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Merge bitcoin#852: Add sage script for generating scalar_split_lambda…
… constants 329a2e0 sage: Add script for generating scalar_split_lambda constants (Tim Ruffing) f554dfc sage: Reorganize files (Tim Ruffing) Pull request description: ACKs for top commit: sipa: ACK 329a2e0 Tree-SHA512: d41fe5eba332f48af0b800778aa076925c4e8e95ec21c4371a500ddd6088b6d52961bdb93f7ce2b127e18095667dbb966a0d14191177f0d0e78dfaf55271d5e2
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""" Generates the constants used in secp256k1_scalar_split_lambda. | ||
See the comments for secp256k1_scalar_split_lambda in src/scalar_impl.h for detailed explanations. | ||
""" | ||
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load("secp256k1_params.sage") | ||
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def inf_norm(v): | ||
"""Returns the infinity norm of a vector.""" | ||
return max(map(abs, v)) | ||
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def gauss_reduction(i1, i2): | ||
v1, v2 = i1.copy(), i2.copy() | ||
while True: | ||
if inf_norm(v2) < inf_norm(v1): | ||
v1, v2 = v2, v1 | ||
# This is essentially | ||
# m = round((v1[0]*v2[0] + v1[1]*v2[1]) / (inf_norm(v1)**2)) | ||
# (rounding to the nearest integer) without relying on floating point arithmetic. | ||
m = ((v1[0]*v2[0] + v1[1]*v2[1]) + (inf_norm(v1)**2) // 2) // (inf_norm(v1)**2) | ||
if m == 0: | ||
return v1, v2 | ||
v2[0] -= m*v1[0] | ||
v2[1] -= m*v1[1] | ||
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def find_split_constants_gauss(): | ||
"""Find constants for secp256k1_scalar_split_lamdba using gauss reduction.""" | ||
(v11, v12), (v21, v22) = gauss_reduction([0, N], [1, int(LAMBDA)]) | ||
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# We use related vectors in secp256k1_scalar_split_lambda. | ||
A1, B1 = -v21, -v11 | ||
A2, B2 = v22, -v21 | ||
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return A1, B1, A2, B2 | ||
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def find_split_constants_explicit_tof(): | ||
"""Find constants for secp256k1_scalar_split_lamdba using the trace of Frobenius. | ||
See Benjamin Smith: "Easy scalar decompositions for efficient scalar multiplication on | ||
elliptic curves and genus 2 Jacobians" (https://eprint.iacr.org/2013/672), Example 2 | ||
""" | ||
assert P % 3 == 1 # The paper says P % 3 == 2 but that appears to be a mistake, see [10]. | ||
assert C.j_invariant() == 0 | ||
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t = C.trace_of_frobenius() | ||
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c = Integer(sqrt((4*P - t**2)/3)) | ||
A1 = Integer((t - c)/2 - 1) | ||
B1 = c | ||
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A2 = Integer((t + c)/2 - 1) | ||
B2 = Integer(1 - (t - c)/2) | ||
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# We use a negated b values in secp256k1_scalar_split_lambda. | ||
B1, B2 = -B1, -B2 | ||
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return A1, B1, A2, B2 | ||
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A1, B1, A2, B2 = find_split_constants_explicit_tof() | ||
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# For extra fun, use an independent method to recompute the constants. | ||
assert (A1, B1, A2, B2) == find_split_constants_gauss() | ||
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# PHI : Z[l] -> Z_n where phi(a + b*l) == a + b*lambda mod n. | ||
def PHI(a,b): | ||
return Z(a + LAMBDA*b) | ||
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# Check that (A1, B1) and (A2, B2) are in the kernel of PHI. | ||
assert PHI(A1, B1) == Z(0) | ||
assert PHI(A2, B2) == Z(0) | ||
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# Check that the parallelogram generated by (A1, A2) and (B1, B2) | ||
# is a fundamental domain by containing exactly N points. | ||
# Since the LHS is the determinant and N != 0, this also checks that | ||
# (A1, A2) and (B1, B2) are linearly independent. By the previous | ||
# assertions, (A1, A2) and (B1, B2) are a basis of the kernel. | ||
assert A1*B2 - B1*A2 == N | ||
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# Check that their components are short enough. | ||
assert (A1 + A2)/2 < sqrt(N) | ||
assert B1 < sqrt(N) | ||
assert B2 < sqrt(N) | ||
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G1 = round((2**384)*B2/N) | ||
G2 = round((2**384)*(-B1)/N) | ||
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def rnddiv2(v): | ||
if v & 1: | ||
v += 1 | ||
return v >> 1 | ||
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def scalar_lambda_split(k): | ||
"""Equivalent to secp256k1_scalar_lambda_split().""" | ||
c1 = rnddiv2((k * G1) >> 383) | ||
c2 = rnddiv2((k * G2) >> 383) | ||
c1 = (c1 * -B1) % N | ||
c2 = (c2 * -B2) % N | ||
r2 = (c1 + c2) % N | ||
r1 = (k + r2 * -LAMBDA) % N | ||
return (r1, r2) | ||
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# The result of scalar_lambda_split can depend on the representation of k (mod n). | ||
SPECIAL = (2**383) // G2 + 1 | ||
assert scalar_lambda_split(SPECIAL) != scalar_lambda_split(SPECIAL + N) | ||
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print(' A1 =', hex(A1)) | ||
print(' -B1 =', hex(-B1)) | ||
print(' A2 =', hex(A2)) | ||
print(' -B2 =', hex(-B2)) | ||
print(' =', hex(Z(-B2))) | ||
print(' -LAMBDA =', hex(-LAMBDA)) | ||
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print(' G1 =', hex(G1)) | ||
print(' G2 =', hex(G2)) |
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"""Prime order of finite field underlying secp256k1 (2^256 - 2^32 - 977)""" | ||
P = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F | ||
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"""Finite field underlying secp256k1""" | ||
F = FiniteField(P) | ||
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"""Elliptic curve secp256k1: y^2 = x^3 + 7""" | ||
C = EllipticCurve([F(0), F(7)]) | ||
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"""Base point of secp256k1""" | ||
G = C.lift_x(0x79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798) | ||
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"""Prime order of secp256k1""" | ||
N = C.order() | ||
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"""Finite field of scalars of secp256k1""" | ||
Z = FiniteField(N) | ||
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""" Beta value of secp256k1 non-trivial endomorphism: lambda * (x, y) = (beta * x, y)""" | ||
BETA = F(2)^((P-1)/3) | ||
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""" Lambda value of secp256k1 non-trivial endomorphism: lambda * (x, y) = (beta * x, y)""" | ||
LAMBDA = Z(3)^((N-1)/3) | ||
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assert is_prime(P) | ||
assert is_prime(N) | ||
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assert BETA != F(1) | ||
assert BETA^3 == F(1) | ||
assert BETA^2 + BETA + 1 == 0 | ||
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assert LAMBDA != Z(1) | ||
assert LAMBDA^3 == Z(1) | ||
assert LAMBDA^2 + LAMBDA + 1 == 0 | ||
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assert Integer(LAMBDA)*G == C(BETA*G[0], G[1]) |