1  """ Generates the constants used in secp256k1_scalar_split_lambda.
  2  
  3  See the comments for secp256k1_scalar_split_lambda in src/scalar_impl.h for detailed explanations.
  4  """
  5  
  6  load("secp256k1_params.sage")
  7  
  8  def inf_norm(v):
  9      """Returns the infinity norm of a vector."""
 10      return max(map(abs, v))
 11  
 12  def 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  
 26  def 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  
 36  def 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  
 59  A1, B1, A2, B2 = find_split_constants_explicit_tof()
 60  
 61  # For extra fun, use an independent method to recompute the constants.
 62  assert (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.
 65  def PHI(a,b):
 66      return Z(a + LAMBDA*b)
 67  
 68  # Check that (A1, B1) and (A2, B2) are in the kernel of PHI.
 69  assert PHI(A1, B1) == Z(0)
 70  assert 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.
 77  assert A1*B2 - B1*A2 == N
 78  
 79  # Check that their components are short enough.
 80  assert (A1 + A2)/2 < sqrt(N)
 81  assert B1 < sqrt(N)
 82  assert 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).
 89  A_EIS, B_EIS = -B1, A1
 90  assert A_EIS**2 + A_EIS*B_EIS + B_EIS**2 == N
 91  assert Z(B_EIS / A_EIS) == LAMBDA
 92  
 93  G1 = round((2**384)*B2/N)
 94  G2 = round((2**384)*(-B1)/N)
 95  
 96  def rnddiv2(v):
 97      if v & 1:
 98          v += 1
 99      return v >> 1
100  
101  def 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).
112  SPECIAL = (2**383) // G2 + 1
113  assert scalar_lambda_split(SPECIAL) != scalar_lambda_split(SPECIAL + N)
114  
115  print('  A1     =', hex(A1))
116  print(' -B1     =', hex(-B1))
117  print('  A2     =', hex(A2))
118  print(' -B2     =', hex(-B2))
119  print('         =', hex(Z(-B2)))
120  print(' -LAMBDA =', hex(-LAMBDA))
121  
122  print('  G1     =', hex(G1))
123  print('  G2     =', hex(G2))