I: int64
V: blst_scalar
proc fromBytesBE(a: array[32, byte]): blst_scalar =
## Convert data to blst native form
blst_scalar_from_bendian(result, a)
doAssert(blst_scalar_fr_check(result).bool)
proc fromBytesBE(a: openArray[byte]): blst_scalar =
## Convert data to blst native form
var b: array[32, byte]
doAssert(a.len <= b.len)
let d = b.len - a.len
for i in 0 ..< a.len:
b[i+d] = a[i]
blst_scalar_from_bendian(result, b)
doAssert(blst_scalar_fr_check(result).bool)
proc getSector(f: File, blockid: int64, sectorid: int64, spb: int64): ZChar =
## Read file sector at given <blockid, sectorid> postion
f.setFilePos((blockid * spb + sectorid) * sizeof(result))
let r = f.readBytes(result, 0, sizeof(result))
proc rndScalar(): blst_scalar =
## Generate random scalar within the subroup order r
var scal{.noInit.}: array[32, byte]
var scalar{.noInit.}: blst_scalar
while true:
for val in scal.mitems:
val = byte Rng.instance.rand(0xFF)
scalar.blst_scalar_from_bendian(scal)
if blst_scalar_fr_check(scalar).bool:
break
return scalar
proc rndP2(): (blst_p2, blst_scalar) =
## Generate random point on G2
var x{.noInit.}: blst_p2
x.blst_p2_from_affine(BLS12_381_G2) # init from generator
let scalar = rndScalar()
x.blst_p2_mult(x, scalar, 255)
return (x, scalar)
proc rndP1(): (blst_p1, blst_scalar) =
## Generate random point on G1
var x{.noInit.}: blst_p1
x.blst_p1_from_affine(BLS12_381_G1) # init from generator
let scalar = rndScalar()
x.blst_p1_mult(x, scalar, 255)
return (x, scalar)
proc posKeygen(): (blst_p2, blst_scalar) =
## Generate POS key pair
rndP2()
proc keygen*(): (PublicKey, SecretKey) =
## Generate key pair for signing metadata and for POS tags
var pk: PublicKey
var sk: SecretKey
var ikm: array[32, byte]
for b in ikm.mitems:
b = byte Rng.instance.rand(0xFF)
doAssert ikm.keyGen(pk.signkey, sk.signkey)
(pk.key, sk.key) = posKeygen()
return (pk, sk)
proc split(f: File, s: int64): int64 =
## Calculate number of blocks for a file
let size = f.getFileSize()
let n = ((size - 1) div (s * sizeof(ZChar))) + 1
echo "File size=", size, " bytes",
", blocks=", n,
", sectors/block=", $s,
", sectorsize=", $sizeof(ZChar), " bytes"
return n
proc hashToG1[T: byte|char](msg: openArray[T]): blst_p1 =
## Hash to curve with Dagger specific domain separation
const dst = "DAGGER-PROOF-OF-CONCEPT"
result.blst_hash_to_g1(msg, dst, aug = "")
proc hashNameI(name: array[namelen, byte], i: int64): blst_p1 =
## Calculate unique filname and block index based hash
# # naive implementation, hashing a long string representation
# # such as "[255, 242, 23]1"
# return hashToG1($name & $i)
# more compact and faster implementation
var namei: array[sizeof(name) + sizeof(int64), byte]
namei[0..sizeof(name)-1] = name
bigEndian64(addr(namei[sizeof(name)]), unsafeAddr(i))
return hashToG1(namei)
proc generateAuthenticatorNaive(i: int64, s: int64, t: TauZero, f: File, ssk: SecretKey): blst_p1 =
## Naive implementation of authenticator as in the S&W paper.
## With the paper's multiplicative notation:
## \sigmai=\(H(file||i)\cdot\prod{j=0}^{s-1}{uj^{m[i][j]}})^{\alpha}
var sum: blst_p1
for j in 0 ..< s:
var prod: blst_p1
prod.blst_p1_mult(t.u[j], fromBytesBE(getSector(f, i, j, s)), 255)
sum.blst_p1_add_or_double(sum, prod)
blst_p1_add_or_double(result, hashNameI(t.name, i), sum)
result.blst_p1_mult(result, ssk.key, 255)
proc generateAuthenticatorOpt(i: int64, s: int64, t: TauZero, ubase: openArray[blst_scalar], f: File, ssk: SecretKey): blst_p1 =
## Optimized implementation of authenticator generation
## This implementation is reduces the number of scalar multiplications
## from s+1 to 1+1 , using knowledge about the scalars (r_j)
## used to generate u_j as u_j = g^{r_j}
##
## With the paper's multiplicative notation, we use:
## (H(file||i)\cdot g^{\sum{j=0}^{s-1}{r_j \cdot m[i][j]}})^{\alpha}
var sum: blst_fr
var sums: blst_scalar
for j in 0 ..< s:
var a, b, x: blst_fr
a.blst_fr_from_scalar(ubase[j])
b.blst_fr_from_scalar(fromBytesBE(getSector(f, i, j, s)))
x.blst_fr_mul(a, b)
sum.blst_fr_add(sum, x)
sums.blst_scalar_from_fr(sum)
result.blst_p1_from_affine(BLS12_381_G1)
result.blst_p1_mult(result, sums, 255)
result.blst_p1_add_or_double(result, hashNameI(t.name, i))
result.blst_p1_mult(result, ssk.key, 255)
proc generateAuthenticator(i: int64, s: int64, t: TauZero, ubase: openArray[blst_scalar], f: File, ssk: SecretKey): blst_p1 =
## Wrapper to select tag generator implementation
# let a = generateAuthenticatorNaive(i, s, t, f, ssk)
let b = generateAuthenticatorOpt(i, s, t, ubase, f, ssk)
# doAssert(a.blst_p1_is_equal(b).bool)
return b
proc setup*(ssk: SecretKey, s:int64, filename: string): (Tau, seq[blst_p1]) =
## Set up the POR scheme by generating tags and metadata
let file = open(filename)
let n = split(file, s)
var t = TauZero(n: n)
# generate a random name
for i in 0 ..< 512 :
t.name[i] = byte Rng.instance.rand(0xFF)
# generate the coefficient vector for combining sectors of a block: U
var ubase: seq[blst_scalar]
for i in 0 ..< s :
let (u, ub) = rndP1()
t.u.add(u)
ubase.add(ub)
#TODO: a better bytearray conversion of TauZero for the signature might be needed
# the current conversion using $t might be architecture dependent and not unique
let signature = sign(ssk.signkey, $t)
let tau = Tau(t: t, signature: signature.exportRaw())
#generate sigmas
var sigmas: seq[blst_p1]
for i in 0 ..< n :
sigmas.add(generateAuthenticator(i, s, t, ubase, file, ssk))
file.close()
result = (tau, sigmas)
proc generateQuery*(tau: Tau, spk: PublicKey, l: int): seq[QElement] =
## Generata a random BLS query of given sizxe
let n = tau.t.n # number of blocks
for i in 0 ..< l :
var q: QElement
q.I = Rng.instance.rand(n-1) #TODO: dedup
q.V = rndScalar() #TODO: fix range
result.add(q)
proc generateProof*(q: openArray[QElement], authenticators: openArray[blst_p1], spk: PublicKey, s: int64, filename: string): (seq[blst_scalar], blst_p1) =
## Generata BLS proofs for a given query
let file = open(filename)
var mu: seq[blst_scalar]
for j in 0 ..< s :
var muj: blst_fr
for qelem in q :
var x, v, sector: blst_fr
let sect = fromBytesBE(getSector(file, qelem.I, j, s))
sector.blst_fr_from_scalar(sect)
v.blst_fr_from_scalar(qelem.V)
x.blst_fr_mul(v, sector)
muj.blst_fr_add(muj, x)
var mujs: blst_scalar
mujs.blst_scalar_from_fr(muj)
mu.add(mujs)
var sigma: blst_p1
for qelem in q:
var prod: blst_p1
prod.blst_p1_mult(authenticators[qelem.I], qelem.V, 255)
sigma.blst_p1_add_or_double(sigma, prod)
file.close()
return (mu, sigma)
proc pairing(a: blst_p1, b: blst_p2): blst_fp12 =
## Calculate pairing G_1,G_2 -> G_T
var aa: blst_p1_affine
var bb: blst_p2_affine
blst_p1_to_affine(aa, a)
blst_p2_to_affine(bb, b)
var l: blst_fp12
blst_miller_loop(l, bb, aa)
blst_final_exp(result, l)
proc verifyPairingsNaive(a1: blst_p1, a2: blst_p2, b1: blst_p1, b2: blst_p2) : bool =
let e1 = pairing(a1, a2)
let e2 = pairing(b1, b2)
return e1 == e2
proc verifyPairingsNeg(a1: blst_p1, a2: blst_p2, b1: blst_p1, b2: blst_p2) : bool =
## Faster pairing verification using 2 miller loops but ony one final exponentiation
## based on https://github.com/benjaminion/c-kzg/blob/main/src/bls12_381.c
var
loop0, loop1, gt_point: blst_fp12
aa1, bb1: blst_p1_affine
aa2, bb2: blst_p2_affine
var a1neg = a1
blst_p1_cneg(a1neg, 1)
blst_p1_to_affine(aa1, a1neg)
blst_p1_to_affine(bb1, b1)
blst_p2_to_affine(aa2, a2)
blst_p2_to_affine(bb2, b2)
blst_miller_loop(loop0, aa2, aa1)
blst_miller_loop(loop1, bb2, bb1)
blst_fp12_mul(gt_point, loop0, loop1)
blst_final_exp(gt_point, gt_point)
return blst_fp12_is_one(gt_point).bool
proc verifyPairings(a1: blst_p1, a2: blst_p2, b1: blst_p1, b2: blst_p2) : bool =
## Wrapper to select verify pairings implementation
verifyPairingsNaive(a1, a2, b1, b2)
#verifyPairingsNeg(a1, a2, b1, b2)
proc verifyProof*(tau: Tau, q: openArray[QElement], mus: openArray[blst_scalar], sigma: blst_p1, spk: PublicKey): bool =
## Verify a BLS proof given a query
# verify signature on Tau
var signature: Signature
if not signature.fromBytes(tau.signature):
return false
if not verify(spk.signkey, $tau.t, signature):
return false
var first: blst_p1
for qelem in q :
var prod: blst_p1
prod.blst_p1_mult(hashNameI(tau.t.name, qelem.I), qelem.V, 255)
first.blst_p1_add_or_double(first, prod)
doAssert(blst_p1_on_curve(first).bool)
let us = tau.t.u
var second: blst_p1
for j in 0 ..< len(us) :
var prod: blst_p1
prod.blst_p1_mult(us[j], mus[j], 255)
second.blst_p1_add_or_double(second, prod)
doAssert(blst_p1_on_curve(second).bool)
var sum: blst_p1
sum.blst_p1_add_or_double(first, second)
var g{.noInit.}: blst_p2
g.blst_p2_from_affine(BLS12_381_G2)
return verifyPairings(sum, spk.key, sigma, g)