-
Notifications
You must be signed in to change notification settings - Fork 13
/
ec.py
176 lines (146 loc) · 4.99 KB
/
ec.py
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
"""
By Willem Hengeveld <[email protected]>
Elliptic curve operations
"""
class WeierstrassCurve:
"""
WeierstrassCurve implements a point on a elliptic curve of form
y^2 = x^3 + A*x + B
with 4a^3+27b^2 != 0 (mod p)
"""
class Point:
"""
represent a value in the WeierstrassCurve
this class forwards all operations to the WeierstrassCurve class
"""
def __init__(self, curve, x, y):
self.curve = curve
self.x = x
self.y = y
# Point + Point
def __add__(self, rhs): return self.curve.add(self, rhs)
def __sub__(self, rhs): return self.curve.sub(self, rhs)
# Point * int or Point * Value
def __mul__(self, rhs): return self.curve.mul(self, rhs)
def __rmul__(self, lhs): return self.curve.mul(self, lhs)
def __div__(self, rhs): return self.curve.div(self, rhs)
def __truediv__(self, rhs): return self.__div__(rhs)
def __floordiv__(self, rhs): return self.__div__(rhs)
def __eq__(self, rhs): return self.curve.eq(self, rhs)
def __ne__(self, rhs): return not (self==rhs)
def __le__(self, rhs): raise Exception("points are not ordered")
def __lt__(self, rhs): raise Exception("points are not ordered")
def __ge__(self, rhs): raise Exception("points are not ordered")
def __gt__(self, rhs): raise Exception("points are not ordered")
def __hash__(self): return int(self.x+self.y)
def __str__(self): return "(%s,%s)" % (self.x, self.y)
def __neg__(self): return self.curve.neg(self)
def __nonzero__(self): return self.curve.nonzero(self)
def __bool__(self): return self.__nonzero__() != 0
def isoncurve(self):
return self.curve.isoncurve(self)
def __init__(self, field, a, b):
self.field = field
self.a = field.value(a)
self.b = field.value(b)
def __str__(self): return "Weierstrass(%s;%s;%s)" % (self.field, self.a, self.b)
def add(self, p, q):
"""
perform elliptic curve addition
"""
if not p: return q
if not q: return p
# calculate the slope of the intersection line
if p==q:
if not p:
return self.zero()
l = (3* p.x**2 + self.a) // (2* p.y)
elif p.x==q.x: # implies: p.y == -q.y
return self.zero()
else:
l = (p.y-q.y)//(p.x-q.x)
# calculate the intersection point
x = l**2 - ( p.x + q.x )
y = l*(p.x-x)-p.y
return self.point(x,y)
# subtraction is : a - b = a + -b
def sub(self, lhs, rhs): return lhs + -rhs
# scalar multiplication is implemented like repeated addition
def mul(self, pt, scalar):
scalar = int(scalar)
ispos = True
if scalar<0:
ispos = False
scalar = -scalar
accumulator = self.zero()
shifter = pt
while scalar != 0:
bit = scalar % 2
if bit:
accumulator += shifter
shifter += shifter
scalar //= 2
if not ispos:
accumulator = -accumulator
return accumulator
def div(self, pt, scalar):
"""
scalar division: P / a = P * (1/a)
scalar is assumed to be of type FiniteField(grouporder)
"""
return pt * (1//scalar)
def eq(self, lhs, rhs): return lhs.x==rhs.x and lhs.y==rhs.y
def neg(self, pt):
if not pt:
return pt
return self.point(pt.x, -pt.y)
def nonzero(self, pt):
return not (pt.x is None and pt.y is None)
def zero(self):
"""
Return the additive identity point ( aka '0' )
P + 0 = P
"""
return self.point(None, None)
def point(self, x, y):
"""
construct a point from 2 values
"""
return WeierstrassCurve.Point(self, self.coord(x), self.coord(y))
def coord(self, x):
if x is None:
return None
return self.field.value(x)
def isoncurve(self, p):
"""
verifies if a point is on the curve
"""
a, b = self.a, self.b
x, y = p.x, p.y
return not p or (y**2 == x**3 + a*x + b)
def decompress(self, x, flag):
"""
calculate the y coordinate given only the x value.
there are 2 possible solutions, use 'flag' to select.
"""
x = self.coord(x)
a, b = self.a, self.b
ysquare = x**3 + a*x + b
y = ysquare.sqrt(flag)
if y is None:
return
return self.point(x, y)
def decompressy(self, y, flag):
"""
calculate the x coordinate given only the y value.
there are 3 possible solutions, use 'flag' to select.
"""
y = self.coord(y)
if self.a:
# works only for a==0
return
xcube = y**2-self.b
x = xcube.cubert(flag)
if x is None:
return
return self.point(x, y)