ecp_nistp521.c

/* crypto/ec/ecp_nistp521.c */
/*
 * Written by Adam Langley (Google) for the OpenSSL project
 */
/* Copyright 2011 Google Inc.
 *
 * Licensed under the Apache License, Version 2.0 (the "License");
 *
 * you may not use this file except in compliance with the License.
 * You may obtain a copy of the License at
 *
 *     http://www.apache.org/licenses/LICENSE-2.0
 *
 *  Unless required by applicable law or agreed to in writing, software
 *  distributed under the License is distributed on an "AS IS" BASIS,
 *  WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 *  See the License for the specific language governing permissions and
 *  limitations under the License.
 */

/*
 * 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/opensslconf.h>
#ifndef OPENSSL_NO_EC_NISTP_64_GCC_128

#ifndef OPENSSL_SYS_VMS
#include <stdint.h>
#else
#include <inttypes.h>
#endif

#include <string.h>
#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 */
#else
  #error "Need GCC 3.1 or later to define type uint128_t"
#endif

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

/* The underlying field.
 *
 * P521 operates over GF(2^521-1). We can serialise 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 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*) &in[7]) >> 2) & bottom58bits;
	out[2] = (*((limb*) &in[14]) >> 4) & bottom58bits;
	out[3] = (*((limb*) &in[21]) >> 6) & bottom58bits;
	out[4] = (*((limb*) &in[29])) & bottom58bits;
	out[5] = (*((limb*) &in[36]) >> 2) & bottom58bits;
	out[6] = (*((limb*) &in[43]) >> 4) & bottom58bits;
	out[7] = (*((limb*) &in[50]) >> 6) & bottom58bits;
	out[8] = (*((limb*) &in[58])) & bottom57bits;
	}

/* felem_to_bin66 takes an felem and serialises 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*) &out[7])) |= in[1] << 2;
	(*((limb*) &out[14])) |= in[2] << 4;
	(*((limb*) &out[21])) |= in[3] << 6;
	(*((limb*) &out[29])) = in[4];
	(*((limb*) &out[36])) |= in[5] << 2;
	(*((limb*) &out[43])) |= in[6] << 4;
	(*((limb*) &out[50])) |= in[7] << 6;
	(*((limb*) &out[58])) = in[8];
	}

/* 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)
		{
		ECerr(EC_F_BN_TO_FELEM, EC_R_BIGNUM_OUT_OF_RANGE);
		return 0;
		}
	if (BN_is_negative(bn))
		{
		ECerr(EC_F_BN_TO_FELEM, EC_R_BIGNUM_OUT_OF_RANGE);
		return 0;
		}
	num_bytes = BN_bn2bin(bn, b_in);
	flip_endian(b_out, b_in, num_bytes);
	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_in, b_out;
	felem_to_bin66(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 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 0 mod p before subtracting. */
	static const limb two63m6 = (((limb)1) << 62) - (((limb)1) << 5);
	static const limb two63m5 = (((limb)1) << 62) - (((limb)1) << 4);

	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(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(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 */
	}

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_assign(ftmp4, ftmp3);
	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 = ((s64) 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 = ((s64) is_p) >> 63;

	is_zero |= is_p;
	return is_zero;
	}

static int felem_is_zero_int(const felem 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 = ((s64) 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 = ((s64) 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 calcuates 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 */