diff --git a/bls-sigs.sage b/bls-sigs.sage new file mode 100644 index 0000000..7229fe8 --- /dev/null +++ b/bls-sigs.sage @@ -0,0 +1,63 @@ +# toy implementation of BLS signatures + +load("bls12-381.sage") +from hashlib import sha256 + +def hash(m): + h_output = sha256(str(m).encode('utf-8')) + return int(h_output.hexdigest(), 16) + +def hash_to_point(m): + # WARNING this hash-to-point approach should not be used! + h = hash(m) + return G2 * h + + +pairing = Pairing() + +class Signer: + def __init__(self): + self.sk = F1.random_element() + self.pk = self.sk * G1 + + def sign(self, m): + H = hash_to_point(m) + return self.sk * H + +def verify(pk, s, m): + H = hash_to_point(m) + return pairing.pair(G1, s) == pairing.pair(pk, H) + +def aggr(points): + R = 0 + for i in range(len(points)): + R = R + points[i] + return R + + +m = 1234 + +# single signature & verification +user0 = Signer() +s = user0.sign(m) +v = verify(user0.pk, s, m) +assert v + + +# BLS signature aggregation +n = 10 +users = [None]*n +pks = [None]*n +sigs = [None]*n +for i in range(n): + users[i] = Signer() + pks[i] = users[i].pk + sigs[i] = users[i].sign(m) + +# aggregate sigs & pks +s_aggr = aggr(sigs) +pk_aggr = aggr(pks) + +# verify aggregated signature +v = verify(pk_aggr, s_aggr, m) +assert v diff --git a/bls12-381.sage b/bls12-381.sage new file mode 100644 index 0000000..9bbcac4 --- /dev/null +++ b/bls12-381.sage @@ -0,0 +1,63 @@ +# The code of this file has been adapted from: +# https://github.com/osirislab/CSAW-CTF-2021-Finals/blob/main/crypto/aBoLiSh_taBLeS/chal.sage +# +# ## Example of usage: +# load("bls12-381.sage") +# pairing = Pairing() +# assert pairing.pair(G1 * 3, G2 * 2) == pairing.pair(G1, G2)^6 + + + +# BLS12-381 Parameters +# https://github.com/zkcrypto/bls12_381 +p = 0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2a0f6b0f6241eabfffeb153ffffb9feffffffffaaab +r = 0x73eda753299d7d483339d80809a1d80553bda402fffe5bfeffffffff00000001 +h1 = 0x396c8c005555e1568c00aaab0000aaab +h2 = 0x5d543a95414e7f1091d50792876a202cd91de4547085abaa68a205b2e5a7ddfa628f1cb4d9e82ef21537e293a6691ae1616ec6e786f0c70cf1c38e31c7238e5 + +# Define base fields +F1 = GF(p) +F2. = GF(p^2, x, x^2 + 1) +F12. = GF(p^12, x, x^12 - 2*x^6 + 2) + +# Define the Elliptic Curves +E1 = EllipticCurve(F1, [0, 4]) +E2 = EllipticCurve(F2, [0, 4*(1 + u)]) +E12 = EllipticCurve(F12, [0, 4]) + +# Generator of order r in E1 / F1 +G1x = 0x17f1d3a73197d7942695638c4fa9ac0fc3688c4f9774b905a14e3a3f171bac586c55e83ff97a1aeffb3af00adb22c6bb +G1y = 0x8b3f481e3aaa0f1a09e30ed741d8ae4fcf5e095d5d00af600db18cb2c04b3edd03cc744a2888ae40caa232946c5e7e1 +G1 = E1(G1x, G1y) + +# Generator of order r in E2 / F2 +G2x0 = 0x24aa2b2f08f0a91260805272dc51051c6e47ad4fa403b02b4510b647ae3d1770bac0326a805bbefd48056c8c121bdb8 +G2x1 = 0x13e02b6052719f607dacd3a088274f65596bd0d09920b61ab5da61bbdc7f5049334cf11213945d57e5ac7d055d042b7e +G2y0 = 0xce5d527727d6e118cc9cdc6da2e351aadfd9baa8cbdd3a76d429a695160d12c923ac9cc3baca289e193548608b82801 +G2y1 = 0x606c4a02ea734cc32acd2b02bc28b99cb3e287e85a763af267492ab572e99ab3f370d275cec1da1aaa9075ff05f79be +G2 = E2(G2x0 + u*G2x1, G2y0 + u*G2y1) + + +class Pairing(): + def lift_E1_to_E12(self, P): + """ + Lift point on E/F_q to E/F_{q^12} using the natural lift + """ + assert P.curve() == E1, "Attempting to lift a point from the wrong curve." + return E12(P) + + def lift_E2_to_E12(self, P): + """ + Lift point on E/F_{q^2} to E/F_{q_12} through the sextic twist + """ + assert P.curve() == E2, "Attempting to lift a point from the wrong curve." + xs, ys = [c.polynomial().coefficients() for c in (h2*P).xy()] + nx = F12(xs[0] - xs[1] + w ^ 6*xs[1]) + ny = F12(ys[0] - ys[1] + w ^ 6*ys[1]) + return E12(nx / (w ^ 2), ny / (w ^ 3)) + + def pair(self, A, B): + A = self.lift_E1_to_E12(A) + B = self.lift_E2_to_E12(B) + return A.ate_pairing(B, r, 12, E12.trace_of_frobenius()) +