/* crypto/ec/ecp_nistputil.c */

/*

 * Written by Bodo Moeller for the OpenSSL project.

 */

/* 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 <openssl/opensslconf.h>

#ifndef OPENSSL_NO_EC_NISTP_64_GCC_128

/*

 * Common utility functions for ecp_nistp224.c, ecp_nistp256.c, ecp_nistp521.c.

 */

# include <stddef.h>

# include "ec_lcl.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(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;

}

#else

static void *dummy = &dummy;

#endif