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 #include <openssl/ec.h> 18 19 #include "internal.h" 20 21 // This function looks at 5+1 scalar bits (5 current, 1 adjacent less 22 // significant bit), and recodes them into a signed digit for use in fast point 23 // multiplication: the use of signed rather than unsigned digits means that 24 // fewer points need to be precomputed, given that point inversion is easy (a 25 // precomputed point dP makes -dP available as well). 26 // 27 // BACKGROUND: 28 // 29 // Signed digits for multiplication were introduced by Booth ("A signed binary 30 // multiplication technique", Quart. Journ. Mech. and Applied Math., vol. IV, 31 // pt. 2 (1951), pp. 236-240), in that case for multiplication of integers. 32 // Booth's original encoding did not generally improve the density of nonzero 33 // digits over the binary representation, and was merely meant to simplify the 34 // handling of signed factors given in two's complement; but it has since been 35 // shown to be the basis of various signed-digit representations that do have 36 // further advantages, including the wNAF, using the following general 37 // approach: 38 // 39 // (1) Given a binary representation 40 // 41 // b_k ... b_2 b_1 b_0, 42 // 43 // of a nonnegative integer (b_k in {0, 1}), rewrite it in digits 0, 1, -1 44 // by using bit-wise subtraction as follows: 45 // 46 // b_k b_(k-1) ... b_2 b_1 b_0 47 // - b_k ... b_3 b_2 b_1 b_0 48 // ------------------------------------- 49 // s_k b_(k-1) ... s_3 s_2 s_1 s_0 50 // 51 // A left-shift followed by subtraction of the original value yields a new 52 // representation of the same value, using signed bits s_i = b_(i+1) - b_i. 53 // This representation from Booth's paper has since appeared in the 54 // literature under a variety of different names including "reversed binary 55 // form", "alternating greedy expansion", "mutual opposite form", and 56 // "sign-alternating {+-1}-representation". 57 // 58 // An interesting property is that among the nonzero bits, values 1 and -1 59 // strictly alternate. 60 // 61 // (2) Various window schemes can be applied to the Booth representation of 62 // integers: for example, right-to-left sliding windows yield the wNAF 63 // (a signed-digit encoding independently discovered by various researchers 64 // in the 1990s), and left-to-right sliding windows yield a left-to-right 65 // equivalent of the wNAF (independently discovered by various researchers 66 // around 2004). 67 // 68 // To prevent leaking information through side channels in point multiplication, 69 // we need to recode the given integer into a regular pattern: sliding windows 70 // as in wNAFs won't do, we need their fixed-window equivalent -- which is a few 71 // decades older: we'll be using the so-called "modified Booth encoding" due to 72 // MacSorley ("High-speed arithmetic in binary computers", Proc. IRE, vol. 49 73 // (1961), pp. 67-91), in a radix-2^5 setting. That is, we always combine five 74 // signed bits into a signed digit: 75 // 76 // s_(4j + 4) s_(4j + 3) s_(4j + 2) s_(4j + 1) s_(4j) 77 // 78 // The sign-alternating property implies that the resulting digit values are 79 // integers from -16 to 16. 80 // 81 // Of course, we don't actually need to compute the signed digits s_i as an 82 // intermediate step (that's just a nice way to see how this scheme relates 83 // to the wNAF): a direct computation obtains the recoded digit from the 84 // six bits b_(4j + 4) ... b_(4j - 1). 85 // 86 // This function takes those five bits as an integer (0 .. 63), writing the 87 // recoded digit to *sign (0 for positive, 1 for negative) and *digit (absolute 88 // value, in the range 0 .. 8). Note that this integer essentially provides the 89 // input bits "shifted to the left" by one position: for example, the input to 90 // compute the least significant recoded digit, given that there's no bit b_-1, 91 // has to be b_4 b_3 b_2 b_1 b_0 0. 92 void ec_GFp_nistp_recode_scalar_bits(uint8_t *sign, uint8_t *digit, 93 uint8_t in) { 94 uint8_t s, d; 95 96 s = ~((in >> 5) - 1); /* sets all bits to MSB(in), 'in' seen as 97 * 6-bit value */ 98 d = (1 << 6) - in - 1; 99 d = (d & s) | (in & ~s); 100 d = (d >> 1) + (d & 1); 101 102 *sign = s & 1; 103 *digit = d; 104 } 105
