203 lines
5.3 KiB
Python
203 lines
5.3 KiB
Python
# Constantine
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# Copyright (c) 2018-2019 Status Research & Development GmbH
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# Copyright (c) 2020-Present Mamy André-Ratsimbazafy
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# Licensed and distributed under either of
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# * MIT license (license terms in the root directory or at http://opensource.org/licenses/MIT).
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# * Apache v2 license (license terms in the root directory or at http://www.apache.org/licenses/LICENSE-2.0).
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# at your option. This file may not be copied, modified, or distributed except according to those terms.
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# ############################################################
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#
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# BN254 GLV Endomorphism
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# Lattice Decomposition
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#
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# ############################################################
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# Parameters
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u = -(2^63 + 2^62 + 2^60 + 2^57 + 2^48 + 2^16)
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p = (u - 1)^2 * (u^4 - u^2 + 1)//3 + u
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r = u^4 - u^2 + 1
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cofactor = Integer('0x396c8c005555e1568c00aaab0000aaab')
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print('p : ' + p.hex())
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print('r : ' + r.hex())
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# Cube root of unity (mod r) formula for any BLS12 curves
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lambda1_r = u^2 - 1
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assert lambda1_r^3 % r == 1
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print('λᵩ1 : ' + lambda1_r.hex())
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print('λᵩ1+r: ' + (lambda1_r+r).hex())
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lambda2_r = u^4
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assert lambda2_r^3 % r == 1
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print('λᵩ2 : ' + lambda2_r.hex())
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# Finite fields
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F = GF(p)
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# K2.<u> = PolynomialRing(F)
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# F2.<beta> = F.extension(u^2+9)
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# K6.<v> = PolynomialRing(F2)
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# F6.<eta> = F2.extension(v^3-beta)
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# K12.<w> = PolynomialRing(F6)
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# K12.<gamma> = F6.extension(w^2-eta)
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# Curves
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b = 4
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G1 = EllipticCurve(F, [0, b])
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# G2 = EllipticCurve(F2, [0, b*beta])
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(phi1, phi2) = (root for root in GF(p)(1).nth_root(3, all=True) if root != 1)
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print('𝜑1 :' + Integer(phi1).hex())
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print('𝜑2 :' + Integer(phi2).hex())
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assert phi1^3 % p == 1
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assert phi2^3 % p == 1
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# Test generator
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set_random_seed(1337)
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# Check
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def checkEndo():
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Prand = G1.random_point()
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assert Prand != G1([0, 1, 0]) # Infinity
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# Clear cofactor
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P = Prand * cofactor
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(Px, Py, Pz) = P
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Qendo1 = G1([Px*phi1 % p, Py, Pz])
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Qendo2 = G1([Px*phi2 % p, Py, Pz])
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Q1 = lambda1_r * P
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Q2 = lambda2_r * P
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assert P != Q1
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assert P != Q2
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assert (F(Px)*F(phi1))^3 == F(Px)^3
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assert (F(Px)*F(phi2))^3 == F(Px)^3
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assert Q1 == Qendo2
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assert Q2 == Qendo2
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print('Endomorphism OK with 𝜑2')
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checkEndo()
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# Lattice
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b = [
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[u^2-1, -1],
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[1, u^2]
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]
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# Babai rounding
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ahat = [u^2, 1]
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v = int(r).bit_length()
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v = int(((v + 64 - 1) // 64) * 64) # round to next multiple of 64
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l = [Integer(a << v) // r for a in ahat]
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print('𝛼\u03051: ' + l[0].hex())
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print('𝛼\u03052: ' + l[1].hex())
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def getGLV2_decomp(scalar):
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a0 = (l[0] * scalar) >> v
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a1 = (l[1] * scalar) >> v
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k0 = scalar - a0 * b[0][0] - a1 * b[1][0]
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k1 = 0 - a0 * b[0][1] - a1 * b[1][1]
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assert int(k0).bit_length() <= (int(r).bit_length() + 1) // 2
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assert int(k1).bit_length() <= (int(r).bit_length() + 1) // 2
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assert scalar == (k0 + k1 * (lambda1_r % r)) % r
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assert scalar == (k0 + k1 * (lambda2_r % r)) % r
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return k0, k1
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def recodeScalars(k):
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m = 2
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L = ((int(r).bit_length() + m-1) // m) + 1 # l = ⌈log2 r/m⌉ + 1
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b = [[0] * L, [0] * L]
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b[0][L-1] = 0
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for i in range(0, L-1): # l-2 inclusive
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b[0][i] = 1 - ((k[0] >> (i+1)) & 1)
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for j in range(1, m):
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for i in range(0, L):
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b[j][i] = k[j] & 1
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k[j] = k[j]//2 + (b[j][i] & b[0][i])
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return b
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def buildLut(P0, P1):
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m = 2
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lut = [0] * (1 << (m-1))
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lut[0] = P0
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lut[1] = P0 + P1
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return lut
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def pointToString(P):
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(Px, Py, Pz) = P
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return '(x: ' + Integer(Px).hex() + ', y: ' + Integer(Py).hex() + ', z: ' + Integer(Pz).hex() + ')'
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def scalarMulGLV(scalar, P0):
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m = 2
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L = ((int(r).bit_length() + m-1) // m) + 1 # l = ⌈log2 r/m⌉ + 1
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print('L: ' + str(L))
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print('scalar: ' + Integer(scalar).hex())
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k0, k1 = getGLV2_decomp(scalar)
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print('k0: ' + k0.hex())
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print('k1: ' + k1.hex())
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P1 = (lambda1_r % r) * P0
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(Px, Py, Pz) = P0
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P1_endo = G1([Px*phi2 % p, Py, Pz])
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assert P1 == P1_endo
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expected = scalar * P0
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decomp = k0*P0 + k1*P1
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assert expected == decomp
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print('------ recode scalar -----------')
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even = k0 & 1 == 0
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if even:
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k0 += 1
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b = recodeScalars([k0, k1])
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print('b0: ' + str(list(reversed(b[0]))))
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print('b1: ' + str(list(reversed(b[1]))))
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print('------------ lut ---------------')
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lut = buildLut(P0, P1)
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print('------------ mul ---------------')
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# b[0][L-1] is always 0
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Q = lut[b[1][L-1]]
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for i in range(L-2, -1, -1):
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Q *= 2
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Q += (1 - 2 * b[0][i]) * lut[b[1][i]]
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if even:
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Q -= P0
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print('final Q: ' + pointToString(Q))
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print('expected: ' + pointToString(expected))
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assert Q == expected # TODO debug
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# Test generator
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set_random_seed(1337)
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for i in range(1):
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print('---------------------------------------')
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# scalar = randrange(r) # Pick an integer below curve order
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# P = G1.random_point()
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scalar = Integer('0xf7e60a832eb77ac47374bc93251360d6c81c21add62767ff816caf11a20d8db')
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P = G1([
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Integer('0xf9679bb02ee7f352fff6a6467a5e563ec8dd38c86a48abd9e8f7f241f1cdd29d54bc3ddea3a33b62e0d7ce22f3d244a'),
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Integer('0x50189b992cf856846b30e52205ff9ef72dc081e9680726586231cbc29a81a162120082585f401e00382d5c86fb1083f'),
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Integer(1)
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])
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scalarMulGLV(scalar, P)
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