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
|
package p751
import (
. "github.com/cloudflare/sidh/internal/isogeny"
cpu "github.com/cloudflare/sidh/internal/utils"
)
const (
// SIDH public key byte size
P751_PublicKeySize = 564
// SIDH shared secret byte size.
P751_SharedSecretSize = 188
// Max size of secret key for 2-torsion group, corresponds to 2^e2
P751_SecretBitLenA = 372
// Size of secret key for 3-torsion group, corresponds to floor(log_2(3^e3))
P751_SecretBitLenB = 378
// P751 bytelen ceil(751/8)
P751_Bytelen = 94
// Size of a compuatation strategy for 2-torsion group
strategySizeA = 185
// Size of a compuatation strategy for 3-torsion group
strategySizeB = 238
// Number of 64-bit limbs used to store Fp element
NumWords = 12
)
// CPU Capabilities. Those flags are referred by assembly code. According to
// https://github.com/golang/go/issues/28230, variables referred from the
// assembly must be in the same package.
// We declare them variables not constants in order to facilitate testing.
var (
// Signals support for MULX which is in BMI2
HasBMI2 = cpu.X86.HasBMI2
// Signals support for ADX and BMI2
HasADXandBMI2 = cpu.X86.HasBMI2 && cpu.X86.HasADX
)
// The x-coordinate of PA
var P751_affine_PA = Fp2Element{
A: FpElement{
0xC2FC08CEAB50AD8B, 0x1D7D710F55E457B1, 0xE8738D92953DCD6E,
0xBAA7EBEE8A3418AA, 0xC9A288345F03F46F, 0xC8D18D167CFE2616,
0x02043761F6B1C045, 0xAA1975E13180E7E9, 0x9E13D3FDC6690DE6,
0x3A024640A3A3BB4F, 0x4E5AD44E6ACBBDAE, 0x0000544BEB561DAD,
},
B: FpElement{
0xE6CC41D21582E411, 0x07C2ECB7C5DF400A, 0xE8E34B521432AEC4,
0x50761E2AB085167D, 0x032CFBCAA6094B3C, 0x6C522F5FDF9DDD71,
0x1319217DC3A1887D, 0xDC4FB25803353A86, 0x362C8D7B63A6AB09,
0x39DCDFBCE47EA488, 0x4C27C99A2C28D409, 0x00003CB0075527C4,
},
}
// The x-coordinate of QA
var P751_affine_QA = Fp2Element{
A: FpElement{
0xD56FE52627914862, 0x1FAD60DC96B5BAEA, 0x01E137D0BF07AB91,
0x404D3E9252161964, 0x3C5385E4CD09A337, 0x4476426769E4AF73,
0x9790C6DB989DFE33, 0xE06E1C04D2AA8B5E, 0x38C08185EDEA73B9,
0xAA41F678A4396CA6, 0x92B9259B2229E9A0, 0x00002F9326818BE0,
},
B: FpElement{
0x0000000000000000, 0x0000000000000000, 0x0000000000000000,
0x0000000000000000, 0x0000000000000000, 0x0000000000000000,
0x0000000000000000, 0x0000000000000000, 0x0000000000000000,
0x0000000000000000, 0x0000000000000000, 0x0000000000000000,
},
}
// The x-coordinate of RA = PA-QA
var P751_affine_RA = Fp2Element{
A: FpElement{
0x0BB84441DFFD19B3, 0x84B4DEA99B48C18E, 0x692DE648AD313805,
0xE6D72761B6DFAEE0, 0x223975C672C3058D, 0xA0FDE0C3CBA26FDC,
0xA5326132A922A3CA, 0xCA5E7F5D5EA96FA4, 0x127C7EFE33FFA8C6,
0x4749B1567E2A23C4, 0x2B7DF5B4AF413BFA, 0x0000656595B9623C,
},
B: FpElement{
0xED78C17F1EC71BE8, 0xF824D6DF753859B1, 0x33A10839B2A8529F,
0xFC03E9E25FDEA796, 0xC4708A8054DF1762, 0x4034F2EC034C6467,
0xABFB70FBF06ECC79, 0xDABE96636EC108B7, 0x49CBCFB090605FD3,
0x20B89711819A45A7, 0xFB8E1590B2B0F63E, 0x0000556A5F964AB2,
},
}
// The x-coordinate of PB
var P751_affine_PB = Fp2Element{
A: FpElement{
0xCFB6D71EF867AB0B, 0x4A5FDD76E9A45C76, 0x38B1EE69194B1F03,
0xF6E7B18A7761F3F0, 0xFCF01A486A52C84C, 0xCBE2F63F5AA75466,
0x6487BCE837B5E4D6, 0x7747F5A8C622E9B8, 0x4CBFE1E4EE6AEBBA,
0x8A8616A13FA91512, 0x53DB980E1579E0A5, 0x000058FEBFF3BE69,
},
B: FpElement{
0xA492034E7C075CC3, 0x677BAF00B04AA430, 0x3AAE0C9A755C94C8,
0x1DC4B064E9EBB08B, 0x3684EDD04E826C66, 0x9BAA6CB661F01B22,
0x20285A00AD2EFE35, 0xDCE95ABD0497065F, 0x16C7FBB3778E3794,
0x26B3AC29CEF25AAF, 0xFB3C28A31A30AC1D, 0x000046ED190624EE,
},
}
// The x-coordinate of QB
var P751_affine_QB = Fp2Element{
A: FpElement{
0xF1A8C9ED7B96C4AB, 0x299429DA5178486E, 0xEF4926F20CD5C2F4,
0x683B2E2858B4716A, 0xDDA2FBCC3CAC3EEB, 0xEC055F9F3A600460,
0xD5A5A17A58C3848B, 0x4652D836F42EAED5, 0x2F2E71ED78B3A3B3,
0xA771C057180ADD1D, 0xC780A5D2D835F512, 0x0000114EA3B55AC1,
},
B: FpElement{
0x0000000000000000, 0x0000000000000000, 0x0000000000000000,
0x0000000000000000, 0x0000000000000000, 0x0000000000000000,
0x0000000000000000, 0x0000000000000000, 0x0000000000000000,
0x0000000000000000, 0x0000000000000000, 0x0000000000000000,
},
}
// The x-coordinate of RB = PB - QB
var P751_affine_RB = Fp2Element{
A: FpElement{
0x1C0D6733769D0F31, 0xF084C3086E2659D1, 0xE23D5DA27BCBD133,
0xF38EC9A8D5864025, 0x6426DC781B3B645B, 0x4B24E8E3C9FB03EE,
0x6432792F9D2CEA30, 0x7CC8E8B1AE76E857, 0x7F32BFB626BB8963,
0xB9F05995B48D7B74, 0x4D71200A7D67E042, 0x0000228457AF0637,
},
B: FpElement{
0x4AE37E7D8F72BD95, 0xDD2D504B3E993488, 0x5D14E7FA1ECB3C3E,
0x127610CEB75D6350, 0x255B4B4CAC446B11, 0x9EA12336C1F70CAF,
0x79FA68A2147BC2F8, 0x11E895CFDADBBC49, 0xE4B9D3C4D6356C18,
0x44B25856A67F951C, 0x5851541F61308D0B, 0x00002FFD994F7E4C,
},
}
// 2-torsion group computation strategy
var P751_AliceIsogenyStrategy = [strategySizeA]uint32{
0x50, 0x30, 0x1B, 0x0F, 0x08, 0x04, 0x02, 0x01, 0x01, 0x02,
0x01, 0x01, 0x04, 0x02, 0x01, 0x01, 0x02, 0x01, 0x01, 0x07,
0x04, 0x02, 0x01, 0x01, 0x02, 0x01, 0x01, 0x03, 0x02, 0x01,
0x01, 0x01, 0x01, 0x0C, 0x07, 0x04, 0x02, 0x01, 0x01, 0x02,
0x01, 0x01, 0x03, 0x02, 0x01, 0x01, 0x01, 0x01, 0x05, 0x03,
0x02, 0x01, 0x01, 0x01, 0x01, 0x02, 0x01, 0x01, 0x01, 0x15,
0x0C, 0x07, 0x04, 0x02, 0x01, 0x01, 0x02, 0x01, 0x01, 0x03,
0x02, 0x01, 0x01, 0x01, 0x01, 0x05, 0x03, 0x02, 0x01, 0x01,
0x01, 0x01, 0x02, 0x01, 0x01, 0x01, 0x09, 0x05, 0x03, 0x02,
0x01, 0x01, 0x01, 0x01, 0x02, 0x01, 0x01, 0x01, 0x04, 0x02,
0x01, 0x01, 0x01, 0x02, 0x01, 0x01, 0x21, 0x14, 0x0C, 0x07,
0x04, 0x02, 0x01, 0x01, 0x02, 0x01, 0x01, 0x03, 0x02, 0x01,
0x01, 0x01, 0x01, 0x05, 0x03, 0x02, 0x01, 0x01, 0x01, 0x01,
0x02, 0x01, 0x01, 0x01, 0x08, 0x05, 0x03, 0x02, 0x01, 0x01,
0x01, 0x01, 0x02, 0x01, 0x01, 0x01, 0x04, 0x02, 0x01, 0x01,
0x02, 0x01, 0x01, 0x10, 0x08, 0x04, 0x02, 0x01, 0x01, 0x01,
0x02, 0x01, 0x01, 0x04, 0x02, 0x01, 0x01, 0x02, 0x01, 0x01,
0x08, 0x04, 0x02, 0x01, 0x01, 0x02, 0x01, 0x01, 0x04, 0x02,
0x01, 0x01, 0x02, 0x01, 0x01}
// 3-torsion group computation strategy
var P751_BobIsogenyStrategy = [strategySizeB]uint32{
0x70, 0x3F, 0x20, 0x10, 0x08, 0x04, 0x02, 0x01, 0x01, 0x02,
0x01, 0x01, 0x04, 0x02, 0x01, 0x01, 0x02, 0x01, 0x01, 0x08,
0x04, 0x02, 0x01, 0x01, 0x02, 0x01, 0x01, 0x04, 0x02, 0x01,
0x01, 0x02, 0x01, 0x01, 0x10, 0x08, 0x04, 0x02, 0x01, 0x01,
0x02, 0x01, 0x01, 0x04, 0x02, 0x01, 0x01, 0x02, 0x01, 0x01,
0x08, 0x04, 0x02, 0x01, 0x01, 0x02, 0x01, 0x01, 0x04, 0x02,
0x01, 0x01, 0x02, 0x01, 0x01, 0x1F, 0x10, 0x08, 0x04, 0x02,
0x01, 0x01, 0x02, 0x01, 0x01, 0x04, 0x02, 0x01, 0x01, 0x02,
0x01, 0x01, 0x08, 0x04, 0x02, 0x01, 0x01, 0x02, 0x01, 0x01,
0x04, 0x02, 0x01, 0x01, 0x02, 0x01, 0x01, 0x0F, 0x08, 0x04,
0x02, 0x01, 0x01, 0x02, 0x01, 0x01, 0x04, 0x02, 0x01, 0x01,
0x02, 0x01, 0x01, 0x07, 0x04, 0x02, 0x01, 0x01, 0x02, 0x01,
0x01, 0x03, 0x02, 0x01, 0x01, 0x01, 0x01, 0x31, 0x1F, 0x10,
0x08, 0x04, 0x02, 0x01, 0x01, 0x02, 0x01, 0x01, 0x04, 0x02,
0x01, 0x01, 0x02, 0x01, 0x01, 0x08, 0x04, 0x02, 0x01, 0x01,
0x02, 0x01, 0x01, 0x04, 0x02, 0x01, 0x01, 0x02, 0x01, 0x01,
0x0F, 0x08, 0x04, 0x02, 0x01, 0x01, 0x02, 0x01, 0x01, 0x04,
0x02, 0x01, 0x01, 0x02, 0x01, 0x01, 0x07, 0x04, 0x02, 0x01,
0x01, 0x02, 0x01, 0x01, 0x03, 0x02, 0x01, 0x01, 0x01, 0x01,
0x15, 0x0C, 0x08, 0x04, 0x02, 0x01, 0x01, 0x02, 0x01, 0x01,
0x04, 0x02, 0x01, 0x01, 0x02, 0x01, 0x01, 0x05, 0x03, 0x02,
0x01, 0x01, 0x01, 0x01, 0x02, 0x01, 0x01, 0x01, 0x09, 0x05,
0x03, 0x02, 0x01, 0x01, 0x01, 0x01, 0x02, 0x01, 0x01, 0x01,
0x04, 0x02, 0x01, 0x01, 0x01, 0x02, 0x01, 0x01}
// Used internally by this package. Not consts as Go doesn't allow arrays to be consts
// -------------------------------
// p751
var p751 = FpElement{
0xffffffffffffffff, 0xffffffffffffffff, 0xffffffffffffffff,
0xffffffffffffffff, 0xffffffffffffffff, 0xeeafffffffffffff,
0xe3ec968549f878a8, 0xda959b1a13f7cc76, 0x084e9867d6ebe876,
0x8562b5045cb25748, 0x0e12909f97badc66, 0x00006fe5d541f71c}
// 2*p751
var p751x2 = FpElement{
0xFFFFFFFFFFFFFFFE, 0xFFFFFFFFFFFFFFFF, 0xFFFFFFFFFFFFFFFF,
0xFFFFFFFFFFFFFFFF, 0xFFFFFFFFFFFFFFFF, 0xDD5FFFFFFFFFFFFF,
0xC7D92D0A93F0F151, 0xB52B363427EF98ED, 0x109D30CFADD7D0ED,
0x0AC56A08B964AE90, 0x1C25213F2F75B8CD, 0x0000DFCBAA83EE38}
// p751 + 1
var p751p1 = FpElement{
0x0000000000000000, 0x0000000000000000, 0x0000000000000000,
0x0000000000000000, 0x0000000000000000, 0xeeb0000000000000,
0xe3ec968549f878a8, 0xda959b1a13f7cc76, 0x084e9867d6ebe876,
0x8562b5045cb25748, 0x0e12909f97badc66, 0x00006fe5d541f71c}
// R^2 = (2^768)^2 mod p
var p751R2 = FpElement{
2535603850726686808, 15780896088201250090, 6788776303855402382,
17585428585582356230, 5274503137951975249, 2266259624764636289,
11695651972693921304, 13072885652150159301, 4908312795585420432,
6229583484603254826, 488927695601805643, 72213483953973}
// 1*R mod p
var P751_OneFp2 = Fp2Element{
A: FpElement{
0x249ad, 0x0, 0x0, 0x0, 0x0, 0x8310000000000000, 0x5527b1e4375c6c66, 0x697797bf3f4f24d0, 0xc89db7b2ac5c4e2e, 0x4ca4b439d2076956, 0x10f7926c7512c7e9, 0x2d5b24bce5e2},
}
// 1/2 * R mod p
var P751_HalfFp2 = Fp2Element{
A: FpElement{
0x00000000000124D6, 0x0000000000000000, 0x0000000000000000,
0x0000000000000000, 0x0000000000000000, 0xB8E0000000000000,
0x9C8A2434C0AA7287, 0xA206996CA9A378A3, 0x6876280D41A41B52,
0xE903B49F175CE04F, 0x0F8511860666D227, 0x00004EA07CFF6E7F},
}
|
