diff --git a/src/secp256k1/sage/gen_exhaustive_groups.sage b/src/secp256k1/sage/gen_exhaustive_groups.sage --- a/src/secp256k1/sage/gen_exhaustive_groups.sage +++ b/src/secp256k1/sage/gen_exhaustive_groups.sage @@ -1,9 +1,4 @@ -# Define field size and field -P = 2^256 - 2^32 - 977 -F = GF(P) -BETA = F(0x7ae96a2b657c07106e64479eac3434e99cf0497512f58995c1396c28719501ee) - -assert(BETA != F(1) and BETA^3 == F(1)) +load("secp256k1_params.sage") orders_done = set() results = {} diff --git a/src/secp256k1/sage/gen_split_lambda_constants.sage b/src/secp256k1/sage/gen_split_lambda_constants.sage new file mode 100644 --- /dev/null +++ b/src/secp256k1/sage/gen_split_lambda_constants.sage @@ -0,0 +1,114 @@ +""" 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. +""" + +load("secp256k1_params.sage") + +def inf_norm(v): + """Returns the infinity norm of a vector.""" + return max(map(abs, v)) + +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] + +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)]) + + # We use related vectors in secp256k1_scalar_split_lambda. + A1, B1 = -v21, -v11 + A2, B2 = v22, -v21 + + return A1, B1, A2, B2 + +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 + + t = C.trace_of_frobenius() + + c = Integer(sqrt((4*P - t**2)/3)) + A1 = Integer((t - c)/2 - 1) + B1 = c + + A2 = Integer((t + c)/2 - 1) + B2 = Integer(1 - (t - c)/2) + + # We use a negated b values in secp256k1_scalar_split_lambda. + B1, B2 = -B1, -B2 + + return A1, B1, A2, B2 + +A1, B1, A2, B2 = find_split_constants_explicit_tof() + +# For extra fun, use an independent method to recompute the constants. +assert (A1, B1, A2, B2) == find_split_constants_gauss() + +# PHI : Z[l] -> Z_n where phi(a + b*l) == a + b*lambda mod n. +def PHI(a,b): + return Z(a + LAMBDA*b) + +# 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) + +# 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 + +# Check that their components are short enough. +assert (A1 + A2)/2 < sqrt(N) +assert B1 < sqrt(N) +assert B2 < sqrt(N) + +G1 = round((2**384)*B2/N) +G2 = round((2**384)*(-B1)/N) + +def rnddiv2(v): + if v & 1: + v += 1 + return v >> 1 + +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) + +# 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) + +print(' A1 =', hex(A1)) +print(' -B1 =', hex(-B1)) +print(' A2 =', hex(A2)) +print(' -B2 =', hex(-B2)) +print(' =', hex(Z(-B2))) +print(' -LAMBDA =', hex(-LAMBDA)) + +print(' G1 =', hex(G1)) +print(' G2 =', hex(G2)) diff --git a/src/secp256k1/sage/secp256k1.sage b/src/secp256k1/sage/prove_group_implementations.sage rename from src/secp256k1/sage/secp256k1.sage rename to src/secp256k1/sage/prove_group_implementations.sage diff --git a/src/secp256k1/sage/secp256k1_params.sage b/src/secp256k1/sage/secp256k1_params.sage new file mode 100644 --- /dev/null +++ b/src/secp256k1/sage/secp256k1_params.sage @@ -0,0 +1,36 @@ +"""Prime order of finite field underlying secp256k1 (2^256 - 2^32 - 977)""" +P = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F + +"""Finite field underlying secp256k1""" +F = FiniteField(P) + +"""Elliptic curve secp256k1: y^2 = x^3 + 7""" +C = EllipticCurve([F(0), F(7)]) + +"""Base point of secp256k1""" +G = C.lift_x(0x79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798) + +"""Prime order of secp256k1""" +N = C.order() + +"""Finite field of scalars of secp256k1""" +Z = FiniteField(N) + +""" Beta value of secp256k1 non-trivial endomorphism: lambda * (x, y) = (beta * x, y)""" +BETA = F(2)^((P-1)/3) + +""" Lambda value of secp256k1 non-trivial endomorphism: lambda * (x, y) = (beta * x, y)""" +LAMBDA = Z(3)^((N-1)/3) + +assert is_prime(P) +assert is_prime(N) + +assert BETA != F(1) +assert BETA^3 == F(1) +assert BETA^2 + BETA + 1 == 0 + +assert LAMBDA != Z(1) +assert LAMBDA^3 == Z(1) +assert LAMBDA^2 + LAMBDA + 1 == 0 + +assert Integer(LAMBDA)*G == C(BETA*G[0], G[1])