constantine/sage/curve_family_bn.sage

# Constantine
# Copyright (c) 2018-2019    Status Research & Development GmbH
# Copyright (c) 2020-Present Mamy André-Ratsimbazafy
# Licensed and distributed under either of
#   * MIT license (license terms in the root directory or at http://opensource.org/licenses/MIT).
#   * Apache v2 license (license terms in the root directory or at http://www.apache.org/licenses/LICENSE-2.0).
# at your option. This file may not be copied, modified, or distributed except according to those terms.

# ############################################################
#
#                    BN Curves parameters
#                  (Barreto-Naehrig curves)
#
# ############################################################
#
# This module derives a BN curve parameters from
# its base parameterx

def compute_curve_characteristic(x_str):
  x = sage_eval(x_str)
  p = 36*x^4 + 36*x^3 + 24*x^2 + 6*x + 1
  r = 36*x^4 + 36*x^3 + 18*x^2 + 6*x + 1
  t = 6*x^2 + 1

  print(f'BN family - {p.nbits()} bits')
  print('  Prime modulus p:     0x' + p.hex())
  print('  Curve order r:       0x' + r.hex())
  print('  trace t:             0x' + t.hex())
  print('  Parameterx:       ' + x_str)
  if x < 0:
      print('  Parameterx (hex): -0x' + (-x).hex())
  else:
      print('  Parameterx (hex):  0x' +x.hex())

  print(f'  p mod 3:           ' + str(p % 3))
  print(f'  p mod 4:           ' + str(p % 4))
  print(f'  p mod 8:           ' + str(p % 8))
  print(f'  p mod 12:          ' + str(p % 12))
  print(f'  p mod 16:          ' + str(p % 16))

  print()

  print(f'  p^2 mod 3:           ' + str(p^2 % 3))
  print(f'  p^2 mod 4:           ' + str(p^2 % 4))
  print(f'  p^2 mod 8:           ' + str(p^2 % 8))
  print(f'  p^2 mod 12:          ' + str(p^2 % 12))
  print(f'  p^2 mod 16:          ' + str(p^2 % 16))

  print()

  print(f'  Endomorphism-based acceleration when p mod 3 == 1')
  print(f'    Endomorphism can be field multiplication by one of the non-trivial cube root of unity 𝜑')
  print(f'      Rationale:')
  print(f'        curve equation is y² = x³ + b, and y² = (x𝜑)³ + b <=> y² = x³ + b (with 𝜑³ == 1) so we are still on the curve')
  print(f'        this means that multiplying by 𝜑 the x-coordinate is equivalent to a scalar multiplication by some λᵩ')
  print(f'        with λᵩ² + λᵩ + 1 ≡ 0 (mod r) and 𝜑² + 𝜑 + 1 ≡ 0 (mod p), see below.')
  print(f'        Hence we have a 2 dimensional decomposition of the scalar multiplication')
  print(f'        i.e. For any [s]P, we can find a corresponding [k1]P + [k2][λᵩ]P with [λᵩ]P being a simple field multiplication by 𝜑')
  print(f'      Finding cube roots:')
  print(f'         x³−1=0 <=> (x−1)(x²+x+1) = 0, if x != 1, x solves (x²+x+1) = 0 <=> x = (-1±√3)/2')
  print(f'         cube roots of unity 𝜑 (mod p): ' + str(['0x' + Integer(root).hex() for root in GF(p)(1).nth_root(3, all=True)]))
  print(f'         cube roots of unity λᵩ (mod r): ' + str(['0x' + Integer(root).hex() for root in GF(r)(1).nth_root(3, all=True)]))
  print(f'    GLV-2 decomposition of s into (k1, k2) on G1')
  print(f'      (k1, k2) = (s, 0) - 𝛼1 b1 - 𝛼2 b2')
  print(f'      𝛼i = 𝛼\u0302i * s / r')
  print(f'        Lattice b1: ' + str(['0x' + b.hex() for b in [2*x+1, 6*x^2+4*x+1]]))
  print(f'        Lattice b2: ' + str(['0x' + b.hex() for b in [6*x^2+2*x, -2*x-1]]))

  # Babai rounding
  ahat1 = 2*x+1
  ahat2 = 6*x^2+4*x+1
  # We want a1 = ahat1 * s/r with m = 2 (for a 2-dim decomposition) and r the curve order
  # To handle rounding errors we instead multiply by
  # 𝜈 = (2^WordBitWidth)^w (i.e. the same as the R magic constant for Montgomery arithmetic)
  # with 𝜈 > r and w minimal so that 𝜈 > r
  # a1 = ahat1*𝜈/r * s/𝜈
  v = int(r).bit_length()
  print(f'      r.bit_length(): {v}')
  v = int(((v + 64 - 1) // 64) * 64) # round to next multiple of 64
  print(f'      𝜈 > r, 𝜈: 2^{v}')
  print(f'      Babai roundings')
  print(f'        𝛼\u03021: ' + '0x' + ahat1.hex())
  print(f'        𝛼\u03022: ' + '0x' + ahat2.hex())
  print(f'      Handle rounding errors')
  print(f'        𝛼1 = 𝛼\u03021 * s / r with 𝛼1 = (𝛼\u03021 * 𝜈/r) * s/𝜈')
  print(f'        𝛼2 = 𝛼\u03022 * s / r with 𝛼2 = (𝛼\u03022 * 𝜈/r) * s/𝜈')
  print(f'        -----------------------------------------------------')
  l1 = Integer(ahat1 << v) // r
  l2 = Integer(ahat2 << v) // r
  print(f'        𝛼1 = (0x{l1.hex()} * s) >> {v}')
  print(f'        𝛼2 = (0x{l2.hex()} * s) >> {v}')

if __name__ == "__main__":
  # Usage
  # sage sage/curve_family_bn.sage '-(2^62 + 2^55 + 1)'
  # sage sage/curve_family_bn.sage 4965661367192848881

  from argparse import ArgumentParser

  parser = ArgumentParser()
  parser.add_argument("curve_param",nargs="+")
  args = parser.parse_args()

  compute_curve_characteristic(args.curve_param[0])