Adam Langley | e9ada86 | 2015-05-11 17:20:37 -0700 | [diff] [blame] | 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 | |
Adam Langley | e9ada86 | 2015-05-11 17:20:37 -0700 | [diff] [blame] | 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 */ |