1/*
2 * Copyright 2011-2016 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#include <openssl/opensslconf.h>
27#ifdef OPENSSL_NO_EC_NISTP_64_GCC_128
28NON_EMPTY_TRANSLATION_UNIT
29#else
30
31/*
32 * Common utility functions for ecp_nistp224.c, ecp_nistp256.c, ecp_nistp521.c.
33 */
34
35# include <stddef.h>
36# include "ec_local.h"
37
38/*
39 * Convert an array of points into affine coordinates. (If the point at
40 * infinity is found (Z = 0), it remains unchanged.) This function is
41 * essentially an equivalent to EC_POINTs_make_affine(), but works with the
42 * internal representation of points as used by ecp_nistp###.c rather than
43 * with (BIGNUM-based) EC_POINT data structures. point_array is the
44 * input/output buffer ('num' points in projective form, i.e. three
45 * coordinates each), based on an internal representation of field elements
46 * of size 'felem_size'. tmp_felems needs to point to a temporary array of
47 * 'num'+1 field elements for storage of intermediate values.
48 */
49void ec_GFp_nistp_points_make_affine_internal(size_t num, void *point_array,
50 size_t felem_size,
51 void *tmp_felems,
52 void (*felem_one) (void *out),
53 int (*felem_is_zero) (const void
54 *in),
55 void (*felem_assign) (void *out,
56 const void
57 *in),
58 void (*felem_square) (void *out,
59 const void
60 *in),
61 void (*felem_mul) (void *out,
62 const void
63 *in1,
64 const void
65 *in2),
66 void (*felem_inv) (void *out,
67 const void
68 *in),
69 void (*felem_contract) (void
70 *out,
71 const
72 void
73 *in))
74{
75 int i = 0;
76
77# define tmp_felem(I) (&((char *)tmp_felems)[(I) * felem_size])
78# define X(I) (&((char *)point_array)[3*(I) * felem_size])
79# define Y(I) (&((char *)point_array)[(3*(I) + 1) * felem_size])
80# define Z(I) (&((char *)point_array)[(3*(I) + 2) * felem_size])
81
82 if (!felem_is_zero(Z(0)))
83 felem_assign(tmp_felem(0), Z(0));
84 else
85 felem_one(tmp_felem(0));
86 for (i = 1; i < (int)num; i++) {
87 if (!felem_is_zero(Z(i)))
88 felem_mul(tmp_felem(i), tmp_felem(i - 1), Z(i));
89 else
90 felem_assign(tmp_felem(i), tmp_felem(i - 1));
91 }
92 /*
93 * Now each tmp_felem(i) is the product of Z(0) .. Z(i), skipping any
94 * zero-valued factors: if Z(i) = 0, we essentially pretend that Z(i) = 1
95 */
96
97 felem_inv(tmp_felem(num - 1), tmp_felem(num - 1));
98 for (i = num - 1; i >= 0; i--) {
99 if (i > 0)
100 /*
101 * tmp_felem(i-1) is the product of Z(0) .. Z(i-1), tmp_felem(i)
102 * is the inverse of the product of Z(0) .. Z(i)
103 */
104 /* 1/Z(i) */
105 felem_mul(tmp_felem(num), tmp_felem(i - 1), tmp_felem(i));
106 else
107 felem_assign(tmp_felem(num), tmp_felem(0)); /* 1/Z(0) */
108
109 if (!felem_is_zero(Z(i))) {
110 if (i > 0)
111 /*
112 * For next iteration, replace tmp_felem(i-1) by its inverse
113 */
114 felem_mul(tmp_felem(i - 1), tmp_felem(i), Z(i));
115
116 /*
117 * Convert point (X, Y, Z) into affine form (X/(Z^2), Y/(Z^3), 1)
118 */
119 felem_square(Z(i), tmp_felem(num)); /* 1/(Z^2) */
120 felem_mul(X(i), X(i), Z(i)); /* X/(Z^2) */
121 felem_mul(Z(i), Z(i), tmp_felem(num)); /* 1/(Z^3) */
122 felem_mul(Y(i), Y(i), Z(i)); /* Y/(Z^3) */
123 felem_contract(X(i), X(i));
124 felem_contract(Y(i), Y(i));
125 felem_one(Z(i));
126 } else {
127 if (i > 0)
128 /*
129 * For next iteration, replace tmp_felem(i-1) by its inverse
130 */
131 felem_assign(tmp_felem(i - 1), tmp_felem(i));
132 }
133 }
134}
135
136/*-
137 * This function looks at 5+1 scalar bits (5 current, 1 adjacent less
138 * significant bit), and recodes them into a signed digit for use in fast point
139 * multiplication: the use of signed rather than unsigned digits means that
140 * fewer points need to be precomputed, given that point inversion is easy
141 * (a precomputed point dP makes -dP available as well).
142 *
143 * BACKGROUND:
144 *
145 * Signed digits for multiplication were introduced by Booth ("A signed binary
146 * multiplication technique", Quart. Journ. Mech. and Applied Math., vol. IV,
147 * pt. 2 (1951), pp. 236-240), in that case for multiplication of integers.
148 * Booth's original encoding did not generally improve the density of nonzero
149 * digits over the binary representation, and was merely meant to simplify the
150 * handling of signed factors given in two's complement; but it has since been
151 * shown to be the basis of various signed-digit representations that do have
152 * further advantages, including the wNAF, using the following general approach:
153 *
154 * (1) Given a binary representation
155 *
156 * b_k ... b_2 b_1 b_0,
157 *
158 * of a nonnegative integer (b_k in {0, 1}), rewrite it in digits 0, 1, -1
159 * by using bit-wise subtraction as follows:
160 *
161 * b_k b_(k-1) ... b_2 b_1 b_0
162 * - b_k ... b_3 b_2 b_1 b_0
163 * -----------------------------------------
164 * s_(k+1) s_k ... s_3 s_2 s_1 s_0
165 *
166 * A left-shift followed by subtraction of the original value yields a new
167 * representation of the same value, using signed bits s_i = b_(i-1) - b_i.
168 * This representation from Booth's paper has since appeared in the
169 * literature under a variety of different names including "reversed binary
170 * form", "alternating greedy expansion", "mutual opposite form", and
171 * "sign-alternating {+-1}-representation".
172 *
173 * An interesting property is that among the nonzero bits, values 1 and -1
174 * strictly alternate.
175 *
176 * (2) Various window schemes can be applied to the Booth representation of
177 * integers: for example, right-to-left sliding windows yield the wNAF
178 * (a signed-digit encoding independently discovered by various researchers
179 * in the 1990s), and left-to-right sliding windows yield a left-to-right
180 * equivalent of the wNAF (independently discovered by various researchers
181 * around 2004).
182 *
183 * To prevent leaking information through side channels in point multiplication,
184 * we need to recode the given integer into a regular pattern: sliding windows
185 * as in wNAFs won't do, we need their fixed-window equivalent -- which is a few
186 * decades older: we'll be using the so-called "modified Booth encoding" due to
187 * MacSorley ("High-speed arithmetic in binary computers", Proc. IRE, vol. 49
188 * (1961), pp. 67-91), in a radix-2^5 setting. That is, we always combine five
189 * signed bits into a signed digit:
190 *
191 * s_(5j + 4) s_(5j + 3) s_(5j + 2) s_(5j + 1) s_(5j)
192 *
193 * The sign-alternating property implies that the resulting digit values are
194 * integers from -16 to 16.
195 *
196 * Of course, we don't actually need to compute the signed digits s_i as an
197 * intermediate step (that's just a nice way to see how this scheme relates
198 * to the wNAF): a direct computation obtains the recoded digit from the
199 * six bits b_(5j + 4) ... b_(5j - 1).
200 *
201 * This function takes those six bits as an integer (0 .. 63), writing the
202 * recoded digit to *sign (0 for positive, 1 for negative) and *digit (absolute
203 * value, in the range 0 .. 16). Note that this integer essentially provides
204 * the input bits "shifted to the left" by one position: for example, the input
205 * to compute the least significant recoded digit, given that there's no bit
206 * b_-1, has to be b_4 b_3 b_2 b_1 b_0 0.
207 *
208 */
209void ec_GFp_nistp_recode_scalar_bits(unsigned char *sign,
210 unsigned char *digit, unsigned char in)
211{
212 unsigned char s, d;
213
214 s = ~((in >> 5) - 1); /* sets all bits to MSB(in), 'in' seen as
215 * 6-bit value */
216 d = (1 << 6) - in - 1;
217 d = (d & s) | (in & ~s);
218 d = (d >> 1) + (d & 1);
219
220 *sign = s & 1;
221 *digit = d;
222}
223#endif
224