update libressl to v2.7.4

[unleashed.git] / lib / libcrypto / ec / ecp_nistp256.c

/* $OpenBSD: ecp_nistp256.c,v 1.18 2017/05/02 03:59:44 deraadt Exp $ */
/*
 * Written by Adam Langley (Google) for the OpenSSL project
 */
/*
 * Copyright (c) 2011 Google Inc.
 *
 * Permission to use, copy, modify, and distribute this software for any
 * purpose with or without fee is hereby granted, provided that the above
 * copyright notice and this permission notice appear in all copies.
 *
 * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
 * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
 * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
 * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
 * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
 * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
 * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
 */

/*
 * A 64-bit implementation of the NIST P-256 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 <stdint.h>
#include <string.h>

#include <openssl/opensslconf.h>

#ifndef OPENSSL_NO_EC_NISTP_64_GCC_128

#include <openssl/err.h>
#include "ec_lcl.h"

#if defined(__GNUC__) && (__GNUC__ > 3 || (__GNUC__ == 3 && __GNUC_MINOR__ >= 1))
  /* even with gcc, the typedef won't work for 32-bit platforms */
  typedef __uint128_t uint128_t; /* nonstandard; implemented by gcc on 64-bit platforms */
  typedef __int128_t int128_t;
#else
  #error "Need GCC 3.1 or later to define type uint128_t"
#endif

typedef uint8_t u8;
typedef uint32_t u32;
typedef uint64_t u64;
typedef int64_t s64;

/* The underlying field.
 *
 * P256 operates over GF(2^256-2^224+2^192+2^96-1). We can serialise an element
 * of this field into 32 bytes. We call this an felem_bytearray. */

typedef u8 felem_bytearray[32];

/* These are the parameters of P256, taken from FIPS 186-3, page 86. These
 * values are big-endian. */
static const felem_bytearray nistp256_curve_params[5] = {
        {0xff, 0xff, 0xff, 0xff, 0x00, 0x00, 0x00, 0x01,       /* p */
         0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
         0x00, 0x00, 0x00, 0x00, 0xff, 0xff, 0xff, 0xff,
         0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff},
        {0xff, 0xff, 0xff, 0xff, 0x00, 0x00, 0x00, 0x01,       /* a = -3 */
         0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00,
         0x00, 0x00, 0x00, 0x00, 0xff, 0xff, 0xff, 0xff,
         0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xfc},      /* b */
        {0x5a, 0xc6, 0x35, 0xd8, 0xaa, 0x3a, 0x93, 0xe7,
         0xb3, 0xeb, 0xbd, 0x55, 0x76, 0x98, 0x86, 0xbc,
         0x65, 0x1d, 0x06, 0xb0, 0xcc, 0x53, 0xb0, 0xf6,
         0x3b, 0xce, 0x3c, 0x3e, 0x27, 0xd2, 0x60, 0x4b},
        {0x6b, 0x17, 0xd1, 0xf2, 0xe1, 0x2c, 0x42, 0x47,       /* x */
         0xf8, 0xbc, 0xe6, 0xe5, 0x63, 0xa4, 0x40, 0xf2,
         0x77, 0x03, 0x7d, 0x81, 0x2d, 0xeb, 0x33, 0xa0,
         0xf4, 0xa1, 0x39, 0x45, 0xd8, 0x98, 0xc2, 0x96},
        {0x4f, 0xe3, 0x42, 0xe2, 0xfe, 0x1a, 0x7f, 0x9b,       /* y */
         0x8e, 0xe7, 0xeb, 0x4a, 0x7c, 0x0f, 0x9e, 0x16,
         0x2b, 0xce, 0x33, 0x57, 0x6b, 0x31, 0x5e, 0xce,
         0xcb, 0xb6, 0x40, 0x68, 0x37, 0xbf, 0x51, 0xf5}
};

/* The representation of field elements.
 * ------------------------------------
 *
 * We represent field elements with either four 128-bit values, eight 128-bit
 * values, or four 64-bit values. The field element represented is:
 *   v[0]*2^0 + v[1]*2^64 + v[2]*2^128 + v[3]*2^192  (mod p)
 * or:
 *   v[0]*2^0 + v[1]*2^64 + v[2]*2^128 + ... + v[8]*2^512  (mod p)
 *
 * 128-bit values are called 'limbs'. Since the limbs are spaced only 64 bits
 * apart, but are 128-bits wide, the most significant bits of each limb overlap
 * with the least significant bits of the next.
 *
 * A field element with four limbs is an 'felem'. One with eight limbs is a
 * 'longfelem'
 *
 * A field element with four, 64-bit values is called a 'smallfelem'. Small
 * values are used as intermediate values before multiplication.
 */

#define NLIMBS 4

typedef uint128_t limb;
typedef limb felem[NLIMBS];
typedef limb longfelem[NLIMBS * 2];
typedef u64 smallfelem[NLIMBS];

/* This is the value of the prime as four 64-bit words, little-endian. */
static const u64 kPrime[4] = {0xfffffffffffffffful, 0xffffffff, 0, 0xffffffff00000001ul};
static const limb bottom32bits = 0xffffffff;
static const u64 bottom63bits = 0x7ffffffffffffffful;

/* bin32_to_felem takes a little-endian byte array and converts it into felem
 * form. This assumes that the CPU is little-endian. */
static void 
bin32_to_felem(felem out, const u8 in[32])
{
        out[0] = *((u64 *) & in[0]);
        out[1] = *((u64 *) & in[8]);
        out[2] = *((u64 *) & in[16]);
        out[3] = *((u64 *) & in[24]);
}

/* smallfelem_to_bin32 takes a smallfelem and serialises into a little endian,
 * 32 byte array. This assumes that the CPU is little-endian. */
static void 
smallfelem_to_bin32(u8 out[32], const smallfelem in)
{
        *((u64 *) & out[0]) = in[0];
        *((u64 *) & out[8]) = in[1];
        *((u64 *) & out[16]) = in[2];
        *((u64 *) & out[24]) = in[3];
}

/* To preserve endianness when using BN_bn2bin and BN_bin2bn */
static void 
flip_endian(u8 * out, const u8 * in, unsigned len)
{
        unsigned i;
        for (i = 0; i < len; ++i)
                out[i] = in[len - 1 - i];
}

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

        /* BN_bn2bin eats leading zeroes */
        memset(b_out, 0, sizeof b_out);
        num_bytes = BN_num_bytes(bn);
        if (num_bytes > sizeof b_out) {
                ECerror(EC_R_BIGNUM_OUT_OF_RANGE);
                return 0;
        }
        if (BN_is_negative(bn)) {
                ECerror(EC_R_BIGNUM_OUT_OF_RANGE);
                return 0;
        }
        num_bytes = BN_bn2bin(bn, b_in);
        flip_endian(b_out, b_in, num_bytes);
        bin32_to_felem(out, b_out);
        return 1;
}

/* felem_to_BN converts an felem into an OpenSSL BIGNUM */
static BIGNUM *
smallfelem_to_BN(BIGNUM * out, const smallfelem in)
{
        felem_bytearray b_in, b_out;
        smallfelem_to_bin32(b_in, in);
        flip_endian(b_out, b_in, sizeof b_out);
        return BN_bin2bn(b_out, sizeof b_out, out);
}


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

static void 
smallfelem_one(smallfelem out)
{
        out[0] = 1;
        out[1] = 0;
        out[2] = 0;
        out[3] = 0;
}

static void 
smallfelem_assign(smallfelem out, const smallfelem in)
{
        out[0] = in[0];
        out[1] = in[1];
        out[2] = in[2];
        out[3] = in[3];
}

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];
}

/* felem_sum sets out = out + in. */
static void 
felem_sum(felem out, const felem in)
{
        out[0] += in[0];
        out[1] += in[1];
        out[2] += in[2];
        out[3] += in[3];
}

/* felem_small_sum sets out = out + in. */
static void 
felem_small_sum(felem out, const smallfelem in)
{
        out[0] += in[0];
        out[1] += in[1];
        out[2] += in[2];
        out[3] += in[3];
}

/* felem_scalar sets out = out * scalar */
static void 
felem_scalar(felem out, const u64 scalar)
{
        out[0] *= scalar;
        out[1] *= scalar;
        out[2] *= scalar;
        out[3] *= scalar;
}

/* longfelem_scalar sets out = out * scalar */
static void 
longfelem_scalar(longfelem out, const u64 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;
}

#define two105m41m9 (((limb)1) << 105) - (((limb)1) << 41) - (((limb)1) << 9)
#define two105 (((limb)1) << 105)
#define two105m41p9 (((limb)1) << 105) - (((limb)1) << 41) + (((limb)1) << 9)

/* zero105 is 0 mod p */
static const felem zero105 = {two105m41m9, two105, two105m41p9, two105m41p9};

/* smallfelem_neg sets |out| to |-small|
 * On exit:
 *   out[i] < out[i] + 2^105
 */
static void 
smallfelem_neg(felem out, const smallfelem small)
{
        /* In order to prevent underflow, we subtract from 0 mod p. */
        out[0] = zero105[0] - small[0];
        out[1] = zero105[1] - small[1];
        out[2] = zero105[2] - small[2];
        out[3] = zero105[3] - small[3];
}

/* felem_diff subtracts |in| from |out|
 * On entry:
 *   in[i] < 2^104
 * On exit:
 *   out[i] < out[i] + 2^105
 */
static void 
felem_diff(felem out, const felem in)
{
        /* In order to prevent underflow, we add 0 mod p before subtracting. */
        out[0] += zero105[0];
        out[1] += zero105[1];
        out[2] += zero105[2];
        out[3] += zero105[3];

        out[0] -= in[0];
        out[1] -= in[1];
        out[2] -= in[2];
        out[3] -= in[3];
}

#define two107m43m11 (((limb)1) << 107) - (((limb)1) << 43) - (((limb)1) << 11)
#define two107 (((limb)1) << 107)
#define two107m43p11 (((limb)1) << 107) - (((limb)1) << 43) + (((limb)1) << 11)

/* zero107 is 0 mod p */
static const felem zero107 = {two107m43m11, two107, two107m43p11, two107m43p11};

/* An alternative felem_diff for larger inputs |in|
 * felem_diff_zero107 subtracts |in| from |out|
 * On entry:
 *   in[i] < 2^106
 * On exit:
 *   out[i] < out[i] + 2^107
 */
static void 
felem_diff_zero107(felem out, const felem in)
{
        /* In order to prevent underflow, we add 0 mod p before subtracting. */
        out[0] += zero107[0];
        out[1] += zero107[1];
        out[2] += zero107[2];
        out[3] += zero107[3];

        out[0] -= in[0];
        out[1] -= in[1];
        out[2] -= in[2];
        out[3] -= in[3];
}

/* longfelem_diff subtracts |in| from |out|
 * On entry:
 *   in[i] < 7*2^67
 * On exit:
 *   out[i] < out[i] + 2^70 + 2^40
 */
static void 
longfelem_diff(longfelem out, const longfelem in)
{
        static const limb two70m8p6 = (((limb) 1) << 70) - (((limb) 1) << 8) + (((limb) 1) << 6);
        static const limb two70p40 = (((limb) 1) << 70) + (((limb) 1) << 40);
        static const limb two70 = (((limb) 1) << 70);
        static const limb two70m40m38p6 = (((limb) 1) << 70) - (((limb) 1) << 40) - (((limb) 1) << 38) + (((limb) 1) << 6);
        static const limb two70m6 = (((limb) 1) << 70) - (((limb) 1) << 6);

        /* add 0 mod p to avoid underflow */
        out[0] += two70m8p6;
        out[1] += two70p40;
        out[2] += two70;
        out[3] += two70m40m38p6;
        out[4] += two70m6;
        out[5] += two70m6;
        out[6] += two70m6;
        out[7] += two70m6;

        /* in[i] < 7*2^67 < 2^70 - 2^40 - 2^38 + 2^6 */
        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];
}

#define two64m0 (((limb)1) << 64) - 1
#define two110p32m0 (((limb)1) << 110) + (((limb)1) << 32) - 1
#define two64m46 (((limb)1) << 64) - (((limb)1) << 46)
#define two64m32 (((limb)1) << 64) - (((limb)1) << 32)

/* zero110 is 0 mod p */
static const felem zero110 = {two64m0, two110p32m0, two64m46, two64m32};

/* felem_shrink converts an felem into a smallfelem. The result isn't quite
 * minimal as the value may be greater than p.
 *
 * On entry:
 *   in[i] < 2^109
 * On exit:
 *   out[i] < 2^64
 */
static void 
felem_shrink(smallfelem out, const felem in)
{
        felem tmp;
        u64 a, b, mask;
        s64 high, low;
        static const u64 kPrime3Test = 0x7fffffff00000001ul;    /* 2^63 - 2^32 + 1 */

        /* Carry 2->3 */
        tmp[3] = zero110[3] + in[3] + ((u64) (in[2] >> 64));
        /* tmp[3] < 2^110 */

        tmp[2] = zero110[2] + (u64) in[2];
        tmp[0] = zero110[0] + in[0];
        tmp[1] = zero110[1] + in[1];
        /* tmp[0] < 2**110, tmp[1] < 2^111, tmp[2] < 2**65 */

        /*
         * We perform two partial reductions where we eliminate the high-word
         * of tmp[3]. We don't update the other words till the end.
         */
        a = tmp[3] >> 64;       /* a < 2^46 */
        tmp[3] = (u64) tmp[3];
        tmp[3] -= a;
        tmp[3] += ((limb) a) << 32;
        /* tmp[3] < 2^79 */

        b = a;
        a = tmp[3] >> 64;       /* a < 2^15 */
        b += a;                 /* b < 2^46 + 2^15 < 2^47 */
        tmp[3] = (u64) tmp[3];
        tmp[3] -= a;
        tmp[3] += ((limb) a) << 32;
        /* tmp[3] < 2^64 + 2^47 */

        /*
         * This adjusts the other two words to complete the two partial
         * reductions.
         */
        tmp[0] += b;
        tmp[1] -= (((limb) b) << 32);

        /*
         * In order to make space in tmp[3] for the carry from 2 -> 3, we
         * conditionally subtract kPrime if tmp[3] is large enough.
         */
        high = tmp[3] >> 64;
        /* As tmp[3] < 2^65, high is either 1 or 0 */
        high <<= 63;
        high >>= 63;
        /*
         * high is: all ones   if the high word of tmp[3] is 1 all zeros  if
         * the high word of tmp[3] if 0
         */
        low = tmp[3];
        mask = low >> 63;
        /*
         * mask is: all ones   if the MSB of low is 1 all zeros  if the MSB
         * of low if 0
         */
        low &= bottom63bits;
        low -= kPrime3Test;
        /* if low was greater than kPrime3Test then the MSB is zero */
        low = ~low;
        low >>= 63;
        /*
         * low is: all ones   if low was > kPrime3Test all zeros  if low was
         * <= kPrime3Test
         */
        mask = (mask & low) | high;
        tmp[0] -= mask & kPrime[0];
        tmp[1] -= mask & kPrime[1];
        /* kPrime[2] is zero, so omitted */
        tmp[3] -= mask & kPrime[3];
        /* tmp[3] < 2**64 - 2**32 + 1 */

        tmp[1] += ((u64) (tmp[0] >> 64));
        tmp[0] = (u64) tmp[0];
        tmp[2] += ((u64) (tmp[1] >> 64));
        tmp[1] = (u64) tmp[1];
        tmp[3] += ((u64) (tmp[2] >> 64));
        tmp[2] = (u64) tmp[2];
        /* tmp[i] < 2^64 */

        out[0] = tmp[0];
        out[1] = tmp[1];
        out[2] = tmp[2];
        out[3] = tmp[3];
}

/* smallfelem_expand converts a smallfelem to an felem */
static void 
smallfelem_expand(felem out, const smallfelem in)
{
        out[0] = in[0];
        out[1] = in[1];
        out[2] = in[2];
        out[3] = in[3];
}

/* smallfelem_square sets |out| = |small|^2
 * On entry:
 *   small[i] < 2^64
 * On exit:
 *   out[i] < 7 * 2^64 < 2^67
 */
static void 
smallfelem_square(longfelem out, const smallfelem small)
{
        limb a;
        u64 high, low;

        a = ((uint128_t) small[0]) * small[0];
        low = a;
        high = a >> 64;
        out[0] = low;
        out[1] = high;

        a = ((uint128_t) small[0]) * small[1];
        low = a;
        high = a >> 64;
        out[1] += low;
        out[1] += low;
        out[2] = high;

        a = ((uint128_t) small[0]) * small[2];
        low = a;
        high = a >> 64;
        out[2] += low;
        out[2] *= 2;
        out[3] = high;

        a = ((uint128_t) small[0]) * small[3];
        low = a;
        high = a >> 64;
        out[3] += low;
        out[4] = high;

        a = ((uint128_t) small[1]) * small[2];
        low = a;
        high = a >> 64;
        out[3] += low;
        out[3] *= 2;
        out[4] += high;

        a = ((uint128_t) small[1]) * small[1];
        low = a;
        high = a >> 64;
        out[2] += low;
        out[3] += high;

        a = ((uint128_t) small[1]) * small[3];
        low = a;
        high = a >> 64;
        out[4] += low;
        out[4] *= 2;
        out[5] = high;

        a = ((uint128_t) small[2]) * small[3];
        low = a;
        high = a >> 64;
        out[5] += low;
        out[5] *= 2;
        out[6] = high;
        out[6] += high;

        a = ((uint128_t) small[2]) * small[2];
        low = a;
        high = a >> 64;
        out[4] += low;
        out[5] += high;

        a = ((uint128_t) small[3]) * small[3];
        low = a;
        high = a >> 64;
        out[6] += low;
        out[7] = high;
}

/* felem_square sets |out| = |in|^2
 * On entry:
 *   in[i] < 2^109
 * On exit:
 *   out[i] < 7 * 2^64 < 2^67
 */
static void 
felem_square(longfelem out, const felem in)
{
        u64 small[4];
        felem_shrink(small, in);
        smallfelem_square(out, small);
}

/* smallfelem_mul sets |out| = |small1| * |small2|
 * On entry:
 *   small1[i] < 2^64
 *   small2[i] < 2^64
 * On exit:
 *   out[i] < 7 * 2^64 < 2^67
 */
static void 
smallfelem_mul(longfelem out, const smallfelem small1, const smallfelem small2)
{
        limb a;
        u64 high, low;

        a = ((uint128_t) small1[0]) * small2[0];
        low = a;
        high = a >> 64;
        out[0] = low;
        out[1] = high;


        a = ((uint128_t) small1[0]) * small2[1];
        low = a;
        high = a >> 64;
        out[1] += low;
        out[2] = high;

        a = ((uint128_t) small1[1]) * small2[0];
        low = a;
        high = a >> 64;
        out[1] += low;
        out[2] += high;


        a = ((uint128_t) small1[0]) * small2[2];
        low = a;
        high = a >> 64;
        out[2] += low;
        out[3] = high;

        a = ((uint128_t) small1[1]) * small2[1];
        low = a;
        high = a >> 64;
        out[2] += low;
        out[3] += high;

        a = ((uint128_t) small1[2]) * small2[0];
        low = a;
        high = a >> 64;
        out[2] += low;
        out[3] += high;


        a = ((uint128_t) small1[0]) * small2[3];
        low = a;
        high = a >> 64;
        out[3] += low;
        out[4] = high;

        a = ((uint128_t) small1[1]) * small2[2];
        low = a;
        high = a >> 64;
        out[3] += low;
        out[4] += high;

        a = ((uint128_t) small1[2]) * small2[1];
        low = a;
        high = a >> 64;
        out[3] += low;
        out[4] += high;

        a = ((uint128_t) small1[3]) * small2[0];
        low = a;
        high = a >> 64;
        out[3] += low;
        out[4] += high;


        a = ((uint128_t) small1[1]) * small2[3];
        low = a;
        high = a >> 64;
        out[4] += low;
        out[5] = high;

        a = ((uint128_t) small1[2]) * small2[2];
        low = a;
        high = a >> 64;
        out[4] += low;
        out[5] += high;

        a = ((uint128_t) small1[3]) * small2[1];
        low = a;
        high = a >> 64;
        out[4] += low;
        out[5] += high;


        a = ((uint128_t) small1[2]) * small2[3];
        low = a;
        high = a >> 64;
        out[5] += low;
        out[6] = high;

        a = ((uint128_t) small1[3]) * small2[2];
        low = a;
        high = a >> 64;
        out[5] += low;
        out[6] += high;


        a = ((uint128_t) small1[3]) * small2[3];
        low = a;
        high = a >> 64;
        out[6] += low;
        out[7] = high;
}

/* felem_mul sets |out| = |in1| * |in2|
 * On entry:
 *   in1[i] < 2^109
 *   in2[i] < 2^109
 * On exit:
 *   out[i] < 7 * 2^64 < 2^67
 */
static void 
felem_mul(longfelem out, const felem in1, const felem in2)
{
        smallfelem small1, small2;
        felem_shrink(small1, in1);
        felem_shrink(small2, in2);
        smallfelem_mul(out, small1, small2);
}

/* felem_small_mul sets |out| = |small1| * |in2|
 * On entry:
 *   small1[i] < 2^64
 *   in2[i] < 2^109
 * On exit:
 *   out[i] < 7 * 2^64 < 2^67
 */
static void 
felem_small_mul(longfelem out, const smallfelem small1, const felem in2)
{
        smallfelem small2;
        felem_shrink(small2, in2);
        smallfelem_mul(out, small1, small2);
}

#define two100m36m4 (((limb)1) << 100) - (((limb)1) << 36) - (((limb)1) << 4)
#define two100 (((limb)1) << 100)
#define two100m36p4 (((limb)1) << 100) - (((limb)1) << 36) + (((limb)1) << 4)
/* zero100 is 0 mod p */
static const felem zero100 = {two100m36m4, two100, two100m36p4, two100m36p4};

/* Internal function for the different flavours of felem_reduce.
 * felem_reduce_ reduces the higher coefficients in[4]-in[7].
 * On entry:
 *   out[0] >= in[6] + 2^32*in[6] + in[7] + 2^32*in[7]
 *   out[1] >= in[7] + 2^32*in[4]
 *   out[2] >= in[5] + 2^32*in[5]
 *   out[3] >= in[4] + 2^32*in[5] + 2^32*in[6]
 * On exit:
 *   out[0] <= out[0] + in[4] + 2^32*in[5]
 *   out[1] <= out[1] + in[5] + 2^33*in[6]
 *   out[2] <= out[2] + in[7] + 2*in[6] + 2^33*in[7]
 *   out[3] <= out[3] + 2^32*in[4] + 3*in[7]
 */
static void 
felem_reduce_(felem out, const longfelem in)
{
        int128_t c;
        /* combine common terms from below */
        c = in[4] + (in[5] << 32);
        out[0] += c;
        out[3] -= c;

        c = in[5] - in[7];
        out[1] += c;
        out[2] -= c;

        /* the remaining terms */
        /* 256: [(0,1),(96,-1),(192,-1),(224,1)] */
        out[1] -= (in[4] << 32);
        out[3] += (in[4] << 32);

        /* 320: [(32,1),(64,1),(128,-1),(160,-1),(224,-1)] */
        out[2] -= (in[5] << 32);

        /* 384: [(0,-1),(32,-1),(96,2),(128,2),(224,-1)] */
        out[0] -= in[6];
        out[0] -= (in[6] << 32);
        out[1] += (in[6] << 33);
        out[2] += (in[6] * 2);
        out[3] -= (in[6] << 32);

        /* 448: [(0,-1),(32,-1),(64,-1),(128,1),(160,2),(192,3)] */
        out[0] -= in[7];
        out[0] -= (in[7] << 32);
        out[2] += (in[7] << 33);
        out[3] += (in[7] * 3);
}

/* felem_reduce converts a longfelem into an felem.
 * To be called directly after felem_square or felem_mul.
 * On entry:
 *   in[0] < 2^64, in[1] < 3*2^64, in[2] < 5*2^64, in[3] < 7*2^64
 *   in[4] < 7*2^64, in[5] < 5*2^64, in[6] < 3*2^64, in[7] < 2*64
 * On exit:
 *   out[i] < 2^101
 */
static void 
felem_reduce(felem out, const longfelem in)
{
        out[0] = zero100[0] + in[0];
        out[1] = zero100[1] + in[1];
        out[2] = zero100[2] + in[2];
        out[3] = zero100[3] + in[3];

        felem_reduce_(out, in);

        /*
         * out[0] > 2^100 - 2^36 - 2^4 - 3*2^64 - 3*2^96 - 2^64 - 2^96 > 0
         * out[1] > 2^100 - 2^64 - 7*2^96 > 0 out[2] > 2^100 - 2^36 + 2^4 -
         * 5*2^64 - 5*2^96 > 0 out[3] > 2^100 - 2^36 + 2^4 - 7*2^64 - 5*2^96
         * - 3*2^96 > 0
         * 
         * out[0] < 2^100 + 2^64 + 7*2^64 + 5*2^96 < 2^101 out[1] < 2^100 +
         * 3*2^64 + 5*2^64 + 3*2^97 < 2^101 out[2] < 2^100 + 5*2^64 + 2^64 +
         * 3*2^65 + 2^97 < 2^101 out[3] < 2^100 + 7*2^64 + 7*2^96 + 3*2^64 <
         * 2^101
         */
}

/* felem_reduce_zero105 converts a larger longfelem into an felem.
 * On entry:
 *   in[0] < 2^71
 * On exit:
 *   out[i] < 2^106
 */
static void 
felem_reduce_zero105(felem out, const longfelem in)
{
        out[0] = zero105[0] + in[0];
        out[1] = zero105[1] + in[1];
        out[2] = zero105[2] + in[2];
        out[3] = zero105[3] + in[3];

        felem_reduce_(out, in);

        /*
         * out[0] > 2^105 - 2^41 - 2^9 - 2^71 - 2^103 - 2^71 - 2^103 > 0
         * out[1] > 2^105 - 2^71 - 2^103 > 0 out[2] > 2^105 - 2^41 + 2^9 -
         * 2^71 - 2^103 > 0 out[3] > 2^105 - 2^41 + 2^9 - 2^71 - 2^103 -
         * 2^103 > 0
         * 
         * out[0] < 2^105 + 2^71 + 2^71 + 2^103 < 2^106 out[1] < 2^105 + 2^71 +
         * 2^71 + 2^103 < 2^106 out[2] < 2^105 + 2^71 + 2^71 + 2^71 + 2^103 <
         * 2^106 out[3] < 2^105 + 2^71 + 2^103 + 2^71 < 2^106
         */
}

/* subtract_u64 sets *result = *result - v and *carry to one if the subtraction
 * underflowed. */
static void 
subtract_u64(u64 * result, u64 * carry, u64 v)
{
        uint128_t r = *result;
        r -= v;
        *carry = (r >> 64) & 1;
        *result = (u64) r;
}

/* felem_contract converts |in| to its unique, minimal representation.
 * On entry:
 *   in[i] < 2^109
 */
static void 
felem_contract(smallfelem out, const felem in)
{
        unsigned i;
        u64 all_equal_so_far = 0, result = 0, carry;

        felem_shrink(out, in);
        /* small is minimal except that the value might be > p */

        all_equal_so_far--;
        /*
         * We are doing a constant time test if out >= kPrime. We need to
         * compare each u64, from most-significant to least significant. For
         * each one, if all words so far have been equal (m is all ones) then
         * a non-equal result is the answer. Otherwise we continue.
         */
        for (i = 3; i < 4; i--) {
                u64 equal;
                uint128_t a = ((uint128_t) kPrime[i]) - out[i];
                /*
                 * if out[i] > kPrime[i] then a will underflow and the high
                 * 64-bits will all be set.
                 */
                result |= all_equal_so_far & ((u64) (a >> 64));

                /*
                 * if kPrime[i] == out[i] then |equal| will be all zeros and
                 * the decrement will make it all ones.
                 */
                equal = kPrime[i] ^ out[i];
                equal--;
                equal &= equal << 32;
                equal &= equal << 16;
                equal &= equal << 8;
                equal &= equal << 4;
                equal &= equal << 2;
                equal &= equal << 1;
                equal = ((s64) equal) >> 63;

                all_equal_so_far &= equal;
        }

        /*
         * if all_equal_so_far is still all ones then the two values are
         * equal and so out >= kPrime is true.
         */
        result |= all_equal_so_far;

        /* if out >= kPrime then we subtract kPrime. */
        subtract_u64(&out[0], &carry, result & kPrime[0]);
        subtract_u64(&out[1], &carry, carry);
        subtract_u64(&out[2], &carry, carry);
        subtract_u64(&out[3], &carry, carry);

        subtract_u64(&out[1], &carry, result & kPrime[1]);
        subtract_u64(&out[2], &carry, carry);
        subtract_u64(&out[3], &carry, carry);

        subtract_u64(&out[2], &carry, result & kPrime[2]);
        subtract_u64(&out[3], &carry, carry);

        subtract_u64(&out[3], &carry, result & kPrime[3]);
}

static void 
smallfelem_square_contract(smallfelem out, const smallfelem in)
{
        longfelem longtmp;
        felem tmp;

        smallfelem_square(longtmp, in);
        felem_reduce(tmp, longtmp);
        felem_contract(out, tmp);
}

static void 
smallfelem_mul_contract(smallfelem out, const smallfelem in1, const smallfelem in2)
{
        longfelem longtmp;
        felem tmp;

        smallfelem_mul(longtmp, in1, in2);
        felem_reduce(tmp, longtmp);
        felem_contract(out, tmp);
}

/* felem_is_zero returns a limb with all bits set if |in| == 0 (mod p) and 0
 * otherwise.
 * On entry:
 *   small[i] < 2^64
 */
static limb 
smallfelem_is_zero(const smallfelem small)
{
        limb result;
        u64 is_p;

        u64 is_zero = small[0] | small[1] | small[2] | small[3];
        is_zero--;
        is_zero &= is_zero << 32;
        is_zero &= is_zero << 16;
        is_zero &= is_zero << 8;
        is_zero &= is_zero << 4;
        is_zero &= is_zero << 2;
        is_zero &= is_zero << 1;
        is_zero = ((s64) is_zero) >> 63;

        is_p = (small[0] ^ kPrime[0]) |
            (small[1] ^ kPrime[1]) |
            (small[2] ^ kPrime[2]) |
            (small[3] ^ kPrime[3]);
        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 = ((s64) is_p) >> 63;

        is_zero |= is_p;

        result = is_zero;
        result |= ((limb) is_zero) << 64;
        return result;
}

static int 
smallfelem_is_zero_int(const smallfelem small)
{
        return (int) (smallfelem_is_zero(small) & ((limb) 1));
}

/* 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;
        /* each e_I will hold |in|^{2^I - 1} */
        felem e2, e4, e8, e16, e32, e64;
        longfelem 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(e2, ftmp);
        felem_square(tmp, ftmp);
        felem_reduce(ftmp, tmp);/* 2^3 - 2^1 */
        felem_square(tmp, ftmp);
        felem_reduce(ftmp, tmp);/* 2^4 - 2^2 */
        felem_mul(tmp, ftmp, e2);
        felem_reduce(ftmp, tmp);/* 2^4 - 2^0 */
        felem_assign(e4, ftmp);
        felem_square(tmp, ftmp);
        felem_reduce(ftmp, tmp);/* 2^5 - 2^1 */
        felem_square(tmp, ftmp);
        felem_reduce(ftmp, tmp);/* 2^6 - 2^2 */
        felem_square(tmp, ftmp);
        felem_reduce(ftmp, tmp);/* 2^7 - 2^3 */
        felem_square(tmp, ftmp);
        felem_reduce(ftmp, tmp);/* 2^8 - 2^4 */
        felem_mul(tmp, ftmp, e4);
        felem_reduce(ftmp, tmp);/* 2^8 - 2^0 */
        felem_assign(e8, ftmp);
        for (i = 0; i < 8; i++) {
                felem_square(tmp, ftmp);
                felem_reduce(ftmp, tmp);
        }                       /* 2^16 - 2^8 */
        felem_mul(tmp, ftmp, e8);
        felem_reduce(ftmp, tmp);/* 2^16 - 2^0 */
        felem_assign(e16, ftmp);
        for (i = 0; i < 16; i++) {
                felem_square(tmp, ftmp);
                felem_reduce(ftmp, tmp);
        }                       /* 2^32 - 2^16 */
        felem_mul(tmp, ftmp, e16);
        felem_reduce(ftmp, tmp);/* 2^32 - 2^0 */
        felem_assign(e32, ftmp);
        for (i = 0; i < 32; i++) {
                felem_square(tmp, ftmp);
                felem_reduce(ftmp, tmp);
        }                       /* 2^64 - 2^32 */
        felem_assign(e64, ftmp);
        felem_mul(tmp, ftmp, in);
        felem_reduce(ftmp, tmp);/* 2^64 - 2^32 + 2^0 */
        for (i = 0; i < 192; i++) {
                felem_square(tmp, ftmp);
                felem_reduce(ftmp, tmp);
        }                       /* 2^256 - 2^224 + 2^192 */

        felem_mul(tmp, e64, e32);
        felem_reduce(ftmp2, tmp);       /* 2^64 - 2^0 */
        for (i = 0; i < 16; i++) {
                felem_square(tmp, ftmp2);
                felem_reduce(ftmp2, tmp);
        }                       /* 2^80 - 2^16 */
        felem_mul(tmp, ftmp2, e16);
        felem_reduce(ftmp2, tmp);       /* 2^80 - 2^0 */
        for (i = 0; i < 8; i++) {
                felem_square(tmp, ftmp2);
                felem_reduce(ftmp2, tmp);
        }                       /* 2^88 - 2^8 */
        felem_mul(tmp, ftmp2, e8);
        felem_reduce(ftmp2, tmp);       /* 2^88 - 2^0 */
        for (i = 0; i < 4; i++) {
                felem_square(tmp, ftmp2);
                felem_reduce(ftmp2, tmp);
        }                       /* 2^92 - 2^4 */
        felem_mul(tmp, ftmp2, e4);
        felem_reduce(ftmp2, tmp);       /* 2^92 - 2^0 */
        felem_square(tmp, ftmp2);
        felem_reduce(ftmp2, tmp);       /* 2^93 - 2^1 */
        felem_square(tmp, ftmp2);
        felem_reduce(ftmp2, tmp);       /* 2^94 - 2^2 */
        felem_mul(tmp, ftmp2, e2);
        felem_reduce(ftmp2, tmp);       /* 2^94 - 2^0 */
        felem_square(tmp, ftmp2);
        felem_reduce(ftmp2, tmp);       /* 2^95 - 2^1 */
        felem_square(tmp, ftmp2);
        felem_reduce(ftmp2, tmp);       /* 2^96 - 2^2 */
        felem_mul(tmp, ftmp2, in);
        felem_reduce(ftmp2, tmp);       /* 2^96 - 3 */

        felem_mul(tmp, ftmp2, ftmp);
        felem_reduce(out, tmp); /* 2^256 - 2^224 + 2^192 + 2^96 - 3 */
}

static void 
smallfelem_inv_contract(smallfelem out, const smallfelem in)
{
        felem tmp;

        smallfelem_expand(tmp, in);
        felem_inv(tmp, tmp);
        felem_contract(out, tmp);
}

/* 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)
{
        longfelem tmp, tmp2;
        felem delta, gamma, beta, alpha, ftmp, ftmp2;
        smallfelem small1, small2;

        felem_assign(ftmp, x_in);
        /* ftmp[i] < 2^106 */
        felem_assign(ftmp2, x_in);
        /* ftmp2[i] < 2^106 */

        /* delta = z^2 */
        felem_square(tmp, z_in);
        felem_reduce(delta, tmp);
        /* delta[i] < 2^101 */

        /* gamma = y^2 */
        felem_square(tmp, y_in);
        felem_reduce(gamma, tmp);
        /* gamma[i] < 2^101 */
        felem_shrink(small1, gamma);

        /* beta = x*gamma */
        felem_small_mul(tmp, small1, x_in);
        felem_reduce(beta, tmp);
        /* beta[i] < 2^101 */

        /* alpha = 3*(x-delta)*(x+delta) */
        felem_diff(ftmp, delta);
        /* ftmp[i] < 2^105 + 2^106 < 2^107 */
        felem_sum(ftmp2, delta);
        /* ftmp2[i] < 2^105 + 2^106 < 2^107 */
        felem_scalar(ftmp2, 3);
        /* ftmp2[i] < 3 * 2^107 < 2^109 */
        felem_mul(tmp, ftmp, ftmp2);
        felem_reduce(alpha, tmp);
        /* alpha[i] < 2^101 */
        felem_shrink(small2, alpha);

        /* x' = alpha^2 - 8*beta */
        smallfelem_square(tmp, small2);
        felem_reduce(x_out, tmp);
        felem_assign(ftmp, beta);
        felem_scalar(ftmp, 8);
        /* ftmp[i] < 8 * 2^101 = 2^104 */
        felem_diff(x_out, ftmp);
        /* x_out[i] < 2^105 + 2^101 < 2^106 */

        /* z' = (y + z)^2 - gamma - delta */
        felem_sum(delta, gamma);
        /* delta[i] < 2^101 + 2^101 = 2^102 */
        felem_assign(ftmp, y_in);
        felem_sum(ftmp, z_in);
        /* ftmp[i] < 2^106 + 2^106 = 2^107 */
        felem_square(tmp, ftmp);
        felem_reduce(z_out, tmp);
        felem_diff(z_out, delta);
        /* z_out[i] < 2^105 + 2^101 < 2^106 */

        /* y' = alpha*(4*beta - x') - 8*gamma^2 */
        felem_scalar(beta, 4);
        /* beta[i] < 4 * 2^101 = 2^103 */
        felem_diff_zero107(beta, x_out);
        /* beta[i] < 2^107 + 2^103 < 2^108 */
        felem_small_mul(tmp, small2, beta);
        /* tmp[i] < 7 * 2^64 < 2^67 */
        smallfelem_square(tmp2, small1);
        /* tmp2[i] < 7 * 2^64 */
        longfelem_scalar(tmp2, 8);
        /* tmp2[i] < 8 * 7 * 2^64 = 7 * 2^67 */
        longfelem_diff(tmp, tmp2);
        /* tmp[i] < 2^67 + 2^70 + 2^40 < 2^71 */
        felem_reduce_zero105(y_out, tmp);
        /* y_out[i] < 2^106 */
}

/* point_double_small is the same as point_double, except that it operates on
 * smallfelems */
static void
point_double_small(smallfelem x_out, smallfelem y_out, smallfelem z_out,
    const smallfelem x_in, const smallfelem y_in, const smallfelem z_in)
{
        felem felem_x_out, felem_y_out, felem_z_out;
        felem felem_x_in, felem_y_in, felem_z_in;

        smallfelem_expand(felem_x_in, x_in);
        smallfelem_expand(felem_y_in, y_in);
        smallfelem_expand(felem_z_in, z_in);
        point_double(felem_x_out, felem_y_out, felem_z_out,
            felem_x_in, felem_y_in, felem_z_in);
        felem_shrink(x_out, felem_x_out);
        felem_shrink(y_out, felem_y_out);
        felem_shrink(z_out, felem_z_out);
}

/* 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;
        }
}

/* copy_small_conditional copies in to out iff mask is all ones. */
static void
copy_small_conditional(felem out, const smallfelem in, limb mask)
{
        unsigned i;
        const u64 mask64 = mask;
        for (i = 0; i < NLIMBS; ++i) {
                out[i] = ((limb) (in[i] & mask64)) | (out[i] & ~mask);
        }
}

/* point_add calcuates (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). This case never
 * happens during single point multiplication, so there is no timing leak for
 * ECDH or ECDSA signing. */
static void 
point_add(felem x3, felem y3, felem z3,
    const felem x1, const felem y1, const felem z1,
    const int mixed, const smallfelem x2, const smallfelem y2, const smallfelem z2)
{
        felem ftmp, ftmp2, ftmp3, ftmp4, ftmp5, ftmp6, x_out, y_out, z_out;
        longfelem tmp, tmp2;
        smallfelem small1, small2, small3, small4, small5;
        limb x_equal, y_equal, z1_is_zero, z2_is_zero;

        felem_shrink(small3, z1);

        z1_is_zero = smallfelem_is_zero(small3);
        z2_is_zero = smallfelem_is_zero(z2);

        /* ftmp = z1z1 = z1**2 */
        smallfelem_square(tmp, small3);
        felem_reduce(ftmp, tmp);
        /* ftmp[i] < 2^101 */
        felem_shrink(small1, ftmp);

        if (!mixed) {
                /* ftmp2 = z2z2 = z2**2 */
                smallfelem_square(tmp, z2);
                felem_reduce(ftmp2, tmp);
                /* ftmp2[i] < 2^101 */
                felem_shrink(small2, ftmp2);

                felem_shrink(small5, x1);

                /* u1 = ftmp3 = x1*z2z2 */
                smallfelem_mul(tmp, small5, small2);
                felem_reduce(ftmp3, tmp);
                /* ftmp3[i] < 2^101 */

                /* ftmp5 = z1 + z2 */
                felem_assign(ftmp5, z1);
                felem_small_sum(ftmp5, z2);
                /* ftmp5[i] < 2^107 */

                /* ftmp5 = (z1 + z2)**2 - (z1z1 + z2z2) = 2z1z2 */
                felem_square(tmp, ftmp5);
                felem_reduce(ftmp5, tmp);
                /* ftmp2 = z2z2 + z1z1 */
                felem_sum(ftmp2, ftmp);
                /* ftmp2[i] < 2^101 + 2^101 = 2^102 */
                felem_diff(ftmp5, ftmp2);
                /* ftmp5[i] < 2^105 + 2^101 < 2^106 */

                /* ftmp2 = z2 * z2z2 */
                smallfelem_mul(tmp, small2, z2);
                felem_reduce(ftmp2, tmp);

                /* s1 = ftmp2 = y1 * z2**3 */
                felem_mul(tmp, y1, ftmp2);
                felem_reduce(ftmp6, tmp);
                /* ftmp6[i] < 2^101 */
        } else {
                /* We'll assume z2 = 1 (special case z2 = 0 is handled later) */

                /* u1 = ftmp3 = x1*z2z2 */
                felem_assign(ftmp3, x1);
                /* ftmp3[i] < 2^106 */

                /* ftmp5 = 2z1z2 */
                felem_assign(ftmp5, z1);
                felem_scalar(ftmp5, 2);
                /* ftmp5[i] < 2*2^106 = 2^107 */

                /* s1 = ftmp2 = y1 * z2**3 */
                felem_assign(ftmp6, y1);
                /* ftmp6[i] < 2^106 */
        }

        /* u2 = x2*z1z1 */
        smallfelem_mul(tmp, x2, small1);
        felem_reduce(ftmp4, tmp);

        /* h = ftmp4 = u2 - u1 */
        felem_diff_zero107(ftmp4, ftmp3);
        /* ftmp4[i] < 2^107 + 2^101 < 2^108 */
        felem_shrink(small4, ftmp4);

        x_equal = smallfelem_is_zero(small4);

        /* z_out = ftmp5 * h */
        felem_small_mul(tmp, small4, ftmp5);
        felem_reduce(z_out, tmp);
        /* z_out[i] < 2^101 */

        /* ftmp = z1 * z1z1 */
        smallfelem_mul(tmp, small1, small3);
        felem_reduce(ftmp, tmp);

        /* s2 = tmp = y2 * z1**3 */
        felem_small_mul(tmp, y2, ftmp);
        felem_reduce(ftmp5, tmp);

        /* r = ftmp5 = (s2 - s1)*2 */
        felem_diff_zero107(ftmp5, ftmp6);
        /* ftmp5[i] < 2^107 + 2^107 = 2^108 */
        felem_scalar(ftmp5, 2);
        /* ftmp5[i] < 2^109 */
        felem_shrink(small1, ftmp5);
        y_equal = smallfelem_is_zero(small1);

        if (x_equal && y_equal && !z1_is_zero && !z2_is_zero) {
                point_double(x3, y3, z3, x1, y1, z1);
                return;
        }
        /* I = ftmp = (2h)**2 */
        felem_assign(ftmp, ftmp4);
        felem_scalar(ftmp, 2);
        /* ftmp[i] < 2*2^108 = 2^109 */
        felem_square(tmp, ftmp);
        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 */
        smallfelem_square(tmp, small1);
        felem_reduce(x_out, tmp);
        felem_assign(ftmp3, ftmp4);
        felem_scalar(ftmp4, 2);
        felem_sum(ftmp4, ftmp2);
        /* ftmp4[i] < 2*2^101 + 2^101 < 2^103 */
        felem_diff(x_out, ftmp4);
        /* x_out[i] < 2^105 + 2^101 */

        /* y_out = r(V-x_out) - 2 * s1 * J */
        felem_diff_zero107(ftmp3, x_out);
        /* ftmp3[i] < 2^107 + 2^101 < 2^108 */
        felem_small_mul(tmp, small1, ftmp3);
        felem_mul(tmp2, ftmp6, ftmp2);
        longfelem_scalar(tmp2, 2);
        /* tmp2[i] < 2*2^67 = 2^68 */
        longfelem_diff(tmp, tmp2);
        /* tmp[i] < 2^67 + 2^70 + 2^40 < 2^71 */
        felem_reduce_zero105(y_out, tmp);
        /* y_out[i] < 2^106 */

        copy_small_conditional(x_out, x2, z1_is_zero);
        copy_conditional(x_out, x1, z2_is_zero);
        copy_small_conditional(y_out, y2, z1_is_zero);
        copy_conditional(y_out, y1, z2_is_zero);
        copy_small_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);
}

/* point_add_small is the same as point_add, except that it operates on
 * smallfelems */
static void 
point_add_small(smallfelem x3, smallfelem y3, smallfelem z3,
    smallfelem x1, smallfelem y1, smallfelem z1,
    smallfelem x2, smallfelem y2, smallfelem z2)
{
        felem felem_x3, felem_y3, felem_z3;
        felem felem_x1, felem_y1, felem_z1;
        smallfelem_expand(felem_x1, x1);
        smallfelem_expand(felem_y1, y1);
        smallfelem_expand(felem_z1, z1);
        point_add(felem_x3, felem_y3, felem_z3, felem_x1, felem_y1, felem_z1, 0, x2, y2, z2);
        felem_shrink(x3, felem_x3);
        felem_shrink(y3, felem_y3);
        felem_shrink(z3, felem_z3);
}

/* 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 2 * 16 elements, starting with the following:
 * index | bits    | point
 * ------+---------+------------------------------
 *     0 | 0 0 0 0 | 0G
 *     1 | 0 0 0 1 | 1G
 *     2 | 0 0 1 0 | 2^64G
 *     3 | 0 0 1 1 | (2^64 + 1)G
 *     4 | 0 1 0 0 | 2^128G
 *     5 | 0 1 0 1 | (2^128 + 1)G
 *     6 | 0 1 1 0 | (2^128 + 2^64)G
 *     7 | 0 1 1 1 | (2^128 + 2^64 + 1)G
 *     8 | 1 0 0 0 | 2^192G
 *     9 | 1 0 0 1 | (2^192 + 1)G
 *    10 | 1 0 1 0 | (2^192 + 2^64)G
 *    11 | 1 0 1 1 | (2^192 + 2^64 + 1)G
 *    12 | 1 1 0 0 | (2^192 + 2^128)G
 *    13 | 1 1 0 1 | (2^192 + 2^128 + 1)G
 *    14 | 1 1 1 0 | (2^192 + 2^128 + 2^64)G
 *    15 | 1 1 1 1 | (2^192 + 2^128 + 2^64 + 1)G
 * followed by a copy of this with each element multiplied by 2^32.
 *
 * The reason for this is so that we can clock bits into four different
 * locations when doing simple scalar multiplies against the base point,
 * and then another four locations using the second 16 elements.
 *
 * Tables for other points have table[i] = iG for i in 0 .. 16. */

/* gmul is the table of precomputed base points */
static const smallfelem gmul[2][16][3] =
{{{{0, 0, 0, 0},
{0, 0, 0, 0},
{0, 0, 0, 0}},
{{0xf4a13945d898c296, 0x77037d812deb33a0, 0xf8bce6e563a440f2, 0x6b17d1f2e12c4247},
{0xcbb6406837bf51f5, 0x2bce33576b315ece, 0x8ee7eb4a7c0f9e16, 0x4fe342e2fe1a7f9b},
{1, 0, 0, 0}},
{{0x90e75cb48e14db63, 0x29493baaad651f7e, 0x8492592e326e25de, 0x0fa822bc2811aaa5},
{0xe41124545f462ee7, 0x34b1a65050fe82f5, 0x6f4ad4bcb3df188b, 0xbff44ae8f5dba80d},
{1, 0, 0, 0}},
{{0x93391ce2097992af, 0xe96c98fd0d35f1fa, 0xb257c0de95e02789, 0x300a4bbc89d6726f},
{0xaa54a291c08127a0, 0x5bb1eeada9d806a5, 0x7f1ddb25ff1e3c6f, 0x72aac7e0d09b4644},
{1, 0, 0, 0}},
{{0x57c84fc9d789bd85, 0xfc35ff7dc297eac3, 0xfb982fd588c6766e, 0x447d739beedb5e67},
{0x0c7e33c972e25b32, 0x3d349b95a7fae500, 0xe12e9d953a4aaff7, 0x2d4825ab834131ee},
{1, 0, 0, 0}},
{{0x13949c932a1d367f, 0xef7fbd2b1a0a11b7, 0xddc6068bb91dfc60, 0xef9519328a9c72ff},
{0x196035a77376d8a8, 0x23183b0895ca1740, 0xc1ee9807022c219c, 0x611e9fc37dbb2c9b},
{1, 0, 0, 0}},
{{0xcae2b1920b57f4bc, 0x2936df5ec6c9bc36, 0x7dea6482e11238bf, 0x550663797b51f5d8},
{0x44ffe216348a964c, 0x9fb3d576dbdefbe1, 0x0afa40018d9d50e5, 0x157164848aecb851},
{1, 0, 0, 0}},
{{0xe48ecafffc5cde01, 0x7ccd84e70d715f26, 0xa2e8f483f43e4391, 0xeb5d7745b21141ea},
{0xcac917e2731a3479, 0x85f22cfe2844b645, 0x0990e6a158006cee, 0xeafd72ebdbecc17b},
{1, 0, 0, 0}},
{{0x6cf20ffb313728be, 0x96439591a3c6b94a, 0x2736ff8344315fc5, 0xa6d39677a7849276},
{0xf2bab833c357f5f4, 0x824a920c2284059b, 0x66b8babd2d27ecdf, 0x674f84749b0b8816},
{1, 0, 0, 0}},
{{0x2df48c04677c8a3e, 0x74e02f080203a56b, 0x31855f7db8c7fedb, 0x4e769e7672c9ddad},
{0xa4c36165b824bbb0, 0xfb9ae16f3b9122a5, 0x1ec0057206947281, 0x42b99082de830663},
{1, 0, 0, 0}},
{{0x6ef95150dda868b9, 0xd1f89e799c0ce131, 0x7fdc1ca008a1c478, 0x78878ef61c6ce04d},
{0x9c62b9121fe0d976, 0x6ace570ebde08d4f, 0xde53142c12309def, 0xb6cb3f5d7b72c321},
{1, 0, 0, 0}},
{{0x7f991ed2c31a3573, 0x5b82dd5bd54fb496, 0x595c5220812ffcae, 0x0c88bc4d716b1287},
{0x3a57bf635f48aca8, 0x7c8181f4df2564f3, 0x18d1b5b39c04e6aa, 0xdd5ddea3f3901dc6},
{1, 0, 0, 0}},
{{0xe96a79fb3e72ad0c, 0x43a0a28c42ba792f, 0xefe0a423083e49f3, 0x68f344af6b317466},
{0xcdfe17db3fb24d4a, 0x668bfc2271f5c626, 0x604ed93c24d67ff3, 0x31b9c405f8540a20},
{1, 0, 0, 0}},
{{0xd36b4789a2582e7f, 0x0d1a10144ec39c28, 0x663c62c3edbad7a0, 0x4052bf4b6f461db9},
{0x235a27c3188d25eb, 0xe724f33999bfcc5b, 0x862be6bd71d70cc8, 0xfecf4d5190b0fc61},
{1, 0, 0, 0}},
{{0x74346c10a1d4cfac, 0xafdf5cc08526a7a4, 0x123202a8f62bff7a, 0x1eddbae2c802e41a},
{0x8fa0af2dd603f844, 0x36e06b7e4c701917, 0x0c45f45273db33a0, 0x43104d86560ebcfc},
{1, 0, 0, 0}},
{{0x9615b5110d1d78e5, 0x66b0de3225c4744b, 0x0a4a46fb6aaf363a, 0xb48e26b484f7a21c},
{0x06ebb0f621a01b2d, 0xc004e4048b7b0f98, 0x64131bcdfed6f668, 0xfac015404d4d3dab},
{1, 0, 0, 0}}},
{{{0, 0, 0, 0},
{0, 0, 0, 0},
{0, 0, 0, 0}},
{{0x3a5a9e22185a5943, 0x1ab919365c65dfb6, 0x21656b32262c71da, 0x7fe36b40af22af89},
{0xd50d152c699ca101, 0x74b3d5867b8af212, 0x9f09f40407dca6f1, 0xe697d45825b63624},
{1, 0, 0, 0}},
{{0xa84aa9397512218e, 0xe9a521b074ca0141, 0x57880b3a18a2e902, 0x4a5b506612a677a6},
{0x0beada7a4c4f3840, 0x626db15419e26d9d, 0xc42604fbe1627d40, 0xeb13461ceac089f1},
{1, 0, 0, 0}},
{{0xf9faed0927a43281, 0x5e52c4144103ecbc, 0xc342967aa815c857, 0x0781b8291c6a220a},
{0x5a8343ceeac55f80, 0x88f80eeee54a05e3, 0x97b2a14f12916434, 0x690cde8df0151593},
{1, 0, 0, 0}},
{{0xaee9c75df7f82f2a, 0x9e4c35874afdf43a, 0xf5622df437371326, 0x8a535f566ec73617},
{0xc5f9a0ac223094b7, 0xcde533864c8c7669, 0x37e02819085a92bf, 0x0455c08468b08bd7},
{1, 0, 0, 0}},
{{0x0c0a6e2c9477b5d9, 0xf9a4bf62876dc444, 0x5050a949b6cdc279, 0x06bada7ab77f8276},
{0xc8b4aed1ea48dac9, 0xdebd8a4b7ea1070f, 0x427d49101366eb70, 0x5b476dfd0e6cb18a},
{1, 0, 0, 0}},
{{0x7c5c3e44278c340a, 0x4d54606812d66f3b, 0x29a751b1ae23c5d8, 0x3e29864e8a2ec908},
{0x142d2a6626dbb850, 0xad1744c4765bd780, 0x1f150e68e322d1ed, 0x239b90ea3dc31e7e},
{1, 0, 0, 0}},
{{0x78c416527a53322a, 0x305dde6709776f8e, 0xdbcab759f8862ed4, 0x820f4dd949f72ff7},
{0x6cc544a62b5debd4, 0x75be5d937b4e8cc4, 0x1b481b1b215c14d3, 0x140406ec783a05ec},
{1, 0, 0, 0}},
{{0x6a703f10e895df07, 0xfd75f3fa01876bd8, 0xeb5b06e70ce08ffe, 0x68f6b8542783dfee},
{0x90c76f8a78712655, 0xcf5293d2f310bf7f, 0xfbc8044dfda45028, 0xcbe1feba92e40ce6},
{1, 0, 0, 0}},
{{0xe998ceea4396e4c1, 0xfc82ef0b6acea274, 0x230f729f2250e927, 0xd0b2f94d2f420109},
{0x4305adddb38d4966, 0x10b838f8624c3b45, 0x7db2636658954e7a, 0x971459828b0719e5},
{1, 0, 0, 0}},
{{0x4bd6b72623369fc9, 0x57f2929e53d0b876, 0xc2d5cba4f2340687, 0x961610004a866aba},
{0x49997bcd2e407a5e, 0x69ab197d92ddcb24, 0x2cf1f2438fe5131c, 0x7acb9fadcee75e44},
{1, 0, 0, 0}},
{{0x254e839423d2d4c0, 0xf57f0c917aea685b, 0xa60d880f6f75aaea, 0x24eb9acca333bf5b},
{0xe3de4ccb1cda5dea, 0xfeef9341c51a6b4f, 0x743125f88bac4c4d, 0x69f891c5acd079cc},
{1, 0, 0, 0}},
{{0xeee44b35702476b5, 0x7ed031a0e45c2258, 0xb422d1e7bd6f8514, 0xe51f547c5972a107},
{0xa25bcd6fc9cf343d, 0x8ca922ee097c184e, 0xa62f98b3a9fe9a06, 0x1c309a2b25bb1387},
{1, 0, 0, 0}},
{{0x9295dbeb1967c459, 0xb00148833472c98e, 0xc504977708011828, 0x20b87b8aa2c4e503},
{0x3063175de057c277, 0x1bd539338fe582dd, 0x0d11adef5f69a044, 0xf5c6fa49919776be},
{1, 0, 0, 0}},
{{0x8c944e760fd59e11, 0x3876cba1102fad5f, 0xa454c3fad83faa56, 0x1ed7d1b9332010b9},
{0xa1011a270024b889, 0x05e4d0dcac0cd344, 0x52b520f0eb6a2a24, 0x3a2b03f03217257a},
{1, 0, 0, 0}},
{{0xf20fc2afdf1d043d, 0xf330240db58d5a62, 0xfc7d229ca0058c3b, 0x15fee545c78dd9f6},
{0x501e82885bc98cda, 0x41ef80e5d046ac04, 0x557d9f49461210fb, 0x4ab5b6b2b8753f81},
{1, 0, 0, 0}}}};

/* select_point selects the |idx|th point from a precomputation table and
 * copies it to out. */
static void 
select_point(const u64 idx, unsigned int size, const smallfelem pre_comp[16][3], smallfelem out[3])
{
        unsigned i, j;
        u64 *outlimbs = &out[0][0];
        memset(outlimbs, 0, 3 * sizeof(smallfelem));

        for (i = 0; i < size; i++) {
                const u64 *inlimbs = (u64 *) & pre_comp[i][0][0];
                u64 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) || (i >= 256))
                return 0;
        return (in[i >> 3] >> (i & 7)) & 1;
}

/* Interleaved point multiplication using precomputed point multiples:
 * The small point multiples 0*P, 1*P, ..., 17*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 smallfelem pre_comp[][17][3], const smallfelem g_pre_comp[2][16][3])
{
        int i, skip;
        unsigned num, gen_mul = (g_scalar != NULL);
        felem nq[3], ftmp;
        smallfelem tmp[3];
        u64 bits;
        u8 sign, digit;

        /* set nq to the point at infinity */
        memset(nq, 0, 3 * sizeof(felem));

        /*
         * Loop over all scalars msb-to-lsb, interleaving additions of
         * multiples of the generator (two in each of the last 32 rounds) and
         * additions of other points multiples (every 5th round).
         */
        skip = 1;               /* save two point operations in the first
                                 * round */
        for (i = (num_points ? 255 : 31); 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 <= 31)) {
                        /* first, look 32 bits upwards */
                        bits = get_bit(g_scalar, i + 224) << 3;
                        bits |= get_bit(g_scalar, i + 160) << 2;
                        bits |= get_bit(g_scalar, i + 96) << 1;
                        bits |= get_bit(g_scalar, i + 32);
                        /* select the point to add, in constant time */
                        select_point(bits, 16, g_pre_comp[1], tmp);

                        if (!skip) {
                                point_add(nq[0], nq[1], nq[2],
                                    nq[0], nq[1], nq[2],
                                    1 /* mixed */ , tmp[0], tmp[1], tmp[2]);
                        } else {
                                smallfelem_expand(nq[0], tmp[0]);
                                smallfelem_expand(nq[1], tmp[1]);
                                smallfelem_expand(nq[2], tmp[2]);
                                skip = 0;
                        }

                        /* second, look at the current position */
                        bits = get_bit(g_scalar, i + 192) << 3;
                        bits |= get_bit(g_scalar, i + 128) << 2;
                        bits |= get_bit(g_scalar, i + 64) << 1;
                        bits |= get_bit(g_scalar, i);
                        /* select the point to add, in constant time */
                        select_point(bits, 16, g_pre_comp[0], tmp);
                        point_add(nq[0], nq[1], nq[2],
                            nq[0], nq[1], nq[2],
                            1 /* mixed */ , tmp[0], tmp[1], tmp[2]);
                }
                /* 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);
                                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);
                                smallfelem_neg(ftmp, tmp[1]);   /* (X, -Y, Z) is the
                                                                 * negative point */
                                copy_small_conditional(ftmp, tmp[1], (((limb) sign) - 1));
                                felem_contract(tmp[1], ftmp);

                                if (!skip) {
                                        point_add(nq[0], nq[1], nq[2],
                                            nq[0], nq[1], nq[2],
                                            mixed, tmp[0], tmp[1], tmp[2]);
                                } else {
                                        smallfelem_expand(nq[0], tmp[0]);
                                        smallfelem_expand(nq[1], tmp[1]);
                                        smallfelem_expand(nq[2], tmp[2]);
                                        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. */
typedef struct {
        smallfelem g_pre_comp[2][16][3];
        int references;
} NISTP256_PRE_COMP;

const EC_METHOD *
EC_GFp_nistp256_method(void)
{
        static const EC_METHOD ret = {
                .flags = EC_FLAGS_DEFAULT_OCT,
                .field_type = NID_X9_62_prime_field,
                .group_init = ec_GFp_nistp256_group_init,
                .group_finish = ec_GFp_simple_group_finish,
                .group_clear_finish = ec_GFp_simple_group_clear_finish,
                .group_copy = ec_GFp_nist_group_copy,
                .group_set_curve = ec_GFp_nistp256_group_set_curve,
                .group_get_curve = ec_GFp_simple_group_get_curve,
                .group_get_degree = ec_GFp_simple_group_get_degree,
                .group_check_discriminant =
                ec_GFp_simple_group_check_discriminant,
                .point_init = ec_GFp_simple_point_init,
                .point_finish = ec_GFp_simple_point_finish,
                .point_clear_finish = ec_GFp_simple_point_clear_finish,
                .point_copy = ec_GFp_simple_point_copy,
                .point_set_to_infinity = ec_GFp_simple_point_set_to_infinity,
                .point_set_Jprojective_coordinates_GFp =
                ec_GFp_simple_set_Jprojective_coordinates_GFp,
                .point_get_Jprojective_coordinates_GFp =
                ec_GFp_simple_get_Jprojective_coordinates_GFp,
                .point_set_affine_coordinates =
                ec_GFp_simple_point_set_affine_coordinates,
                .point_get_affine_coordinates =
                ec_GFp_nistp256_point_get_affine_coordinates,
                .add = ec_GFp_simple_add,
                .dbl = ec_GFp_simple_dbl,
                .invert = ec_GFp_simple_invert,
                .is_at_infinity = ec_GFp_simple_is_at_infinity,
                .is_on_curve = ec_GFp_simple_is_on_curve,
                .point_cmp = ec_GFp_simple_cmp,
                .make_affine = ec_GFp_simple_make_affine,
                .points_make_affine = ec_GFp_simple_points_make_affine,
                .mul = ec_GFp_nistp256_points_mul,
                .precompute_mult = ec_GFp_nistp256_precompute_mult,
                .have_precompute_mult = ec_GFp_nistp256_have_precompute_mult,
                .field_mul = ec_GFp_nist_field_mul,
                .field_sqr = ec_GFp_nist_field_sqr
        };

        return &ret;
}

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

static NISTP256_PRE_COMP *
nistp256_pre_comp_new()
{
        NISTP256_PRE_COMP *ret = NULL;
        ret = malloc(sizeof *ret);
        if (!ret) {
                ECerror(ERR_R_MALLOC_FAILURE);
                return ret;
        }
        memset(ret->g_pre_comp, 0, sizeof(ret->g_pre_comp));
        ret->references = 1;
        return ret;
}

static void *
nistp256_pre_comp_dup(void *src_)
{
        NISTP256_PRE_COMP *src = src_;

        /* no need to actually copy, these objects never change! */
        CRYPTO_add(&src->references, 1, CRYPTO_LOCK_EC_PRE_COMP);

        return src_;
}

static void 
nistp256_pre_comp_free(void *pre_)
{
        int i;
        NISTP256_PRE_COMP *pre = pre_;

        if (!pre)
                return;

        i = CRYPTO_add(&pre->references, -1, CRYPTO_LOCK_EC_PRE_COMP);
        if (i > 0)
                return;

        free(pre);
}

static void 
nistp256_pre_comp_clear_free(void *pre_)
{
        int i;
        NISTP256_PRE_COMP *pre = pre_;

        if (!pre)
                return;

        i = CRYPTO_add(&pre->references, -1, CRYPTO_LOCK_EC_PRE_COMP);
        if (i > 0)
                return;

        freezero(pre, sizeof *pre);
}

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

int 
ec_GFp_nistp256_group_init(EC_GROUP * group)
{
        int ret;
        ret = ec_GFp_simple_group_init(group);
        group->a_is_minus3 = 1;
        return ret;
}

int 
ec_GFp_nistp256_group_set_curve(EC_GROUP * group, const BIGNUM * p,
    const BIGNUM * a, const BIGNUM * b, BN_CTX * ctx)
{
        int ret = 0;
        BN_CTX *new_ctx = NULL;
        BIGNUM *curve_p, *curve_a, *curve_b;

        if (ctx == NULL)
                if ((ctx = new_ctx = BN_CTX_new()) == NULL)
                        return 0;
        BN_CTX_start(ctx);
        if (((curve_p = BN_CTX_get(ctx)) == NULL) ||
            ((curve_a = BN_CTX_get(ctx)) == NULL) ||
            ((curve_b = BN_CTX_get(ctx)) == NULL))
                goto err;
        BN_bin2bn(nistp256_curve_params[0], sizeof(felem_bytearray), curve_p);
        BN_bin2bn(nistp256_curve_params[1], sizeof(felem_bytearray), curve_a);
        BN_bin2bn(nistp256_curve_params[2], sizeof(felem_bytearray), curve_b);
        if ((BN_cmp(curve_p, p)) || (BN_cmp(curve_a, a)) ||
            (BN_cmp(curve_b, b))) {
                ECerror(EC_R_WRONG_CURVE_PARAMETERS);
                goto err;
        }
        group->field_mod_func = BN_nist_mod_256;
        ret = ec_GFp_simple_group_set_curve(group, p, a, b, ctx);
err:
        BN_CTX_end(ctx);
        BN_CTX_free(new_ctx);
        return ret;
}

/* Takes the Jacobian coordinates (X, Y, Z) of a point and returns
 * (X', Y') = (X/Z^2, Y/Z^3) */
int 
ec_GFp_nistp256_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;
        smallfelem x_out, y_out;
        longfelem tmp;

        if (EC_POINT_is_at_infinity(group, point) > 0) {
                ECerror(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 (!smallfelem_to_BN(x, x_out)) {
                        ECerror(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 (!smallfelem_to_BN(y, y_out)) {
                        ECerror(ERR_R_BN_LIB);
                        return 0;
                }
        }
        return 1;
}

static void 
make_points_affine(size_t num, smallfelem points[ /* num */ ][3], smallfelem tmp_smallfelems[ /* num+1 */ ])
{
        /*
         * Runs in constant time, unless an input is the point at infinity
         * (which normally shouldn't happen).
         */
        ec_GFp_nistp_points_make_affine_internal(
            num,
            points,
            sizeof(smallfelem),
            tmp_smallfelems,
            (void (*) (void *)) smallfelem_one,
            (int (*) (const void *)) smallfelem_is_zero_int,
            (void (*) (void *, const void *)) smallfelem_assign,
            (void (*) (void *, const void *)) smallfelem_square_contract,
            (void (*) (void *, const void *, const void *)) smallfelem_mul_contract,
            (void (*) (void *, const void *)) smallfelem_inv_contract,
            (void (*) (void *, const void *)) smallfelem_assign /* nothing to 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 
ec_GFp_nistp256_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;
        BN_CTX *new_ctx = NULL;
        BIGNUM *x, *y, *z, *tmp_scalar;
        felem_bytearray g_secret;
        felem_bytearray *secrets = NULL;
        smallfelem(*pre_comp)[17][3] = NULL;
        smallfelem *tmp_smallfelems = NULL;
        felem_bytearray tmp;
        unsigned i, num_bytes;
        int have_pre_comp = 0;
        size_t num_points = num;
        smallfelem x_in, y_in, z_in;
        felem x_out, y_out, z_out;
        NISTP256_PRE_COMP *pre = NULL;
        const smallfelem(*g_pre_comp)[16][3] = NULL;
        EC_POINT *generator = NULL;
        const EC_POINT *p = NULL;
        const BIGNUM *p_scalar = NULL;

        if (ctx == NULL)
                if ((ctx = new_ctx = BN_CTX_new()) == NULL)
                        return 0;
        BN_CTX_start(ctx);
        if (((x = BN_CTX_get(ctx)) == NULL) ||
            ((y = BN_CTX_get(ctx)) == NULL) ||
            ((z = BN_CTX_get(ctx)) == NULL) ||
            ((tmp_scalar = BN_CTX_get(ctx)) == NULL))
                goto err;

        if (scalar != NULL) {
                pre = EC_EX_DATA_get_data(group->extra_data,
                    nistp256_pre_comp_dup, nistp256_pre_comp_free,
                    nistp256_pre_comp_clear_free);
                if (pre)
                        /* we have precomputation, try to use it */
                        g_pre_comp = (const smallfelem(*)[16][3]) pre->g_pre_comp;
                else
                        /* try to use the standard precomputation */
                        g_pre_comp = &gmul[0];
                generator = EC_POINT_new(group);
                if (generator == NULL)
                        goto err;
                /* get the generator from precomputation */
                if (!smallfelem_to_BN(x, g_pre_comp[0][1][0]) ||
                    !smallfelem_to_BN(y, g_pre_comp[0][1][1]) ||
                    !smallfelem_to_BN(z, g_pre_comp[0][1][2])) {
                        ECerror(ERR_R_BN_LIB);
                        goto err;
                }
                if (!EC_POINT_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 >= 3) {
                        /*
                         * unless we precompute multiples for just one or two
                         * points, converting those into affine form is time
                         * well spent
                         */
                        mixed = 1;
                }
                secrets = calloc(num_points, sizeof(felem_bytearray));
                pre_comp = calloc(num_points, 17 * 3 * sizeof(smallfelem));
                if (mixed) {
                        /* XXX should do more int overflow checking */
                        tmp_smallfelems = reallocarray(NULL,
                            (num_points * 17 + 1), sizeof(smallfelem));
                }
                if ((secrets == NULL) || (pre_comp == NULL) || (mixed && (tmp_smallfelems == NULL))) {
                        ECerror(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 < 2^256 */
                                if ((BN_num_bits(p_scalar) > 256) || (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)) {
                                                ECerror(ERR_R_BN_LIB);
                                                goto err;
                                        }
                                        num_bytes = BN_bn2bin(tmp_scalar, tmp);
                                } else
                                        num_bytes = BN_bn2bin(p_scalar, tmp);
                                flip_endian(secrets[i], tmp, num_bytes);
                                /* 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;
                                felem_shrink(pre_comp[i][1][0], x_out);
                                felem_shrink(pre_comp[i][1][1], y_out);
                                felem_shrink(pre_comp[i][1][2], z_out);
                                for (j = 2; j <= 16; ++j) {
                                        if (j & 1) {
                                                point_add_small(
                                                    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],
                                                    pre_comp[i][j - 1][0], pre_comp[i][j - 1][1], pre_comp[i][j - 1][2]);
                                        } else {
                                                point_double_small(
                                                    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_smallfelems);
        }
        /* the scalar for the generator */
        if ((scalar != NULL) && (have_pre_comp)) {
                memset(g_secret, 0, sizeof(g_secret));
                /* reduce scalar to 0 <= scalar < 2^256 */
                if ((BN_num_bits(scalar) > 256) || (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)) {
                                ECerror(ERR_R_BN_LIB);
                                goto err;
                        }
                        num_bytes = BN_bn2bin(tmp_scalar, tmp);
                } else
                        num_bytes = BN_bn2bin(scalar, tmp);
                flip_endian(g_secret, tmp, num_bytes);
                /* do the multiplication with generator precomputation */
                batch_mul(x_out, y_out, z_out,
                    (const felem_bytearray(*)) secrets, num_points,
                    g_secret,
                    mixed, (const smallfelem(*)[17][3]) pre_comp,
                    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 smallfelem(*)[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 ((!smallfelem_to_BN(x, x_in)) || (!smallfelem_to_BN(y, y_in)) ||
            (!smallfelem_to_BN(z, z_in))) {
                ECerror(ERR_R_BN_LIB);
                goto err;
        }
        ret = EC_POINT_set_Jprojective_coordinates_GFp(group, r, x, y, z, ctx);

err:
        BN_CTX_end(ctx);
        EC_POINT_free(generator);
        BN_CTX_free(new_ctx);
        free(secrets);
        free(pre_comp);
        free(tmp_smallfelems);
        return ret;
}

int 
ec_GFp_nistp256_precompute_mult(EC_GROUP * group, BN_CTX * ctx)
{
        int ret = 0;
        NISTP256_PRE_COMP *pre = NULL;
        int i, j;
        BN_CTX *new_ctx = NULL;
        BIGNUM *x, *y;
        EC_POINT *generator = NULL;
        smallfelem tmp_smallfelems[32];
        felem x_tmp, y_tmp, z_tmp;

        /* throw away old precomputation */
        EC_EX_DATA_free_data(&group->extra_data, nistp256_pre_comp_dup,
            nistp256_pre_comp_free, nistp256_pre_comp_clear_free);
        if (ctx == NULL)
                if ((ctx = new_ctx = BN_CTX_new()) == NULL)
                        return 0;
        BN_CTX_start(ctx);
        if (((x = BN_CTX_get(ctx)) == NULL) ||
            ((y = BN_CTX_get(ctx)) == NULL))
                goto err;
        /* get the generator */
        if (group->generator == NULL)
                goto err;
        generator = EC_POINT_new(group);
        if (generator == NULL)
                goto err;
        BN_bin2bn(nistp256_curve_params[3], sizeof(felem_bytearray), x);
        BN_bin2bn(nistp256_curve_params[4], sizeof(felem_bytearray), y);
        if (!EC_POINT_set_affine_coordinates_GFp(group, generator, x, y, ctx))
                goto err;
        if ((pre = nistp256_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));
                ret = 1;
                goto err;
        }
        if ((!BN_to_felem(x_tmp, &group->generator->X)) ||
            (!BN_to_felem(y_tmp, &group->generator->Y)) ||
            (!BN_to_felem(z_tmp, &group->generator->Z)))
                goto err;
        felem_shrink(pre->g_pre_comp[0][1][0], x_tmp);
        felem_shrink(pre->g_pre_comp[0][1][1], y_tmp);
        felem_shrink(pre->g_pre_comp[0][1][2], z_tmp);
        /*
         * compute 2^64*G, 2^128*G, 2^192*G for the first table, 2^32*G,
         * 2^96*G, 2^160*G, 2^224*G for the second one
         */
        for (i = 1; i <= 8; i <<= 1) {
                point_double_small(
                    pre->g_pre_comp[1][i][0], pre->g_pre_comp[1][i][1], pre->g_pre_comp[1][i][2],
                    pre->g_pre_comp[0][i][0], pre->g_pre_comp[0][i][1], pre->g_pre_comp[0][i][2]);
                for (j = 0; j < 31; ++j) {
                        point_double_small(
                            pre->g_pre_comp[1][i][0], pre->g_pre_comp[1][i][1], pre->g_pre_comp[1][i][2],
                            pre->g_pre_comp[1][i][0], pre->g_pre_comp[1][i][1], pre->g_pre_comp[1][i][2]);
                }
                if (i == 8)
                        break;
                point_double_small(
                    pre->g_pre_comp[0][2 * i][0], pre->g_pre_comp[0][2 * i][1], pre->g_pre_comp[0][2 * i][2],
                    pre->g_pre_comp[1][i][0], pre->g_pre_comp[1][i][1], pre->g_pre_comp[1][i][2]);
                for (j = 0; j < 31; ++j) {
                        point_double_small(
                            pre->g_pre_comp[0][2 * i][0], pre->g_pre_comp[0][2 * i][1], pre->g_pre_comp[0][2 * i][2],
                            pre->g_pre_comp[0][2 * i][0], pre->g_pre_comp[0][2 * i][1], pre->g_pre_comp[0][2 * i][2]);
                }
        }
        for (i = 0; i < 2; i++) {
                /* g_pre_comp[i][0] is the point at infinity */
                memset(pre->g_pre_comp[i][0], 0, sizeof(pre->g_pre_comp[i][0]));
                /* the remaining multiples */
                /* 2^64*G + 2^128*G resp. 2^96*G + 2^160*G */
                point_add_small(
                    pre->g_pre_comp[i][6][0], pre->g_pre_comp[i][6][1], pre->g_pre_comp[i][6][2],
                    pre->g_pre_comp[i][4][0], pre->g_pre_comp[i][4][1], pre->g_pre_comp[i][4][2],
                    pre->g_pre_comp[i][2][0], pre->g_pre_comp[i][2][1], pre->g_pre_comp[i][2][2]);
                /* 2^64*G + 2^192*G resp. 2^96*G + 2^224*G */
                point_add_small(
                    pre->g_pre_comp[i][10][0], pre->g_pre_comp[i][10][1], pre->g_pre_comp[i][10][2],
                    pre->g_pre_comp[i][8][0], pre->g_pre_comp[i][8][1], pre->g_pre_comp[i][8][2],
                    pre->g_pre_comp[i][2][0], pre->g_pre_comp[i][2][1], pre->g_pre_comp[i][2][2]);
                /* 2^128*G + 2^192*G resp. 2^160*G + 2^224*G */
                point_add_small(
                    pre->g_pre_comp[i][12][0], pre->g_pre_comp[i][12][1], pre->g_pre_comp[i][12][2],
                    pre->g_pre_comp[i][8][0], pre->g_pre_comp[i][8][1], pre->g_pre_comp[i][8][2],
                    pre->g_pre_comp[i][4][0], pre->g_pre_comp[i][4][1], pre->g_pre_comp[i][4][2]);
                /*
                 * 2^64*G + 2^128*G + 2^192*G resp. 2^96*G + 2^160*G +
                 * 2^224*G
                 */
                point_add_small(
                    pre->g_pre_comp[i][14][0], pre->g_pre_comp[i][14][1], pre->g_pre_comp[i][14][2],
                    pre->g_pre_comp[i][12][0], pre->g_pre_comp[i][12][1], pre->g_pre_comp[i][12][2],
                    pre->g_pre_comp[i][2][0], pre->g_pre_comp[i][2][1], pre->g_pre_comp[i][2][2]);
                for (j = 1; j < 8; ++j) {
                        /* odd multiples: add G resp. 2^32*G */
                        point_add_small(
                            pre->g_pre_comp[i][2 * j + 1][0], pre->g_pre_comp[i][2 * j + 1][1], pre->g_pre_comp[i][2 * j + 1][2],
                            pre->g_pre_comp[i][2 * j][0], pre->g_pre_comp[i][2 * j][1], pre->g_pre_comp[i][2 * j][2],
                            pre->g_pre_comp[i][1][0], pre->g_pre_comp[i][1][1], pre->g_pre_comp[i][1][2]);
                }
        }
        make_points_affine(31, &(pre->g_pre_comp[0][1]), tmp_smallfelems);

        if (!EC_EX_DATA_set_data(&group->extra_data, pre, nistp256_pre_comp_dup,
                nistp256_pre_comp_free, nistp256_pre_comp_clear_free))
                goto err;
        ret = 1;
        pre = NULL;
err:
        BN_CTX_end(ctx);
        EC_POINT_free(generator);
        BN_CTX_free(new_ctx);
        nistp256_pre_comp_free(pre);
        return ret;
}

int 
ec_GFp_nistp256_have_precompute_mult(const EC_GROUP * group)
{
        if (EC_EX_DATA_get_data(group->extra_data, nistp256_pre_comp_dup,
                nistp256_pre_comp_free, nistp256_pre_comp_clear_free)
            != NULL)
                return 1;
        else
                return 0;
}
#endif