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