[331] | 1 | /*
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| 2 | * Copyright 2011-2016 The OpenSSL Project Authors. All Rights Reserved.
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| 3 | *
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| 4 | * Licensed under the OpenSSL license (the "License"). You may not use
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| 5 | * this file except in compliance with the License. You can obtain a copy
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| 6 | * in the file LICENSE in the source distribution or at
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| 7 | * https://www.openssl.org/source/license.html
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| 8 | */
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| 9 |
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| 10 | /* Copyright 2011 Google Inc.
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| 11 | *
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| 12 | * Licensed under the Apache License, Version 2.0 (the "License");
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| 13 | *
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| 14 | * you may not use this file except in compliance with the License.
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| 15 | * You may obtain a copy of the License at
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| 16 | *
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| 17 | * http://www.apache.org/licenses/LICENSE-2.0
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| 18 | *
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| 19 | * Unless required by applicable law or agreed to in writing, software
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| 20 | * distributed under the License is distributed on an "AS IS" BASIS,
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| 21 | * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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| 22 | * See the License for the specific language governing permissions and
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| 23 | * limitations under the License.
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| 24 | */
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| 25 |
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| 26 | #include <openssl/opensslconf.h>
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| 27 | #ifdef OPENSSL_NO_EC_NISTP_64_GCC_128
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| 28 | NON_EMPTY_TRANSLATION_UNIT
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| 29 | #else
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| 30 |
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| 31 | /*
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| 32 | * Common utility functions for ecp_nistp224.c, ecp_nistp256.c, ecp_nistp521.c.
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| 33 | */
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| 34 |
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| 35 | # include <stddef.h>
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| 36 | # include "ec_lcl.h"
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| 37 |
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| 38 | /*
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| 39 | * Convert an array of points into affine coordinates. (If the point at
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| 40 | * infinity is found (Z = 0), it remains unchanged.) This function is
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| 41 | * essentially an equivalent to EC_POINTs_make_affine(), but works with the
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| 42 | * internal representation of points as used by ecp_nistp###.c rather than
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| 43 | * with (BIGNUM-based) EC_POINT data structures. point_array is the
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| 44 | * input/output buffer ('num' points in projective form, i.e. three
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| 45 | * coordinates each), based on an internal representation of field elements
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| 46 | * of size 'felem_size'. tmp_felems needs to point to a temporary array of
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| 47 | * 'num'+1 field elements for storage of intermediate values.
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| 48 | */
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| 49 | void ec_GFp_nistp_points_make_affine_internal(size_t num, void *point_array,
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| 50 | size_t felem_size,
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| 51 | void *tmp_felems,
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| 52 | void (*felem_one) (void *out),
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| 53 | int (*felem_is_zero) (const void
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| 54 | *in),
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| 55 | void (*felem_assign) (void *out,
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| 56 | const void
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| 57 | *in),
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| 58 | void (*felem_square) (void *out,
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| 59 | const void
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| 60 | *in),
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| 61 | void (*felem_mul) (void *out,
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| 62 | const void
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| 63 | *in1,
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| 64 | const void
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| 65 | *in2),
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| 66 | void (*felem_inv) (void *out,
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| 67 | const void
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| 68 | *in),
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| 69 | void (*felem_contract) (void
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| 70 | *out,
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| 71 | const
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| 72 | void
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| 73 | *in))
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| 74 | {
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| 75 | int i = 0;
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| 76 |
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| 77 | # define tmp_felem(I) (&((char *)tmp_felems)[(I) * felem_size])
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| 78 | # define X(I) (&((char *)point_array)[3*(I) * felem_size])
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| 79 | # define Y(I) (&((char *)point_array)[(3*(I) + 1) * felem_size])
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| 80 | # define Z(I) (&((char *)point_array)[(3*(I) + 2) * felem_size])
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| 81 |
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| 82 | if (!felem_is_zero(Z(0)))
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| 83 | felem_assign(tmp_felem(0), Z(0));
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| 84 | else
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| 85 | felem_one(tmp_felem(0));
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| 86 | for (i = 1; i < (int)num; i++) {
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| 87 | if (!felem_is_zero(Z(i)))
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| 88 | felem_mul(tmp_felem(i), tmp_felem(i - 1), Z(i));
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| 89 | else
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| 90 | felem_assign(tmp_felem(i), tmp_felem(i - 1));
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| 91 | }
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| 92 | /*
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| 93 | * Now each tmp_felem(i) is the product of Z(0) .. Z(i), skipping any
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| 94 | * zero-valued factors: if Z(i) = 0, we essentially pretend that Z(i) = 1
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| 95 | */
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| 96 |
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| 97 | felem_inv(tmp_felem(num - 1), tmp_felem(num - 1));
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| 98 | for (i = num - 1; i >= 0; i--) {
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| 99 | if (i > 0)
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| 100 | /*
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| 101 | * tmp_felem(i-1) is the product of Z(0) .. Z(i-1), tmp_felem(i)
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| 102 | * is the inverse of the product of Z(0) .. Z(i)
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| 103 | */
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| 104 | /* 1/Z(i) */
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| 105 | felem_mul(tmp_felem(num), tmp_felem(i - 1), tmp_felem(i));
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| 106 | else
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| 107 | felem_assign(tmp_felem(num), tmp_felem(0)); /* 1/Z(0) */
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| 108 |
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| 109 | if (!felem_is_zero(Z(i))) {
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| 110 | if (i > 0)
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| 111 | /*
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| 112 | * For next iteration, replace tmp_felem(i-1) by its inverse
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| 113 | */
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| 114 | felem_mul(tmp_felem(i - 1), tmp_felem(i), Z(i));
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| 115 |
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| 116 | /*
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| 117 | * Convert point (X, Y, Z) into affine form (X/(Z^2), Y/(Z^3), 1)
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| 118 | */
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| 119 | felem_square(Z(i), tmp_felem(num)); /* 1/(Z^2) */
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| 120 | felem_mul(X(i), X(i), Z(i)); /* X/(Z^2) */
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| 121 | felem_mul(Z(i), Z(i), tmp_felem(num)); /* 1/(Z^3) */
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| 122 | felem_mul(Y(i), Y(i), Z(i)); /* Y/(Z^3) */
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| 123 | felem_contract(X(i), X(i));
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| 124 | felem_contract(Y(i), Y(i));
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| 125 | felem_one(Z(i));
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| 126 | } else {
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| 127 | if (i > 0)
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| 128 | /*
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| 129 | * For next iteration, replace tmp_felem(i-1) by its inverse
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| 130 | */
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| 131 | felem_assign(tmp_felem(i - 1), tmp_felem(i));
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| 132 | }
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| 133 | }
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| 134 | }
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| 135 |
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| 136 | /*-
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| 137 | * This function looks at 5+1 scalar bits (5 current, 1 adjacent less
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| 138 | * significant bit), and recodes them into a signed digit for use in fast point
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| 139 | * multiplication: the use of signed rather than unsigned digits means that
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| 140 | * fewer points need to be precomputed, given that point inversion is easy
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| 141 | * (a precomputed point dP makes -dP available as well).
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| 142 | *
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| 143 | * BACKGROUND:
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| 144 | *
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| 145 | * Signed digits for multiplication were introduced by Booth ("A signed binary
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| 146 | * multiplication technique", Quart. Journ. Mech. and Applied Math., vol. IV,
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| 147 | * pt. 2 (1951), pp. 236-240), in that case for multiplication of integers.
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| 148 | * Booth's original encoding did not generally improve the density of nonzero
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| 149 | * digits over the binary representation, and was merely meant to simplify the
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| 150 | * handling of signed factors given in two's complement; but it has since been
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| 151 | * shown to be the basis of various signed-digit representations that do have
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| 152 | * further advantages, including the wNAF, using the following general approach:
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| 153 | *
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| 154 | * (1) Given a binary representation
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| 155 | *
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| 156 | * b_k ... b_2 b_1 b_0,
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| 157 | *
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| 158 | * of a nonnegative integer (b_k in {0, 1}), rewrite it in digits 0, 1, -1
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| 159 | * by using bit-wise subtraction as follows:
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| 160 | *
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| 161 | * b_k b_(k-1) ... b_2 b_1 b_0
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| 162 | * - b_k ... b_3 b_2 b_1 b_0
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| 163 | * -------------------------------------
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| 164 | * s_k b_(k-1) ... s_3 s_2 s_1 s_0
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| 165 | *
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| 166 | * A left-shift followed by subtraction of the original value yields a new
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| 167 | * representation of the same value, using signed bits s_i = b_(i+1) - b_i.
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| 168 | * This representation from Booth's paper has since appeared in the
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| 169 | * literature under a variety of different names including "reversed binary
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| 170 | * form", "alternating greedy expansion", "mutual opposite form", and
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| 171 | * "sign-alternating {+-1}-representation".
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| 172 | *
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| 173 | * An interesting property is that among the nonzero bits, values 1 and -1
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| 174 | * strictly alternate.
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| 175 | *
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| 176 | * (2) Various window schemes can be applied to the Booth representation of
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| 177 | * integers: for example, right-to-left sliding windows yield the wNAF
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| 178 | * (a signed-digit encoding independently discovered by various researchers
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| 179 | * in the 1990s), and left-to-right sliding windows yield a left-to-right
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| 180 | * equivalent of the wNAF (independently discovered by various researchers
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| 181 | * around 2004).
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| 182 | *
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| 183 | * To prevent leaking information through side channels in point multiplication,
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| 184 | * we need to recode the given integer into a regular pattern: sliding windows
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| 185 | * as in wNAFs won't do, we need their fixed-window equivalent -- which is a few
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| 186 | * decades older: we'll be using the so-called "modified Booth encoding" due to
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| 187 | * MacSorley ("High-speed arithmetic in binary computers", Proc. IRE, vol. 49
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| 188 | * (1961), pp. 67-91), in a radix-2^5 setting. That is, we always combine five
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| 189 | * signed bits into a signed digit:
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| 190 | *
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| 191 | * s_(4j + 4) s_(4j + 3) s_(4j + 2) s_(4j + 1) s_(4j)
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| 192 | *
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| 193 | * The sign-alternating property implies that the resulting digit values are
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| 194 | * integers from -16 to 16.
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| 195 | *
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| 196 | * Of course, we don't actually need to compute the signed digits s_i as an
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| 197 | * intermediate step (that's just a nice way to see how this scheme relates
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| 198 | * to the wNAF): a direct computation obtains the recoded digit from the
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| 199 | * six bits b_(4j + 4) ... b_(4j - 1).
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| 200 | *
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| 201 | * This function takes those five bits as an integer (0 .. 63), writing the
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| 202 | * recoded digit to *sign (0 for positive, 1 for negative) and *digit (absolute
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| 203 | * value, in the range 0 .. 8). Note that this integer essentially provides the
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| 204 | * input bits "shifted to the left" by one position: for example, the input to
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| 205 | * compute the least significant recoded digit, given that there's no bit b_-1,
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| 206 | * has to be b_4 b_3 b_2 b_1 b_0 0.
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| 207 | *
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| 208 | */
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| 209 | void ec_GFp_nistp_recode_scalar_bits(unsigned char *sign,
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| 210 | unsigned char *digit, unsigned char in)
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| 211 | {
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| 212 | unsigned char s, d;
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| 213 |
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| 214 | s = ~((in >> 5) - 1); /* sets all bits to MSB(in), 'in' seen as
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| 215 | * 6-bit value */
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| 216 | d = (1 << 6) - in - 1;
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| 217 | d = (d & s) | (in & ~s);
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| 218 | d = (d >> 1) + (d & 1);
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| 219 |
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| 220 | *sign = s & 1;
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| 221 | *digit = d;
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| 222 | }
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| 223 | #endif
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