376 lines
11 KiB
Python
376 lines
11 KiB
Python
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# 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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# BLS12-381 GLS Endomorphism
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# Lattice Decomposition
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#
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# ############################################################
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# Parameters
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x = 3 * 2^46 * (7 * 13 * 499) + 1
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p = (x - 1)^2 * (x^4 - x^2 + 1)//3 + x
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r = x^4 - x^2 + 1
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t = x+1
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print(' Prime modulus p: 0x' + p.hex())
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print(' Curve order r: 0x' + r.hex())
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print(' trace t: 0x' + t.hex())
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# Finite fields
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Fp = GF(p)
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K2.<u> = PolynomialRing(Fp)
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Fp2.<beta> = Fp.extension(u^2+5)
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SNR = Fp2([0, 1]) # Sextic Non-Residue for Sextic Twist
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# Curves
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b = 1
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G1 = EllipticCurve(Fp, [0, b])
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G2 = EllipticCurve(Fp2, [0, b/SNR])
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# https://crypto.stackexchange.com/questions/64064/order-of-twisted-curve-in-pairings
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# https://math.stackexchange.com/questions/144194/how-to-find-the-order-of-elliptic-curve-over-finite-field-extension
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cofactorG1 = G1.order() // r
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cofactorG2 = G2.order() // r
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print('')
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print('cofactor G1: ' + cofactorG1.hex())
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print('cofactor G2: ' + cofactorG2.hex())
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print('')
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# Frobenius constants (D type: use SNR, M type use 1/SNR)
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FrobConst_psi = SNR^((p-1)/6)
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FrobConst_psi_2 = FrobConst_psi * FrobConst_psi
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FrobConst_psi_3 = FrobConst_psi_2 * FrobConst_psi
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FrobConst_psi2_2 = FrobConst_psi_2 * FrobConst_psi_2^p
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FrobConst_psi2_3 = FrobConst_psi_3 * FrobConst_psi_3^p
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def psi(P):
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(Px, Py, Pz) = P
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return G2([
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FrobConst_psi_2 * Px.frobenius(1),
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FrobConst_psi_3 * Py.frobenius(1)
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# Pz.frobenius() - Always 1 after extract
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])
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def psi2(P):
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(Px, Py, Pz) = P
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return G2([
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FrobConst_psi2_2 * Px.frobenius(2),
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FrobConst_psi2_3 * Py.frobenius(2)
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# Pz - Always 1 after extract
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])
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def clearCofactorG2(P):
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return cofactorG2 * P
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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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P = G2.random_point()
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P = clearCofactorG2(P)
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(Px, Py, Pz) = P
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# Galbraith-Lin-Scott, 2008, Theorem 1
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assert psi(psi(P)) - t*psi(P) + p*P == G2([0, 1, 0])
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# Galbraith-Scott, 2008, Lemma 1
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# k-th cyclotomic polynomial with k = 12
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assert psi2(psi2(P)) - psi2(P) + P == G2([0, 1, 0])
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assert p % r == (t-1) % r
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# assert (p^4 - p^2 + 1) % r == 0
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assert ((t-1)^4 - (t-1)^2 + 1) % r == 0
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assert (t-1)*P == (p % r)*P
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assert (t-1)*P == psi(P)
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print('Endomorphism OK')
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checkEndo()
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def subgroup_check(P):
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ppP = psi2(P)
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assert x * psi(ppP) - ppP + P == G2([0,1,0])
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# Decomposition generated by LLL-algorithm and Babai rounding
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# to solve the Shortest (Basis) Vector Problem
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#
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# TODO: This lattice is generating wrong result
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# Lattice from Guide to Pairing-Based Cryptography
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# Lat = [
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# [ x, 1, 0, 0],
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# [ 0, x, 1, 0],
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# [ 0, 0, x, 1],
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# [ 1, 0,-1, x]
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# ]
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# ahat = [x*(x^2+1), -(x^2+1), x, -1]
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# Lattice from my own LLL+Babai rounding routines
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Lat = Matrix([
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[-x, 1, 0, 0],
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[ 0,-x, 1, 0],
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[ 0, 0,-x, 1],
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[ 1, 0,-1,-x]
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])
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# print('Lat: ' + str(Lat))
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ahat = vector([r, 0, 0, 0]) * Lat.inverse()
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# print('ahat: ' + str(ahat))
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n = int(r).bit_length()
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n = int(((n + 64 - 1) // 64) * 64) # round to next multiple of 64
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v = [Integer(int(a) << n) // r for a in ahat]
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def pretty_print_lattice(Lat):
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latHex = [['0x' + x.hex() if x >= 0 else '-0x' + (-x).hex() for x in vec] for vec in Lat]
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maxlen = max([len(cell) for row in latHex for cell in row])
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for row in latHex:
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row = ' '.join(cell.rjust(maxlen + 2) for cell in row)
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print(row)
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print('\nLattice')
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pretty_print_lattice(Lat)
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print('\nbasis:')
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print(' 𝛼\u03050: 0x' + v[0].hex())
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print(' 𝛼\u03051: 0x' + v[1].hex())
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print(' 𝛼\u03052: 0x' + v[2].hex())
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print(' 𝛼\u03053: 0x' + v[3].hex())
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print('')
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maxInfNorm = abs(x + 2)
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print('\nmax infinity norm:')
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print(' ||(a0, a1, a2, a3)||∞ ≤ 0x' + str(maxInfNorm.hex()))
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print(' infinity norm bitlength: ' + str(int(maxInfNorm).bit_length()))
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# Contrary to Faz2013 paper, we use the max infinity norm
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# to properly dimension our recoding instead of ⌈log2 r/m⌉ + 1
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# which fails for some inputs
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# +1 for signed column
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# Optional +1 for handling negative miniscalars
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L = int(maxInfNorm).bit_length() + 1
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L += 1
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lambda1 = (t-1) % r
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lambda2 = lambda1^2 % r
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lambda3 = lambda1^3 % r
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def getGLV2_decomp(scalar):
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maxLen = (int(r).bit_length() + 3) // 4 + 1
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maxLen += 1 # Deal with negative miniscalars
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a0 = (v[0] * scalar) >> n
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a1 = (v[1] * scalar) >> n
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a2 = (v[2] * scalar) >> n
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a3 = (v[3] * scalar) >> n
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k0 = scalar - a0 * Lat[0][0] - a1 * Lat[1][0] - a2 * Lat[2][0] - a3 * Lat[3][0]
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k1 = 0 - a0 * Lat[0][1] - a1 * Lat[1][1] - a2 * Lat[2][1] - a3 * Lat[3][1]
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k2 = 0 - a0 * Lat[0][2] - a1 * Lat[1][2] - a2 * Lat[2][2] - a3 * Lat[3][2]
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k3 = 0 - a0 * Lat[0][3] - a1 * Lat[1][3] - a2 * Lat[2][3] - a3 * Lat[3][3]
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print("k0.bitlength(): " + str(int(k0).bit_length()))
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print("k1.bitlength(): " + str(int(k1).bit_length()))
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print("k2.bitlength(): " + str(int(k2).bit_length()))
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print("k3.bitlength(): " + str(int(k3).bit_length()))
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print('k0: ' + k0.hex())
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print('k1: ' + k1.hex())
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print('k2: ' + k2.hex())
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print('k3: ' + k3.hex())
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assert scalar == (k0 + k1*lambda1 + k2*lambda2 + k3*lambda3) % r
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assert int(k0).bit_length() <= maxLen
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assert int(k1).bit_length() <= maxLen
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assert int(k2).bit_length() <= maxLen
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assert int(k3).bit_length() <= maxLen
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return k0, k1, k2, k3
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def recodeScalars(k):
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m = 4
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b = [[0] * L, [0] * L, [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 clearBit(v, bit):
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return v & ~int(1 << bit)
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def buildLut(P0, P_endos):
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m = 4
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assert len(P_endos) == m-1
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lut = [0] * (1 << (m-1))
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lut[0] = P0
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lutS = [''] * (1 << (m-1))
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lutS[0] = 'P0'
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endoS = ['P1', 'P2', 'P3']
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for u in range(1, 1 << (m-1)):
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msb = u.bit_length() - 1
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idx = clearBit(u, msb)
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lut[u] = lut[clearBit(u, msb)] + P_endos[msb]
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lutS[u] = lutS[clearBit(u, msb)] + ' + ' + endoS[msb]
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print('LUT: ' + str(lutS))
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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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vPx = vector(Px)
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vPy = vector(Py)
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result = 'Point(\n'
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result += ' Px: ' + Integer(vPx[0]).hex() + ' + β * ' + Integer(vPx[1]).hex() + '\n'
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result += ' Py: ' + Integer(vPy[0]).hex() + ' + β * ' + Integer(vPy[1]).hex() + '\n'
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result += ')'
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return result
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def getIndex(glvRecoding, bit):
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m = 4
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index = 0
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for k in range(1, m):
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index |= ((glvRecoding[k][bit] & 1) << (k-1))
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return index
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def updateFactors(factors, recoded, bit):
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index = getIndex(recoded, bit)
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if recoded[0][bit] == 0: # Positive
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factors[0] += 1
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factors[1] += (index >> 0) & 1
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factors[2] += (index >> 1) & 1
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factors[3] += (index >> 2) & 1
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else:
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factors[0] -= 1
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factors[1] -= (index >> 0) & 1
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factors[2] -= (index >> 1) & 1
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factors[3] -= (index >> 2) & 1
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def doubleFactors(factors):
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for i in range(len(factors)):
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factors[i] *= 2
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def printFactors(factors):
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print('Multiplication done: ')
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for i in range(len(factors)):
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print(f' f{i}: {factors[i].hex()}')
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def scalarMulEndo(scalar, P0):
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m = 4
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print('L: ' + str(L))
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print('scalar: ' + Integer(scalar).hex())
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k0, k1, k2, k3 = getGLV2_decomp(scalar)
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P1 = psi(P0)
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P2 = psi2(P0)
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P3 = psi(P2)
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expected = scalar * P0
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decomp = k0*P0 + k1*P1 + k2*P2 + k3*P3
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print('expected: ' + pointToString(expected))
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print('decomp: ' + pointToString(decomp))
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assert expected == decomp
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# Alternative to adding an extra bit
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# to deal with miniscalars
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# if k0 < 0: k0 = -k0; P0 = -P0
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# if k1 < 0: k1 = -k1; P1 = -P1
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# if k2 < 0: k2 = -k2; P2 = -P2
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# if k3 < 0: k3 = -k3; P3 = -P3
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# assert expected == k0*P0 + k1*P1 + k2*P2 + k3*P3
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# Somehow the recoding doesn't cope with first scalar being negative
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if k0 < 0:
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k0 = -k0
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P0 = -P0
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print('------ recode scalar -----------')
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even = k0 & 1 == 0
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print('was even: ' + str(even))
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if even:
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k0 += 1
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b = recodeScalars([k0, k1, k2, k3])
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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('b2: ' + str(list(reversed(b[2]))))
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print('b3: ' + str(list(reversed(b[3]))))
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print('------------ lut ---------------')
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lut = buildLut(P0, [P1, P2, P3])
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print('------------ mul ---------------')
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# b[0][L-1] is always 0
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print(f'L-1: {getIndex(b, L-1)}')
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print(f'L-2: {getIndex(b, L-2)}')
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|
print(f'L-3: {getIndex(b, L-3)}')
|
|||
|
|
print(f'L-4: {getIndex(b, L-4)}')
|
|||
|
|
print(f'L-5: {getIndex(b, L-5)}')
|
|||
|
|
print(f'L-6: {getIndex(b, L-6)}')
|
|||
|
|
|
|||
|
|
factors = [0, 0, 0, 0] # Track the decomposed scalar applied (debugging)
|
|||
|
|
updateFactors(factors, b, L-1)
|
|||
|
|
|
|||
|
|
Q = lut[getIndex(b, L-1)]
|
|||
|
|
for bit in range(L-2, -1, -1):
|
|||
|
|
Q *= 2
|
|||
|
|
Q += (1 - 2 * b[0][bit]) * lut[getIndex(b, bit)]
|
|||
|
|
|
|||
|
|
doubleFactors(factors)
|
|||
|
|
updateFactors(factors, b, bit)
|
|||
|
|
|
|||
|
|
if even:
|
|||
|
|
Q -= P0
|
|||
|
|
print('----')
|
|||
|
|
print('final Q: ' + pointToString(Q))
|
|||
|
|
print('expected: ' + pointToString(expected))
|
|||
|
|
print('----')
|
|||
|
|
printFactors(factors)
|
|||
|
|
print('Mul expected:')
|
|||
|
|
print(' k0: ' + k0.hex())
|
|||
|
|
print(' k1: ' + k1.hex())
|
|||
|
|
print(' k2: ' + k2.hex())
|
|||
|
|
print(' k3: ' + k3.hex())
|
|||
|
|
|
|||
|
|
assert Q == expected
|
|||
|
|
|
|||
|
|
# Test generator
|
|||
|
|
set_random_seed(1337)
|
|||
|
|
|
|||
|
|
for i in range(1):
|
|||
|
|
print('---------------------------------------')
|
|||
|
|
scalar = randrange(r) # Pick an integer below curve order
|
|||
|
|
# P = G2.random_point()
|
|||
|
|
# P = clearCofactorG2(P)
|
|||
|
|
|
|||
|
|
scalar = Integer('0x9d432eb58ec68bbc09d10961451d99c7796fb2f795eca603d6feaf3e2a1634b')
|
|||
|
|
P = G2([
|
|||
|
|
Fp2([Integer('0x267401f3ef554fe74ae131d56a10edf14ae40192654901b4618d2bf7af22e77c2a9b79e407348dbd4aad13ca73b33a'),
|
|||
|
|
Integer('0x12dcca838f46a3e0418e5dd8b978362757a16bfd78f0b77f4a1916ace353938389ae3ea228d0eb5020a0aaa58884aec')]),
|
|||
|
|
Fp2([Integer('0x11799118d2e054aabd9f74c0843fecbdc1c0d56f61c61c5854c2507ae2416e48a6b2cd3bc8bf7495a4d3d8270eafe2b'),
|
|||
|
|
Integer('0x823b9f8fb9f8297734a14359fa2c2a0de275e7e638197eaaaa7cff28f9cb3101bdabb570016672455f1ecae625e294')])
|
|||
|
|
])
|
|||
|
|
|
|||
|
|
subgroup_check(P)
|
|||
|
|
scalarMulEndo(scalar, P)
|