/* 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 <openssl/base.h>
#include <assert.h>
#include <openssl/bn.h>
#include <openssl/type_check.h>
#include "../internal.h"
#if defined(__cplusplus)
extern "C" {
#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_size_t`) 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. */
OPENSSL_COMPILE_ASSERT(sizeof(size_t) == sizeof(BN_ULONG),
size_t_and_bn_ulong_are_different_sizes);
static inline void booth_recode(BN_ULONG *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_size_t|,
* but 'in' seen as (`w+1`)-bit value. */
BN_ULONG 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_size_t(s & 1);
*digit = d;
}
void gfp_little_endian_bytes_from_scalar(uint8_t str[], size_t str_len,
const BN_ULONG scalar[],
size_t num_limbs);
#if defined(__cplusplus)
}
#endif
#endif /* OPENSSL_HEADER_EC_ECP_NISTZ_H */