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