1 /* Copyright (c) 2015, Google Inc. 2 * 3 * Permission to use, copy, modify, and/or distribute this software for any 4 * purpose with or without fee is hereby granted, provided that the above 5 * copyright notice and this permission notice appear in all copies. 6 * 7 * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES 8 * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF 9 * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY 10 * SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES 11 * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN ACTION 12 * OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF OR IN 13 * CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE. */ 14 15 #include <openssl/base.h> 16 17 18 #if defined(OPENSSL_64_BIT) && !defined(OPENSSL_WINDOWS) 19 20 #include <openssl/ec.h> 21 22 #include "internal.h" 23 24 /* Convert an array of points into affine coordinates. (If the point at 25 * infinity is found (Z = 0), it remains unchanged.) This function is 26 * essentially an equivalent to EC_POINTs_make_affine(), but works with the 27 * internal representation of points as used by ecp_nistp###.c rather than 28 * with (BIGNUM-based) EC_POINT data structures. point_array is the 29 * input/output buffer ('num' points in projective form, i.e. three 30 * coordinates each), based on an internal representation of field elements 31 * of size 'felem_size'. tmp_felems needs to point to a temporary array of 32 * 'num'+1 field elements for storage of intermediate values. */ 33 void ec_GFp_nistp_points_make_affine_internal( 34 size_t num, void *point_array, size_t felem_size, void *tmp_felems, 35 void (*felem_one)(void *out), int (*felem_is_zero)(const void *in), 36 void (*felem_assign)(void *out, const void *in), 37 void (*felem_square)(void *out, const void *in), 38 void (*felem_mul)(void *out, const void *in1, const void *in2), 39 void (*felem_inv)(void *out, const void *in), 40 void (*felem_contract)(void *out, const void *in)) { 41 int i = 0; 42 43 #define tmp_felem(I) (&((char *)tmp_felems)[(I)*felem_size]) 44 #define X(I) (&((char *)point_array)[3 * (I)*felem_size]) 45 #define Y(I) (&((char *)point_array)[(3 * (I) + 1) * felem_size]) 46 #define Z(I) (&((char *)point_array)[(3 * (I) + 2) * felem_size]) 47 48 if (!felem_is_zero(Z(0))) { 49 felem_assign(tmp_felem(0), Z(0)); 50 } else { 51 felem_one(tmp_felem(0)); 52 } 53 54 for (i = 1; i < (int)num; i++) { 55 if (!felem_is_zero(Z(i))) { 56 felem_mul(tmp_felem(i), tmp_felem(i - 1), Z(i)); 57 } else { 58 felem_assign(tmp_felem(i), tmp_felem(i - 1)); 59 } 60 } 61 /* Now each tmp_felem(i) is the product of Z(0) .. Z(i), skipping any 62 * zero-valued factors: if Z(i) = 0, we essentially pretend that Z(i) = 1. */ 63 64 felem_inv(tmp_felem(num - 1), tmp_felem(num - 1)); 65 for (i = num - 1; i >= 0; i--) { 66 if (i > 0) { 67 /* tmp_felem(i-1) is the product of Z(0) .. Z(i-1), tmp_felem(i) 68 * is the inverse of the product of Z(0) .. Z(i). */ 69 /* 1/Z(i) */ 70 felem_mul(tmp_felem(num), tmp_felem(i - 1), tmp_felem(i)); 71 } else { 72 felem_assign(tmp_felem(num), tmp_felem(0)); /* 1/Z(0) */ 73 } 74 75 if (!felem_is_zero(Z(i))) { 76 if (i > 0) { 77 /* For next iteration, replace tmp_felem(i-1) by its inverse. */ 78 felem_mul(tmp_felem(i - 1), tmp_felem(i), Z(i)); 79 } 80 81 /* Convert point (X, Y, Z) into affine form (X/(Z^2), Y/(Z^3), 1). */ 82 felem_square(Z(i), tmp_felem(num)); /* 1/(Z^2) */ 83 felem_mul(X(i), X(i), Z(i)); /* X/(Z^2) */ 84 felem_mul(Z(i), Z(i), tmp_felem(num)); /* 1/(Z^3) */ 85 felem_mul(Y(i), Y(i), Z(i)); /* Y/(Z^3) */ 86 felem_contract(X(i), X(i)); 87 felem_contract(Y(i), Y(i)); 88 felem_one(Z(i)); 89 } else { 90 if (i > 0) { 91 /* For next iteration, replace tmp_felem(i-1) by its inverse. */ 92 felem_assign(tmp_felem(i - 1), tmp_felem(i)); 93 } 94 } 95 } 96 } 97 98 /* This function looks at 5+1 scalar bits (5 current, 1 adjacent less 99 * significant bit), and recodes them into a signed digit for use in fast point 100 * multiplication: the use of signed rather than unsigned digits means that 101 * fewer points need to be precomputed, given that point inversion is easy (a 102 * precomputed point dP makes -dP available as well). 103 * 104 * BACKGROUND: 105 * 106 * Signed digits for multiplication were introduced by Booth ("A signed binary 107 * multiplication technique", Quart. Journ. Mech. and Applied Math., vol. IV, 108 * pt. 2 (1951), pp. 236-240), in that case for multiplication of integers. 109 * Booth's original encoding did not generally improve the density of nonzero 110 * digits over the binary representation, and was merely meant to simplify the 111 * handling of signed factors given in two's complement; but it has since been 112 * shown to be the basis of various signed-digit representations that do have 113 * further advantages, including the wNAF, using the following general 114 * approach: 115 * 116 * (1) Given a binary representation 117 * 118 * b_k ... b_2 b_1 b_0, 119 * 120 * of a nonnegative integer (b_k in {0, 1}), rewrite it in digits 0, 1, -1 121 * by using bit-wise subtraction as follows: 122 * 123 * b_k b_(k-1) ... b_2 b_1 b_0 124 * - b_k ... b_3 b_2 b_1 b_0 125 * ------------------------------------- 126 * s_k b_(k-1) ... s_3 s_2 s_1 s_0 127 * 128 * A left-shift followed by subtraction of the original value yields a new 129 * representation of the same value, using signed bits s_i = b_(i+1) - b_i. 130 * This representation from Booth's paper has since appeared in the 131 * literature under a variety of different names including "reversed binary 132 * form", "alternating greedy expansion", "mutual opposite form", and 133 * "sign-alternating {+-1}-representation". 134 * 135 * An interesting property is that among the nonzero bits, values 1 and -1 136 * strictly alternate. 137 * 138 * (2) Various window schemes can be applied to the Booth representation of 139 * integers: for example, right-to-left sliding windows yield the wNAF 140 * (a signed-digit encoding independently discovered by various researchers 141 * in the 1990s), and left-to-right sliding windows yield a left-to-right 142 * equivalent of the wNAF (independently discovered by various researchers 143 * around 2004). 144 * 145 * To prevent leaking information through side channels in point multiplication, 146 * we need to recode the given integer into a regular pattern: sliding windows 147 * as in wNAFs won't do, we need their fixed-window equivalent -- which is a few 148 * decades older: we'll be using the so-called "modified Booth encoding" due to 149 * MacSorley ("High-speed arithmetic in binary computers", Proc. IRE, vol. 49 150 * (1961), pp. 67-91), in a radix-2^5 setting. That is, we always combine five 151 * signed bits into a signed digit: 152 * 153 * s_(4j + 4) s_(4j + 3) s_(4j + 2) s_(4j + 1) s_(4j) 154 * 155 * The sign-alternating property implies that the resulting digit values are 156 * integers from -16 to 16. 157 * 158 * Of course, we don't actually need to compute the signed digits s_i as an 159 * intermediate step (that's just a nice way to see how this scheme relates 160 * to the wNAF): a direct computation obtains the recoded digit from the 161 * six bits b_(4j + 4) ... b_(4j - 1). 162 * 163 * This function takes those five bits as an integer (0 .. 63), writing the 164 * recoded digit to *sign (0 for positive, 1 for negative) and *digit (absolute 165 * value, in the range 0 .. 8). Note that this integer essentially provides the 166 * input bits "shifted to the left" by one position: for example, the input to 167 * compute the least significant recoded digit, given that there's no bit b_-1, 168 * has to be b_4 b_3 b_2 b_1 b_0 0. */ 169 void ec_GFp_nistp_recode_scalar_bits(uint8_t *sign, uint8_t *digit, 170 uint8_t in) { 171 uint8_t s, d; 172 173 s = ~((in >> 5) - 1); /* sets all bits to MSB(in), 'in' seen as 174 * 6-bit value */ 175 d = (1 << 6) - in - 1; 176 d = (d & s) | (in & ~s); 177 d = (d >> 1) + (d & 1); 178 179 *sign = s & 1; 180 *digit = d; 181 } 182 183 #endif /* 64_BIT && !WINDOWS */ 184
