Hash-to-Curve BLS12-381 G1 (#189)
* Skeleton of hash to curve for BLS12-381 G1 * Remove isodegree parameter * Fix polynomial evaluation of hashToG1 * Optimize hash_to_curve and add bench for hash to G1 * slight optim of jacobian isomap + v7 test vectors
This commit is contained in:
Mamy Ratsimbazafy
GitHub
parent
bde4f97b56
commit
65eedd1cf7
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@ -35,6 +35,21 @@ template bench(op: string, C: static Curve, iters: int, body: untyped): untyped
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measure(iters, startTime, stopTime, startClk, stopClk, body)
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report(op, $C, startTime, stopTime, startClk, stopClk, iters)
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proc bench_BLS12_381_hash_to_G1(iters: int) =
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const dst = "BLS_SIG_BLS12381G1-SHA256-SSWU-RO_POP_"
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let msg = "Mr F was here"
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var P: ECP_ShortW_Prj[Fp[BLS12_381], G1]
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bench("Hash to G1 (Draft #11)", BLS12_381, iters):
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sha256.hashToCurve(
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k = 128,
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output = P,
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augmentation = "",
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message = msg,
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domainSepTag = dst
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)
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proc bench_BLS12_381_hash_to_G2(iters: int) =
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const dst = "BLS_SIG_BLS12381G2-SHA256-SSWU-RO_POP_"
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let msg = "Mr F was here"
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@ -50,7 +65,25 @@ proc bench_BLS12_381_hash_to_G2(iters: int) =
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domainSepTag = dst
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)
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proc bench_BLS12_381_proj_aff_conversion(iters: int) =
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proc bench_BLS12_381_G1_proj_aff_conversion(iters: int) =
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const dst = "BLS_SIG_BLS12381G1-SHA256-SSWU-RO_POP_"
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let msg = "Mr F was here"
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var P: ECP_ShortW_Prj[Fp[BLS12_381], G1]
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var Paff: ECP_ShortW_Aff[Fp[BLS12_381], G1]
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sha256.hashToCurve(
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k = 128,
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output = P,
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augmentation = "",
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message = msg,
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domainSepTag = dst
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)
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bench("G1 Proj->Affine conversion (for pairing)", BLS12_381, iters):
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Paff.affine(P)
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proc bench_BLS12_381_G2_proj_aff_conversion(iters: int) =
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const dst = "BLS_SIG_BLS12381G2-SHA256-SSWU-RO_POP_"
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let msg = "Mr F was here"
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@ -65,15 +98,17 @@ proc bench_BLS12_381_proj_aff_conversion(iters: int) =
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domainSepTag = dst
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)
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bench("Proj->Affine conversion (for pairing)", BLS12_381, iters):
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bench("G2 Proj->Affine conversion (for pairing)", BLS12_381, iters):
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Paff.affine(P)
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const Iters = 1000
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proc main() =
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separator()
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bench_BLS12_381_hash_to_G1(Iters)
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bench_BLS12_381_hash_to_G2(Iters)
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bench_BLS12_381_proj_aff_conversion(Iters)
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bench_BLS12_381_G1_proj_aff_conversion(Iters)
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bench_BLS12_381_G2_proj_aff_conversion(Iters)
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separator()
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main()
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@ -25,25 +25,9 @@ import
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# No exceptions allowed
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{.push raises: [].}
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func poly_eval_horner[F](r: var F, x: F, poly: openarray[F]) =
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## Fast polynomial evaluation using Horner's rule
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## The polynomial k₀ + k₁ x + k₂ x² + k₃ x³ + ... + kₙ xⁿ
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## MUST be stored in order
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## [k₀, k₁, k₂, k₃, ..., kₙ]
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##
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## Assuming a degree n = 3 polynomial
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## Horner's rule rewrites its evaluation as
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## ((k₃ x + k₂)x + k₁) x + k₀
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## which is n additions and n multiplications,
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## the lowest complexity of general polynomial evaluation algorithm.
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r = poly[^1] # TODO: optim when poly[^1] == 1
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for i in countdown(poly.len-2, 0):
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r *= x
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r += poly[i]
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func poly_eval_horner_scaled[F; D, N: static int](
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r: var F, xn: F,
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xd_pow: array[D, F], poly: array[N, F]) =
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xd_pow: array[D, F], poly: static array[N, F], numPolyLen: static int) =
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## Fast polynomial evaluation using Horner's rule
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## Result is scaled by xd^N with N the polynomial degree
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## to avoid finite field division
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@ -66,25 +50,33 @@ func poly_eval_horner_scaled[F; D, N: static int](
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## ((k₃ xn + k₂ xd) xn + k₁ xd²) xn + k₀ xd³
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##
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## avoiding expensive divisions
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r = poly[^1] # TODO: optim when poly[^1] == 1
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for i in countdown(N-2, 0):
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var t: F
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# i = N-2
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when poly[^1].isOne().bool:
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block:
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r.prod(poly[N-2], xd_pow[0])
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r += xn
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else:
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block:
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var t{.noInit.}: F
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r.prod(poly[N-1], xn)
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t.prod(poly[N-2], xd_pow[0])
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r += t
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for i in countdown(N-3, 0):
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var t{.noInit.}: F
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r *= xn
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t.prod(poly[i], xd_pow[N-2-i])
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t.prod(poly[i], xd_pow[(N-2-i)])
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r += t
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const
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poly_degree = N-1 # [1, x, x², x³] of length 4
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isodegree = D # Isogeny degree
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static: doAssert isodegree - poly_degree >= 0
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when isodegree - poly_degree > 0:
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when N-numPolyLen < 0:
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# Missing scaling factor
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r *= xd_pow[isodegree - poly_degree - 1]
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r *= xd_pow[0]
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func h2c_isogeny_map[F](
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rxn, rxd, ryn, ryd: var F,
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xn, xd, yn: F, isodegree: static int) =
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xn, xd, yn: F,
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G: static Subgroup) =
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## Given G2, the target prime order subgroup of E2,
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## this function maps an element of
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## E'2 a curve isogenous to E2
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@ -99,29 +91,41 @@ func h2c_isogeny_map[F](
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## (rx, ry) with rx = rxn/rxd and ry = ryn/ryd
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# xd^e with e in [1, N], for example [xd, xd², xd³]
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static: doAssert isodegree >= 2
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var xd_pow{.noInit.}: array[isodegree, F]
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const maxdegree = max([
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h2cIsomapPoly(F.C, G, xnum).len,
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h2cIsomapPoly(F.C, G, xden).len,
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h2cIsomapPoly(F.C, G, ynum).len,
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h2cIsomapPoly(F.C, G, yden).len,
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])
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var xd_pow{.noInit.}: array[maxdegree, F]
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xd_pow[0] = xd
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xd_pow[1].square(xd_pow[0])
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for i in 2 ..< xd_pow.len:
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xd_pow[i].prod(xd_pow[i-1], xd_pow[0])
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const xnLen = h2cIsomapPoly(F.C, G, xnum).len
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const ynLen = h2cIsomapPoly(F.C, G, ynum).len
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rxn.poly_eval_horner_scaled(
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xn, xd_pow,
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h2cIsomapPoly(F.C, G2, isodegree, xnum)
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h2cIsomapPoly(F.C, G, xnum),
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xnLen
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)
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rxd.poly_eval_horner_scaled(
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xn, xd_pow,
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h2cIsomapPoly(F.C, G2, isodegree, xden)
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h2cIsomapPoly(F.C, G, xden),
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xnLen
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)
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ryn.poly_eval_horner_scaled(
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xn, xd_pow,
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h2cIsomapPoly(F.C, G2, isodegree, ynum)
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h2cIsomapPoly(F.C, G, ynum),
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ynLen
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)
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ryd.poly_eval_horner_scaled(
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xn, xd_pow,
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h2cIsomapPoly(F.C, G2, isodegree, yden)
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h2cIsomapPoly(F.C, G, yden),
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ynLen
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)
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# y coordinate is y' * poly_yNum(x)
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@ -129,7 +133,7 @@ func h2c_isogeny_map[F](
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func h2c_isogeny_map*[F; G: static Subgroup](
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r: var ECP_ShortW_Prj[F, G],
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xn, xd, yn: F, isodegree: static int) =
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xn, xd, yn: F) =
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## Given G2, the target prime order subgroup of E2,
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## this function maps an element of
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## E'2 a curve isogenous to E2
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@ -151,7 +155,8 @@ func h2c_isogeny_map*[F; G: static Subgroup](
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rxd = r.z,
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ryn = r.y,
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ryd = t,
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xn, xd, yn, isodegree
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xn, xd, yn,
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G
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)
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# Now convert to projective coordinates
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@ -163,7 +168,7 @@ func h2c_isogeny_map*[F; G: static Subgroup](
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func h2c_isogeny_map*[F; G: static Subgroup](
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r: var ECP_ShortW_Jac[F, G],
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xn, xd, yn: F, isodegree: static int) =
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xn, xd, yn: F) =
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## Given G2, the target prime order subgroup of E2,
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## this function maps an element of
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## E'2 a curve isogenous to E2
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@ -184,7 +189,8 @@ func h2c_isogeny_map*[F; G: static Subgroup](
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h2c_isogeny_map(
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rxn, rxd,
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ryn, ryd,
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xn, xd, yn, isodegree
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xn, xd, yn,
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G
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)
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# Now convert to jacobian coordinates
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@ -200,8 +206,7 @@ func h2c_isogeny_map*[F; G: static Subgroup](
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func h2c_isogeny_map*[F; G: static Subgroup](
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r: var ECP_ShortW_Jac[F, G],
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P: ECP_ShortW_Jac[F, G],
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isodegree: static int) =
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P: ECP_ShortW_Jac[F, G]) =
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## Map P in isogenous curve E'2
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## to r in E2
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##
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@ -218,29 +223,47 @@ func h2c_isogeny_map*[F; G: static Subgroup](
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var yn{.noInit.}, yd{.noInit.}: F
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# Z²^e with e in [1, N], for example [Z², Z⁴, Z⁶]
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static: doAssert isodegree >= 2
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var ZZpow{.noInit.}: array[isodegree, F]
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const maxdegree = max([
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h2cIsomapPoly(F.C, G, xnum).len,
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h2cIsomapPoly(F.C, G, xden).len,
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h2cIsomapPoly(F.C, G, ynum).len,
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h2cIsomapPoly(F.C, G, yden).len,
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])
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var ZZpow{.noInit.}: array[maxdegree, F]
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ZZpow[0].square(P.z)
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ZZpow[1].square(ZZpow[0])
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for i in 2 ..< ZZpow.len:
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ZZpow[i].prod(ZZpow[i-1], ZZpow[0])
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# ZZpow[1].square(ZZpow[0])
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# for i in 2 ..< ZZpow.len:
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# ZZpow[i].prod(ZZpow[i-1], ZZpow[0])
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staticFor i, 1, ZZpow.len:
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when bool(i and 1): # is odd
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ZZpow[i].square(ZZpow[(i-1) shr 1])
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else:
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ZZpow[i].prod(ZZpow[(i-1) shr 1], ZZpow[((i-1) shr 1) + 1])
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const xnLen = h2cIsomapPoly(F.C, G, xnum).len
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const ynLen = h2cIsomapPoly(F.C, G, ynum).len
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xn.poly_eval_horner_scaled(
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P.x, ZZpow,
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h2cIsomapPoly(F.C, G2, isodegree, xnum)
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h2cIsomapPoly(F.C, G, xnum),
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xnLen
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)
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xd.poly_eval_horner_scaled(
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P.x, ZZpow,
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h2cIsomapPoly(F.C, G2, isodegree, xden)
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h2cIsomapPoly(F.C, G, xden),
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xnLen
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)
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yn.poly_eval_horner_scaled(
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P.x, ZZpow,
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h2cIsomapPoly(F.C, G2, isodegree, ynum)
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h2cIsomapPoly(F.C, G, ynum),
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ynLen
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)
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yd.poly_eval_horner_scaled(
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P.x, ZZpow,
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h2cIsomapPoly(F.C, G2, isodegree, yden)
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h2cIsomapPoly(F.C, G, yden),
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ynLen
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)
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# yn = y' * poly_yNum(x) = yZ³ * poly_yNum(x)
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@ -230,3 +230,86 @@ func mapToIsoCurve_sswuG2_opt9mod16*[C: static Curve](
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y.cneg(e1 xor e2)
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# yd.setOne()
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func mapToIsoCurve_sswuG1_opt3mod4*[C: static Curve](
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xn, xd, yn: var Fp[C],
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u: Fp[C], xd3: var Fp[C]) =
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## Given G1, the target prime order subgroup of E1 we want to hash to,
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## this function maps any field element of Fp to E'1
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## a curve isogenous to E1 using the Simplified Shallue-van de Woestijne method.
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##
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## This requires p² ≡ 3 (mod 4).
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##
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## Input:
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## - u, an Fp element
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## Output:
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## - (xn, xd, yn, yd) such that (x', y') = (xn/xd, yn/yd)
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## is a point of E'1
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## - yd is implied to be 1
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## Scratchspace:
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## - xd3 is temporary scratchspace that will hold xd³
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## after execution (which might be useful for Jacobian coordinate conversion)
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#
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# Paper: https://eprint.iacr.org/2019/403
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# Spec: https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-hash-to-curve-11#appendix-G.2.1
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# Sage: https://github.com/cfrg/draft-irtf-cfrg-hash-to-curve/blob/f7dd3761/poc/sswu_opt_3mod4.sage#L33-L76
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# BLST: https://github.com/supranational/blst/blob/v0.3.4/src/map_to_g1.c#L322-L365
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# Formal verification: https://github.com/GaloisInc/BLST-Verification/blob/8e2efde4/spec/implementation/HashToG1.cry
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var
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uu {.noInit.}, tv2 {.noInit.}: Fp[C]
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tv4 {.noInit.}, x2n {.noInit.}, gx1 {.noInit.}: Fp[C]
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y2 {.noInit.}: Fp[C]
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e1, e2: SecretBool
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# Aliases
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template y: untyped = yn
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template x1n: untyped = xn
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template y1: untyped = yn
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template Zuu: untyped = x2n
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template gxd: untyped = xd3
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# x numerators
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uu.square(u) # uu = u²
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Zuu.prod(uu, h2cConst(C, G1, Z)) # Zuu = Z * uu
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tv2.square(Zuu) # tv2 = Zuu²
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tv2 += Zuu # tv2 = tv2 + Zuu
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x1n.setOne()
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x1n += tv2 # x1n = tv2 + 1
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x1n *= h2cConst(C, G1, Bprime_E1) # x1n = x1n * B'
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x2n.prod(Zuu, x1n) # x2n = Zuu * x1n
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# x denumerator
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xd.prod(tv2, h2cConst(C, G1, minus_A)) # xd = -A * tv2
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e1 = xd.isZero() # e1 = xd == 0
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xd.ccopy(h2cConst(C, G1, ZmulA), e1) # If xd == 0, set xd = Z*A
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# y numerators
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tv2.square(xd)
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gxd.prod(xd, tv2) # gxd = xd³
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tv2 *= h2CConst(C, G1, Aprime_E1)
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gx1.square(x1n)
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gx1 += tv2 # x1n² + A * xd²
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gx1 *= x1n # x1n³ + A * x1n * xd²
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tv2.prod(gxd, h2cConst(C, G1, Bprime_E1))
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gx1 += tv2 # gx1 = x1n³ + A * x1n * xd² + B * xd³
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tv4.square(gxd) # tv4 = gxd²
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tv2.prod(gx1, gxd) # tv2 = gx1 * gxd
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tv4 *= tv2 # tv4 = gx1 * gxd³
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# Start searching for sqrt(gx1)
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e2 = y1.invsqrt_if_square(tv4) # y1 = tv4^c1 = (gx1 * gxd³)^((p²-9)/16)
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y1 *= tv2 # y1 *= gx1*gxd
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y2.prod(y1, h2cConst(C, G1, sqrt_minus_Z3))
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y2 *= uu
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y2 *= u
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# Choose numerators
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xn.ccopy(x2n, not e2) # xn = e2 ? x1n : x2n
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yn.ccopy(y2, not e2) # yn = e2 ? y1 : y2
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e1 = sgn0(u)
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e2 = sgn0(y)
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y.cneg(e1 xor e2)
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# yd.setOne()
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@ -34,8 +34,27 @@ import
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|||
# Map to curve
|
||||
# ----------------------------------------------------------------
|
||||
|
||||
func mapToIsoCurve_sswuG2_opt9mod16[F; G: static Subgroup](
|
||||
r: var ECP_ShortW_Jac[F, G],
|
||||
func mapToIsoCurve_sswuG1_opt3mod4[F](
|
||||
r: var ECP_ShortW_Jac[F, G1],
|
||||
u: F) =
|
||||
var
|
||||
xn{.noInit.}, xd{.noInit.}: F
|
||||
yn{.noInit.}: F
|
||||
xd3{.noInit.}: F
|
||||
|
||||
mapToIsoCurve_sswuG1_opt3mod4(
|
||||
xn, xd,
|
||||
yn,
|
||||
u, xd3
|
||||
)
|
||||
|
||||
# Convert to Jacobian
|
||||
r.z = xd # Z = xd
|
||||
r.x.prod(xn, xd) # X = xZ² = xn/xd * xd² = xn*xd
|
||||
r.y.prod(yn, xd3) # Y = yZ³ = yn * xd³
|
||||
|
||||
func mapToIsoCurve_sswuG2_opt9mod16[F](
|
||||
r: var ECP_ShortW_Jac[F, G2],
|
||||
u: F) =
|
||||
var
|
||||
xn{.noInit.}, xd{.noInit.}: F
|
||||
|
|
@ -59,29 +78,30 @@ func mapToCurve[F; G: static Subgroup](
|
|||
## Map an element of the
|
||||
## finite or extension field F
|
||||
## to an elliptic curve E
|
||||
|
||||
when F.C == BLS12_381 and F is Fp2:
|
||||
|
||||
when F.C.getCoefA() * F.C.getCoefB() == 0:
|
||||
# https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-hash-to-curve-11#section-6.6.3
|
||||
# Simplified Shallue-van de Woestijne-Ulas method for AB == 0
|
||||
|
||||
# 1. Map to E' isogenous to E
|
||||
when F is Fp and F.C.hasP3mod4_primeModulus():
|
||||
mapToIsoCurve_sswuG1_opt3mod4(
|
||||
xn, xd,
|
||||
yn,
|
||||
u, xd3
|
||||
)
|
||||
elif F is Fp2 and F.C.hasP3mod4_primeModulus():
|
||||
# p ≡ 3 (mod 4) => p² ≡ 9 (mod 16)
|
||||
mapToIsoCurve_sswuG2_opt9mod16(
|
||||
xn, xd,
|
||||
yn,
|
||||
u, xd3
|
||||
)
|
||||
else:
|
||||
{.error: "Not implemented".}
|
||||
|
||||
# 1. Map to E'2 isogenous to E2
|
||||
var
|
||||
xn{.noInit.}, xd{.noInit.}: F
|
||||
yn{.noInit.}: F
|
||||
xd3{.noInit.}: F
|
||||
|
||||
mapToIsoCurve_sswuG2_opt9mod16(
|
||||
xn, xd,
|
||||
yn,
|
||||
u, xd3
|
||||
)
|
||||
|
||||
# 2. Map from E'2 to E2
|
||||
r.h2c_isogeny_map(
|
||||
xn, xd,
|
||||
yn,
|
||||
isodegree = 3
|
||||
)
|
||||
# 2. Map from E'1 to E1
|
||||
r.h2c_isogeny_map(xn, xd, yn)
|
||||
else:
|
||||
{.error: "Not implemented".}
|
||||
|
||||
|
|
@ -106,15 +126,27 @@ func mapToCurve_fusedAdd[F; G: static Subgroup](
|
|||
# So we use jacobian coordinates for computation on isogenies.
|
||||
|
||||
var P0{.noInit.}, P1{.noInit.}: ECP_ShortW_Jac[F, G]
|
||||
when F.C == BLS12_381 and F is Fp2:
|
||||
# 1. Map to E'2 isogenous to E2
|
||||
P0.mapToIsoCurve_sswuG2_opt9mod16(u0)
|
||||
P1.mapToIsoCurve_sswuG2_opt9mod16(u1)
|
||||
|
||||
P0.sum(P0, P1, h2CConst(F.C, G2, Aprime_E2))
|
||||
when F.C.getCoefA() * F.C.getCoefB() == 0:
|
||||
# https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-hash-to-curve-11#section-6.6.3
|
||||
# Simplified Shallue-van de Woestijne-Ulas method for AB == 0
|
||||
|
||||
# 1. Map to E' isogenous to E
|
||||
when F is Fp and F.C.hasP3mod4_primeModulus():
|
||||
# 1. Map to E'1 isogenous to E1
|
||||
P0.mapToIsoCurve_sswuG1_opt3mod4(u0)
|
||||
P1.mapToIsoCurve_sswuG1_opt3mod4(u1)
|
||||
P0.sum(P0, P1, h2CConst(F.C, G1, Aprime_E1))
|
||||
elif F is Fp2 and F.C.hasP3mod4_primeModulus():
|
||||
# p ≡ 3 (mod 4) => p² ≡ 9 (mod 16)
|
||||
# 1. Map to E'2 isogenous to E2
|
||||
P0.mapToIsoCurve_sswuG2_opt9mod16(u0)
|
||||
P1.mapToIsoCurve_sswuG2_opt9mod16(u1)
|
||||
P0.sum(P0, P1, h2CConst(F.C, G2, Aprime_E2))
|
||||
else:
|
||||
{.error: "Not implemented".}
|
||||
|
||||
# 2. Map from E'2 to E2
|
||||
r.h2c_isogeny_map(P0, isodegree = 3)
|
||||
r.h2c_isogeny_map(P0)
|
||||
else:
|
||||
{.error: "Not implemented".}
|
||||
|
||||
|
|
|
|||
|
|
@ -25,10 +25,6 @@ import
|
|||
# Specialized routine for p ≡ 3 (mod 4)
|
||||
# ------------------------------------------------------------
|
||||
|
||||
func hasP3mod4_primeModulus*(C: static Curve): static bool =
|
||||
## Returns true iff p ≡ 3 (mod 4)
|
||||
(BaseType(C.Mod.limbs[0]) and 3) == 3
|
||||
|
||||
func invsqrt_p3mod4(r: var Fp, a: Fp) =
|
||||
## Compute the inverse square root of ``a``
|
||||
##
|
||||
|
|
@ -58,10 +54,6 @@ func invsqrt_p3mod4(r: var Fp, a: Fp) =
|
|||
# Specialized routine for p ≡ 5 (mod 8)
|
||||
# ------------------------------------------------------------
|
||||
|
||||
func hasP5mod8_primeModulus*(C: static Curve): static bool =
|
||||
## Returns true iff p ≡ 5 (mod 8)
|
||||
(BaseType(C.Mod.limbs[0]) and 7) == 5
|
||||
|
||||
func invsqrt_p5mod8(r: var Fp, a: Fp) =
|
||||
## Compute the inverse square root of ``a``
|
||||
##
|
||||
|
|
|
|||
|
|
@ -37,4 +37,16 @@ template matchingBigInt*(C: static Curve): untyped =
|
|||
|
||||
template matchingLimbs2x*(C: Curve): untyped =
|
||||
const N2 = wordsRequired(getCurveBitwidth(C)) * 2 # TODO upstream, not precomputing N2 breaks semcheck
|
||||
array[N2, SecretWord] # TODO upstream, using Limbs[N2] breaks semcheck
|
||||
array[N2, SecretWord] # TODO upstream, using Limbs[N2] breaks semcheck
|
||||
|
||||
func hasP3mod4_primeModulus*(C: static Curve): static bool =
|
||||
## Returns true iff p ≡ 3 (mod 4)
|
||||
(BaseType(C.Mod.limbs[0]) and 3) == 3
|
||||
|
||||
func hasP5mod8_primeModulus*(C: static Curve): static bool =
|
||||
## Returns true iff p ≡ 5 (mod 8)
|
||||
(BaseType(C.Mod.limbs[0]) and 7) == 5
|
||||
|
||||
func hasP9mod16_primeModulus*(C: static Curve): static bool =
|
||||
## Returns true iff p ≡ 9 (mod 16)
|
||||
(BaseType(C.Mod.limbs[0]) and 15) == 9
|
||||
|
|
@ -0,0 +1,280 @@
|
|||
# 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
|
||||
../config/curves,
|
||||
../io/io_fields
|
||||
|
||||
# Hash-to-Curve map to isogenous BLS12-381 E'1 constants
|
||||
# -----------------------------------------------------------------
|
||||
#
|
||||
# y² = x³ + A'*x + B' with p ≡ 3 (mod 4) the BLS12-381 characteristic (base modulus)
|
||||
#
|
||||
# Hardcoding from spec:
|
||||
# - https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-hash-to-curve-14#section-8.8.2
|
||||
# - https://github.com/cfrg/draft-irtf-cfrg-hash-to-curve/blob/f7dd3761/poc/sswu_opt_3mod4.sage#L126-L132
|
||||
|
||||
const BLS12_381_h2c_G1_Aprime_E1* = Fp[BLS12_381].fromHex(
|
||||
"0x144698a3b8e9433d693a02c96d4982b0ea985383ee66a8d8e8981aefd881ac98936f8da0e0f97f5cf428082d584c1d")
|
||||
const BLS12_381_h2c_G1_Bprime_E1* = Fp[BLS12_381].fromHex(
|
||||
"0x12e2908d11688030018b12e8753eee3b2016c1f0f24f4070a0b9c14fcef35ef55a23215a316ceaa5d1cc48e98e172be0")
|
||||
const BLS12_381_h2c_G1_Z* = Fp[BLS12_381].fromHex(
|
||||
"0xb")
|
||||
const BLS12_381_h2c_G1_minus_A* = Fp[BLS12_381].fromHex(
|
||||
"0x19eccb5195c6fd570db26db379de6354b38cb3316f96ac168e483a8606d8747786189071107306805d0ad7f7d2a75e8e")
|
||||
const BLS12_381_h2c_G1_ZmulA* = Fp[BLS12_381].fromHex(
|
||||
"0xdf088f08f205e3a3857e1ea7b2289d9a148b96ab3e694151fe89284e4d926a8e55cb15e9aab878fe7db859f2cb453f")
|
||||
const BLS12_381_h2c_G1_sqrt_minus_Z3* = Fp[BLS12_381].fromHex(
|
||||
"0x3d689d1e0e762cef9f2bec6130316806b4c80eda6fc10ce77ae83eab1ea8b8b8a407c9c6db195e06f2dbeabc2baeff5")
|
||||
|
||||
# Hash-to-Curve 11-isogeny map BLS12-381 E'1 constants
|
||||
# -----------------------------------------------------------------
|
||||
#
|
||||
# The polynomials map a point (x', y') on the isogenous curve E'1
|
||||
# to (x, y) on E1, represented as (xnum/xden, y' * ynum/yden)
|
||||
|
||||
const BLS12_381_h2c_G1_isogeny_map_xnum* = [
|
||||
# Polynomial k₀ + k₁ x + k₂ x² + k₃ x³ + ... + kₙ xⁿ
|
||||
# The polynomial is stored as an array of coefficients ordered from k₀ to kₙ
|
||||
|
||||
# 1
|
||||
Fp[BLS12_381].fromHex(
|
||||
"0x11a05f2b1e833340b809101dd99815856b303e88a2d7005ff2627b56cdb4e2c85610c2d5f2e62d6eaeac1662734649b7"
|
||||
),
|
||||
# x
|
||||
Fp[BLS12_381].fromHex(
|
||||
"0x17294ed3e943ab2f0588bab22147a81c7c17e75b2f6a8417f565e33c70d1e86b4838f2a6f318c356e834eef1b3cb83bb"
|
||||
),
|
||||
# x²
|
||||
Fp[BLS12_381].fromHex(
|
||||
"0xd54005db97678ec1d1048c5d10a9a1bce032473295983e56878e501ec68e25c958c3e3d2a09729fe0179f9dac9edcb0"
|
||||
),
|
||||
# x³
|
||||
Fp[BLS12_381].fromHex(
|
||||
"0x1778e7166fcc6db74e0609d307e55412d7f5e4656a8dbf25f1b33289f1b330835336e25ce3107193c5b388641d9b6861"
|
||||
),
|
||||
# x⁴
|
||||
Fp[BLS12_381].fromHex(
|
||||
"0xe99726a3199f4436642b4b3e4118e5499db995a1257fb3f086eeb65982fac18985a286f301e77c451154ce9ac8895d9"
|
||||
),
|
||||
# x⁵
|
||||
Fp[BLS12_381].fromHex(
|
||||
"0x1630c3250d7313ff01d1201bf7a74ab5db3cb17dd952799b9ed3ab9097e68f90a0870d2dcae73d19cd13c1c66f652983"
|
||||
),
|
||||
# x⁶
|
||||
Fp[BLS12_381].fromHex(
|
||||
"0xd6ed6553fe44d296a3726c38ae652bfb11586264f0f8ce19008e218f9c86b2a8da25128c1052ecaddd7f225a139ed84"
|
||||
),
|
||||
# x⁷
|
||||
Fp[BLS12_381].fromHex(
|
||||
"0x17b81e7701abdbe2e8743884d1117e53356de5ab275b4db1a682c62ef0f2753339b7c8f8c8f475af9ccb5618e3f0c88e"
|
||||
),
|
||||
# x⁸
|
||||
Fp[BLS12_381].fromHex(
|
||||
"0x80d3cf1f9a78fc47b90b33563be990dc43b756ce79f5574a2c596c928c5d1de4fa295f296b74e956d71986a8497e317"
|
||||
),
|
||||
# x⁹
|
||||
Fp[BLS12_381].fromHex(
|
||||
"0x169b1f8e1bcfa7c42e0c37515d138f22dd2ecb803a0c5c99676314baf4bb1b7fa3190b2edc0327797f241067be390c9e"
|
||||
),
|
||||
# x¹⁰
|
||||
Fp[BLS12_381].fromHex(
|
||||
"0x10321da079ce07e272d8ec09d2565b0dfa7dccdde6787f96d50af36003b14866f69b771f8c285decca67df3f1605fb7b"
|
||||
),
|
||||
# x¹¹
|
||||
Fp[BLS12_381].fromHex(
|
||||
"0x6e08c248e260e70bd1e962381edee3d31d79d7e22c837bc23c0bf1bc24c6b68c24b1b80b64d391fa9c8ba2e8ba2d229"
|
||||
)
|
||||
]
|
||||
const BLS12_381_h2c_G1_isogeny_map_xden* = [
|
||||
# Polynomial k₀ + k₁ x + k₂ x² + k₃ x³ + ... + kₙ xⁿ
|
||||
# The polynomial is stored as an array of coefficients ordered from k₀ to kₙ
|
||||
|
||||
# 1
|
||||
Fp[BLS12_381].fromHex(
|
||||
"0x8ca8d548cff19ae18b2e62f4bd3fa6f01d5ef4ba35b48ba9c9588617fc8ac62b558d681be343df8993cf9fa40d21b1c"
|
||||
),
|
||||
# x
|
||||
Fp[BLS12_381].fromHex(
|
||||
"0x12561a5deb559c4348b4711298e536367041e8ca0cf0800c0126c2588c48bf5713daa8846cb026e9e5c8276ec82b3bff"
|
||||
),
|
||||
# x²
|
||||
Fp[BLS12_381].fromHex(
|
||||
"0xb2962fe57a3225e8137e629bff2991f6f89416f5a718cd1fca64e00b11aceacd6a3d0967c94fedcfcc239ba5cb83e19"
|
||||
),
|
||||
# x³
|
||||
Fp[BLS12_381].fromHex(
|
||||
"0x3425581a58ae2fec83aafef7c40eb545b08243f16b1655154cca8abc28d6fd04976d5243eecf5c4130de8938dc62cd8"
|
||||
),
|
||||
# x⁴
|
||||
Fp[BLS12_381].fromHex(
|
||||
"0x13a8e162022914a80a6f1d5f43e7a07dffdfc759a12062bb8d6b44e833b306da9bd29ba81f35781d539d395b3532a21e"
|
||||
),
|
||||
# x⁵
|
||||
Fp[BLS12_381].fromHex(
|
||||
"0xe7355f8e4e667b955390f7f0506c6e9395735e9ce9cad4d0a43bcef24b8982f7400d24bc4228f11c02df9a29f6304a5"
|
||||
),
|
||||
# x⁶
|
||||
Fp[BLS12_381].fromHex(
|
||||
"0x772caacf16936190f3e0c63e0596721570f5799af53a1894e2e073062aede9cea73b3538f0de06cec2574496ee84a3a"
|
||||
),
|
||||
# x⁷
|
||||
Fp[BLS12_381].fromHex(
|
||||
"0x14a7ac2a9d64a8b230b3f5b074cf01996e7f63c21bca68a81996e1cdf9822c580fa5b9489d11e2d311f7d99bbdcc5a5e"
|
||||
),
|
||||
# x⁸
|
||||
Fp[BLS12_381].fromHex(
|
||||
"0xa10ecf6ada54f825e920b3dafc7a3cce07f8d1d7161366b74100da67f39883503826692abba43704776ec3a79a1d641"
|
||||
),
|
||||
# x⁹
|
||||
Fp[BLS12_381].fromHex(
|
||||
"0x95fc13ab9e92ad4476d6e3eb3a56680f682b4ee96f7d03776df533978f31c1593174e4b4b7865002d6384d168ecdd0a"
|
||||
),
|
||||
# x¹⁰
|
||||
Fp[BLS12_381].fromHex(
|
||||
"0x1"
|
||||
)
|
||||
]
|
||||
const BLS12_381_h2c_G1_isogeny_map_ynum* = [
|
||||
# Polynomial k₀ + k₁ x + k₂ x² + k₃ x³ + ... + kₙ xⁿ
|
||||
# The polynomial is stored as an array of coefficients ordered from k₀ to kₙ
|
||||
|
||||
# y
|
||||
Fp[BLS12_381].fromHex(
|
||||
"0x90d97c81ba24ee0259d1f094980dcfa11ad138e48a869522b52af6c956543d3cd0c7aee9b3ba3c2be9845719707bb33"
|
||||
),
|
||||
# x*y
|
||||
Fp[BLS12_381].fromHex(
|
||||
"0x134996a104ee5811d51036d776fb46831223e96c254f383d0f906343eb67ad34d6c56711962fa8bfe097e75a2e41c696"
|
||||
),
|
||||
# x²*y
|
||||
Fp[BLS12_381].fromHex(
|
||||
"0xcc786baa966e66f4a384c86a3b49942552e2d658a31ce2c344be4b91400da7d26d521628b00523b8dfe240c72de1f6"
|
||||
),
|
||||
# x³*y
|
||||
Fp[BLS12_381].fromHex(
|
||||
"0x1f86376e8981c217898751ad8746757d42aa7b90eeb791c09e4a3ec03251cf9de405aba9ec61deca6355c77b0e5f4cb"
|
||||
),
|
||||
# x⁴*y
|
||||
Fp[BLS12_381].fromHex(
|
||||
"0x8cc03fdefe0ff135caf4fe2a21529c4195536fbe3ce50b879833fd221351adc2ee7f8dc099040a841b6daecf2e8fedb"
|
||||
),
|
||||
# x⁵*y
|
||||
Fp[BLS12_381].fromHex(
|
||||
"0x16603fca40634b6a2211e11db8f0a6a074a7d0d4afadb7bd76505c3d3ad5544e203f6326c95a807299b23ab13633a5f0"
|
||||
),
|
||||
# x⁶*y
|
||||
Fp[BLS12_381].fromHex(
|
||||
"0x4ab0b9bcfac1bbcb2c977d027796b3ce75bb8ca2be184cb5231413c4d634f3747a87ac2460f415ec961f8855fe9d6f2"
|
||||
),
|
||||
# x⁷*y
|
||||
Fp[BLS12_381].fromHex(
|
||||
"0x987c8d5333ab86fde9926bd2ca6c674170a05bfe3bdd81ffd038da6c26c842642f64550fedfe935a15e4ca31870fb29"
|
||||
),
|
||||
# x⁸*y
|
||||
Fp[BLS12_381].fromHex(
|
||||
"0x9fc4018bd96684be88c9e221e4da1bb8f3abd16679dc26c1e8b6e6a1f20cabe69d65201c78607a360370e577bdba587"
|
||||
),
|
||||
# x⁹*y
|
||||
Fp[BLS12_381].fromHex(
|
||||
"0xe1bba7a1186bdb5223abde7ada14a23c42a0ca7915af6fe06985e7ed1e4d43b9b3f7055dd4eba6f2bafaaebca731c30"
|
||||
),
|
||||
# x¹⁰*y
|
||||
Fp[BLS12_381].fromHex(
|
||||
"0x19713e47937cd1be0dfd0b8f1d43fb93cd2fcbcb6caf493fd1183e416389e61031bf3a5cce3fbafce813711ad011c132"
|
||||
),
|
||||
# x¹¹*y
|
||||
Fp[BLS12_381].fromHex(
|
||||
"0x18b46a908f36f6deb918c143fed2edcc523559b8aaf0c2462e6bfe7f911f643249d9cdf41b44d606ce07c8a4d0074d8e"
|
||||
),
|
||||
# x¹²*y
|
||||
Fp[BLS12_381].fromHex(
|
||||
"0xb182cac101b9399d155096004f53f447aa7b12a3426b08ec02710e807b4633f06c851c1919211f20d4c04f00b971ef8"
|
||||
),
|
||||
# x¹³*y
|
||||
Fp[BLS12_381].fromHex(
|
||||
"0x245a394ad1eca9b72fc00ae7be315dc757b3b080d4c158013e6632d3c40659cc6cf90ad1c232a6442d9d3f5db980133"
|
||||
),
|
||||
# x¹⁴*y
|
||||
Fp[BLS12_381].fromHex(
|
||||
"0x5c129645e44cf1102a159f748c4a3fc5e673d81d7e86568d9ab0f5d396a7ce46ba1049b6579afb7866b1e715475224b"
|
||||
),
|
||||
# x¹⁵*y
|
||||
Fp[BLS12_381].fromHex(
|
||||
"0x15e6be4e990f03ce4ea50b3b42df2eb5cb181d8f84965a3957add4fa95af01b2b665027efec01c7704b456be69c8b604"
|
||||
),
|
||||
]
|
||||
const BLS12_381_h2c_G1_isogeny_map_yden* = [
|
||||
# Polynomial k₀ + k₁ x + k₂ x² + k₃ x³ + ... + kₙ xⁿ
|
||||
# The polynomial is stored as an array of coefficients ordered from k₀ to kₙ
|
||||
|
||||
# 1
|
||||
Fp[BLS12_381].fromHex(
|
||||
"0x16112c4c3a9c98b252181140fad0eae9601a6de578980be6eec3232b5be72e7a07f3688ef60c206d01479253b03663c1"
|
||||
),
|
||||
# x
|
||||
Fp[BLS12_381].fromHex(
|
||||
"0x1962d75c2381201e1a0cbd6c43c348b885c84ff731c4d59ca4a10356f453e01f78a4260763529e3532f6102c2e49a03d"
|
||||
),
|
||||
# x²
|
||||
Fp[BLS12_381].fromHex(
|
||||
"0x58df3306640da276faaae7d6e8eb15778c4855551ae7f310c35a5dd279cd2eca6757cd636f96f891e2538b53dbf67f2"
|
||||
),
|
||||
# x³
|
||||
Fp[BLS12_381].fromHex(
|
||||
"0x16b7d288798e5395f20d23bf89edb4d1d115c5dbddbcd30e123da489e726af41727364f2c28297ada8d26d98445f5416"
|
||||
),
|
||||
# x⁴
|
||||
Fp[BLS12_381].fromHex(
|
||||
"0xbe0e079545f43e4b00cc912f8228ddcc6d19c9f0f69bbb0542eda0fc9dec916a20b15dc0fd2ededda39142311a5001d"
|
||||
),
|
||||
# x⁵
|
||||
Fp[BLS12_381].fromHex(
|
||||
"0x8d9e5297186db2d9fb266eaac783182b70152c65550d881c5ecd87b6f0f5a6449f38db9dfa9cce202c6477faaf9b7ac"
|
||||
),
|
||||
# x⁶
|
||||
Fp[BLS12_381].fromHex(
|
||||
"0x166007c08a99db2fc3ba8734ace9824b5eecfdfa8d0cf8ef5dd365bc400a0051d5fa9c01a58b1fb93d1a1399126a775c"
|
||||
),
|
||||
# x⁷
|
||||
Fp[BLS12_381].fromHex(
|
||||
"0x16a3ef08be3ea7ea03bcddfabba6ff6ee5a4375efa1f4fd7feb34fd206357132b920f5b00801dee460ee415a15812ed9"
|
||||
),
|
||||
# x⁸
|
||||
Fp[BLS12_381].fromHex(
|
||||
"0x1866c8ed336c61231a1be54fd1d74cc4f9fb0ce4c6af5920abc5750c4bf39b4852cfe2f7bb9248836b233d9d55535d4a"
|
||||
),
|
||||
# x⁹
|
||||
Fp[BLS12_381].fromHex(
|
||||
"0x167a55cda70a6e1cea820597d94a84903216f763e13d87bb5308592e7ea7d4fbc7385ea3d529b35e346ef48bb8913f55"
|
||||
),
|
||||
# x¹⁰
|
||||
Fp[BLS12_381].fromHex(
|
||||
"0x4d2f259eea405bd48f010a01ad2911d9c6dd039bb61a6290e591b36e636a5c871a5c29f4f83060400f8b49cba8f6aa8"
|
||||
),
|
||||
# x¹¹
|
||||
Fp[BLS12_381].fromHex(
|
||||
"0xaccbb67481d033ff5852c1e48c50c477f94ff8aefce42d28c0f9a88cea7913516f968986f7ebbea9684b529e2561092"
|
||||
),
|
||||
# x¹²
|
||||
Fp[BLS12_381].fromHex(
|
||||
"0xad6b9514c767fe3c3613144b45f1496543346d98adf02267d5ceef9a00d9b8693000763e3b90ac11e99b138573345cc"
|
||||
),
|
||||
# x¹³
|
||||
Fp[BLS12_381].fromHex(
|
||||
"0x2660400eb2e4f3b628bdd0d53cd76f2bf565b94e72927c1cb748df27942480e420517bd8714cc80d1fadc1326ed06f7"
|
||||
),
|
||||
# x¹⁴
|
||||
Fp[BLS12_381].fromHex(
|
||||
"0xe0fa1d816ddc03e6b24255e0d7819c171c40f65e273b853324efcd6356caa205ca2f570f13497804415473a1d634b8f"
|
||||
),
|
||||
# x¹⁵
|
||||
Fp[BLS12_381].fromHex(
|
||||
"0x1"
|
||||
)
|
||||
]
|
||||
|
|
@ -55,7 +55,7 @@ const BLS12_381_h2c_G2_inv_norm_inv_Z3* = Fp[BLS12_381].fromHex( # 1/||1/Z³||
|
|||
# The polynomials map a point (x', y') on the isogenous curve E'2
|
||||
# to (x, y) on E2, represented as (xnum/xden, y' * ynum/yden)
|
||||
|
||||
const BLS12_381_h2c_G2_3_isogeny_map_xnum* = [
|
||||
const BLS12_381_h2c_G2_isogeny_map_xnum* = [
|
||||
# Polynomial k₀ + k₁ x + k₂ x² + k₃ x³ + ... + kₙ xⁿ
|
||||
# The polynomial is stored as an array of coefficients ordered from k₀ to kₙ
|
||||
|
||||
|
|
@ -69,18 +69,18 @@ const BLS12_381_h2c_G2_3_isogeny_map_xnum* = [
|
|||
"0x0",
|
||||
"0x11560bf17baa99bc32126fced787c88f984f87adf7ae0c7f9a208c6b4f20a4181472aaa9cb8d555526a9ffffffffc71a"
|
||||
),
|
||||
# x^2
|
||||
# x²
|
||||
Fp2[BLS12_381].fromHex(
|
||||
"0x11560bf17baa99bc32126fced787c88f984f87adf7ae0c7f9a208c6b4f20a4181472aaa9cb8d555526a9ffffffffc71e",
|
||||
"0x8ab05f8bdd54cde190937e76bc3e447cc27c3d6fbd7063fcd104635a790520c0a395554e5c6aaaa9354ffffffffe38d"
|
||||
),
|
||||
# x^3
|
||||
# x³
|
||||
Fp2[BLS12_381].fromHex(
|
||||
"0x171d6541fa38ccfaed6dea691f5fb614cb14b4e7f4e810aa22d6108f142b85757098e38d0f671c7188e2aaaaaaaa5ed1",
|
||||
"0x0"
|
||||
)
|
||||
]
|
||||
const BLS12_381_h2c_G2_3_isogeny_map_xden* = [
|
||||
const BLS12_381_h2c_G2_isogeny_map_xden* = [
|
||||
# Polynomial k₀ + k₁ x + k₂ x² + k₃ x³ + ... + kₙ xⁿ
|
||||
# The polynomial is stored as an array of coefficients ordered from k₀ to kₙ
|
||||
|
||||
|
|
@ -94,13 +94,13 @@ const BLS12_381_h2c_G2_3_isogeny_map_xden* = [
|
|||
"0xc",
|
||||
"0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2a0f6b0f6241eabfffeb153ffffb9feffffffffaa9f"
|
||||
),
|
||||
# x^2
|
||||
# x²
|
||||
Fp2[BLS12_381].fromHex(
|
||||
"0x1",
|
||||
"0x0"
|
||||
)
|
||||
]
|
||||
const BLS12_381_h2c_G2_3_isogeny_map_ynum* = [
|
||||
const BLS12_381_h2c_G2_isogeny_map_ynum* = [
|
||||
# Polynomial k₀ + k₁ x + k₂ x² + k₃ x³ + ... + kₙ xⁿ
|
||||
# The polynomial is stored as an array of coefficients ordered from k₀ to kₙ
|
||||
|
||||
|
|
@ -114,18 +114,18 @@ const BLS12_381_h2c_G2_3_isogeny_map_ynum* = [
|
|||
"0x0",
|
||||
"0x5c759507e8e333ebb5b7a9a47d7ed8532c52d39fd3a042a88b58423c50ae15d5c2638e343d9c71c6238aaaaaaaa97be"
|
||||
),
|
||||
# x^2*y
|
||||
# x²*y
|
||||
Fp2[BLS12_381].fromHex(
|
||||
"0x11560bf17baa99bc32126fced787c88f984f87adf7ae0c7f9a208c6b4f20a4181472aaa9cb8d555526a9ffffffffc71c",
|
||||
"0x8ab05f8bdd54cde190937e76bc3e447cc27c3d6fbd7063fcd104635a790520c0a395554e5c6aaaa9354ffffffffe38f"
|
||||
),
|
||||
# x^3*y
|
||||
# x³*y
|
||||
Fp2[BLS12_381].fromHex(
|
||||
"0x124c9ad43b6cf79bfbf7043de3811ad0761b0f37a1e26286b0e977c69aa274524e79097a56dc4bd9e1b371c71c718b10",
|
||||
"0x0"
|
||||
)
|
||||
]
|
||||
const BLS12_381_h2c_G2_3_isogeny_map_yden* = [
|
||||
const BLS12_381_h2c_G2_isogeny_map_yden* = [
|
||||
# Polynomial k₀ + k₁ x + k₂ x² + k₃ x³ + ... + kₙ xⁿ
|
||||
# The polynomial is stored as an array of coefficients ordered from k₀ to kₙ
|
||||
|
||||
|
|
@ -139,12 +139,12 @@ const BLS12_381_h2c_G2_3_isogeny_map_yden* = [
|
|||
"0x0",
|
||||
"0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2a0f6b0f6241eabfffeb153ffffb9feffffffffa9d3"
|
||||
),
|
||||
# x^2
|
||||
# x²
|
||||
Fp2[BLS12_381].fromHex(
|
||||
"0x12",
|
||||
"0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2a0f6b0f6241eabfffeb153ffffb9feffffffffaa99"
|
||||
),
|
||||
# x^3
|
||||
# x³
|
||||
Fp2[BLS12_381].fromHex(
|
||||
"0x1",
|
||||
"0x0"
|
||||
|
|
@ -9,7 +9,9 @@
|
|||
import
|
||||
std/macros,
|
||||
../config/curves,
|
||||
./bls12_381_g2_hash_to_curve
|
||||
../elliptic/ec_shortweierstrass_affine,
|
||||
./bls12_381_hash_to_curve_g1,
|
||||
./bls12_381_hash_to_curve_g2
|
||||
|
||||
{.experimental: "dynamicBindSym".}
|
||||
|
||||
|
|
@ -19,11 +21,9 @@ macro h2cConst*(C: static Curve, group, value: untyped): untyped =
|
|||
return bindSym($C & "_h2c_" & $group & "_" & $value)
|
||||
|
||||
macro h2cIsomapPoly*(C: static Curve,
|
||||
group: untyped,
|
||||
isodegree: static int,
|
||||
group: static Subgroup,
|
||||
value: untyped): untyped =
|
||||
## Get an isogeny map polynomial
|
||||
## for mapping to a elliptic curve group (G1 or G2)
|
||||
return bindSym($C & "_h2c_" &
|
||||
$group & "_" & $isodegree &
|
||||
"_isogeny_map_" & $value)
|
||||
$group & "_isogeny_map_" & $value)
|
||||
|
|
|
|||
|
|
@ -25,7 +25,7 @@ export
|
|||
func clearCofactor*[ECP](P: var ECP) {.inline.} =
|
||||
## Clear the cofactor of a point on the curve
|
||||
## From a point on the curve, returns a point on the subgroup of order r
|
||||
when ECP.F.C in {BN254_Nogami, BN254_SNarks, BLS12_377, BLS12_381}:
|
||||
when ECP.F.C in {BN254_Nogami, BN254_Snarks, BLS12_377, BLS12_381}:
|
||||
P.clearCofactorFast()
|
||||
else:
|
||||
P.clearCofactorReference()
|
||||
|
|
|
|||
|
|
@ -13,6 +13,7 @@
|
|||
# ############################################################
|
||||
|
||||
import
|
||||
./arithmetic,
|
||||
elliptic/[
|
||||
ec_shortweierstrass_affine,
|
||||
ec_shortweierstrass_jacobian,
|
||||
|
|
|
|||
|
|
@ -465,6 +465,9 @@ func appendHex*(dst: var string, big: BigInt, order: static Endianness = bigEndi
|
|||
# 2 Convert canonical uint to hex
|
||||
dst.add bytes.nativeEndianToHex(order)
|
||||
|
||||
func toHex*(a: openArray[byte]): string =
|
||||
nativeEndianToHex(a, system.cpuEndian)
|
||||
|
||||
func toHex*(big: BigInt, order: static Endianness = bigEndian): string =
|
||||
## Stringify an int to hex.
|
||||
## Note. Leading zeros are not removed.
|
||||
|
|
|
|||
|
|
@ -115,6 +115,127 @@ def find_z_sswu(F, A, B):
|
|||
return Z_cand
|
||||
ctr += 1
|
||||
|
||||
# BLS12-381 G1
|
||||
# ---------------------------------------------------------
|
||||
# Hardcoding from spec:
|
||||
# - https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-hash-to-curve-11#section-8.8.1
|
||||
# - https://github.com/cfrg/draft-irtf-cfrg-hash-to-curve/blob/f7dd3761/poc/sswu_opt_3mod4.sage#L126-L132
|
||||
|
||||
def genBLS12381G1_H2C_constants(curve_config):
|
||||
curve_name = 'BLS12_381'
|
||||
|
||||
# ------------------------------------------
|
||||
p = curve_config[curve_name]['field']['modulus']
|
||||
Fp = GF(p)
|
||||
K.<u> = PolynomialRing(Fp)
|
||||
# ------------------------------------------
|
||||
|
||||
# Hash to curve isogenous curve parameters
|
||||
# y² = x³ + A'*x + B'
|
||||
|
||||
print('\n----> Hash-to-Curve map to isogenous BLS12-381 E\'1 <----\n')
|
||||
buf = inspect.cleandoc(f"""
|
||||
# Hash-to-Curve map to isogenous BLS12-381 E'1 constants
|
||||
# -----------------------------------------------------------------
|
||||
#
|
||||
# y² = x³ + A'*x + B' with p ≡ 3 (mod 4) the BLS12-381 characteristic (base modulus)
|
||||
#
|
||||
# Hardcoding from spec:
|
||||
# - https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-hash-to-curve-11#section-8.8.1
|
||||
# - https://github.com/cfrg/draft-irtf-cfrg-hash-to-curve/blob/f7dd3761/poc/sswu_opt_3mod4.sage#L126-L132
|
||||
""")
|
||||
buf += '\n\n'
|
||||
|
||||
# Base constants
|
||||
Aprime_E1 = Fp('0x144698a3b8e9433d693a02c96d4982b0ea985383ee66a8d8e8981aefd881ac98936f8da0e0f97f5cf428082d584c1d')
|
||||
Bprime_E1 = Fp('0x12e2908d11688030018b12e8753eee3b2016c1f0f24f4070a0b9c14fcef35ef55a23215a316ceaa5d1cc48e98e172be0')
|
||||
Z = Fp(11)
|
||||
# Extra
|
||||
minus_A = -Aprime_E1
|
||||
ZmulA = Z * Aprime_E1
|
||||
sqrt_minus_Z3 = sqrt(-Z^3)
|
||||
|
||||
buf += f'const {curve_name}_h2c_G1_Aprime_E1* = '
|
||||
buf += field_to_nim(Aprime_E1, 'Fp', curve_name)
|
||||
buf += '\n'
|
||||
|
||||
buf += f'const {curve_name}_h2c_G1_Bprime_E1* = '
|
||||
buf += field_to_nim(Bprime_E1, 'Fp', curve_name)
|
||||
buf += '\n'
|
||||
|
||||
buf += f'const {curve_name}_h2c_G1_Z* = '
|
||||
buf += field_to_nim(Z, 'Fp', curve_name)
|
||||
buf += '\n'
|
||||
|
||||
buf += f'const {curve_name}_h2c_G1_minus_A* = '
|
||||
buf += field_to_nim(minus_A, 'Fp', curve_name)
|
||||
buf += '\n'
|
||||
|
||||
buf += f'const {curve_name}_h2c_G1_ZmulA* = '
|
||||
buf += field_to_nim(ZmulA, 'Fp', curve_name)
|
||||
buf += '\n'
|
||||
|
||||
buf += f'const {curve_name}_h2c_G1_sqrt_minus_Z3* = '
|
||||
buf += field_to_nim(sqrt_minus_Z3, 'Fp', curve_name)
|
||||
buf += '\n'
|
||||
|
||||
return buf
|
||||
|
||||
def genBLS12381G1_H2C_isogeny_map(curve_config):
|
||||
curve_name = 'BLS12_381'
|
||||
|
||||
# Hash to curve isogenous curve parameters
|
||||
# y² = x³ + A'*x + B'
|
||||
|
||||
print('\n----> Hash-to-Curve 3-isogeny map BLS12-381 E\'1 constants <----\n')
|
||||
buf = inspect.cleandoc(f"""
|
||||
# Hash-to-Curve 11-isogeny map BLS12-381 E'1 constants
|
||||
# -----------------------------------------------------------------
|
||||
#
|
||||
# The polynomials map a point (x', y') on the isogenous curve E'2
|
||||
# to (x, y) on E2, represented as (xnum/xden, y' * ynum/yden)
|
||||
|
||||
""")
|
||||
buf += '\n\n'
|
||||
|
||||
p = curve_config[curve_name]['field']['modulus']
|
||||
Fp = GF(p)
|
||||
|
||||
# Base constants - E1
|
||||
A = curve_config[curve_name]['curve']['a']
|
||||
B = curve_config[curve_name]['curve']['b']
|
||||
E1 = EllipticCurve(Fp, [A, B])
|
||||
|
||||
# Base constants - Isogenous curve E'1, degree 11
|
||||
Aprime_E1 = Fp('0x144698a3b8e9433d693a02c96d4982b0ea985383ee66a8d8e8981aefd881ac98936f8da0e0f97f5cf428082d584c1d')
|
||||
Bprime_E1 = Fp('0x12e2908d11688030018b12e8753eee3b2016c1f0f24f4070a0b9c14fcef35ef55a23215a316ceaa5d1cc48e98e172be0')
|
||||
Eprime1 = EllipticCurve(Fp, [Aprime_E1, Bprime_E1])
|
||||
|
||||
iso = EllipticCurveIsogeny(E=E1, kernel=None, codomain=Eprime1, degree=11).dual()
|
||||
if (- iso.rational_maps()[1])(1, 1) > iso.rational_maps()[1](1, 1):
|
||||
iso.switch_sign()
|
||||
|
||||
(xm, ym) = iso.rational_maps()
|
||||
maps = (xm.numerator(), xm.denominator(), ym.numerator(), ym.denominator())
|
||||
|
||||
buf += dump_poly(
|
||||
'BLS12_381_h2c_G1_11_isogeny_map_xnum',
|
||||
xm.numerator(), 'Fp', curve_name)
|
||||
buf += '\n'
|
||||
buf += dump_poly(
|
||||
'BLS12_381_h2c_G1_11_isogeny_map_xden',
|
||||
xm.denominator(), 'Fp', curve_name)
|
||||
buf += '\n'
|
||||
buf += dump_poly(
|
||||
'BLS12_381_h2c_G1_11_isogeny_map_ynum',
|
||||
ym.numerator(), 'Fp', curve_name)
|
||||
buf += '\n'
|
||||
buf += dump_poly(
|
||||
'BLS12_381_h2c_G1_11_isogeny_map_yden',
|
||||
ym.denominator(), 'Fp', curve_name)
|
||||
|
||||
return buf
|
||||
|
||||
# BLS12-381 G2
|
||||
# ---------------------------------------------------------
|
||||
# Hardcoding from spec:
|
||||
|
|
@ -303,12 +424,32 @@ if __name__ == "__main__":
|
|||
curve = args.curve[0]
|
||||
group = args.curve[1]
|
||||
|
||||
if curve == 'BLS12_381' and group == 'G2':
|
||||
if curve == 'BLS12_381' and group == 'G1':
|
||||
h2c = genBLS12381G1_H2C_constants(Curves)
|
||||
h2c += '\n\n'
|
||||
h2c += genBLS12381G1_H2C_isogeny_map(Curves)
|
||||
|
||||
with open(f'{curve.lower()}_hash_to_curve_g1.nim', 'w') as f:
|
||||
f.write(copyright())
|
||||
f.write('\n\n')
|
||||
|
||||
f.write(inspect.cleandoc("""
|
||||
import
|
||||
../config/curves,
|
||||
../io/io_fields
|
||||
"""))
|
||||
|
||||
f.write('\n\n')
|
||||
f.write(h2c)
|
||||
|
||||
print(f'Successfully created {curve.lower()}_hash_to_curve_g1.nim')
|
||||
|
||||
elif curve == 'BLS12_381' and group == 'G2':
|
||||
h2c = genBLS12381G2_H2C_constants(Curves)
|
||||
h2c += '\n\n'
|
||||
h2c += genBLS12381G2_H2C_isogeny_map(Curves)
|
||||
|
||||
with open(f'{curve.lower()}_g2_hash_to_curve.nim', 'w') as f:
|
||||
with open(f'{curve.lower()}_hash_to_curve_g2.nim', 'w') as f:
|
||||
f.write(copyright())
|
||||
f.write('\n\n')
|
||||
|
||||
|
|
@ -321,7 +462,7 @@ if __name__ == "__main__":
|
|||
f.write('\n\n')
|
||||
f.write(h2c)
|
||||
|
||||
print(f'Successfully created {curve.lower()}_g2_hash_to_curve.nim')
|
||||
print(f'Successfully created {curve.lower()}_hash_to_curve_g2.nim')
|
||||
else:
|
||||
raise ValueError(
|
||||
curve + group +
|
||||
|
|
|
|||
|
|
@ -143,6 +143,12 @@ echo "Hash-to-curve" & '\n'
|
|||
|
||||
# Hash-to-curve v8 to latest
|
||||
# https://github.com/cfrg/draft-irtf-cfrg-hash-to-curve/blob/draft-irtf-cfrg-hash-to-curve-10/poc/vectors/BLS12381G2_XMD:SHA-256_SSWU_RO_.json
|
||||
run_hash_to_curve_test(
|
||||
ECP_ShortW_Prj[Fp[BLS12_381], G1],
|
||||
"v8",
|
||||
"tv_h2c_v8_BLS12_381_hash_to_G1_SHA256_SSWU_RO.json"
|
||||
)
|
||||
|
||||
run_hash_to_curve_test(
|
||||
ECP_ShortW_Prj[Fp2[BLS12_381], G2],
|
||||
"v8",
|
||||
|
|
@ -151,6 +157,12 @@ run_hash_to_curve_test(
|
|||
|
||||
# Hash-to-curve v7 (different domain separation tag)
|
||||
# https://github.com/cfrg/draft-irtf-cfrg-hash-to-curve/blob/draft-irtf-cfrg-hash-to-curve-07/poc/vectors/BLS12381G2_XMD:SHA-256_SSWU_RO_.json
|
||||
run_hash_to_curve_test(
|
||||
ECP_ShortW_Prj[Fp[BLS12_381], G1],
|
||||
"v7",
|
||||
"tv_h2c_v7_BLS12_381_hash_to_G1_SHA256_SSWU_RO.json"
|
||||
)
|
||||
|
||||
run_hash_to_curve_test(
|
||||
ECP_ShortW_Prj[Fp2[BLS12_381], G2],
|
||||
"v7",
|
||||
|
|
|
|||
|
|
@ -0,0 +1,96 @@
|
|||
{
|
||||
"L": "0x40",
|
||||
"Z": "0xb",
|
||||
"ciphersuite": "BLS12381G1_XMD:SHA-256_SSWU_RO_",
|
||||
"curve": "BLS12381G1",
|
||||
"dst": "BLS12381G1_XMD:SHA-256_SSWU_RO_TESTGEN",
|
||||
"expand": "XMD",
|
||||
"field": {
|
||||
"m": "0x1",
|
||||
"p": "0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2a0f6b0f6241eabfffeb153ffffb9feffffffffaaab"
|
||||
},
|
||||
"hash": "sha256",
|
||||
"k": "0x80",
|
||||
"map": {
|
||||
"name": "SSWU"
|
||||
},
|
||||
"randomOracle": true,
|
||||
"vectors": [
|
||||
{
|
||||
"P": {
|
||||
"x": "0x0576730ab036cbac1d95b38dca905586f28d0a59048db4e8778782d89bff856ddef89277ead5a21e2975c4a6e3d8c79e",
|
||||
"y": "0x1273e568bebf1864393c517f999b87c1eaa1b8432f95aea8160cd981b5b05d8cd4a7cf00103b6ef87f728e4b547dd7ae"
|
||||
},
|
||||
"Q0": {
|
||||
"x": "0x0b63f31bcc08df890f35ee362c8538fac22cf22637aa2ba22d9c85bc1bda995926ab690d86830bf8ae06f4d537ccf6d7",
|
||||
"y": "0x0666f3763cc7b223ab237e313f6474c9a3c2f5ed985ee8d1faa0928b4b428ec1a366226125ce8f415edb3f706e71d80e"
|
||||
},
|
||||
"Q1": {
|
||||
"x": "0x0362c0f9d6cf4b73309a16b439d096b3ead588ab03cff57daf56fe747ab6d7774d5bfc0bd0a55bbeb0f05ec25cc191f6",
|
||||
"y": "0x18d279b38babbd69aa176031655d138a731c049385aeef6eff3bf80e45ebcad0a941cdfc135e9ea1690a25eb6eac38e5"
|
||||
},
|
||||
"msg": "",
|
||||
"u": [
|
||||
"0x0633af2b38973d1cfb6e905292c41f209fe52e5be989b5e0d32c06a0e3c23e4843927cb8289b440f3cde0da46dc9ba0d",
|
||||
"0x022474974e47d74c495de648eff1c8e4fabbae0d8ce3e30e3d1a5f9386cdf2582f78df056342d59ccca34321d93ef13d"
|
||||
]
|
||||
},
|
||||
{
|
||||
"P": {
|
||||
"x": "0x061daf0cc00d8912dac1d4cf5a7c32fca97f8b3bf3f805121888e5eb89f77f9a9f406569027ac6d0e61b1229f42c43d6",
|
||||
"y": "0x0de1601e5ba02cb637c1d35266f5700acee9850796dc88e860d022d7b9e7e3dce5950952e97861e5bb16d215c87f030d"
|
||||
},
|
||||
"Q0": {
|
||||
"x": "0x0e8334d819ca7fad50979a487e0bc95cb1410914f1d760f842fc3dd0102755e7ca81b0356da7b9771ab11bf50efbca67",
|
||||
"y": "0x120397edf7002610f907c2d4ecfcc4e817f1f8915becb5919510796bf595d854048461662ad960347216b00dfc79db38"
|
||||
},
|
||||
"Q1": {
|
||||
"x": "0x013e1240e4da2abda009e263089cb8e57f1b24d0d1df09f644cc9c9a8b3fde7d154c7f1b0895a0af22b902a8140fb3ce",
|
||||
"y": "0x0d6a9f75f2088dcac8f8ec0ab94bf2dac23b7b832bf23c91f9241f753c5831054b058192351a972347cb19806e78477d"
|
||||
},
|
||||
"msg": "abc",
|
||||
"u": [
|
||||
"0x07df547923a0c77ddc4fea1a8a2eb156aef1746d5452239a55a378c5d3590e0b75cddff0eef2a9214a41923f2be27b55",
|
||||
"0x0f95fd8f00e25c3073ec07f249a7d527e580f01a6986158aeb064ed831d544fb9c5dbceb6604c908db5430d8f3d1c4f3"
|
||||
]
|
||||
},
|
||||
{
|
||||
"P": {
|
||||
"x": "0x0fb3455436843e76079c7cf3dfef75e5a104dfe257a29a850c145568d500ad31ccfe79be9ae0ea31a722548070cf98cd",
|
||||
"y": "0x177989f7e2c751658df1b26943ee829d3ebcf131d8f805571712f3a7527ee5334ecff8a97fc2a50cea86f5e6212e9a57"
|
||||
},
|
||||
"Q0": {
|
||||
"x": "0x03ff794b445b926906b2fa710ba5db9f7b8689429a1630ab672854b5ba1a7c59bf3667d64aa63824a8798dcb631bfa9a",
|
||||
"y": "0x1581711cffadabb6136f4bf57749e04b92787c7486da6b6da1fa758655c9af275b23540370d9f3987a100f0d3dc8e6db"
|
||||
},
|
||||
"Q1": {
|
||||
"x": "0x133bdea6715b4ef780693cd0055025b221becc8e04506a776484590df9b43af62ef402778a9c98ec540bc293e9741565",
|
||||
"y": "0x0d953a5bdb2d16e62bfeab742e70ea64fddd83e8210b2416d40a02b0d90986fd0d00a3d77751ac467964ecc037dc284a"
|
||||
},
|
||||
"msg": "abcdef0123456789",
|
||||
"u": [
|
||||
"0x0a5d2ed6108aa08d652ab61af11c12d8750ed179cb962779c7b5393f219ad4b78b7b252a2896a341ad451e93f1904fb0",
|
||||
"0x17d6cd69f4bd29b85c550539a296c76ced075d9d39a81f4cdc2804a7184ff9ea4a5a85dac4a2a61e317894d0fba55740"
|
||||
]
|
||||
},
|
||||
{
|
||||
"P": {
|
||||
"x": "0x0514af2137c1ae1d78d5cb97ee606ea142824c199f0f25ac463a0c78200de57640d34686521d3e9cf6b3721834f8a038",
|
||||
"y": "0x047a85d6898416a0899e26219bca7c4f0fa682717199de196b02b95eaf9fb55456ac3b810e78571a1b7f5692b7c58ab6"
|
||||
},
|
||||
"Q0": {
|
||||
"x": "0x17dc55e956f1e24c800fa6a61ccba179fbba6bbe27e96ab16b862defa4782d567f8733f1ee39acd7b665ba0204318d7f",
|
||||
"y": "0x0252510413c32677817c4ee5f84e4c8f66721e489913a50eac550f41ad48b01b763ec9eed5bd68bcac76131ca9ebd741"
|
||||
},
|
||||
"Q1": {
|
||||
"x": "0x03d2f5444d39ed19b6087cd684d2a72795038d1ffe9ab120f7e5ce41fb48af76bba3eb4efc8a696418fcb3c5cfdbfd94",
|
||||
"y": "0x061a1fe191cbf373b74d5f642148722160b5524dd2c06a49c5e4d5480966de5b0854d53cbea144b482fa687eaf9fdc66"
|
||||
},
|
||||
"msg": "a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa",
|
||||
"u": [
|
||||
"0x111e524a9da7ae49a1cf6b03f5bc9d374f16951dcba59d03529a94afd4e5ba171fb1dffa373d13993503d594abd1b5ed",
|
||||
"0x006ad90d8d5101c88db3923376f2a33ff922ba39342d1a5462785796b6ebdec10dedb5e9cd9ff8e611af939f7f617844"
|
||||
]
|
||||
}
|
||||
]
|
||||
}
|
||||
|
|
@ -0,0 +1,115 @@
|
|||
{
|
||||
"L": "0x40",
|
||||
"Z": "0xb",
|
||||
"ciphersuite": "BLS12381G1_XMD:SHA-256_SSWU_RO_",
|
||||
"curve": "BLS12-381 G1",
|
||||
"dst": "QUUX-V01-CS02-with-BLS12381G1_XMD:SHA-256_SSWU_RO_",
|
||||
"expand": "XMD",
|
||||
"field": {
|
||||
"m": "0x1",
|
||||
"p": "0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2a0f6b0f6241eabfffeb153ffffb9feffffffffaaab"
|
||||
},
|
||||
"hash": "sha256",
|
||||
"k": "0x80",
|
||||
"map": {
|
||||
"name": "SSWU"
|
||||
},
|
||||
"randomOracle": true,
|
||||
"vectors": [
|
||||
{
|
||||
"P": {
|
||||
"x": "0x052926add2207b76ca4fa57a8734416c8dc95e24501772c814278700eed6d1e4e8cf62d9c09db0fac349612b759e79a1",
|
||||
"y": "0x08ba738453bfed09cb546dbb0783dbb3a5f1f566ed67bb6be0e8c67e2e81a4cc68ee29813bb7994998f3eae0c9c6a265"
|
||||
},
|
||||
"Q0": {
|
||||
"x": "0x11a3cce7e1d90975990066b2f2643b9540fa40d6137780df4e753a8054d07580db3b7f1f03396333d4a359d1fe3766fe",
|
||||
"y": "0x0eeaf6d794e479e270da10fdaf768db4c96b650a74518fc67b04b03927754bac66f3ac720404f339ecdcc028afa091b7"
|
||||
},
|
||||
"Q1": {
|
||||
"x": "0x160003aaf1632b13396dbad518effa00fff532f604de1a7fc2082ff4cb0afa2d63b2c32da1bef2bf6c5ca62dc6b72f9c",
|
||||
"y": "0x0d8bb2d14e20cf9f6036152ed386d79189415b6d015a20133acb4e019139b94e9c146aaad5817f866c95d609a361735e"
|
||||
},
|
||||
"msg": "",
|
||||
"u": [
|
||||
"0x0ba14bd907ad64a016293ee7c2d276b8eae71f25a4b941eece7b0d89f17f75cb3ae5438a614fb61d6835ad59f29c564f",
|
||||
"0x019b9bd7979f12657976de2884c7cce192b82c177c80e0ec604436a7f538d231552f0d96d9f7babe5fa3b19b3ff25ac9"
|
||||
]
|
||||
},
|
||||
{
|
||||
"P": {
|
||||
"x": "0x03567bc5ef9c690c2ab2ecdf6a96ef1c139cc0b2f284dca0a9a7943388a49a3aee664ba5379a7655d3c68900be2f6903",
|
||||
"y": "0x0b9c15f3fe6e5cf4211f346271d7b01c8f3b28be689c8429c85b67af215533311f0b8dfaaa154fa6b88176c229f2885d"
|
||||
},
|
||||
"Q0": {
|
||||
"x": "0x125435adce8e1cbd1c803e7123f45392dc6e326d292499c2c45c5865985fd74fe8f042ecdeeec5ecac80680d04317d80",
|
||||
"y": "0x0e8828948c989126595ee30e4f7c931cbd6f4570735624fd25aef2fa41d3f79cfb4b4ee7b7e55a8ce013af2a5ba20bf2"
|
||||
},
|
||||
"Q1": {
|
||||
"x": "0x11def93719829ecda3b46aa8c31fc3ac9c34b428982b898369608e4f042babee6c77ab9218aad5c87ba785481eff8ae4",
|
||||
"y": "0x0007c9cef122ccf2efd233d6eb9bfc680aa276652b0661f4f820a653cec1db7ff69899f8e52b8e92b025a12c822a6ce6"
|
||||
},
|
||||
"msg": "abc",
|
||||
"u": [
|
||||
"0x0d921c33f2bad966478a03ca35d05719bdf92d347557ea166e5bba579eea9b83e9afa5c088573c2281410369fbd32951",
|
||||
"0x003574a00b109ada2f26a37a91f9d1e740dffd8d69ec0c35e1e9f4652c7dba61123e9dd2e76c655d956e2b3462611139"
|
||||
]
|
||||
},
|
||||
{
|
||||
"P": {
|
||||
"x": "0x11e0b079dea29a68f0383ee94fed1b940995272407e3bb916bbf268c263ddd57a6a27200a784cbc248e84f357ce82d98",
|
||||
"y": "0x03a87ae2caf14e8ee52e51fa2ed8eefe80f02457004ba4d486d6aa1f517c0889501dc7413753f9599b099ebcbbd2d709"
|
||||
},
|
||||
"Q0": {
|
||||
"x": "0x08834484878c217682f6d09a4b51444802fdba3d7f2df9903a0ddadb92130ebbfa807fffa0eabf257d7b48272410afff",
|
||||
"y": "0x0b318f7ecf77f45a0f038e62d7098221d2dbbca2a394164e2e3fe953dc714ac2cde412d8f2d7f0c03b259e6795a2508e"
|
||||
},
|
||||
"Q1": {
|
||||
"x": "0x158418ed6b27e2549f05531a8281b5822b31c3bf3144277fbb977f8d6e2694fedceb7011b3c2b192f23e2a44b2bd106e",
|
||||
"y": "0x1879074f344471fac5f839e2b4920789643c075792bec5af4282c73f7941cda5aa77b00085eb10e206171b9787c4169f"
|
||||
},
|
||||
"msg": "abcdef0123456789",
|
||||
"u": [
|
||||
"0x062d1865eb80ebfa73dcfc45db1ad4266b9f3a93219976a3790ab8d52d3e5f1e62f3b01795e36834b17b70e7b76246d4",
|
||||
"0x0cdc3e2f271f29c4ff75020857ce6c5d36008c9b48385ea2f2bf6f96f428a3deb798aa033cd482d1cdc8b30178b08e3a"
|
||||
]
|
||||
},
|
||||
{
|
||||
"P": {
|
||||
"x": "0x15f68eaa693b95ccb85215dc65fa81038d69629f70aeee0d0f677cf22285e7bf58d7cb86eefe8f2e9bc3f8cb84fac488",
|
||||
"y": "0x1807a1d50c29f430b8cafc4f8638dfeeadf51211e1602a5f184443076715f91bb90a48ba1e370edce6ae1062f5e6dd38"
|
||||
},
|
||||
"Q0": {
|
||||
"x": "0x0cbd7f84ad2c99643fea7a7ac8f52d63d66cefa06d9a56148e58b984b3dd25e1f41ff47154543343949c64f88d48a710",
|
||||
"y": "0x052c00e4ed52d000d94881a5638ae9274d3efc8bc77bc0e5c650de04a000b2c334a9e80b85282a00f3148dfdface0865"
|
||||
},
|
||||
"Q1": {
|
||||
"x": "0x06493fb68f0d513af08be0372f849436a787e7b701ae31cb964d968021d6ba6bd7d26a38aaa5a68e8c21a6b17dc8b579",
|
||||
"y": "0x02e98f2ccf5802b05ffaac7c20018bc0c0b2fd580216c4aa2275d2909dc0c92d0d0bdc979226adeb57a29933536b6bb4"
|
||||
},
|
||||
"msg": "q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq",
|
||||
"u": [
|
||||
"0x010476f6a060453c0b1ad0b628f3e57c23039ee16eea5e71bb87c3b5419b1255dc0e5883322e563b84a29543823c0e86",
|
||||
"0x0b1a912064fb0554b180e07af7e787f1f883a0470759c03c1b6509eb8ce980d1670305ae7b928226bb58fdc0a419f46e"
|
||||
]
|
||||
},
|
||||
{
|
||||
"P": {
|
||||
"x": "0x082aabae8b7dedb0e78aeb619ad3bfd9277a2f77ba7fad20ef6aabdc6c31d19ba5a6d12283553294c1825c4b3ca2dcfe",
|
||||
"y": "0x05b84ae5a942248eea39e1d91030458c40153f3b654ab7872d779ad1e942856a20c438e8d99bc8abfbf74729ce1f7ac8"
|
||||
},
|
||||
"Q0": {
|
||||
"x": "0x0cf97e6dbd0947857f3e578231d07b309c622ade08f2c08b32ff372bd90db19467b2563cc997d4407968d4ac80e154f8",
|
||||
"y": "0x127f0cddf2613058101a5701f4cb9d0861fd6c2a1b8e0afe194fccf586a3201a53874a2761a9ab6d7220c68661a35ab3"
|
||||
},
|
||||
"Q1": {
|
||||
"x": "0x092f1acfa62b05f95884c6791fba989bbe58044ee6355d100973bf9553ade52b47929264e6ae770fb264582d8dce512a",
|
||||
"y": "0x028e6d0169a72cfedb737be45db6c401d3adfb12c58c619c82b93a5dfcccef12290de530b0480575ddc8397cda0bbebf"
|
||||
},
|
||||
"msg": "a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa",
|
||||
"u": [
|
||||
"0x0a8ffa7447f6be1c5a2ea4b959c9454b431e29ccc0802bc052413a9c5b4f9aac67a93431bd480d15be1e057c8a08e8c6",
|
||||
"0x05d487032f602c90fa7625dbafe0f4a49ef4a6b0b33d7bb349ff4cf5410d297fd6241876e3e77b651cfc8191e40a68b7"
|
||||
]
|
||||
}
|
||||
]
|
||||
}
|
||||
Loading…
Reference in New Issue