#! /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].
#
# 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.
# Modified extensively as part of python-ecdsa.

from __future__ import division

try:
    from gmpy2 import mpz

    GMPY = True
except ImportError:  # pragma: no branch
    try:
        from gmpy import mpz

        GMPY = True
    except ImportError:
        GMPY = False


from six import python_2_unicode_compatible
from . import numbertheory


@python_2_unicode_compatible
class CurveFp(object):
    """Elliptic Curve over the field of integers modulo a prime."""

    if GMPY:  # pragma: no branch

        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:  # pragma: no branch

        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):
        """Return True if other is an identical curve, False otherwise.

        Note: the value of the cofactor of the curve is not taken into account
        when comparing curves, as it's derived from the base point and
        intrinsic curve characteristic (but it's complex to compute),
        only the prime and curve parameters are considered.
        """
        if isinstance(other, CurveFp):
            return (
                self.__p == other.__p
                and self.__a == other.__a
                and self.__b == other.__b
            )
        return NotImplemented

    def __ne__(self, other):
        """Return False if other is an identical curve, True otherwise."""
        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 generation for it
        """
        self.__curve = curve
        if GMPY:  # pragma: no branch
            self.__coords = (mpz(x), mpz(y), mpz(z))
            self.__order = order and mpz(order)
        else:  # pragma: no branch
            self.__coords = (x, y, z)
            self.__order = order
        self.__generator = generator
        self.__precompute = []

    def _maybe_precompute(self):
        if not self.__generator or self.__precompute:
            return

        # since this code will execute just once, and it's fully deterministic,
        # depend on atomicity of the last assignment to switch from empty
        # self.__precompute to filled one and just ignore the unlikely
        # situation when two threads execute it at the same time (as it won't
        # lead to inconsistent __precompute)
        order = self.__order
        assert order
        precompute = []
        i = 1
        order *= 2
        coord_x, coord_y, coord_z = self.__coords
        doubler = PointJacobi(self.__curve, coord_x, coord_y, coord_z, order)
        order *= 2
        precompute.append((doubler.x(), doubler.y()))

        while i < order:
            i *= 2
            doubler = doubler.double().scale()
            precompute.append((doubler.x(), doubler.y()))

        self.__precompute = precompute

    def __getstate__(self):
        # while this code can execute at the same time as _maybe_precompute()
        # is updating the __precompute or scale() is updating the __coords,
        # there is no requirement for consistency between __coords and
        # __precompute
        state = self.__dict__.copy()
        return state

    def __setstate__(self, state):
        self.__dict__.update(state)

    def __eq__(self, other):
        """Compare for equality two points with each-other.

        Note: only points that lay on the same curve can be equal.
        """
        x1, y1, z1 = self.__coords
        if other is INFINITY:
            return not y1 or not z1
        if isinstance(other, Point):
            x2, y2, z2 = other.x(), other.y(), 1
        elif isinstance(other, PointJacobi):
            x2, y2, z2 = other.__coords
        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 __ne__(self, other):
        """Compare for inequality two points with each-other."""
        return not self == other

    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.
        """
        x, _, z = self.__coords
        if z == 1:
            return x
        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.
        """
        _, y, z = self.__coords
        if z == 1:
            return y
        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.
        """
        x, y, z = self.__coords
        if z == 1:
            return self

        # scaling is deterministic, so even if two threads execute the below
        # code at the same time, they will set __coords to the same value
        p = self.__curve.p()
        z_inv = numbertheory.inverse_mod(z, p)
        zz_inv = z_inv * z_inv % p
        x = x * zz_inv % p
        y = y * zz_inv * z_inv % p
        self.__coords = (x, y, 1)
        return self

    def to_affine(self):
        """Return point in affine form."""
        _, y, z = self.__coords
        if not y or not z:
            return INFINITY
        self.scale()
        x, y, z = self.__coords
        return Point(self.__curve, x, 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
        )

    # please 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 Y1 or 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."""
        X1, Y1, Z1 = self.__coords

        if not Y1:
            return INFINITY

        p, a = self.__curve.p(), self.__curve.a()

        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()
        X1, Y1, Z1 = self.__coords
        X2, Y2, Z2 = other.__coords

        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 -= 4
                ret.append(nd)
                mult -= nd
            else:
                ret.append(0)
            mult //= 2
        return ret

    def __mul__(self, other):
        """Multiply point by an integer."""
        if not self.__coords[1] 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()
        X2, Y2, _ = self.__coords
        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)

    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 == 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 and no point doublings)
        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

        # (X3, Y3, Z3) is the accumulator
        X3, Y3, Z3 = 0, 0, 1
        p, a = self.__curve.p(), self.__curve.a()

        # as we have 6 unique points to work with, we can't scale all of them,
        # but do scale the ones that are used most often
        self.scale()
        X1, Y1, Z1 = self.__coords
        other.scale()
        X2, Y2, Z2 = other.__coords

        _double = self._double
        _add = self._add

        # with NAF we have 3 options: no add, subtract, add
        # so with 2 points, we have 9 combinations:
        # 0, -A, +A, -B, -A-B, +A-B, +B, -A+B, +A+B
        # so we need 4 combined points:
        mAmB_X, mAmB_Y, mAmB_Z = _add(X1, -Y1, Z1, X2, -Y2, Z2, p)
        pAmB_X, pAmB_Y, pAmB_Z = _add(X1, Y1, Z1, X2, -Y2, Z2, p)
        mApB_X, mApB_Y, mApB_Z = _add(X1, -Y1, Z1, X2, Y2, Z2, p)
        pApB_X, pApB_Y, pApB_Z = _add(X1, Y1, Z1, X2, Y2, Z2, p)
        # when the self and other sum to infinity, we need to add them
        # one by one to get correct result but as that's very unlikely to
        # happen in regular operation, we don't need to optimise this case
        if not pApB_Y or not pApB_Z:
            return self * self_mul + other * other_mul

        # gmp object creation has cumulatively higher overhead than the
        # speedup we get from calculating the NAF using gmp so ensure use
        # of int()
        self_naf = list(reversed(self._naf(int(self_mul))))
        other_naf = list(reversed(self._naf(int(other_mul))))
        # ensure that the lists are the same length (zip() will truncate
        # longer one otherwise)
        if len(self_naf) < len(other_naf):
            self_naf = [0] * (len(other_naf) - len(self_naf)) + self_naf
        elif len(self_naf) > len(other_naf):
            other_naf = [0] * (len(self_naf) - len(other_naf)) + other_naf

        for A, B in zip(self_naf, other_naf):
            X3, Y3, Z3 = _double(X3, Y3, Z3, p, a)

            # conditions ordered from most to least likely
            if A == 0:
                if B == 0:
                    pass
                elif B < 0:
                    X3, Y3, Z3 = _add(X3, Y3, Z3, X2, -Y2, Z2, p)
                else:
                    assert B > 0
                    X3, Y3, Z3 = _add(X3, Y3, Z3, X2, Y2, Z2, p)
            elif A < 0:
                if B == 0:
                    X3, Y3, Z3 = _add(X3, Y3, Z3, X1, -Y1, Z1, p)
                elif B < 0:
                    X3, Y3, Z3 = _add(X3, Y3, Z3, mAmB_X, mAmB_Y, mAmB_Z, p)
                else:
                    assert B > 0
                    X3, Y3, Z3 = _add(X3, Y3, Z3, mApB_X, mApB_Y, mApB_Z, p)
            else:
                assert A > 0
                if B == 0:
                    X3, Y3, Z3 = _add(X3, Y3, Z3, X1, Y1, Z1, p)
                elif B < 0:
                    X3, Y3, Z3 = _add(X3, Y3, Z3, pAmB_X, pAmB_Y, pAmB_Z, p)
                else:
                    assert B > 0
                    X3, Y3, Z3 = _add(X3, Y3, Z3, pApB_X, pApB_Y, pApB_Z, p)

        if not Y3 or not Z3:
            return INFINITY

        return PointJacobi(self.__curve, X3, Y3, Z3, self.__order)

    def __neg__(self):
        """Return negated point."""
        x, y, z = self.__coords
        return PointJacobi(self.__curve, x, -y, z, self.__order)


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.

        Note: only points that lay on the same curve can be equal.
        """
        if isinstance(other, Point):
            return (
                self.__curve == other.__curve
                and self.__x == other.__x
                and self.__y == other.__y
            )
        return NotImplemented

    def __ne__(self, other):
        """Returns False if points are identical, True otherwise."""
        return not self == other

    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)
