1 /* crypto/ec/ec_mult.c */
2 /* ====================================================================
3 * Copyright (c) 1998-2001 The OpenSSL Project. All rights reserved.
5 * Redistribution and use in source and binary forms, with or without
6 * modification, are permitted provided that the following conditions
9 * 1. Redistributions of source code must retain the above copyright
10 * notice, this list of conditions and the following disclaimer.
12 * 2. Redistributions in binary form must reproduce the above copyright
13 * notice, this list of conditions and the following disclaimer in
14 * the documentation and/or other materials provided with the
17 * 3. All advertising materials mentioning features or use of this
18 * software must display the following acknowledgment:
19 * "This product includes software developed by the OpenSSL Project
20 * for use in the OpenSSL Toolkit. (http://www.openssl.org/)"
22 * 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to
23 * endorse or promote products derived from this software without
24 * prior written permission. For written permission, please contact
25 * openssl-core@openssl.org.
27 * 5. Products derived from this software may not be called "OpenSSL"
28 * nor may "OpenSSL" appear in their names without prior written
29 * permission of the OpenSSL Project.
31 * 6. Redistributions of any form whatsoever must retain the following
33 * "This product includes software developed by the OpenSSL Project
34 * for use in the OpenSSL Toolkit (http://www.openssl.org/)"
36 * THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY
37 * EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
38 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
39 * PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE OpenSSL PROJECT OR
40 * ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
41 * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
42 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
43 * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
44 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
45 * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
46 * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED
47 * OF THE POSSIBILITY OF SUCH DAMAGE.
48 * ====================================================================
50 * This product includes cryptographic software written by Eric Young
51 * (eay@cryptsoft.com). This product includes software written by Tim
52 * Hudson (tjh@cryptsoft.com).
56 #include <openssl/err.h>
61 /* TODO: width-m NAFs */
63 /* TODO: optional Lim-Lee precomputation for the generator */
66 #define EC_window_bits_for_scalar_size(b) \
73 /* For window size 'w' (w >= 2), we compute the odd multiples
75 * This accounts for 2^(w-1) point additions (neglecting constants),
76 * each of which requires 16 field multiplications (4 squarings
77 * and 12 general multiplications) in the case of curves defined
78 * over GF(p), which are the only curves we have so far.
80 * Converting these precomputed points into affine form takes
81 * three field multiplications for inverting Z and one squaring
82 * and three multiplications for adjusting X and Y, i.e.
83 * 7 multiplications in total (1 squaring and 6 general multiplications),
84 * again except for constants.
86 * The average number of windows for a 'b' bit scalar is roughly
88 * Each of these windows (except possibly for the first one, but
89 * we are ignoring constants anyway) requires one point addition.
90 * As the precomputed table stores points in affine form, these
91 * additions take only 11 field multiplications each (3 squarings
92 * and 8 general multiplications).
94 * So the total workload, except for constants, is
96 * 2^(w-1)*[5 squarings + 18 multiplications]
97 * + (b/(w+1))*[3 squarings + 8 multiplications]
99 * If we assume that 10 squarings are as costly as 9 multiplications,
100 * our task is to find the 'w' that, given 'b', minimizes
102 * 2^(w-1)*(5*9 + 18*10) + (b/(w+1))*(3*9 + 8*10)
103 * = 2^(w-1)*225 + (b/(w+1))*107.
105 * Thus optimal window sizes should be roughly as follows:
107 * w >= 6 if b >= 1414
108 * w = 5 if 1413 >= b >= 505
109 * w = 4 if 504 >= b >= 169
110 * w = 3 if 168 >= b >= 51
111 * w = 2 if 50 >= b >= 13
114 * If we assume instead that squarings are exactly as costly as
115 * multiplications, we have to minimize
116 * 2^(w-1)*23 + (b/(w+1))*11.
118 * This gives us the following (nearly unchanged) table of optimal
121 * w >= 6 if b >= 1406
122 * w = 5 if 1405 >= b >= 502
123 * w = 4 if 501 >= b >= 168
124 * w = 3 if 167 >= b >= 51
125 * w = 2 if 50 >= b >= 13
128 * Note that neither table tries to take into account memory usage
129 * (allocation overhead, code locality etc.). Actual timings with
130 * NIST curves P-192, P-224, and P-256 with scalars of 192, 224,
131 * and 256 bits, respectively, show that w = 3 (instead of 4) is
132 * preferrable; timings with NIST curve P-384 and 384-bit scalars
133 * confirm that w = 4 is optimal for this case; and timings with
134 * NIST curve P-521 and 521-bit scalars show that w = 4 (instead
135 * of 5) is preferrable. So we generously round up all the
136 * boundaries and use the following table:
138 * w >= 6 if b >= 2000
139 * w = 5 if 1999 >= b >= 800
140 * w = 4 if 799 >= b >= 300
141 * w = 3 if 299 >= b >= 70
142 * w = 2 if 69 >= b >= 20
149 * \sum scalars[i]*points[i]
152 * is included in the addition if scalar != NULL
154 int EC_POINTs_mul(const EC_GROUP *group, EC_POINT *r, const BIGNUM *scalar,
155 size_t num, const EC_POINT *points[], const BIGNUM *scalars[], BN_CTX *ctx)
157 BN_CTX *new_ctx = NULL;
158 EC_POINT *generator = NULL;
159 EC_POINT *tmp = NULL;
163 int r_is_at_infinity = 1;
165 size_t *wsize = NULL; /* individual window sizes */
166 unsigned long *wbits = NULL; /* individual window contents */
167 int *wpos = NULL; /* position of bottom bit of current individual windows
168 * (wpos[i] is valid if wbits[i] != 0) */
170 EC_POINT **val = NULL; /* precomputation */
172 EC_POINT ***val_sub = NULL; /* pointers to sub-arrays of 'val' */
177 generator = EC_GROUP_get0_generator(group);
178 if (generator == NULL)
180 ECerr(EC_F_EC_POINTS_MUL, EC_R_UNDEFINED_GENERATOR);
185 for (i = 0; i < num; i++)
187 if (group->meth != points[i]->meth)
189 ECerr(EC_F_EC_POINTS_MUL, EC_R_INCOMPATIBLE_OBJECTS);
194 totalnum = num + (scalar != NULL);
196 wsize = OPENSSL_malloc(totalnum * sizeof wsize[0]);
197 wbits = OPENSSL_malloc(totalnum * sizeof wbits[0]);
198 wpos = OPENSSL_malloc(totalnum * sizeof wpos[0]);
199 if (wsize == NULL || wbits == NULL || wpos == NULL) goto err;
201 /* num_val := total number of points to precompute */
203 for (i = 0; i < totalnum; i++)
207 bits = i < num ? BN_num_bits(scalars[i]) : BN_num_bits(scalar);
208 wsize[i] = EC_window_bits_for_scalar_size(bits);
209 num_val += 1u << (wsize[i] - 1);
216 /* all precomputed points go into a single array 'val',
217 * 'val_sub[i]' is a pointer to the subarray for the i-th point */
218 val = OPENSSL_malloc((num_val + 1) * sizeof val[0]);
219 if (val == NULL) goto err;
220 val[num_val] = NULL; /* pivot element */
222 val_sub = OPENSSL_malloc(totalnum * sizeof val_sub[0]);
223 if (val_sub == NULL) goto err;
225 /* allocate points for precomputation */
227 for (i = 0; i < totalnum; i++)
230 for (j = 0; j < (1u << (wsize[i] - 1)); j++)
232 *v = EC_POINT_new(group);
233 if (*v == NULL) goto err;
237 if (!(v == val + num_val))
239 ECerr(EC_F_EC_POINTS_MUL, ERR_R_INTERNAL_ERROR);
245 ctx = new_ctx = BN_CTX_new();
250 tmp = EC_POINT_new(group);
251 if (tmp == NULL) goto err;
253 /* prepare precomputed values:
254 * val_sub[i][0] := points[i]
255 * val_sub[i][1] := 3 * points[i]
256 * val_sub[i][2] := 5 * points[i]
259 for (i = 0; i < totalnum; i++)
263 if (!EC_POINT_copy(val_sub[i][0], points[i])) goto err;
266 if (!EC_POINT_invert(group, val_sub[i][0], ctx)) goto err;
271 if (!EC_POINT_copy(val_sub[i][0], generator)) goto err;
274 if (!EC_POINT_invert(group, val_sub[i][0], ctx)) goto err;
280 if (!EC_POINT_dbl(group, tmp, val_sub[i][0], ctx)) goto err;
281 for (j = 1; j < (1u << (wsize[i] - 1)); j++)
283 if (!EC_POINT_add(group, val_sub[i][j], val_sub[i][j - 1], tmp, ctx)) goto err;
288 #if 1 /* optional; EC_window_bits_for_scalar_size assumes we do this step */
289 if (!EC_POINTs_make_affine(group, num_val, val, ctx)) goto err;
292 r_is_at_infinity = 1;
294 for (k = max_bits - 1; k >= 0; k--)
296 if (!r_is_at_infinity)
298 if (!EC_POINT_dbl(group, r, r, ctx)) goto err;
301 for (i = 0; i < totalnum; i++)
307 s = i < num ? scalars[i] : scalar;
309 if (BN_is_bit_set(s, k))
311 /* look at bits k - wsize[i] + 1 .. k for this window */
312 t = k - wsize[i] + 1;
313 while (!BN_is_bit_set(s, t)) /* BN_is_bit_set is false for t < 0 */
317 for (t = k - 1; t >= wpos[i]; t--)
320 if (BN_is_bit_set(s, t))
323 /* now wbits[i] is the odd bit pattern at bits wpos[i] .. k */
327 if ((wbits[i] != 0) && (wpos[i] == k))
329 if (r_is_at_infinity)
331 if (!EC_POINT_copy(r, val_sub[i][wbits[i] >> 1])) goto err;
332 r_is_at_infinity = 0;
336 if (!EC_POINT_add(group, r, r, val_sub[i][wbits[i] >> 1], ctx)) goto err;
343 if (r_is_at_infinity)
344 if (!EC_POINT_set_to_infinity(group, r)) goto err;
350 BN_CTX_free(new_ctx);
361 for (v = val; *v != NULL; v++)
362 EC_POINT_clear_free(*v);
368 OPENSSL_free(val_sub);
374 int EC_POINT_mul(const EC_GROUP *group, EC_POINT *r, const BIGNUM *g_scalar, const EC_POINT *point, const BIGNUM *p_scalar, BN_CTX *ctx)
376 const EC_POINT *points[1];
377 const BIGNUM *scalars[1];
380 scalars[0] = p_scalar;
382 return EC_POINTs_mul(group, r, g_scalar, (point != NULL && p_scalar != NULL), points, scalars, ctx);
386 int EC_GROUP_precompute_mult(EC_GROUP *group, BN_CTX *ctx)
388 const EC_POINT *generator;
389 BN_CTX *new_ctx = NULL;
393 generator = EC_GROUP_get0_generator(group);
394 if (generator == NULL)
396 ECerr(EC_F_EC_GROUP_PRECOMPUTE_MULT, EC_R_UNDEFINED_GENERATOR);
402 ctx = new_ctx = BN_CTX_new();
408 order = BN_CTX_get(ctx);
409 if (order == NULL) goto err;
411 if (!EC_GROUP_get_order(group, order, ctx)) return 0;
412 if (BN_is_zero(order))
414 ECerr(EC_F_EC_GROUP_PRECOMPUTE_MULT, EC_R_UNKNOWN_ORDER);
425 BN_CTX_free(new_ctx);