blob: 5bd97a4071e03ea774696d02ffbbbb9253c46a13 [file] [log] [blame]
/* Copyright (c) 2015, Google Inc.
*
* Permission to use, copy, modify, and/or distribute this software for any
* purpose with or without fee is hereby granted, provided that the above
* copyright notice and this permission notice appear in all copies.
*
* THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
* WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
* MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY
* SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
* WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN ACTION
* OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF OR IN
* CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE. */
#ifndef OPENSSL_HEADER_EC_ECP_NISTZ_H
#define OPENSSL_HEADER_EC_ECP_NISTZ_H
#include <GFp/base.h>
#include "../../limbs/limbs.h"
#if defined(__GNUC__)
#pragma GCC diagnostic push
#pragma GCC diagnostic ignored "-Wconversion"
#pragma GCC diagnostic ignored "-Wsign-conversion"
#endif
// This function looks at `w + 1` scalar bits (`w` current, 1 adjacent less
// significant bit), and recodes them into a signed digit for use in fast point
// multiplication: the use of signed rather than unsigned digits means that
// fewer points need to be precomputed, given that point inversion is easy (a
// precomputed point dP makes -dP available as well).
//
// BACKGROUND:
//
// Signed digits for multiplication were introduced by Booth ("A signed binary
// multiplication technique", Quart. Journ. Mech. and Applied Math., vol. IV,
// pt. 2 (1951), pp. 236-240), in that case for multiplication of integers.
// Booth's original encoding did not generally improve the density of nonzero
// digits over the binary representation, and was merely meant to simplify the
// handling of signed factors given in two's complement; but it has since been
// shown to be the basis of various signed-digit representations that do have
// further advantages, including the wNAF, using the following general
// approach:
//
// (1) Given a binary representation
//
// b_k ... b_2 b_1 b_0,
//
// of a nonnegative integer (b_k in {0, 1}), rewrite it in digits 0, 1, -1
// by using bit-wise subtraction as follows:
//
// b_k b_(k-1) ... b_2 b_1 b_0
// - b_k ... b_3 b_2 b_1 b_0
// -------------------------------------
// s_k b_(k-1) ... s_3 s_2 s_1 s_0
//
// A left-shift followed by subtraction of the original value yields a new
// representation of the same value, using signed bits s_i = b_(i+1) - b_i.
// This representation from Booth's paper has since appeared in the
// literature under a variety of different names including "reversed binary
// form", "alternating greedy expansion", "mutual opposite form", and
// "sign-alternating {+-1}-representation".
//
// An interesting property is that among the nonzero bits, values 1 and -1
// strictly alternate.
//
// (2) Various window schemes can be applied to the Booth representation of
// integers: for example, right-to-left sliding windows yield the wNAF
// (a signed-digit encoding independently discovered by various researchers
// in the 1990s), and left-to-right sliding windows yield a left-to-right
// equivalent of the wNAF (independently discovered by various researchers
// around 2004).
//
// To prevent leaking information through side channels in point multiplication,
// we need to recode the given integer into a regular pattern: sliding windows
// as in wNAFs won't do, we need their fixed-window equivalent -- which is a few
// decades older: we'll be using the so-called "modified Booth encoding" due to
// MacSorley ("High-speed arithmetic in binary computers", Proc. IRE, vol. 49
// (1961), pp. 67-91), in a radix-2**w setting. That is, we always combine `w`
// signed bits into a signed digit, e.g. (for `w == 5`):
//
// s_(4j + 4) s_(4j + 3) s_(4j + 2) s_(4j + 1) s_(4j)
//
// The sign-alternating property implies that the resulting digit values are
// integers from `-2**(w-1)` to `2**(w-1)`, e.g. -16 to 16 for `w == 5`.
//
// Of course, we don't actually need to compute the signed digits s_i as an
// intermediate step (that's just a nice way to see how this scheme relates
// to the wNAF): a direct computation obtains the recoded digit from the
// six bits b_(4j + 4) ... b_(4j - 1).
//
// This function takes those `w` bits as an integer, writing the recoded digit
// to |*is_negative| (a mask for `constant_time_select_s`) and |*digit|
// (absolute value, in the range 0 .. 2**(w-1). Note that this integer
// essentially provides the input bits "shifted to the left" by one position.
// For example, the input to compute the least significant recoded digit, given
// that there's no bit b_-1, has to be b_4 b_3 b_2 b_1 b_0 0.
static inline void booth_recode(Limb *is_negative, unsigned *digit,
unsigned in, unsigned w) {
assert(w >= 2);
assert(w <= 7);
// Set all bits of `s` to MSB(in), similar to |constant_time_msb_s|,
// but 'in' seen as (`w+1`)-bit value.
Limb s = ~((in >> w) - 1);
unsigned d;
d = (1 << (w + 1)) - in - 1;
d = (d & s) | (in & ~s);
d = (d >> 1) + (d & 1);
*is_negative = constant_time_is_nonzero_w(s & 1);
*digit = d;
}
#if defined(__GNUC__)
#pragma GCC diagnostic pop
#endif
void gfp_little_endian_bytes_from_scalar(uint8_t str[], size_t str_len,
const Limb scalar[],
size_t num_limbs);
#endif // OPENSSL_HEADER_EC_ECP_NISTZ_H