ecp_nistp224.c

/* crypto/ec/ecp_nistp224.c */
/*
 * Written by Emilia Kasper (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-224 elliptic curve point multiplication
 *
 * Inspired by Daniel J. Bernstein's public domain nistp224 implementation
 * and Adam Langley's public domain 64-bit C implementation of curve25519
 */

#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;


/******************************************************************************/
/*		    INTERNAL REPRESENTATION OF FIELD ELEMENTS
 *
 * Field elements are represented as a_0 + 2^56*a_1 + 2^112*a_2 + 2^168*a_3
 * using 64-bit coefficients called 'limbs',
 * and sometimes (for multiplication results) as
 * b_0 + 2^56*b_1 + 2^112*b_2 + 2^168*b_3 + 2^224*b_4 + 2^280*b_5 + 2^336*b_6
 * using 128-bit coefficients called 'widelimbs'.
 * A 4-limb representation is an 'felem';
 * a 7-widelimb representation is a 'widefelem'.
 * Even within felems, bits of adjacent limbs overlap, and we don't always
 * reduce the representations: we ensure that inputs to each felem
 * multiplication satisfy a_i < 2^60, so outputs satisfy b_i < 4*2^60*2^60,
 * and fit into a 128-bit word without overflow. The coefficients are then
 * again partially reduced to obtain an felem satisfying a_i < 2^57.
 * We only reduce to the unique minimal representation at the end of the
 * computation.
 */

typedef uint64_t limb;
typedef uint128_t widelimb;

typedef limb felem[4];
typedef widelimb widefelem[7];

/* Field element represented as a byte arrary.
 * 28*8 = 224 bits is also the group order size for the elliptic curve,
 * and we also use this type for scalars for point multiplication.
  */
typedef u8 felem_bytearray[28];

static const felem_bytearray nistp224_curve_params[5] = {
	{0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,    /* p */
	 0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0x00,0x00,0x00,0x00,
	 0x00,0x00,0x00,0x00,0x00,0x00,0x00,0x01},
	{0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,    /* a */
	 0xFF,0xFF,0xFF,0xFF,0xFF,0xFE,0xFF,0xFF,0xFF,0xFF,
	 0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFE},
	{0xB4,0x05,0x0A,0x85,0x0C,0x04,0xB3,0xAB,0xF5,0x41,    /* b */
	 0x32,0x56,0x50,0x44,0xB0,0xB7,0xD7,0xBF,0xD8,0xBA,
	 0x27,0x0B,0x39,0x43,0x23,0x55,0xFF,0xB4},
	{0xB7,0x0E,0x0C,0xBD,0x6B,0xB4,0xBF,0x7F,0x32,0x13,    /* x */
	 0x90,0xB9,0x4A,0x03,0xC1,0xD3,0x56,0xC2,0x11,0x22,
	 0x34,0x32,0x80,0xD6,0x11,0x5C,0x1D,0x21},
	{0xbd,0x37,0x63,0x88,0xb5,0xf7,0x23,0xfb,0x4c,0x22,    /* y */
	 0xdf,0xe6,0xcd,0x43,0x75,0xa0,0x5a,0x07,0x47,0x64,
	 0x44,0xd5,0x81,0x99,0x85,0x00,0x7e,0x34}
};

/* Precomputed multiples of the standard generator
 * Points are given in coordinates (X, Y, Z) where Z normally is 1
 * (0 for the point at infinity).
 * For each field element, slice a_0 is word 0, etc.
 *
 * The 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 | 2^56G
 *     3 | 0 0 1 1 | (2^56 + 1)G
 *     4 | 0 1 0 0 | 2^112G
 *     5 | 0 1 0 1 | (2^112 + 1)G
 *     6 | 0 1 1 0 | (2^112 + 2^56)G
 *     7 | 0 1 1 1 | (2^112 + 2^56 + 1)G
 *     8 | 1 0 0 0 | 2^168G
 *     9 | 1 0 0 1 | (2^168 + 1)G
 *    10 | 1 0 1 0 | (2^168 + 2^56)G
 *    11 | 1 0 1 1 | (2^168 + 2^56 + 1)G
 *    12 | 1 1 0 0 | (2^168 + 2^112)G
 *    13 | 1 1 0 1 | (2^168 + 2^112 + 1)G
 *    14 | 1 1 1 0 | (2^168 + 2^112 + 2^56)G
 *    15 | 1 1 1 1 | (2^168 + 2^112 + 2^56 + 1)G
 * followed by a copy of this with each element multiplied by 2^28.
 *
 * 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.
 */
static const felem gmul[2][16][3] =
{{{{0, 0, 0, 0},
   {0, 0, 0, 0},
   {0, 0, 0, 0}},
  {{0x3280d6115c1d21, 0xc1d356c2112234, 0x7f321390b94a03, 0xb70e0cbd6bb4bf},
   {0xd5819985007e34, 0x75a05a07476444, 0xfb4c22dfe6cd43, 0xbd376388b5f723},
   {1, 0, 0, 0}},
  {{0xfd9675666ebbe9, 0xbca7664d40ce5e, 0x2242df8d8a2a43, 0x1f49bbb0f99bc5},
   {0x29e0b892dc9c43, 0xece8608436e662, 0xdc858f185310d0, 0x9812dd4eb8d321},
   {1, 0, 0, 0}},
  {{0x6d3e678d5d8eb8, 0x559eed1cb362f1, 0x16e9a3bbce8a3f, 0xeedcccd8c2a748},
   {0xf19f90ed50266d, 0xabf2b4bf65f9df, 0x313865468fafec, 0x5cb379ba910a17},
   {1, 0, 0, 0}},
  {{0x0641966cab26e3, 0x91fb2991fab0a0, 0xefec27a4e13a0b, 0x0499aa8a5f8ebe},
   {0x7510407766af5d, 0x84d929610d5450, 0x81d77aae82f706, 0x6916f6d4338c5b},
   {1, 0, 0, 0}},
  {{0xea95ac3b1f15c6, 0x086000905e82d4, 0xdd323ae4d1c8b1, 0x932b56be7685a3},
   {0x9ef93dea25dbbf, 0x41665960f390f0, 0xfdec76dbe2a8a7, 0x523e80f019062a},
   {1, 0, 0, 0}},
  {{0x822fdd26732c73, 0xa01c83531b5d0f, 0x363f37347c1ba4, 0xc391b45c84725c},
   {0xbbd5e1b2d6ad24, 0xddfbcde19dfaec, 0xc393da7e222a7f, 0x1efb7890ede244},
   {1, 0, 0, 0}},
  {{0x4c9e90ca217da1, 0xd11beca79159bb, 0xff8d33c2c98b7c, 0x2610b39409f849},
   {0x44d1352ac64da0, 0xcdbb7b2c46b4fb, 0x966c079b753c89, 0xfe67e4e820b112},
   {1, 0, 0, 0}},
  {{0xe28cae2df5312d, 0xc71b61d16f5c6e, 0x79b7619a3e7c4c, 0x05c73240899b47},
   {0x9f7f6382c73e3a, 0x18615165c56bda, 0x641fab2116fd56, 0x72855882b08394},
   {1, 0, 0, 0}},
  {{0x0469182f161c09, 0x74a98ca8d00fb5, 0xb89da93489a3e0, 0x41c98768fb0c1d},
   {0xe5ea05fb32da81, 0x3dce9ffbca6855, 0x1cfe2d3fbf59e6, 0x0e5e03408738a7},
   {1, 0, 0, 0}},
  {{0xdab22b2333e87f, 0x4430137a5dd2f6, 0xe03ab9f738beb8, 0xcb0c5d0dc34f24},
   {0x764a7df0c8fda5, 0x185ba5c3fa2044, 0x9281d688bcbe50, 0xc40331df893881},
   {1, 0, 0, 0}},
  {{0xb89530796f0f60, 0xade92bd26909a3, 0x1a0c83fb4884da, 0x1765bf22a5a984},
   {0x772a9ee75db09e, 0x23bc6c67cec16f, 0x4c1edba8b14e2f, 0xe2a215d9611369},
   {1, 0, 0, 0}},
  {{0x571e509fb5efb3, 0xade88696410552, 0xc8ae85fada74fe, 0x6c7e4be83bbde3},
   {0xff9f51160f4652, 0xb47ce2495a6539, 0xa2946c53b582f4, 0x286d2db3ee9a60},
   {1, 0, 0, 0}},
  {{0x40bbd5081a44af, 0x0995183b13926c, 0xbcefba6f47f6d0, 0x215619e9cc0057},
   {0x8bc94d3b0df45e, 0xf11c54a3694f6f, 0x8631b93cdfe8b5, 0xe7e3f4b0982db9},
   {1, 0, 0, 0}},
  {{0xb17048ab3e1c7b, 0xac38f36ff8a1d8, 0x1c29819435d2c6, 0xc813132f4c07e9},
   {0x2891425503b11f, 0x08781030579fea, 0xf5426ba5cc9674, 0x1e28ebf18562bc},
   {1, 0, 0, 0}},
  {{0x9f31997cc864eb, 0x06cd91d28b5e4c, 0xff17036691a973, 0xf1aef351497c58},
   {0xdd1f2d600564ff, 0xdead073b1402db, 0x74a684435bd693, 0xeea7471f962558},
   {1, 0, 0, 0}}},
 {{{0, 0, 0, 0},
   {0, 0, 0, 0},
   {0, 0, 0, 0}},
  {{0x9665266dddf554, 0x9613d78b60ef2d, 0xce27a34cdba417, 0xd35ab74d6afc31},
   {0x85ccdd22deb15e, 0x2137e5783a6aab, 0xa141cffd8c93c6, 0x355a1830e90f2d},
   {1, 0, 0, 0}},
  {{0x1a494eadaade65, 0xd6da4da77fe53c, 0xe7992996abec86, 0x65c3553c6090e3},
   {0xfa610b1fb09346, 0xf1c6540b8a4aaf, 0xc51a13ccd3cbab, 0x02995b1b18c28a},
   {1, 0, 0, 0}},
  {{0x7874568e7295ef, 0x86b419fbe38d04, 0xdc0690a7550d9a, 0xd3966a44beac33},
   {0x2b7280ec29132f, 0xbeaa3b6a032df3, 0xdc7dd88ae41200, 0xd25e2513e3a100},
   {1, 0, 0, 0}},
  {{0x924857eb2efafd, 0xac2bce41223190, 0x8edaa1445553fc, 0x825800fd3562d5},
   {0x8d79148ea96621, 0x23a01c3dd9ed8d, 0xaf8b219f9416b5, 0xd8db0cc277daea},
   {1, 0, 0, 0}},
  {{0x76a9c3b1a700f0, 0xe9acd29bc7e691, 0x69212d1a6b0327, 0x6322e97fe154be},
   {0x469fc5465d62aa, 0x8d41ed18883b05, 0x1f8eae66c52b88, 0xe4fcbe9325be51},
   {1, 0, 0, 0}},
  {{0x825fdf583cac16, 0x020b857c7b023a, 0x683c17744b0165, 0x14ffd0a2daf2f1},
   {0x323b36184218f9, 0x4944ec4e3b47d4, 0xc15b3080841acf, 0x0bced4b01a28bb},
   {1, 0, 0, 0}},
  {{0x92ac22230df5c4, 0x52f33b4063eda8, 0xcb3f19870c0c93, 0x40064f2ba65233},
   {0xfe16f0924f8992, 0x012da25af5b517, 0x1a57bb24f723a6, 0x06f8bc76760def},
   {1, 0, 0, 0}},
  {{0x4a7084f7817cb9, 0xbcab0738ee9a78, 0x3ec11e11d9c326, 0xdc0fe90e0f1aae},
   {0xcf639ea5f98390, 0x5c350aa22ffb74, 0x9afae98a4047b7, 0x956ec2d617fc45},
   {1, 0, 0, 0}},
  {{0x4306d648c1be6a, 0x9247cd8bc9a462, 0xf5595e377d2f2e, 0xbd1c3caff1a52e},
   {0x045e14472409d0, 0x29f3e17078f773, 0x745a602b2d4f7d, 0x191837685cdfbb},
   {1, 0, 0, 0}},
  {{0x5b6ee254a8cb79, 0x4953433f5e7026, 0xe21faeb1d1def4, 0xc4c225785c09de},
   {0x307ce7bba1e518, 0x31b125b1036db8, 0x47e91868839e8f, 0xc765866e33b9f3},
   {1, 0, 0, 0}},
  {{0x3bfece24f96906, 0x4794da641e5093, 0xde5df64f95db26, 0x297ecd89714b05},
   {0x701bd3ebb2c3aa, 0x7073b4f53cb1d5, 0x13c5665658af16, 0x9895089d66fe58},
   {1, 0, 0, 0}},
  {{0x0fef05f78c4790, 0x2d773633b05d2e, 0x94229c3a951c94, 0xbbbd70df4911bb},
   {0xb2c6963d2c1168, 0x105f47a72b0d73, 0x9fdf6111614080, 0x7b7e94b39e67b0},
   {1, 0, 0, 0}},
  {{0xad1a7d6efbe2b3, 0xf012482c0da69d, 0x6b3bdf12438345, 0x40d7558d7aa4d9},
   {0x8a09fffb5c6d3d, 0x9a356e5d9ffd38, 0x5973f15f4f9b1c, 0xdcd5f59f63c3ea},
   {1, 0, 0, 0}},
  {{0xacf39f4c5ca7ab, 0x4c8071cc5fd737, 0xc64e3602cd1184, 0x0acd4644c9abba},
   {0x6c011a36d8bf6e, 0xfecd87ba24e32a, 0x19f6f56574fad8, 0x050b204ced9405},
   {1, 0, 0, 0}},
  {{0xed4f1cae7d9a96, 0x5ceef7ad94c40a, 0x778e4a3bf3ef9b, 0x7405783dc3b55e},
   {0x32477c61b6e8c6, 0xb46a97570f018b, 0x91176d0a7e95d1, 0x3df90fbc4c7d0e},
   {1, 0, 0, 0}}}};

/* Precomputation for the group generator. */
typedef struct {
	felem g_pre_comp[2][16][3];
	int references;
} NISTP224_PRE_COMP;

const EC_METHOD *EC_GFp_nistp224_method(void)
	{
	static const EC_METHOD ret = {
		EC_FLAGS_DEFAULT_OCT,
		NID_X9_62_prime_field,
		ec_GFp_nistp224_group_init,
		ec_GFp_simple_group_finish,
		ec_GFp_simple_group_clear_finish,
		ec_GFp_nist_group_copy,
		ec_GFp_nistp224_group_set_curve,
		ec_GFp_simple_group_get_curve,
		ec_GFp_simple_group_get_degree,
		ec_GFp_simple_group_check_discriminant,
		ec_GFp_simple_point_init,
		ec_GFp_simple_point_finish,
		ec_GFp_simple_point_clear_finish,
		ec_GFp_simple_point_copy,
		ec_GFp_simple_point_set_to_infinity,
		ec_GFp_simple_set_Jprojective_coordinates_GFp,
		ec_GFp_simple_get_Jprojective_coordinates_GFp,
		ec_GFp_simple_point_set_affine_coordinates,
		ec_GFp_nistp224_point_get_affine_coordinates,
		0 /* point_set_compressed_coordinates */,
		0 /* point2oct */,
		0 /* oct2point */,
		ec_GFp_simple_add,
		ec_GFp_simple_dbl,
		ec_GFp_simple_invert,
		ec_GFp_simple_is_at_infinity,
		ec_GFp_simple_is_on_curve,
		ec_GFp_simple_cmp,
		ec_GFp_simple_make_affine,
		ec_GFp_simple_points_make_affine,
		ec_GFp_nistp224_points_mul,
		ec_GFp_nistp224_precompute_mult,
		ec_GFp_nistp224_have_precompute_mult,
		ec_GFp_nist_field_mul,
		ec_GFp_nist_field_sqr,
		0 /* field_div */,
		0 /* field_encode */,
		0 /* field_decode */,
		0 /* field_set_to_one */ };

	return &ret;
	}

/* Helper functions to convert field elements to/from internal representation */
static void bin28_to_felem(felem out, const u8 in[28])
	{
	out[0] = *((const uint64_t *)(in)) & 0x00ffffffffffffff;
	out[1] = (*((const uint64_t *)(in+7))) & 0x00ffffffffffffff;
	out[2] = (*((const uint64_t *)(in+14))) & 0x00ffffffffffffff;
	out[3] = (*((const uint64_t *)(in+21))) & 0x00ffffffffffffff;
	}

static void felem_to_bin28(u8 out[28], const felem in)
	{
	unsigned i;
	for (i = 0; i < 7; ++i)
		{
		out[i]	  = in[0]>>(8*i);
		out[i+7]  = in[1]>>(8*i);
		out[i+14] = in[2]>>(8*i);
		out[i+21] = in[3]>>(8*i);
		}
	}

/* 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];
	}

/* From OpenSSL BIGNUM to internal representation */
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);
	bin28_to_felem(out, b_out);
	return 1;
	}

/* From internal representation to OpenSSL BIGNUM */
static BIGNUM *felem_to_BN(BIGNUM *out, const felem in)
	{
	felem_bytearray b_in, b_out;
	felem_to_bin28(b_in, in);
	flip_endian(b_out, b_in, sizeof b_out);
	return BN_bin2bn(b_out, sizeof b_out, out);
	}

/******************************************************************************/
/*				FIELD OPERATIONS
 *
 * Field operations, using the internal representation of field elements.
 * NB! These operations are specific to our point multiplication and cannot be
 * expected to be correct in general - e.g., multiplication with a large scalar
 * will cause an overflow.
 *
 */

static void felem_one(felem out)
	{
	out[0] = 1;
	out[1] = 0;
	out[2] = 0;
	out[3] = 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];
	}

/* Sum two field elements: 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];
	}

/* Get negative value: out = -in */
/* Assumes in[i] < 2^57 */
static void felem_neg(felem out, const felem in)
	{
	static const limb two58p2 = (((limb) 1) << 58) + (((limb) 1) << 2);
	static const limb two58m2 = (((limb) 1) << 58) - (((limb) 1) << 2);
	static const limb two58m42m2 = (((limb) 1) << 58) -
	    (((limb) 1) << 42) - (((limb) 1) << 2);

	/* Set to 0 mod 2^224-2^96+1 to ensure out > in */
	out[0] = two58p2 - in[0];
	out[1] = two58m42m2 - in[1];
	out[2] = two58m2 - in[2];
	out[3] = two58m2 - in[3];
	}

/* Subtract field elements: out -= in */
/* Assumes in[i] < 2^57 */
static void felem_diff(felem out, const felem in)
	{
	static const limb two58p2 = (((limb) 1) << 58) + (((limb) 1) << 2);
	static const limb two58m2 = (((limb) 1) << 58) - (((limb) 1) << 2);
	static const limb two58m42m2 = (((limb) 1) << 58) -
	    (((limb) 1) << 42) - (((limb) 1) << 2);

	/* Add 0 mod 2^224-2^96+1 to ensure out > in */
	out[0] += two58p2;
	out[1] += two58m42m2;
	out[2] += two58m2;
	out[3] += two58m2;

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

/* Subtract in unreduced 128-bit mode: out -= in */
/* Assumes in[i] < 2^119 */
static void widefelem_diff(widefelem out, const widefelem in)
	{
	static const widelimb two120 = ((widelimb) 1) << 120;
	static const widelimb two120m64 = (((widelimb) 1) << 120) -
		(((widelimb) 1) << 64);
	static const widelimb two120m104m64 = (((widelimb) 1) << 120) -
		(((widelimb) 1) << 104) - (((widelimb) 1) << 64);

	/* Add 0 mod 2^224-2^96+1 to ensure out > in */
	out[0] += two120;
	out[1] += two120m64;
	out[2] += two120m64;
	out[3] += two120;
	out[4] += two120m104m64;
	out[5] += two120m64;
	out[6] += two120m64;

	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];
	}

/* Subtract in mixed mode: out128 -= in64 */
/* in[i] < 2^63 */
static void felem_diff_128_64(widefelem out, const felem in)
	{
	static const widelimb two64p8 = (((widelimb) 1) << 64) +
		(((widelimb) 1) << 8);
	static const widelimb two64m8 = (((widelimb) 1) << 64) -
		(((widelimb) 1) << 8);
	static const widelimb two64m48m8 = (((widelimb) 1) << 64) -
		(((widelimb) 1) << 48) - (((widelimb) 1) << 8);

	/* Add 0 mod 2^224-2^96+1 to ensure out > in */
	out[0] += two64p8;
	out[1] += two64m48m8;
	out[2] += two64m8;
	out[3] += two64m8;

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

/* Multiply a field element by a scalar: out = out * scalar
 * The scalars we actually use are small, so results fit without overflow */
static void felem_scalar(felem out, const limb scalar)
	{
	out[0] *= scalar;
	out[1] *= scalar;
	out[2] *= scalar;
	out[3] *= scalar;
	}

/* Multiply an unreduced field element by a scalar: out = out * scalar
 * The scalars we actually use are small, so results fit without overflow */
static void widefelem_scalar(widefelem out, const widelimb scalar)
	{
	out[0] *= scalar;
	out[1] *= scalar;
	out[2] *= scalar;
	out[3] *= scalar;
	out[4] *= scalar;
	out[5] *= scalar;
	out[6] *= scalar;
	}

/* Square a field element: out = in^2 */
static void felem_square(widefelem out, const felem in)
	{
	limb tmp0, tmp1, tmp2;
	tmp0 = 2 * in[0]; tmp1 = 2 * in[1]; tmp2 = 2 * in[2];
	out[0] = ((widelimb) in[0]) * in[0];
	out[1] = ((widelimb) in[0]) * tmp1;
	out[2] = ((widelimb) in[0]) * tmp2 + ((widelimb) in[1]) * in[1];
	out[3] = ((widelimb) in[3]) * tmp0 +
		((widelimb) in[1]) * tmp2;
	out[4] = ((widelimb) in[3]) * tmp1 + ((widelimb) in[2]) * in[2];
	out[5] = ((widelimb) in[3]) * tmp2;
	out[6] = ((widelimb) in[3]) * in[3];
	}

/* Multiply two field elements: out = in1 * in2 */
static void felem_mul(widefelem out, const felem in1, const felem in2)
	{
	out[0] = ((widelimb) in1[0]) * in2[0];
	out[1] = ((widelimb) in1[0]) * in2[1] + ((widelimb) in1[1]) * in2[0];
	out[2] = ((widelimb) in1[0]) * in2[2] + ((widelimb) in1[1]) * in2[1] +
		((widelimb) in1[2]) * in2[0];
	out[3] = ((widelimb) in1[0]) * in2[3] + ((widelimb) in1[1]) * in2[2] +
		((widelimb) in1[2]) * in2[1] + ((widelimb) in1[3]) * in2[0];
	out[4] = ((widelimb) in1[1]) * in2[3] + ((widelimb) in1[2]) * in2[2] +
		((widelimb) in1[3]) * in2[1];
	out[5] = ((widelimb) in1[2]) * in2[3] + ((widelimb) in1[3]) * in2[2];
	out[6] = ((widelimb) in1[3]) * in2[3];
	}

/* Reduce seven 128-bit coefficients to four 64-bit coefficients.
 * Requires in[i] < 2^126,
 * ensures out[0] < 2^56, out[1] < 2^56, out[2] < 2^56, out[3] <= 2^56 + 2^16 */
static void felem_reduce(felem out, const widefelem in)
	{
	static const widelimb two127p15 = (((widelimb) 1) << 127) +
		(((widelimb) 1) << 15);
	static const widelimb two127m71 = (((widelimb) 1) << 127) -
		(((widelimb) 1) << 71);
	static const widelimb two127m71m55 = (((widelimb) 1) << 127) -
		(((widelimb) 1) << 71) - (((widelimb) 1) << 55);
	widelimb output[5];

	/* Add 0 mod 2^224-2^96+1 to ensure all differences are positive */
	output[0] = in[0] + two127p15;
	output[1] = in[1] + two127m71m55;
	output[2] = in[2] + two127m71;
	output[3] = in[3];
	output[4] = in[4];

	/* Eliminate in[4], in[5], in[6] */
	output[4] += in[6] >> 16;
	output[3] += (in[6] & 0xffff) << 40;
	output[2] -= in[6];

	output[3] += in[5] >> 16;
	output[2] += (in[5] & 0xffff) << 40;
	output[1] -= in[5];

	output[2] += output[4] >> 16;
	output[1] += (output[4] & 0xffff) << 40;
	output[0] -= output[4];

	/* Carry 2 -> 3 -> 4 */
	output[3] += output[2] >> 56;
	output[2] &= 0x00ffffffffffffff;

	output[4] = output[3] >> 56;
	output[3] &= 0x00ffffffffffffff;

	/* Now output[2] < 2^56, output[3] < 2^56, output[4] < 2^72 */

	/* Eliminate output[4] */
	output[2] += output[4] >> 16;
	/* output[2] < 2^56 + 2^56 = 2^57 */
	output[1] += (output[4] & 0xffff) << 40;
	output[0] -= output[4];

	/* Carry 0 -> 1 -> 2 -> 3 */
	output[1] += output[0] >> 56;
	out[0] = output[0] & 0x00ffffffffffffff;

	output[2] += output[1] >> 56;
	/* output[2] < 2^57 + 2^72 */
	out[1] = output[1] & 0x00ffffffffffffff;
	output[3] += output[2] >> 56;
	/* output[3] <= 2^56 + 2^16 */
	out[2] = output[2] & 0x00ffffffffffffff;

	/* out[0] < 2^56, out[1] < 2^56, out[2] < 2^56,
	 * out[3] <= 2^56 + 2^16 (due to final carry),
	 * so out < 2*p */
	out[3] = output[3];
	}

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

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

/* Reduce to unique minimal representation.
 * Requires 0 <= in < 2*p (always call felem_reduce first) */
static void felem_contract(felem out, const felem in)
	{
	static const int64_t two56 = ((limb) 1) << 56;
	/* 0 <= in < 2*p, p = 2^224 - 2^96 + 1 */
	/* if in > p , reduce in = in - 2^224 + 2^96 - 1 */
	int64_t tmp[4], a;
	tmp[0] = in[0];
	tmp[1] = in[1];
	tmp[2] = in[2];
	tmp[3] = in[3];
	/* Case 1: a = 1 iff in >= 2^224 */
	a = (in[3] >> 56);
	tmp[0] -= a;
	tmp[1] += a << 40;
	tmp[3] &= 0x00ffffffffffffff;
	/* Case 2: a = 0 iff p <= in < 2^224, i.e.,
	 * the high 128 bits are all 1 and the lower part is non-zero */
	a = ((in[3] & in[2] & (in[1] | 0x000000ffffffffff)) + 1) |
		(((int64_t)(in[0] + (in[1] & 0x000000ffffffffff)) - 1) >> 63);
	a &= 0x00ffffffffffffff;
	/* turn a into an all-one mask (if a = 0) or an all-zero mask */
	a = (a - 1) >> 63;
	/* subtract 2^224 - 2^96 + 1 if a is all-one*/
	tmp[3] &= a ^ 0xffffffffffffffff;
	tmp[2] &= a ^ 0xffffffffffffffff;
	tmp[1] &= (a ^ 0xffffffffffffffff) | 0x000000ffffffffff;
	tmp[0] -= 1 & a;

	/* eliminate negative coefficients: if tmp[0] is negative, tmp[1] must
	 * be non-zero, so we only need one step */
	a = tmp[0] >> 63;
	tmp[0] += two56 & a;
	tmp[1] -= 1 & a;

	/* carry 1 -> 2 -> 3 */
	tmp[2] += tmp[1] >> 56;
	tmp[1] &= 0x00ffffffffffffff;

	tmp[3] += tmp[2] >> 56;
	tmp[2] &= 0x00ffffffffffffff;

	/* Now 0 <= out < p */
	out[0] = tmp[0];
	out[1] = tmp[1];
	out[2] = tmp[2];
	out[3] = tmp[3];
	}

/* Zero-check: returns 1 if input is 0, and 0 otherwise.
 * We know that field elements are reduced to in < 2^225,
 * so we only need to check three cases: 0, 2^224 - 2^96 + 1,
 * and 2^225 - 2^97 + 2 */
static limb felem_is_zero(const felem in)
	{
	limb zero, two224m96p1, two225m97p2;

	zero = in[0] | in[1] | in[2] | in[3];
	zero = (((int64_t)(zero) - 1) >> 63) & 1;
	two224m96p1 = (in[0] ^ 1) | (in[1] ^ 0x00ffff0000000000)
		| (in[2] ^ 0x00ffffffffffffff) | (in[3] ^ 0x00ffffffffffffff);
	two224m96p1 = (((int64_t)(two224m96p1) - 1) >> 63) & 1;
	two225m97p2 = (in[0] ^ 2) | (in[1] ^ 0x00fffe0000000000)
		| (in[2] ^ 0x00ffffffffffffff) | (in[3] ^ 0x01ffffffffffffff);
	two225m97p2 = (((int64_t)(two225m97p2) - 1) >> 63) & 1;
	return (zero | two224m96p1 | two225m97p2);
	}

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

/* Invert a field element */
/* Computation chain copied from djb's code */
static void felem_inv(felem out, const felem in)
	{
	felem ftmp, ftmp2, ftmp3, ftmp4;
	widefelem tmp;
	unsigned i;

	felem_square(tmp, in); felem_reduce(ftmp, tmp);		/* 2 */
	felem_mul(tmp, in, ftmp); felem_reduce(ftmp, tmp);	/* 2^2 - 1 */
	felem_square(tmp, ftmp); felem_reduce(ftmp, tmp);	/* 2^3 - 2 */
	felem_mul(tmp, in, ftmp); felem_reduce(ftmp, tmp);	/* 2^3 - 1 */
	felem_square(tmp, ftmp); felem_reduce(ftmp2, tmp);	/* 2^4 - 2 */
	felem_square(tmp, ftmp2); felem_reduce(ftmp2, tmp);	/* 2^5 - 4 */
	felem_square(tmp, ftmp2); felem_reduce(ftmp2, tmp);	/* 2^6 - 8 */
	felem_mul(tmp, ftmp2, ftmp); felem_reduce(ftmp, tmp);	/* 2^6 - 1 */
	felem_square(tmp, ftmp); felem_reduce(ftmp2, tmp);	/* 2^7 - 2 */
	for (i = 0; i < 5; ++i)					/* 2^12 - 2^6 */
		{
		felem_square(tmp, ftmp2); felem_reduce(ftmp2, tmp);
		}
	felem_mul(tmp, ftmp2, ftmp); felem_reduce(ftmp2, tmp);	/* 2^12 - 1 */
	felem_square(tmp, ftmp2); felem_reduce(ftmp3, tmp);	/* 2^13 - 2 */
	for (i = 0; i < 11; ++i)				/* 2^24 - 2^12 */
		{
		felem_square(tmp, ftmp3); felem_reduce(ftmp3, tmp);
		}
	felem_mul(tmp, ftmp3, ftmp2); felem_reduce(ftmp2, tmp); /* 2^24 - 1 */
	felem_square(tmp, ftmp2); felem_reduce(ftmp3, tmp);	/* 2^25 - 2 */
	for (i = 0; i < 23; ++i)				/* 2^48 - 2^24 */
		{
		felem_square(tmp, ftmp3); felem_reduce(ftmp3, tmp);
		}
	felem_mul(tmp, ftmp3, ftmp2); felem_reduce(ftmp3, tmp); /* 2^48 - 1 */
	felem_square(tmp, ftmp3); felem_reduce(ftmp4, tmp);	/* 2^49 - 2 */
	for (i = 0; i < 47; ++i)				/* 2^96 - 2^48 */
		{
		felem_square(tmp, ftmp4); felem_reduce(ftmp4, tmp);
		}
	felem_mul(tmp, ftmp3, ftmp4); felem_reduce(ftmp3, tmp); /* 2^96 - 1 */
	felem_square(tmp, ftmp3); felem_reduce(ftmp4, tmp);	/* 2^97 - 2 */
	for (i = 0; i < 23; ++i)				/* 2^120 - 2^24 */
		{
		felem_square(tmp, ftmp4); felem_reduce(ftmp4, tmp);
		}
	felem_mul(tmp, ftmp2, ftmp4); felem_reduce(ftmp2, tmp); /* 2^120 - 1 */
	for (i = 0; i < 6; ++i)					/* 2^126 - 2^6 */
		{
		felem_square(tmp, ftmp2); felem_reduce(ftmp2, tmp);
		}
	felem_mul(tmp, ftmp2, ftmp); felem_reduce(ftmp, tmp);	/* 2^126 - 1 */
	felem_square(tmp, ftmp); felem_reduce(ftmp, tmp);	/* 2^127 - 2 */
	felem_mul(tmp, ftmp, in); felem_reduce(ftmp, tmp);	/* 2^127 - 1 */
	for (i = 0; i < 97; ++i)				/* 2^224 - 2^97 */
		{
		felem_square(tmp, ftmp); felem_reduce(ftmp, tmp);
		}
	felem_mul(tmp, ftmp, ftmp3); felem_reduce(out, tmp);	/* 2^224 - 2^96 - 1 */
	}

/* Copy in constant time:
 * if icopy == 1, copy in to out,
 * if icopy == 0, copy out to itself. */
static void
copy_conditional(felem out, const felem in, limb icopy)
	{
	unsigned i;
	/* icopy is a (64-bit) 0 or 1, so copy is either all-zero or all-one */
	const limb copy = -icopy;
	for (i = 0; i < 4; ++i)
		{
		const limb tmp = copy & (in[i] ^ out[i]);
		out[i] ^= tmp;
		}
	}

/******************************************************************************/
/*			 ELLIPTIC CURVE POINT OPERATIONS
 *
 * Points are represented in Jacobian projective coordinates:
 * (X, Y, Z) corresponds to the affine point (X/Z^2, Y/Z^3),
 * or to the point at infinity if Z == 0.
 *
 */

/* Double an elliptic curve point:
 * (X', Y', Z') = 2 * (X, Y, Z), where
 * X' = (3 * (X - Z^2) * (X + Z^2))^2 - 8 * X * Y^2
 * Y' = 3 * (X - Z^2) * (X + Z^2) * (4 * X * Y^2 - X') - 8 * Y^2
 * Z' = (Y + Z)^2 - Y^2 - Z^2 = 2 * Y * Z
 * 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)
	{
	widefelem 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);

	/* gamma = y^2 */
	felem_square(tmp, y_in);
	felem_reduce(gamma, tmp);

	/* beta = x*gamma */
	felem_mul(tmp, x_in, gamma);
	felem_reduce(beta, tmp);

	/* alpha = 3*(x-delta)*(x+delta) */
	felem_diff(ftmp, delta);
	/* ftmp[i] < 2^57 + 2^58 + 2 < 2^59 */
	felem_sum(ftmp2, delta);
	/* ftmp2[i] < 2^57 + 2^57 = 2^58 */
	felem_scalar(ftmp2, 3);
	/* ftmp2[i] < 3 * 2^58 < 2^60 */
	felem_mul(tmp, ftmp, ftmp2);
	/* tmp[i] < 2^60 * 2^59 * 4 = 2^121 */
	felem_reduce(alpha, tmp);

	/* x' = alpha^2 - 8*beta */
	felem_square(tmp, alpha);
	/* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
	felem_assign(ftmp, beta);
	felem_scalar(ftmp, 8);
	/* ftmp[i] < 8 * 2^57 = 2^60 */
	felem_diff_128_64(tmp, ftmp);
	/* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */
	felem_reduce(x_out, tmp);

	/* z' = (y + z)^2 - gamma - delta */
	felem_sum(delta, gamma);
	/* delta[i] < 2^57 + 2^57 = 2^58 */
	felem_assign(ftmp, y_in);
	felem_sum(ftmp, z_in);
	/* ftmp[i] < 2^57 + 2^57 = 2^58 */
	felem_square(tmp, ftmp);
	/* tmp[i] < 4 * 2^58 * 2^58 = 2^118 */
	felem_diff_128_64(tmp, delta);
	/* tmp[i] < 2^118 + 2^64 + 8 < 2^119 */
	felem_reduce(z_out, tmp);

	/* y' = alpha*(4*beta - x') - 8*gamma^2 */
	felem_scalar(beta, 4);
	/* beta[i] < 4 * 2^57 = 2^59 */
	felem_diff(beta, x_out);
	/* beta[i] < 2^59 + 2^58 + 2 < 2^60 */
	felem_mul(tmp, alpha, beta);
	/* tmp[i] < 4 * 2^57 * 2^60 = 2^119 */
	felem_square(tmp2, gamma);
	/* tmp2[i] < 4 * 2^57 * 2^57 = 2^116 */
	widefelem_scalar(tmp2, 8);
	/* tmp2[i] < 8 * 2^116 = 2^119 */
	widefelem_diff(tmp, tmp2);
	/* tmp[i] < 2^119 + 2^120 < 2^121 */
	felem_reduce(y_out, tmp);
	}

/* Add two elliptic curve points:
 * (X_1, Y_1, Z_1) + (X_2, Y_2, Z_2) = (X_3, Y_3, Z_3), where
 * X_3 = (Z_1^3 * Y_2 - Z_2^3 * Y_1)^2 - (Z_1^2 * X_2 - Z_2^2 * X_1)^3 -
 * 2 * Z_2^2 * X_1 * (Z_1^2 * X_2 - Z_2^2 * X_1)^2
 * Y_3 = (Z_1^3 * Y_2 - Z_2^3 * Y_1) * (Z_2^2 * X_1 * (Z_1^2 * X_2 - Z_2^2 * X_1)^2 - X_3) -
 *        Z_2^3 * Y_1 * (Z_1^2 * X_2 - Z_2^2 * X_1)^3
 * Z_3 = (Z_1^2 * X_2 - Z_2^2 * X_1) * (Z_1 * Z_2)
 *
 * This runs faster if 'mixed' is set, which requires Z_2 = 1 or Z_2 = 0.
 */

/* This function is not entirely constant-time:
 * it 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 felem x2, const felem y2, const felem z2)
	{
	felem ftmp, ftmp2, ftmp3, ftmp4, ftmp5, x_out, y_out, z_out;
	widefelem tmp, tmp2;
	limb z1_is_zero, z2_is_zero, x_equal, y_equal;

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

		/* ftmp4 = z2^3 */
		felem_mul(tmp, ftmp2, z2);
		felem_reduce(ftmp4, tmp);

		/* ftmp4 = z2^3*y1 */
		felem_mul(tmp2, ftmp4, y1);
		felem_reduce(ftmp4, tmp2);

		/* ftmp2 = z2^2*x1 */
		felem_mul(tmp2, ftmp2, x1);
		felem_reduce(ftmp2, tmp2);
		}
	else
		{
		/* We'll assume z2 = 1 (special case z2 = 0 is handled later) */

		/* ftmp4 = z2^3*y1 */
		felem_assign(ftmp4, y1);

		/* ftmp2 = z2^2*x1 */
		felem_assign(ftmp2, x1);
		}

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

	/* ftmp3 = z1^3 */
	felem_mul(tmp, ftmp, z1);
	felem_reduce(ftmp3, tmp);

	/* tmp = z1^3*y2 */
	felem_mul(tmp, ftmp3, y2);
	/* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */

	/* ftmp3 = z1^3*y2 - z2^3*y1 */
	felem_diff_128_64(tmp, ftmp4);
	/* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */
	felem_reduce(ftmp3, tmp);

	/* tmp = z1^2*x2 */
	felem_mul(tmp, ftmp, x2);
	/* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */

	/* ftmp = z1^2*x2 - z2^2*x1 */
	felem_diff_128_64(tmp, ftmp2);
	/* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */
	felem_reduce(ftmp, tmp);

	/* the formulae are incorrect if the points are equal
	 * so we check for this and do doubling if this happens */
	x_equal = felem_is_zero(ftmp);
	y_equal = felem_is_zero(ftmp3);
	z1_is_zero = felem_is_zero(z1);
	z2_is_zero = felem_is_zero(z2);
	/* In affine coordinates, (X_1, Y_1) == (X_2, Y_2) */
	if (x_equal && y_equal && !z1_is_zero && !z2_is_zero)
		{
		point_double(x3, y3, z3, x1, y1, z1);
		return;
		}

	/* ftmp5 = z1*z2 */
	if (!mixed)
		{
		felem_mul(tmp, z1, z2);
		felem_reduce(ftmp5, tmp);
		}
	else
		{
		/* special case z2 = 0 is handled later */
		felem_assign(ftmp5, z1);
		}

	/* z_out = (z1^2*x2 - z2^2*x1)*(z1*z2) */
	felem_mul(tmp, ftmp, ftmp5);
	felem_reduce(z_out, tmp);

	/* ftmp = (z1^2*x2 - z2^2*x1)^2 */
	felem_assign(ftmp5, ftmp);
	felem_square(tmp, ftmp);
	felem_reduce(ftmp, tmp);

	/* ftmp5 = (z1^2*x2 - z2^2*x1)^3 */
	felem_mul(tmp, ftmp, ftmp5);
	felem_reduce(ftmp5, tmp);

	/* ftmp2 = z2^2*x1*(z1^2*x2 - z2^2*x1)^2 */
	felem_mul(tmp, ftmp2, ftmp);
	felem_reduce(ftmp2, tmp);

	/* tmp = z2^3*y1*(z1^2*x2 - z2^2*x1)^3 */
	felem_mul(tmp, ftmp4, ftmp5);
	/* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */

	/* tmp2 = (z1^3*y2 - z2^3*y1)^2 */
	felem_square(tmp2, ftmp3);
	/* tmp2[i] < 4 * 2^57 * 2^57 < 2^116 */

	/* tmp2 = (z1^3*y2 - z2^3*y1)^2 - (z1^2*x2 - z2^2*x1)^3 */
	felem_diff_128_64(tmp2, ftmp5);
	/* tmp2[i] < 2^116 + 2^64 + 8 < 2^117 */

	/* ftmp5 = 2*z2^2*x1*(z1^2*x2 - z2^2*x1)^2 */
	felem_assign(ftmp5, ftmp2);
	felem_scalar(ftmp5, 2);
	/* ftmp5[i] < 2 * 2^57 = 2^58 */

	/* x_out = (z1^3*y2 - z2^3*y1)^2 - (z1^2*x2 - z2^2*x1)^3 -
	   2*z2^2*x1*(z1^2*x2 - z2^2*x1)^2 */
	felem_diff_128_64(tmp2, ftmp5);
	/* tmp2[i] < 2^117 + 2^64 + 8 < 2^118 */
	felem_reduce(x_out, tmp2);

	/* ftmp2 = z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out */
	felem_diff(ftmp2, x_out);
	/* ftmp2[i] < 2^57 + 2^58 + 2 < 2^59 */

	/* tmp2 = (z1^3*y2 - z2^3*y1)*(z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out) */
	felem_mul(tmp2, ftmp3, ftmp2);
	/* tmp2[i] < 4 * 2^57 * 2^59 = 2^118 */

	/* y_out = (z1^3*y2 - z2^3*y1)*(z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out) -
	   z2^3*y1*(z1^2*x2 - z2^2*x1)^3 */
	widefelem_diff(tmp2, tmp);
	/* tmp2[i] < 2^118 + 2^120 < 2^121 */
	felem_reduce(y_out, tmp2);

	/* the result (x_out, y_out, z_out) is incorrect if one of the inputs is
	 * the point at infinity, so we need to check for this separately */

	/* if point 1 is at infinity, copy point 2 to output, and vice versa */
	copy_conditional(x_out, x2, z1_is_zero);