Inside Bitcoin

1""" Generates the constants used in secp256k1_scalar_split_lambda.
2
3See the comments for secp256k1_scalar_split_lambda in src/scalar_impl.h for detailed explanations.
4"""
5
6load("secp256k1_params.sage")
7
8def inf_norm(v):
9    """Returns the infinity norm of a vector."""
10    return max(map(abs, v))
11
12def gauss_reduction(i1, i2):
13    v1, v2 = i1.copy(), i2.copy()
14    while True:
15        if inf_norm(v2) < inf_norm(v1):
16            v1, v2 = v2, v1
17        # This is essentially
18        #    m = round((v1[0]*v2[0] + v1[1]*v2[1]) / (inf_norm(v1)**2))
19        # (rounding to the nearest integer) without relying on floating point arithmetic.
20        m = ((v1[0]*v2[0] + v1[1]*v2[1]) + (inf_norm(v1)**2) // 2) // (inf_norm(v1)**2)
21        if m == 0:
22            return v1, v2
23        v2[0] -= m*v1[0]
24        v2[1] -= m*v1[1]
25
26def find_split_constants_gauss():
27    """Find constants for secp256k1_scalar_split_lamdba using gauss reduction."""
28    (v11, v12), (v21, v22) = gauss_reduction([0, N], [1, int(LAMBDA)])
29
30    # We use related vectors in secp256k1_scalar_split_lambda.
31    A1, B1 = -v21, -v11
32    A2, B2 = v22, -v21
33
34    return A1, B1, A2, B2
35
36def find_split_constants_explicit_tof():
37    """Find constants for secp256k1_scalar_split_lamdba using the trace of Frobenius.
38
39    See Benjamin Smith: "Easy scalar decompositions for efficient scalar multiplication on
40    elliptic curves and genus 2 Jacobians" (https://eprint.iacr.org/2013/672), Example 2
41    """
42    assert P % 3 == 1 # The paper says P % 3 == 2 but that appears to be a mistake, see [10].
43    assert C.j_invariant() == 0
44
45    t = C.trace_of_frobenius()
46
47    c = Integer(sqrt((4*P - t**2)/3))
48    A1 = Integer((t - c)/2 - 1)
49    B1 = c
50
51    A2 = Integer((t + c)/2 - 1)
52    B2 = Integer(1 - (t - c)/2)
53
54    # We use a negated b values in secp256k1_scalar_split_lambda.
55    B1, B2 = -B1, -B2
56
57    return A1, B1, A2, B2
58
59A1, B1, A2, B2 = find_split_constants_explicit_tof()
60
61# For extra fun, use an independent method to recompute the constants.
62assert (A1, B1, A2, B2) == find_split_constants_gauss()
63
64# PHI : Z[l] -> Z_n where phi(a + b*l) == a + b*lambda mod n.
65def PHI(a,b):
66    return Z(a + LAMBDA*b)
67
68# Check that (A1, B1) and (A2, B2) are in the kernel of PHI.
69assert PHI(A1, B1) == Z(0)
70assert PHI(A2, B2) == Z(0)
71
72# Check that the parallelogram generated by (A1, A2) and (B1, B2)
73# is a fundamental domain by containing exactly N points.
74# Since the LHS is the determinant and N != 0, this also checks that
75# (A1, A2) and (B1, B2) are linearly independent. By the previous
76# assertions, (A1, A2) and (B1, B2) are a basis of the kernel.
77assert A1*B2 - B1*A2 == N
78
79# Check that their components are short enough.
80assert (A1 + A2)/2 < sqrt(N)
81assert B1 < sqrt(N)
82assert B2 < sqrt(N)
83
84# Verify connection to Eisenstein integers Z[w] where w = (-1 + sqrt(-3))/2.
85# The group order N factors as N = pi * conj(pi) in Z[w], where pi = A - B*w
86# is an Eisenstein prime with norm A^2 + A*B + B^2. The GLV endomorphism
87# eigenvalue LAMBDA equals B/A mod N, which is the image of w^2 under the
88# isomorphism Z[w]/(pi) -> Z/NZ (since w -> A/B and (A/B)^2 = B/A in Z/NZ).
89A_EIS, B_EIS = -B1, A1
90assert A_EIS**2 + A_EIS*B_EIS + B_EIS**2 == N
91assert Z(B_EIS / A_EIS) == LAMBDA
92
93G1 = round((2**384)*B2/N)
94G2 = round((2**384)*(-B1)/N)
95
96def rnddiv2(v):
97    if v & 1:
98        v += 1
99    return v >> 1
100
101def scalar_lambda_split(k):
102    """Equivalent to secp256k1_scalar_lambda_split()."""
103    c1 = rnddiv2((k * G1) >> 383)
104    c2 = rnddiv2((k * G2) >> 383)
105    c1 = (c1 * -B1) % N
106    c2 = (c2 * -B2) % N
107    r2 = (c1 + c2) % N
108    r1 = (k + r2 * -LAMBDA) % N
109    return (r1, r2)
110
111# The result of scalar_lambda_split can depend on the representation of k (mod n).
112SPECIAL = (2**383) // G2 + 1
113assert scalar_lambda_split(SPECIAL) != scalar_lambda_split(SPECIAL + N)
114
115print('  A1     =', hex(A1))
116print(' -B1     =', hex(-B1))
117print('  A2     =', hex(A2))
118print(' -B2     =', hex(-B2))
119print('         =', hex(Z(-B2)))
120print(' -LAMBDA =', hex(-LAMBDA))
121
122print('  G1     =', hex(G1))
123print('  G2     =', hex(G2))
124