1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
|
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
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).union(coordinate_model.parameters).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(f"Parameter {name} has wrong modulus.")
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 not isinstance(point.coordinate_model, AffineCoordinateModel):
raise ValueError("Coordinate model of point is not affine.")
if point.coordinate_model.curve_model != self.model:
raise ValueError("Curve model of point does not match the curve.")
locls = {var + str(i + 1): point.coords[var]
for i, point in enumerate(points) for var in point.coords}
locls.update(self.parameters)
for line in formulas:
exec(compile(line, "", mode="exec"), None, locls)
if not isinstance(locls["x"], Mod):
locls["x"] = Mod(locls["x"], self.prime)
if not isinstance(locls["y"], Mod):
locls["y"] = Mod(locls["y"], self.prime)
return Point(AffineCoordinateModel(self.model), x=locls["x"], y=locls["y"])
def affine_add(self, one: Point, other: Point) -> Point:
"""
Add two affine points using the affine addition formula.
Handles the case of point at infinity gracefully.
"""
if isinstance(one, InfinityPoint):
return other
if isinstance(other, InfinityPoint):
return one
if one == other:
return self.affine_double(one)
return self._execute_base_formulas(self.model.base_addition, one, other)
def affine_double(self, one: Point) -> Point:
"""
Double an affine point using the affine doubling formula.
Handles the case of point at infinity gracefully.
"""
if isinstance(one, InfinityPoint):
return one
return self._execute_base_formulas(self.model.base_doubling, one)
def affine_negate(self, one: Point) -> Point:
"""
Negate an affine point using the affine negation formula.
Handles the case of point at infinity gracefully.
"""
if isinstance(one, InfinityPoint):
return one
return self._execute_base_formulas(self.model.base_negation, one)
def affine_multiply(self, point: Point, scalar: int) -> Point:
"""
Multiply an affine point by a scalar using the affine doubling and addition formulas.
Handles the case of point at infinity gracefully.
"""
if isinstance(point, InfinityPoint):
return point
if not isinstance(point.coordinate_model, AffineCoordinateModel):
raise ValueError("Coordinate model of point is not affine.")
if point.coordinate_model.curve_model != self.model:
raise ValueError("Curve model of point does not match the curve.")
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]:
"""
Get the neutral point in affine form, if it has one, otherwise `None`.
:return: The affine neutral point or `None`.
"""
if not self.neutral_is_affine:
return None
locls = {**self.parameters}
for line in self.model.base_neutral:
exec(compile(line, "", mode="exec"), None, locls)
if not isinstance(locls["x"], Mod):
locls["x"] = Mod(locls["x"], self.prime)
if not isinstance(locls["y"], Mod):
locls["y"] = Mod(locls["y"], self.prime)
return Point(AffineCoordinateModel(self.model), x=locls["x"], y=locls["y"])
@property
def neutral_is_affine(self):
"""Whether the neutral 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})"
|
