315 lines
9.2 KiB
Python
315 lines
9.2 KiB
Python
# coding: utf-8
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"""
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Classes and objects to represent prime-field elliptic curves and points on them.
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Exports the following items:
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- PrimeCurve()
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- PrimePoint()
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- SECP192R1_CURVE
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- SECP192R1_BASE_POINT
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- SECP224R1_CURVE
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- SECP224R1_BASE_POINT
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- SECP256R1_CURVE
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- SECP256R1_BASE_POINT
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- SECP384R1_CURVE
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- SECP384R1_BASE_POINT
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- SECP521R1_CURVE
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- SECP521R1_BASE_POINT
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The curve constants are all PrimeCurve() objects and the base point constants
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are all PrimePoint() objects.
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Some of the following source code is derived from
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http://webpages.charter.net/curryfans/peter/downloads.html, but has been heavily
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modified to fit into this projects lint settings. The original project license
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is listed below:
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Copyright (c) 2014 Peter Pearson
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Permission is hereby granted, free of charge, to any person obtaining a copy
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of this software and associated documentation files (the "Software"), to deal
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in the Software without restriction, including without limitation the rights
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to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
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copies of the Software, and to permit persons to whom the Software is
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furnished to do so, subject to the following conditions:
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The above copyright notice and this permission notice shall be included in
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all copies or substantial portions of the Software.
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THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
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IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
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FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
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AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
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LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
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OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
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THE SOFTWARE.
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"""
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from __future__ import unicode_literals, division, absolute_import, print_function
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from ._int import inverse_mod
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class PrimeCurve():
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"""
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Elliptic curve over a prime field. Characteristic two field curves are not
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supported.
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"""
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def __init__(self, p, a, b):
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"""
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The curve of points satisfying y^2 = x^3 + a*x + b (mod p)
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:param p:
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The prime number as an integer
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:param a:
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The component a as an integer
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:param b:
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The component b as an integer
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"""
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self.p = p
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self.a = a
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self.b = b
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def contains(self, point):
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"""
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:param point:
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A Point object
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:return:
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Boolean if the point is on this curve
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"""
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y2 = point.y * point.y
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x3 = point.x * point.x * point.x
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return (y2 - (x3 + self.a * point.x + self.b)) % self.p == 0
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class PrimePoint():
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"""
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A point on a prime-field elliptic curve
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"""
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def __init__(self, curve, x, y, order=None):
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"""
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:param curve:
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A PrimeCurve object
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:param x:
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The x coordinate of the point as an integer
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:param y:
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The y coordinate of the point as an integer
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:param order:
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The order of the point, as an integer - optional
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"""
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self.curve = curve
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self.x = x
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self.y = y
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self.order = order
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# self.curve is allowed to be None only for INFINITY:
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if self.curve:
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if not self.curve.contains(self):
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raise ValueError('Invalid EC point')
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if self.order:
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if self * self.order != INFINITY:
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raise ValueError('Invalid EC point')
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def __cmp__(self, other):
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"""
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:param other:
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A PrimePoint object
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:return:
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0 if identical, 1 otherwise
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"""
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if self.curve == other.curve and self.x == other.x and self.y == other.y:
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return 0
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else:
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return 1
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def __add__(self, other):
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"""
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:param other:
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A PrimePoint object
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:return:
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A PrimePoint object
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"""
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# X9.62 B.3:
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if other == INFINITY:
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return self
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if self == INFINITY:
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return other
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assert self.curve == other.curve
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if self.x == other.x:
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if (self.y + other.y) % self.curve.p == 0:
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return INFINITY
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else:
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return self.double()
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p = self.curve.p
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l_ = ((other.y - self.y) * inverse_mod(other.x - self.x, p)) % p
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x3 = (l_ * l_ - self.x - other.x) % p
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y3 = (l_ * (self.x - x3) - self.y) % p
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return PrimePoint(self.curve, x3, y3)
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def __mul__(self, other):
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"""
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:param other:
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An integer to multiple the Point by
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:return:
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A PrimePoint object
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"""
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def leftmost_bit(x):
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assert x > 0
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result = 1
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while result <= x:
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result = 2 * result
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return result // 2
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e = other
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if self.order:
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e = e % self.order
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if e == 0:
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return INFINITY
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if self == INFINITY:
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return INFINITY
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assert e > 0
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# From X9.62 D.3.2:
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e3 = 3 * e
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negative_self = PrimePoint(self.curve, self.x, -self.y, self.order)
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i = leftmost_bit(e3) // 2
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result = self
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# print "Multiplying %s by %d (e3 = %d):" % ( self, other, e3 )
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while i > 1:
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result = result.double()
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if (e3 & i) != 0 and (e & i) == 0:
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result = result + self
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if (e3 & i) == 0 and (e & i) != 0:
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result = result + negative_self
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# print ". . . i = %d, result = %s" % ( i, result )
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i = i // 2
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return result
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def __rmul__(self, other):
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"""
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:param other:
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An integer to multiple the Point by
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:return:
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A PrimePoint object
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"""
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return self * other
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def double(self):
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"""
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:return:
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A PrimePoint object that is twice this point
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"""
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# X9.62 B.3:
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p = self.curve.p
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a = self.curve.a
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l_ = ((3 * self.x * self.x + a) * inverse_mod(2 * self.y, p)) % p
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x3 = (l_ * l_ - 2 * self.x) % p
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y3 = (l_ * (self.x - x3) - self.y) % p
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return PrimePoint(self.curve, x3, y3)
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# This one point is the Point At Infinity for all purposes:
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INFINITY = PrimePoint(None, None, None)
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# NIST Curve P-192:
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SECP192R1_CURVE = PrimeCurve(
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6277101735386680763835789423207666416083908700390324961279,
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-3,
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0x64210519e59c80e70fa7e9ab72243049feb8deecc146b9b1
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)
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SECP192R1_BASE_POINT = PrimePoint(
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SECP192R1_CURVE,
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0x188da80eb03090f67cbf20eb43a18800f4ff0afd82ff1012,
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0x07192b95ffc8da78631011ed6b24cdd573f977a11e794811,
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6277101735386680763835789423176059013767194773182842284081
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)
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# NIST Curve P-224:
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SECP224R1_CURVE = PrimeCurve(
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26959946667150639794667015087019630673557916260026308143510066298881,
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-3,
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0xb4050a850c04b3abf54132565044b0b7d7bfd8ba270b39432355ffb4
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)
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SECP224R1_BASE_POINT = PrimePoint(
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SECP224R1_CURVE,
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0xb70e0cbd6bb4bf7f321390b94a03c1d356c21122343280d6115c1d21,
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0xbd376388b5f723fb4c22dfe6cd4375a05a07476444d5819985007e34,
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26959946667150639794667015087019625940457807714424391721682722368061
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)
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# NIST Curve P-256:
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SECP256R1_CURVE = PrimeCurve(
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115792089210356248762697446949407573530086143415290314195533631308867097853951,
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-3,
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0x5ac635d8aa3a93e7b3ebbd55769886bc651d06b0cc53b0f63bce3c3e27d2604b
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)
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SECP256R1_BASE_POINT = PrimePoint(
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SECP256R1_CURVE,
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0x6b17d1f2e12c4247f8bce6e563a440f277037d812deb33a0f4a13945d898c296,
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0x4fe342e2fe1a7f9b8ee7eb4a7c0f9e162bce33576b315ececbb6406837bf51f5,
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115792089210356248762697446949407573529996955224135760342422259061068512044369
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)
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# NIST Curve P-384:
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SECP384R1_CURVE = PrimeCurve(
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39402006196394479212279040100143613805079739270465446667948293404245721771496870329047266088258938001861606973112319, # noqa
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-3,
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0xb3312fa7e23ee7e4988e056be3f82d19181d9c6efe8141120314088f5013875ac656398d8a2ed19d2a85c8edd3ec2aef
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)
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SECP384R1_BASE_POINT = PrimePoint(
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SECP384R1_CURVE,
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0xaa87ca22be8b05378eb1c71ef320ad746e1d3b628ba79b9859f741e082542a385502f25dbf55296c3a545e3872760ab7,
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0x3617de4a96262c6f5d9e98bf9292dc29f8f41dbd289a147ce9da3113b5f0b8c00a60b1ce1d7e819d7a431d7c90ea0e5f,
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39402006196394479212279040100143613805079739270465446667946905279627659399113263569398956308152294913554433653942643
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)
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# NIST Curve P-521:
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SECP521R1_CURVE = PrimeCurve(
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6864797660130609714981900799081393217269435300143305409394463459185543183397656052122559640661454554977296311391480858037121987999716643812574028291115057151, # noqa
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-3,
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0x051953eb9618e1c9a1f929a21a0b68540eea2da725b99b315f3b8b489918ef109e156193951ec7e937b1652c0bd3bb1bf073573df883d2c34f1ef451fd46b503f00 # noqa
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)
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SECP521R1_BASE_POINT = PrimePoint(
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SECP521R1_CURVE,
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0xc6858e06b70404e9cd9e3ecb662395b4429c648139053fb521f828af606b4d3dbaa14b5e77efe75928fe1dc127a2ffa8de3348b3c1856a429bf97e7e31c2e5bd66, # noqa
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0x11839296a789a3bc0045c8a5fb42c7d1bd998f54449579b446817afbd17273e662c97ee72995ef42640c550b9013fad0761353c7086a272c24088be94769fd16650, # noqa
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6864797660130609714981900799081393217269435300143305409394463459185543183397655394245057746333217197532963996371363321113864768612440380340372808892707005449 # noqa
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)
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