1 /**********************************************************************
2 * Copyright (c) 2013, 2014 Pieter Wuille *
3 * Distributed under the MIT software license, see the accompanying *
4 * file COPYING or http://www.opensource.org/licenses/mit-license.php.*
5 **********************************************************************/
6
7 #ifndef SECP256K1_FIELD_IMPL_H
8 #define SECP256K1_FIELD_IMPL_H
9
10 #if defined HAVE_CONFIG_H
11 #include "libsecp256k1-config.h"
12 #endif
13
14 #include "util.h"
15
16 #if defined(USE_FIELD_10X26)
17 #include "field_10x26_impl.h"
18 #elif defined(USE_FIELD_5X52)
19 #include "field_5x52_impl.h"
20 #else
21 #error "Please select field implementation"
22 #endif
23
secp256k1_fe_equal(const secp256k1_fe * a,const secp256k1_fe * b)24 SECP256K1_INLINE static int secp256k1_fe_equal(const secp256k1_fe *a, const secp256k1_fe *b) {
25 secp256k1_fe na;
26 secp256k1_fe_negate(&na, a, 1);
27 secp256k1_fe_add(&na, b);
28 return secp256k1_fe_normalizes_to_zero(&na);
29 }
30
secp256k1_fe_equal_var(const secp256k1_fe * a,const secp256k1_fe * b)31 SECP256K1_INLINE static int secp256k1_fe_equal_var(const secp256k1_fe *a, const secp256k1_fe *b) {
32 secp256k1_fe na;
33 secp256k1_fe_negate(&na, a, 1);
34 secp256k1_fe_add(&na, b);
35 return secp256k1_fe_normalizes_to_zero_var(&na);
36 }
37
secp256k1_fe_sqrt(secp256k1_fe * r,const secp256k1_fe * a)38 static int secp256k1_fe_sqrt(secp256k1_fe *r, const secp256k1_fe *a) {
39 /** Given that p is congruent to 3 mod 4, we can compute the square root of
40 * a mod p as the (p+1)/4'th power of a.
41 *
42 * As (p+1)/4 is an even number, it will have the same result for a and for
43 * (-a). Only one of these two numbers actually has a square root however,
44 * so we test at the end by squaring and comparing to the input.
45 * Also because (p+1)/4 is an even number, the computed square root is
46 * itself always a square (a ** ((p+1)/4) is the square of a ** ((p+1)/8)).
47 */
48 secp256k1_fe x2, x3, x6, x9, x11, x22, x44, x88, x176, x220, x223, t1;
49 int j;
50
51 /** The binary representation of (p + 1)/4 has 3 blocks of 1s, with lengths in
52 * { 2, 22, 223 }. Use an addition chain to calculate 2^n - 1 for each block:
53 * 1, [2], 3, 6, 9, 11, [22], 44, 88, 176, 220, [223]
54 */
55
56 secp256k1_fe_sqr(&x2, a);
57 secp256k1_fe_mul(&x2, &x2, a);
58
59 secp256k1_fe_sqr(&x3, &x2);
60 secp256k1_fe_mul(&x3, &x3, a);
61
62 x6 = x3;
63 for (j=0; j<3; j++) {
64 secp256k1_fe_sqr(&x6, &x6);
65 }
66 secp256k1_fe_mul(&x6, &x6, &x3);
67
68 x9 = x6;
69 for (j=0; j<3; j++) {
70 secp256k1_fe_sqr(&x9, &x9);
71 }
72 secp256k1_fe_mul(&x9, &x9, &x3);
73
74 x11 = x9;
75 for (j=0; j<2; j++) {
76 secp256k1_fe_sqr(&x11, &x11);
77 }
78 secp256k1_fe_mul(&x11, &x11, &x2);
79
80 x22 = x11;
81 for (j=0; j<11; j++) {
82 secp256k1_fe_sqr(&x22, &x22);
83 }
84 secp256k1_fe_mul(&x22, &x22, &x11);
85
86 x44 = x22;
87 for (j=0; j<22; j++) {
88 secp256k1_fe_sqr(&x44, &x44);
89 }
90 secp256k1_fe_mul(&x44, &x44, &x22);
91
92 x88 = x44;
93 for (j=0; j<44; j++) {
94 secp256k1_fe_sqr(&x88, &x88);
95 }
96 secp256k1_fe_mul(&x88, &x88, &x44);
97
98 x176 = x88;
99 for (j=0; j<88; j++) {
100 secp256k1_fe_sqr(&x176, &x176);
101 }
102 secp256k1_fe_mul(&x176, &x176, &x88);
103
104 x220 = x176;
105 for (j=0; j<44; j++) {
106 secp256k1_fe_sqr(&x220, &x220);
107 }
108 secp256k1_fe_mul(&x220, &x220, &x44);
109
110 x223 = x220;
111 for (j=0; j<3; j++) {
112 secp256k1_fe_sqr(&x223, &x223);
113 }
114 secp256k1_fe_mul(&x223, &x223, &x3);
115
116 /* The final result is then assembled using a sliding window over the blocks. */
117
118 t1 = x223;
119 for (j=0; j<23; j++) {
120 secp256k1_fe_sqr(&t1, &t1);
121 }
122 secp256k1_fe_mul(&t1, &t1, &x22);
123 for (j=0; j<6; j++) {
124 secp256k1_fe_sqr(&t1, &t1);
125 }
126 secp256k1_fe_mul(&t1, &t1, &x2);
127 secp256k1_fe_sqr(&t1, &t1);
128 secp256k1_fe_sqr(r, &t1);
129
130 /* Check that a square root was actually calculated */
131
132 secp256k1_fe_sqr(&t1, r);
133 return secp256k1_fe_equal(&t1, a);
134 }
135
secp256k1_fe_inv(secp256k1_fe * r,const secp256k1_fe * a)136 static void secp256k1_fe_inv(secp256k1_fe *r, const secp256k1_fe *a) {
137 secp256k1_fe x2, x3, x6, x9, x11, x22, x44, x88, x176, x220, x223, t1;
138 int j;
139
140 /** The binary representation of (p - 2) has 5 blocks of 1s, with lengths in
141 * { 1, 2, 22, 223 }. Use an addition chain to calculate 2^n - 1 for each block:
142 * [1], [2], 3, 6, 9, 11, [22], 44, 88, 176, 220, [223]
143 */
144
145 secp256k1_fe_sqr(&x2, a);
146 secp256k1_fe_mul(&x2, &x2, a);
147
148 secp256k1_fe_sqr(&x3, &x2);
149 secp256k1_fe_mul(&x3, &x3, a);
150
151 x6 = x3;
152 for (j=0; j<3; j++) {
153 secp256k1_fe_sqr(&x6, &x6);
154 }
155 secp256k1_fe_mul(&x6, &x6, &x3);
156
157 x9 = x6;
158 for (j=0; j<3; j++) {
159 secp256k1_fe_sqr(&x9, &x9);
160 }
161 secp256k1_fe_mul(&x9, &x9, &x3);
162
163 x11 = x9;
164 for (j=0; j<2; j++) {
165 secp256k1_fe_sqr(&x11, &x11);
166 }
167 secp256k1_fe_mul(&x11, &x11, &x2);
168
169 x22 = x11;
170 for (j=0; j<11; j++) {
171 secp256k1_fe_sqr(&x22, &x22);
172 }
173 secp256k1_fe_mul(&x22, &x22, &x11);
174
175 x44 = x22;
176 for (j=0; j<22; j++) {
177 secp256k1_fe_sqr(&x44, &x44);
178 }
179 secp256k1_fe_mul(&x44, &x44, &x22);
180
181 x88 = x44;
182 for (j=0; j<44; j++) {
183 secp256k1_fe_sqr(&x88, &x88);
184 }
185 secp256k1_fe_mul(&x88, &x88, &x44);
186
187 x176 = x88;
188 for (j=0; j<88; j++) {
189 secp256k1_fe_sqr(&x176, &x176);
190 }
191 secp256k1_fe_mul(&x176, &x176, &x88);
192
193 x220 = x176;
194 for (j=0; j<44; j++) {
195 secp256k1_fe_sqr(&x220, &x220);
196 }
197 secp256k1_fe_mul(&x220, &x220, &x44);
198
199 x223 = x220;
200 for (j=0; j<3; j++) {
201 secp256k1_fe_sqr(&x223, &x223);
202 }
203 secp256k1_fe_mul(&x223, &x223, &x3);
204
205 /* The final result is then assembled using a sliding window over the blocks. */
206
207 t1 = x223;
208 for (j=0; j<23; j++) {
209 secp256k1_fe_sqr(&t1, &t1);
210 }
211 secp256k1_fe_mul(&t1, &t1, &x22);
212 for (j=0; j<5; j++) {
213 secp256k1_fe_sqr(&t1, &t1);
214 }
215 secp256k1_fe_mul(&t1, &t1, a);
216 for (j=0; j<3; j++) {
217 secp256k1_fe_sqr(&t1, &t1);
218 }
219 secp256k1_fe_mul(&t1, &t1, &x2);
220 for (j=0; j<2; j++) {
221 secp256k1_fe_sqr(&t1, &t1);
222 }
223 secp256k1_fe_mul(r, a, &t1);
224 }
225
secp256k1_fe_inv_var(secp256k1_fe * r,const secp256k1_fe * a)226 static void secp256k1_fe_inv_var(secp256k1_fe *r, const secp256k1_fe *a) {
227 #if defined(USE_FIELD_INV_BUILTIN)
228 secp256k1_fe_inv(r, a);
229 #elif defined(USE_FIELD_INV_NUM)
230 secp256k1_num n, m;
231 static const secp256k1_fe negone = SECP256K1_FE_CONST(
232 0xFFFFFFFFUL, 0xFFFFFFFFUL, 0xFFFFFFFFUL, 0xFFFFFFFFUL,
233 0xFFFFFFFFUL, 0xFFFFFFFFUL, 0xFFFFFFFEUL, 0xFFFFFC2EUL
234 );
235 /* secp256k1 field prime, value p defined in "Standards for Efficient Cryptography" (SEC2) 2.7.1. */
236 static const unsigned char prime[32] = {
237 0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,
238 0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,
239 0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,
240 0xFF,0xFF,0xFF,0xFE,0xFF,0xFF,0xFC,0x2F
241 };
242 unsigned char b[32];
243 int res;
244 secp256k1_fe c = *a;
245 secp256k1_fe_normalize_var(&c);
246 secp256k1_fe_get_b32(b, &c);
247 secp256k1_num_set_bin(&n, b, 32);
248 secp256k1_num_set_bin(&m, prime, 32);
249 secp256k1_num_mod_inverse(&n, &n, &m);
250 secp256k1_num_get_bin(b, 32, &n);
251 res = secp256k1_fe_set_b32(r, b);
252 (void)res;
253 VERIFY_CHECK(res);
254 /* Verify the result is the (unique) valid inverse using non-GMP code. */
255 secp256k1_fe_mul(&c, &c, r);
256 secp256k1_fe_add(&c, &negone);
257 CHECK(secp256k1_fe_normalizes_to_zero_var(&c));
258 #else
259 #error "Please select field inverse implementation"
260 #endif
261 }
262
secp256k1_fe_inv_all_var(secp256k1_fe * r,const secp256k1_fe * a,size_t len)263 static void secp256k1_fe_inv_all_var(secp256k1_fe *r, const secp256k1_fe *a, size_t len) {
264 secp256k1_fe u;
265 size_t i;
266 if (len < 1) {
267 return;
268 }
269
270 VERIFY_CHECK((r + len <= a) || (a + len <= r));
271
272 r[0] = a[0];
273
274 i = 0;
275 while (++i < len) {
276 secp256k1_fe_mul(&r[i], &r[i - 1], &a[i]);
277 }
278
279 secp256k1_fe_inv_var(&u, &r[--i]);
280
281 while (i > 0) {
282 size_t j = i--;
283 secp256k1_fe_mul(&r[j], &r[i], &u);
284 secp256k1_fe_mul(&u, &u, &a[j]);
285 }
286
287 r[0] = u;
288 }
289
secp256k1_fe_is_quad_var(const secp256k1_fe * a)290 static int secp256k1_fe_is_quad_var(const secp256k1_fe *a) {
291 #ifndef USE_NUM_NONE
292 unsigned char b[32];
293 secp256k1_num n;
294 secp256k1_num m;
295 /* secp256k1 field prime, value p defined in "Standards for Efficient Cryptography" (SEC2) 2.7.1. */
296 static const unsigned char prime[32] = {
297 0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,
298 0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,
299 0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,
300 0xFF,0xFF,0xFF,0xFE,0xFF,0xFF,0xFC,0x2F
301 };
302
303 secp256k1_fe c = *a;
304 secp256k1_fe_normalize_var(&c);
305 secp256k1_fe_get_b32(b, &c);
306 secp256k1_num_set_bin(&n, b, 32);
307 secp256k1_num_set_bin(&m, prime, 32);
308 return secp256k1_num_jacobi(&n, &m) >= 0;
309 #else
310 secp256k1_fe r;
311 return secp256k1_fe_sqrt(&r, a);
312 #endif
313 }
314
315 #endif /* SECP256K1_FIELD_IMPL_H */
316