crypto/ec/util-64.c

/* 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. */

#include <openssl/base.h>


#if defined(OPENSSL_64_BIT) && !defined(OPENSSL_WINDOWS)

#include <openssl/ec.h>

#include "internal.h"

/* Convert an array of points into affine coordinates. (If the point at
 * infinity is found (Z = 0), it remains unchanged.) This function is
 * essentially an equivalent to EC_POINTs_make_affine(), but works with the
 * internal representation of points as used by ecp_nistp###.c rather than
 * with (BIGNUM-based) EC_POINT data structures. point_array is the
 * input/output buffer ('num' points in projective form, i.e. three
 * coordinates each), based on an internal representation of field elements
 * of size 'felem_size'. tmp_felems needs to point to a temporary array of
 * 'num'+1 field elements for storage of intermediate values. */
void ec_GFp_nistp_points_make_affine_internal(
    size_t num, void *point_array, size_t felem_size, void *tmp_felems,
    void (*felem_one)(void *out), int (*felem_is_zero)(const void *in),
    void (*felem_assign)(void *out, const void *in),
    void (*felem_square)(void *out, const void *in),
    void (*felem_mul)(void *out, const void *in1, const void *in2),
    void (*felem_inv)(void *out, const void *in),
    void (*felem_contract)(void *out, const void *in)) {
  int i = 0;

#define tmp_felem(I) (&((char *)tmp_felems)[(I)*felem_size])
#define X(I) (&((char *)point_array)[3 * (I)*felem_size])
#define Y(I) (&((char *)point_array)[(3 * (I) + 1) * felem_size])
#define Z(I) (&((char *)point_array)[(3 * (I) + 2) * felem_size])

  if (!felem_is_zero(Z(0))) {
    felem_assign(tmp_felem(0), Z(0));
  } else {
    felem_one(tmp_felem(0));
  }

  for (i = 1; i < (int)num; i++) {
    if (!felem_is_zero(Z(i))) {
      felem_mul(tmp_felem(i), tmp_felem(i - 1), Z(i));
    } else {
      felem_assign(tmp_felem(i), tmp_felem(i - 1));
    }
  }
  /* Now each tmp_felem(i) is the product of Z(0) .. Z(i), skipping any
   * zero-valued factors: if Z(i) = 0, we essentially pretend that Z(i) = 1. */

  felem_inv(tmp_felem(num - 1), tmp_felem(num - 1));
  for (i = num - 1; i >= 0; i--) {
    if (i > 0) {
      /* tmp_felem(i-1) is the product of Z(0) .. Z(i-1), tmp_felem(i)
       * is the inverse of the product of Z(0) .. Z(i). */
      /* 1/Z(i) */
      felem_mul(tmp_felem(num), tmp_felem(i - 1), tmp_felem(i));
    } else {
      felem_assign(tmp_felem(num), tmp_felem(0)); /* 1/Z(0) */
    }

    if (!felem_is_zero(Z(i))) {
      if (i > 0) {
        /* For next iteration, replace tmp_felem(i-1) by its inverse. */
        felem_mul(tmp_felem(i - 1), tmp_felem(i), Z(i));
      }

      /* Convert point (X, Y, Z) into affine form (X/(Z^2), Y/(Z^3), 1). */
      felem_square(Z(i), tmp_felem(num));    /* 1/(Z^2) */
      felem_mul(X(i), X(i), Z(i));           /* X/(Z^2) */
      felem_mul(Z(i), Z(i), tmp_felem(num)); /* 1/(Z^3) */
      felem_mul(Y(i), Y(i), Z(i));           /* Y/(Z^3) */
      felem_contract(X(i), X(i));
      felem_contract(Y(i), Y(i));
      felem_one(Z(i));
    } else {
      if (i > 0) {
        /* For next iteration, replace tmp_felem(i-1) by its inverse. */
        felem_assign(tmp_felem(i - 1), tmp_felem(i));
      }
    }
  }
}

/* This function looks at 5+1 scalar bits (5 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-2^5 setting.  That is, we always combine five
 * signed bits into a signed digit:
 *
 *       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 -16 to 16.
 *
 * 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 five bits as an integer (0 .. 63), writing the
 * recoded digit to *sign (0 for positive, 1 for negative) and *digit (absolute
 * value, in the range 0 .. 8).  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. */
void ec_GFp_nistp_recode_scalar_bits(uint8_t *sign, uint8_t *digit,
                                     uint8_t in) {
  uint8_t s, d;

  s = ~((in >> 5) - 1); /* sets all bits to MSB(in), 'in' seen as
                          * 6-bit value */
  d = (1 << 6) - in - 1;
  d = (d & s) | (in & ~s);
  d = (d >> 1) + (d & 1);

  *sign = s & 1;
  *digit = d;
}

#endif  /* 64_BIT && !WINDOWS */