/*
* Copyright 2011-2019 The OpenSSL Project Authors. All Rights Reserved.
*
* Licensed under the OpenSSL license (the "License"). You may not use
* this file except in compliance with the License. You can obtain a copy
* in the file LICENSE in the source distribution or at
* https://www.openssl.org/source/license.html
*/
/* Copyright 2011 Google Inc.
*
* Licensed under the Apache License, Version 2.0 (the "License");
*
* you may not use this file except in compliance with the License.
* You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
#include
#ifdef OPENSSL_NO_EC_NISTP_64_GCC_128
NON_EMPTY_TRANSLATION_UNIT
#else
/*
* Common utility functions for ecp_nistp224.c, ecp_nistp256.c, ecp_nistp521.c.
*/
# include
# include "ec_local.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+1) s_k ... 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_(5j + 4) s_(5j + 3) s_(5j + 2) s_(5j + 1) s_(5j)
*
* 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_(5j + 4) ... b_(5j - 1).
*
* This function takes those six 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 .. 16). 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(unsigned char *sign,
unsigned char *digit, unsigned char in)
{
unsigned char 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