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- /* 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 */
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