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#! /usr/bin/env python
#
# Implementation of elliptic curves, for cryptographic applications.
#
# This module doesn't provide any way to choose a random elliptic
# curve, nor to verify that an elliptic curve was chosen randomly,
# because one can simply use NIST's standard curves.
#
# Notes from X9.62-1998 (draft):
# Nomenclature:
# - Q is a public key.
# The "Elliptic Curve Domain Parameters" include:
# - q is the "field size", which in our case equals p.
# - p is a big prime.
# - G is a point of prime order (5.1.1.1).
# - n is the order of G (5.1.1.1).
# Public-key validation (5.2.2):
# - Verify that Q is not the point at infinity.
# - Verify that X_Q and Y_Q are in [0,p-1].
# - Verify that Q is on the curve.
# - Verify that nQ is the point at infinity.
# Signature generation (5.3):
# - Pick random k from [1,n-1].
# Signature checking (5.4.2):
# - Verify that r and s are in [1,n-1].
#
# Version of 2008.11.25.
#
# Revision history:
# 2005.12.31 - Initial version.
# 2008.11.25 - Change CurveFp.is_on to contains_point.
#
# Written in 2005 by Peter Pearson and placed in the public domain.
from __future__ import division
from .six import print_
from . import numbertheory
class CurveFp( object ):
"""Elliptic Curve over the field of integers modulo a prime."""
def __init__( self, p, a, b ):
"""The curve of points satisfying y^2 = x^3 + a*x + b (mod p)."""
self.__p = p
self.__a = a
self.__b = b
def p( self ):
return self.__p
def a( self ):
return self.__a
def b( self ):
return self.__b
def contains_point( self, x, y ):
"""Is the point (x,y) on this curve?"""
return ( y * y - ( x * x * x + self.__a * x + self.__b ) ) % self.__p == 0
class Point( object ):
"""A point on an elliptic curve. Altering x and y is forbidding,
but they can be read by the x() and y() methods."""
def __init__( self, curve, x, y, order = None ):
"""curve, x, y, order; order (optional) is the order of this point."""
self.__curve = curve
self.__x = x
self.__y = y
self.__order = order
# self.curve is allowed to be None only for INFINITY:
if self.__curve: assert self.__curve.contains_point( x, y )
if order: assert self * order == INFINITY
def __eq__( self, other ):
"""Return True if the points are identical, False otherwise."""
if self.__curve == other.__curve \
and self.__x == other.__x \
and self.__y == other.__y:
return True
else:
return False
def __add__( self, other ):
"""Add one point to another point."""
# X9.62 B.3:
if other == INFINITY: return self
if self == INFINITY: return other
assert self.__curve == other.__curve
if self.__x == other.__x:
if ( self.__y + other.__y ) % self.__curve.p() == 0:
return INFINITY
else:
return self.double()
p = self.__curve.p()
l = ( ( other.__y - self.__y ) * \
numbertheory.inverse_mod( other.__x - self.__x, p ) ) % p
x3 = ( l * l - self.__x - other.__x ) % p
y3 = ( l * ( self.__x - x3 ) - self.__y ) % p
return Point( self.__curve, x3, y3 )
def __mul__( self, other ):
"""Multiply a point by an integer."""
def leftmost_bit( x ):
assert x > 0
result = 1
while result <= x: result = 2 * result
return result // 2
e = other
if self.__order: e = e % self.__order
if e == 0: return INFINITY
if self == INFINITY: return INFINITY
assert e > 0
# From X9.62 D.3.2:
e3 = 3 * e
negative_self = Point( self.__curve, self.__x, -self.__y, self.__order )
i = leftmost_bit( e3 ) // 2
result = self
# print_("Multiplying %s by %d (e3 = %d):" % ( self, other, e3 ))
while i > 1:
result = result.double()
if ( e3 & i ) != 0 and ( e & i ) == 0: result = result + self
if ( e3 & i ) == 0 and ( e & i ) != 0: result = result + negative_self
# print_(". . . i = %d, result = %s" % ( i, result ))
i = i // 2
return result
def __rmul__( self, other ):
"""Multiply a point by an integer."""
return self * other
def __str__( self ):
if self == INFINITY: return "infinity"
return "(%d,%d)" % ( self.__x, self.__y )
def double( self ):
"""Return a new point that is twice the old."""
if self == INFINITY:
return INFINITY
# X9.62 B.3:
p = self.__curve.p()
a = self.__curve.a()
l = ( ( 3 * self.__x * self.__x + a ) * \
numbertheory.inverse_mod( 2 * self.__y, p ) ) % p
x3 = ( l * l - 2 * self.__x ) % p
y3 = ( l * ( self.__x - x3 ) - self.__y ) % p
return Point( self.__curve, x3, y3 )
def x( self ):
return self.__x
def y( self ):
return self.__y
def curve( self ):
return self.__curve
def order( self ):
return self.__order
# This one point is the Point At Infinity for all purposes:
INFINITY = Point( None, None, None )
def __main__():
class FailedTest(Exception): pass
def test_add( c, x1, y1, x2, y2, x3, y3 ):
"""We expect that on curve c, (x1,y1) + (x2, y2 ) = (x3, y3)."""
p1 = Point( c, x1, y1 )
p2 = Point( c, x2, y2 )
p3 = p1 + p2
print_("%s + %s = %s" % ( p1, p2, p3 ), end=' ')
if p3.x() != x3 or p3.y() != y3:
raise FailedTest("Failure: should give (%d,%d)." % ( x3, y3 ))
else:
print_(" Good.")
def test_double( c, x1, y1, x3, y3 ):
"""We expect that on curve c, 2*(x1,y1) = (x3, y3)."""
p1 = Point( c, x1, y1 )
p3 = p1.double()
print_("%s doubled = %s" % ( p1, p3 ), end=' ')
if p3.x() != x3 or p3.y() != y3:
raise FailedTest("Failure: should give (%d,%d)." % ( x3, y3 ))
else:
print_(" Good.")
def test_double_infinity( c ):
"""We expect that on curve c, 2*INFINITY = INFINITY."""
p1 = INFINITY
p3 = p1.double()
print_("%s doubled = %s" % ( p1, p3 ), end=' ')
if p3.x() != INFINITY.x() or p3.y() != INFINITY.y():
raise FailedTest("Failure: should give (%d,%d)." % ( INFINITY.x(), INFINITY.y() ))
else:
print_(" Good.")
def test_multiply( c, x1, y1, m, x3, y3 ):
"""We expect that on curve c, m*(x1,y1) = (x3,y3)."""
p1 = Point( c, x1, y1 )
p3 = p1 * m
print_("%s * %d = %s" % ( p1, m, p3 ), end=' ')
if p3.x() != x3 or p3.y() != y3:
raise FailedTest("Failure: should give (%d,%d)." % ( x3, y3 ))
else:
print_(" Good.")
# A few tests from X9.62 B.3:
c = CurveFp( 23, 1, 1 )
test_add( c, 3, 10, 9, 7, 17, 20 )
test_double( c, 3, 10, 7, 12 )
test_add( c, 3, 10, 3, 10, 7, 12 ) # (Should just invoke double.)
test_multiply( c, 3, 10, 2, 7, 12 )
test_double_infinity(c)
# From X9.62 I.1 (p. 96):
g = Point( c, 13, 7, 7 )
check = INFINITY
for i in range( 7 + 1 ):
p = ( i % 7 ) * g
print_("%s * %d = %s, expected %s . . ." % ( g, i, p, check ), end=' ')
if p == check:
print_(" Good.")
else:
raise FailedTest("Bad.")
check = check + g
# NIST Curve P-192:
p = 6277101735386680763835789423207666416083908700390324961279
r = 6277101735386680763835789423176059013767194773182842284081
#s = 0x3045ae6fc8422f64ed579528d38120eae12196d5L
c = 0x3099d2bbbfcb2538542dcd5fb078b6ef5f3d6fe2c745de65
b = 0x64210519e59c80e70fa7e9ab72243049feb8deecc146b9b1
Gx = 0x188da80eb03090f67cbf20eb43a18800f4ff0afd82ff1012
Gy = 0x07192b95ffc8da78631011ed6b24cdd573f977a11e794811
c192 = CurveFp( p, -3, b )
p192 = Point( c192, Gx, Gy, r )
# Checking against some sample computations presented
# in X9.62:
d = 651056770906015076056810763456358567190100156695615665659
Q = d * p192
if Q.x() != 0x62B12D60690CDCF330BABAB6E69763B471F994DD702D16A5:
raise FailedTest("p192 * d came out wrong.")
else:
print_("p192 * d came out right.")
k = 6140507067065001063065065565667405560006161556565665656654
R = k * p192
if R.x() != 0x885052380FF147B734C330C43D39B2C4A89F29B0F749FEAD \
or R.y() != 0x9CF9FA1CBEFEFB917747A3BB29C072B9289C2547884FD835:
raise FailedTest("k * p192 came out wrong.")
else:
print_("k * p192 came out right.")
u1 = 2563697409189434185194736134579731015366492496392189760599
u2 = 6266643813348617967186477710235785849136406323338782220568
temp = u1 * p192 + u2 * Q
if temp.x() != 0x885052380FF147B734C330C43D39B2C4A89F29B0F749FEAD \
or temp.y() != 0x9CF9FA1CBEFEFB917747A3BB29C072B9289C2547884FD835:
raise FailedTest("u1 * p192 + u2 * Q came out wrong.")
else:
print_("u1 * p192 + u2 * Q came out right.")
if __name__ == "__main__":
__main__()