2 * Copyright 2011-2020 The OpenSSL Project Authors. All Rights Reserved.
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
10 /* Copyright 2011 Google Inc.
12 * Licensed under the Apache License, Version 2.0 (the "License");
14 * you may not use this file except in compliance with the License.
15 * You may obtain a copy of the License at
17 * http://www.apache.org/licenses/LICENSE-2.0
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.
27 * ECDSA low level APIs are deprecated for public use, but still ok for
30 #include "internal/deprecated.h"
32 #include <openssl/opensslconf.h>
35 * Common utility functions for ecp_nistp224.c, ecp_nistp256.c, ecp_nistp521.c.
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.
52 void ec_GFp_nistp_points_make_affine_internal(size_t num, void *point_array,
55 void (*felem_one) (void *out),
56 int (*felem_is_zero) (const void
58 void (*felem_assign) (void *out,
61 void (*felem_square) (void *out,
64 void (*felem_mul) (void *out,
69 void (*felem_inv) (void *out,
72 void (*felem_contract) (void
80 #define tmp_felem(I) (&((char *)tmp_felems)[(I) * felem_size])
81 #define X(I) (&((char *)point_array)[3*(I) * felem_size])
82 #define Y(I) (&((char *)point_array)[(3*(I) + 1) * felem_size])
83 #define Z(I) (&((char *)point_array)[(3*(I) + 2) * felem_size])
85 if (!felem_is_zero(Z(0)))
86 felem_assign(tmp_felem(0), Z(0));
88 felem_one(tmp_felem(0));
89 for (i = 1; i < (int)num; i++) {
90 if (!felem_is_zero(Z(i)))
91 felem_mul(tmp_felem(i), tmp_felem(i - 1), Z(i));
93 felem_assign(tmp_felem(i), tmp_felem(i - 1));
96 * Now each tmp_felem(i) is the product of Z(0) .. Z(i), skipping any
97 * zero-valued factors: if Z(i) = 0, we essentially pretend that Z(i) = 1
100 felem_inv(tmp_felem(num - 1), tmp_felem(num - 1));
101 for (i = num - 1; i >= 0; i--) {
104 * tmp_felem(i-1) is the product of Z(0) .. Z(i-1), tmp_felem(i)
105 * is the inverse of the product of Z(0) .. Z(i)
108 felem_mul(tmp_felem(num), tmp_felem(i - 1), tmp_felem(i));
110 felem_assign(tmp_felem(num), tmp_felem(0)); /* 1/Z(0) */
112 if (!felem_is_zero(Z(i))) {
115 * For next iteration, replace tmp_felem(i-1) by its inverse
117 felem_mul(tmp_felem(i - 1), tmp_felem(i), Z(i));
120 * Convert point (X, Y, Z) into affine form (X/(Z^2), Y/(Z^3), 1)
122 felem_square(Z(i), tmp_felem(num)); /* 1/(Z^2) */
123 felem_mul(X(i), X(i), Z(i)); /* X/(Z^2) */
124 felem_mul(Z(i), Z(i), tmp_felem(num)); /* 1/(Z^3) */
125 felem_mul(Y(i), Y(i), Z(i)); /* Y/(Z^3) */
126 felem_contract(X(i), X(i));
127 felem_contract(Y(i), Y(i));
132 * For next iteration, replace tmp_felem(i-1) by its inverse
134 felem_assign(tmp_felem(i - 1), tmp_felem(i));
140 * This function looks at 5+1 scalar bits (5 current, 1 adjacent less
141 * significant bit), and recodes them into a signed digit for use in fast point
142 * multiplication: the use of signed rather than unsigned digits means that
143 * fewer points need to be precomputed, given that point inversion is easy
144 * (a precomputed point dP makes -dP available as well).
148 * Signed digits for multiplication were introduced by Booth ("A signed binary
149 * multiplication technique", Quart. Journ. Mech. and Applied Math., vol. IV,
150 * pt. 2 (1951), pp. 236-240), in that case for multiplication of integers.
151 * Booth's original encoding did not generally improve the density of nonzero
152 * digits over the binary representation, and was merely meant to simplify the
153 * handling of signed factors given in two's complement; but it has since been
154 * shown to be the basis of various signed-digit representations that do have
155 * further advantages, including the wNAF, using the following general approach:
157 * (1) Given a binary representation
159 * b_k ... b_2 b_1 b_0,
161 * of a nonnegative integer (b_k in {0, 1}), rewrite it in digits 0, 1, -1
162 * by using bit-wise subtraction as follows:
164 * b_k b_(k-1) ... b_2 b_1 b_0
165 * - b_k ... b_3 b_2 b_1 b_0
166 * -----------------------------------------
167 * s_(k+1) s_k ... s_3 s_2 s_1 s_0
169 * A left-shift followed by subtraction of the original value yields a new
170 * representation of the same value, using signed bits s_i = b_(i-1) - b_i.
171 * This representation from Booth's paper has since appeared in the
172 * literature under a variety of different names including "reversed binary
173 * form", "alternating greedy expansion", "mutual opposite form", and
174 * "sign-alternating {+-1}-representation".
176 * An interesting property is that among the nonzero bits, values 1 and -1
177 * strictly alternate.
179 * (2) Various window schemes can be applied to the Booth representation of
180 * integers: for example, right-to-left sliding windows yield the wNAF
181 * (a signed-digit encoding independently discovered by various researchers
182 * in the 1990s), and left-to-right sliding windows yield a left-to-right
183 * equivalent of the wNAF (independently discovered by various researchers
186 * To prevent leaking information through side channels in point multiplication,
187 * we need to recode the given integer into a regular pattern: sliding windows
188 * as in wNAFs won't do, we need their fixed-window equivalent -- which is a few
189 * decades older: we'll be using the so-called "modified Booth encoding" due to
190 * MacSorley ("High-speed arithmetic in binary computers", Proc. IRE, vol. 49
191 * (1961), pp. 67-91), in a radix-2^5 setting. That is, we always combine five
192 * signed bits into a signed digit:
194 * s_(5j + 4) s_(5j + 3) s_(5j + 2) s_(5j + 1) s_(5j)
196 * The sign-alternating property implies that the resulting digit values are
197 * integers from -16 to 16.
199 * Of course, we don't actually need to compute the signed digits s_i as an
200 * intermediate step (that's just a nice way to see how this scheme relates
201 * to the wNAF): a direct computation obtains the recoded digit from the
202 * six bits b_(5j + 4) ... b_(5j - 1).
204 * This function takes those six bits as an integer (0 .. 63), writing the
205 * recoded digit to *sign (0 for positive, 1 for negative) and *digit (absolute
206 * value, in the range 0 .. 16). Note that this integer essentially provides
207 * the input bits "shifted to the left" by one position: for example, the input
208 * to compute the least significant recoded digit, given that there's no bit
209 * b_-1, has to be b_4 b_3 b_2 b_1 b_0 0.
212 void ec_GFp_nistp_recode_scalar_bits(unsigned char *sign,
213 unsigned char *digit, unsigned char in)
217 s = ~((in >> 5) - 1); /* sets all bits to MSB(in), 'in' seen as
219 d = (1 << 6) - in - 1;
220 d = (d & s) | (in & ~s);
221 d = (d >> 1) + (d & 1);