291 lines
8.3 KiB
Python
291 lines
8.3 KiB
Python
#! /usr/bin/env python
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#
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# Implementation of elliptic curves, for cryptographic applications.
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#
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# This module doesn't provide any way to choose a random elliptic
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# curve, nor to verify that an elliptic curve was chosen randomly,
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# because one can simply use NIST's standard curves.
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#
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# Notes from X9.62-1998 (draft):
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# Nomenclature:
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# - Q is a public key.
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# The "Elliptic Curve Domain Parameters" include:
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# - q is the "field size", which in our case equals p.
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# - p is a big prime.
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# - G is a point of prime order (5.1.1.1).
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# - n is the order of G (5.1.1.1).
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# Public-key validation (5.2.2):
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# - Verify that Q is not the point at infinity.
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# - Verify that X_Q and Y_Q are in [0,p-1].
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# - Verify that Q is on the curve.
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# - Verify that nQ is the point at infinity.
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# Signature generation (5.3):
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# - Pick random k from [1,n-1].
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# Signature checking (5.4.2):
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# - Verify that r and s are in [1,n-1].
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#
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# Version of 2008.11.25.
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#
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# Revision history:
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# 2005.12.31 - Initial version.
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# 2008.11.25 - Change CurveFp.is_on to contains_point.
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#
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# Written in 2005 by Peter Pearson and placed in the public domain.
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import numbertheory
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class CurveFp( object ):
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"""Elliptic Curve over the field of integers modulo a prime."""
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def __init__( self, p, a, b ):
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"""The curve of points satisfying y^2 = x^3 + a*x + b (mod p)."""
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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 p( self ):
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return self.__p
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def a( self ):
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return self.__a
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def b( self ):
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return self.__b
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def contains_point( self, x, y ):
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"""Is the point (x,y) on this curve?"""
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return ( y * y - ( x * x * x + self.__a * x + self.__b ) ) % self.__p == 0
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class Point( object ):
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"""A point on an elliptic curve. Altering x and y is forbidding,
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but they can be read by the x() and y() methods."""
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def __init__( self, curve, x, y, order = None ):
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"""curve, x, y, order; order (optional) is the order of this point."""
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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: assert self.__curve.contains_point( x, y )
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if order: assert self * order == INFINITY
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def __cmp__( self, other ):
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"""Return 0 if the points are identical, 1 otherwise."""
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if self.__curve == other.__curve \
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and self.__x == other.__x \
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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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"""Add one point to another point."""
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# X9.62 B.3:
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if other == INFINITY: return self
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if self == INFINITY: 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 ) * \
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numbertheory.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 Point( self.__curve, x3, y3 )
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def __mul__( self, other ):
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"""Multiply a point by an integer."""
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def leftmost_bit( x ):
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assert x > 0
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result = 1L
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while result <= x: result = 2 * result
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return result // 2
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e = other
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if self.__order: e = e % self.__order
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if e == 0: return INFINITY
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if self == INFINITY: 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 = Point( 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: result = result + self
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if ( e3 & i ) == 0 and ( e & i ) != 0: 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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"""Multiply a point by an integer."""
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return self * other
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def __str__( self ):
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if self == INFINITY: return "infinity"
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return "(%d,%d)" % ( self.__x, self.__y )
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def double( self ):
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"""Return a new point that is twice the old."""
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if self == INFINITY:
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return INFINITY
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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 ) * \
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numbertheory.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 Point( self.__curve, x3, y3 )
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def x( self ):
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return self.__x
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def y( self ):
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return self.__y
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def curve( self ):
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return self.__curve
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def order( self ):
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return self.__order
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# This one point is the Point At Infinity for all purposes:
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INFINITY = Point( None, None, None )
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def __main__():
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class FailedTest(Exception): pass
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def test_add( c, x1, y1, x2, y2, x3, y3 ):
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"""We expect that on curve c, (x1,y1) + (x2, y2 ) = (x3, y3)."""
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p1 = Point( c, x1, y1 )
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p2 = Point( c, x2, y2 )
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p3 = p1 + p2
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print "%s + %s = %s" % ( p1, p2, p3 ),
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if p3.x() != x3 or p3.y() != y3:
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raise FailedTest("Failure: should give (%d,%d)." % ( x3, y3 ))
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else:
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print " Good."
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def test_double( c, x1, y1, x3, y3 ):
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"""We expect that on curve c, 2*(x1,y1) = (x3, y3)."""
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p1 = Point( c, x1, y1 )
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p3 = p1.double()
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print "%s doubled = %s" % ( p1, p3 ),
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if p3.x() != x3 or p3.y() != y3:
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raise FailedTest("Failure: should give (%d,%d)." % ( x3, y3 ))
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else:
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print " Good."
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def test_double_infinity( c ):
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"""We expect that on curve c, 2*INFINITY = INFINITY."""
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p1 = INFINITY
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p3 = p1.double()
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print "%s doubled = %s" % ( p1, p3 ),
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if p3.x() != INFINITY.x() or p3.y() != INFINITY.y():
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raise FailedTest("Failure: should give (%d,%d)." % ( INFINITY.x(), INFINITY.y() ))
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else:
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print " Good."
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def test_multiply( c, x1, y1, m, x3, y3 ):
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"""We expect that on curve c, m*(x1,y1) = (x3,y3)."""
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p1 = Point( c, x1, y1 )
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p3 = p1 * m
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print "%s * %d = %s" % ( p1, m, p3 ),
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if p3.x() != x3 or p3.y() != y3:
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raise FailedTest("Failure: should give (%d,%d)." % ( x3, y3 ))
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else:
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print " Good."
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# A few tests from X9.62 B.3:
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c = CurveFp( 23, 1, 1 )
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test_add( c, 3, 10, 9, 7, 17, 20 )
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test_double( c, 3, 10, 7, 12 )
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test_add( c, 3, 10, 3, 10, 7, 12 ) # (Should just invoke double.)
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test_multiply( c, 3, 10, 2, 7, 12 )
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test_double_infinity(c)
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# From X9.62 I.1 (p. 96):
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g = Point( c, 13, 7, 7 )
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check = INFINITY
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for i in range( 7 + 1 ):
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p = ( i % 7 ) * g
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print "%s * %d = %s, expected %s . . ." % ( g, i, p, check ),
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if p == check:
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print " Good."
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else:
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raise FailedTest("Bad.")
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check = check + g
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# NIST Curve P-192:
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p = 6277101735386680763835789423207666416083908700390324961279L
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r = 6277101735386680763835789423176059013767194773182842284081L
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#s = 0x3045ae6fc8422f64ed579528d38120eae12196d5L
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c = 0x3099d2bbbfcb2538542dcd5fb078b6ef5f3d6fe2c745de65L
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b = 0x64210519e59c80e70fa7e9ab72243049feb8deecc146b9b1L
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Gx = 0x188da80eb03090f67cbf20eb43a18800f4ff0afd82ff1012L
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Gy = 0x07192b95ffc8da78631011ed6b24cdd573f977a11e794811L
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c192 = CurveFp( p, -3, b )
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p192 = Point( c192, Gx, Gy, r )
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# Checking against some sample computations presented
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# in X9.62:
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d = 651056770906015076056810763456358567190100156695615665659L
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Q = d * p192
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if Q.x() != 0x62B12D60690CDCF330BABAB6E69763B471F994DD702D16A5L:
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raise FailedTest("p192 * d came out wrong.")
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else:
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print "p192 * d came out right."
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k = 6140507067065001063065065565667405560006161556565665656654L
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R = k * p192
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if R.x() != 0x885052380FF147B734C330C43D39B2C4A89F29B0F749FEADL \
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or R.y() != 0x9CF9FA1CBEFEFB917747A3BB29C072B9289C2547884FD835L:
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raise FailedTest("k * p192 came out wrong.")
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else:
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print "k * p192 came out right."
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u1 = 2563697409189434185194736134579731015366492496392189760599L
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u2 = 6266643813348617967186477710235785849136406323338782220568L
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temp = u1 * p192 + u2 * Q
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if temp.x() != 0x885052380FF147B734C330C43D39B2C4A89F29B0F749FEADL \
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or temp.y() != 0x9CF9FA1CBEFEFB917747A3BB29C072B9289C2547884FD835L:
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raise FailedTest("u1 * p192 + u2 * Q came out wrong.")
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else:
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print "u1 * p192 + u2 * Q came out right."
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if __name__ == "__main__":
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__main__()
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