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
|
/* This Source Code Form is subject to the terms of the Mozilla Public
* License, v. 2.0. If a copy of the MPL was not distributed with this
* file, You can obtain one at http://mozilla.org/MPL/2.0/. */
#include "ecp.h"
#include "mplogic.h"
#include <stdlib.h>
/* Checks if point P(px, py) is at infinity. Uses affine coordinates. */
mp_err
ec_GFp_pt_is_inf_aff(const mp_int *px, const mp_int *py)
{
if ((mp_cmp_z(px) == 0) && (mp_cmp_z(py) == 0)) {
return MP_YES;
} else {
return MP_NO;
}
}
/* Sets P(px, py) to be the point at infinity. Uses affine coordinates. */
mp_err
ec_GFp_pt_set_inf_aff(mp_int *px, mp_int *py)
{
mp_zero(px);
mp_zero(py);
return MP_OKAY;
}
/* Computes R = P + Q based on IEEE P1363 A.10.1. Elliptic curve points P,
* Q, and R can all be identical. Uses affine coordinates. Assumes input
* is already field-encoded using field_enc, and returns output that is
* still field-encoded. */
mp_err
ec_GFp_pt_add_aff(const mp_int *px, const mp_int *py, const mp_int *qx,
const mp_int *qy, mp_int *rx, mp_int *ry,
const ECGroup *group)
{
mp_err res = MP_OKAY;
mp_int lambda, temp, tempx, tempy;
MP_DIGITS(&lambda) = 0;
MP_DIGITS(&temp) = 0;
MP_DIGITS(&tempx) = 0;
MP_DIGITS(&tempy) = 0;
MP_CHECKOK(mp_init(&lambda));
MP_CHECKOK(mp_init(&temp));
MP_CHECKOK(mp_init(&tempx));
MP_CHECKOK(mp_init(&tempy));
/* if P = inf, then R = Q */
if (ec_GFp_pt_is_inf_aff(px, py) == 0) {
MP_CHECKOK(mp_copy(qx, rx));
MP_CHECKOK(mp_copy(qy, ry));
res = MP_OKAY;
goto CLEANUP;
}
/* if Q = inf, then R = P */
if (ec_GFp_pt_is_inf_aff(qx, qy) == 0) {
MP_CHECKOK(mp_copy(px, rx));
MP_CHECKOK(mp_copy(py, ry));
res = MP_OKAY;
goto CLEANUP;
}
/* if px != qx, then lambda = (py-qy) / (px-qx) */
if (mp_cmp(px, qx) != 0) {
MP_CHECKOK(group->meth->field_sub(py, qy, &tempy, group->meth));
MP_CHECKOK(group->meth->field_sub(px, qx, &tempx, group->meth));
MP_CHECKOK(group->meth->field_div(&tempy, &tempx, &lambda, group->meth));
} else {
/* if py != qy or qy = 0, then R = inf */
if (((mp_cmp(py, qy) != 0)) || (mp_cmp_z(qy) == 0)) {
mp_zero(rx);
mp_zero(ry);
res = MP_OKAY;
goto CLEANUP;
}
/* lambda = (3qx^2+a) / (2qy) */
MP_CHECKOK(group->meth->field_sqr(qx, &tempx, group->meth));
MP_CHECKOK(mp_set_int(&temp, 3));
if (group->meth->field_enc) {
MP_CHECKOK(group->meth->field_enc(&temp, &temp, group->meth));
}
MP_CHECKOK(group->meth->field_mul(&tempx, &temp, &tempx, group->meth));
MP_CHECKOK(group->meth->field_add(&tempx, &group->curvea, &tempx, group->meth));
MP_CHECKOK(mp_set_int(&temp, 2));
if (group->meth->field_enc) {
MP_CHECKOK(group->meth->field_enc(&temp, &temp, group->meth));
}
MP_CHECKOK(group->meth->field_mul(qy, &temp, &tempy, group->meth));
MP_CHECKOK(group->meth->field_div(&tempx, &tempy, &lambda, group->meth));
}
/* rx = lambda^2 - px - qx */
MP_CHECKOK(group->meth->field_sqr(&lambda, &tempx, group->meth));
MP_CHECKOK(group->meth->field_sub(&tempx, px, &tempx, group->meth));
MP_CHECKOK(group->meth->field_sub(&tempx, qx, &tempx, group->meth));
/* ry = (x1-x2) * lambda - y1 */
MP_CHECKOK(group->meth->field_sub(qx, &tempx, &tempy, group->meth));
MP_CHECKOK(group->meth->field_mul(&tempy, &lambda, &tempy, group->meth));
MP_CHECKOK(group->meth->field_sub(&tempy, qy, &tempy, group->meth));
MP_CHECKOK(mp_copy(&tempx, rx));
MP_CHECKOK(mp_copy(&tempy, ry));
CLEANUP:
mp_clear(&lambda);
mp_clear(&temp);
mp_clear(&tempx);
mp_clear(&tempy);
return res;
}
/* Computes R = P - Q. Elliptic curve points P, Q, and R can all be
* identical. Uses affine coordinates. Assumes input is already
* field-encoded using field_enc, and returns output that is still
* field-encoded. */
mp_err
ec_GFp_pt_sub_aff(const mp_int *px, const mp_int *py, const mp_int *qx,
const mp_int *qy, mp_int *rx, mp_int *ry,
const ECGroup *group)
{
mp_err res = MP_OKAY;
mp_int nqy;
MP_DIGITS(&nqy) = 0;
MP_CHECKOK(mp_init(&nqy));
/* nqy = -qy */
MP_CHECKOK(group->meth->field_neg(qy, &nqy, group->meth));
res = group->point_add(px, py, qx, &nqy, rx, ry, group);
CLEANUP:
mp_clear(&nqy);
return res;
}
/* Computes R = 2P. Elliptic curve points P and R can be identical. Uses
* affine coordinates. Assumes input is already field-encoded using
* field_enc, and returns output that is still field-encoded. */
mp_err
ec_GFp_pt_dbl_aff(const mp_int *px, const mp_int *py, mp_int *rx,
mp_int *ry, const ECGroup *group)
{
return ec_GFp_pt_add_aff(px, py, px, py, rx, ry, group);
}
/* by default, this routine is unused and thus doesn't need to be compiled */
#ifdef ECL_ENABLE_GFP_PT_MUL_AFF
/* Computes R = nP based on IEEE P1363 A.10.3. Elliptic curve points P and
* R can be identical. Uses affine coordinates. Assumes input is already
* field-encoded using field_enc, and returns output that is still
* field-encoded. */
mp_err
ec_GFp_pt_mul_aff(const mp_int *n, const mp_int *px, const mp_int *py,
mp_int *rx, mp_int *ry, const ECGroup *group)
{
mp_err res = MP_OKAY;
mp_int k, k3, qx, qy, sx, sy;
int b1, b3, i, l;
MP_DIGITS(&k) = 0;
MP_DIGITS(&k3) = 0;
MP_DIGITS(&qx) = 0;
MP_DIGITS(&qy) = 0;
MP_DIGITS(&sx) = 0;
MP_DIGITS(&sy) = 0;
MP_CHECKOK(mp_init(&k));
MP_CHECKOK(mp_init(&k3));
MP_CHECKOK(mp_init(&qx));
MP_CHECKOK(mp_init(&qy));
MP_CHECKOK(mp_init(&sx));
MP_CHECKOK(mp_init(&sy));
/* if n = 0 then r = inf */
if (mp_cmp_z(n) == 0) {
mp_zero(rx);
mp_zero(ry);
res = MP_OKAY;
goto CLEANUP;
}
/* Q = P, k = n */
MP_CHECKOK(mp_copy(px, &qx));
MP_CHECKOK(mp_copy(py, &qy));
MP_CHECKOK(mp_copy(n, &k));
/* if n < 0 then Q = -Q, k = -k */
if (mp_cmp_z(n) < 0) {
MP_CHECKOK(group->meth->field_neg(&qy, &qy, group->meth));
MP_CHECKOK(mp_neg(&k, &k));
}
#ifdef ECL_DEBUG /* basic double and add method */
l = mpl_significant_bits(&k) - 1;
MP_CHECKOK(mp_copy(&qx, &sx));
MP_CHECKOK(mp_copy(&qy, &sy));
for (i = l - 1; i >= 0; i--) {
/* S = 2S */
MP_CHECKOK(group->point_dbl(&sx, &sy, &sx, &sy, group));
/* if k_i = 1, then S = S + Q */
if (mpl_get_bit(&k, i) != 0) {
MP_CHECKOK(group->point_add(&sx, &sy, &qx, &qy, &sx, &sy, group));
}
}
#else /* double and add/subtract method from \
* standard */
/* k3 = 3 * k */
MP_CHECKOK(mp_set_int(&k3, 3));
MP_CHECKOK(mp_mul(&k, &k3, &k3));
/* S = Q */
MP_CHECKOK(mp_copy(&qx, &sx));
MP_CHECKOK(mp_copy(&qy, &sy));
/* l = index of high order bit in binary representation of 3*k */
l = mpl_significant_bits(&k3) - 1;
/* for i = l-1 downto 1 */
for (i = l - 1; i >= 1; i--) {
/* S = 2S */
MP_CHECKOK(group->point_dbl(&sx, &sy, &sx, &sy, group));
b3 = MP_GET_BIT(&k3, i);
b1 = MP_GET_BIT(&k, i);
/* if k3_i = 1 and k_i = 0, then S = S + Q */
if ((b3 == 1) && (b1 == 0)) {
MP_CHECKOK(group->point_add(&sx, &sy, &qx, &qy, &sx, &sy, group));
/* if k3_i = 0 and k_i = 1, then S = S - Q */
} else if ((b3 == 0) && (b1 == 1)) {
MP_CHECKOK(group->point_sub(&sx, &sy, &qx, &qy, &sx, &sy, group));
}
}
#endif
/* output S */
MP_CHECKOK(mp_copy(&sx, rx));
MP_CHECKOK(mp_copy(&sy, ry));
CLEANUP:
mp_clear(&k);
mp_clear(&k3);
mp_clear(&qx);
mp_clear(&qy);
mp_clear(&sx);
mp_clear(&sy);
return res;
}
#endif
/* Validates a point on a GFp curve. */
mp_err
ec_GFp_validate_point(const mp_int *px, const mp_int *py, const ECGroup *group)
{
mp_err res = MP_NO;
mp_int accl, accr, tmp, pxt, pyt;
MP_DIGITS(&accl) = 0;
MP_DIGITS(&accr) = 0;
MP_DIGITS(&tmp) = 0;
MP_DIGITS(&pxt) = 0;
MP_DIGITS(&pyt) = 0;
MP_CHECKOK(mp_init(&accl));
MP_CHECKOK(mp_init(&accr));
MP_CHECKOK(mp_init(&tmp));
MP_CHECKOK(mp_init(&pxt));
MP_CHECKOK(mp_init(&pyt));
/* 1: Verify that publicValue is not the point at infinity */
if (ec_GFp_pt_is_inf_aff(px, py) == MP_YES) {
res = MP_NO;
goto CLEANUP;
}
/* 2: Verify that the coordinates of publicValue are elements
* of the field.
*/
if ((MP_SIGN(px) == MP_NEG) || (mp_cmp(px, &group->meth->irr) >= 0) ||
(MP_SIGN(py) == MP_NEG) || (mp_cmp(py, &group->meth->irr) >= 0)) {
res = MP_NO;
goto CLEANUP;
}
/* 3: Verify that publicValue is on the curve. */
if (group->meth->field_enc) {
group->meth->field_enc(px, &pxt, group->meth);
group->meth->field_enc(py, &pyt, group->meth);
} else {
MP_CHECKOK(mp_copy(px, &pxt));
MP_CHECKOK(mp_copy(py, &pyt));
}
/* left-hand side: y^2 */
MP_CHECKOK(group->meth->field_sqr(&pyt, &accl, group->meth));
/* right-hand side: x^3 + a*x + b = (x^2 + a)*x + b by Horner's rule */
MP_CHECKOK(group->meth->field_sqr(&pxt, &tmp, group->meth));
MP_CHECKOK(group->meth->field_add(&tmp, &group->curvea, &tmp, group->meth));
MP_CHECKOK(group->meth->field_mul(&tmp, &pxt, &accr, group->meth));
MP_CHECKOK(group->meth->field_add(&accr, &group->curveb, &accr, group->meth));
/* check LHS - RHS == 0 */
MP_CHECKOK(group->meth->field_sub(&accl, &accr, &accr, group->meth));
if (mp_cmp_z(&accr) != 0) {
res = MP_NO;
goto CLEANUP;
}
/* 4: Verify that the order of the curve times the publicValue
* is the point at infinity.
*/
MP_CHECKOK(ECPoint_mul(group, &group->order, px, py, &pxt, &pyt));
if (ec_GFp_pt_is_inf_aff(&pxt, &pyt) != MP_YES) {
res = MP_NO;
goto CLEANUP;
}
res = MP_YES;
CLEANUP:
mp_clear(&accl);
mp_clear(&accr);
mp_clear(&tmp);
mp_clear(&pxt);
mp_clear(&pyt);
return res;
}
|
