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 /* This function looks at 5+1 scalar bits (5 current, 1 adjacent less 25 * significant bit), and recodes them into a signed digit for use in fast point 26 * multiplication: the use of signed rather than unsigned digits means that 27 * fewer points need to be precomputed, given that point inversion is easy (a 28 * precomputed point dP makes -dP available as well). 29 * 30 * BACKGROUND: 31 * 32 * Signed digits for multiplication were introduced by Booth ("A signed binary 33 * multiplication technique", Quart. Journ. Mech. and Applied Math., vol. IV, 34 * pt. 2 (1951), pp. 236-240), in that case for multiplication of integers. 35 * Booth's original encoding did not generally improve the density of nonzero 36 * digits over the binary representation, and was merely meant to simplify the 37 * handling of signed factors given in two's complement; but it has since been 38 * shown to be the basis of various signed-digit representations that do have 39 * further advantages, including the wNAF, using the following general 40 * approach: 41 * 42 * (1) Given a binary representation 43 * 44 * b_k ... b_2 b_1 b_0, 45 * 46 * of a nonnegative integer (b_k in {0, 1}), rewrite it in digits 0, 1, -1 47 * by using bit-wise subtraction as follows: 48 * 49 * b_k b_(k-1) ... b_2 b_1 b_0 50 * - b_k ... b_3 b_2 b_1 b_0 51 * ------------------------------------- 52 * s_k b_(k-1) ... s_3 s_2 s_1 s_0 53 * 54 * A left-shift followed by subtraction of the original value yields a new 55 * representation of the same value, using signed bits s_i = b_(i+1) - b_i. 56 * This representation from Booth's paper has since appeared in the 57 * literature under a variety of different names including "reversed binary 58 * form", "alternating greedy expansion", "mutual opposite form", and 59 * "sign-alternating {+-1}-representation". 60 * 61 * An interesting property is that among the nonzero bits, values 1 and -1 62 * strictly alternate. 63 * 64 * (2) Various window schemes can be applied to the Booth representation of 65 * integers: for example, right-to-left sliding windows yield the wNAF 66 * (a signed-digit encoding independently discovered by various researchers 67 * in the 1990s), and left-to-right sliding windows yield a left-to-right 68 * equivalent of the wNAF (independently discovered by various researchers 69 * around 2004). 70 * 71 * To prevent leaking information through side channels in point multiplication, 72 * we need to recode the given integer into a regular pattern: sliding windows 73 * as in wNAFs won't do, we need their fixed-window equivalent -- which is a few 74 * decades older: we'll be using the so-called "modified Booth encoding" due to 75 * MacSorley ("High-speed arithmetic in binary computers", Proc. IRE, vol. 49 76 * (1961), pp. 67-91), in a radix-2^5 setting. That is, we always combine five 77 * signed bits into a signed digit: 78 * 79 * s_(4j + 4) s_(4j + 3) s_(4j + 2) s_(4j + 1) s_(4j) 80 * 81 * The sign-alternating property implies that the resulting digit values are 82 * integers from -16 to 16. 83 * 84 * Of course, we don't actually need to compute the signed digits s_i as an 85 * intermediate step (that's just a nice way to see how this scheme relates 86 * to the wNAF): a direct computation obtains the recoded digit from the 87 * six bits b_(4j + 4) ... b_(4j - 1). 88 * 89 * This function takes those five bits as an integer (0 .. 63), writing the 90 * recoded digit to *sign (0 for positive, 1 for negative) and *digit (absolute 91 * value, in the range 0 .. 8). Note that this integer essentially provides the 92 * input bits "shifted to the left" by one position: for example, the input to 93 * compute the least significant recoded digit, given that there's no bit b_-1, 94 * has to be b_4 b_3 b_2 b_1 b_0 0. */ 95 void ec_GFp_nistp_recode_scalar_bits(uint8_t *sign, uint8_t *digit, 96 uint8_t in) { 97 uint8_t s, d; 98 99 s = ~((in >> 5) - 1); /* sets all bits to MSB(in), 'in' seen as 100 * 6-bit value */ 101 d = (1 << 6) - in - 1; 102 d = (d & s) | (in & ~s); 103 d = (d >> 1) + (d & 1); 104 105 *sign = s & 1; 106 *digit = d; 107 } 108 109 #endif /* 64_BIT && !WINDOWS */ 110
