1 /* This Source Code Form is subject to the terms of the Mozilla Public
2 * License, v. 2.0. If a copy of the MPL was not distributed with this
3 * file, You can obtain one at http://mozilla.org/MPL/2.0/. */
4
5 #include "ecp.h"
6 #include "mplogic.h"
7 #include <stdlib.h>
8
9 /* Checks if point P(px, py) is at infinity. Uses affine coordinates. */
10 mp_err
ec_GFp_pt_is_inf_aff(const mp_int * px,const mp_int * py)11 ec_GFp_pt_is_inf_aff(const mp_int *px, const mp_int *py)
12 {
13
14 if ((mp_cmp_z(px) == 0) && (mp_cmp_z(py) == 0)) {
15 return MP_YES;
16 } else {
17 return MP_NO;
18 }
19 }
20
21 /* Sets P(px, py) to be the point at infinity. Uses affine coordinates. */
22 mp_err
ec_GFp_pt_set_inf_aff(mp_int * px,mp_int * py)23 ec_GFp_pt_set_inf_aff(mp_int *px, mp_int *py)
24 {
25 mp_zero(px);
26 mp_zero(py);
27 return MP_OKAY;
28 }
29
30 /* Computes R = P + Q based on IEEE P1363 A.10.1. Elliptic curve points P,
31 * Q, and R can all be identical. Uses affine coordinates. Assumes input
32 * is already field-encoded using field_enc, and returns output that is
33 * still field-encoded. */
34 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)35 ec_GFp_pt_add_aff(const mp_int *px, const mp_int *py, const mp_int *qx,
36 const mp_int *qy, mp_int *rx, mp_int *ry,
37 const ECGroup *group)
38 {
39 mp_err res = MP_OKAY;
40 mp_int lambda, temp, tempx, tempy;
41
42 MP_DIGITS(&lambda) = 0;
43 MP_DIGITS(&temp) = 0;
44 MP_DIGITS(&tempx) = 0;
45 MP_DIGITS(&tempy) = 0;
46 MP_CHECKOK(mp_init(&lambda));
47 MP_CHECKOK(mp_init(&temp));
48 MP_CHECKOK(mp_init(&tempx));
49 MP_CHECKOK(mp_init(&tempy));
50 /* if P = inf, then R = Q */
51 if (ec_GFp_pt_is_inf_aff(px, py) == 0) {
52 MP_CHECKOK(mp_copy(qx, rx));
53 MP_CHECKOK(mp_copy(qy, ry));
54 res = MP_OKAY;
55 goto CLEANUP;
56 }
57 /* if Q = inf, then R = P */
58 if (ec_GFp_pt_is_inf_aff(qx, qy) == 0) {
59 MP_CHECKOK(mp_copy(px, rx));
60 MP_CHECKOK(mp_copy(py, ry));
61 res = MP_OKAY;
62 goto CLEANUP;
63 }
64 /* if px != qx, then lambda = (py-qy) / (px-qx) */
65 if (mp_cmp(px, qx) != 0) {
66 MP_CHECKOK(group->meth->field_sub(py, qy, &tempy, group->meth));
67 MP_CHECKOK(group->meth->field_sub(px, qx, &tempx, group->meth));
68 MP_CHECKOK(group->meth->field_div(&tempy, &tempx, &lambda, group->meth));
69 } else {
70 /* if py != qy or qy = 0, then R = inf */
71 if (((mp_cmp(py, qy) != 0)) || (mp_cmp_z(qy) == 0)) {
72 mp_zero(rx);
73 mp_zero(ry);
74 res = MP_OKAY;
75 goto CLEANUP;
76 }
77 /* lambda = (3qx^2+a) / (2qy) */
78 MP_CHECKOK(group->meth->field_sqr(qx, &tempx, group->meth));
79 MP_CHECKOK(mp_set_int(&temp, 3));
80 if (group->meth->field_enc) {
81 MP_CHECKOK(group->meth->field_enc(&temp, &temp, group->meth));
82 }
83 MP_CHECKOK(group->meth->field_mul(&tempx, &temp, &tempx, group->meth));
84 MP_CHECKOK(group->meth->field_add(&tempx, &group->curvea, &tempx, group->meth));
85 MP_CHECKOK(mp_set_int(&temp, 2));
86 if (group->meth->field_enc) {
87 MP_CHECKOK(group->meth->field_enc(&temp, &temp, group->meth));
88 }
89 MP_CHECKOK(group->meth->field_mul(qy, &temp, &tempy, group->meth));
90 MP_CHECKOK(group->meth->field_div(&tempx, &tempy, &lambda, group->meth));
91 }
92 /* rx = lambda^2 - px - qx */
93 MP_CHECKOK(group->meth->field_sqr(&lambda, &tempx, group->meth));
94 MP_CHECKOK(group->meth->field_sub(&tempx, px, &tempx, group->meth));
95 MP_CHECKOK(group->meth->field_sub(&tempx, qx, &tempx, group->meth));
96 /* ry = (x1-x2) * lambda - y1 */
97 MP_CHECKOK(group->meth->field_sub(qx, &tempx, &tempy, group->meth));
98 MP_CHECKOK(group->meth->field_mul(&tempy, &lambda, &tempy, group->meth));
99 MP_CHECKOK(group->meth->field_sub(&tempy, qy, &tempy, group->meth));
100 MP_CHECKOK(mp_copy(&tempx, rx));
101 MP_CHECKOK(mp_copy(&tempy, ry));
102
103 CLEANUP:
104 mp_clear(&lambda);
105 mp_clear(&temp);
106 mp_clear(&tempx);
107 mp_clear(&tempy);
108 return res;
109 }
110
111 /* Computes R = P - Q. Elliptic curve points P, Q, and R can all be
112 * identical. Uses affine coordinates. Assumes input is already
113 * field-encoded using field_enc, and returns output that is still
114 * field-encoded. */
115 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)116 ec_GFp_pt_sub_aff(const mp_int *px, const mp_int *py, const mp_int *qx,
117 const mp_int *qy, mp_int *rx, mp_int *ry,
118 const ECGroup *group)
119 {
120 mp_err res = MP_OKAY;
121 mp_int nqy;
122
123 MP_DIGITS(&nqy) = 0;
124 MP_CHECKOK(mp_init(&nqy));
125 /* nqy = -qy */
126 MP_CHECKOK(group->meth->field_neg(qy, &nqy, group->meth));
127 res = group->point_add(px, py, qx, &nqy, rx, ry, group);
128 CLEANUP:
129 mp_clear(&nqy);
130 return res;
131 }
132
133 /* Computes R = 2P. Elliptic curve points P and R can be identical. Uses
134 * affine coordinates. Assumes input is already field-encoded using
135 * field_enc, and returns output that is still field-encoded. */
136 mp_err
ec_GFp_pt_dbl_aff(const mp_int * px,const mp_int * py,mp_int * rx,mp_int * ry,const ECGroup * group)137 ec_GFp_pt_dbl_aff(const mp_int *px, const mp_int *py, mp_int *rx,
138 mp_int *ry, const ECGroup *group)
139 {
140 return ec_GFp_pt_add_aff(px, py, px, py, rx, ry, group);
141 }
142
143 /* by default, this routine is unused and thus doesn't need to be compiled */
144 #ifdef ECL_ENABLE_GFP_PT_MUL_AFF
145 /* Computes R = nP based on IEEE P1363 A.10.3. Elliptic curve points P and
146 * R can be identical. Uses affine coordinates. Assumes input is already
147 * field-encoded using field_enc, and returns output that is still
148 * field-encoded. */
149 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)150 ec_GFp_pt_mul_aff(const mp_int *n, const mp_int *px, const mp_int *py,
151 mp_int *rx, mp_int *ry, const ECGroup *group)
152 {
153 mp_err res = MP_OKAY;
154 mp_int k, k3, qx, qy, sx, sy;
155 int b1, b3, i, l;
156
157 MP_DIGITS(&k) = 0;
158 MP_DIGITS(&k3) = 0;
159 MP_DIGITS(&qx) = 0;
160 MP_DIGITS(&qy) = 0;
161 MP_DIGITS(&sx) = 0;
162 MP_DIGITS(&sy) = 0;
163 MP_CHECKOK(mp_init(&k));
164 MP_CHECKOK(mp_init(&k3));
165 MP_CHECKOK(mp_init(&qx));
166 MP_CHECKOK(mp_init(&qy));
167 MP_CHECKOK(mp_init(&sx));
168 MP_CHECKOK(mp_init(&sy));
169
170 /* if n = 0 then r = inf */
171 if (mp_cmp_z(n) == 0) {
172 mp_zero(rx);
173 mp_zero(ry);
174 res = MP_OKAY;
175 goto CLEANUP;
176 }
177 /* Q = P, k = n */
178 MP_CHECKOK(mp_copy(px, &qx));
179 MP_CHECKOK(mp_copy(py, &qy));
180 MP_CHECKOK(mp_copy(n, &k));
181 /* if n < 0 then Q = -Q, k = -k */
182 if (mp_cmp_z(n) < 0) {
183 MP_CHECKOK(group->meth->field_neg(&qy, &qy, group->meth));
184 MP_CHECKOK(mp_neg(&k, &k));
185 }
186 #ifdef ECL_DEBUG /* basic double and add method */
187 l = mpl_significant_bits(&k) - 1;
188 MP_CHECKOK(mp_copy(&qx, &sx));
189 MP_CHECKOK(mp_copy(&qy, &sy));
190 for (i = l - 1; i >= 0; i--) {
191 /* S = 2S */
192 MP_CHECKOK(group->point_dbl(&sx, &sy, &sx, &sy, group));
193 /* if k_i = 1, then S = S + Q */
194 if (mpl_get_bit(&k, i) != 0) {
195 MP_CHECKOK(group->point_add(&sx, &sy, &qx, &qy, &sx, &sy, group));
196 }
197 }
198 #else /* double and add/subtract method from \
199 * standard */
200 /* k3 = 3 * k */
201 MP_CHECKOK(mp_set_int(&k3, 3));
202 MP_CHECKOK(mp_mul(&k, &k3, &k3));
203 /* S = Q */
204 MP_CHECKOK(mp_copy(&qx, &sx));
205 MP_CHECKOK(mp_copy(&qy, &sy));
206 /* l = index of high order bit in binary representation of 3*k */
207 l = mpl_significant_bits(&k3) - 1;
208 /* for i = l-1 downto 1 */
209 for (i = l - 1; i >= 1; i--) {
210 /* S = 2S */
211 MP_CHECKOK(group->point_dbl(&sx, &sy, &sx, &sy, group));
212 b3 = MP_GET_BIT(&k3, i);
213 b1 = MP_GET_BIT(&k, i);
214 /* if k3_i = 1 and k_i = 0, then S = S + Q */
215 if ((b3 == 1) && (b1 == 0)) {
216 MP_CHECKOK(group->point_add(&sx, &sy, &qx, &qy, &sx, &sy, group));
217 /* if k3_i = 0 and k_i = 1, then S = S - Q */
218 } else if ((b3 == 0) && (b1 == 1)) {
219 MP_CHECKOK(group->point_sub(&sx, &sy, &qx, &qy, &sx, &sy, group));
220 }
221 }
222 #endif
223 /* output S */
224 MP_CHECKOK(mp_copy(&sx, rx));
225 MP_CHECKOK(mp_copy(&sy, ry));
226
227 CLEANUP:
228 mp_clear(&k);
229 mp_clear(&k3);
230 mp_clear(&qx);
231 mp_clear(&qy);
232 mp_clear(&sx);
233 mp_clear(&sy);
234 return res;
235 }
236 #endif
237
238 /* Validates a point on a GFp curve. */
239 mp_err
ec_GFp_validate_point(const mp_int * px,const mp_int * py,const ECGroup * group)240 ec_GFp_validate_point(const mp_int *px, const mp_int *py, const ECGroup *group)
241 {
242 mp_err res = MP_NO;
243 mp_int accl, accr, tmp, pxt, pyt;
244
245 MP_DIGITS(&accl) = 0;
246 MP_DIGITS(&accr) = 0;
247 MP_DIGITS(&tmp) = 0;
248 MP_DIGITS(&pxt) = 0;
249 MP_DIGITS(&pyt) = 0;
250 MP_CHECKOK(mp_init(&accl));
251 MP_CHECKOK(mp_init(&accr));
252 MP_CHECKOK(mp_init(&tmp));
253 MP_CHECKOK(mp_init(&pxt));
254 MP_CHECKOK(mp_init(&pyt));
255
256 /* 1: Verify that publicValue is not the point at infinity */
257 if (ec_GFp_pt_is_inf_aff(px, py) == MP_YES) {
258 res = MP_NO;
259 goto CLEANUP;
260 }
261 /* 2: Verify that the coordinates of publicValue are elements
262 * of the field.
263 */
264 if ((MP_SIGN(px) == MP_NEG) || (mp_cmp(px, &group->meth->irr) >= 0) ||
265 (MP_SIGN(py) == MP_NEG) || (mp_cmp(py, &group->meth->irr) >= 0)) {
266 res = MP_NO;
267 goto CLEANUP;
268 }
269 /* 3: Verify that publicValue is on the curve. */
270 if (group->meth->field_enc) {
271 group->meth->field_enc(px, &pxt, group->meth);
272 group->meth->field_enc(py, &pyt, group->meth);
273 } else {
274 MP_CHECKOK(mp_copy(px, &pxt));
275 MP_CHECKOK(mp_copy(py, &pyt));
276 }
277 /* left-hand side: y^2 */
278 MP_CHECKOK(group->meth->field_sqr(&pyt, &accl, group->meth));
279 /* right-hand side: x^3 + a*x + b = (x^2 + a)*x + b by Horner's rule */
280 MP_CHECKOK(group->meth->field_sqr(&pxt, &tmp, group->meth));
281 MP_CHECKOK(group->meth->field_add(&tmp, &group->curvea, &tmp, group->meth));
282 MP_CHECKOK(group->meth->field_mul(&tmp, &pxt, &accr, group->meth));
283 MP_CHECKOK(group->meth->field_add(&accr, &group->curveb, &accr, group->meth));
284 /* check LHS - RHS == 0 */
285 MP_CHECKOK(group->meth->field_sub(&accl, &accr, &accr, group->meth));
286 if (mp_cmp_z(&accr) != 0) {
287 res = MP_NO;
288 goto CLEANUP;
289 }
290 /* 4: Verify that the order of the curve times the publicValue
291 * is the point at infinity.
292 */
293 MP_CHECKOK(ECPoint_mul(group, &group->order, px, py, &pxt, &pyt));
294 if (ec_GFp_pt_is_inf_aff(&pxt, &pyt) != MP_YES) {
295 res = MP_NO;
296 goto CLEANUP;
297 }
298
299 res = MP_YES;
300
301 CLEANUP:
302 mp_clear(&accl);
303 mp_clear(&accr);
304 mp_clear(&tmp);
305 mp_clear(&pxt);
306 mp_clear(&pyt);
307 return res;
308 }
309