399 lines
11 KiB
Python
399 lines
11 KiB
Python
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from sympy.core.numbers import oo
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from sympy.core.relational import Eq
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from sympy.core.symbol import symbols
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from sympy.polys.domains import FiniteField, QQ, RationalField, FF
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from sympy.solvers.solvers import solve
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from sympy.utilities.iterables import is_sequence
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from sympy.utilities.misc import as_int
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from .factor_ import divisors
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from .residue_ntheory import polynomial_congruence
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class EllipticCurve:
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"""
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Create the following Elliptic Curve over domain.
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`y^{2} + a_{1} x y + a_{3} y = x^{3} + a_{2} x^{2} + a_{4} x + a_{6}`
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The default domain is ``QQ``. If no coefficient ``a1``, ``a2``, ``a3``,
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it create curve as following form.
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`y^{2} = x^{3} + a_{4} x + a_{6}`
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Examples
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========
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References
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==========
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.. [1] J. Silverman "A Friendly Introduction to Number Theory" Third Edition
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.. [2] https://mathworld.wolfram.com/EllipticDiscriminant.html
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.. [3] G. Hardy, E. Wright "An Introduction to the Theory of Numbers" Sixth Edition
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"""
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def __init__(self, a4, a6, a1=0, a2=0, a3=0, modulus = 0):
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if modulus == 0:
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domain = QQ
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else:
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domain = FF(modulus)
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a1, a2, a3, a4, a6 = map(domain.convert, (a1, a2, a3, a4, a6))
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self._domain = domain
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self.modulus = modulus
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# Calculate discriminant
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b2 = a1**2 + 4 * a2
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b4 = 2 * a4 + a1 * a3
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b6 = a3**2 + 4 * a6
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b8 = a1**2 * a6 + 4 * a2 * a6 - a1 * a3 * a4 + a2 * a3**2 - a4**2
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self._b2, self._b4, self._b6, self._b8 = b2, b4, b6, b8
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self._discrim = -b2**2 * b8 - 8 * b4**3 - 27 * b6**2 + 9 * b2 * b4 * b6
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self._a1 = a1
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self._a2 = a2
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self._a3 = a3
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self._a4 = a4
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self._a6 = a6
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x, y, z = symbols('x y z')
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self.x, self.y, self.z = x, y, z
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self._eq = Eq(y**2*z + a1*x*y*z + a3*y*z**2, x**3 + a2*x**2*z + a4*x*z**2 + a6*z**3)
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if isinstance(self._domain, FiniteField):
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self._rank = 0
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elif isinstance(self._domain, RationalField):
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self._rank = None
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def __call__(self, x, y, z=1):
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return EllipticCurvePoint(x, y, z, self)
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def __contains__(self, point):
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if is_sequence(point):
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if len(point) == 2:
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z1 = 1
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else:
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z1 = point[2]
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x1, y1 = point[:2]
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elif isinstance(point, EllipticCurvePoint):
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x1, y1, z1 = point.x, point.y, point.z
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else:
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raise ValueError('Invalid point.')
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if self.characteristic == 0 and z1 == 0:
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return True
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return self._eq.subs({self.x: x1, self.y: y1, self.z: z1})
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def __repr__(self):
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return 'E({}): {}'.format(self._domain, self._eq)
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def minimal(self):
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"""
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Return minimal Weierstrass equation.
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Examples
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========
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>>> from sympy.ntheory.elliptic_curve import EllipticCurve
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>>> e1 = EllipticCurve(-10, -20, 0, -1, 1)
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>>> e1.minimal()
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E(QQ): Eq(y**2*z, x**3 - 13392*x*z**2 - 1080432*z**3)
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"""
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char = self.characteristic
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if char == 2:
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return self
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if char == 3:
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return EllipticCurve(self._b4/2, self._b6/4, a2=self._b2/4, modulus=self.modulus)
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c4 = self._b2**2 - 24*self._b4
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c6 = -self._b2**3 + 36*self._b2*self._b4 - 216*self._b6
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return EllipticCurve(-27*c4, -54*c6, modulus=self.modulus)
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def points(self):
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"""
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Return points of curve over Finite Field.
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Examples
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========
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>>> from sympy.ntheory.elliptic_curve import EllipticCurve
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>>> e2 = EllipticCurve(1, 1, 1, 1, 1, modulus=5)
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>>> e2.points()
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{(0, 2), (1, 4), (2, 0), (2, 2), (3, 0), (3, 1), (4, 0)}
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"""
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char = self.characteristic
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all_pt = set()
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if char >= 1:
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for i in range(char):
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congruence_eq = ((self._eq.lhs - self._eq.rhs).subs({self.x: i, self.z: 1}))
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sol = polynomial_congruence(congruence_eq, char)
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for num in sol:
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all_pt.add((i, num))
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return all_pt
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else:
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raise ValueError("Infinitely many points")
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def points_x(self, x):
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"Returns points on with curve where xcoordinate = x"
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pt = []
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if self._domain == QQ:
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for y in solve(self._eq.subs(self.x, x)):
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pt.append((x, y))
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congruence_eq = ((self._eq.lhs - self._eq.rhs).subs({self.x: x, self.z: 1}))
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for y in polynomial_congruence(congruence_eq, self.characteristic):
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pt.append((x, y))
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return pt
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def torsion_points(self):
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"""
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Return torsion points of curve over Rational number.
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Return point objects those are finite order.
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According to Nagell-Lutz theorem, torsion point p(x, y)
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x and y are integers, either y = 0 or y**2 is divisor
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of discriminent. According to Mazur's theorem, there are
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at most 15 points in torsion collection.
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Examples
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========
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>>> from sympy.ntheory.elliptic_curve import EllipticCurve
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>>> e2 = EllipticCurve(-43, 166)
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>>> sorted(e2.torsion_points())
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[(-5, -16), (-5, 16), O, (3, -8), (3, 8), (11, -32), (11, 32)]
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"""
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if self.characteristic > 0:
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raise ValueError("No torsion point for Finite Field.")
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l = [EllipticCurvePoint.point_at_infinity(self)]
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for xx in solve(self._eq.subs({self.y: 0, self.z: 1})):
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if xx.is_rational:
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l.append(self(xx, 0))
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for i in divisors(self.discriminant, generator=True):
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j = int(i**.5)
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if j**2 == i:
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for xx in solve(self._eq.subs({self.y: j, self.z: 1})):
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if not xx.is_rational:
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continue
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p = self(xx, j)
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if p.order() != oo:
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l.extend([p, -p])
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return l
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@property
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def characteristic(self):
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"""
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Return domain characteristic.
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Examples
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========
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>>> from sympy.ntheory.elliptic_curve import EllipticCurve
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>>> e2 = EllipticCurve(-43, 166)
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>>> e2.characteristic
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0
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"""
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return self._domain.characteristic()
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@property
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def discriminant(self):
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"""
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Return curve discriminant.
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Examples
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========
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>>> from sympy.ntheory.elliptic_curve import EllipticCurve
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>>> e2 = EllipticCurve(0, 17)
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>>> e2.discriminant
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-124848
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"""
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return int(self._discrim)
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@property
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def is_singular(self):
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"""
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Return True if curve discriminant is equal to zero.
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"""
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return self.discriminant == 0
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@property
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def j_invariant(self):
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"""
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Return curve j-invariant.
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Examples
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========
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>>> from sympy.ntheory.elliptic_curve import EllipticCurve
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>>> e1 = EllipticCurve(-2, 0, 0, 1, 1)
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>>> e1.j_invariant
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1404928/389
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"""
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c4 = self._b2**2 - 24*self._b4
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return self._domain.to_sympy(c4**3 / self._discrim)
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@property
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def order(self):
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"""
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Number of points in Finite field.
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Examples
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========
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>>> from sympy.ntheory.elliptic_curve import EllipticCurve
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>>> e2 = EllipticCurve(1, 0, modulus=19)
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>>> e2.order
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19
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"""
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if self.characteristic == 0:
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raise NotImplementedError("Still not implemented")
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return len(self.points())
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@property
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def rank(self):
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"""
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Number of independent points of infinite order.
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For Finite field, it must be 0.
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"""
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if self._rank is not None:
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return self._rank
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raise NotImplementedError("Still not implemented")
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class EllipticCurvePoint:
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"""
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Point of Elliptic Curve
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Examples
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========
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>>> from sympy.ntheory.elliptic_curve import EllipticCurve
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>>> e1 = EllipticCurve(-17, 16)
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>>> p1 = e1(0, -4, 1)
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>>> p2 = e1(1, 0)
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>>> p1 + p2
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(15, -56)
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>>> e3 = EllipticCurve(-1, 9)
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>>> e3(1, -3) * 3
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(664/169, 17811/2197)
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>>> (e3(1, -3) * 3).order()
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oo
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>>> e2 = EllipticCurve(-2, 0, 0, 1, 1)
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>>> p = e2(-1,1)
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>>> q = e2(0, -1)
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>>> p+q
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(4, 8)
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>>> p-q
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(1, 0)
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>>> 3*p-5*q
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(328/361, -2800/6859)
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"""
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@staticmethod
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def point_at_infinity(curve):
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return EllipticCurvePoint(0, 1, 0, curve)
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def __init__(self, x, y, z, curve):
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dom = curve._domain.convert
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self.x = dom(x)
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self.y = dom(y)
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self.z = dom(z)
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self._curve = curve
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self._domain = self._curve._domain
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if not self._curve.__contains__(self):
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raise ValueError("The curve does not contain this point")
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def __add__(self, p):
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if self.z == 0:
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return p
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if p.z == 0:
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return self
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x1, y1 = self.x/self.z, self.y/self.z
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x2, y2 = p.x/p.z, p.y/p.z
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a1 = self._curve._a1
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a2 = self._curve._a2
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a3 = self._curve._a3
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a4 = self._curve._a4
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a6 = self._curve._a6
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if x1 != x2:
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slope = (y1 - y2) / (x1 - x2)
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yint = (y1 * x2 - y2 * x1) / (x2 - x1)
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else:
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if (y1 + y2) == 0:
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return self.point_at_infinity(self._curve)
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slope = (3 * x1**2 + 2*a2*x1 + a4 - a1*y1) / (a1 * x1 + a3 + 2 * y1)
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yint = (-x1**3 + a4*x1 + 2*a6 - a3*y1) / (a1*x1 + a3 + 2*y1)
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x3 = slope**2 + a1*slope - a2 - x1 - x2
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y3 = -(slope + a1) * x3 - yint - a3
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return self._curve(x3, y3, 1)
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def __lt__(self, other):
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return (self.x, self.y, self.z) < (other.x, other.y, other.z)
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def __mul__(self, n):
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n = as_int(n)
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r = self.point_at_infinity(self._curve)
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if n == 0:
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return r
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if n < 0:
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return -self * -n
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p = self
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while n:
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if n & 1:
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r = r + p
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n >>= 1
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p = p + p
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return r
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def __rmul__(self, n):
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return self * n
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def __neg__(self):
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return EllipticCurvePoint(self.x, -self.y - self._curve._a1*self.x - self._curve._a3, self.z, self._curve)
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def __repr__(self):
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if self.z == 0:
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return 'O'
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dom = self._curve._domain
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try:
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return '({}, {})'.format(dom.to_sympy(self.x), dom.to_sympy(self.y))
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except TypeError:
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pass
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return '({}, {})'.format(self.x, self.y)
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def __sub__(self, other):
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return self.__add__(-other)
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def order(self):
|
||
|
|
"""
|
||
|
|
Return point order n where nP = 0.
|
||
|
|
|
||
|
|
"""
|
||
|
|
if self.z == 0:
|
||
|
|
return 1
|
||
|
|
if self.y == 0: # P = -P
|
||
|
|
return 2
|
||
|
|
p = self * 2
|
||
|
|
if p.y == -self.y: # 2P = -P
|
||
|
|
return 3
|
||
|
|
i = 2
|
||
|
|
if self._domain != QQ:
|
||
|
|
while int(p.x) == p.x and int(p.y) == p.y:
|
||
|
|
p = self + p
|
||
|
|
i += 1
|
||
|
|
if p.z == 0:
|
||
|
|
return i
|
||
|
|
return oo
|
||
|
|
while p.x.numerator == p.x and p.y.numerator == p.y:
|
||
|
|
p = self + p
|
||
|
|
i += 1
|
||
|
|
if i > 12:
|
||
|
|
return oo
|
||
|
|
if p.z == 0:
|
||
|
|
return i
|
||
|
|
return oo
|