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
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
|
/********************************************************************************************
* Supersingular Isogeny Key Encapsulation Library
*
* Abstract: core functions over GF(p) and GF(p^2)
*********************************************************************************************/
#include "sike_r1_namespace.h"
#include "P503_internal_r1.h"
__inline void fpcopy(const felm_t a, felm_t c)
{ // Copy a field element, c = a.
unsigned int i;
for (i = 0; i < NWORDS_FIELD; i++)
c[i] = a[i];
}
__inline void fpzero(felm_t a)
{ // Zero a field element, a = 0.
unsigned int i;
for (i = 0; i < NWORDS_FIELD; i++)
a[i] = 0;
}
void to_mont(const felm_t a, felm_t mc)
{ // Conversion to Montgomery representation,
// mc = a*R^2*R^(-1) mod p = a*R mod p, where a in [0, p-1].
// The Montgomery constant R^2 mod p is the global value "Montgomery_R2".
fpmul_mont(a, (const digit_t*)&Montgomery_R2, mc);
}
void from_mont(const felm_t ma, felm_t c)
{ // Conversion from Montgomery representation to standard representation,
// c = ma*R^(-1) mod p = a mod p, where ma in [0, p-1].
digit_t one[NWORDS_FIELD] = {0};
one[0] = 1;
fpmul_mont(ma, one, c);
fpcorrection(c);
}
void copy_words(const digit_t* a, digit_t* c, const unsigned int nwords)
{ // Copy wordsize digits, c = a, where lng(a) = nwords.
unsigned int i;
for (i = 0; i < nwords; i++) {
c[i] = a[i];
}
}
void fpmul_mont(const felm_t ma, const felm_t mb, felm_t mc)
{ // Multiprecision multiplication, c = a*b mod p.
dfelm_t temp = {0};
mp_mul(ma, mb, temp, NWORDS_FIELD);
rdc_mont(temp, mc);
}
void fpsqr_mont(const felm_t ma, felm_t mc)
{ // Multiprecision squaring, c = a^2 mod p.
dfelm_t temp = {0};
mp_mul(ma, ma, temp, NWORDS_FIELD);
rdc_mont(temp, mc);
}
void fpinv_mont(felm_t a)
{ // Field inversion using Montgomery arithmetic, a = a^(-1)*R mod p.
felm_t tt;
fpcopy(a, tt);
fpinv_chain_mont(tt);
fpsqr_mont(tt, tt);
fpsqr_mont(tt, tt);
fpmul_mont(a, tt, a);
}
void fp2copy(const f2elm_t *a, f2elm_t *c)
{ // Copy a GF(p^2) element, c = a.
fpcopy(a->e[0], c->e[0]);
fpcopy(a->e[1], c->e[1]);
}
void fp2neg(f2elm_t *a)
{ // GF(p^2) negation, a = -a in GF(p^2).
fpneg(a->e[0]);
fpneg(a->e[1]);
}
__inline void fp2add(const f2elm_t *a, const f2elm_t *b, f2elm_t *c)
{ // GF(p^2) addition, c = a+b in GF(p^2).
fpadd(a->e[0], b->e[0], c->e[0]);
fpadd(a->e[1], b->e[1], c->e[1]);
}
__inline void fp2sub(const f2elm_t *a, const f2elm_t *b, f2elm_t *c)
{ // GF(p^2) subtraction, c = a-b in GF(p^2).
fpsub(a->e[0], b->e[0], c->e[0]);
fpsub(a->e[1], b->e[1], c->e[1]);
}
void fp2div2(const f2elm_t *a, f2elm_t *c)
{ // GF(p^2) division by two, c = a/2 in GF(p^2).
fpdiv2(a->e[0], c->e[0]);
fpdiv2(a->e[1], c->e[1]);
}
void fp2correction(f2elm_t *a)
{ // Modular correction, a = a in GF(p^2).
fpcorrection(a->e[0]);
fpcorrection(a->e[1]);
}
__inline static void mp_addfast(const digit_t* a, const digit_t* b, digit_t* c)
{ // Multiprecision addition, c = a+b.
mp_add(a, b, c, NWORDS_FIELD);
}
__inline static void mp_addfastx2(const digit_t* a, const digit_t* b, digit_t* c)
{ // Double-length multiprecision addition, c = a+b.
mp_add(a, b, c, 2*NWORDS_FIELD);
}
void fp2sqr_mont(const f2elm_t *a, f2elm_t *c)
{ // GF(p^2) squaring using Montgomery arithmetic, c = a^2 in GF(p^2).
// Inputs: a = a0+a1*i, where a0, a1 are in [0, 2*p-1]
// Output: c = c0+c1*i, where c0, c1 are in [0, 2*p-1]
felm_t t1, t2, t3;
mp_addfast(a->e[0], a->e[1], t1); // t1 = a0+a1
fpsub(a->e[0], a->e[1], t2); // t2 = a0-a1
mp_addfast(a->e[0], a->e[0], t3); // t3 = 2a0
fpmul_mont(t1, t2, c->e[0]); // c0 = (a0+a1)(a0-a1)
fpmul_mont(t3, a->e[1], c->e[1]); // c1 = 2a0*a1
}
unsigned int mp_sub(const digit_t* a, const digit_t* b, digit_t* c, const unsigned int nwords)
{ // Multiprecision subtraction, c = a-b, where lng(a) = lng(b) = nwords. Returns the borrow bit.
unsigned int i, borrow = 0;
for (i = 0; i < nwords; i++) {
SUBC(borrow, a[i], b[i], borrow, c[i]);
}
return borrow;
}
__inline static digit_t mp_subfast(const digit_t* a, const digit_t* b, digit_t* c)
{ // Multiprecision subtraction, c = a-b, where lng(a) = lng(b) = 2*NWORDS_FIELD.
// If c < 0 then returns mask = 0xFF..F, else mask = 0x00..0
return (0 - (digit_t)mp_sub(a, b, c, 2*NWORDS_FIELD));
}
void fp2mul_mont(const f2elm_t *a, const f2elm_t *b, f2elm_t *c)
{ // GF(p^2) multiplication using Montgomery arithmetic, c = a*b in GF(p^2).
// Inputs: a = a0+a1*i and b = b0+b1*i, where a0, a1, b0, b1 are in [0, 2*p-1]
// Output: c = c0+c1*i, where c0, c1 are in [0, 2*p-1]
felm_t t1, t2;
dfelm_t tt1, tt2, tt3;
digit_t mask;
unsigned int i, borrow = 0;
mp_mul(a->e[0], b->e[0], tt1, NWORDS_FIELD); // tt1 = a0*b0
mp_mul(a->e[1], b->e[1], tt2, NWORDS_FIELD); // tt2 = a1*b1
mp_addfast(a->e[0], a->e[1], t1); // t1 = a0+a1
mp_addfast(b->e[0], b->e[1], t2); // t2 = b0+b1
mask = mp_subfast(tt1, tt2, tt3); // tt3 = a0*b0 - a1*b1. If tt3 < 0 then mask = 0xFF..F, else if tt3 >= 0 then mask = 0x00..0
for (i = 0; i < NWORDS_FIELD; i++) {
ADDC(borrow, tt3[NWORDS_FIELD+i], ((const digit_t*)PRIME)[i] & mask, borrow, tt3[NWORDS_FIELD+i]);
}
rdc_mont(tt3, c->e[0]); // c[0] = a0*b0 - a1*b1
mp_addfastx2(tt1, tt2, tt1); // tt1 = a0*b0 + a1*b1
mp_mul(t1, t2, tt2, NWORDS_FIELD); // tt2 = (a0+a1)*(b0+b1)
mp_subfast(tt2, tt1, tt2); // tt2 = (a0+a1)*(b0+b1) - a0*b0 - a1*b1
rdc_mont(tt2, c->e[1]); // c[1] = (a0+a1)*(b0+b1) - a0*b0 - a1*b1
}
void fpinv_chain_mont(felm_t a)
{ // Chain to compute a^(p-3)/4 using Montgomery arithmetic.
unsigned int i, j;
felm_t t[15], tt;
// Precomputed table
fpsqr_mont(a, tt);
fpmul_mont(a, tt, t[0]);
for (i = 0; i <= 13; i++) fpmul_mont(t[i], tt, t[i+1]);
fpcopy(a, tt);
for (i = 0; i < 8; i++) fpsqr_mont(tt, tt);
fpmul_mont(a, tt, tt);
for (i = 0; i < 5; i++) fpsqr_mont(tt, tt);
fpmul_mont(t[8], tt, tt);
for (i = 0; i < 5; i++) fpsqr_mont(tt, tt);
fpmul_mont(t[6], tt, tt);
for (i = 0; i < 6; i++) fpsqr_mont(tt, tt);
fpmul_mont(t[9], tt, tt);
for (i = 0; i < 7; i++) fpsqr_mont(tt, tt);
fpmul_mont(t[0], tt, tt);
for (i = 0; i < 7; i++) fpsqr_mont(tt, tt);
fpmul_mont(a, tt, tt);
for (i = 0; i < 7; i++) fpsqr_mont(tt, tt);
fpmul_mont(t[6], tt, tt);
for (i = 0; i < 7; i++) fpsqr_mont(tt, tt);
fpmul_mont(t[2], tt, tt);
for (i = 0; i < 5; i++) fpsqr_mont(tt, tt);
fpmul_mont(t[8], tt, tt);
for (i = 0; i < 7; i++) fpsqr_mont(tt, tt);
fpmul_mont(a, tt, tt);
for (i = 0; i < 8; i++) fpsqr_mont(tt, tt);
fpmul_mont(t[10], tt, tt);
for (i = 0; i < 5; i++) fpsqr_mont(tt, tt);
fpmul_mont(t[0], tt, tt);
for (i = 0; i < 6; i++) fpsqr_mont(tt, tt);
fpmul_mont(t[10], tt, tt);
for (i = 0; i < 5; i++) fpsqr_mont(tt, tt);
fpmul_mont(t[10], tt, tt);
for (i = 0; i < 5; i++) fpsqr_mont(tt, tt);
fpmul_mont(t[5], tt, tt);
for (i = 0; i < 5; i++) fpsqr_mont(tt, tt);
fpmul_mont(t[2], tt, tt);
for (i = 0; i < 5; i++) fpsqr_mont(tt, tt);
fpmul_mont(t[6], tt, tt);
for (i = 0; i < 5; i++) fpsqr_mont(tt, tt);
fpmul_mont(t[3], tt, tt);
for (i = 0; i < 6; i++) fpsqr_mont(tt, tt);
fpmul_mont(t[5], tt, tt);
for (i = 0; i < 12; i++) fpsqr_mont(tt, tt);
fpmul_mont(t[12], tt, tt);
for (i = 0; i < 5; i++) fpsqr_mont(tt, tt);
fpmul_mont(t[8], tt, tt);
for (i = 0; i < 5; i++) fpsqr_mont(tt, tt);
fpmul_mont(t[6], tt, tt);
for (i = 0; i < 5; i++) fpsqr_mont(tt, tt);
fpmul_mont(t[12], tt, tt);
for (i = 0; i < 6; i++) fpsqr_mont(tt, tt);
fpmul_mont(t[11], tt, tt);
for (i = 0; i < 8; i++) fpsqr_mont(tt, tt);
fpmul_mont(t[6], tt, tt);
for (i = 0; i < 5; i++) fpsqr_mont(tt, tt);
fpmul_mont(t[5], tt, tt);
for (i = 0; i < 5; i++) fpsqr_mont(tt, tt);
fpmul_mont(t[14], tt, tt);
for (i = 0; i < 7; i++) fpsqr_mont(tt, tt);
fpmul_mont(t[14], tt, tt);
for (i = 0; i < 5; i++) fpsqr_mont(tt, tt);
fpmul_mont(t[5], tt, tt);
for (i = 0; i < 5; i++) fpsqr_mont(tt, tt);
fpmul_mont(t[6], tt, tt);
for (i = 0; i < 8; i++) fpsqr_mont(tt, tt);
fpmul_mont(t[8], tt, tt);
for (i = 0; i < 5; i++) fpsqr_mont(tt, tt);
fpmul_mont(a, tt, tt);
for (i = 0; i < 8; i++) fpsqr_mont(tt, tt);
fpmul_mont(t[4], tt, tt);
for (i = 0; i < 5; i++) fpsqr_mont(tt, tt);
fpmul_mont(t[6], tt, tt);
for (i = 0; i < 5; i++) fpsqr_mont(tt, tt);
fpmul_mont(t[5], tt, tt);
for (i = 0; i < 8; i++) fpsqr_mont(tt, tt);
fpmul_mont(t[7], tt, tt);
for (i = 0; i < 5; i++) fpsqr_mont(tt, tt);
fpmul_mont(a, tt, tt);
for (i = 0; i < 5; i++) fpsqr_mont(tt, tt);
fpmul_mont(t[0], tt, tt);
for (i = 0; i < 5; i++) fpsqr_mont(tt, tt);
fpmul_mont(t[11], tt, tt);
for (i = 0; i < 5; i++) fpsqr_mont(tt, tt);
fpmul_mont(t[13], tt, tt);
for (i = 0; i < 8; i++) fpsqr_mont(tt, tt);
fpmul_mont(t[1], tt, tt);
for (i = 0; i < 6; i++) fpsqr_mont(tt, tt);
fpmul_mont(t[10], tt, tt);
for (j = 0; j < 49; j++) {
for (i = 0; i < 5; i++) fpsqr_mont(tt, tt);
fpmul_mont(t[14], tt, tt);
}
fpcopy(tt, a);
}
void fp2inv_mont(f2elm_t *a)
{// GF(p^2) inversion using Montgomery arithmetic, a = (a0-i*a1)/(a0^2+a1^2).
f2elm_t t1;
fpsqr_mont(a->e[0], t1.e[0]); // t10 = a0^2
fpsqr_mont(a->e[1], t1.e[1]); // t11 = a1^2
fpadd(t1.e[0], t1.e[1], t1.e[0]); // t10 = a0^2+a1^2
fpinv_mont(t1.e[0]); // t10 = (a0^2+a1^2)^-1
fpneg(a->e[1]); // a = a0-i*a1
fpmul_mont(a->e[0], t1.e[0], a->e[0]);
fpmul_mont(a->e[1], t1.e[0], a->e[1]); // a = (a0-i*a1)*(a0^2+a1^2)^-1
}
void to_fp2mont(const f2elm_t *a, f2elm_t *mc)
{ // Conversion of a GF(p^2) element to Montgomery representation,
// mc_i = a_i*R^2*R^(-1) = a_i*R in GF(p^2).
to_mont(a->e[0], mc->e[0]);
to_mont(a->e[1], mc->e[1]);
}
void from_fp2mont(const f2elm_t *ma, f2elm_t *c)
{ // Conversion of a GF(p^2) element from Montgomery representation to standard representation,
// c_i = ma_i*R^(-1) = a_i in GF(p^2).
from_mont(ma->e[0], c->e[0]);
from_mont(ma->e[1], c->e[1]);
}
unsigned int is_felm_zero(const felm_t x)
{ // Is x = 0? return 1 (TRUE) if condition is true, 0 (FALSE) otherwise.
// SECURITY NOTE: This function does not run in constant-time.
for (unsigned int i = 0; i < NWORDS_FIELD; i++) {
if (x[i] != 0) {
return 0;
}
}
return 1;
}
unsigned int mp_add(const digit_t* a, const digit_t* b, digit_t* c, const unsigned int nwords)
{ // Multiprecision addition, c = a+b, where lng(a) = lng(b) = nwords. Returns the carry bit.
unsigned int i, carry = 0;
for (i = 0; i < nwords; i++) {
ADDC(carry, a[i], b[i], carry, c[i]);
}
return carry;
}
void mp_shiftr1(digit_t* x, const unsigned int nwords)
{ // Multiprecision right shift by one.
unsigned int i;
for (i = 0; i < nwords-1; i++) {
SHIFTR(x[i+1], x[i], 1, x[i], RADIX);
}
x[nwords-1] >>= 1;
}
|
