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1 | /* |
2 | * Written by Bodo Moeller for the OpenSSL project. | |
3 | */ | |
4 | /* Copyright 2011 Google Inc. | |
5 | * | |
6 | * Licensed under the Apache License, Version 2.0 (the "License"); | |
7 | * | |
8 | * you may not use this file except in compliance with the License. | |
9 | * You may obtain a copy of the License at | |
10 | * | |
11 | * http://www.apache.org/licenses/LICENSE-2.0 | |
12 | * | |
13 | * Unless required by applicable law or agreed to in writing, software | |
14 | * distributed under the License is distributed on an "AS IS" BASIS, | |
15 | * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. | |
16 | * See the License for the specific language governing permissions and | |
17 | * limitations under the License. | |
18 | */ | |
19 | ||
e0d6132b BM |
20 | #include <openssl/opensslconf.h> |
21 | #ifndef OPENSSL_NO_EC_NISTP_64_GCC_128 | |
3e00b4c9 BM |
22 | |
23 | /* | |
24 | * Common utility functions for ecp_nistp224.c, ecp_nistp256.c, ecp_nistp521.c. | |
25 | */ | |
26 | ||
0f113f3e MC |
27 | # include <stddef.h> |
28 | # include "ec_lcl.h" | |
3e00b4c9 | 29 | |
0f113f3e MC |
30 | /* |
31 | * Convert an array of points into affine coordinates. (If the point at | |
32 | * infinity is found (Z = 0), it remains unchanged.) This function is | |
33 | * essentially an equivalent to EC_POINTs_make_affine(), but works with the | |
34 | * internal representation of points as used by ecp_nistp###.c rather than | |
35 | * with (BIGNUM-based) EC_POINT data structures. point_array is the | |
36 | * input/output buffer ('num' points in projective form, i.e. three | |
37 | * coordinates each), based on an internal representation of field elements | |
38 | * of size 'felem_size'. tmp_felems needs to point to a temporary array of | |
39 | * 'num'+1 field elements for storage of intermediate values. | |
3e00b4c9 BM |
40 | */ |
41 | void ec_GFp_nistp_points_make_affine_internal(size_t num, void *point_array, | |
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42 | size_t felem_size, |
43 | void *tmp_felems, | |
44 | void (*felem_one) (void *out), | |
45 | int (*felem_is_zero) (const void | |
46 | *in), | |
47 | void (*felem_assign) (void *out, | |
48 | const void | |
49 | *in), | |
50 | void (*felem_square) (void *out, | |
51 | const void | |
52 | *in), | |
53 | void (*felem_mul) (void *out, | |
54 | const void | |
55 | *in1, | |
56 | const void | |
57 | *in2), | |
58 | void (*felem_inv) (void *out, | |
59 | const void | |
60 | *in), | |
61 | void (*felem_contract) (void | |
62 | *out, | |
63 | const | |
64 | void | |
65 | *in)) | |
66 | { | |
67 | int i = 0; | |
3e00b4c9 | 68 | |
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69 | # define tmp_felem(I) (&((char *)tmp_felems)[(I) * felem_size]) |
70 | # define X(I) (&((char *)point_array)[3*(I) * felem_size]) | |
71 | # define Y(I) (&((char *)point_array)[(3*(I) + 1) * felem_size]) | |
72 | # define Z(I) (&((char *)point_array)[(3*(I) + 2) * felem_size]) | |
3e00b4c9 | 73 | |
0f113f3e MC |
74 | if (!felem_is_zero(Z(0))) |
75 | felem_assign(tmp_felem(0), Z(0)); | |
76 | else | |
77 | felem_one(tmp_felem(0)); | |
78 | for (i = 1; i < (int)num; i++) { | |
79 | if (!felem_is_zero(Z(i))) | |
80 | felem_mul(tmp_felem(i), tmp_felem(i - 1), Z(i)); | |
81 | else | |
82 | felem_assign(tmp_felem(i), tmp_felem(i - 1)); | |
83 | } | |
84 | /* | |
85 | * Now each tmp_felem(i) is the product of Z(0) .. Z(i), skipping any | |
86 | * zero-valued factors: if Z(i) = 0, we essentially pretend that Z(i) = 1 | |
87 | */ | |
3e00b4c9 | 88 | |
0f113f3e MC |
89 | felem_inv(tmp_felem(num - 1), tmp_felem(num - 1)); |
90 | for (i = num - 1; i >= 0; i--) { | |
91 | if (i > 0) | |
92 | /* | |
93 | * tmp_felem(i-1) is the product of Z(0) .. Z(i-1), tmp_felem(i) | |
94 | * is the inverse of the product of Z(0) .. Z(i) | |
95 | */ | |
96 | /* 1/Z(i) */ | |
97 | felem_mul(tmp_felem(num), tmp_felem(i - 1), tmp_felem(i)); | |
98 | else | |
99 | felem_assign(tmp_felem(num), tmp_felem(0)); /* 1/Z(0) */ | |
3e00b4c9 | 100 | |
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101 | if (!felem_is_zero(Z(i))) { |
102 | if (i > 0) | |
103 | /* | |
104 | * For next iteration, replace tmp_felem(i-1) by its inverse | |
105 | */ | |
106 | felem_mul(tmp_felem(i - 1), tmp_felem(i), Z(i)); | |
3e00b4c9 | 107 | |
0f113f3e MC |
108 | /* |
109 | * Convert point (X, Y, Z) into affine form (X/(Z^2), Y/(Z^3), 1) | |
110 | */ | |
111 | felem_square(Z(i), tmp_felem(num)); /* 1/(Z^2) */ | |
112 | felem_mul(X(i), X(i), Z(i)); /* X/(Z^2) */ | |
113 | felem_mul(Z(i), Z(i), tmp_felem(num)); /* 1/(Z^3) */ | |
114 | felem_mul(Y(i), Y(i), Z(i)); /* Y/(Z^3) */ | |
115 | felem_contract(X(i), X(i)); | |
116 | felem_contract(Y(i), Y(i)); | |
117 | felem_one(Z(i)); | |
118 | } else { | |
119 | if (i > 0) | |
120 | /* | |
121 | * For next iteration, replace tmp_felem(i-1) by its inverse | |
122 | */ | |
123 | felem_assign(tmp_felem(i - 1), tmp_felem(i)); | |
124 | } | |
125 | } | |
126 | } | |
3e00b4c9 | 127 | |
1d97c843 | 128 | /*- |
3e00b4c9 BM |
129 | * This function looks at 5+1 scalar bits (5 current, 1 adjacent less |
130 | * significant bit), and recodes them into a signed digit for use in fast point | |
131 | * multiplication: the use of signed rather than unsigned digits means that | |
132 | * fewer points need to be precomputed, given that point inversion is easy | |
133 | * (a precomputed point dP makes -dP available as well). | |
134 | * | |
135 | * BACKGROUND: | |
136 | * | |
137 | * Signed digits for multiplication were introduced by Booth ("A signed binary | |
138 | * multiplication technique", Quart. Journ. Mech. and Applied Math., vol. IV, | |
139 | * pt. 2 (1951), pp. 236-240), in that case for multiplication of integers. | |
140 | * Booth's original encoding did not generally improve the density of nonzero | |
141 | * digits over the binary representation, and was merely meant to simplify the | |
142 | * handling of signed factors given in two's complement; but it has since been | |
143 | * shown to be the basis of various signed-digit representations that do have | |
144 | * further advantages, including the wNAF, using the following general approach: | |
145 | * | |
146 | * (1) Given a binary representation | |
147 | * | |
148 | * b_k ... b_2 b_1 b_0, | |
149 | * | |
150 | * of a nonnegative integer (b_k in {0, 1}), rewrite it in digits 0, 1, -1 | |
151 | * by using bit-wise subtraction as follows: | |
152 | * | |
153 | * b_k b_(k-1) ... b_2 b_1 b_0 | |
154 | * - b_k ... b_3 b_2 b_1 b_0 | |
155 | * ------------------------------------- | |
156 | * s_k b_(k-1) ... s_3 s_2 s_1 s_0 | |
157 | * | |
158 | * A left-shift followed by subtraction of the original value yields a new | |
159 | * representation of the same value, using signed bits s_i = b_(i+1) - b_i. | |
160 | * This representation from Booth's paper has since appeared in the | |
161 | * literature under a variety of different names including "reversed binary | |
162 | * form", "alternating greedy expansion", "mutual opposite form", and | |
163 | * "sign-alternating {+-1}-representation". | |
164 | * | |
165 | * An interesting property is that among the nonzero bits, values 1 and -1 | |
166 | * strictly alternate. | |
167 | * | |
168 | * (2) Various window schemes can be applied to the Booth representation of | |
169 | * integers: for example, right-to-left sliding windows yield the wNAF | |
170 | * (a signed-digit encoding independently discovered by various researchers | |
171 | * in the 1990s), and left-to-right sliding windows yield a left-to-right | |
172 | * equivalent of the wNAF (independently discovered by various researchers | |
173 | * around 2004). | |
174 | * | |
175 | * To prevent leaking information through side channels in point multiplication, | |
176 | * we need to recode the given integer into a regular pattern: sliding windows | |
177 | * as in wNAFs won't do, we need their fixed-window equivalent -- which is a few | |
178 | * decades older: we'll be using the so-called "modified Booth encoding" due to | |
179 | * MacSorley ("High-speed arithmetic in binary computers", Proc. IRE, vol. 49 | |
180 | * (1961), pp. 67-91), in a radix-2^5 setting. That is, we always combine five | |
181 | * signed bits into a signed digit: | |
182 | * | |
183 | * s_(4j + 4) s_(4j + 3) s_(4j + 2) s_(4j + 1) s_(4j) | |
184 | * | |
185 | * The sign-alternating property implies that the resulting digit values are | |
186 | * integers from -16 to 16. | |
187 | * | |
188 | * Of course, we don't actually need to compute the signed digits s_i as an | |
189 | * intermediate step (that's just a nice way to see how this scheme relates | |
190 | * to the wNAF): a direct computation obtains the recoded digit from the | |
191 | * six bits b_(4j + 4) ... b_(4j - 1). | |
192 | * | |
193 | * This function takes those five bits as an integer (0 .. 63), writing the | |
194 | * recoded digit to *sign (0 for positive, 1 for negative) and *digit (absolute | |
195 | * value, in the range 0 .. 8). Note that this integer essentially provides the | |
196 | * input bits "shifted to the left" by one position: for example, the input to | |
197 | * compute the least significant recoded digit, given that there's no bit b_-1, | |
198 | * has to be b_4 b_3 b_2 b_1 b_0 0. | |
199 | * | |
200 | */ | |
0f113f3e MC |
201 | void ec_GFp_nistp_recode_scalar_bits(unsigned char *sign, |
202 | unsigned char *digit, unsigned char in) | |
203 | { | |
204 | unsigned char s, d; | |
3e00b4c9 | 205 | |
0f113f3e MC |
206 | s = ~((in >> 5) - 1); /* sets all bits to MSB(in), 'in' seen as |
207 | * 6-bit value */ | |
208 | d = (1 << 6) - in - 1; | |
209 | d = (d & s) | (in & ~s); | |
210 | d = (d >> 1) + (d & 1); | |
3e00b4c9 | 211 | |
0f113f3e MC |
212 | *sign = s & 1; |
213 | *digit = d; | |
214 | } | |
3e00b4c9 | 215 | #else |
0f113f3e | 216 | static void *dummy = &dummy; |
3e00b4c9 | 217 | #endif |