ring/crypto/ec/ecp_nistz.h

/* 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;
}


#if defined(__cplusplus)
}
#endif

#endif /* OPENSSL_HEADER_EC_ECP_NISTZ_H */