1 | /* Originally written by Bodo Moeller for the OpenSSL project. |
2 | * ==================================================================== |
3 | * Copyright (c) 1998-2005 The OpenSSL Project. All rights reserved. |
4 | * |
5 | * Redistribution and use in source and binary forms, with or without |
6 | * modification, are permitted provided that the following conditions |
7 | * are met: |
8 | * |
9 | * 1. Redistributions of source code must retain the above copyright |
10 | * notice, this list of conditions and the following disclaimer. |
11 | * |
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 |
15 | * distribution. |
16 | * |
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/)" |
21 | * |
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. |
26 | * |
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. |
30 | * |
31 | * 6. Redistributions of any form whatsoever must retain the following |
32 | * acknowledgment: |
33 | * "This product includes software developed by the OpenSSL Project |
34 | * for use in the OpenSSL Toolkit (http://www.openssl.org/)" |
35 | * |
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 | * ==================================================================== |
49 | * |
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). |
53 | * |
54 | */ |
55 | /* ==================================================================== |
56 | * Copyright 2002 Sun Microsystems, Inc. ALL RIGHTS RESERVED. |
57 | * |
58 | * Portions of the attached software ("Contribution") are developed by |
59 | * SUN MICROSYSTEMS, INC., and are contributed to the OpenSSL project. |
60 | * |
61 | * The Contribution is licensed pursuant to the OpenSSL open source |
62 | * license provided above. |
63 | * |
64 | * The elliptic curve binary polynomial software is originally written by |
65 | * Sheueling Chang Shantz and Douglas Stebila of Sun Microsystems |
66 | * Laboratories. */ |
67 | |
68 | #include <openssl/ec.h> |
69 | |
70 | #include <assert.h> |
71 | #include <string.h> |
72 | |
73 | #include <openssl/bn.h> |
74 | #include <openssl/err.h> |
75 | #include <openssl/thread.h> |
76 | |
77 | #include "internal.h" |
78 | #include "../bn/internal.h" |
79 | #include "../../internal.h" |
80 | |
81 | |
82 | // This file implements the wNAF-based interleaving multi-exponentiation method |
83 | // at: |
84 | // http://link.springer.com/chapter/10.1007%2F3-540-45537-X_13 |
85 | // http://www.bmoeller.de/pdf/TI-01-08.multiexp.pdf |
86 | |
87 | void ec_compute_wNAF(const EC_GROUP *group, int8_t *out, |
88 | const EC_SCALAR *scalar, size_t bits, int w) { |
89 | // 'int8_t' can represent integers with absolute values less than 2^7. |
90 | assert(0 < w && w <= 7); |
91 | assert(bits != 0); |
92 | int bit = 1 << w; // 2^w, at most 128 |
93 | int next_bit = bit << 1; // 2^(w+1), at most 256 |
94 | int mask = next_bit - 1; // at most 255 |
95 | |
96 | int window_val = scalar->words[0] & mask; |
97 | for (size_t j = 0; j < bits + 1; j++) { |
98 | assert(0 <= window_val && window_val <= next_bit); |
99 | int digit = 0; |
100 | if (window_val & 1) { |
101 | assert(0 < window_val && window_val < next_bit); |
102 | if (window_val & bit) { |
103 | digit = window_val - next_bit; |
104 | // We know -next_bit < digit < 0 and window_val - digit = next_bit. |
105 | |
106 | // modified wNAF |
107 | if (j + w + 1 >= bits) { |
108 | // special case for generating modified wNAFs: |
109 | // no new bits will be added into window_val, |
110 | // so using a positive digit here will decrease |
111 | // the total length of the representation |
112 | |
113 | digit = window_val & (mask >> 1); |
114 | // We know 0 < digit < bit and window_val - digit = bit. |
115 | } |
116 | } else { |
117 | digit = window_val; |
118 | // We know 0 < digit < bit and window_val - digit = 0. |
119 | } |
120 | |
121 | window_val -= digit; |
122 | |
123 | // Now window_val is 0 or 2^(w+1) in standard wNAF generation. |
124 | // For modified window NAFs, it may also be 2^w. |
125 | // |
126 | // See the comments above for the derivation of each of these bounds. |
127 | assert(window_val == 0 || window_val == next_bit || window_val == bit); |
128 | assert(-bit < digit && digit < bit); |
129 | |
130 | // window_val was odd, so digit is also odd. |
131 | assert(digit & 1); |
132 | } |
133 | |
134 | out[j] = digit; |
135 | |
136 | // Incorporate the next bit. Previously, |window_val| <= |next_bit|, so if |
137 | // we shift and add at most one copy of |bit|, this will continue to hold |
138 | // afterwards. |
139 | window_val >>= 1; |
140 | window_val += |
141 | bit * bn_is_bit_set_words(scalar->words, group->order.width, j + w + 1); |
142 | assert(window_val <= next_bit); |
143 | } |
144 | |
145 | // bits + 1 entries should be sufficient to consume all bits. |
146 | assert(window_val == 0); |
147 | } |
148 | |
149 | // compute_precomp sets |out[i]| to (2*i+1)*p, for i from 0 to |len|. |
150 | static void compute_precomp(const EC_GROUP *group, EC_RAW_POINT *out, |
151 | const EC_RAW_POINT *p, size_t len) { |
152 | ec_GFp_simple_point_copy(&out[0], p); |
153 | EC_RAW_POINT two_p; |
154 | ec_GFp_mont_dbl(group, &two_p, p); |
155 | for (size_t i = 1; i < len; i++) { |
156 | ec_GFp_mont_add(group, &out[i], &out[i - 1], &two_p); |
157 | } |
158 | } |
159 | |
160 | static void lookup_precomp(const EC_GROUP *group, EC_RAW_POINT *out, |
161 | const EC_RAW_POINT *precomp, int digit) { |
162 | if (digit < 0) { |
163 | digit = -digit; |
164 | ec_GFp_simple_point_copy(out, &precomp[digit >> 1]); |
165 | ec_GFp_simple_invert(group, out); |
166 | } else { |
167 | ec_GFp_simple_point_copy(out, &precomp[digit >> 1]); |
168 | } |
169 | } |
170 | |
171 | // EC_WNAF_WINDOW_BITS is the window size to use for |ec_GFp_mont_mul_public|. |
172 | #define EC_WNAF_WINDOW_BITS 4 |
173 | |
174 | // EC_WNAF_TABLE_SIZE is the table size to use for |ec_GFp_mont_mul_public|. |
175 | #define EC_WNAF_TABLE_SIZE (1 << (EC_WNAF_WINDOW_BITS - 1)) |
176 | |
177 | void ec_GFp_mont_mul_public(const EC_GROUP *group, EC_RAW_POINT *r, |
178 | const EC_SCALAR *g_scalar, const EC_RAW_POINT *p, |
179 | const EC_SCALAR *p_scalar) { |
180 | size_t bits = BN_num_bits(&group->order); |
181 | size_t wNAF_len = bits + 1; |
182 | |
183 | int8_t g_wNAF[EC_MAX_BYTES * 8 + 1]; |
184 | EC_RAW_POINT g_precomp[EC_WNAF_TABLE_SIZE]; |
185 | assert(wNAF_len <= OPENSSL_ARRAY_SIZE(g_wNAF)); |
186 | const EC_RAW_POINT *g = &group->generator->raw; |
187 | ec_compute_wNAF(group, g_wNAF, g_scalar, bits, EC_WNAF_WINDOW_BITS); |
188 | compute_precomp(group, g_precomp, g, EC_WNAF_TABLE_SIZE); |
189 | |
190 | int8_t p_wNAF[EC_MAX_BYTES * 8 + 1]; |
191 | EC_RAW_POINT p_precomp[EC_WNAF_TABLE_SIZE]; |
192 | assert(wNAF_len <= OPENSSL_ARRAY_SIZE(p_wNAF)); |
193 | ec_compute_wNAF(group, p_wNAF, p_scalar, bits, EC_WNAF_WINDOW_BITS); |
194 | compute_precomp(group, p_precomp, p, EC_WNAF_TABLE_SIZE); |
195 | |
196 | EC_RAW_POINT tmp; |
197 | int r_is_at_infinity = 1; |
198 | for (size_t k = wNAF_len - 1; k < wNAF_len; k--) { |
199 | if (!r_is_at_infinity) { |
200 | ec_GFp_mont_dbl(group, r, r); |
201 | } |
202 | |
203 | if (g_wNAF[k] != 0) { |
204 | lookup_precomp(group, &tmp, g_precomp, g_wNAF[k]); |
205 | if (r_is_at_infinity) { |
206 | ec_GFp_simple_point_copy(r, &tmp); |
207 | r_is_at_infinity = 0; |
208 | } else { |
209 | ec_GFp_mont_add(group, r, r, &tmp); |
210 | } |
211 | } |
212 | |
213 | if (p_wNAF[k] != 0) { |
214 | lookup_precomp(group, &tmp, p_precomp, p_wNAF[k]); |
215 | if (r_is_at_infinity) { |
216 | ec_GFp_simple_point_copy(r, &tmp); |
217 | r_is_at_infinity = 0; |
218 | } else { |
219 | ec_GFp_mont_add(group, r, r, &tmp); |
220 | } |
221 | } |
222 | } |
223 | |
224 | if (r_is_at_infinity) { |
225 | ec_GFp_simple_point_set_to_infinity(group, r); |
226 | } |
227 | } |
228 | |