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 "ecl-priv.h"
7 #include "mplogic.h"
8 #include <stdlib.h>
9
10 #define MAX_SCRATCH 6
11
12 /* Computes R = 2P. Elliptic curve points P and R can be identical. Uses
13 * Modified Jacobian coordinates.
14 *
15 * Assumes input is already field-encoded using field_enc, and returns
16 * output that is still field-encoded.
17 *
18 */
19 static mp_err
ec_GFp_pt_dbl_jm(const mp_int * px,const mp_int * py,const mp_int * pz,const mp_int * paz4,mp_int * rx,mp_int * ry,mp_int * rz,mp_int * raz4,mp_int scratch[],const ECGroup * group)20 ec_GFp_pt_dbl_jm(const mp_int *px, const mp_int *py, const mp_int *pz,
21 const mp_int *paz4, mp_int *rx, mp_int *ry, mp_int *rz,
22 mp_int *raz4, mp_int scratch[], const ECGroup *group)
23 {
24 mp_err res = MP_OKAY;
25 mp_int *t0, *t1, *M, *S;
26
27 t0 = &scratch[0];
28 t1 = &scratch[1];
29 M = &scratch[2];
30 S = &scratch[3];
31
32 #if MAX_SCRATCH < 4
33 #error "Scratch array defined too small "
34 #endif
35
36 /* Check for point at infinity */
37 if (ec_GFp_pt_is_inf_jac(px, py, pz) == MP_YES) {
38 /* Set r = pt at infinity by setting rz = 0 */
39
40 MP_CHECKOK(ec_GFp_pt_set_inf_jac(rx, ry, rz));
41 goto CLEANUP;
42 }
43
44 /* M = 3 (px^2) + a*(pz^4) */
45 MP_CHECKOK(group->meth->field_sqr(px, t0, group->meth));
46 MP_CHECKOK(group->meth->field_add(t0, t0, M, group->meth));
47 MP_CHECKOK(group->meth->field_add(t0, M, t0, group->meth));
48 MP_CHECKOK(group->meth->field_add(t0, paz4, M, group->meth));
49
50 /* rz = 2 * py * pz */
51 MP_CHECKOK(group->meth->field_mul(py, pz, S, group->meth));
52 MP_CHECKOK(group->meth->field_add(S, S, rz, group->meth));
53
54 /* t0 = 2y^2 , t1 = 8y^4 */
55 MP_CHECKOK(group->meth->field_sqr(py, t0, group->meth));
56 MP_CHECKOK(group->meth->field_add(t0, t0, t0, group->meth));
57 MP_CHECKOK(group->meth->field_sqr(t0, t1, group->meth));
58 MP_CHECKOK(group->meth->field_add(t1, t1, t1, group->meth));
59
60 /* S = 4 * px * py^2 = 2 * px * t0 */
61 MP_CHECKOK(group->meth->field_mul(px, t0, S, group->meth));
62 MP_CHECKOK(group->meth->field_add(S, S, S, group->meth));
63
64 /* rx = M^2 - 2S */
65 MP_CHECKOK(group->meth->field_sqr(M, rx, group->meth));
66 MP_CHECKOK(group->meth->field_sub(rx, S, rx, group->meth));
67 MP_CHECKOK(group->meth->field_sub(rx, S, rx, group->meth));
68
69 /* ry = M * (S - rx) - t1 */
70 MP_CHECKOK(group->meth->field_sub(S, rx, S, group->meth));
71 MP_CHECKOK(group->meth->field_mul(S, M, ry, group->meth));
72 MP_CHECKOK(group->meth->field_sub(ry, t1, ry, group->meth));
73
74 /* ra*z^4 = 2*t1*(apz4) */
75 MP_CHECKOK(group->meth->field_mul(paz4, t1, raz4, group->meth));
76 MP_CHECKOK(group->meth->field_add(raz4, raz4, raz4, group->meth));
77
78 CLEANUP:
79 return res;
80 }
81
82 /* Computes R = P + Q where R is (rx, ry, rz), P is (px, py, pz) and Q is
83 * (qx, qy, 1). Elliptic curve points P, Q, and R can all be identical.
84 * Uses mixed Modified_Jacobian-affine coordinates. Assumes input is
85 * already field-encoded using field_enc, and returns output that is still
86 * field-encoded. */
87 static mp_err
ec_GFp_pt_add_jm_aff(const mp_int * px,const mp_int * py,const mp_int * pz,const mp_int * paz4,const mp_int * qx,const mp_int * qy,mp_int * rx,mp_int * ry,mp_int * rz,mp_int * raz4,mp_int scratch[],const ECGroup * group)88 ec_GFp_pt_add_jm_aff(const mp_int *px, const mp_int *py, const mp_int *pz,
89 const mp_int *paz4, const mp_int *qx,
90 const mp_int *qy, mp_int *rx, mp_int *ry, mp_int *rz,
91 mp_int *raz4, mp_int scratch[], const ECGroup *group)
92 {
93 mp_err res = MP_OKAY;
94 mp_int *A, *B, *C, *D, *C2, *C3;
95
96 A = &scratch[0];
97 B = &scratch[1];
98 C = &scratch[2];
99 D = &scratch[3];
100 C2 = &scratch[4];
101 C3 = &scratch[5];
102
103 #if MAX_SCRATCH < 6
104 #error "Scratch array defined too small "
105 #endif
106
107 /* If either P or Q is the point at infinity, then return the other
108 * point */
109 if (ec_GFp_pt_is_inf_jac(px, py, pz) == MP_YES) {
110 MP_CHECKOK(ec_GFp_pt_aff2jac(qx, qy, rx, ry, rz, group));
111 MP_CHECKOK(group->meth->field_sqr(rz, raz4, group->meth));
112 MP_CHECKOK(group->meth->field_sqr(raz4, raz4, group->meth));
113 MP_CHECKOK(group->meth->field_mul(raz4, &group->curvea, raz4, group->meth));
114 goto CLEANUP;
115 }
116 if (ec_GFp_pt_is_inf_aff(qx, qy) == MP_YES) {
117 MP_CHECKOK(mp_copy(px, rx));
118 MP_CHECKOK(mp_copy(py, ry));
119 MP_CHECKOK(mp_copy(pz, rz));
120 MP_CHECKOK(mp_copy(paz4, raz4));
121 goto CLEANUP;
122 }
123
124 /* A = qx * pz^2, B = qy * pz^3 */
125 MP_CHECKOK(group->meth->field_sqr(pz, A, group->meth));
126 MP_CHECKOK(group->meth->field_mul(A, pz, B, group->meth));
127 MP_CHECKOK(group->meth->field_mul(A, qx, A, group->meth));
128 MP_CHECKOK(group->meth->field_mul(B, qy, B, group->meth));
129
130 /* Check P == Q */
131 if (mp_cmp(A, px) == 0) {
132 if (mp_cmp(B, py) == 0) {
133 /* If Px == Qx && Py == Qy, double P. */
134 return ec_GFp_pt_dbl_jm(px, py, pz, paz4, rx, ry, rz, raz4,
135 scratch, group);
136 }
137 /* If Px == Qx && Py != Qy, return point at infinity. */
138 return ec_GFp_pt_set_inf_jac(rx, ry, rz);
139 }
140
141 /* C = A - px, D = B - py */
142 MP_CHECKOK(group->meth->field_sub(A, px, C, group->meth));
143 MP_CHECKOK(group->meth->field_sub(B, py, D, group->meth));
144
145 /* C2 = C^2, C3 = C^3 */
146 MP_CHECKOK(group->meth->field_sqr(C, C2, group->meth));
147 MP_CHECKOK(group->meth->field_mul(C, C2, C3, group->meth));
148
149 /* rz = pz * C */
150 MP_CHECKOK(group->meth->field_mul(pz, C, rz, group->meth));
151
152 /* C = px * C^2 */
153 MP_CHECKOK(group->meth->field_mul(px, C2, C, group->meth));
154 /* A = D^2 */
155 MP_CHECKOK(group->meth->field_sqr(D, A, group->meth));
156
157 /* rx = D^2 - (C^3 + 2 * (px * C^2)) */
158 MP_CHECKOK(group->meth->field_add(C, C, rx, group->meth));
159 MP_CHECKOK(group->meth->field_add(C3, rx, rx, group->meth));
160 MP_CHECKOK(group->meth->field_sub(A, rx, rx, group->meth));
161
162 /* C3 = py * C^3 */
163 MP_CHECKOK(group->meth->field_mul(py, C3, C3, group->meth));
164
165 /* ry = D * (px * C^2 - rx) - py * C^3 */
166 MP_CHECKOK(group->meth->field_sub(C, rx, ry, group->meth));
167 MP_CHECKOK(group->meth->field_mul(D, ry, ry, group->meth));
168 MP_CHECKOK(group->meth->field_sub(ry, C3, ry, group->meth));
169
170 /* raz4 = a * rz^4 */
171 MP_CHECKOK(group->meth->field_sqr(rz, raz4, group->meth));
172 MP_CHECKOK(group->meth->field_sqr(raz4, raz4, group->meth));
173 MP_CHECKOK(group->meth->field_mul(raz4, &group->curvea, raz4, group->meth));
174 CLEANUP:
175 return res;
176 }
177
178 /* Computes R = nP where R is (rx, ry) and P is the base point. Elliptic
179 * curve points P and R can be identical. Uses mixed Modified-Jacobian
180 * co-ordinates for doubling and Chudnovsky Jacobian coordinates for
181 * additions. Assumes input is already field-encoded using field_enc, and
182 * returns output that is still field-encoded. Uses 5-bit window NAF
183 * method (algorithm 11) for scalar-point multiplication from Brown,
184 * Hankerson, Lopez, Menezes. Software Implementation of the NIST Elliptic
185 * Curves Over Prime Fields. */
186 mp_err
ec_GFp_pt_mul_jm_wNAF(const mp_int * n,const mp_int * px,const mp_int * py,mp_int * rx,mp_int * ry,const ECGroup * group)187 ec_GFp_pt_mul_jm_wNAF(const mp_int *n, const mp_int *px, const mp_int *py,
188 mp_int *rx, mp_int *ry, const ECGroup *group)
189 {
190 mp_err res = MP_OKAY;
191 mp_int precomp[16][2], rz, tpx, tpy;
192 mp_int raz4;
193 mp_int scratch[MAX_SCRATCH];
194 signed char *naf = NULL;
195 int i, orderBitSize;
196
197 MP_DIGITS(&rz) = 0;
198 MP_DIGITS(&raz4) = 0;
199 MP_DIGITS(&tpx) = 0;
200 MP_DIGITS(&tpy) = 0;
201 for (i = 0; i < 16; i++) {
202 MP_DIGITS(&precomp[i][0]) = 0;
203 MP_DIGITS(&precomp[i][1]) = 0;
204 }
205 for (i = 0; i < MAX_SCRATCH; i++) {
206 MP_DIGITS(&scratch[i]) = 0;
207 }
208
209 ARGCHK(group != NULL, MP_BADARG);
210 ARGCHK((n != NULL) && (px != NULL) && (py != NULL), MP_BADARG);
211
212 /* initialize precomputation table */
213 MP_CHECKOK(mp_init(&tpx));
214 MP_CHECKOK(mp_init(&tpy));
215 ;
216 MP_CHECKOK(mp_init(&rz));
217 MP_CHECKOK(mp_init(&raz4));
218
219 for (i = 0; i < 16; i++) {
220 MP_CHECKOK(mp_init(&precomp[i][0]));
221 MP_CHECKOK(mp_init(&precomp[i][1]));
222 }
223 for (i = 0; i < MAX_SCRATCH; i++) {
224 MP_CHECKOK(mp_init(&scratch[i]));
225 }
226
227 /* Set out[8] = P */
228 MP_CHECKOK(mp_copy(px, &precomp[8][0]));
229 MP_CHECKOK(mp_copy(py, &precomp[8][1]));
230
231 /* Set (tpx, tpy) = 2P */
232 MP_CHECKOK(group->point_dbl(&precomp[8][0], &precomp[8][1], &tpx, &tpy,
233 group));
234
235 /* Set 3P, 5P, ..., 15P */
236 for (i = 8; i < 15; i++) {
237 MP_CHECKOK(group->point_add(&precomp[i][0], &precomp[i][1], &tpx, &tpy,
238 &precomp[i + 1][0], &precomp[i + 1][1],
239 group));
240 }
241
242 /* Set -15P, -13P, ..., -P */
243 for (i = 0; i < 8; i++) {
244 MP_CHECKOK(mp_copy(&precomp[15 - i][0], &precomp[i][0]));
245 MP_CHECKOK(group->meth->field_neg(&precomp[15 - i][1], &precomp[i][1],
246 group->meth));
247 }
248
249 /* R = inf */
250 MP_CHECKOK(ec_GFp_pt_set_inf_jac(rx, ry, &rz));
251
252 orderBitSize = mpl_significant_bits(&group->order);
253
254 /* Allocate memory for NAF */
255 naf = (signed char *)malloc(sizeof(signed char) * (orderBitSize + 1));
256 if (naf == NULL) {
257 res = MP_MEM;
258 goto CLEANUP;
259 }
260
261 /* Compute 5NAF */
262 ec_compute_wNAF(naf, orderBitSize, n, 5);
263
264 /* wNAF method */
265 for (i = orderBitSize; i >= 0; i--) {
266 /* R = 2R */
267 ec_GFp_pt_dbl_jm(rx, ry, &rz, &raz4, rx, ry, &rz,
268 &raz4, scratch, group);
269 if (naf[i] != 0) {
270 ec_GFp_pt_add_jm_aff(rx, ry, &rz, &raz4,
271 &precomp[(naf[i] + 15) / 2][0],
272 &precomp[(naf[i] + 15) / 2][1], rx, ry,
273 &rz, &raz4, scratch, group);
274 }
275 }
276
277 /* convert result S to affine coordinates */
278 MP_CHECKOK(ec_GFp_pt_jac2aff(rx, ry, &rz, rx, ry, group));
279
280 CLEANUP:
281 for (i = 0; i < MAX_SCRATCH; i++) {
282 mp_clear(&scratch[i]);
283 }
284 for (i = 0; i < 16; i++) {
285 mp_clear(&precomp[i][0]);
286 mp_clear(&precomp[i][1]);
287 }
288 mp_clear(&tpx);
289 mp_clear(&tpy);
290 mp_clear(&rz);
291 mp_clear(&raz4);
292 free(naf);
293 return res;
294 }
295