/src/openssl31/crypto/ec/ecp_nistp521.c

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/*
 * Copyright 2011-2022 The OpenSSL Project Authors. All Rights Reserved.
 *
 * Licensed under the Apache License 2.0 (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.
 */

/*
 * ECDSA low level APIs are deprecated for public use, but still ok for
 * internal use.
 */
#include "internal/deprecated.h"

/*
 * A 64-bit implementation of the NIST P-521 elliptic curve point multiplication
 *
 * OpenSSL integration was taken from Emilia Kasper's work in ecp_nistp224.c.
 * Otherwise based on Emilia's P224 work, which was inspired by my curve25519
 * work which got its smarts from Daniel J. Bernstein's work on the same.
 */

#include <openssl/e_os2.h>

#include <string.h>
#include <openssl/err.h>
#include "ec_local.h"

#include "internal/numbers.h"

#ifndef INT128_MAX
# error "Your compiler doesn't appear to support 128-bit integer types"
#endif

typedef uint8_t u8;
typedef uint64_t u64;

/*
 * The underlying field. P521 operates over GF(2^521-1). We can serialize an
 * element of this field into 66 bytes where the most significant byte
 * contains only a single bit. We call this an felem_bytearray.
 */

typedef u8 felem_bytearray[66];

/*
 * These are the parameters of P521, taken from FIPS 186-3, section D.1.2.5.
 * These values are big-endian.
 */
static const felem_bytearray nistp521_curve_params[5] = {
    {0x01, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, /* p */
     0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
     0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
     0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
     0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
     0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
     0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
     0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
     0xff, 0xff},
    {0x01, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, /* a = -3 */
     0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
     0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
     0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
     0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
     0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
     0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
     0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
     0xff, 0xfc},
    {0x00, 0x51, 0x95, 0x3e, 0xb9, 0x61, 0x8e, 0x1c, /* b */
     0x9a, 0x1f, 0x92, 0x9a, 0x21, 0xa0, 0xb6, 0x85,
     0x40, 0xee, 0xa2, 0xda, 0x72, 0x5b, 0x99, 0xb3,
     0x15, 0xf3, 0xb8, 0xb4, 0x89, 0x91, 0x8e, 0xf1,
     0x09, 0xe1, 0x56, 0x19, 0x39, 0x51, 0xec, 0x7e,
     0x93, 0x7b, 0x16, 0x52, 0xc0, 0xbd, 0x3b, 0xb1,
     0xbf, 0x07, 0x35, 0x73, 0xdf, 0x88, 0x3d, 0x2c,
     0x34, 0xf1, 0xef, 0x45, 0x1f, 0xd4, 0x6b, 0x50,
     0x3f, 0x00},
    {0x00, 0xc6, 0x85, 0x8e, 0x06, 0xb7, 0x04, 0x04, /* x */
     0xe9, 0xcd, 0x9e, 0x3e, 0xcb, 0x66, 0x23, 0x95,
     0xb4, 0x42, 0x9c, 0x64, 0x81, 0x39, 0x05, 0x3f,
     0xb5, 0x21, 0xf8, 0x28, 0xaf, 0x60, 0x6b, 0x4d,
     0x3d, 0xba, 0xa1, 0x4b, 0x5e, 0x77, 0xef, 0xe7,
     0x59, 0x28, 0xfe, 0x1d, 0xc1, 0x27, 0xa2, 0xff,
     0xa8, 0xde, 0x33, 0x48, 0xb3, 0xc1, 0x85, 0x6a,
     0x42, 0x9b, 0xf9, 0x7e, 0x7e, 0x31, 0xc2, 0xe5,
     0xbd, 0x66},
    {0x01, 0x18, 0x39, 0x29, 0x6a, 0x78, 0x9a, 0x3b, /* y */
     0xc0, 0x04, 0x5c, 0x8a, 0x5f, 0xb4, 0x2c, 0x7d,
     0x1b, 0xd9, 0x98, 0xf5, 0x44, 0x49, 0x57, 0x9b,
     0x44, 0x68, 0x17, 0xaf, 0xbd, 0x17, 0x27, 0x3e,
     0x66, 0x2c, 0x97, 0xee, 0x72, 0x99, 0x5e, 0xf4,
     0x26, 0x40, 0xc5, 0x50, 0xb9, 0x01, 0x3f, 0xad,
     0x07, 0x61, 0x35, 0x3c, 0x70, 0x86, 0xa2, 0x72,
     0xc2, 0x40, 0x88, 0xbe, 0x94, 0x76, 0x9f, 0xd1,
     0x66, 0x50}
};

/*-
 * The representation of field elements.
 * ------------------------------------
 *
 * We represent field elements with nine values. These values are either 64 or
 * 128 bits and the field element represented is:
 *   v[0]*2^0 + v[1]*2^58 + v[2]*2^116 + ... + v[8]*2^464  (mod p)
 * Each of the nine values is called a 'limb'. Since the limbs are spaced only
 * 58 bits apart, but are greater than 58 bits in length, the most significant
 * bits of each limb overlap with the least significant bits of the next.
 *
 * A field element with 64-bit limbs is an 'felem'. One with 128-bit limbs is a
 * 'largefelem' */

#define NLIMBS 9

typedef uint64_t limb;
typedef limb limb_aX __attribute((__aligned__(1)));
typedef limb felem[NLIMBS];
typedef uint128_t largefelem[NLIMBS];

static const limb bottom57bits = 0x1ffffffffffffff;
static const limb bottom58bits = 0x3ffffffffffffff;

/*
 * bin66_to_felem takes a little-endian byte array and converts it into felem
 * form. This assumes that the CPU is little-endian.
 */
static void bin66_to_felem(felem out, const u8 in[66])
{
    out[0] = (*((limb *) & in[0])) & bottom58bits;
    out[1] = (*((limb_aX *) & in[7]) >> 2) & bottom58bits;
    out[2] = (*((limb_aX *) & in[14]) >> 4) & bottom58bits;
    out[3] = (*((limb_aX *) & in[21]) >> 6) & bottom58bits;
    out[4] = (*((limb_aX *) & in[29])) & bottom58bits;
    out[5] = (*((limb_aX *) & in[36]) >> 2) & bottom58bits;
    out[6] = (*((limb_aX *) & in[43]) >> 4) & bottom58bits;
    out[7] = (*((limb_aX *) & in[50]) >> 6) & bottom58bits;
    out[8] = (*((limb_aX *) & in[58])) & bottom57bits;
}

/*
 * felem_to_bin66 takes an felem and serializes into a little endian, 66 byte
 * array. This assumes that the CPU is little-endian.
 */
static void felem_to_bin66(u8 out[66], const felem in)
{
    memset(out, 0, 66);
    (*((limb *) & out[0])) = in[0];
    (*((limb_aX *) & out[7])) |= in[1] << 2;
    (*((limb_aX *) & out[14])) |= in[2] << 4;
    (*((limb_aX *) & out[21])) |= in[3] << 6;
    (*((limb_aX *) & out[29])) = in[4];
    (*((limb_aX *) & out[36])) |= in[5] << 2;
    (*((limb_aX *) & out[43])) |= in[6] << 4;
    (*((limb_aX *) & out[50])) |= in[7] << 6;
    (*((limb_aX *) & out[58])) = in[8];
}

/* BN_to_felem converts an OpenSSL BIGNUM into an felem */
static int BN_to_felem(felem out, const BIGNUM *bn)
{
    felem_bytearray b_out;
    int num_bytes;

    if (BN_is_negative(bn)) {
        ERR_raise(ERR_LIB_EC, EC_R_BIGNUM_OUT_OF_RANGE);
        return 0;
    }
    num_bytes = BN_bn2lebinpad(bn, b_out, sizeof(b_out));
    if (num_bytes < 0) {
        ERR_raise(ERR_LIB_EC, EC_R_BIGNUM_OUT_OF_RANGE);
        return 0;
    }
    bin66_to_felem(out, b_out);
    return 1;
}

/* felem_to_BN converts an felem into an OpenSSL BIGNUM */
static BIGNUM *felem_to_BN(BIGNUM *out, const felem in)
{
    felem_bytearray b_out;
    felem_to_bin66(b_out, in);
    return BN_lebin2bn(b_out, sizeof(b_out), out);
}

/*-
 * Field operations
 * ----------------
 */

static void felem_one(felem out)
{
    out[0] = 1;
    out[1] = 0;
    out[2] = 0;
    out[3] = 0;
    out[4] = 0;
    out[5] = 0;
    out[6] = 0;
    out[7] = 0;
    out[8] = 0;
}

static void felem_assign(felem out, const felem in)
{
    out[0] = in[0];
    out[1] = in[1];
    out[2] = in[2];
    out[3] = in[3];
    out[4] = in[4];
    out[5] = in[5];
    out[6] = in[6];
    out[7] = in[7];
    out[8] = in[8];
}

/* felem_sum64 sets out = out + in. */
static void felem_sum64(felem out, const felem in)
{
    out[0] += in[0];
    out[1] += in[1];
    out[2] += in[2];
    out[3] += in[3];
    out[4] += in[4];
    out[5] += in[5];
    out[6] += in[6];
    out[7] += in[7];
    out[8] += in[8];
}

/* felem_scalar sets out = in * scalar */
static void felem_scalar(felem out, const felem in, limb scalar)
{
    out[0] = in[0] * scalar;
    out[1] = in[1] * scalar;
    out[2] = in[2] * scalar;
    out[3] = in[3] * scalar;
    out[4] = in[4] * scalar;
    out[5] = in[5] * scalar;
    out[6] = in[6] * scalar;
    out[7] = in[7] * scalar;
    out[8] = in[8] * scalar;
}

/* felem_scalar64 sets out = out * scalar */
static void felem_scalar64(felem out, limb scalar)
{
    out[0] *= scalar;
    out[1] *= scalar;
    out[2] *= scalar;
    out[3] *= scalar;
    out[4] *= scalar;
    out[5] *= scalar;
    out[6] *= scalar;
    out[7] *= scalar;
    out[8] *= scalar;
}

/* felem_scalar128 sets out = out * scalar */
static void felem_scalar128(largefelem out, limb scalar)
{
    out[0] *= scalar;
    out[1] *= scalar;
    out[2] *= scalar;
    out[3] *= scalar;
    out[4] *= scalar;
    out[5] *= scalar;
    out[6] *= scalar;
    out[7] *= scalar;
    out[8] *= scalar;
}

/*-
 * felem_neg sets |out| to |-in|
 * On entry:
 *   in[i] < 2^59 + 2^14
 * On exit:
 *   out[i] < 2^62
 */
static void felem_neg(felem out, const felem in)
{
    /* In order to prevent underflow, we subtract from 0 mod p. */
    static const limb two62m3 = (((limb) 1) << 62) - (((limb) 1) << 5);
    static const limb two62m2 = (((limb) 1) << 62) - (((limb) 1) << 4);

    out[0] = two62m3 - in[0];
    out[1] = two62m2 - in[1];
    out[2] = two62m2 - in[2];
    out[3] = two62m2 - in[3];
    out[4] = two62m2 - in[4];
    out[5] = two62m2 - in[5];
    out[6] = two62m2 - in[6];
    out[7] = two62m2 - in[7];
    out[8] = two62m2 - in[8];
}

/*-
 * felem_diff64 subtracts |in| from |out|
 * On entry:
 *   in[i] < 2^59 + 2^14
 * On exit:
 *   out[i] < out[i] + 2^62
 */
static void felem_diff64(felem out, const felem in)
{
    /*
     * In order to prevent underflow, we add 0 mod p before subtracting.
     */
    static const limb two62m3 = (((limb) 1) << 62) - (((limb) 1) << 5);
    static const limb two62m2 = (((limb) 1) << 62) - (((limb) 1) << 4);

    out[0] += two62m3 - in[0];
    out[1] += two62m2 - in[1];
    out[2] += two62m2 - in[2];
    out[3] += two62m2 - in[3];
    out[4] += two62m2 - in[4];
    out[5] += two62m2 - in[5];
    out[6] += two62m2 - in[6];
    out[7] += two62m2 - in[7];
    out[8] += two62m2 - in[8];
}

/*-
 * felem_diff_128_64 subtracts |in| from |out|
 * On entry:
 *   in[i] < 2^62 + 2^17
 * On exit:
 *   out[i] < out[i] + 2^63
 */
static void felem_diff_128_64(largefelem out, const felem in)
{
    /*
     * In order to prevent underflow, we add 64p mod p (which is equivalent
     * to 0 mod p) before subtracting. p is 2^521 - 1, i.e. in binary a 521
     * digit number with all bits set to 1. See "The representation of field
     * elements" comment above for a description of how limbs are used to
     * represent a number. 64p is represented with 8 limbs containing a number
     * with 58 bits set and one limb with a number with 57 bits set.
     */
    static const limb two63m6 = (((limb) 1) << 63) - (((limb) 1) << 6);
    static const limb two63m5 = (((limb) 1) << 63) - (((limb) 1) << 5);

    out[0] += two63m6 - in[0];
    out[1] += two63m5 - in[1];
    out[2] += two63m5 - in[2];
    out[3] += two63m5 - in[3];
    out[4] += two63m5 - in[4];
    out[5] += two63m5 - in[5];
    out[6] += two63m5 - in[6];
    out[7] += two63m5 - in[7];
    out[8] += two63m5 - in[8];
}

/*-
 * felem_diff_128_64 subtracts |in| from |out|
 * On entry:
 *   in[i] < 2^126
 * On exit:
 *   out[i] < out[i] + 2^127 - 2^69
 */
static void felem_diff128(largefelem out, const largefelem in)
{
    /*
     * In order to prevent underflow, we add 0 mod p before subtracting.
     */
    static const uint128_t two127m70 =
        (((uint128_t) 1) << 127) - (((uint128_t) 1) << 70);
    static const uint128_t two127m69 =
        (((uint128_t) 1) << 127) - (((uint128_t) 1) << 69);

    out[0] += (two127m70 - in[0]);
    out[1] += (two127m69 - in[1]);
    out[2] += (two127m69 - in[2]);
    out[3] += (two127m69 - in[3]);
    out[4] += (two127m69 - in[4]);
    out[5] += (two127m69 - in[5]);
    out[6] += (two127m69 - in[6]);
    out[7] += (two127m69 - in[7]);
    out[8] += (two127m69 - in[8]);
}

/*-
 * felem_square sets |out| = |in|^2
 * On entry:
 *   in[i] < 2^62
 * On exit:
 *   out[i] < 17 * max(in[i]) * max(in[i])
 */
static void felem_square_ref(largefelem out, const felem in)
{
    felem inx2, inx4;
    felem_scalar(inx2, in, 2);
    felem_scalar(inx4, in, 4);

    /*-
     * We have many cases were we want to do
     *   in[x] * in[y] +
     *   in[y] * in[x]
     * This is obviously just
     *   2 * in[x] * in[y]
     * However, rather than do the doubling on the 128 bit result, we
     * double one of the inputs to the multiplication by reading from
     * |inx2|
     */

    out[0] = ((uint128_t) in[0]) * in[0];
    out[1] = ((uint128_t) in[0]) * inx2[1];
    out[2] = ((uint128_t) in[0]) * inx2[2] + ((uint128_t) in[1]) * in[1];
    out[3] = ((uint128_t) in[0]) * inx2[3] + ((uint128_t) in[1]) * inx2[2];
    out[4] = ((uint128_t) in[0]) * inx2[4] +
             ((uint128_t) in[1]) * inx2[3] + ((uint128_t) in[2]) * in[2];
    out[5] = ((uint128_t) in[0]) * inx2[5] +
             ((uint128_t) in[1]) * inx2[4] + ((uint128_t) in[2]) * inx2[3];
    out[6] = ((uint128_t) in[0]) * inx2[6] +
             ((uint128_t) in[1]) * inx2[5] +
             ((uint128_t) in[2]) * inx2[4] + ((uint128_t) in[3]) * in[3];
    out[7] = ((uint128_t) in[0]) * inx2[7] +
             ((uint128_t) in[1]) * inx2[6] +
             ((uint128_t) in[2]) * inx2[5] + ((uint128_t) in[3]) * inx2[4];
    out[8] = ((uint128_t) in[0]) * inx2[8] +
             ((uint128_t) in[1]) * inx2[7] +
             ((uint128_t) in[2]) * inx2[6] +
             ((uint128_t) in[3]) * inx2[5] + ((uint128_t) in[4]) * in[4];

    /*
     * The remaining limbs fall above 2^521, with the first falling at 2^522.
     * They correspond to locations one bit up from the limbs produced above
     * so we would have to multiply by two to align them. Again, rather than
     * operate on the 128-bit result, we double one of the inputs to the
     * multiplication. If we want to double for both this reason, and the
     * reason above, then we end up multiplying by four.
     */

    /* 9 */
    out[0] += ((uint128_t) in[1]) * inx4[8] +
              ((uint128_t) in[2]) * inx4[7] +
              ((uint128_t) in[3]) * inx4[6] + ((uint128_t) in[4]) * inx4[5];

    /* 10 */
    out[1] += ((uint128_t) in[2]) * inx4[8] +
              ((uint128_t) in[3]) * inx4[7] +
              ((uint128_t) in[4]) * inx4[6] + ((uint128_t) in[5]) * inx2[5];

    /* 11 */
    out[2] += ((uint128_t) in[3]) * inx4[8] +
              ((uint128_t) in[4]) * inx4[7] + ((uint128_t) in[5]) * inx4[6];

    /* 12 */
    out[3] += ((uint128_t) in[4]) * inx4[8] +
              ((uint128_t) in[5]) * inx4[7] + ((uint128_t) in[6]) * inx2[6];

    /* 13 */
    out[4] += ((uint128_t) in[5]) * inx4[8] + ((uint128_t) in[6]) * inx4[7];

    /* 14 */
    out[5] += ((uint128_t) in[6]) * inx4[8] + ((uint128_t) in[7]) * inx2[7];

    /* 15 */
    out[6] += ((uint128_t) in[7]) * inx4[8];

    /* 16 */
    out[7] += ((uint128_t) in[8]) * inx2[8];
}

/*-
 * felem_mul sets |out| = |in1| * |in2|
 * On entry:
 *   in1[i] < 2^64
 *   in2[i] < 2^63
 * On exit:
 *   out[i] < 17 * max(in1[i]) * max(in2[i])
 */
static void felem_mul_ref(largefelem out, const felem in1, const felem in2)
{
    felem in2x2;
    felem_scalar(in2x2, in2, 2);

    out[0] = ((uint128_t) in1[0]) * in2[0];

    out[1] = ((uint128_t) in1[0]) * in2[1] +
             ((uint128_t) in1[1]) * in2[0];

    out[2] = ((uint128_t) in1[0]) * in2[2] +
             ((uint128_t) in1[1]) * in2[1] +
             ((uint128_t) in1[2]) * in2[0];

    out[3] = ((uint128_t) in1[0]) * in2[3] +
             ((uint128_t) in1[1]) * in2[2] +
             ((uint128_t) in1[2]) * in2[1] +
             ((uint128_t) in1[3]) * in2[0];

    out[4] = ((uint128_t) in1[0]) * in2[4] +
             ((uint128_t) in1[1]) * in2[3] +
             ((uint128_t) in1[2]) * in2[2] +
             ((uint128_t) in1[3]) * in2[1] +
             ((uint128_t) in1[4]) * in2[0];

    out[5] = ((uint128_t) in1[0]) * in2[5] +
             ((uint128_t) in1[1]) * in2[4] +
             ((uint128_t) in1[2]) * in2[3] +
             ((uint128_t) in1[3]) * in2[2] +
             ((uint128_t) in1[4]) * in2[1] +
             ((uint128_t) in1[5]) * in2[0];

    out[6] = ((uint128_t) in1[0]) * in2[6] +
             ((uint128_t) in1[1]) * in2[5] +
             ((uint128_t) in1[2]) * in2[4] +
             ((uint128_t) in1[3]) * in2[3] +
             ((uint128_t) in1[4]) * in2[2] +
             ((uint128_t) in1[5]) * in2[1] +
             ((uint128_t) in1[6]) * in2[0];

    out[7] = ((uint128_t) in1[0]) * in2[7] +
             ((uint128_t) in1[1]) * in2[6] +
             ((uint128_t) in1[2]) * in2[5] +
             ((uint128_t) in1[3]) * in2[4] +
             ((uint128_t) in1[4]) * in2[3] +
             ((uint128_t) in1[5]) * in2[2] +
             ((uint128_t) in1[6]) * in2[1] +
             ((uint128_t) in1[7]) * in2[0];

    out[8] = ((uint128_t) in1[0]) * in2[8] +
             ((uint128_t) in1[1]) * in2[7] +
             ((uint128_t) in1[2]) * in2[6] +
             ((uint128_t) in1[3]) * in2[5] +
             ((uint128_t) in1[4]) * in2[4] +
             ((uint128_t) in1[5]) * in2[3] +
             ((uint128_t) in1[6]) * in2[2] +
             ((uint128_t) in1[7]) * in2[1] +
             ((uint128_t) in1[8]) * in2[0];

    /* See comment in felem_square about the use of in2x2 here */

    out[0] += ((uint128_t) in1[1]) * in2x2[8] +
              ((uint128_t) in1[2]) * in2x2[7] +
              ((uint128_t) in1[3]) * in2x2[6] +
              ((uint128_t) in1[4]) * in2x2[5] +
              ((uint128_t) in1[5]) * in2x2[4] +
              ((uint128_t) in1[6]) * in2x2[3] +
              ((uint128_t) in1[7]) * in2x2[2] +
              ((uint128_t) in1[8]) * in2x2[1];

    out[1] += ((uint128_t) in1[2]) * in2x2[8] +
              ((uint128_t) in1[3]) * in2x2[7] +
              ((uint128_t) in1[4]) * in2x2[6] +
              ((uint128_t) in1[5]) * in2x2[5] +
              ((uint128_t) in1[6]) * in2x2[4] +
              ((uint128_t) in1[7]) * in2x2[3] +
              ((uint128_t) in1[8]) * in2x2[2];

    out[2] += ((uint128_t) in1[3]) * in2x2[8] +
              ((uint128_t) in1[4]) * in2x2[7] +
              ((uint128_t) in1[5]) * in2x2[6] +
              ((uint128_t) in1[6]) * in2x2[5] +
              ((uint128_t) in1[7]) * in2x2[4] +
              ((uint128_t) in1[8]) * in2x2[3];

    out[3] += ((uint128_t) in1[4]) * in2x2[8] +
              ((uint128_t) in1[5]) * in2x2[7] +
              ((uint128_t) in1[6]) * in2x2[6] +
              ((uint128_t) in1[7]) * in2x2[5] +
              ((uint128_t) in1[8]) * in2x2[4];

    out[4] += ((uint128_t) in1[5]) * in2x2[8] +
              ((uint128_t) in1[6]) * in2x2[7] +
              ((uint128_t) in1[7]) * in2x2[6] +
              ((uint128_t) in1[8]) * in2x2[5];

    out[5] += ((uint128_t) in1[6]) * in2x2[8] +
              ((uint128_t) in1[7]) * in2x2[7] +
              ((uint128_t) in1[8]) * in2x2[6];

    out[6] += ((uint128_t) in1[7]) * in2x2[8] +
              ((uint128_t) in1[8]) * in2x2[7];

    out[7] += ((uint128_t) in1[8]) * in2x2[8];
}

static const limb bottom52bits = 0xfffffffffffff;

/*-
 * felem_reduce converts a largefelem to an felem.
 * On entry:
 *   in[i] < 2^128
 * On exit:
 *   out[i] < 2^59 + 2^14
 */
static void felem_reduce(felem out, const largefelem in)
{
    u64 overflow1, overflow2;

    out[0] = ((limb) in[0]) & bottom58bits;
    out[1] = ((limb) in[1]) & bottom58bits;
    out[2] = ((limb) in[2]) & bottom58bits;
    out[3] = ((limb) in[3]) & bottom58bits;
    out[4] = ((limb) in[4]) & bottom58bits;
    out[5] = ((limb) in[5]) & bottom58bits;
    out[6] = ((limb) in[6]) & bottom58bits;
    out[7] = ((limb) in[7]) & bottom58bits;
    out[8] = ((limb) in[8]) & bottom58bits;

    /* out[i] < 2^58 */

    out[1] += ((limb) in[0]) >> 58;
    out[1] += (((limb) (in[0] >> 64)) & bottom52bits) << 6;
    /*-
     * out[1] < 2^58 + 2^6 + 2^58
     *        = 2^59 + 2^6
     */
    out[2] += ((limb) (in[0] >> 64)) >> 52;

    out[2] += ((limb) in[1]) >> 58;
    out[2] += (((limb) (in[1] >> 64)) & bottom52bits) << 6;
    out[3] += ((limb) (in[1] >> 64)) >> 52;

    out[3] += ((limb) in[2]) >> 58;
    out[3] += (((limb) (in[2] >> 64)) & bottom52bits) << 6;
    out[4] += ((limb) (in[2] >> 64)) >> 52;

    out[4] += ((limb) in[3]) >> 58;
    out[4] += (((limb) (in[3] >> 64)) & bottom52bits) << 6;
    out[5] += ((limb) (in[3] >> 64)) >> 52;

    out[5] += ((limb) in[4]) >> 58;
    out[5] += (((limb) (in[4] >> 64)) & bottom52bits) << 6;
    out[6] += ((limb) (in[4] >> 64)) >> 52;

    out[6] += ((limb) in[5]) >> 58;
    out[6] += (((limb) (in[5] >> 64)) & bottom52bits) << 6;
    out[7] += ((limb) (in[5] >> 64)) >> 52;

    out[7] += ((limb) in[6]) >> 58;
    out[7] += (((limb) (in[6] >> 64)) & bottom52bits) << 6;
    out[8] += ((limb) (in[6] >> 64)) >> 52;

    out[8] += ((limb) in[7]) >> 58;
    out[8] += (((limb) (in[7] >> 64)) & bottom52bits) << 6;
    /*-
     * out[x > 1] < 2^58 + 2^6 + 2^58 + 2^12
     *            < 2^59 + 2^13
     */
    overflow1 = ((limb) (in[7] >> 64)) >> 52;

    overflow1 += ((limb) in[8]) >> 58;
    overflow1 += (((limb) (in[8] >> 64)) & bottom52bits) << 6;
    overflow2 = ((limb) (in[8] >> 64)) >> 52;

    overflow1 <<= 1;            /* overflow1 < 2^13 + 2^7 + 2^59 */
    overflow2 <<= 1;            /* overflow2 < 2^13 */

    out[0] += overflow1;        /* out[0] < 2^60 */
    out[1] += overflow2;        /* out[1] < 2^59 + 2^6 + 2^13 */

    out[1] += out[0] >> 58;
    out[0] &= bottom58bits;
    /*-
     * out[0] < 2^58
     * out[1] < 2^59 + 2^6 + 2^13 + 2^2
     *        < 2^59 + 2^14
     */
}

#if defined(ECP_NISTP521_ASM)
void felem_square_wrapper(largefelem out, const felem in);
void felem_mul_wrapper(largefelem out, const felem in1, const felem in2);

static void (*felem_square_p)(largefelem out, const felem in) =
    felem_square_wrapper;
static void (*felem_mul_p)(largefelem out, const felem in1, const felem in2) =
    felem_mul_wrapper;

void p521_felem_square(largefelem out, const felem in);
void p521_felem_mul(largefelem out, const felem in1, const felem in2);

# if defined(_ARCH_PPC64)
#  include "crypto/ppc_arch.h"
# endif

void felem_select(void)
{
# if defined(_ARCH_PPC64)
    if ((OPENSSL_ppccap_P & PPC_MADD300) && (OPENSSL_ppccap_P & PPC_ALTIVEC)) {
        felem_square_p = p521_felem_square;
        felem_mul_p = p521_felem_mul;

        return;
    }
# endif

    /* Default */
    felem_square_p = felem_square_ref;
    felem_mul_p = felem_mul_ref;
}

void felem_square_wrapper(largefelem out, const felem in)
{
    felem_select();
    felem_square_p(out, in);
}

void felem_mul_wrapper(largefelem out, const felem in1, const felem in2)
{
    felem_select();
    felem_mul_p(out, in1, in2);
}

# define felem_square felem_square_p
# define felem_mul felem_mul_p
#else
# define felem_square felem_square_ref
# define felem_mul felem_mul_ref
#endif

static void felem_square_reduce(felem out, const felem in)
{
    largefelem tmp;
    felem_square(tmp, in);
    felem_reduce(out, tmp);
}

static void felem_mul_reduce(felem out, const felem in1, const felem in2)
{
    largefelem tmp;
    felem_mul(tmp, in1, in2);
    felem_reduce(out, tmp);
}

/*-
 * felem_inv calculates |out| = |in|^{-1}
 *
 * Based on Fermat's Little Theorem:
 *   a^p = a (mod p)
 *   a^{p-1} = 1 (mod p)
 *   a^{p-2} = a^{-1} (mod p)
 */
static void felem_inv(felem out, const felem in)
{
    felem ftmp, ftmp2, ftmp3, ftmp4;
    largefelem tmp;
    unsigned i;

    felem_square(tmp, in);
    felem_reduce(ftmp, tmp);    /* 2^1 */
    felem_mul(tmp, in, ftmp);
    felem_reduce(ftmp, tmp);    /* 2^2 - 2^0 */
    felem_assign(ftmp2, ftmp);
    felem_square(tmp, ftmp);
    felem_reduce(ftmp, tmp);    /* 2^3 - 2^1 */
    felem_mul(tmp, in, ftmp);
    felem_reduce(ftmp, tmp);    /* 2^3 - 2^0 */
    felem_square(tmp, ftmp);
    felem_reduce(ftmp, tmp);    /* 2^4 - 2^1 */

    felem_square(tmp, ftmp2);
    felem_reduce(ftmp3, tmp);   /* 2^3 - 2^1 */
    felem_square(tmp, ftmp3);
    felem_reduce(ftmp3, tmp);   /* 2^4 - 2^2 */
    felem_mul(tmp, ftmp3, ftmp2);
    felem_reduce(ftmp3, tmp);   /* 2^4 - 2^0 */

    felem_assign(ftmp2, ftmp3);
    felem_square(tmp, ftmp3);
    felem_reduce(ftmp3, tmp);   /* 2^5 - 2^1 */
    felem_square(tmp, ftmp3);
    felem_reduce(ftmp3, tmp);   /* 2^6 - 2^2 */
    felem_square(tmp, ftmp3);
    felem_reduce(ftmp3, tmp);   /* 2^7 - 2^3 */
    felem_square(tmp, ftmp3);
    felem_reduce(ftmp3, tmp);   /* 2^8 - 2^4 */
    felem_mul(tmp, ftmp3, ftmp);
    felem_reduce(ftmp4, tmp);   /* 2^8 - 2^1 */
    felem_square(tmp, ftmp4);
    felem_reduce(ftmp4, tmp);   /* 2^9 - 2^2 */
    felem_mul(tmp, ftmp3, ftmp2);
    felem_reduce(ftmp3, tmp);   /* 2^8 - 2^0 */
    felem_assign(ftmp2, ftmp3);

    for (i = 0; i < 8; i++) {
        felem_square(tmp, ftmp3);
        felem_reduce(ftmp3, tmp); /* 2^16 - 2^8 */
    }
    felem_mul(tmp, ftmp3, ftmp2);
    felem_reduce(ftmp3, tmp);   /* 2^16 - 2^0 */
    felem_assign(ftmp2, ftmp3);

    for (i = 0; i < 16; i++) {
        felem_square(tmp, ftmp3);
        felem_reduce(ftmp3, tmp); /* 2^32 - 2^16 */
    }
    felem_mul(tmp, ftmp3, ftmp2);
    felem_reduce(ftmp3, tmp);   /* 2^32 - 2^0 */
    felem_assign(ftmp2, ftmp3);

    for (i = 0; i < 32; i++) {
        felem_square(tmp, ftmp3);
        felem_reduce(ftmp3, tmp); /* 2^64 - 2^32 */
    }
    felem_mul(tmp, ftmp3, ftmp2);
    felem_reduce(ftmp3, tmp);   /* 2^64 - 2^0 */
    felem_assign(ftmp2, ftmp3);

    for (i = 0; i < 64; i++) {
        felem_square(tmp, ftmp3);
        felem_reduce(ftmp3, tmp); /* 2^128 - 2^64 */
    }
    felem_mul(tmp, ftmp3, ftmp2);
    felem_reduce(ftmp3, tmp);   /* 2^128 - 2^0 */
    felem_assign(ftmp2, ftmp3);

    for (i = 0; i < 128; i++) {
        felem_square(tmp, ftmp3);
        felem_reduce(ftmp3, tmp); /* 2^256 - 2^128 */
    }
    felem_mul(tmp, ftmp3, ftmp2);
    felem_reduce(ftmp3, tmp);   /* 2^256 - 2^0 */
    felem_assign(ftmp2, ftmp3);

    for (i = 0; i < 256; i++) {
        felem_square(tmp, ftmp3);
        felem_reduce(ftmp3, tmp); /* 2^512 - 2^256 */
    }
    felem_mul(tmp, ftmp3, ftmp2);
    felem_reduce(ftmp3, tmp);   /* 2^512 - 2^0 */

    for (i = 0; i < 9; i++) {
        felem_square(tmp, ftmp3);
        felem_reduce(ftmp3, tmp); /* 2^521 - 2^9 */
    }
    felem_mul(tmp, ftmp3, ftmp4);
    felem_reduce(ftmp3, tmp);   /* 2^512 - 2^2 */
    felem_mul(tmp, ftmp3, in);
    felem_reduce(out, tmp);     /* 2^512 - 3 */
}

/* This is 2^521-1, expressed as an felem */
static const felem kPrime = {
    0x03ffffffffffffff, 0x03ffffffffffffff, 0x03ffffffffffffff,
    0x03ffffffffffffff, 0x03ffffffffffffff, 0x03ffffffffffffff,
    0x03ffffffffffffff, 0x03ffffffffffffff, 0x01ffffffffffffff
};

/*-
 * felem_is_zero returns a limb with all bits set if |in| == 0 (mod p) and 0
 * otherwise.
 * On entry:
 *   in[i] < 2^59 + 2^14
 */
static limb felem_is_zero(const felem in)
{
    felem ftmp;
    limb is_zero, is_p;
    felem_assign(ftmp, in);

    ftmp[0] += ftmp[8] >> 57;
    ftmp[8] &= bottom57bits;
    /* ftmp[8] < 2^57 */
    ftmp[1] += ftmp[0] >> 58;
    ftmp[0] &= bottom58bits;
    ftmp[2] += ftmp[1] >> 58;
    ftmp[1] &= bottom58bits;
    ftmp[3] += ftmp[2] >> 58;
    ftmp[2] &= bottom58bits;
    ftmp[4] += ftmp[3] >> 58;
    ftmp[3] &= bottom58bits;
    ftmp[5] += ftmp[4] >> 58;
    ftmp[4] &= bottom58bits;
    ftmp[6] += ftmp[5] >> 58;
    ftmp[5] &= bottom58bits;
    ftmp[7] += ftmp[6] >> 58;
    ftmp[6] &= bottom58bits;
    ftmp[8] += ftmp[7] >> 58;
    ftmp[7] &= bottom58bits;
    /* ftmp[8] < 2^57 + 4 */

    /*
     * The ninth limb of 2*(2^521-1) is 0x03ffffffffffffff, which is greater
     * than our bound for ftmp[8]. Therefore we only have to check if the
     * zero is zero or 2^521-1.
     */

    is_zero = 0;
    is_zero |= ftmp[0];
    is_zero |= ftmp[1];
    is_zero |= ftmp[2];
    is_zero |= ftmp[3];
    is_zero |= ftmp[4];
    is_zero |= ftmp[5];
    is_zero |= ftmp[6];
    is_zero |= ftmp[7];
    is_zero |= ftmp[8];

    is_zero--;
    /*
     * We know that ftmp[i] < 2^63, therefore the only way that the top bit
     * can be set is if is_zero was 0 before the decrement.
     */
    is_zero = 0 - (is_zero >> 63);

    is_p = ftmp[0] ^ kPrime[0];
    is_p |= ftmp[1] ^ kPrime[1];
    is_p |= ftmp[2] ^ kPrime[2];
    is_p |= ftmp[3] ^ kPrime[3];
    is_p |= ftmp[4] ^ kPrime[4];
    is_p |= ftmp[5] ^ kPrime[5];
    is_p |= ftmp[6] ^ kPrime[6];
    is_p |= ftmp[7] ^ kPrime[7];
    is_p |= ftmp[8] ^ kPrime[8];

    is_p--;
    is_p = 0 - (is_p >> 63);

    is_zero |= is_p;
    return is_zero;
}

static int felem_is_zero_int(const void *in)
{
    return (int)(felem_is_zero(in) & ((limb) 1));
}

/*-
 * felem_contract converts |in| to its unique, minimal representation.
 * On entry:
 *   in[i] < 2^59 + 2^14
 */
static void felem_contract(felem out, const felem in)
{
    limb is_p, is_greater, sign;
    static const limb two58 = ((limb) 1) << 58;

    felem_assign(out, in);

    out[0] += out[8] >> 57;
    out[8] &= bottom57bits;
    /* out[8] < 2^57 */
    out[1] += out[0] >> 58;
    out[0] &= bottom58bits;
    out[2] += out[1] >> 58;
    out[1] &= bottom58bits;
    out[3] += out[2] >> 58;
    out[2] &= bottom58bits;
    out[4] += out[3] >> 58;
    out[3] &= bottom58bits;
    out[5] += out[4] >> 58;
    out[4] &= bottom58bits;
    out[6] += out[5] >> 58;
    out[5] &= bottom58bits;
    out[7] += out[6] >> 58;
    out[6] &= bottom58bits;
    out[8] += out[7] >> 58;
    out[7] &= bottom58bits;
    /* out[8] < 2^57 + 4 */

    /*
     * If the value is greater than 2^521-1 then we have to subtract 2^521-1
     * out. See the comments in felem_is_zero regarding why we don't test for
     * other multiples of the prime.
     */

    /*
     * First, if |out| is equal to 2^521-1, we subtract it out to get zero.
     */

    is_p = out[0] ^ kPrime[0];
    is_p |= out[1] ^ kPrime[1];
    is_p |= out[2] ^ kPrime[2];
    is_p |= out[3] ^ kPrime[3];
    is_p |= out[4] ^ kPrime[4];
    is_p |= out[5] ^ kPrime[5];
    is_p |= out[6] ^ kPrime[6];
    is_p |= out[7] ^ kPrime[7];
    is_p |= out[8] ^ kPrime[8];

    is_p--;
    is_p &= is_p << 32;
    is_p &= is_p << 16;
    is_p &= is_p << 8;
    is_p &= is_p << 4;
    is_p &= is_p << 2;
    is_p &= is_p << 1;
    is_p = 0 - (is_p >> 63);
    is_p = ~is_p;

    /* is_p is 0 iff |out| == 2^521-1 and all ones otherwise */

    out[0] &= is_p;
    out[1] &= is_p;
    out[2] &= is_p;
    out[3] &= is_p;
    out[4] &= is_p;
    out[5] &= is_p;
    out[6] &= is_p;
    out[7] &= is_p;
    out[8] &= is_p;

    /*
     * In order to test that |out| >= 2^521-1 we need only test if out[8] >>
     * 57 is greater than zero as (2^521-1) + x >= 2^522
     */
    is_greater = out[8] >> 57;
    is_greater |= is_greater << 32;
    is_greater |= is_greater << 16;
    is_greater |= is_greater << 8;
    is_greater |= is_greater << 4;
    is_greater |= is_greater << 2;
    is_greater |= is_greater << 1;
    is_greater = 0 - (is_greater >> 63);

    out[0] -= kPrime[0] & is_greater;
    out[1] -= kPrime[1] & is_greater;
    out[2] -= kPrime[2] & is_greater;
    out[3] -= kPrime[3] & is_greater;
    out[4] -= kPrime[4] & is_greater;
    out[5] -= kPrime[5] & is_greater;
    out[6] -= kPrime[6] & is_greater;
    out[7] -= kPrime[7] & is_greater;
    out[8] -= kPrime[8] & is_greater;

    /* Eliminate negative coefficients */
    sign = -(out[0] >> 63);
    out[0] += (two58 & sign);
    out[1] -= (1 & sign);
    sign = -(out[1] >> 63);
    out[1] += (two58 & sign);
    out[2] -= (1 & sign);
    sign = -(out[2] >> 63);
    out[2] += (two58 & sign);
    out[3] -= (1 & sign);
    sign = -(out[3] >> 63);
    out[3] += (two58 & sign);
    out[4] -= (1 & sign);
    sign = -(out[4] >> 63);
    out[4] += (two58 & sign);
    out[5] -= (1 & sign);
    sign = -(out[0] >> 63);
    out[5] += (two58 & sign);
    out[6] -= (1 & sign);
    sign = -(out[6] >> 63);
    out[6] += (two58 & sign);
    out[7] -= (1 & sign);
    sign = -(out[7] >> 63);
    out[7] += (two58 & sign);
    out[8] -= (1 & sign);
    sign = -(out[5] >> 63);
    out[5] += (two58 & sign);
    out[6] -= (1 & sign);
    sign = -(out[6] >> 63);
    out[6] += (two58 & sign);
    out[7] -= (1 & sign);
    sign = -(out[7] >> 63);
    out[7] += (two58 & sign);
    out[8] -= (1 & sign);
}

/*-
 * Group operations
 * ----------------
 *
 * Building on top of the field operations we have the operations on the
 * elliptic curve group itself. Points on the curve are represented in Jacobian
 * coordinates */

/*-
 * point_double calculates 2*(x_in, y_in, z_in)
 *
 * The method is taken from:
 *   http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#doubling-dbl-2001-b
 *
 * Outputs can equal corresponding inputs, i.e., x_out == x_in is allowed.
 * while x_out == y_in is not (maybe this works, but it's not tested). */
static void
point_double(felem x_out, felem y_out, felem z_out,
             const felem x_in, const felem y_in, const felem z_in)
{
    largefelem tmp, tmp2;
    felem delta, gamma, beta, alpha, ftmp, ftmp2;

    felem_assign(ftmp, x_in);
    felem_assign(ftmp2, x_in);

    /* delta = z^2 */
    felem_square(tmp, z_in);
    felem_reduce(delta, tmp);   /* delta[i] < 2^59 + 2^14 */

    /* gamma = y^2 */
    felem_square(tmp, y_in);
    felem_reduce(gamma, tmp);   /* gamma[i] < 2^59 + 2^14 */

    /* beta = x*gamma */
    felem_mul(tmp, x_in, gamma);
    felem_reduce(beta, tmp);    /* beta[i] < 2^59 + 2^14 */

    /* alpha = 3*(x-delta)*(x+delta) */
    felem_diff64(ftmp, delta);
    /* ftmp[i] < 2^61 */
    felem_sum64(ftmp2, delta);
    /* ftmp2[i] < 2^60 + 2^15 */
    felem_scalar64(ftmp2, 3);
    /* ftmp2[i] < 3*2^60 + 3*2^15 */
    felem_mul(tmp, ftmp, ftmp2);
    /*-
     * tmp[i] < 17(3*2^121 + 3*2^76)
     *        = 61*2^121 + 61*2^76
     *        < 64*2^121 + 64*2^76
     *        = 2^127 + 2^82
     *        < 2^128
     */
    felem_reduce(alpha, tmp);

    /* x' = alpha^2 - 8*beta */
    felem_square(tmp, alpha);
    /*
     * tmp[i] < 17*2^120 < 2^125
     */
    felem_assign(ftmp, beta);
    felem_scalar64(ftmp, 8);
    /* ftmp[i] < 2^62 + 2^17 */
    felem_diff_128_64(tmp, ftmp);
    /* tmp[i] < 2^125 + 2^63 + 2^62 + 2^17 */
    felem_reduce(x_out, tmp);

    /* z' = (y + z)^2 - gamma - delta */
    felem_sum64(delta, gamma);
    /* delta[i] < 2^60 + 2^15 */
    felem_assign(ftmp, y_in);
    felem_sum64(ftmp, z_in);
    /* ftmp[i] < 2^60 + 2^15 */
    felem_square(tmp, ftmp);
    /*
     * tmp[i] < 17(2^122) < 2^127
     */
    felem_diff_128_64(tmp, delta);
    /* tmp[i] < 2^127 + 2^63 */
    felem_reduce(z_out, tmp);

    /* y' = alpha*(4*beta - x') - 8*gamma^2 */
    felem_scalar64(beta, 4);
    /* beta[i] < 2^61 + 2^16 */
    felem_diff64(beta, x_out);
    /* beta[i] < 2^61 + 2^60 + 2^16 */
    felem_mul(tmp, alpha, beta);
    /*-
     * tmp[i] < 17*((2^59 + 2^14)(2^61 + 2^60 + 2^16))
     *        = 17*(2^120 + 2^75 + 2^119 + 2^74 + 2^75 + 2^30)
     *        = 17*(2^120 + 2^119 + 2^76 + 2^74 + 2^30)
     *        < 2^128
     */
    felem_square(tmp2, gamma);
    /*-
     * tmp2[i] < 17*(2^59 + 2^14)^2
     *         = 17*(2^118 + 2^74 + 2^28)
     */
    felem_scalar128(tmp2, 8);
    /*-
     * tmp2[i] < 8*17*(2^118 + 2^74 + 2^28)
     *         = 2^125 + 2^121 + 2^81 + 2^77 + 2^35 + 2^31
     *         < 2^126
     */
    felem_diff128(tmp, tmp2);
    /*-
     * tmp[i] < 2^127 - 2^69 + 17(2^120 + 2^119 + 2^76 + 2^74 + 2^30)
     *        = 2^127 + 2^124 + 2^122 + 2^120 + 2^118 + 2^80 + 2^78 + 2^76 +
     *          2^74 + 2^69 + 2^34 + 2^30
     *        < 2^128
     */
    felem_reduce(y_out, tmp);
}

/* copy_conditional copies in to out iff mask is all ones. */
static void copy_conditional(felem out, const felem in, limb mask)
{
    unsigned i;
    for (i = 0; i < NLIMBS; ++i) {
        const limb tmp = mask & (in[i] ^ out[i]);
        out[i] ^= tmp;
    }
}

/*-
 * point_add calculates (x1, y1, z1) + (x2, y2, z2)
 *
 * The method is taken from
 *   http://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#addition-add-2007-bl,
 * adapted for mixed addition (z2 = 1, or z2 = 0 for the point at infinity).
 *
 * This function includes a branch for checking whether the two input points
 * are equal (while not equal to the point at infinity). See comment below
 * on constant-time.
 */
static void point_add(felem x3, felem y3, felem z3,
                      const felem x1, const felem y1, const felem z1,
                      const int mixed, const felem x2, const felem y2,
                      const felem z2)
{
    felem ftmp, ftmp2, ftmp3, ftmp4, ftmp5, ftmp6, x_out, y_out, z_out;
    largefelem tmp, tmp2;
    limb x_equal, y_equal, z1_is_zero, z2_is_zero;
    limb points_equal;

    z1_is_zero = felem_is_zero(z1);
    z2_is_zero = felem_is_zero(z2);

    /* ftmp = z1z1 = z1**2 */
    felem_square(tmp, z1);
    felem_reduce(ftmp, tmp);

    if (!mixed) {
        /* ftmp2 = z2z2 = z2**2 */
        felem_square(tmp, z2);
        felem_reduce(ftmp2, tmp);

        /* u1 = ftmp3 = x1*z2z2 */
        felem_mul(tmp, x1, ftmp2);
        felem_reduce(ftmp3, tmp);

        /* ftmp5 = z1 + z2 */
        felem_assign(ftmp5, z1);
        felem_sum64(ftmp5, z2);
        /* ftmp5[i] < 2^61 */

        /* ftmp5 = (z1 + z2)**2 - z1z1 - z2z2 = 2*z1z2 */
        felem_square(tmp, ftmp5);
        /* tmp[i] < 17*2^122 */
        felem_diff_128_64(tmp, ftmp);
        /* tmp[i] < 17*2^122 + 2^63 */
        felem_diff_128_64(tmp, ftmp2);
        /* tmp[i] < 17*2^122 + 2^64 */
        felem_reduce(ftmp5, tmp);

        /* ftmp2 = z2 * z2z2 */
        felem_mul(tmp, ftmp2, z2);
        felem_reduce(ftmp2, tmp);

        /* s1 = ftmp6 = y1 * z2**3 */
        felem_mul(tmp, y1, ftmp2);
        felem_reduce(ftmp6, tmp);
    } else {
        /*
         * We'll assume z2 = 1 (special case z2 = 0 is handled later)
         */

        /* u1 = ftmp3 = x1*z2z2 */
        felem_assign(ftmp3, x1);

        /* ftmp5 = 2*z1z2 */
        felem_scalar(ftmp5, z1, 2);

        /* s1 = ftmp6 = y1 * z2**3 */
        felem_assign(ftmp6, y1);
    }

    /* u2 = x2*z1z1 */
    felem_mul(tmp, x2, ftmp);
    /* tmp[i] < 17*2^120 */

    /* h = ftmp4 = u2 - u1 */
    felem_diff_128_64(tmp, ftmp3);
    /* tmp[i] < 17*2^120 + 2^63 */
    felem_reduce(ftmp4, tmp);

    x_equal = felem_is_zero(ftmp4);

    /* z_out = ftmp5 * h */
    felem_mul(tmp, ftmp5, ftmp4);
    felem_reduce(z_out, tmp);

    /* ftmp = z1 * z1z1 */
    felem_mul(tmp, ftmp, z1);
    felem_reduce(ftmp, tmp);

    /* s2 = tmp = y2 * z1**3 */
    felem_mul(tmp, y2, ftmp);
    /* tmp[i] < 17*2^120 */

    /* r = ftmp5 = (s2 - s1)*2 */
    felem_diff_128_64(tmp, ftmp6);
    /* tmp[i] < 17*2^120 + 2^63 */
    felem_reduce(ftmp5, tmp);
    y_equal = felem_is_zero(ftmp5);
    felem_scalar64(ftmp5, 2);
    /* ftmp5[i] < 2^61 */

    /*
     * The formulae are incorrect if the points are equal, in affine coordinates
     * (X_1, Y_1) == (X_2, Y_2), so we check for this and do doubling if this
     * happens.
     *
     * We use bitwise operations to avoid potential side-channels introduced by
     * the short-circuiting behaviour of boolean operators.
     *
     * The special case of either point being the point at infinity (z1 and/or
     * z2 are zero), is handled separately later on in this function, so we
     * avoid jumping to point_double here in those special cases.
     *
     * Notice the comment below on the implications of this branching for timing
     * leaks and why it is considered practically irrelevant.
     */
    points_equal = (x_equal & y_equal & (~z1_is_zero) & (~z2_is_zero));

    if (points_equal) {
        /*
         * This is obviously not constant-time but it will almost-never happen
         * for ECDH / ECDSA. The case where it can happen is during scalar-mult
         * where the intermediate value gets very close to the group order.
         * Since |ossl_ec_GFp_nistp_recode_scalar_bits| produces signed digits
         * for the scalar, it's possible for the intermediate value to be a small
         * negative multiple of the base point, and for the final signed digit
         * to be the same value. We believe that this only occurs for the scalar
         * 1fffffffffffffffffffffffffffffffffffffffffffffffffffffffffff
         * ffffffa51868783bf2f966b7fcc0148f709a5d03bb5c9b8899c47aebb6fb
         * 71e913863f7, in that case the penultimate intermediate is -9G and
         * the final digit is also -9G. Since this only happens for a single
         * scalar, the timing leak is irrelevant. (Any attacker who wanted to
         * check whether a secret scalar was that exact value, can already do
         * so.)
         */
        point_double(x3, y3, z3, x1, y1, z1);
        return;
    }

    /* I = ftmp = (2h)**2 */
    felem_assign(ftmp, ftmp4);
    felem_scalar64(ftmp, 2);
    /* ftmp[i] < 2^61 */
    felem_square(tmp, ftmp);
    /* tmp[i] < 17*2^122 */
    felem_reduce(ftmp, tmp);

    /* J = ftmp2 = h * I */
    felem_mul(tmp, ftmp4, ftmp);
    felem_reduce(ftmp2, tmp);

    /* V = ftmp4 = U1 * I */
    felem_mul(tmp, ftmp3, ftmp);
    felem_reduce(ftmp4, tmp);

    /* x_out = r**2 - J - 2V */
    felem_square(tmp, ftmp5);
    /* tmp[i] < 17*2^122 */
    felem_diff_128_64(tmp, ftmp2);
    /* tmp[i] < 17*2^122 + 2^63 */
    felem_assign(ftmp3, ftmp4);
    felem_scalar64(ftmp4, 2);
    /* ftmp4[i] < 2^61 */
    felem_diff_128_64(tmp, ftmp4);
    /* tmp[i] < 17*2^122 + 2^64 */
    felem_reduce(x_out, tmp);

    /* y_out = r(V-x_out) - 2 * s1 * J */
    felem_diff64(ftmp3, x_out);
    /*
     * ftmp3[i] < 2^60 + 2^60 = 2^61
     */
    felem_mul(tmp, ftmp5, ftmp3);
    /* tmp[i] < 17*2^122 */
    felem_mul(tmp2, ftmp6, ftmp2);
    /* tmp2[i] < 17*2^120 */
    felem_scalar128(tmp2, 2);
    /* tmp2[i] < 17*2^121 */
    felem_diff128(tmp, tmp2);
        /*-
         * tmp[i] < 2^127 - 2^69 + 17*2^122
         *        = 2^126 - 2^122 - 2^6 - 2^2 - 1
         *        < 2^127
         */
    felem_reduce(y_out, tmp);

    copy_conditional(x_out, x2, z1_is_zero);
    copy_conditional(x_out, x1, z2_is_zero);
    copy_conditional(y_out, y2, z1_is_zero);
    copy_conditional(y_out, y1, z2_is_zero);
    copy_conditional(z_out, z2, z1_is_zero);
    copy_conditional(z_out, z1, z2_is_zero);
    felem_assign(x3, x_out);
    felem_assign(y3, y_out);
    felem_assign(z3, z_out);
}

/*-
 * Base point pre computation
 * --------------------------
 *
 * Two different sorts of precomputed tables are used in the following code.
 * Each contain various points on the curve, where each point is three field
 * elements (x, y, z).
 *
 * For the base point table, z is usually 1 (0 for the point at infinity).
 * This table has 16 elements:
 * index | bits    | point
 * ------+---------+------------------------------
 *     0 | 0 0 0 0 | 0G
 *     1 | 0 0 0 1 | 1G
 *     2 | 0 0 1 0 | 2^130G
 *     3 | 0 0 1 1 | (2^130 + 1)G
 *     4 | 0 1 0 0 | 2^260G
 *     5 | 0 1 0 1 | (2^260 + 1)G
 *     6 | 0 1 1 0 | (2^260 + 2^130)G
 *     7 | 0 1 1 1 | (2^260 + 2^130 + 1)G
 *     8 | 1 0 0 0 | 2^390G
 *     9 | 1 0 0 1 | (2^390 + 1)G
 *    10 | 1 0 1 0 | (2^390 + 2^130)G
 *    11 | 1 0 1 1 | (2^390 + 2^130 + 1)G
 *    12 | 1 1 0 0 | (2^390 + 2^260)G
 *    13 | 1 1 0 1 | (2^390 + 2^260 + 1)G
 *    14 | 1 1 1 0 | (2^390 + 2^260 + 2^130)G
 *    15 | 1 1 1 1 | (2^390 + 2^260 + 2^130 + 1)G
 *
 * The reason for this is so that we can clock bits into four different
 * locations when doing simple scalar multiplies against the base point.
 *
 * Tables for other points have table[i] = iG for i in 0 .. 16. */

/* gmul is the table of precomputed base points */
static const felem gmul[16][3] = {
{{0, 0, 0, 0, 0, 0, 0, 0, 0},
 {0, 0, 0, 0, 0, 0, 0, 0, 0},
 {0, 0, 0, 0, 0, 0, 0, 0, 0}},
{{0x017e7e31c2e5bd66, 0x022cf0615a90a6fe, 0x00127a2ffa8de334,
  0x01dfbf9d64a3f877, 0x006b4d3dbaa14b5e, 0x014fed487e0a2bd8,
  0x015b4429c6481390, 0x03a73678fb2d988e, 0x00c6858e06b70404},
 {0x00be94769fd16650, 0x031c21a89cb09022, 0x039013fad0761353,
  0x02657bd099031542, 0x03273e662c97ee72, 0x01e6d11a05ebef45,
  0x03d1bd998f544495, 0x03001172297ed0b1, 0x011839296a789a3b},
 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
{{0x0373faacbc875bae, 0x00f325023721c671, 0x00f666fd3dbde5ad,
  0x01a6932363f88ea7, 0x01fc6d9e13f9c47b, 0x03bcbffc2bbf734e,
  0x013ee3c3647f3a92, 0x029409fefe75d07d, 0x00ef9199963d85e5},
 {0x011173743ad5b178, 0x02499c7c21bf7d46, 0x035beaeabb8b1a58,
  0x00f989c4752ea0a3, 0x0101e1de48a9c1a3, 0x01a20076be28ba6c,
  0x02f8052e5eb2de95, 0x01bfe8f82dea117c, 0x0160074d3c36ddb7},
 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
{{0x012f3fc373393b3b, 0x03d3d6172f1419fa, 0x02adc943c0b86873,
  0x00d475584177952b, 0x012a4d1673750ee2, 0x00512517a0f13b0c,
  0x02b184671a7b1734, 0x0315b84236f1a50a, 0x00a4afc472edbdb9},
 {0x00152a7077f385c4, 0x03044007d8d1c2ee, 0x0065829d61d52b52,
  0x00494ff6b6631d0d, 0x00a11d94d5f06bcf, 0x02d2f89474d9282e,
  0x0241c5727c06eeb9, 0x0386928710fbdb9d, 0x01f883f727b0dfbe},
 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
{{0x019b0c3c9185544d, 0x006243a37c9d97db, 0x02ee3cbe030a2ad2,
  0x00cfdd946bb51e0d, 0x0271c00932606b91, 0x03f817d1ec68c561,
  0x03f37009806a369c, 0x03c1f30baf184fd5, 0x01091022d6d2f065},
 {0x0292c583514c45ed, 0x0316fca51f9a286c, 0x00300af507c1489a,
  0x0295f69008298cf1, 0x02c0ed8274943d7b, 0x016509b9b47a431e,
  0x02bc9de9634868ce, 0x005b34929bffcb09, 0x000c1a0121681524},
 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
{{0x0286abc0292fb9f2, 0x02665eee9805b3f7, 0x01ed7455f17f26d6,
  0x0346355b83175d13, 0x006284944cd0a097, 0x0191895bcdec5e51,
  0x02e288370afda7d9, 0x03b22312bfefa67a, 0x01d104d3fc0613fe},
 {0x0092421a12f7e47f, 0x0077a83fa373c501, 0x03bd25c5f696bd0d,
  0x035c41e4d5459761, 0x01ca0d1742b24f53, 0x00aaab27863a509c,
  0x018b6de47df73917, 0x025c0b771705cd01, 0x01fd51d566d760a7},
 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
{{0x01dd92ff6b0d1dbd, 0x039c5e2e8f8afa69, 0x0261ed13242c3b27,
  0x0382c6e67026e6a0, 0x01d60b10be2089f9, 0x03c15f3dce86723f,
  0x03c764a32d2a062d, 0x017307eac0fad056, 0x018207c0b96c5256},
 {0x0196a16d60e13154, 0x03e6ce74c0267030, 0x00ddbf2b4e52a5aa,
  0x012738241bbf31c8, 0x00ebe8dc04685a28, 0x024c2ad6d380d4a2,
  0x035ee062a6e62d0e, 0x0029ed74af7d3a0f, 0x00eef32aec142ebd},
 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
{{0x00c31ec398993b39, 0x03a9f45bcda68253, 0x00ac733c24c70890,
  0x00872b111401ff01, 0x01d178c23195eafb, 0x03bca2c816b87f74,
  0x0261a9af46fbad7a, 0x0324b2a8dd3d28f9, 0x00918121d8f24e23},
 {0x032bc8c1ca983cd7, 0x00d869dfb08fc8c6, 0x01693cb61fce1516,
  0x012a5ea68f4e88a8, 0x010869cab88d7ae3, 0x009081ad277ceee1,
  0x033a77166d064cdc, 0x03955235a1fb3a95, 0x01251a4a9b25b65e},
 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
{{0x00148a3a1b27f40b, 0x0123186df1b31fdc, 0x00026e7beaad34ce,
  0x01db446ac1d3dbba, 0x0299c1a33437eaec, 0x024540610183cbb7,
  0x0173bb0e9ce92e46, 0x02b937e43921214b, 0x01ab0436a9bf01b5},
 {0x0383381640d46948, 0x008dacbf0e7f330f, 0x03602122bcc3f318,
  0x01ee596b200620d6, 0x03bd0585fda430b3, 0x014aed77fd123a83,
  0x005ace749e52f742, 0x0390fe041da2b842, 0x0189a8ceb3299242},
 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
{{0x012a19d6b3282473, 0x00c0915918b423ce, 0x023a954eb94405ae,
  0x00529f692be26158, 0x0289fa1b6fa4b2aa, 0x0198ae4ceea346ef,
  0x0047d8cdfbdedd49, 0x00cc8c8953f0f6b8, 0x001424abbff49203},
 {0x0256732a1115a03a, 0x0351bc38665c6733, 0x03f7b950fb4a6447,
  0x000afffa94c22155, 0x025763d0a4dab540, 0x000511e92d4fc283,
  0x030a7e9eda0ee96c, 0x004c3cd93a28bf0a, 0x017edb3a8719217f},
 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
{{0x011de5675a88e673, 0x031d7d0f5e567fbe, 0x0016b2062c970ae5,
  0x03f4a2be49d90aa7, 0x03cef0bd13822866, 0x03f0923dcf774a6c,
  0x0284bebc4f322f72, 0x016ab2645302bb2c, 0x01793f95dace0e2a},
 {0x010646e13527a28f, 0x01ca1babd59dc5e7, 0x01afedfd9a5595df,
  0x01f15785212ea6b1, 0x0324e5d64f6ae3f4, 0x02d680f526d00645,
  0x0127920fadf627a7, 0x03b383f75df4f684, 0x0089e0057e783b0a},
 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
{{0x00f334b9eb3c26c6, 0x0298fdaa98568dce, 0x01c2d24843a82292,
  0x020bcb24fa1b0711, 0x02cbdb3d2b1875e6, 0x0014907598f89422,
  0x03abe3aa43b26664, 0x02cbf47f720bc168, 0x0133b5e73014b79b},
 {0x034aab5dab05779d, 0x00cdc5d71fee9abb, 0x0399f16bd4bd9d30,
  0x03582fa592d82647, 0x02be1cdfb775b0e9, 0x0034f7cea32e94cb,
  0x0335a7f08f56f286, 0x03b707e9565d1c8b, 0x0015c946ea5b614f},
 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
{{0x024676f6cff72255, 0x00d14625cac96378, 0x00532b6008bc3767,
  0x01fc16721b985322, 0x023355ea1b091668, 0x029de7afdc0317c3,
  0x02fc8a7ca2da037c, 0x02de1217d74a6f30, 0x013f7173175b73bf},
 {0x0344913f441490b5, 0x0200f9e272b61eca, 0x0258a246b1dd55d2,
  0x03753db9ea496f36, 0x025e02937a09c5ef, 0x030cbd3d14012692,
  0x01793a67e70dc72a, 0x03ec1d37048a662e, 0x006550f700c32a8d},
 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
{{0x00d3f48a347eba27, 0x008e636649b61bd8, 0x00d3b93716778fb3,
  0x004d1915757bd209, 0x019d5311a3da44e0, 0x016d1afcbbe6aade,
  0x0241bf5f73265616, 0x0384672e5d50d39b, 0x005009fee522b684},
 {0x029b4fab064435fe, 0x018868ee095bbb07, 0x01ea3d6936cc92b8,
  0x000608b00f78a2f3, 0x02db911073d1c20f, 0x018205938470100a,
  0x01f1e4964cbe6ff2, 0x021a19a29eed4663, 0x01414485f42afa81},
 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
{{0x01612b3a17f63e34, 0x03813992885428e6, 0x022b3c215b5a9608,
  0x029b4057e19f2fcb, 0x0384059a587af7e6, 0x02d6400ace6fe610,
  0x029354d896e8e331, 0x00c047ee6dfba65e, 0x0037720542e9d49d},
 {0x02ce9eed7c5e9278, 0x0374ed703e79643b, 0x01316c54c4072006,
  0x005aaa09054b2ee8, 0x002824000c840d57, 0x03d4eba24771ed86,
  0x0189c50aabc3bdae, 0x0338c01541e15510, 0x00466d56e38eed42},
 {1, 0, 0, 0, 0, 0, 0, 0, 0}},
{{0x007efd8330ad8bd6, 0x02465ed48047710b, 0x0034c6606b215e0c,
  0x016ae30c53cbf839, 0x01fa17bd37161216, 0x018ead4e61ce8ab9,
  0x005482ed5f5dee46, 0x037543755bba1d7f, 0x005e5ac7e70a9d0f},
 {0x0117e1bb2fdcb2a2, 0x03deea36249f40c4, 0x028d09b4a6246cb7,
  0x03524b8855bcf756, 0x023d7d109d5ceb58, 0x0178e43e3223ef9c,
  0x0154536a0c6e966a, 0x037964d1286ee9fe, 0x0199bcd90e125055},
 {1, 0, 0, 0, 0, 0, 0, 0, 0}}
};

/*
 * select_point selects the |idx|th point from a precomputation table and
 * copies it to out.
 */
 /* pre_comp below is of the size provided in |size| */
static void select_point(const limb idx, unsigned int size,
                         const felem pre_comp[][3], felem out[3])
{
    unsigned i, j;
    limb *outlimbs = &out[0][0];

    memset(out, 0, sizeof(*out) * 3);

    for (i = 0; i < size; i++) {
        const limb *inlimbs = &pre_comp[i][0][0];
        limb mask = i ^ idx;
        mask |= mask >> 4;
        mask |= mask >> 2;
        mask |= mask >> 1;
        mask &= 1;
        mask--;
        for (j = 0; j < NLIMBS * 3; j++)
            outlimbs[j] |= inlimbs[j] & mask;
    }
}

/* get_bit returns the |i|th bit in |in| */
static char get_bit(const felem_bytearray in, int i)
{
    if (i < 0)
        return 0;
    return (in[i >> 3] >> (i & 7)) & 1;
}

/*
 * Interleaved point multiplication using precomputed point multiples: The
 * small point multiples 0*P, 1*P, ..., 16*P are in pre_comp[], the scalars
 * in scalars[]. If g_scalar is non-NULL, we also add this multiple of the
 * generator, using certain (large) precomputed multiples in g_pre_comp.
 * Output point (X, Y, Z) is stored in x_out, y_out, z_out
 */
static void batch_mul(felem x_out, felem y_out, felem z_out,
                      const felem_bytearray scalars[],
                      const unsigned num_points, const u8 *g_scalar,
                      const int mixed, const felem pre_comp[][17][3],
                      const felem g_pre_comp[16][3])
{
    int i, skip;
    unsigned num, gen_mul = (g_scalar != NULL);
    felem nq[3], tmp[4];
    limb bits;
    u8 sign, digit;

    /* set nq to the point at infinity */
    memset(nq, 0, sizeof(nq));

    /*
     * Loop over all scalars msb-to-lsb, interleaving additions of multiples
     * of the generator (last quarter of rounds) and additions of other
     * points multiples (every 5th round).
     */
    skip = 1;                   /* save two point operations in the first
                                 * round */
    for (i = (num_points ? 520 : 130); i >= 0; --i) {
        /* double */
        if (!skip)
            point_double(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2]);

        /* add multiples of the generator */
        if (gen_mul && (i <= 130)) {
            bits = get_bit(g_scalar, i + 390) << 3;
            if (i < 130) {
                bits |= get_bit(g_scalar, i + 260) << 2;
                bits |= get_bit(g_scalar, i + 130) << 1;
                bits |= get_bit(g_scalar, i);
            }
            /* select the point to add, in constant time */
            select_point(bits, 16, g_pre_comp, tmp);
            if (!skip) {
                /* The 1 argument below is for "mixed" */
                point_add(nq[0], nq[1], nq[2],
                          nq[0], nq[1], nq[2], 1, tmp[0], tmp[1], tmp[2]);
            } else {
                memcpy(nq, tmp, 3 * sizeof(felem));
                skip = 0;
            }
        }

        /* do other additions every 5 doublings */
        if (num_points && (i % 5 == 0)) {
            /* loop over all scalars */
            for (num = 0; num < num_points; ++num) {
                bits = get_bit(scalars[num], i + 4) << 5;
                bits |= get_bit(scalars[num], i + 3) << 4;
                bits |= get_bit(scalars[num], i + 2) << 3;
                bits |= get_bit(scalars[num], i + 1) << 2;
                bits |= get_bit(scalars[num], i) << 1;
                bits |= get_bit(scalars[num], i - 1);
                ossl_ec_GFp_nistp_recode_scalar_bits(&sign, &digit, bits);

                /*
                 * select the point to add or subtract, in constant time
                 */
                select_point(digit, 17, pre_comp[num], tmp);
                felem_neg(tmp[3], tmp[1]); /* (X, -Y, Z) is the negative
                                            * point */
                copy_conditional(tmp[1], tmp[3], (-(limb) sign));

                if (!skip) {
                    point_add(nq[0], nq[1], nq[2],
                              nq[0], nq[1], nq[2],
                              mixed, tmp[0], tmp[1], tmp[2]);
                } else {
                    memcpy(nq, tmp, 3 * sizeof(felem));
                    skip = 0;
                }
            }
        }
    }
    felem_assign(x_out, nq[0]);
    felem_assign(y_out, nq[1]);
    felem_assign(z_out, nq[2]);
}

/* Precomputation for the group generator. */
struct nistp521_pre_comp_st {
    felem g_pre_comp[16][3];
    CRYPTO_REF_COUNT references;
    CRYPTO_RWLOCK *lock;
};

const EC_METHOD *EC_GFp_nistp521_method(void)
{
    static const EC_METHOD ret = {
        EC_FLAGS_DEFAULT_OCT,
        NID_X9_62_prime_field,
        ossl_ec_GFp_nistp521_group_init,
        ossl_ec_GFp_simple_group_finish,
        ossl_ec_GFp_simple_group_clear_finish,
        ossl_ec_GFp_nist_group_copy,
        ossl_ec_GFp_nistp521_group_set_curve,
        ossl_ec_GFp_simple_group_get_curve,
        ossl_ec_GFp_simple_group_get_degree,
        ossl_ec_group_simple_order_bits,
        ossl_ec_GFp_simple_group_check_discriminant,
        ossl_ec_GFp_simple_point_init,
        ossl_ec_GFp_simple_point_finish,
        ossl_ec_GFp_simple_point_clear_finish,
        ossl_ec_GFp_simple_point_copy,
        ossl_ec_GFp_simple_point_set_to_infinity,
        ossl_ec_GFp_simple_point_set_affine_coordinates,
        ossl_ec_GFp_nistp521_point_get_affine_coordinates,
        0 /* point_set_compressed_coordinates */ ,
        0 /* point2oct */ ,
        0 /* oct2point */ ,
        ossl_ec_GFp_simple_add,
        ossl_ec_GFp_simple_dbl,
        ossl_ec_GFp_simple_invert,
        ossl_ec_GFp_simple_is_at_infinity,
        ossl_ec_GFp_simple_is_on_curve,
        ossl_ec_GFp_simple_cmp,
        ossl_ec_GFp_simple_make_affine,
        ossl_ec_GFp_simple_points_make_affine,
        ossl_ec_GFp_nistp521_points_mul,
        ossl_ec_GFp_nistp521_precompute_mult,
        ossl_ec_GFp_nistp521_have_precompute_mult,
        ossl_ec_GFp_nist_field_mul,
        ossl_ec_GFp_nist_field_sqr,
        0 /* field_div */ ,
        ossl_ec_GFp_simple_field_inv,
        0 /* field_encode */ ,
        0 /* field_decode */ ,
        0,                      /* field_set_to_one */
        ossl_ec_key_simple_priv2oct,
        ossl_ec_key_simple_oct2priv,
        0, /* set private */
        ossl_ec_key_simple_generate_key,
        ossl_ec_key_simple_check_key,
        ossl_ec_key_simple_generate_public_key,
        0, /* keycopy */
        0, /* keyfinish */
        ossl_ecdh_simple_compute_key,
        ossl_ecdsa_simple_sign_setup,
        ossl_ecdsa_simple_sign_sig,
        ossl_ecdsa_simple_verify_sig,
        0, /* field_inverse_mod_ord */
        0, /* blind_coordinates */
        0, /* ladder_pre */
        0, /* ladder_step */
        0  /* ladder_post */
    };

    return &ret;
}

/******************************************************************************/
/*
 * FUNCTIONS TO MANAGE PRECOMPUTATION
 */

static NISTP521_PRE_COMP *nistp521_pre_comp_new(void)
{
    NISTP521_PRE_COMP *ret = OPENSSL_zalloc(sizeof(*ret));

    if (ret == NULL) {
        ERR_raise(ERR_LIB_EC, ERR_R_MALLOC_FAILURE);
        return ret;
    }

    ret->references = 1;

    ret->lock = CRYPTO_THREAD_lock_new();
    if (ret->lock == NULL) {
        ERR_raise(ERR_LIB_EC, ERR_R_MALLOC_FAILURE);
        OPENSSL_free(ret);
        return NULL;
    }
    return ret;
}

NISTP521_PRE_COMP *EC_nistp521_pre_comp_dup(NISTP521_PRE_COMP *p)
{
    int i;
    if (p != NULL)
        CRYPTO_UP_REF(&p->references, &i, p->lock);
    return p;
}

void EC_nistp521_pre_comp_free(NISTP521_PRE_COMP *p)
{
    int i;

    if (p == NULL)
        return;

    CRYPTO_DOWN_REF(&p->references, &i, p->lock);
    REF_PRINT_COUNT("EC_nistp521", p);
    if (i > 0)
        return;
    REF_ASSERT_ISNT(i < 0);

    CRYPTO_THREAD_lock_free(p->lock);
    OPENSSL_free(p);
}

/******************************************************************************/
/*
 * OPENSSL EC_METHOD FUNCTIONS
 */

int ossl_ec_GFp_nistp521_group_init(EC_GROUP *group)
{
    int ret;
    ret = ossl_ec_GFp_simple_group_init(group);
    group->a_is_minus3 = 1;
    return ret;
}

int ossl_ec_GFp_nistp521_group_set_curve(EC_GROUP *group, const BIGNUM *p,
                                         const BIGNUM *a, const BIGNUM *b,
                                         BN_CTX *ctx)
{
    int ret = 0;
    BIGNUM *curve_p, *curve_a, *curve_b;
#ifndef FIPS_MODULE
    BN_CTX *new_ctx = NULL;

    if (ctx == NULL)
        ctx = new_ctx = BN_CTX_new();
#endif
    if (ctx == NULL)
        return 0;

    BN_CTX_start(ctx);
    curve_p = BN_CTX_get(ctx);
    curve_a = BN_CTX_get(ctx);
    curve_b = BN_CTX_get(ctx);
    if (curve_b == NULL)
        goto err;
    BN_bin2bn(nistp521_curve_params[0], sizeof(felem_bytearray), curve_p);
    BN_bin2bn(nistp521_curve_params[1], sizeof(felem_bytearray), curve_a);
    BN_bin2bn(nistp521_curve_params[2], sizeof(felem_bytearray), curve_b);
    if ((BN_cmp(curve_p, p)) || (BN_cmp(curve_a, a)) || (BN_cmp(curve_b, b))) {
        ERR_raise(ERR_LIB_EC, EC_R_WRONG_CURVE_PARAMETERS);
        goto err;
    }
    group->field_mod_func = BN_nist_mod_521;
    ret = ossl_ec_GFp_simple_group_set_curve(group, p, a, b, ctx);
 err:
    BN_CTX_end(ctx);
#ifndef FIPS_MODULE
    BN_CTX_free(new_ctx);
#endif
    return ret;
}

/*
 * Takes the Jacobian coordinates (X, Y, Z) of a point and returns (X', Y') =
 * (X/Z^2, Y/Z^3)
 */
int ossl_ec_GFp_nistp521_point_get_affine_coordinates(const EC_GROUP *group,
                                                      const EC_POINT *point,
                                                      BIGNUM *x, BIGNUM *y,
                                                      BN_CTX *ctx)
{
    felem z1, z2, x_in, y_in, x_out, y_out;
    largefelem tmp;

    if (EC_POINT_is_at_infinity(group, point)) {
        ERR_raise(ERR_LIB_EC, EC_R_POINT_AT_INFINITY);
        return 0;
    }
    if ((!BN_to_felem(x_in, point->X)) || (!BN_to_felem(y_in, point->Y)) ||
        (!BN_to_felem(z1, point->Z)))
        return 0;
    felem_inv(z2, z1);
    felem_square(tmp, z2);
    felem_reduce(z1, tmp);
    felem_mul(tmp, x_in, z1);
    felem_reduce(x_in, tmp);
    felem_contract(x_out, x_in);
    if (x != NULL) {
        if (!felem_to_BN(x, x_out)) {
            ERR_raise(ERR_LIB_EC, ERR_R_BN_LIB);
            return 0;
        }
    }
    felem_mul(tmp, z1, z2);
    felem_reduce(z1, tmp);
    felem_mul(tmp, y_in, z1);
    felem_reduce(y_in, tmp);
    felem_contract(y_out, y_in);
    if (y != NULL) {
        if (!felem_to_BN(y, y_out)) {
            ERR_raise(ERR_LIB_EC, ERR_R_BN_LIB);
            return 0;
        }
    }
    return 1;
}

/* points below is of size |num|, and tmp_felems is of size |num+1/ */
static void make_points_affine(size_t num, felem points[][3],
                               felem tmp_felems[])
{
    /*
     * Runs in constant time, unless an input is the point at infinity (which
     * normally shouldn't happen).
     */
    ossl_ec_GFp_nistp_points_make_affine_internal(num,
                                                  points,
                                                  sizeof(felem),
                                                  tmp_felems,
                                                  (void (*)(void *))felem_one,
                                                  felem_is_zero_int,
                                                  (void (*)(void *, const void *))
                                                  felem_assign,
                                                  (void (*)(void *, const void *))
                                                  felem_square_reduce, (void (*)
                                                                        (void *,
                                                                         const void
                                                                         *,
                                                                         const void
                                                                         *))
                                                  felem_mul_reduce,
                                                  (void (*)(void *, const void *))
                                                  felem_inv,
                                                  (void (*)(void *, const void *))
                                                  felem_contract);
}

/*
 * Computes scalar*generator + \sum scalars[i]*points[i], ignoring NULL
 * values Result is stored in r (r can equal one of the inputs).
 */
int ossl_ec_GFp_nistp521_points_mul(const EC_GROUP *group, EC_POINT *r,
                                    const BIGNUM *scalar, size_t num,
                                    const EC_POINT *points[],
                                    const BIGNUM *scalars[], BN_CTX *ctx)
{
    int ret = 0;
    int j;
    int mixed = 0;
    BIGNUM *x, *y, *z, *tmp_scalar;
    felem_bytearray g_secret;
    felem_bytearray *secrets = NULL;
    felem (*pre_comp)[17][3] = NULL;
    felem *tmp_felems = NULL;
    unsigned i;
    int num_bytes;
    int have_pre_comp = 0;
    size_t num_points = num;
    felem x_in, y_in, z_in, x_out, y_out, z_out;
    NISTP521_PRE_COMP *pre = NULL;
    felem(*g_pre_comp)[3] = NULL;
    EC_POINT *generator = NULL;
    const EC_POINT *p = NULL;
    const BIGNUM *p_scalar = NULL;

    BN_CTX_start(ctx);
    x = BN_CTX_get(ctx);
    y = BN_CTX_get(ctx);
    z = BN_CTX_get(ctx);
    tmp_scalar = BN_CTX_get(ctx);
    if (tmp_scalar == NULL)
        goto err;

    if (scalar != NULL) {
        pre = group->pre_comp.nistp521;
        if (pre)
            /* we have precomputation, try to use it */
            g_pre_comp = &pre->g_pre_comp[0];
        else
            /* try to use the standard precomputation */
            g_pre_comp = (felem(*)[3]) gmul;
        generator = EC_POINT_new(group);
        if (generator == NULL)
            goto err;
        /* get the generator from precomputation */
        if (!felem_to_BN(x, g_pre_comp[1][0]) ||
            !felem_to_BN(y, g_pre_comp[1][1]) ||
            !felem_to_BN(z, g_pre_comp[1][2])) {
            ERR_raise(ERR_LIB_EC, ERR_R_BN_LIB);
            goto err;
        }
        if (!ossl_ec_GFp_simple_set_Jprojective_coordinates_GFp(group,
                                                                generator,
                                                                x, y, z, ctx))
            goto err;
        if (0 == EC_POINT_cmp(group, generator, group->generator, ctx))
            /* precomputation matches generator */
            have_pre_comp = 1;
        else
            /*
             * we don't have valid precomputation: treat the generator as a
             * random point
             */
            num_points++;
    }

    if (num_points > 0) {
        if (num_points >= 2) {
            /*
             * unless we precompute multiples for just one point, converting
             * those into affine form is time well spent
             */
            mixed = 1;
        }
        secrets = OPENSSL_zalloc(sizeof(*secrets) * num_points);
        pre_comp = OPENSSL_zalloc(sizeof(*pre_comp) * num_points);
        if (mixed)
            tmp_felems =
                OPENSSL_malloc(sizeof(*tmp_felems) * (num_points * 17 + 1));
        if ((secrets == NULL) || (pre_comp == NULL)
            || (mixed && (tmp_felems == NULL))) {
            ERR_raise(ERR_LIB_EC, ERR_R_MALLOC_FAILURE);
            goto err;
        }

        /*
         * we treat NULL scalars as 0, and NULL points as points at infinity,
         * i.e., they contribute nothing to the linear combination
         */
        for (i = 0; i < num_points; ++i) {
            if (i == num) {
                /*
                 * we didn't have a valid precomputation, so we pick the
                 * generator
                 */
                p = EC_GROUP_get0_generator(group);
                p_scalar = scalar;
            } else {
                /* the i^th point */
                p = points[i];
                p_scalar = scalars[i];
            }
            if ((p_scalar != NULL) && (p != NULL)) {
                /* reduce scalar to 0 <= scalar < 2^521 */
                if ((BN_num_bits(p_scalar) > 521)
                    || (BN_is_negative(p_scalar))) {
                    /*
                     * this is an unusual input, and we don't guarantee
                     * constant-timeness
                     */
                    if (!BN_nnmod(tmp_scalar, p_scalar, group->order, ctx)) {
                        ERR_raise(ERR_LIB_EC, ERR_R_BN_LIB);
                        goto err;
                    }
                    num_bytes = BN_bn2lebinpad(tmp_scalar,
                                               secrets[i], sizeof(secrets[i]));
                } else {
                    num_bytes = BN_bn2lebinpad(p_scalar,
                                               secrets[i], sizeof(secrets[i]));
                }
                if (num_bytes < 0) {
                    ERR_raise(ERR_LIB_EC, ERR_R_BN_LIB);
                    goto err;
                }
                /* precompute multiples */
                if ((!BN_to_felem(x_out, p->X)) ||
                    (!BN_to_felem(y_out, p->Y)) ||
                    (!BN_to_felem(z_out, p->Z)))
                    goto err;
                memcpy(pre_comp[i][1][0], x_out, sizeof(felem));
                memcpy(pre_comp[i][1][1], y_out, sizeof(felem));
                memcpy(pre_comp[i][1][2], z_out, sizeof(felem));
                for (j = 2; j <= 16; ++j) {
                    if (j & 1) {
                        point_add(pre_comp[i][j][0], pre_comp[i][j][1],
                                  pre_comp[i][j][2], pre_comp[i][1][0],
                                  pre_comp[i][1][1], pre_comp[i][1][2], 0,
                                  pre_comp[i][j - 1][0],
                                  pre_comp[i][j - 1][1],
                                  pre_comp[i][j - 1][2]);
                    } else {
                        point_double(pre_comp[i][j][0], pre_comp[i][j][1],
                                     pre_comp[i][j][2], pre_comp[i][j / 2][0],
                                     pre_comp[i][j / 2][1],
                                     pre_comp[i][j / 2][2]);
                    }
                }
            }
        }
        if (mixed)
            make_points_affine(num_points * 17, pre_comp[0], tmp_felems);
    }

    /* the scalar for the generator */
    if ((scalar != NULL) && (have_pre_comp)) {
        memset(g_secret, 0, sizeof(g_secret));
        /* reduce scalar to 0 <= scalar < 2^521 */
        if ((BN_num_bits(scalar) > 521) || (BN_is_negative(scalar))) {
            /*
             * this is an unusual input, and we don't guarantee
             * constant-timeness
             */
            if (!BN_nnmod(tmp_scalar, scalar, group->order, ctx)) {
                ERR_raise(ERR_LIB_EC, ERR_R_BN_LIB);
                goto err;
            }
            num_bytes = BN_bn2lebinpad(tmp_scalar, g_secret, sizeof(g_secret));
        } else {
            num_bytes = BN_bn2lebinpad(scalar, g_secret, sizeof(g_secret));
        }
        /* do the multiplication with generator precomputation */
        batch_mul(x_out, y_out, z_out,
                  (const felem_bytearray(*))secrets, num_points,
                  g_secret,
                  mixed, (const felem(*)[17][3])pre_comp,
                  (const felem(*)[3])g_pre_comp);
    } else {
        /* do the multiplication without generator precomputation */
        batch_mul(x_out, y_out, z_out,
                  (const felem_bytearray(*))secrets, num_points,
                  NULL, mixed, (const felem(*)[17][3])pre_comp, NULL);
    }
    /* reduce the output to its unique minimal representation */
    felem_contract(x_in, x_out);
    felem_contract(y_in, y_out);
    felem_contract(z_in, z_out);
    if ((!felem_to_BN(x, x_in)) || (!felem_to_BN(y, y_in)) ||
        (!felem_to_BN(z, z_in))) {
        ERR_raise(ERR_LIB_EC, ERR_R_BN_LIB);
        goto err;
    }
    ret = ossl_ec_GFp_simple_set_Jprojective_coordinates_GFp(group, r, x, y, z,
                                                             ctx);

 err:
    BN_CTX_end(ctx);
    EC_POINT_free(generator);
    OPENSSL_free(secrets);
    OPENSSL_free(pre_comp);
    OPENSSL_free(tmp_felems);
    return ret;
}

int ossl_ec_GFp_nistp521_precompute_mult(EC_GROUP *group, BN_CTX *ctx)
{
    int ret = 0;
    NISTP521_PRE_COMP *pre = NULL;
    int i, j;
    BIGNUM *x, *y;
    EC_POINT *generator = NULL;
    felem tmp_felems[16];
#ifndef FIPS_MODULE
    BN_CTX *new_ctx = NULL;
#endif

    /* throw away old precomputation */
    EC_pre_comp_free(group);

#ifndef FIPS_MODULE
    if (ctx == NULL)
        ctx = new_ctx = BN_CTX_new();
#endif
    if (ctx == NULL)
        return 0;

    BN_CTX_start(ctx);
    x = BN_CTX_get(ctx);
    y = BN_CTX_get(ctx);
    if (y == NULL)
        goto err;
    /* get the generator */
    if (group->generator == NULL)
        goto err;
    generator = EC_POINT_new(group);
    if (generator == NULL)
        goto err;
    BN_bin2bn(nistp521_curve_params[3], sizeof(felem_bytearray), x);
    BN_bin2bn(nistp521_curve_params[4], sizeof(felem_bytearray), y);
    if (!EC_POINT_set_affine_coordinates(group, generator, x, y, ctx))
        goto err;
    if ((pre = nistp521_pre_comp_new()) == NULL)
        goto err;
    /*
     * if the generator is the standard one, use built-in precomputation
     */
    if (0 == EC_POINT_cmp(group, generator, group->generator, ctx)) {
        memcpy(pre->g_pre_comp, gmul, sizeof(pre->g_pre_comp));
        goto done;
    }
    if ((!BN_to_felem(pre->g_pre_comp[1][0], group->generator->X)) ||
        (!BN_to_felem(pre->g_pre_comp[1][1], group->generator->Y)) ||
        (!BN_to_felem(pre->g_pre_comp[1][2], group->generator->Z)))
        goto err;
    /* compute 2^130*G, 2^260*G, 2^390*G */
    for (i = 1; i <= 4; i <<= 1) {
        point_double(pre->g_pre_comp[2 * i][0], pre->g_pre_comp[2 * i][1],
                     pre->g_pre_comp[2 * i][2], pre->g_pre_comp[i][0],
                     pre->g_pre_comp[i][1], pre->g_pre_comp[i][2]);
        for (j = 0; j < 129; ++j) {
            point_double(pre->g_pre_comp[2 * i][0],
                         pre->g_pre_comp[2 * i][1],
                         pre->g_pre_comp[2 * i][2],
                         pre->g_pre_comp[2 * i][0],
                         pre->g_pre_comp[2 * i][1],
                         pre->g_pre_comp[2 * i][2]);
        }
    }
    /* g_pre_comp[0] is the point at infinity */
    memset(pre->g_pre_comp[0], 0, sizeof(pre->g_pre_comp[0]));
    /* the remaining multiples */
    /* 2^130*G + 2^260*G */
    point_add(pre->g_pre_comp[6][0], pre->g_pre_comp[6][1],
              pre->g_pre_comp[6][2], pre->g_pre_comp[4][0],
              pre->g_pre_comp[4][1], pre->g_pre_comp[4][2],
              0, pre->g_pre_comp[2][0], pre->g_pre_comp[2][1],
              pre->g_pre_comp[2][2]);
    /* 2^130*G + 2^390*G */
    point_add(pre->g_pre_comp[10][0], pre->g_pre_comp[10][1],
              pre->g_pre_comp[10][2], pre->g_pre_comp[8][0],
              pre->g_pre_comp[8][1], pre->g_pre_comp[8][2],
              0, pre->g_pre_comp[2][0], pre->g_pre_comp[2][1],
              pre->g_pre_comp[2][2]);
    /* 2^260*G + 2^390*G */
    point_add(pre->g_pre_comp[12][0], pre->g_pre_comp[12][1],
              pre->g_pre_comp[12][2], pre->g_pre_comp[8][0],
              pre->g_pre_comp[8][1], pre->g_pre_comp[8][2],
              0, pre->g_pre_comp[4][0], pre->g_pre_comp[4][1],
              pre->g_pre_comp[4][2]);
    /* 2^130*G + 2^260*G + 2^390*G */
    point_add(pre->g_pre_comp[14][0], pre->g_pre_comp[14][1],
              pre->g_pre_comp[14][2], pre->g_pre_comp[12][0],
              pre->g_pre_comp[12][1], pre->g_pre_comp[12][2],
              0, pre->g_pre_comp[2][0], pre->g_pre_comp[2][1],
              pre->g_pre_comp[2][2]);
    for (i = 1; i < 8; ++i) {
        /* odd multiples: add G */
        point_add(pre->g_pre_comp[2 * i + 1][0],
                  pre->g_pre_comp[2 * i + 1][1],
                  pre->g_pre_comp[2 * i + 1][2], pre->g_pre_comp[2 * i][0],
                  pre->g_pre_comp[2 * i][1], pre->g_pre_comp[2 * i][2], 0,
                  pre->g_pre_comp[1][0], pre->g_pre_comp[1][1],
                  pre->g_pre_comp[1][2]);
    }
    make_points_affine(15, &(pre->g_pre_comp[1]), tmp_felems);

 done:
    SETPRECOMP(group, nistp521, pre);
    ret = 1;
    pre = NULL;
 err:
    BN_CTX_end(ctx);
    EC_POINT_free(generator);
#ifndef FIPS_MODULE
    BN_CTX_free(new_ctx);
#endif
    EC_nistp521_pre_comp_free(pre);
    return ret;
}

int ossl_ec_GFp_nistp521_have_precompute_mult(const EC_GROUP *group)
{
    return HAVEPRECOMP(group, nistp521);
}

Coverage Report

Created: 2025-06-13 06:58