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. */
ec_GFp_nistp_recode_scalar_bits(uint8_t * sign,uint8_t * digit,uint8_t in)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