105 lines
4.8 KiB
C
105 lines
4.8 KiB
C
/* Copyright (c) 2015, Google Inc.
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*
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* Permission to use, copy, modify, and/or distribute this software for any
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* purpose with or without fee is hereby granted, provided that the above
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* copyright notice and this permission notice appear in all copies.
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*
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* THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
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* WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
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* MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY
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* SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
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* WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN ACTION
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* OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF OR IN
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* CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE. */
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#include <openssl/base.h>
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#include <openssl/ec.h>
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#include "internal.h"
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// This function looks at 5+1 scalar bits (5 current, 1 adjacent less
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// significant bit), and recodes them into a signed digit for use in fast point
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// multiplication: the use of signed rather than unsigned digits means that
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// fewer points need to be precomputed, given that point inversion is easy (a
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// precomputed point dP makes -dP available as well).
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//
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// BACKGROUND:
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//
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// Signed digits for multiplication were introduced by Booth ("A signed binary
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// multiplication technique", Quart. Journ. Mech. and Applied Math., vol. IV,
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// pt. 2 (1951), pp. 236-240), in that case for multiplication of integers.
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// Booth's original encoding did not generally improve the density of nonzero
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// digits over the binary representation, and was merely meant to simplify the
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// handling of signed factors given in two's complement; but it has since been
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// shown to be the basis of various signed-digit representations that do have
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// further advantages, including the wNAF, using the following general
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// approach:
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//
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// (1) Given a binary representation
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//
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// b_k ... b_2 b_1 b_0,
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//
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// of a nonnegative integer (b_k in {0, 1}), rewrite it in digits 0, 1, -1
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// by using bit-wise subtraction as follows:
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//
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// b_k b_(k-1) ... b_2 b_1 b_0
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// - b_k ... b_3 b_2 b_1 b_0
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// -------------------------------------
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// s_k b_(k-1) ... s_3 s_2 s_1 s_0
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//
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// A left-shift followed by subtraction of the original value yields a new
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// representation of the same value, using signed bits s_i = b_(i+1) - b_i.
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// This representation from Booth's paper has since appeared in the
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// literature under a variety of different names including "reversed binary
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// form", "alternating greedy expansion", "mutual opposite form", and
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// "sign-alternating {+-1}-representation".
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//
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// An interesting property is that among the nonzero bits, values 1 and -1
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// strictly alternate.
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//
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// (2) Various window schemes can be applied to the Booth representation of
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// integers: for example, right-to-left sliding windows yield the wNAF
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// (a signed-digit encoding independently discovered by various researchers
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// in the 1990s), and left-to-right sliding windows yield a left-to-right
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// equivalent of the wNAF (independently discovered by various researchers
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// around 2004).
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//
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// To prevent leaking information through side channels in point multiplication,
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// we need to recode the given integer into a regular pattern: sliding windows
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// as in wNAFs won't do, we need their fixed-window equivalent -- which is a few
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// decades older: we'll be using the so-called "modified Booth encoding" due to
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// MacSorley ("High-speed arithmetic in binary computers", Proc. IRE, vol. 49
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// (1961), pp. 67-91), in a radix-2^5 setting. That is, we always combine five
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// signed bits into a signed digit:
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//
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// s_(4j + 4) s_(4j + 3) s_(4j + 2) s_(4j + 1) s_(4j)
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//
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// The sign-alternating property implies that the resulting digit values are
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// integers from -16 to 16.
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//
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// Of course, we don't actually need to compute the signed digits s_i as an
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// intermediate step (that's just a nice way to see how this scheme relates
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// to the wNAF): a direct computation obtains the recoded digit from the
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// six bits b_(4j + 4) ... b_(4j - 1).
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//
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// This function takes those five bits as an integer (0 .. 63), writing the
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// recoded digit to *sign (0 for positive, 1 for negative) and *digit (absolute
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// value, in the range 0 .. 8). Note that this integer essentially provides the
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// input bits "shifted to the left" by one position: for example, the input to
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// compute the least significant recoded digit, given that there's no bit b_-1,
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// has to be b_4 b_3 b_2 b_1 b_0 0.
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void ec_GFp_nistp_recode_scalar_bits(uint8_t *sign, uint8_t *digit,
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uint8_t in) {
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uint8_t s, d;
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s = ~((in >> 5) - 1); /* sets all bits to MSB(in), 'in' seen as
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* 6-bit value */
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d = (1 << 6) - in - 1;
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d = (d & s) | (in & ~s);
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d = (d >> 1) + (d & 1);
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*sign = s & 1;
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*digit = d;
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}
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