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from ast import Module
from copy import copy
from typing import MutableMapping, Union, List, Optional
from public import public
from .coordinates import CoordinateModel, AffineCoordinateModel
from .mod import Mod
from .model import CurveModel, ShortWeierstrassModel
from .point import Point, InfinityPoint
@public
class EllipticCurve(object):
"""An elliptic curve."""
model: CurveModel
coordinate_model: CoordinateModel
prime: int
parameters: MutableMapping[str, Mod]
neutral: Point
def __init__(self, model: CurveModel, coordinate_model: CoordinateModel,
prime: int, neutral: Point, parameters: MutableMapping[str, Union[Mod, int]]):
if coordinate_model not in model.coordinates.values() and not isinstance(coordinate_model, AffineCoordinateModel):
raise ValueError
if set(model.parameter_names).symmetric_difference(parameters.keys()):
raise ValueError
self.model = model
self.coordinate_model = coordinate_model
self.prime = prime
self.parameters = {}
for name, value in parameters.items():
if isinstance(value, Mod):
if value.n != prime:
raise ValueError
else:
value = Mod(value, prime)
self.parameters[name] = value
self.neutral = neutral
def _execute_base_formulas(self, formulas: List[Module], *points: Point) -> Point:
for point in points:
if point.coordinate_model.curve_model != self.model:
raise ValueError
if not isinstance(point.coordinate_model, AffineCoordinateModel):
raise ValueError
locals = {var + str(i + 1): point.coords[var]
for i, point in enumerate(points) for var in point.coords}
locals.update(self.parameters)
for line in formulas:
exec(compile(line, "", mode="exec"), None, locals)
if not isinstance(locals["x"], Mod):
locals["x"] = Mod(locals["x"], self.prime)
if not isinstance(locals["y"], Mod):
locals["y"] = Mod(locals["y"], self.prime)
return Point(AffineCoordinateModel(self.model), x=locals["x"], y=locals["y"])
def affine_add(self, one: Point, other: Point) -> Point:
return self._execute_base_formulas(self.model.base_addition, one, other)
def affine_double(self, one: Point) -> Point:
return self._execute_base_formulas(self.model.base_doubling, one)
def affine_negate(self, one: Point) -> Point:
return self._execute_base_formulas(self.model.base_negation, one)
def affine_multiply(self, point: Point, scalar: int) -> Point:
if point.coordinate_model.curve_model != self.model:
raise ValueError
if not isinstance(point.coordinate_model, AffineCoordinateModel):
raise ValueError
q = copy(point)
r = copy(point)
for i in range(scalar.bit_length() - 2, -1, -1):
r = self.affine_double(r)
if scalar & (1 << i) != 0:
r = self.affine_add(r, q)
return r
@property
def affine_neutral(self) -> Optional[Point]:
if not self.neutral_is_affine:
return None
locals = {**self.parameters}
for line in self.model.base_neutral:
exec(compile(line, "", mode="exec"), None, locals)
if not isinstance(locals["x"], Mod):
locals["x"] = Mod(locals["x"], self.prime)
if not isinstance(locals["y"], Mod):
locals["y"] = Mod(locals["y"], self.prime)
return Point(AffineCoordinateModel(self.model), x=locals["x"], y=locals["y"])
@property
def neutral_is_affine(self):
"""Whether the neurtal point is an affine point."""
return bool(self.model.base_neutral)
def is_neutral(self, point: Point) -> bool:
"""Check whether the point is the neutral point."""
return self.neutral == point
def is_on_curve(self, point: Point) -> bool:
"""Check whether the point is on the curve."""
if point.coordinate_model.curve_model != self.model:
return False
if self.is_neutral(point):
return True
loc = {**self.parameters, **point.to_affine().coords}
return eval(compile(self.model.equation, "", mode="eval"), loc)
def to_affine(self) -> "EllipticCurve":
"""Convert this curve into the affine coordinate model, if possible."""
coord_model = AffineCoordinateModel(self.model)
return EllipticCurve(self.model, coord_model, self.prime, self.neutral.to_affine(), self.parameters) # type: ignore[arg-type]
def decode_point(self, encoded: bytes) -> Point:
"""Decode a point encoded as a sequence of bytes (ANSI X9.62)."""
if encoded[0] == 0x00 and len(encoded) == 1:
return InfinityPoint(self.coordinate_model)
coord_len = (self.prime.bit_length() + 7) // 8
if encoded[0] in (0x04, 0x06):
data = encoded[1:]
if len(data) != coord_len * len(self.coordinate_model.variables):
raise ValueError("Encoded point has bad length")
coords = {}
for var in sorted(self.coordinate_model.variables):
coords[var] = Mod(int.from_bytes(data[:coord_len], "big"), self.prime)
data = data[coord_len:]
return Point(self.coordinate_model, **coords)
elif encoded[0] in (0x02, 0x03):
if isinstance(self.coordinate_model, AffineCoordinateModel):
data = encoded[1:]
if len(data) != coord_len:
raise ValueError("Encoded point has bad length")
x = Mod(int.from_bytes(data, "big"), self.prime)
loc = {**self.parameters, "x": x}
rhs = eval(compile(self.model.ysquared, "", mode="eval"), loc)
if not rhs.is_residue():
raise ValueError("Point not on curve")
sqrt = rhs.sqrt()
yp = encoded[0] & 0x01
if int(sqrt) & 0x01 == yp:
y = sqrt
else:
y = -sqrt
return Point(self.coordinate_model, x=x, y=y)
else:
raise NotImplementedError
else:
raise ValueError(f"Wrong encoding type: {hex(encoded[0])}, should be one of 0x04, 0x06, 0x02, 0x03 or 0x00")
def affine_random(self) -> Point:
"""Generate a random affine point on the curve."""
while True:
x = Mod.random(self.prime)
loc = {**self.parameters, "x":x}
ysquared = eval(compile(self.model.ysquared, "", mode="eval"), loc)
if ysquared.is_residue():
y = ysquared.sqrt()
b = Mod.random(2)
if b == 1:
y = -y
return Point(AffineCoordinateModel(self.model), x=x, y=y)
def __eq__(self, other):
if not isinstance(other, EllipticCurve):
return False
return self.model == other.model and self.coordinate_model == other.coordinate_model and self.prime == other.prime and self.parameters == other.parameters
def __str__(self):
return "EllipticCurve"
def __repr__(self):
params = ", ".join((f"{key}={val}" for key, val in self.parameters.items()))
return f"{self.__class__.__name__}([{params}] on {self.model} using {self.coordinate_model})"
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