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Diffstat (limited to 'frozen_deps/ecdsa/ellipticcurve.py')
| -rw-r--r-- | frozen_deps/ecdsa/ellipticcurve.py | 847 |
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diff --git a/frozen_deps/ecdsa/ellipticcurve.py b/frozen_deps/ecdsa/ellipticcurve.py new file mode 100644 index 0000000..25565df --- /dev/null +++ b/frozen_deps/ecdsa/ellipticcurve.py @@ -0,0 +1,847 @@ +#! /usr/bin/env python +# -*- coding: utf-8 -*- +# +# 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 + +try: + from gmpy2 import mpz + + GMPY = True +except ImportError: + try: + from gmpy import mpz + + GMPY = True + except ImportError: + GMPY = False + + +from six import python_2_unicode_compatible +from . import numbertheory +from ._rwlock import RWLock + + +@python_2_unicode_compatible +class CurveFp(object): + """Elliptic Curve over the field of integers modulo a prime.""" + + if GMPY: + + def __init__(self, p, a, b, h=None): + """ + The curve of points satisfying y^2 = x^3 + a*x + b (mod p). + + h is an integer that is the cofactor of the elliptic curve domain + parameters; it is the number of points satisfying the elliptic + curve equation divided by the order of the base point. It is used + for selection of efficient algorithm for public point verification. + """ + self.__p = mpz(p) + self.__a = mpz(a) + self.__b = mpz(b) + # h is not used in calculations and it can be None, so don't use + # gmpy with it + self.__h = h + + else: + + def __init__(self, p, a, b, h=None): + """ + The curve of points satisfying y^2 = x^3 + a*x + b (mod p). + + h is an integer that is the cofactor of the elliptic curve domain + parameters; it is the number of points satisfying the elliptic + curve equation divided by the order of the base point. It is used + for selection of efficient algorithm for public point verification. + """ + self.__p = p + self.__a = a + self.__b = b + self.__h = h + + def __eq__(self, other): + if isinstance(other, CurveFp): + """Return True if the curves are identical, False otherwise.""" + return ( + self.__p == other.__p + and self.__a == other.__a + and self.__b == other.__b + ) + return NotImplemented + + def __ne__(self, other): + return not (self == other) + + def __hash__(self): + return hash((self.__p, self.__a, self.__b)) + + def p(self): + return self.__p + + def a(self): + return self.__a + + def b(self): + return self.__b + + def cofactor(self): + return self.__h + + def contains_point(self, x, y): + """Is the point (x,y) on this curve?""" + return (y * y - ((x * x + self.__a) * x + self.__b)) % self.__p == 0 + + def __str__(self): + return "CurveFp(p=%d, a=%d, b=%d, h=%d)" % ( + self.__p, + self.__a, + self.__b, + self.__h, + ) + + +class PointJacobi(object): + """ + Point on an elliptic curve. Uses Jacobi coordinates. + + In Jacobian coordinates, there are three parameters, X, Y and Z. + They correspond to affine parameters 'x' and 'y' like so: + + x = X / Z² + y = Y / Z³ + """ + + def __init__(self, curve, x, y, z, order=None, generator=False): + """ + Initialise a point that uses Jacobi representation internally. + + :param CurveFp curve: curve on which the point resides + :param int x: the X parameter of Jacobi representation (equal to x when + converting from affine coordinates + :param int y: the Y parameter of Jacobi representation (equal to y when + converting from affine coordinates + :param int z: the Z parameter of Jacobi representation (equal to 1 when + converting from affine coordinates + :param int order: the point order, must be non zero when using + generator=True + :param bool generator: the point provided is a curve generator, as + such, it will be commonly used with scalar multiplication. This will + cause to precompute multiplication table for it + """ + self.__curve = curve + # since it's generally better (faster) to use scaled points vs unscaled + # ones, use writer-biased RWLock for locking: + self._update_lock = RWLock() + if GMPY: + self.__x = mpz(x) + self.__y = mpz(y) + self.__z = mpz(z) + self.__order = order and mpz(order) + else: + self.__x = x + self.__y = y + self.__z = z + self.__order = order + self.__generator = generator + self.__precompute = [] + + def _maybe_precompute(self): + if self.__generator: + # since we lack promotion of read-locks to write-locks, we do a + # "acquire-read-lock, check, acquire-write-lock plus recheck" cycle + try: + self._update_lock.reader_acquire() + if self.__precompute: + return + finally: + self._update_lock.reader_release() + + try: + self._update_lock.writer_acquire() + if self.__precompute: + return + order = self.__order + assert order + i = 1 + order *= 2 + doubler = PointJacobi( + self.__curve, self.__x, self.__y, self.__z, order + ) + order *= 2 + self.__precompute.append((doubler.x(), doubler.y())) + + while i < order: + i *= 2 + doubler = doubler.double().scale() + self.__precompute.append((doubler.x(), doubler.y())) + + finally: + self._update_lock.writer_release() + + def __getstate__(self): + try: + self._update_lock.reader_acquire() + state = self.__dict__.copy() + finally: + self._update_lock.reader_release() + del state["_update_lock"] + return state + + def __setstate__(self, state): + self.__dict__.update(state) + self._update_lock = RWLock() + + def __eq__(self, other): + """Compare two points with each-other.""" + try: + self._update_lock.reader_acquire() + if other is INFINITY: + return not self.__y or not self.__z + x1, y1, z1 = self.__x, self.__y, self.__z + finally: + self._update_lock.reader_release() + if isinstance(other, Point): + x2, y2, z2 = other.x(), other.y(), 1 + elif isinstance(other, PointJacobi): + try: + other._update_lock.reader_acquire() + x2, y2, z2 = other.__x, other.__y, other.__z + finally: + other._update_lock.reader_release() + else: + return NotImplemented + if self.__curve != other.curve(): + return False + p = self.__curve.p() + + zz1 = z1 * z1 % p + zz2 = z2 * z2 % p + + # compare the fractions by bringing them to the same denominator + # depend on short-circuit to save 4 multiplications in case of + # inequality + return (x1 * zz2 - x2 * zz1) % p == 0 and ( + y1 * zz2 * z2 - y2 * zz1 * z1 + ) % p == 0 + + def order(self): + """Return the order of the point. + + None if it is undefined. + """ + return self.__order + + def curve(self): + """Return curve over which the point is defined.""" + return self.__curve + + def x(self): + """ + Return affine x coordinate. + + This method should be used only when the 'y' coordinate is not needed. + It's computationally more efficient to use `to_affine()` and then + call x() and y() on the returned instance. Or call `scale()` + and then x() and y() on the returned instance. + """ + try: + self._update_lock.reader_acquire() + if self.__z == 1: + return self.__x + x = self.__x + z = self.__z + finally: + self._update_lock.reader_release() + p = self.__curve.p() + z = numbertheory.inverse_mod(z, p) + return x * z ** 2 % p + + def y(self): + """ + Return affine y coordinate. + + This method should be used only when the 'x' coordinate is not needed. + It's computationally more efficient to use `to_affine()` and then + call x() and y() on the returned instance. Or call `scale()` + and then x() and y() on the returned instance. + """ + try: + self._update_lock.reader_acquire() + if self.__z == 1: + return self.__y + y = self.__y + z = self.__z + finally: + self._update_lock.reader_release() + p = self.__curve.p() + z = numbertheory.inverse_mod(z, p) + return y * z ** 3 % p + + def scale(self): + """ + Return point scaled so that z == 1. + + Modifies point in place, returns self. + """ + try: + self._update_lock.reader_acquire() + if self.__z == 1: + return self + finally: + self._update_lock.reader_release() + + try: + self._update_lock.writer_acquire() + # scaling already scaled point is safe (as inverse of 1 is 1) and + # quick so we don't need to optimise for the unlikely event when + # two threads hit the lock at the same time + p = self.__curve.p() + z_inv = numbertheory.inverse_mod(self.__z, p) + zz_inv = z_inv * z_inv % p + self.__x = self.__x * zz_inv % p + self.__y = self.__y * zz_inv * z_inv % p + # we are setting the z last so that the check above will return + # true only after all values were already updated + self.__z = 1 + finally: + self._update_lock.writer_release() + return self + + def to_affine(self): + """Return point in affine form.""" + if not self.__y or not self.__z: + return INFINITY + self.scale() + # after point is scaled, it's immutable, so no need to perform locking + return Point(self.__curve, self.__x, self.__y, self.__order) + + @staticmethod + def from_affine(point, generator=False): + """Create from an affine point. + + :param bool generator: set to True to make the point to precalculate + multiplication table - useful for public point when verifying many + signatures (around 100 or so) or for generator points of a curve. + """ + return PointJacobi( + point.curve(), point.x(), point.y(), 1, point.order(), generator + ) + + # plese note that all the methods that use the equations from hyperelliptic + # are formatted in a way to maximise performance. + # Things that make code faster: multiplying instead of taking to the power + # (`xx = x * x; xxxx = xx * xx % p` is faster than `xxxx = x**4 % p` and + # `pow(x, 4, p)`), + # multiple assignments at the same time (`x1, x2 = self.x1, self.x2` is + # faster than `x1 = self.x1; x2 = self.x2`), + # similarly, sometimes the `% p` is skipped if it makes the calculation + # faster and the result of calculation is later reduced modulo `p` + + def _double_with_z_1(self, X1, Y1, p, a): + """Add a point to itself with z == 1.""" + # after: + # http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian.html#doubling-mdbl-2007-bl + XX, YY = X1 * X1 % p, Y1 * Y1 % p + if not YY: + return 0, 0, 1 + YYYY = YY * YY % p + S = 2 * ((X1 + YY) ** 2 - XX - YYYY) % p + M = 3 * XX + a + T = (M * M - 2 * S) % p + # X3 = T + Y3 = (M * (S - T) - 8 * YYYY) % p + Z3 = 2 * Y1 % p + return T, Y3, Z3 + + def _double(self, X1, Y1, Z1, p, a): + """Add a point to itself, arbitrary z.""" + if Z1 == 1: + return self._double_with_z_1(X1, Y1, p, a) + if not Z1: + return 0, 0, 1 + # after: + # http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian.html#doubling-dbl-2007-bl + XX, YY = X1 * X1 % p, Y1 * Y1 % p + if not YY: + return 0, 0, 1 + YYYY = YY * YY % p + ZZ = Z1 * Z1 % p + S = 2 * ((X1 + YY) ** 2 - XX - YYYY) % p + M = (3 * XX + a * ZZ * ZZ) % p + T = (M * M - 2 * S) % p + # X3 = T + Y3 = (M * (S - T) - 8 * YYYY) % p + Z3 = ((Y1 + Z1) ** 2 - YY - ZZ) % p + + return T, Y3, Z3 + + def double(self): + """Add a point to itself.""" + if not self.__y: + return INFINITY + + p, a = self.__curve.p(), self.__curve.a() + + try: + self._update_lock.reader_acquire() + X1, Y1, Z1 = self.__x, self.__y, self.__z + finally: + self._update_lock.reader_release() + + X3, Y3, Z3 = self._double(X1, Y1, Z1, p, a) + + if not Y3 or not Z3: + return INFINITY + return PointJacobi(self.__curve, X3, Y3, Z3, self.__order) + + def _add_with_z_1(self, X1, Y1, X2, Y2, p): + """add points when both Z1 and Z2 equal 1""" + # after: + # http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian.html#addition-mmadd-2007-bl + H = X2 - X1 + HH = H * H + I = 4 * HH % p + J = H * I + r = 2 * (Y2 - Y1) + if not H and not r: + return self._double_with_z_1(X1, Y1, p, self.__curve.a()) + V = X1 * I + X3 = (r ** 2 - J - 2 * V) % p + Y3 = (r * (V - X3) - 2 * Y1 * J) % p + Z3 = 2 * H % p + return X3, Y3, Z3 + + def _add_with_z_eq(self, X1, Y1, Z1, X2, Y2, p): + """add points when Z1 == Z2""" + # after: + # http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian.html#addition-zadd-2007-m + A = (X2 - X1) ** 2 % p + B = X1 * A % p + C = X2 * A + D = (Y2 - Y1) ** 2 % p + if not A and not D: + return self._double(X1, Y1, Z1, p, self.__curve.a()) + X3 = (D - B - C) % p + Y3 = ((Y2 - Y1) * (B - X3) - Y1 * (C - B)) % p + Z3 = Z1 * (X2 - X1) % p + return X3, Y3, Z3 + + def _add_with_z2_1(self, X1, Y1, Z1, X2, Y2, p): + """add points when Z2 == 1""" + # after: + # http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian.html#addition-madd-2007-bl + Z1Z1 = Z1 * Z1 % p + U2, S2 = X2 * Z1Z1 % p, Y2 * Z1 * Z1Z1 % p + H = (U2 - X1) % p + HH = H * H % p + I = 4 * HH % p + J = H * I + r = 2 * (S2 - Y1) % p + if not r and not H: + return self._double_with_z_1(X2, Y2, p, self.__curve.a()) + V = X1 * I + X3 = (r * r - J - 2 * V) % p + Y3 = (r * (V - X3) - 2 * Y1 * J) % p + Z3 = ((Z1 + H) ** 2 - Z1Z1 - HH) % p + return X3, Y3, Z3 + + def _add_with_z_ne(self, X1, Y1, Z1, X2, Y2, Z2, p): + """add points with arbitrary z""" + # after: + # http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian.html#addition-add-2007-bl + Z1Z1 = Z1 * Z1 % p + Z2Z2 = Z2 * Z2 % p + U1 = X1 * Z2Z2 % p + U2 = X2 * Z1Z1 % p + S1 = Y1 * Z2 * Z2Z2 % p + S2 = Y2 * Z1 * Z1Z1 % p + H = U2 - U1 + I = 4 * H * H % p + J = H * I % p + r = 2 * (S2 - S1) % p + if not H and not r: + return self._double(X1, Y1, Z1, p, self.__curve.a()) + V = U1 * I + X3 = (r * r - J - 2 * V) % p + Y3 = (r * (V - X3) - 2 * S1 * J) % p + Z3 = ((Z1 + Z2) ** 2 - Z1Z1 - Z2Z2) * H % p + + return X3, Y3, Z3 + + def __radd__(self, other): + """Add other to self.""" + return self + other + + def _add(self, X1, Y1, Z1, X2, Y2, Z2, p): + """add two points, select fastest method.""" + if not Y1 or not Z1: + return X2, Y2, Z2 + if not Y2 or not Z2: + return X1, Y1, Z1 + if Z1 == Z2: + if Z1 == 1: + return self._add_with_z_1(X1, Y1, X2, Y2, p) + return self._add_with_z_eq(X1, Y1, Z1, X2, Y2, p) + if Z1 == 1: + return self._add_with_z2_1(X2, Y2, Z2, X1, Y1, p) + if Z2 == 1: + return self._add_with_z2_1(X1, Y1, Z1, X2, Y2, p) + return self._add_with_z_ne(X1, Y1, Z1, X2, Y2, Z2, p) + + def __add__(self, other): + """Add two points on elliptic curve.""" + if self == INFINITY: + return other + if other == INFINITY: + return self + if isinstance(other, Point): + other = PointJacobi.from_affine(other) + if self.__curve != other.__curve: + raise ValueError("The other point is on different curve") + + p = self.__curve.p() + try: + self._update_lock.reader_acquire() + X1, Y1, Z1 = self.__x, self.__y, self.__z + finally: + self._update_lock.reader_release() + try: + other._update_lock.reader_acquire() + X2, Y2, Z2 = other.__x, other.__y, other.__z + finally: + other._update_lock.reader_release() + X3, Y3, Z3 = self._add(X1, Y1, Z1, X2, Y2, Z2, p) + + if not Y3 or not Z3: + return INFINITY + return PointJacobi(self.__curve, X3, Y3, Z3, self.__order) + + def __rmul__(self, other): + """Multiply point by an integer.""" + return self * other + + def _mul_precompute(self, other): + """Multiply point by integer with precomputation table.""" + X3, Y3, Z3, p = 0, 0, 1, self.__curve.p() + _add = self._add + for X2, Y2 in self.__precompute: + if other % 2: + if other % 4 >= 2: + other = (other + 1) // 2 + X3, Y3, Z3 = _add(X3, Y3, Z3, X2, -Y2, 1, p) + else: + other = (other - 1) // 2 + X3, Y3, Z3 = _add(X3, Y3, Z3, X2, Y2, 1, p) + else: + other //= 2 + + if not Y3 or not Z3: + return INFINITY + return PointJacobi(self.__curve, X3, Y3, Z3, self.__order) + + @staticmethod + def _naf(mult): + """Calculate non-adjacent form of number.""" + ret = [] + while mult: + if mult % 2: + nd = mult % 4 + if nd >= 2: + nd = nd - 4 + ret += [nd] + mult -= nd + else: + ret += [0] + mult //= 2 + return ret + + def __mul__(self, other): + """Multiply point by an integer.""" + if not self.__y or not other: + return INFINITY + if other == 1: + return self + if self.__order: + # order*2 as a protection for Minerva + other = other % (self.__order * 2) + self._maybe_precompute() + if self.__precompute: + return self._mul_precompute(other) + + self = self.scale() + # once scaled, point is immutable, not need to lock + X2, Y2 = self.__x, self.__y + X3, Y3, Z3 = 0, 0, 1 + p, a = self.__curve.p(), self.__curve.a() + _double = self._double + _add = self._add + # since adding points when at least one of them is scaled + # is quicker, reverse the NAF order + for i in reversed(self._naf(other)): + X3, Y3, Z3 = _double(X3, Y3, Z3, p, a) + if i < 0: + X3, Y3, Z3 = _add(X3, Y3, Z3, X2, -Y2, 1, p) + elif i > 0: + X3, Y3, Z3 = _add(X3, Y3, Z3, X2, Y2, 1, p) + + if not Y3 or not Z3: + return INFINITY + + return PointJacobi(self.__curve, X3, Y3, Z3, self.__order) + + @staticmethod + def _leftmost_bit(x): + """Return integer with the same magnitude as x but only one bit set""" + assert x > 0 + result = 1 + while result <= x: + result = 2 * result + return result // 2 + + def mul_add(self, self_mul, other, other_mul): + """ + Do two multiplications at the same time, add results. + + calculates self*self_mul + other*other_mul + """ + if other is INFINITY or other_mul == 0: + return self * self_mul + if self_mul == 0: + return other * other_mul + if not isinstance(other, PointJacobi): + other = PointJacobi.from_affine(other) + # when the points have precomputed answers, then multiplying them alone + # is faster (as it uses NAF) + self._maybe_precompute() + other._maybe_precompute() + if self.__precompute and other.__precompute: + return self * self_mul + other * other_mul + + if self.__order: + self_mul = self_mul % self.__order + other_mul = other_mul % self.__order + + i = self._leftmost_bit(max(self_mul, other_mul)) * 2 + X3, Y3, Z3 = 0, 0, 1 + p, a = self.__curve.p(), self.__curve.a() + self = self.scale() + # after scaling, point is immutable, no need for locking + X1, Y1 = self.__x, self.__y + other = other.scale() + X2, Y2 = other.__x, other.__y + both = self + other + if both is INFINITY: + X4, Y4 = 0, 0 + else: + both.scale() + X4, Y4 = both.__x, both.__y + _double = self._double + _add = self._add + while i > 1: + X3, Y3, Z3 = _double(X3, Y3, Z3, p, a) + i = i // 2 + + if self_mul & i and other_mul & i: + X3, Y3, Z3 = _add(X3, Y3, Z3, X4, Y4, 1, p) + elif self_mul & i: + X3, Y3, Z3 = _add(X3, Y3, Z3, X1, Y1, 1, p) + elif other_mul & i: + X3, Y3, Z3 = _add(X3, Y3, Z3, X2, Y2, 1, p) + + if not Y3 or not Z3: + return INFINITY + + return PointJacobi(self.__curve, X3, Y3, Z3, self.__order) + + def __neg__(self): + """Return negated point.""" + try: + self._update_lock.reader_acquire() + return PointJacobi( + self.__curve, self.__x, -self.__y, self.__z, self.__order + ) + finally: + self._update_lock.reader_release() + + +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 + if GMPY: + self.__x = x and mpz(x) + self.__y = y and mpz(y) + self.__order = order and mpz(order) + else: + 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) + # for curves with cofactor 1, all points that are on the curve are + # scalar multiples of the base point, so performing multiplication is + # not necessary to verify that. See Section 3.2.2.1 of SEC 1 v2 + if curve and curve.cofactor() != 1 and order: + assert self * order == INFINITY + + def __eq__(self, other): + """Return True if the points are identical, False otherwise.""" + if isinstance(other, Point): + return ( + self.__curve == other.__curve + and self.__x == other.__x + and self.__y == other.__y + ) + return NotImplemented + + def __neg__(self): + return Point(self.__curve, self.__x, self.__curve.p() - self.__y) + + def __add__(self, other): + """Add one point to another point.""" + + # X9.62 B.3: + + if not isinstance(other, Point): + return NotImplemented + 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 e == 0 or (self.__order and e % self.__order == 0): + return INFINITY + if self == INFINITY: + return INFINITY + if e < 0: + return (-self) * (-e) + + # 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) |