xref: /illumos-gate/usr/src/common/crypto/ecc/ecp_jac.c (revision c40a6cd7)
1 /*
2  * ***** BEGIN LICENSE BLOCK *****
3  * Version: MPL 1.1/GPL 2.0/LGPL 2.1
4  *
5  * The contents of this file are subject to the Mozilla Public License Version
6  * 1.1 (the "License"); you may not use this file except in compliance with
7  * the License. You may obtain a copy of the License at
8  * http://www.mozilla.org/MPL/
9  *
10  * Software distributed under the License is distributed on an "AS IS" basis,
11  * WITHOUT WARRANTY OF ANY KIND, either express or implied. See the License
12  * for the specific language governing rights and limitations under the
13  * License.
14  *
15  * The Original Code is the elliptic curve math library for prime field curves.
16  *
17  * The Initial Developer of the Original Code is
18  * Sun Microsystems, Inc.
19  * Portions created by the Initial Developer are Copyright (C) 2003
20  * the Initial Developer. All Rights Reserved.
21  *
22  * Contributor(s):
23  *   Sheueling Chang-Shantz <sheueling.chang@sun.com>,
24  *   Stephen Fung <fungstep@hotmail.com>, and
25  *   Douglas Stebila <douglas@stebila.ca>, Sun Microsystems Laboratories.
26  *   Bodo Moeller <moeller@cdc.informatik.tu-darmstadt.de>,
27  *   Nils Larsch <nla@trustcenter.de>, and
28  *   Lenka Fibikova <fibikova@exp-math.uni-essen.de>, the OpenSSL Project
29  *
30  * Alternatively, the contents of this file may be used under the terms of
31  * either the GNU General Public License Version 2 or later (the "GPL"), or
32  * the GNU Lesser General Public License Version 2.1 or later (the "LGPL"),
33  * in which case the provisions of the GPL or the LGPL are applicable instead
34  * of those above. If you wish to allow use of your version of this file only
35  * under the terms of either the GPL or the LGPL, and not to allow others to
36  * use your version of this file under the terms of the MPL, indicate your
37  * decision by deleting the provisions above and replace them with the notice
38  * and other provisions required by the GPL or the LGPL. If you do not delete
39  * the provisions above, a recipient may use your version of this file under
40  * the terms of any one of the MPL, the GPL or the LGPL.
41  *
42  * ***** END LICENSE BLOCK ***** */
43 /*
44  * Copyright 2007 Sun Microsystems, Inc.  All rights reserved.
45  * Use is subject to license terms.
46  *
47  * Sun elects to use this software under the MPL license.
48  */
49 
50 #include "ecp.h"
51 #include "mplogic.h"
52 #ifndef _KERNEL
53 #include <stdlib.h>
54 #endif
55 #ifdef ECL_DEBUG
56 #include <assert.h>
57 #endif
58 
59 /* Converts a point P(px, py) from affine coordinates to Jacobian
60  * projective coordinates R(rx, ry, rz). Assumes input is already
61  * field-encoded using field_enc, and returns output that is still
62  * field-encoded. */
63 mp_err
ec_GFp_pt_aff2jac(const mp_int * px,const mp_int * py,mp_int * rx,mp_int * ry,mp_int * rz,const ECGroup * group)64 ec_GFp_pt_aff2jac(const mp_int *px, const mp_int *py, mp_int *rx,
65 				  mp_int *ry, mp_int *rz, const ECGroup *group)
66 {
67 	mp_err res = MP_OKAY;
68 
69 	if (ec_GFp_pt_is_inf_aff(px, py) == MP_YES) {
70 		MP_CHECKOK(ec_GFp_pt_set_inf_jac(rx, ry, rz));
71 	} else {
72 		MP_CHECKOK(mp_copy(px, rx));
73 		MP_CHECKOK(mp_copy(py, ry));
74 		MP_CHECKOK(mp_set_int(rz, 1));
75 		if (group->meth->field_enc) {
76 			MP_CHECKOK(group->meth->field_enc(rz, rz, group->meth));
77 		}
78 	}
79   CLEANUP:
80 	return res;
81 }
82 
83 /* Converts a point P(px, py, pz) from Jacobian projective coordinates to
84  * affine coordinates R(rx, ry).  P and R can share x and y coordinates.
85  * Assumes input is already field-encoded using field_enc, and returns
86  * output that is still field-encoded. */
87 mp_err
ec_GFp_pt_jac2aff(const mp_int * px,const mp_int * py,const mp_int * pz,mp_int * rx,mp_int * ry,const ECGroup * group)88 ec_GFp_pt_jac2aff(const mp_int *px, const mp_int *py, const mp_int *pz,
89 				  mp_int *rx, mp_int *ry, const ECGroup *group)
90 {
91 	mp_err res = MP_OKAY;
92 	mp_int z1, z2, z3;
93 
94 	MP_DIGITS(&z1) = 0;
95 	MP_DIGITS(&z2) = 0;
96 	MP_DIGITS(&z3) = 0;
97 	MP_CHECKOK(mp_init(&z1, FLAG(px)));
98 	MP_CHECKOK(mp_init(&z2, FLAG(px)));
99 	MP_CHECKOK(mp_init(&z3, FLAG(px)));
100 
101 	/* if point at infinity, then set point at infinity and exit */
102 	if (ec_GFp_pt_is_inf_jac(px, py, pz) == MP_YES) {
103 		MP_CHECKOK(ec_GFp_pt_set_inf_aff(rx, ry));
104 		goto CLEANUP;
105 	}
106 
107 	/* transform (px, py, pz) into (px / pz^2, py / pz^3) */
108 	if (mp_cmp_d(pz, 1) == 0) {
109 		MP_CHECKOK(mp_copy(px, rx));
110 		MP_CHECKOK(mp_copy(py, ry));
111 	} else {
112 		MP_CHECKOK(group->meth->field_div(NULL, pz, &z1, group->meth));
113 		MP_CHECKOK(group->meth->field_sqr(&z1, &z2, group->meth));
114 		MP_CHECKOK(group->meth->field_mul(&z1, &z2, &z3, group->meth));
115 		MP_CHECKOK(group->meth->field_mul(px, &z2, rx, group->meth));
116 		MP_CHECKOK(group->meth->field_mul(py, &z3, ry, group->meth));
117 	}
118 
119   CLEANUP:
120 	mp_clear(&z1);
121 	mp_clear(&z2);
122 	mp_clear(&z3);
123 	return res;
124 }
125 
126 /* Checks if point P(px, py, pz) is at infinity. Uses Jacobian
127  * coordinates. */
128 mp_err
ec_GFp_pt_is_inf_jac(const mp_int * px,const mp_int * py,const mp_int * pz)129 ec_GFp_pt_is_inf_jac(const mp_int *px, const mp_int *py, const mp_int *pz)
130 {
131 	return mp_cmp_z(pz);
132 }
133 
134 /* Sets P(px, py, pz) to be the point at infinity.  Uses Jacobian
135  * coordinates. */
136 mp_err
ec_GFp_pt_set_inf_jac(mp_int * px,mp_int * py,mp_int * pz)137 ec_GFp_pt_set_inf_jac(mp_int *px, mp_int *py, mp_int *pz)
138 {
139 	mp_zero(pz);
140 	return MP_OKAY;
141 }
142 
143 /* Computes R = P + Q where R is (rx, ry, rz), P is (px, py, pz) and Q is
144  * (qx, qy, 1).  Elliptic curve points P, Q, and R can all be identical.
145  * Uses mixed Jacobian-affine coordinates. Assumes input is already
146  * field-encoded using field_enc, and returns output that is still
147  * field-encoded. Uses equation (2) from Brown, Hankerson, Lopez, and
148  * Menezes. Software Implementation of the NIST Elliptic Curves Over Prime
149  * Fields. */
150 mp_err
ec_GFp_pt_add_jac_aff(const mp_int * px,const mp_int * py,const mp_int * pz,const mp_int * qx,const mp_int * qy,mp_int * rx,mp_int * ry,mp_int * rz,const ECGroup * group)151 ec_GFp_pt_add_jac_aff(const mp_int *px, const mp_int *py, const mp_int *pz,
152 					  const mp_int *qx, const mp_int *qy, mp_int *rx,
153 					  mp_int *ry, mp_int *rz, const ECGroup *group)
154 {
155 	mp_err res = MP_OKAY;
156 	mp_int A, B, C, D, C2, C3;
157 
158 	MP_DIGITS(&A) = 0;
159 	MP_DIGITS(&B) = 0;
160 	MP_DIGITS(&C) = 0;
161 	MP_DIGITS(&D) = 0;
162 	MP_DIGITS(&C2) = 0;
163 	MP_DIGITS(&C3) = 0;
164 	MP_CHECKOK(mp_init(&A, FLAG(px)));
165 	MP_CHECKOK(mp_init(&B, FLAG(px)));
166 	MP_CHECKOK(mp_init(&C, FLAG(px)));
167 	MP_CHECKOK(mp_init(&D, FLAG(px)));
168 	MP_CHECKOK(mp_init(&C2, FLAG(px)));
169 	MP_CHECKOK(mp_init(&C3, FLAG(px)));
170 
171 	/* If either P or Q is the point at infinity, then return the other
172 	 * point */
173 	if (ec_GFp_pt_is_inf_jac(px, py, pz) == MP_YES) {
174 		MP_CHECKOK(ec_GFp_pt_aff2jac(qx, qy, rx, ry, rz, group));
175 		goto CLEANUP;
176 	}
177 	if (ec_GFp_pt_is_inf_aff(qx, qy) == MP_YES) {
178 		MP_CHECKOK(mp_copy(px, rx));
179 		MP_CHECKOK(mp_copy(py, ry));
180 		MP_CHECKOK(mp_copy(pz, rz));
181 		goto CLEANUP;
182 	}
183 
184 	/* A = qx * pz^2, B = qy * pz^3 */
185 	MP_CHECKOK(group->meth->field_sqr(pz, &A, group->meth));
186 	MP_CHECKOK(group->meth->field_mul(&A, pz, &B, group->meth));
187 	MP_CHECKOK(group->meth->field_mul(&A, qx, &A, group->meth));
188 	MP_CHECKOK(group->meth->field_mul(&B, qy, &B, group->meth));
189 
190 	/* C = A - px, D = B - py */
191 	MP_CHECKOK(group->meth->field_sub(&A, px, &C, group->meth));
192 	MP_CHECKOK(group->meth->field_sub(&B, py, &D, group->meth));
193 
194 	/* C2 = C^2, C3 = C^3 */
195 	MP_CHECKOK(group->meth->field_sqr(&C, &C2, group->meth));
196 	MP_CHECKOK(group->meth->field_mul(&C, &C2, &C3, group->meth));
197 
198 	/* rz = pz * C */
199 	MP_CHECKOK(group->meth->field_mul(pz, &C, rz, group->meth));
200 
201 	/* C = px * C^2 */
202 	MP_CHECKOK(group->meth->field_mul(px, &C2, &C, group->meth));
203 	/* A = D^2 */
204 	MP_CHECKOK(group->meth->field_sqr(&D, &A, group->meth));
205 
206 	/* rx = D^2 - (C^3 + 2 * (px * C^2)) */
207 	MP_CHECKOK(group->meth->field_add(&C, &C, rx, group->meth));
208 	MP_CHECKOK(group->meth->field_add(&C3, rx, rx, group->meth));
209 	MP_CHECKOK(group->meth->field_sub(&A, rx, rx, group->meth));
210 
211 	/* C3 = py * C^3 */
212 	MP_CHECKOK(group->meth->field_mul(py, &C3, &C3, group->meth));
213 
214 	/* ry = D * (px * C^2 - rx) - py * C^3 */
215 	MP_CHECKOK(group->meth->field_sub(&C, rx, ry, group->meth));
216 	MP_CHECKOK(group->meth->field_mul(&D, ry, ry, group->meth));
217 	MP_CHECKOK(group->meth->field_sub(ry, &C3, ry, group->meth));
218 
219   CLEANUP:
220 	mp_clear(&A);
221 	mp_clear(&B);
222 	mp_clear(&C);
223 	mp_clear(&D);
224 	mp_clear(&C2);
225 	mp_clear(&C3);
226 	return res;
227 }
228 
229 /* Computes R = 2P.  Elliptic curve points P and R can be identical.  Uses
230  * Jacobian coordinates.
231  *
232  * Assumes input is already field-encoded using field_enc, and returns
233  * output that is still field-encoded.
234  *
235  * This routine implements Point Doubling in the Jacobian Projective
236  * space as described in the paper "Efficient elliptic curve exponentiation
237  * using mixed coordinates", by H. Cohen, A Miyaji, T. Ono.
238  */
239 mp_err
ec_GFp_pt_dbl_jac(const mp_int * px,const mp_int * py,const mp_int * pz,mp_int * rx,mp_int * ry,mp_int * rz,const ECGroup * group)240 ec_GFp_pt_dbl_jac(const mp_int *px, const mp_int *py, const mp_int *pz,
241 				  mp_int *rx, mp_int *ry, mp_int *rz, const ECGroup *group)
242 {
243 	mp_err res = MP_OKAY;
244 	mp_int t0, t1, M, S;
245 
246 	MP_DIGITS(&t0) = 0;
247 	MP_DIGITS(&t1) = 0;
248 	MP_DIGITS(&M) = 0;
249 	MP_DIGITS(&S) = 0;
250 	MP_CHECKOK(mp_init(&t0, FLAG(px)));
251 	MP_CHECKOK(mp_init(&t1, FLAG(px)));
252 	MP_CHECKOK(mp_init(&M, FLAG(px)));
253 	MP_CHECKOK(mp_init(&S, FLAG(px)));
254 
255 	if (ec_GFp_pt_is_inf_jac(px, py, pz) == MP_YES) {
256 		MP_CHECKOK(ec_GFp_pt_set_inf_jac(rx, ry, rz));
257 		goto CLEANUP;
258 	}
259 
260 	if (mp_cmp_d(pz, 1) == 0) {
261 		/* M = 3 * px^2 + a */
262 		MP_CHECKOK(group->meth->field_sqr(px, &t0, group->meth));
263 		MP_CHECKOK(group->meth->field_add(&t0, &t0, &M, group->meth));
264 		MP_CHECKOK(group->meth->field_add(&t0, &M, &t0, group->meth));
265 		MP_CHECKOK(group->meth->
266 				   field_add(&t0, &group->curvea, &M, group->meth));
267 	} else if (mp_cmp_int(&group->curvea, -3, FLAG(px)) == 0) {
268 		/* M = 3 * (px + pz^2) * (px - pz^2) */
269 		MP_CHECKOK(group->meth->field_sqr(pz, &M, group->meth));
270 		MP_CHECKOK(group->meth->field_add(px, &M, &t0, group->meth));
271 		MP_CHECKOK(group->meth->field_sub(px, &M, &t1, group->meth));
272 		MP_CHECKOK(group->meth->field_mul(&t0, &t1, &M, group->meth));
273 		MP_CHECKOK(group->meth->field_add(&M, &M, &t0, group->meth));
274 		MP_CHECKOK(group->meth->field_add(&t0, &M, &M, group->meth));
275 	} else {
276 		/* M = 3 * (px^2) + a * (pz^4) */
277 		MP_CHECKOK(group->meth->field_sqr(px, &t0, group->meth));
278 		MP_CHECKOK(group->meth->field_add(&t0, &t0, &M, group->meth));
279 		MP_CHECKOK(group->meth->field_add(&t0, &M, &t0, group->meth));
280 		MP_CHECKOK(group->meth->field_sqr(pz, &M, group->meth));
281 		MP_CHECKOK(group->meth->field_sqr(&M, &M, group->meth));
282 		MP_CHECKOK(group->meth->
283 				   field_mul(&M, &group->curvea, &M, group->meth));
284 		MP_CHECKOK(group->meth->field_add(&M, &t0, &M, group->meth));
285 	}
286 
287 	/* rz = 2 * py * pz */
288 	/* t0 = 4 * py^2 */
289 	if (mp_cmp_d(pz, 1) == 0) {
290 		MP_CHECKOK(group->meth->field_add(py, py, rz, group->meth));
291 		MP_CHECKOK(group->meth->field_sqr(rz, &t0, group->meth));
292 	} else {
293 		MP_CHECKOK(group->meth->field_add(py, py, &t0, group->meth));
294 		MP_CHECKOK(group->meth->field_mul(&t0, pz, rz, group->meth));
295 		MP_CHECKOK(group->meth->field_sqr(&t0, &t0, group->meth));
296 	}
297 
298 	/* S = 4 * px * py^2 = px * (2 * py)^2 */
299 	MP_CHECKOK(group->meth->field_mul(px, &t0, &S, group->meth));
300 
301 	/* rx = M^2 - 2 * S */
302 	MP_CHECKOK(group->meth->field_add(&S, &S, &t1, group->meth));
303 	MP_CHECKOK(group->meth->field_sqr(&M, rx, group->meth));
304 	MP_CHECKOK(group->meth->field_sub(rx, &t1, rx, group->meth));
305 
306 	/* ry = M * (S - rx) - 8 * py^4 */
307 	MP_CHECKOK(group->meth->field_sqr(&t0, &t1, group->meth));
308 	if (mp_isodd(&t1)) {
309 		MP_CHECKOK(mp_add(&t1, &group->meth->irr, &t1));
310 	}
311 	MP_CHECKOK(mp_div_2(&t1, &t1));
312 	MP_CHECKOK(group->meth->field_sub(&S, rx, &S, group->meth));
313 	MP_CHECKOK(group->meth->field_mul(&M, &S, &M, group->meth));
314 	MP_CHECKOK(group->meth->field_sub(&M, &t1, ry, group->meth));
315 
316   CLEANUP:
317 	mp_clear(&t0);
318 	mp_clear(&t1);
319 	mp_clear(&M);
320 	mp_clear(&S);
321 	return res;
322 }
323 
324 /* by default, this routine is unused and thus doesn't need to be compiled */
325 #ifdef ECL_ENABLE_GFP_PT_MUL_JAC
326 /* Computes R = nP where R is (rx, ry) and P is (px, py). The parameters
327  * a, b and p are the elliptic curve coefficients and the prime that
328  * determines the field GFp.  Elliptic curve points P and R can be
329  * identical.  Uses mixed Jacobian-affine coordinates. Assumes input is
330  * already field-encoded using field_enc, and returns output that is still
331  * field-encoded. Uses 4-bit window method. */
332 mp_err
ec_GFp_pt_mul_jac(const mp_int * n,const mp_int * px,const mp_int * py,mp_int * rx,mp_int * ry,const ECGroup * group)333 ec_GFp_pt_mul_jac(const mp_int *n, const mp_int *px, const mp_int *py,
334 				  mp_int *rx, mp_int *ry, const ECGroup *group)
335 {
336 	mp_err res = MP_OKAY;
337 	mp_int precomp[16][2], rz;
338 	int i, ni, d;
339 
340 	MP_DIGITS(&rz) = 0;
341 	for (i = 0; i < 16; i++) {
342 		MP_DIGITS(&precomp[i][0]) = 0;
343 		MP_DIGITS(&precomp[i][1]) = 0;
344 	}
345 
346 	ARGCHK(group != NULL, MP_BADARG);
347 	ARGCHK((n != NULL) && (px != NULL) && (py != NULL), MP_BADARG);
348 
349 	/* initialize precomputation table */
350 	for (i = 0; i < 16; i++) {
351 		MP_CHECKOK(mp_init(&precomp[i][0]));
352 		MP_CHECKOK(mp_init(&precomp[i][1]));
353 	}
354 
355 	/* fill precomputation table */
356 	mp_zero(&precomp[0][0]);
357 	mp_zero(&precomp[0][1]);
358 	MP_CHECKOK(mp_copy(px, &precomp[1][0]));
359 	MP_CHECKOK(mp_copy(py, &precomp[1][1]));
360 	for (i = 2; i < 16; i++) {
361 		MP_CHECKOK(group->
362 				   point_add(&precomp[1][0], &precomp[1][1],
363 							 &precomp[i - 1][0], &precomp[i - 1][1],
364 							 &precomp[i][0], &precomp[i][1], group));
365 	}
366 
367 	d = (mpl_significant_bits(n) + 3) / 4;
368 
369 	/* R = inf */
370 	MP_CHECKOK(mp_init(&rz));
371 	MP_CHECKOK(ec_GFp_pt_set_inf_jac(rx, ry, &rz));
372 
373 	for (i = d - 1; i >= 0; i--) {
374 		/* compute window ni */
375 		ni = MP_GET_BIT(n, 4 * i + 3);
376 		ni <<= 1;
377 		ni |= MP_GET_BIT(n, 4 * i + 2);
378 		ni <<= 1;
379 		ni |= MP_GET_BIT(n, 4 * i + 1);
380 		ni <<= 1;
381 		ni |= MP_GET_BIT(n, 4 * i);
382 		/* R = 2^4 * R */
383 		MP_CHECKOK(ec_GFp_pt_dbl_jac(rx, ry, &rz, rx, ry, &rz, group));
384 		MP_CHECKOK(ec_GFp_pt_dbl_jac(rx, ry, &rz, rx, ry, &rz, group));
385 		MP_CHECKOK(ec_GFp_pt_dbl_jac(rx, ry, &rz, rx, ry, &rz, group));
386 		MP_CHECKOK(ec_GFp_pt_dbl_jac(rx, ry, &rz, rx, ry, &rz, group));
387 		/* R = R + (ni * P) */
388 		MP_CHECKOK(ec_GFp_pt_add_jac_aff
389 				   (rx, ry, &rz, &precomp[ni][0], &precomp[ni][1], rx, ry,
390 					&rz, group));
391 	}
392 
393 	/* convert result S to affine coordinates */
394 	MP_CHECKOK(ec_GFp_pt_jac2aff(rx, ry, &rz, rx, ry, group));
395 
396   CLEANUP:
397 	mp_clear(&rz);
398 	for (i = 0; i < 16; i++) {
399 		mp_clear(&precomp[i][0]);
400 		mp_clear(&precomp[i][1]);
401 	}
402 	return res;
403 }
404 #endif
405 
406 /* Elliptic curve scalar-point multiplication. Computes R(x, y) = k1 * G +
407  * k2 * P(x, y), where G is the generator (base point) of the group of
408  * points on the elliptic curve. Allows k1 = NULL or { k2, P } = NULL.
409  * Uses mixed Jacobian-affine coordinates. Input and output values are
410  * assumed to be NOT field-encoded. Uses algorithm 15 (simultaneous
411  * multiple point multiplication) from Brown, Hankerson, Lopez, Menezes.
412  * Software Implementation of the NIST Elliptic Curves over Prime Fields. */
413 mp_err
ec_GFp_pts_mul_jac(const mp_int * k1,const mp_int * k2,const mp_int * px,const mp_int * py,mp_int * rx,mp_int * ry,const ECGroup * group)414 ec_GFp_pts_mul_jac(const mp_int *k1, const mp_int *k2, const mp_int *px,
415 				   const mp_int *py, mp_int *rx, mp_int *ry,
416 				   const ECGroup *group)
417 {
418 	mp_err res = MP_OKAY;
419 	mp_int precomp[4][4][2];
420 	mp_int rz;
421 	const mp_int *a, *b;
422 	int i, j;
423 	int ai, bi, d;
424 
425 	for (i = 0; i < 4; i++) {
426 		for (j = 0; j < 4; j++) {
427 			MP_DIGITS(&precomp[i][j][0]) = 0;
428 			MP_DIGITS(&precomp[i][j][1]) = 0;
429 		}
430 	}
431 	MP_DIGITS(&rz) = 0;
432 
433 	ARGCHK(group != NULL, MP_BADARG);
434 	ARGCHK(!((k1 == NULL)
435 			 && ((k2 == NULL) || (px == NULL)
436 				 || (py == NULL))), MP_BADARG);
437 
438 	/* if some arguments are not defined used ECPoint_mul */
439 	if (k1 == NULL) {
440 		return ECPoint_mul(group, k2, px, py, rx, ry);
441 	} else if ((k2 == NULL) || (px == NULL) || (py == NULL)) {
442 		return ECPoint_mul(group, k1, NULL, NULL, rx, ry);
443 	}
444 
445 	/* initialize precomputation table */
446 	for (i = 0; i < 4; i++) {
447 		for (j = 0; j < 4; j++) {
448 			MP_CHECKOK(mp_init(&precomp[i][j][0], FLAG(k1)));
449 			MP_CHECKOK(mp_init(&precomp[i][j][1], FLAG(k1)));
450 		}
451 	}
452 
453 	/* fill precomputation table */
454 	/* assign {k1, k2} = {a, b} such that len(a) >= len(b) */
455 	if (mpl_significant_bits(k1) < mpl_significant_bits(k2)) {
456 		a = k2;
457 		b = k1;
458 		if (group->meth->field_enc) {
459 			MP_CHECKOK(group->meth->
460 					   field_enc(px, &precomp[1][0][0], group->meth));
461 			MP_CHECKOK(group->meth->
462 					   field_enc(py, &precomp[1][0][1], group->meth));
463 		} else {
464 			MP_CHECKOK(mp_copy(px, &precomp[1][0][0]));
465 			MP_CHECKOK(mp_copy(py, &precomp[1][0][1]));
466 		}
467 		MP_CHECKOK(mp_copy(&group->genx, &precomp[0][1][0]));
468 		MP_CHECKOK(mp_copy(&group->geny, &precomp[0][1][1]));
469 	} else {
470 		a = k1;
471 		b = k2;
472 		MP_CHECKOK(mp_copy(&group->genx, &precomp[1][0][0]));
473 		MP_CHECKOK(mp_copy(&group->geny, &precomp[1][0][1]));
474 		if (group->meth->field_enc) {
475 			MP_CHECKOK(group->meth->
476 					   field_enc(px, &precomp[0][1][0], group->meth));
477 			MP_CHECKOK(group->meth->
478 					   field_enc(py, &precomp[0][1][1], group->meth));
479 		} else {
480 			MP_CHECKOK(mp_copy(px, &precomp[0][1][0]));
481 			MP_CHECKOK(mp_copy(py, &precomp[0][1][1]));
482 		}
483 	}
484 	/* precompute [*][0][*] */
485 	mp_zero(&precomp[0][0][0]);
486 	mp_zero(&precomp[0][0][1]);
487 	MP_CHECKOK(group->
488 			   point_dbl(&precomp[1][0][0], &precomp[1][0][1],
489 						 &precomp[2][0][0], &precomp[2][0][1], group));
490 	MP_CHECKOK(group->
491 			   point_add(&precomp[1][0][0], &precomp[1][0][1],
492 						 &precomp[2][0][0], &precomp[2][0][1],
493 						 &precomp[3][0][0], &precomp[3][0][1], group));
494 	/* precompute [*][1][*] */
495 	for (i = 1; i < 4; i++) {
496 		MP_CHECKOK(group->
497 				   point_add(&precomp[0][1][0], &precomp[0][1][1],
498 							 &precomp[i][0][0], &precomp[i][0][1],
499 							 &precomp[i][1][0], &precomp[i][1][1], group));
500 	}
501 	/* precompute [*][2][*] */
502 	MP_CHECKOK(group->
503 			   point_dbl(&precomp[0][1][0], &precomp[0][1][1],
504 						 &precomp[0][2][0], &precomp[0][2][1], group));
505 	for (i = 1; i < 4; i++) {
506 		MP_CHECKOK(group->
507 				   point_add(&precomp[0][2][0], &precomp[0][2][1],
508 							 &precomp[i][0][0], &precomp[i][0][1],
509 							 &precomp[i][2][0], &precomp[i][2][1], group));
510 	}
511 	/* precompute [*][3][*] */
512 	MP_CHECKOK(group->
513 			   point_add(&precomp[0][1][0], &precomp[0][1][1],
514 						 &precomp[0][2][0], &precomp[0][2][1],
515 						 &precomp[0][3][0], &precomp[0][3][1], group));
516 	for (i = 1; i < 4; i++) {
517 		MP_CHECKOK(group->
518 				   point_add(&precomp[0][3][0], &precomp[0][3][1],
519 							 &precomp[i][0][0], &precomp[i][0][1],
520 							 &precomp[i][3][0], &precomp[i][3][1], group));
521 	}
522 
523 	d = (mpl_significant_bits(a) + 1) / 2;
524 
525 	/* R = inf */
526 	MP_CHECKOK(mp_init(&rz, FLAG(k1)));
527 	MP_CHECKOK(ec_GFp_pt_set_inf_jac(rx, ry, &rz));
528 
529 	for (i = d - 1; i >= 0; i--) {
530 		ai = MP_GET_BIT(a, 2 * i + 1);
531 		ai <<= 1;
532 		ai |= MP_GET_BIT(a, 2 * i);
533 		bi = MP_GET_BIT(b, 2 * i + 1);
534 		bi <<= 1;
535 		bi |= MP_GET_BIT(b, 2 * i);
536 		/* R = 2^2 * R */
537 		MP_CHECKOK(ec_GFp_pt_dbl_jac(rx, ry, &rz, rx, ry, &rz, group));
538 		MP_CHECKOK(ec_GFp_pt_dbl_jac(rx, ry, &rz, rx, ry, &rz, group));
539 		/* R = R + (ai * A + bi * B) */
540 		MP_CHECKOK(ec_GFp_pt_add_jac_aff
541 				   (rx, ry, &rz, &precomp[ai][bi][0], &precomp[ai][bi][1],
542 					rx, ry, &rz, group));
543 	}
544 
545 	MP_CHECKOK(ec_GFp_pt_jac2aff(rx, ry, &rz, rx, ry, group));
546 
547 	if (group->meth->field_dec) {
548 		MP_CHECKOK(group->meth->field_dec(rx, rx, group->meth));
549 		MP_CHECKOK(group->meth->field_dec(ry, ry, group->meth));
550 	}
551 
552   CLEANUP:
553 	mp_clear(&rz);
554 	for (i = 0; i < 4; i++) {
555 		for (j = 0; j < 4; j++) {
556 			mp_clear(&precomp[i][j][0]);
557 			mp_clear(&precomp[i][j][1]);
558 		}
559 	}
560 	return res;
561 }
562