forked from algorand/bls_sigs_ref
/
fields.py
453 lines (382 loc) · 16.8 KB
/
fields.py
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#!/usr/bin/python
#
# implementation of Fp and Fp^2 operations
#
# This file is derived from `fields.py` in the Chia BLS signatures Python implementation,
# https://github.com/Chia-Network/bls-signatures/
# which is (C) 2018 Chia Network Inc. and licensed under the Apache 2.0 license.
# See copyright notice at the end of this file.
#
# Changes from the original version:
# * Some unneeded functionality was removed and some pylint errors were fixed.
# * added trivial __reversed__ method to Fq to support generic sgn0 impl
# * q -> p in frob_coeffs for consistency with the rest of this library
# * moved sgn0 and sqrt_F2 into this file
#
# Changes (C) 2019 Riad S. Wahby <rsw@cs.stanford.edu>
from copy import deepcopy
from consts import p
# "sign" of x: returns -1 if x is the lexically larger of x and -1 * x, else returns 1
def sgn0(x):
thresh = (p - 1) // 2
sign = 0
for xi in reversed(x):
if xi > thresh:
sign = -1 if sign == 0 else sign
elif xi > 0:
sign = 1 if sign == 0 else sign
sign = 1 if sign == 0 else sign
return sign
class Fq(int):
"""
Represents an element of a finite field mod a prime q.
"""
Q = None
extension = 1
def __new__(cls, Q, x):
ret = super().__new__(cls, x % Q)
ret.Q = Q
return ret
def __neg__(self):
return Fq(self.Q, super().__neg__())
def __add__(self, other):
if not isinstance(other, int):
return NotImplemented
return Fq(self.Q, super().__add__(other))
def __radd__(self, other):
if not isinstance(other, int):
return NotImplemented
return self.__add__(other)
def __sub__(self, other):
if not isinstance(other, int):
return NotImplemented
return Fq(self.Q, super().__sub__(other))
def __rsub__(self, other):
if not isinstance(other, int):
return NotImplemented
return Fq(self.Q, super().__rsub__(other))
def __mul__(self, other):
if not isinstance(other, int):
return NotImplemented
return Fq(self.Q, super().__mul__(other))
def __rmul__(self, other):
return self.__mul__(other)
def __eq__(self, other):
if not isinstance(other, type(self)):
return super().__eq__(other)
return super().__eq__(other) and self.Q == other.Q
def __str__(self):
s = hex(int(self))
s2 = s[0:7] + ".." + s[-5:] if len(s) > 10 else s
return "Fq(" + s2 + ")"
def __repr__(self):
return "Fq(" + hex(int(self)) + ")"
def __pow__(self, other):
if other == 0:
return Fq(self.Q, 1)
if other == 1:
return self
if other % 2 == 0:
return (self * self) ** (other // 2)
return (self * self) ** (other // 2) * self
def qi_power(self, _):
return self
def __invert__(self):
"""
Extended euclidian algorithm for inversion.
"""
x0, x1, y0, y1 = 1, 0, 0, 1
a = int(self.Q)
b = int(self)
while a != 0:
q, b, a = b // a, a, b % a
x0, x1 = x1, x0 - q * x1
y0, y1 = y1, y0 - q * y1
return Fq(self.Q, x0)
def __floordiv__(self, other):
if (isinstance(other, int) and
not isinstance(other, type(self))):
other = Fq(self.Q, other)
return self * ~other
__truediv__ = __floordiv__
def __iter__(self):
yield self
def __reversed__(self):
yield self
def __deepcopy__(self, memo):
return Fq(self.Q, int(self))
@classmethod
def zero(cls, Q):
return Fq(Q, 0)
@classmethod
def one(cls, Q):
return Fq(Q, 1)
@classmethod
def from_fq(cls, _, fq):
return fq
class FieldExtBase(tuple):
"""
Represents an extension of a field (or extension of an extension).
The elements of the tuple can be other FieldExtBase or they can be
Fq elements. For example, Fq2 = (Fq, Fq). Fq12 = (Fq6, Fq6), etc.
"""
extension = None
basefield = None
embedding = None
root = None
Q = None
def __new__(cls, Q, *args):
new_args = args[:]
try:
arg_extension = args[0].extension
args[1].extension # pylint: disable=pointless-statement
except AttributeError:
if len(args) != 2:
raise Exception("Invalid number of arguments")
arg_extension = 1
new_args = [Fq(Q, a) for a in args]
if arg_extension != 1:
if len(args) != cls.embedding:
raise Exception("Invalid number of arguments")
for arg in new_args:
assert arg.extension == arg_extension
assert all(isinstance(arg, cls.basefield
if cls.basefield is not Fq else int)
for arg in new_args)
ret = super().__new__(cls, new_args)
ret.Q = Q
return ret
def __neg__(self):
cls = type(self)
ret = super().__new__(cls, (-x for x in self))
ret.Q = self.Q
ret.root = self.root
return ret
def __add__(self, other):
cls = type(self)
if not isinstance(other, cls):
if type(other) != int and other.extension > self.extension: # pylint: disable=unidiomatic-typecheck
return NotImplemented
other_new = [cls.basefield.zero(self.Q) for _ in self]
other_new[0] = other_new[0] + other
else:
other_new = other
ret = super().__new__(cls, (a + b for a, b in zip(self, other_new)))
ret.Q = self.Q
ret.root = self.root
return ret
def __radd__(self, other):
return self.__add__(other)
def __sub__(self, other):
return self + (-other)
def __rsub__(self, other):
return (-self) + other
def __mul__(self, other):
cls = type(self)
if isinstance(other, int):
ret = super().__new__(cls, (a * other for a in self))
ret.Q = self.Q
ret.root = self.root
return ret
if cls.extension < other.extension:
return NotImplemented
buf = [cls.basefield.zero(self.Q) for _ in self]
for i, x in enumerate(self):
if cls.extension == other.extension:
for j, y in enumerate(other):
if x and y:
if i+j >= self.embedding:
buf[(i + j) % self.embedding] += (x * y *
self.root)
else:
buf[(i + j) % self.embedding] += x * y
else:
if x:
buf[i] = x * other
ret = super().__new__(cls, buf)
ret.Q = self.Q
ret.root = self.root
return ret
def __rmul__(self, other):
return self.__mul__(other)
def __floordiv__(self, other):
return self * ~other
__truediv__ = __floordiv__
def __eq__(self, other):
if not isinstance(other, type(self)):
if isinstance(other, (FieldExtBase, int)):
if (not isinstance(other, FieldExtBase)
or self.extension > other.extension):
for i in range(1, self.embedding):
if self[i] != (type(self.root).zero(self.Q)):
return False
return self[0] == other
return NotImplemented
return NotImplemented
return super().__eq__(other) and self.Q == other.Q
def __lt__(self, other):
# Reverse the order for comparison (3i + 1 > 2i + 7)
return self[::-1].__lt__(other[::-1])
def __neq__(self, other):
return not self.__eq__(other)
def __str__(self):
return ("Fq" + str(self.extension) + "(" + ", ".join([a.__str__()
for a in self])
+ ")")
def __repr__(self):
return ("Fq" + str(self.extension) + "(" + ", ".join([a.__repr__()
for a in self])
+ ")")
def __pow__(self, e):
assert isinstance(e, int) and e >= 0
ans = type(self).one(self.Q)
base = self
ans.root = self.root
while e:
if e & 1:
ans *= base
base *= base
e >>= 1
return ans
def __bool__(self):
return any(x for x in self)
def set_root(self, _root):
self.root = _root
@classmethod
def zero(cls, Q):
return cls.from_fq(Q, Fq(Q, 0))
@classmethod
def one(cls, Q):
return cls.from_fq(Q, Fq(Q, 1))
@classmethod
def from_fq(cls, Q, fq):
y = cls.basefield.from_fq(Q, fq)
z = cls.basefield.zero(Q)
ret = super().__new__(cls,
(z if i else y for i in range(cls.embedding)))
ret.Q = Q
if cls == Fq2:
ret.set_root(Fq(Q, -1))
elif cls == Fq6:
ret.set_root(Fq2(Q, Fq.one(Q), Fq.one(Q)))
elif cls == Fq12:
r = Fq6(Q, Fq2.zero(Q), Fq2.one(Q), Fq2.zero(Q))
ret.set_root(r)
return ret
def __deepcopy__(self, memo):
cls = type(self)
ret = super().__new__(cls, (deepcopy(a, memo) for a in self))
ret.Q = self.Q
ret.root = self.root
return ret
def qi_power(self, i):
cls = type(self)
i %= cls.extension
if i == 0:
return self
ret = super().__new__(cls,
(a.qi_power(i) * frob_coeffs[cls.extension, i, j] if j else a.qi_power(i)
for j, a in enumerate(self)))
ret.Q = self.Q
ret.root = self.root
return ret
class Fq2(FieldExtBase):
# Fq2 is constructed as Fq(u) / (u^2 - i) where i = -1
extension = 2
embedding = 2
basefield = Fq
def __init__(self, Q, *_):
# pylint: disable=super-init-not-called
super().set_root(Fq(Q, -1))
def __invert__(self):
a, b = self
factor = ~(a * a + b * b)
ret = Fq2(self.Q, a * factor, -b * factor)
return ret
def mul_by_nonresidue(self):
# multiply by u + 1
a, b = self
return Fq2(self.Q, a - b, a + b)
# roots of unity, used for computing square roots in Fq2
rv1 = 0x6af0e0437ff400b6831e36d6bd17ffe48395dabc2d3435e77f76e17009241c5ee67992f72ec05f4c81084fbede3cc09
roots_of_unity = (Fq2(p, 1, 0), Fq2(p, 0, 1), Fq2(p, rv1, rv1), Fq2(p, rv1, p - rv1))
del rv1
# sqrt function -- returns None when input is nonsquare
def sqrt_F2(val):
sqrt_cand = pow(val, (p ** 2 + 7) // 16)
ret = None
for root in roots_of_unity:
tmp = sqrt_cand * root
ret = tmp if pow(tmp, 2) == val else ret
return ret
class Fq6(FieldExtBase):
# Fq6 is constructed as Fq2(v) / (v^3 - j) where j = u + 1
extension = 6
embedding = 3
basefield = Fq2
def __init__(self, Q, *_):
# pylint: disable=super-init-not-called
super().set_root(Fq2(Q, Fq.one(Q), Fq.one(Q)))
def __invert__(self):
a, b, c = self
g0 = a*a - b*c.mul_by_nonresidue()
g1 = (c*c).mul_by_nonresidue() - a*b
g2 = b*b - a*c
factor = ~(g0*a + (g1*c + g2*b).mul_by_nonresidue())
# TODO(mariano54): no inverse pylint: disable=fixme
return Fq6(self.Q, g0 * factor, g1 * factor, g2 * factor)
def mul_by_nonresidue(self):
# multiply by v
a, b, c = self
return Fq6(self.Q, c * self.root, a, b)
class Fq12(FieldExtBase):
# Fq12 is constructed as Fq6(w) / (w^2 - k) where k = v
extension = 12
embedding = 2
basefield = Fq6
def __init__(self, Q, *_):
# pylint: disable=super-init-not-called
super().set_root(Fq6(Q, Fq2.zero(Q), Fq2.one(Q), Fq2.zero(Q)))
def __invert__(self):
a, b = self
factor = ~(a*a - (b*b).mul_by_nonresidue())
return Fq12(self.Q, a * factor, -b * factor)
# Frobenius coefficients for raising elements to q**i -th powers
# These are specific to this given q
frob_coeffs = {
(2, 1, 1) : Fq(p, -1),
(6, 1, 1) : Fq2(p, Fq(p, 0x0), Fq(p, 0x1a0111ea397fe699ec02408663d4de85aa0d857d89759ad4897d29650fb85f9b409427eb4f49fffd8bfd00000000aaac)),
(6, 1, 2) : Fq2(p, Fq(p, 0x1a0111ea397fe699ec02408663d4de85aa0d857d89759ad4897d29650fb85f9b409427eb4f49fffd8bfd00000000aaad), Fq(p, 0x0)),
(6, 2, 1) : Fq2(p, Fq(p, 0x5f19672fdf76ce51ba69c6076a0f77eaddb3a93be6f89688de17d813620a00022e01fffffffefffe), Fq(p, 0x0)),
(6, 2, 2) : Fq2(p, Fq(p, 0x1a0111ea397fe699ec02408663d4de85aa0d857d89759ad4897d29650fb85f9b409427eb4f49fffd8bfd00000000aaac), Fq(p, 0x0)),
(6, 3, 1) : Fq2(p, Fq(p, 0x0), Fq(p, 0x1)),
(6, 3, 2) : Fq2(p, Fq(p, 0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2a0f6b0f6241eabfffeb153ffffb9feffffffffaaaa), Fq(p, 0x0)),
(6, 4, 1) : Fq2(p, Fq(p, 0x1a0111ea397fe699ec02408663d4de85aa0d857d89759ad4897d29650fb85f9b409427eb4f49fffd8bfd00000000aaac), Fq(p, 0x0)),
(6, 4, 2) : Fq2(p, Fq(p, 0x5f19672fdf76ce51ba69c6076a0f77eaddb3a93be6f89688de17d813620a00022e01fffffffefffe), Fq(p, 0x0)),
(6, 5, 1) : Fq2(p, Fq(p, 0x0), Fq(p, 0x5f19672fdf76ce51ba69c6076a0f77eaddb3a93be6f89688de17d813620a00022e01fffffffefffe)),
(6, 5, 2) : Fq2(p, Fq(p, 0x5f19672fdf76ce51ba69c6076a0f77eaddb3a93be6f89688de17d813620a00022e01fffffffeffff), Fq(p, 0x0)),
(12, 1, 1) : Fq6(p, Fq2(p, Fq(p, 0x1904d3bf02bb0667c231beb4202c0d1f0fd603fd3cbd5f4f7b2443d784bab9c4f67ea53d63e7813d8d0775ed92235fb8), Fq(p, 0xfc3e2b36c4e03288e9e902231f9fb854a14787b6c7b36fec0c8ec971f63c5f282d5ac14d6c7ec22cf78a126ddc4af3)), Fq2(p, Fq(p, 0x0), Fq(p, 0x0)), Fq2(p, Fq(p, 0x0), Fq(p, 0x0))),
(12, 2, 1) : Fq6(p, Fq2(p, Fq(p, 0x5f19672fdf76ce51ba69c6076a0f77eaddb3a93be6f89688de17d813620a00022e01fffffffeffff), Fq(p, 0x0)), Fq2(p, Fq(p, 0x0), Fq(p, 0x0)), Fq2(p, Fq(p, 0x0), Fq(p, 0x0))),
(12, 3, 1) : Fq6(p, Fq2(p, Fq(p, 0x135203e60180a68ee2e9c448d77a2cd91c3dedd930b1cf60ef396489f61eb45e304466cf3e67fa0af1ee7b04121bdea2), Fq(p, 0x6af0e0437ff400b6831e36d6bd17ffe48395dabc2d3435e77f76e17009241c5ee67992f72ec05f4c81084fbede3cc09)), Fq2(p, Fq(p, 0x0), Fq(p, 0x0)), Fq2(p, Fq(p, 0x0), Fq(p, 0x0))),
(12, 4, 1) : Fq6(p, Fq2(p, Fq(p, 0x5f19672fdf76ce51ba69c6076a0f77eaddb3a93be6f89688de17d813620a00022e01fffffffefffe), Fq(p, 0x0)), Fq2(p, Fq(p, 0x0), Fq(p, 0x0)), Fq2(p, Fq(p, 0x0), Fq(p, 0x0))),
(12, 5, 1) : Fq6(p, Fq2(p, Fq(p, 0x144e4211384586c16bd3ad4afa99cc9170df3560e77982d0db45f3536814f0bd5871c1908bd478cd1ee605167ff82995), Fq(p, 0x5b2cfd9013a5fd8df47fa6b48b1e045f39816240c0b8fee8beadf4d8e9c0566c63a3e6e257f87329b18fae980078116)), Fq2(p, Fq(p, 0x0), Fq(p, 0x0)), Fq2(p, Fq(p, 0x0), Fq(p, 0x0))),
(12, 6, 1) : Fq6(p, Fq2(p, Fq(p, 0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2a0f6b0f6241eabfffeb153ffffb9feffffffffaaaa), Fq(p, 0x0)), Fq2(p, Fq(p, 0x0), Fq(p, 0x0)), Fq2(p, Fq(p, 0x0), Fq(p, 0x0))),
(12, 7, 1) : Fq6(p, Fq2(p, Fq(p, 0xfc3e2b36c4e03288e9e902231f9fb854a14787b6c7b36fec0c8ec971f63c5f282d5ac14d6c7ec22cf78a126ddc4af3), Fq(p, 0x1904d3bf02bb0667c231beb4202c0d1f0fd603fd3cbd5f4f7b2443d784bab9c4f67ea53d63e7813d8d0775ed92235fb8)), Fq2(p, Fq(p, 0x0), Fq(p, 0x0)), Fq2(p, Fq(p, 0x0), Fq(p, 0x0))),
(12, 8, 1) : Fq6(p, Fq2(p, Fq(p, 0x1a0111ea397fe699ec02408663d4de85aa0d857d89759ad4897d29650fb85f9b409427eb4f49fffd8bfd00000000aaac), Fq(p, 0x0)), Fq2(p, Fq(p, 0x0), Fq(p, 0x0)), Fq2(p, Fq(p, 0x0), Fq(p, 0x0))),
(12, 9, 1) : Fq6(p, Fq2(p, Fq(p, 0x6af0e0437ff400b6831e36d6bd17ffe48395dabc2d3435e77f76e17009241c5ee67992f72ec05f4c81084fbede3cc09), Fq(p, 0x135203e60180a68ee2e9c448d77a2cd91c3dedd930b1cf60ef396489f61eb45e304466cf3e67fa0af1ee7b04121bdea2)), Fq2(p, Fq(p, 0x0), Fq(p, 0x0)), Fq2(p, Fq(p, 0x0), Fq(p, 0x0))),
(12, 10, 1) : Fq6(p, Fq2(p, Fq(p, 0x1a0111ea397fe699ec02408663d4de85aa0d857d89759ad4897d29650fb85f9b409427eb4f49fffd8bfd00000000aaad), Fq(p, 0x0)), Fq2(p, Fq(p, 0x0), Fq(p, 0x0)), Fq2(p, Fq(p, 0x0), Fq(p, 0x0))),
(12, 11, 1) : Fq6(p, Fq2(p, Fq(p, 0x5b2cfd9013a5fd8df47fa6b48b1e045f39816240c0b8fee8beadf4d8e9c0566c63a3e6e257f87329b18fae980078116), Fq(p, 0x144e4211384586c16bd3ad4afa99cc9170df3560e77982d0db45f3536814f0bd5871c1908bd478cd1ee605167ff82995)), Fq2(p, Fq(p, 0x0), Fq(p, 0x0)), Fq2(p, Fq(p, 0x0), Fq(p, 0x0))),
}
# Copyright 2018 Chia Network Inc
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
# http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
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