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
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
|
from copy import copy
from public import public
from typing import Mapping, Tuple, Optional, MutableMapping
from pyecsca.ec.naf import naf, wnaf
from .context import Context
from .curve import EllipticCurve
from .formula import (Formula, AdditionFormula, DoublingFormula, ScalingFormula, LadderFormula,
NegationFormula)
from .point import Point
class ScalarMultiplier(object):
curve: EllipticCurve
formulas: Mapping[str, Formula]
context: Context
_point: Point = None
def __init__(self, curve: EllipticCurve, ctx: Context = None, **formulas: Optional[Formula]):
for formula in formulas.values():
if formula is not None and formula.coordinate_model is not curve.coordinate_model:
raise ValueError
self.curve = curve
if ctx:
self.context = ctx
else:
self.context = Context()
self.formulas = dict(filter(lambda pair: pair[1] is not None, formulas.items()))
def _add(self, one: Point, other: Point) -> Point:
if "add" not in self.formulas:
raise NotImplementedError
if one == self.curve.neutral:
return copy(other)
if other == self.curve.neutral:
return copy(one)
return self.context.execute(self.formulas["add"], one, other, **self.curve.parameters)[0]
def _dbl(self, point: Point) -> Point:
if "dbl" not in self.formulas:
raise NotImplementedError
if point == self.curve.neutral:
return copy(point)
return self.context.execute(self.formulas["dbl"], point, **self.curve.parameters)[0]
def _scl(self, point: Point) -> Point:
if "scl" not in self.formulas:
raise NotImplementedError
return self.context.execute(self.formulas["scl"], point, **self.curve.parameters)[0]
def _ladd(self, start: Point, to_dbl: Point, to_add: Point) -> Tuple[Point, ...]:
if "ladd" not in self.formulas:
raise NotImplementedError
return self.context.execute(self.formulas["ladd"], start, to_dbl, to_add,
**self.curve.parameters)
def _neg(self, point: Point) -> Point:
if "neg" not in self.formulas:
raise NotImplementedError
return self.context.execute(self.formulas["neg"], point, **self.curve.parameters)[0]
def init(self, point: Point):
self._point = point
def _init_multiply(self, point: Optional[Point]) -> Point:
if point is None:
if self._point is None:
raise ValueError
else:
if self._point != point:
self.init(point)
return self._point
def multiply(self, scalar: int, point: Optional[Point] = None) -> Point:
raise NotImplementedError
@public
class LTRMultiplier(ScalarMultiplier):
"""
Classic double and add scalar multiplication algorithm, that scans the scalar left-to-right (msb to lsb)
The `always` parameter determines whether the double and add always method is used.
"""
always: bool
def __init__(self, curve: EllipticCurve, add: AdditionFormula, dbl: DoublingFormula,
scl: ScalingFormula = None,
ctx: Context = None, always: bool = False):
super().__init__(curve, ctx, add=add, dbl=dbl, scl=scl)
self.always = always
def multiply(self, scalar: int, point: Optional[Point] = None) -> Point:
q = self._init_multiply(point)
r = copy(self.curve.neutral)
for i in range(scalar.bit_length(), -1, -1):
r = self._dbl(r)
if scalar & (1 << i) != 0:
r = self._add(r, q)
elif self.always:
self._add(r, q)
if "scl" in self.formulas:
r = self._scl(r)
return r
@public
class RTLMultiplier(ScalarMultiplier):
"""
Classic double and add scalar multiplication algorithm, that scans the scalar right-to-left (lsb to msb)
The `always` parameter determines whether the double and add always method is used.
"""
always: bool
def __init__(self, curve: EllipticCurve, add: AdditionFormula, dbl: DoublingFormula,
scl: ScalingFormula = None,
ctx: Context = None, always: bool = False):
super().__init__(curve, ctx, add=add, dbl=dbl, scl=scl)
self.always = always
def multiply(self, scalar: int, point: Optional[Point] = None) -> Point:
q = self._init_multiply(point)
r = copy(self.curve.neutral)
while scalar > 0:
if scalar & 1 != 0:
r = self._add(r, q)
elif self.always:
self._add(r, q)
q = self._dbl(q)
scalar >>= 1
if "scl" in self.formulas:
r = self._scl(r)
return r
class CoronMultiplier(ScalarMultiplier):
"""
Coron's double and add resistant against SPA, from:
Resistance against Differential Power Analysis for Elliptic Curve Cryptosystems
https://link.springer.com/content/pdf/10.1007/3-540-48059-5_25.pdf
"""
def __init__(self, curve: EllipticCurve, add: AdditionFormula, dbl: DoublingFormula,
scl: ScalingFormula = None, ctx: Context = None):
super().__init__(curve, ctx, add=add, dbl=dbl, scl=scl)
def multiply(self, scalar: int, point: Optional[Point] = None):
q = self._init_multiply(point)
p0 = copy(q)
for i in range(scalar.bit_length() - 2, -1, -1):
p0 = self._dbl(p0)
p1 = self._add(p0, q)
if scalar & (1 << i) != 0:
p0 = p1
if "scl" in self.formulas:
p0 = self._scl(p0)
return p0
@public
class LadderMultiplier(ScalarMultiplier):
"""
Montgomery ladder multiplier, using a three input, two output ladder formula.
"""
def __init__(self, curve: EllipticCurve, ladd: LadderFormula, scl: ScalingFormula = None,
ctx: Context = None):
super().__init__(curve, ctx, ladd=ladd, scl=scl)
def multiply(self, scalar: int, point: Optional[Point] = None) -> Point:
q = self._init_multiply(point)
p0 = copy(q)
p1 = self._ladd(self.curve.neutral, q, q)[1]
for i in range(scalar.bit_length() - 1, -1, -1):
if scalar & (1 << i) != 0:
p0, p1 = self._ladd(q, p0, p1)
else:
p1, p0 = self._ladd(q, p1, p0)
if "scl" in self.formulas:
p0 = self._scl(p0)
return p0
@public
class SimpleLadderMultiplier(ScalarMultiplier):
"""
Montgomery ladder multiplier, using addition and doubling formulas.
"""
def __init__(self, curve: EllipticCurve, add: AdditionFormula, dbl: DoublingFormula,
scl: ScalingFormula = None, ctx: Context = None):
super().__init__(curve, ctx, add=add, dbl=dbl, scl=scl)
def multiply(self, scalar: int, point: Optional[Point] = None) -> Point:
q = self._init_multiply(point)
p0 = copy(q)
p1 = self._dbl(q)
for i in range(scalar.bit_length() - 2, -1, -1):
if scalar & (1 << i) != 0:
p0 = self._add(p0, p1)
p1 = self._dbl(p1)
else:
p1 = self._add(p0, p1)
p0 = self._dbl(p0)
if "scl" in self.formulas:
p0 = self._scl(p0)
return p0
@public
class BinaryNAFMultiplier(ScalarMultiplier):
"""
Binary NAF (Non Adjacent Form) multiplier, left-to-right.
"""
_point_neg: Point
def __init__(self, curve: EllipticCurve, add: AdditionFormula, dbl: DoublingFormula,
neg: NegationFormula, scl: ScalingFormula = None, ctx: Context = None):
super().__init__(curve, ctx, add=add, dbl=dbl, neg=neg, scl=scl)
def init(self, point: Point):
super().init(point)
self._point_neg = self._neg(point)
def multiply(self, scalar: int, point: Optional[Point] = None) -> Point:
self._init_multiply(point)
bnaf = naf(scalar)
q = copy(self.curve.neutral)
for val in bnaf:
q = self._dbl(q)
if val == 1:
q = self._add(q, self._point)
if val == -1:
q = self._add(q, self._point_neg)
if "scl" in self.formulas:
q = self._scl(q)
return q
@public
class WindowNAFMultiplier(ScalarMultiplier):
"""
Window NAF (Non Adjacent Form) multiplier, left-to-right.
"""
_points: MutableMapping[int, Point]
_width: int
def __init__(self, curve: EllipticCurve, add: AdditionFormula, dbl: DoublingFormula,
neg: NegationFormula, width: int, scl: ScalingFormula = None, ctx: Context = None):
super().__init__(curve, ctx, add=add, dbl=dbl, neg=neg, scl=scl)
self._width = width
def init(self, point: Point):
self._point = point
self._points = {}
current_point = point
double_point = self._dbl(point)
for i in range(1, (self._width + 1) // 2 + 1):
self._points[2 ** i - 1] = current_point
current_point = self._add(current_point, double_point)
def multiply(self, scalar: int, point: Optional[Point] = None):
self._init_multiply(point)
naf = wnaf(scalar, self._width)
q = copy(self.curve.neutral)
for val in naf:
q = self._dbl(q)
if val > 0:
q = self._add(q, self._points[val])
elif val < 0:
neg = self._neg(self._points[-val])
q = self._add(q, neg)
if "scl" in self.formulas:
q = self._scl(q)
return q
|
