1 /*
2 * Copyright 2011-2021 The OpenSSL Project Authors. All Rights Reserved.
3 *
4 * Licensed under the Apache License 2.0 (the "License"). You may not use
5 * this file except in compliance with the License. You can obtain a copy
6 * in the file LICENSE in the source distribution or at
7 * https://www.openssl.org/source/license.html
8 */
9
10 /* Copyright 2011 Google Inc.
11 *
12 * Licensed under the Apache License, Version 2.0 (the "License");
13 *
14 * you may not use this file except in compliance with the License.
15 * You may obtain a copy of the License at
16 *
17 * http://www.apache.org/licenses/LICENSE-2.0
18 *
19 * Unless required by applicable law or agreed to in writing, software
20 * distributed under the License is distributed on an "AS IS" BASIS,
21 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
22 * See the License for the specific language governing permissions and
23 * limitations under the License.
24 */
25
26 /*
27 * ECDSA low level APIs are deprecated for public use, but still ok for
28 * internal use.
29 */
30 #include "internal/deprecated.h"
31
32 #include <openssl/opensslconf.h>
33
34 /*
35 * Common utility functions for ecp_nistp224.c, ecp_nistp256.c, ecp_nistp521.c.
36 */
37
38 #include <stddef.h>
39 #include "ec_local.h"
40
41 /*
42 * Convert an array of points into affine coordinates. (If the point at
43 * infinity is found (Z = 0), it remains unchanged.) This function is
44 * essentially an equivalent to EC_POINTs_make_affine(), but works with the
45 * internal representation of points as used by ecp_nistp###.c rather than
46 * with (BIGNUM-based) EC_POINT data structures. point_array is the
47 * input/output buffer ('num' points in projective form, i.e. three
48 * coordinates each), based on an internal representation of field elements
49 * of size 'felem_size'. tmp_felems needs to point to a temporary array of
50 * 'num'+1 field elements for storage of intermediate values.
51 */
52 void
ossl_ec_GFp_nistp_points_make_affine_internal(size_t num,void * point_array,size_t felem_size,void * tmp_felems,void (* felem_one)(void * out),int (* felem_is_zero)(const void * in),void (* felem_assign)(void * out,const void * in),void (* felem_square)(void * out,const void * in),void (* felem_mul)(void * out,const void * in1,const void * in2),void (* felem_inv)(void * out,const void * in),void (* felem_contract)(void * out,const void * in))53 ossl_ec_GFp_nistp_points_make_affine_internal(size_t num, void *point_array,
54 size_t felem_size,
55 void *tmp_felems,
56 void (*felem_one) (void *out),
57 int (*felem_is_zero) (const void
58 *in),
59 void (*felem_assign) (void *out,
60 const void
61 *in),
62 void (*felem_square) (void *out,
63 const void
64 *in),
65 void (*felem_mul) (void *out,
66 const void
67 *in1,
68 const void
69 *in2),
70 void (*felem_inv) (void *out,
71 const void
72 *in),
73 void (*felem_contract) (void
74 *out,
75 const
76 void
77 *in))
78 {
79 int i = 0;
80
81 #define tmp_felem(I) (&((char *)tmp_felems)[(I) * felem_size])
82 #define X(I) (&((char *)point_array)[3*(I) * felem_size])
83 #define Y(I) (&((char *)point_array)[(3*(I) + 1) * felem_size])
84 #define Z(I) (&((char *)point_array)[(3*(I) + 2) * felem_size])
85
86 if (!felem_is_zero(Z(0)))
87 felem_assign(tmp_felem(0), Z(0));
88 else
89 felem_one(tmp_felem(0));
90 for (i = 1; i < (int)num; i++) {
91 if (!felem_is_zero(Z(i)))
92 felem_mul(tmp_felem(i), tmp_felem(i - 1), Z(i));
93 else
94 felem_assign(tmp_felem(i), tmp_felem(i - 1));
95 }
96 /*
97 * Now each tmp_felem(i) is the product of Z(0) .. Z(i), skipping any
98 * zero-valued factors: if Z(i) = 0, we essentially pretend that Z(i) = 1
99 */
100
101 felem_inv(tmp_felem(num - 1), tmp_felem(num - 1));
102 for (i = num - 1; i >= 0; i--) {
103 if (i > 0)
104 /*
105 * tmp_felem(i-1) is the product of Z(0) .. Z(i-1), tmp_felem(i)
106 * is the inverse of the product of Z(0) .. Z(i)
107 */
108 /* 1/Z(i) */
109 felem_mul(tmp_felem(num), tmp_felem(i - 1), tmp_felem(i));
110 else
111 felem_assign(tmp_felem(num), tmp_felem(0)); /* 1/Z(0) */
112
113 if (!felem_is_zero(Z(i))) {
114 if (i > 0)
115 /*
116 * For next iteration, replace tmp_felem(i-1) by its inverse
117 */
118 felem_mul(tmp_felem(i - 1), tmp_felem(i), Z(i));
119
120 /*
121 * Convert point (X, Y, Z) into affine form (X/(Z^2), Y/(Z^3), 1)
122 */
123 felem_square(Z(i), tmp_felem(num)); /* 1/(Z^2) */
124 felem_mul(X(i), X(i), Z(i)); /* X/(Z^2) */
125 felem_mul(Z(i), Z(i), tmp_felem(num)); /* 1/(Z^3) */
126 felem_mul(Y(i), Y(i), Z(i)); /* Y/(Z^3) */
127 felem_contract(X(i), X(i));
128 felem_contract(Y(i), Y(i));
129 felem_one(Z(i));
130 } else {
131 if (i > 0)
132 /*
133 * For next iteration, replace tmp_felem(i-1) by its inverse
134 */
135 felem_assign(tmp_felem(i - 1), tmp_felem(i));
136 }
137 }
138 }
139
140 /*-
141 * This function looks at 5+1 scalar bits (5 current, 1 adjacent less
142 * significant bit), and recodes them into a signed digit for use in fast point
143 * multiplication: the use of signed rather than unsigned digits means that
144 * fewer points need to be precomputed, given that point inversion is easy
145 * (a precomputed point dP makes -dP available as well).
146 *
147 * BACKGROUND:
148 *
149 * Signed digits for multiplication were introduced by Booth ("A signed binary
150 * multiplication technique", Quart. Journ. Mech. and Applied Math., vol. IV,
151 * pt. 2 (1951), pp. 236-240), in that case for multiplication of integers.
152 * Booth's original encoding did not generally improve the density of nonzero
153 * digits over the binary representation, and was merely meant to simplify the
154 * handling of signed factors given in two's complement; but it has since been
155 * shown to be the basis of various signed-digit representations that do have
156 * further advantages, including the wNAF, using the following general approach:
157 *
158 * (1) Given a binary representation
159 *
160 * b_k ... b_2 b_1 b_0,
161 *
162 * of a nonnegative integer (b_k in {0, 1}), rewrite it in digits 0, 1, -1
163 * by using bit-wise subtraction as follows:
164 *
165 * b_k b_(k-1) ... b_2 b_1 b_0
166 * - b_k ... b_3 b_2 b_1 b_0
167 * -----------------------------------------
168 * s_(k+1) s_k ... s_3 s_2 s_1 s_0
169 *
170 * A left-shift followed by subtraction of the original value yields a new
171 * representation of the same value, using signed bits s_i = b_(i-1) - b_i.
172 * This representation from Booth's paper has since appeared in the
173 * literature under a variety of different names including "reversed binary
174 * form", "alternating greedy expansion", "mutual opposite form", and
175 * "sign-alternating {+-1}-representation".
176 *
177 * An interesting property is that among the nonzero bits, values 1 and -1
178 * strictly alternate.
179 *
180 * (2) Various window schemes can be applied to the Booth representation of
181 * integers: for example, right-to-left sliding windows yield the wNAF
182 * (a signed-digit encoding independently discovered by various researchers
183 * in the 1990s), and left-to-right sliding windows yield a left-to-right
184 * equivalent of the wNAF (independently discovered by various researchers
185 * around 2004).
186 *
187 * To prevent leaking information through side channels in point multiplication,
188 * we need to recode the given integer into a regular pattern: sliding windows
189 * as in wNAFs won't do, we need their fixed-window equivalent -- which is a few
190 * decades older: we'll be using the so-called "modified Booth encoding" due to
191 * MacSorley ("High-speed arithmetic in binary computers", Proc. IRE, vol. 49
192 * (1961), pp. 67-91), in a radix-2^5 setting. That is, we always combine five
193 * signed bits into a signed digit:
194 *
195 * s_(5j + 4) s_(5j + 3) s_(5j + 2) s_(5j + 1) s_(5j)
196 *
197 * The sign-alternating property implies that the resulting digit values are
198 * integers from -16 to 16.
199 *
200 * Of course, we don't actually need to compute the signed digits s_i as an
201 * intermediate step (that's just a nice way to see how this scheme relates
202 * to the wNAF): a direct computation obtains the recoded digit from the
203 * six bits b_(5j + 4) ... b_(5j - 1).
204 *
205 * This function takes those six bits as an integer (0 .. 63), writing the
206 * recoded digit to *sign (0 for positive, 1 for negative) and *digit (absolute
207 * value, in the range 0 .. 16). Note that this integer essentially provides
208 * the input bits "shifted to the left" by one position: for example, the input
209 * to compute the least significant recoded digit, given that there's no bit
210 * b_-1, has to be b_4 b_3 b_2 b_1 b_0 0.
211 *
212 */
ossl_ec_GFp_nistp_recode_scalar_bits(unsigned char * sign,unsigned char * digit,unsigned char in)213 void ossl_ec_GFp_nistp_recode_scalar_bits(unsigned char *sign,
214 unsigned char *digit, unsigned char in)
215 {
216 unsigned char s, d;
217
218 s = ~((in >> 5) - 1); /* sets all bits to MSB(in), 'in' seen as
219 * 6-bit value */
220 d = (1 << 6) - in - 1;
221 d = (d & s) | (in & ~s);
222 d = (d >> 1) + (d & 1);
223
224 *sign = s & 1;
225 *digit = d;
226 }
227
