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. 92void 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