/* crypto/ec/ec_mult.c */ /* ==================================================================== * Copyright (c) 1998-2001 The OpenSSL Project. All rights reserved. * * Redistribution and use in source and binary forms, with or without * modification, are permitted provided that the following conditions * are met: * * 1. Redistributions of source code must retain the above copyright * notice, this list of conditions and the following disclaimer. * * 2. Redistributions in binary form must reproduce the above copyright * notice, this list of conditions and the following disclaimer in * the documentation and/or other materials provided with the * distribution. * * 3. All advertising materials mentioning features or use of this * software must display the following acknowledgment: * "This product includes software developed by the OpenSSL Project * for use in the OpenSSL Toolkit. (http://www.openssl.org/)" * * 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to * endorse or promote products derived from this software without * prior written permission. For written permission, please contact * openssl-core@openssl.org. * * 5. Products derived from this software may not be called "OpenSSL" * nor may "OpenSSL" appear in their names without prior written * permission of the OpenSSL Project. * * 6. Redistributions of any form whatsoever must retain the following * acknowledgment: * "This product includes software developed by the OpenSSL Project * for use in the OpenSSL Toolkit (http://www.openssl.org/)" * * THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY * EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR * PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE OpenSSL PROJECT OR * ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED * OF THE POSSIBILITY OF SUCH DAMAGE. * ==================================================================== * * This product includes cryptographic software written by Eric Young * (eay@cryptsoft.com). This product includes software written by Tim * Hudson (tjh@cryptsoft.com). * */ #include #include "ec_lcl.h" /* TODO: optional precomputation of multiples of the generator */ #if 1 /* * wNAF-based interleaving multi-exponentation method * () */ /* Determine the width-(w+1) Non-Adjacent Form (wNAF) of 'scalar'. * This is an array r[] of values that are either zero or odd with an * absolute value less than 2^w satisfying * scalar = \sum_j r[j]*2^j * where at most one of any w+1 consecutive digits is non-zero. */ static signed char *compute_wNAF(const BIGNUM *scalar, int w, size_t *ret_len, BN_CTX *ctx) { BIGNUM *c; int ok = 0; signed char *r = NULL; int sign = 1; int bit, next_bit, mask; size_t len = 0, j; BN_CTX_start(ctx); c = BN_CTX_get(ctx); if (c == NULL) goto err; if (w <= 0 || w > 7) /* 'signed char' can represent integers with absolute values less than 2^7 */ { ECerr(EC_F_COMPUTE_WNAF, ERR_R_INTERNAL_ERROR); goto err; } bit = 1 << w; /* at most 128 */ next_bit = bit << 1; /* at most 256 */ mask = next_bit - 1; /* at most 255 */ if (!BN_copy(c, scalar)) goto err; if (c->neg) { sign = -1; c->neg = 0; } len = BN_num_bits(c) + 1; /* wNAF may be one digit longer than binary representation */ r = OPENSSL_malloc(len); if (r == NULL) goto err; j = 0; while (!BN_is_zero(c)) { int u = 0; if (BN_is_odd(c)) { if (c->d == NULL || c->top == 0) { ECerr(EC_F_COMPUTE_WNAF, ERR_R_INTERNAL_ERROR); goto err; } u = c->d[0] & mask; if (u & bit) { u -= next_bit; /* u < 0 */ if (!BN_add_word(c, -u)) goto err; } else { /* u > 0 */ if (!BN_sub_word(c, u)) goto err; } if (u <= -bit || u >= bit || !(u & 1) || c->neg) { ECerr(EC_F_COMPUTE_WNAF, ERR_R_INTERNAL_ERROR); goto err; } } r[j++] = sign * u; if (BN_is_odd(c)) { ECerr(EC_F_COMPUTE_WNAF, ERR_R_INTERNAL_ERROR); goto err; } if (!BN_rshift1(c, c)) goto err; } if (j > len) { ECerr(EC_F_COMPUTE_WNAF, ERR_R_INTERNAL_ERROR); goto err; } len = j; ok = 1; err: BN_CTX_end(ctx); if (!ok) { OPENSSL_free(r); r = NULL; } if (ok) *ret_len = len; return r; } /* TODO: table should be optimised for the wNAF-based implementation */ #define EC_window_bits_for_scalar_size(b) \ ((b) >= 2000 ? 6 : \ (b) >= 800 ? 5 : \ (b) >= 300 ? 4 : \ (b) >= 70 ? 3 : \ (b) >= 20 ? 2 : \ 1) /* Compute * \sum scalars[i]*points[i], * also including * scalar*generator * in the addition if scalar != NULL */ int EC_POINTs_mul(const EC_GROUP *group, EC_POINT *r, const BIGNUM *scalar, size_t num, const EC_POINT *points[], const BIGNUM *scalars[], BN_CTX *ctx) { BN_CTX *new_ctx = NULL; EC_POINT *generator = NULL; EC_POINT *tmp = NULL; size_t totalnum; size_t i, j; int k; int r_is_inverted = 0; int r_is_at_infinity = 1; size_t *wsize = NULL; /* individual window sizes */ signed char **wNAF = NULL; /* individual wNAFs */ size_t *wNAF_len = NULL; size_t max_len = 0; size_t num_val; EC_POINT **val = NULL; /* precomputation */ EC_POINT **v; EC_POINT ***val_sub = NULL; /* pointers to sub-arrays of 'val' */ int ret = 0; if (scalar != NULL) { generator = EC_GROUP_get0_generator(group); if (generator == NULL) { ECerr(EC_F_EC_POINTS_MUL, EC_R_UNDEFINED_GENERATOR); return 0; } } for (i = 0; i < num; i++) { if (group->meth != points[i]->meth) { ECerr(EC_F_EC_POINTS_MUL, EC_R_INCOMPATIBLE_OBJECTS); return 0; } } totalnum = num + (scalar != NULL); wsize = OPENSSL_malloc(totalnum * sizeof wsize[0]); wNAF_len = OPENSSL_malloc(totalnum * sizeof wNAF_len[0]); wNAF = OPENSSL_malloc(totalnum * sizeof wNAF[0] + 1); if (wNAF != NULL) { wNAF[0] = NULL; /* preliminary pivot */ } if (wsize == NULL || wNAF_len == NULL || wNAF == NULL) goto err; /* num_val := total number of points to precompute */ num_val = 0; for (i = 0; i < totalnum; i++) { size_t bits; bits = i < num ? BN_num_bits(scalars[i]) : BN_num_bits(scalar); wsize[i] = EC_window_bits_for_scalar_size(bits); num_val += 1u << (wsize[i] - 1); } /* all precomputed points go into a single array 'val', * 'val_sub[i]' is a pointer to the subarray for the i-th point */ val = OPENSSL_malloc((num_val + 1) * sizeof val[0]); if (val == NULL) goto err; val[num_val] = NULL; /* pivot element */ val_sub = OPENSSL_malloc(totalnum * sizeof val_sub[0]); if (val_sub == NULL) goto err; /* allocate points for precomputation */ v = val; for (i = 0; i < totalnum; i++) { val_sub[i] = v; for (j = 0; j < (1u << (wsize[i] - 1)); j++) { *v = EC_POINT_new(group); if (*v == NULL) goto err; v++; } } if (!(v == val + num_val)) { ECerr(EC_F_EC_POINTS_MUL, ERR_R_INTERNAL_ERROR); goto err; } if (ctx == NULL) { ctx = new_ctx = BN_CTX_new(); if (ctx == NULL) goto err; } tmp = EC_POINT_new(group); if (tmp == NULL) goto err; /* prepare precomputed values: * val_sub[i][0] := points[i] * val_sub[i][1] := 3 * points[i] * val_sub[i][2] := 5 * points[i] * ... */ for (i = 0; i < totalnum; i++) { if (i < num) { if (!EC_POINT_copy(val_sub[i][0], points[i])) goto err; } else { if (!EC_POINT_copy(val_sub[i][0], generator)) goto err; } if (wsize[i] > 1) { if (!EC_POINT_dbl(group, tmp, val_sub[i][0], ctx)) goto err; for (j = 1; j < (1u << (wsize[i] - 1)); j++) { if (!EC_POINT_add(group, val_sub[i][j], val_sub[i][j - 1], tmp, ctx)) goto err; } } wNAF[i + 1] = NULL; /* make sure we always have a pivot */ wNAF[i] = compute_wNAF((i < num ? scalars[i] : scalar), wsize[i], &wNAF_len[i], ctx); if (wNAF[i] == NULL) goto err; if (wNAF_len[i] > max_len) max_len = wNAF_len[i]; } #if 1 /* optional; EC_window_bits_for_scalar_size assumes we do this step */ if (!EC_POINTs_make_affine(group, num_val, val, ctx)) goto err; #endif r_is_at_infinity = 1; for (k = max_len - 1; k >= 0; k--) { if (!r_is_at_infinity) { if (!EC_POINT_dbl(group, r, r, ctx)) goto err; } for (i = 0; i < totalnum; i++) { if (wNAF_len[i] > k) { int digit = wNAF[i][k]; int is_neg; if (digit) { is_neg = digit < 0; if (is_neg) digit = -digit; if (is_neg != r_is_inverted) { if (!r_is_at_infinity) { if (!EC_POINT_invert(group, r, ctx)) goto err; } r_is_inverted = !r_is_inverted; } /* digit > 0 */ if (r_is_at_infinity) { if (!EC_POINT_copy(r, val_sub[i][digit >> 1])) goto err; r_is_at_infinity = 0; } else { if (!EC_POINT_add(group, r, r, val_sub[i][digit >> 1], ctx)) goto err; } } } } } if (r_is_at_infinity) { if (!EC_POINT_set_to_infinity(group, r)) goto err; } else { if (r_is_inverted) if (!EC_POINT_invert(group, r, ctx)) goto err; } ret = 1; err: if (new_ctx != NULL) BN_CTX_free(new_ctx); if (tmp != NULL) EC_POINT_free(tmp); if (wsize != NULL) OPENSSL_free(wsize); if (wNAF_len != NULL) OPENSSL_free(wNAF_len); if (wNAF != NULL) { signed char **w; for (w = wNAF; *w != NULL; w++) OPENSSL_free(*w); OPENSSL_free(wNAF); } if (val != NULL) { for (v = val; *v != NULL; v++) EC_POINT_clear_free(*v); OPENSSL_free(val); } if (val_sub != NULL) { OPENSSL_free(val_sub); } return ret; } #else /* * Basic interleaving multi-exponentation method */ #define EC_window_bits_for_scalar_size(b) \ ((b) >= 2000 ? 6 : \ (b) >= 800 ? 5 : \ (b) >= 300 ? 4 : \ (b) >= 70 ? 3 : \ (b) >= 20 ? 2 : \ 1) /* For window size 'w' (w >= 2), we compute the odd multiples * 1*P .. (2^w-1)*P. * This accounts for 2^(w-1) point additions (neglecting constants), * each of which requires 16 field multiplications (4 squarings * and 12 general multiplications) in the case of curves defined * over GF(p), which are the only curves we have so far. * * Converting these precomputed points into affine form takes * three field multiplications for inverting Z and one squaring * and three multiplications for adjusting X and Y, i.e. * 7 multiplications in total (1 squaring and 6 general multiplications), * again except for constants. * * The average number of windows for a 'b' bit scalar is roughly * b/(w+1). * Each of these windows (except possibly for the first one, but * we are ignoring constants anyway) requires one point addition. * As the precomputed table stores points in affine form, these * additions take only 11 field multiplications each (3 squarings * and 8 general multiplications). * * So the total workload, except for constants, is * * 2^(w-1)*[5 squarings + 18 multiplications] * + (b/(w+1))*[3 squarings + 8 multiplications] * * If we assume that 10 squarings are as costly as 9 multiplications, * our task is to find the 'w' that, given 'b', minimizes * * 2^(w-1)*(5*9 + 18*10) + (b/(w+1))*(3*9 + 8*10) * = 2^(w-1)*225 + (b/(w+1))*107. * * Thus optimal window sizes should be roughly as follows: * * w >= 6 if b >= 1414 * w = 5 if 1413 >= b >= 505 * w = 4 if 504 >= b >= 169 * w = 3 if 168 >= b >= 51 * w = 2 if 50 >= b >= 13 * w = 1 if 12 >= b * * If we assume instead that squarings are exactly as costly as * multiplications, we have to minimize * 2^(w-1)*23 + (b/(w+1))*11. * * This gives us the following (nearly unchanged) table of optimal * windows sizes: * * w >= 6 if b >= 1406 * w = 5 if 1405 >= b >= 502 * w = 4 if 501 >= b >= 168 * w = 3 if 167 >= b >= 51 * w = 2 if 50 >= b >= 13 * w = 1 if 12 >= b * * Note that neither table tries to take into account memory usage * (allocation overhead, code locality etc.). Actual timings with * NIST curves P-192, P-224, and P-256 with scalars of 192, 224, * and 256 bits, respectively, show that w = 3 (instead of 4) is * preferrable; timings with NIST curve P-384 and 384-bit scalars * confirm that w = 4 is optimal for this case; and timings with * NIST curve P-521 and 521-bit scalars show that w = 4 (instead * of 5) is preferrable. So we generously round up all the * boundaries and use the following table: * * w >= 6 if b >= 2000 * w = 5 if 1999 >= b >= 800 * w = 4 if 799 >= b >= 300 * w = 3 if 299 >= b >= 70 * w = 2 if 69 >= b >= 20 * w = 1 if 19 >= b */ int EC_POINTs_mul(const EC_GROUP *group, EC_POINT *r, const BIGNUM *scalar, size_t num, const EC_POINT *points[], const BIGNUM *scalars[], BN_CTX *ctx) { BN_CTX *new_ctx = NULL; EC_POINT *generator = NULL; EC_POINT *tmp = NULL; size_t totalnum; size_t i, j; int k, t; int r_is_at_infinity = 1; size_t max_bits = 0; size_t *wsize = NULL; /* individual window sizes */ unsigned long *wbits = NULL; /* individual window contents */ int *wpos = NULL; /* position of bottom bit of current individual windows * (wpos[i] is valid if wbits[i] != 0) */ size_t num_val; EC_POINT **val = NULL; /* precomputation */ EC_POINT **v; EC_POINT ***val_sub = NULL; /* pointers to sub-arrays of 'val' */ int ret = 0; if (scalar != NULL) { generator = EC_GROUP_get0_generator(group); if (generator == NULL) { ECerr(EC_F_EC_POINTS_MUL, EC_R_UNDEFINED_GENERATOR); return 0; } } for (i = 0; i < num; i++) { if (group->meth != points[i]->meth) { ECerr(EC_F_EC_POINTS_MUL, EC_R_INCOMPATIBLE_OBJECTS); return 0; } } totalnum = num + (scalar != NULL); wsize = OPENSSL_malloc(totalnum * sizeof wsize[0]); wbits = OPENSSL_malloc(totalnum * sizeof wbits[0]); wpos = OPENSSL_malloc(totalnum * sizeof wpos[0]); if (wsize == NULL || wbits == NULL || wpos == NULL) goto err; /* num_val := total number of points to precompute */ num_val = 0; for (i = 0; i < totalnum; i++) { size_t bits; bits = i < num ? BN_num_bits(scalars[i]) : BN_num_bits(scalar); wsize[i] = EC_window_bits_for_scalar_size(bits); num_val += 1u << (wsize[i] - 1); if (bits > max_bits) max_bits = bits; wbits[i] = 0; wpos[i] = 0; } /* all precomputed points go into a single array 'val', * 'val_sub[i]' is a pointer to the subarray for the i-th point */ val = OPENSSL_malloc((num_val + 1) * sizeof val[0]); if (val == NULL) goto err; val[num_val] = NULL; /* pivot element */ val_sub = OPENSSL_malloc(totalnum * sizeof val_sub[0]); if (val_sub == NULL) goto err; /* allocate points for precomputation */ v = val; for (i = 0; i < totalnum; i++) { val_sub[i] = v; for (j = 0; j < (1u << (wsize[i] - 1)); j++) { *v = EC_POINT_new(group); if (*v == NULL) goto err; v++; } } if (!(v == val + num_val)) { ECerr(EC_F_EC_POINTS_MUL, ERR_R_INTERNAL_ERROR); goto err; } if (ctx == NULL) { ctx = new_ctx = BN_CTX_new(); if (ctx == NULL) goto err; } tmp = EC_POINT_new(group); if (tmp == NULL) goto err; /* prepare precomputed values: * val_sub[i][0] := points[i] * val_sub[i][1] := 3 * points[i] * val_sub[i][2] := 5 * points[i] * ... */ for (i = 0; i < totalnum; i++) { if (i < num) { if (!EC_POINT_copy(val_sub[i][0], points[i])) goto err; if (scalars[i]->neg) { if (!EC_POINT_invert(group, val_sub[i][0], ctx)) goto err; } } else { if (!EC_POINT_copy(val_sub[i][0], generator)) goto err; if (scalar->neg) { if (!EC_POINT_invert(group, val_sub[i][0], ctx)) goto err; } } if (wsize[i] > 1) { if (!EC_POINT_dbl(group, tmp, val_sub[i][0], ctx)) goto err; for (j = 1; j < (1u << (wsize[i] - 1)); j++) { if (!EC_POINT_add(group, val_sub[i][j], val_sub[i][j - 1], tmp, ctx)) goto err; } } } #if 1 /* optional; EC_window_bits_for_scalar_size assumes we do this step */ if (!EC_POINTs_make_affine(group, num_val, val, ctx)) goto err; #endif r_is_at_infinity = 1; for (k = max_bits - 1; k >= 0; k--) { if (!r_is_at_infinity) { if (!EC_POINT_dbl(group, r, r, ctx)) goto err; } for (i = 0; i < totalnum; i++) { if (wbits[i] == 0) { const BIGNUM *s; s = i < num ? scalars[i] : scalar; if (BN_is_bit_set(s, k)) { /* look at bits k - wsize[i] + 1 .. k for this window */ t = k - wsize[i] + 1; while (!BN_is_bit_set(s, t)) /* BN_is_bit_set is false for t < 0 */ t++; wpos[i] = t; wbits[i] = 1; for (t = k - 1; t >= wpos[i]; t--) { wbits[i] <<= 1; if (BN_is_bit_set(s, t)) wbits[i]++; } /* now wbits[i] is the odd bit pattern at bits wpos[i] .. k */ } } if ((wbits[i] != 0) && (wpos[i] == k)) { if (r_is_at_infinity) { if (!EC_POINT_copy(r, val_sub[i][wbits[i] >> 1])) goto err; r_is_at_infinity = 0; } else { if (!EC_POINT_add(group, r, r, val_sub[i][wbits[i] >> 1], ctx)) goto err; } wbits[i] = 0; } } } if (r_is_at_infinity) if (!EC_POINT_set_to_infinity(group, r)) goto err; ret = 1; err: if (new_ctx != NULL) BN_CTX_free(new_ctx); if (tmp != NULL) EC_POINT_free(tmp); if (wsize != NULL) OPENSSL_free(wsize); if (wbits != NULL) OPENSSL_free(wbits); if (wpos != NULL) OPENSSL_free(wpos); if (val != NULL) { for (v = val; *v != NULL; v++) EC_POINT_clear_free(*v); OPENSSL_free(val); } if (val_sub != NULL) { OPENSSL_free(val_sub); } return ret; } #endif 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) { const EC_POINT *points[1]; const BIGNUM *scalars[1]; points[0] = point; scalars[0] = p_scalar; return EC_POINTs_mul(group, r, g_scalar, (point != NULL && p_scalar != NULL), points, scalars, ctx); } int EC_GROUP_precompute_mult(EC_GROUP *group, BN_CTX *ctx) { const EC_POINT *generator; BN_CTX *new_ctx = NULL; BIGNUM *order; int ret = 0; generator = EC_GROUP_get0_generator(group); if (generator == NULL) { ECerr(EC_F_EC_GROUP_PRECOMPUTE_MULT, EC_R_UNDEFINED_GENERATOR); return 0; } if (ctx == NULL) { ctx = new_ctx = BN_CTX_new(); if (ctx == NULL) return 0; } BN_CTX_start(ctx); order = BN_CTX_get(ctx); if (order == NULL) goto err; if (!EC_GROUP_get_order(group, order, ctx)) return 0; if (BN_is_zero(order)) { ECerr(EC_F_EC_GROUP_PRECOMPUTE_MULT, EC_R_UNKNOWN_ORDER); goto err; } /* TODO */ ret = 1; err: BN_CTX_end(ctx); if (new_ctx != NULL) BN_CTX_free(new_ctx); return ret; }