1 /* 2 * Copyright 2011-2019 The OpenSSL Project Authors. All Rights Reserved. 3 * 4 * Licensed under the OpenSSL license (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 28 NON_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 */ 49 void 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 */ 209 void 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