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

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 = {}

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))

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])