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Source file src/crypto/elliptic/p256.go

Documentation: crypto/elliptic

  // Copyright 2013 The Go Authors. All rights reserved.
  // Use of this source code is governed by a BSD-style
  // license that can be found in the LICENSE file.
  
  // +build !amd64
  
  package elliptic
  
  // This file contains a constant-time, 32-bit implementation of P256.
  
  import (
  	"math/big"
  )
  
  type p256Curve struct {
  	*CurveParams
  }
  
  var (
  	p256Params *CurveParams
  
  	// RInverse contains 1/R mod p - the inverse of the Montgomery constant
  	// (2**257).
  	p256RInverse *big.Int
  )
  
  func initP256() {
  	// See FIPS 186-3, section D.2.3
  	p256Params = &CurveParams{Name: "P-256"}
  	p256Params.P, _ = new(big.Int).SetString("115792089210356248762697446949407573530086143415290314195533631308867097853951", 10)
  	p256Params.N, _ = new(big.Int).SetString("115792089210356248762697446949407573529996955224135760342422259061068512044369", 10)
  	p256Params.B, _ = new(big.Int).SetString("5ac635d8aa3a93e7b3ebbd55769886bc651d06b0cc53b0f63bce3c3e27d2604b", 16)
  	p256Params.Gx, _ = new(big.Int).SetString("6b17d1f2e12c4247f8bce6e563a440f277037d812deb33a0f4a13945d898c296", 16)
  	p256Params.Gy, _ = new(big.Int).SetString("4fe342e2fe1a7f9b8ee7eb4a7c0f9e162bce33576b315ececbb6406837bf51f5", 16)
  	p256Params.BitSize = 256
  
  	p256RInverse, _ = new(big.Int).SetString("7fffffff00000001fffffffe8000000100000000ffffffff0000000180000000", 16)
  
  	// Arch-specific initialization, i.e. let a platform dynamically pick a P256 implementation
  	initP256Arch()
  }
  
  func (curve p256Curve) Params() *CurveParams {
  	return curve.CurveParams
  }
  
  // p256GetScalar endian-swaps the big-endian scalar value from in and writes it
  // to out. If the scalar is equal or greater than the order of the group, it's
  // reduced modulo that order.
  func p256GetScalar(out *[32]byte, in []byte) {
  	n := new(big.Int).SetBytes(in)
  	var scalarBytes []byte
  
  	if n.Cmp(p256Params.N) >= 0 {
  		n.Mod(n, p256Params.N)
  		scalarBytes = n.Bytes()
  	} else {
  		scalarBytes = in
  	}
  
  	for i, v := range scalarBytes {
  		out[len(scalarBytes)-(1+i)] = v
  	}
  }
  
  func (p256Curve) ScalarBaseMult(scalar []byte) (x, y *big.Int) {
  	var scalarReversed [32]byte
  	p256GetScalar(&scalarReversed, scalar)
  
  	var x1, y1, z1 [p256Limbs]uint32
  	p256ScalarBaseMult(&x1, &y1, &z1, &scalarReversed)
  	return p256ToAffine(&x1, &y1, &z1)
  }
  
  func (p256Curve) ScalarMult(bigX, bigY *big.Int, scalar []byte) (x, y *big.Int) {
  	var scalarReversed [32]byte
  	p256GetScalar(&scalarReversed, scalar)
  
  	var px, py, x1, y1, z1 [p256Limbs]uint32
  	p256FromBig(&px, bigX)
  	p256FromBig(&py, bigY)
  	p256ScalarMult(&x1, &y1, &z1, &px, &py, &scalarReversed)
  	return p256ToAffine(&x1, &y1, &z1)
  }
  
  // Field elements are represented as nine, unsigned 32-bit words.
  //
  // The value of an field element is:
  //   x[0] + (x[1] * 2**29) + (x[2] * 2**57) + ... + (x[8] * 2**228)
  //
  // That is, each limb is alternately 29 or 28-bits wide in little-endian
  // order.
  //
  // This means that a field element hits 2**257, rather than 2**256 as we would
  // like. A 28, 29, ... pattern would cause us to hit 2**256, but that causes
  // problems when multiplying as terms end up one bit short of a limb which
  // would require much bit-shifting to correct.
  //
  // Finally, the values stored in a field element are in Montgomery form. So the
  // value |y| is stored as (y*R) mod p, where p is the P-256 prime and R is
  // 2**257.
  
  const (
  	p256Limbs    = 9
  	bottom29Bits = 0x1fffffff
  )
  
  var (
  	// p256One is the number 1 as a field element.
  	p256One  = [p256Limbs]uint32{2, 0, 0, 0xffff800, 0x1fffffff, 0xfffffff, 0x1fbfffff, 0x1ffffff, 0}
  	p256Zero = [p256Limbs]uint32{0, 0, 0, 0, 0, 0, 0, 0, 0}
  	// p256P is the prime modulus as a field element.
  	p256P = [p256Limbs]uint32{0x1fffffff, 0xfffffff, 0x1fffffff, 0x3ff, 0, 0, 0x200000, 0xf000000, 0xfffffff}
  	// p2562P is the twice prime modulus as a field element.
  	p2562P = [p256Limbs]uint32{0x1ffffffe, 0xfffffff, 0x1fffffff, 0x7ff, 0, 0, 0x400000, 0xe000000, 0x1fffffff}
  )
  
  // p256Precomputed contains precomputed values to aid the calculation of scalar
  // multiples of the base point, G. It's actually two, equal length, tables
  // concatenated.
  //
  // The first table contains (x,y) field element pairs for 16 multiples of the
  // base point, G.
  //
  //   Index  |  Index (binary) | Value
  //       0  |           0000  | 0G (all zeros, omitted)
  //       1  |           0001  | G
  //       2  |           0010  | 2**64G
  //       3  |           0011  | 2**64G + G
  //       4  |           0100  | 2**128G
  //       5  |           0101  | 2**128G + G
  //       6  |           0110  | 2**128G + 2**64G
  //       7  |           0111  | 2**128G + 2**64G + G
  //       8  |           1000  | 2**192G
  //       9  |           1001  | 2**192G + G
  //      10  |           1010  | 2**192G + 2**64G
  //      11  |           1011  | 2**192G + 2**64G + G
  //      12  |           1100  | 2**192G + 2**128G
  //      13  |           1101  | 2**192G + 2**128G + G
  //      14  |           1110  | 2**192G + 2**128G + 2**64G
  //      15  |           1111  | 2**192G + 2**128G + 2**64G + G
  //
  // The second table follows the same style, but the terms are 2**32G,
  // 2**96G, 2**160G, 2**224G.
  //
  // This is ~2KB of data.
  var p256Precomputed = [p256Limbs * 2 * 15 * 2]uint32{
  	0x11522878, 0xe730d41, 0xdb60179, 0x4afe2ff, 0x12883add, 0xcaddd88, 0x119e7edc, 0xd4a6eab, 0x3120bee,
  	0x1d2aac15, 0xf25357c, 0x19e45cdd, 0x5c721d0, 0x1992c5a5, 0xa237487, 0x154ba21, 0x14b10bb, 0xae3fe3,
  	0xd41a576, 0x922fc51, 0x234994f, 0x60b60d3, 0x164586ae, 0xce95f18, 0x1fe49073, 0x3fa36cc, 0x5ebcd2c,
  	0xb402f2f, 0x15c70bf, 0x1561925c, 0x5a26704, 0xda91e90, 0xcdc1c7f, 0x1ea12446, 0xe1ade1e, 0xec91f22,
  	0x26f7778, 0x566847e, 0xa0bec9e, 0x234f453, 0x1a31f21a, 0xd85e75c, 0x56c7109, 0xa267a00, 0xb57c050,
  	0x98fb57, 0xaa837cc, 0x60c0792, 0xcfa5e19, 0x61bab9e, 0x589e39b, 0xa324c5, 0x7d6dee7, 0x2976e4b,
  	0x1fc4124a, 0xa8c244b, 0x1ce86762, 0xcd61c7e, 0x1831c8e0, 0x75774e1, 0x1d96a5a9, 0x843a649, 0xc3ab0fa,
  	0x6e2e7d5, 0x7673a2a, 0x178b65e8, 0x4003e9b, 0x1a1f11c2, 0x7816ea, 0xf643e11, 0x58c43df, 0xf423fc2,
  	0x19633ffa, 0x891f2b2, 0x123c231c, 0x46add8c, 0x54700dd, 0x59e2b17, 0x172db40f, 0x83e277d, 0xb0dd609,
  	0xfd1da12, 0x35c6e52, 0x19ede20c, 0xd19e0c0, 0x97d0f40, 0xb015b19, 0x449e3f5, 0xe10c9e, 0x33ab581,
  	0x56a67ab, 0x577734d, 0x1dddc062, 0xc57b10d, 0x149b39d, 0x26a9e7b, 0xc35df9f, 0x48764cd, 0x76dbcca,
  	0xca4b366, 0xe9303ab, 0x1a7480e7, 0x57e9e81, 0x1e13eb50, 0xf466cf3, 0x6f16b20, 0x4ba3173, 0xc168c33,
  	0x15cb5439, 0x6a38e11, 0x73658bd, 0xb29564f, 0x3f6dc5b, 0x53b97e, 0x1322c4c0, 0x65dd7ff, 0x3a1e4f6,
  	0x14e614aa, 0x9246317, 0x1bc83aca, 0xad97eed, 0xd38ce4a, 0xf82b006, 0x341f077, 0xa6add89, 0x4894acd,
  	0x9f162d5, 0xf8410ef, 0x1b266a56, 0xd7f223, 0x3e0cb92, 0xe39b672, 0x6a2901a, 0x69a8556, 0x7e7c0,
  	0x9b7d8d3, 0x309a80, 0x1ad05f7f, 0xc2fb5dd, 0xcbfd41d, 0x9ceb638, 0x1051825c, 0xda0cf5b, 0x812e881,
  	0x6f35669, 0x6a56f2c, 0x1df8d184, 0x345820, 0x1477d477, 0x1645db1, 0xbe80c51, 0xc22be3e, 0xe35e65a,
  	0x1aeb7aa0, 0xc375315, 0xf67bc99, 0x7fdd7b9, 0x191fc1be, 0x61235d, 0x2c184e9, 0x1c5a839, 0x47a1e26,
  	0xb7cb456, 0x93e225d, 0x14f3c6ed, 0xccc1ac9, 0x17fe37f3, 0x4988989, 0x1a90c502, 0x2f32042, 0xa17769b,
  	0xafd8c7c, 0x8191c6e, 0x1dcdb237, 0x16200c0, 0x107b32a1, 0x66c08db, 0x10d06a02, 0x3fc93, 0x5620023,
  	0x16722b27, 0x68b5c59, 0x270fcfc, 0xfad0ecc, 0xe5de1c2, 0xeab466b, 0x2fc513c, 0x407f75c, 0xbaab133,
  	0x9705fe9, 0xb88b8e7, 0x734c993, 0x1e1ff8f, 0x19156970, 0xabd0f00, 0x10469ea7, 0x3293ac0, 0xcdc98aa,
  	0x1d843fd, 0xe14bfe8, 0x15be825f, 0x8b5212, 0xeb3fb67, 0x81cbd29, 0xbc62f16, 0x2b6fcc7, 0xf5a4e29,
  	0x13560b66, 0xc0b6ac2, 0x51ae690, 0xd41e271, 0xf3e9bd4, 0x1d70aab, 0x1029f72, 0x73e1c35, 0xee70fbc,
  	0xad81baf, 0x9ecc49a, 0x86c741e, 0xfe6be30, 0x176752e7, 0x23d416, 0x1f83de85, 0x27de188, 0x66f70b8,
  	0x181cd51f, 0x96b6e4c, 0x188f2335, 0xa5df759, 0x17a77eb6, 0xfeb0e73, 0x154ae914, 0x2f3ec51, 0x3826b59,
  	0xb91f17d, 0x1c72949, 0x1362bf0a, 0xe23fddf, 0xa5614b0, 0xf7d8f, 0x79061, 0x823d9d2, 0x8213f39,
  	0x1128ae0b, 0xd095d05, 0xb85c0c2, 0x1ecb2ef, 0x24ddc84, 0xe35e901, 0x18411a4a, 0xf5ddc3d, 0x3786689,
  	0x52260e8, 0x5ae3564, 0x542b10d, 0x8d93a45, 0x19952aa4, 0x996cc41, 0x1051a729, 0x4be3499, 0x52b23aa,
  	0x109f307e, 0x6f5b6bb, 0x1f84e1e7, 0x77a0cfa, 0x10c4df3f, 0x25a02ea, 0xb048035, 0xe31de66, 0xc6ecaa3,
  	0x28ea335, 0x2886024, 0x1372f020, 0xf55d35, 0x15e4684c, 0xf2a9e17, 0x1a4a7529, 0xcb7beb1, 0xb2a78a1,
  	0x1ab21f1f, 0x6361ccf, 0x6c9179d, 0xb135627, 0x1267b974, 0x4408bad, 0x1cbff658, 0xe3d6511, 0xc7d76f,
  	0x1cc7a69, 0xe7ee31b, 0x54fab4f, 0x2b914f, 0x1ad27a30, 0xcd3579e, 0xc50124c, 0x50daa90, 0xb13f72,
  	0xb06aa75, 0x70f5cc6, 0x1649e5aa, 0x84a5312, 0x329043c, 0x41c4011, 0x13d32411, 0xb04a838, 0xd760d2d,
  	0x1713b532, 0xbaa0c03, 0x84022ab, 0x6bcf5c1, 0x2f45379, 0x18ae070, 0x18c9e11e, 0x20bca9a, 0x66f496b,
  	0x3eef294, 0x67500d2, 0xd7f613c, 0x2dbbeb, 0xb741038, 0xe04133f, 0x1582968d, 0xbe985f7, 0x1acbc1a,
  	0x1a6a939f, 0x33e50f6, 0xd665ed4, 0xb4b7bd6, 0x1e5a3799, 0x6b33847, 0x17fa56ff, 0x65ef930, 0x21dc4a,
  	0x2b37659, 0x450fe17, 0xb357b65, 0xdf5efac, 0x15397bef, 0x9d35a7f, 0x112ac15f, 0x624e62e, 0xa90ae2f,
  	0x107eecd2, 0x1f69bbe, 0x77d6bce, 0x5741394, 0x13c684fc, 0x950c910, 0x725522b, 0xdc78583, 0x40eeabb,
  	0x1fde328a, 0xbd61d96, 0xd28c387, 0x9e77d89, 0x12550c40, 0x759cb7d, 0x367ef34, 0xae2a960, 0x91b8bdc,
  	0x93462a9, 0xf469ef, 0xb2e9aef, 0xd2ca771, 0x54e1f42, 0x7aaa49, 0x6316abb, 0x2413c8e, 0x5425bf9,
  	0x1bed3e3a, 0xf272274, 0x1f5e7326, 0x6416517, 0xea27072, 0x9cedea7, 0x6e7633, 0x7c91952, 0xd806dce,
  	0x8e2a7e1, 0xe421e1a, 0x418c9e1, 0x1dbc890, 0x1b395c36, 0xa1dc175, 0x1dc4ef73, 0x8956f34, 0xe4b5cf2,
  	0x1b0d3a18, 0x3194a36, 0x6c2641f, 0xe44124c, 0xa2f4eaa, 0xa8c25ba, 0xf927ed7, 0x627b614, 0x7371cca,
  	0xba16694, 0x417bc03, 0x7c0a7e3, 0x9c35c19, 0x1168a205, 0x8b6b00d, 0x10e3edc9, 0x9c19bf2, 0x5882229,
  	0x1b2b4162, 0xa5cef1a, 0x1543622b, 0x9bd433e, 0x364e04d, 0x7480792, 0x5c9b5b3, 0xe85ff25, 0x408ef57,
  	0x1814cfa4, 0x121b41b, 0xd248a0f, 0x3b05222, 0x39bb16a, 0xc75966d, 0xa038113, 0xa4a1769, 0x11fbc6c,
  	0x917e50e, 0xeec3da8, 0x169d6eac, 0x10c1699, 0xa416153, 0xf724912, 0x15cd60b7, 0x4acbad9, 0x5efc5fa,
  	0xf150ed7, 0x122b51, 0x1104b40a, 0xcb7f442, 0xfbb28ff, 0x6ac53ca, 0x196142cc, 0x7bf0fa9, 0x957651,
  	0x4e0f215, 0xed439f8, 0x3f46bd5, 0x5ace82f, 0x110916b6, 0x6db078, 0xffd7d57, 0xf2ecaac, 0xca86dec,
  	0x15d6b2da, 0x965ecc9, 0x1c92b4c2, 0x1f3811, 0x1cb080f5, 0x2d8b804, 0x19d1c12d, 0xf20bd46, 0x1951fa7,
  	0xa3656c3, 0x523a425, 0xfcd0692, 0xd44ddc8, 0x131f0f5b, 0xaf80e4a, 0xcd9fc74, 0x99bb618, 0x2db944c,
  	0xa673090, 0x1c210e1, 0x178c8d23, 0x1474383, 0x10b8743d, 0x985a55b, 0x2e74779, 0x576138, 0x9587927,
  	0x133130fa, 0xbe05516, 0x9f4d619, 0xbb62570, 0x99ec591, 0xd9468fe, 0x1d07782d, 0xfc72e0b, 0x701b298,
  	0x1863863b, 0x85954b8, 0x121a0c36, 0x9e7fedf, 0xf64b429, 0x9b9d71e, 0x14e2f5d8, 0xf858d3a, 0x942eea8,
  	0xda5b765, 0x6edafff, 0xa9d18cc, 0xc65e4ba, 0x1c747e86, 0xe4ea915, 0x1981d7a1, 0x8395659, 0x52ed4e2,
  	0x87d43b7, 0x37ab11b, 0x19d292ce, 0xf8d4692, 0x18c3053f, 0x8863e13, 0x4c146c0, 0x6bdf55a, 0x4e4457d,
  	0x16152289, 0xac78ec2, 0x1a59c5a2, 0x2028b97, 0x71c2d01, 0x295851f, 0x404747b, 0x878558d, 0x7d29aa4,
  	0x13d8341f, 0x8daefd7, 0x139c972d, 0x6b7ea75, 0xd4a9dde, 0xff163d8, 0x81d55d7, 0xa5bef68, 0xb7b30d8,
  	0xbe73d6f, 0xaa88141, 0xd976c81, 0x7e7a9cc, 0x18beb771, 0xd773cbd, 0x13f51951, 0x9d0c177, 0x1c49a78,
  }
  
  // Field element operations:
  
  // nonZeroToAllOnes returns:
  //   0xffffffff for 0 < x <= 2**31
  //   0 for x == 0 or x > 2**31.
  func nonZeroToAllOnes(x uint32) uint32 {
  	return ((x - 1) >> 31) - 1
  }
  
  // p256ReduceCarry adds a multiple of p in order to cancel |carry|,
  // which is a term at 2**257.
  //
  // On entry: carry < 2**3, inout[0,2,...] < 2**29, inout[1,3,...] < 2**28.
  // On exit: inout[0,2,..] < 2**30, inout[1,3,...] < 2**29.
  func p256ReduceCarry(inout *[p256Limbs]uint32, carry uint32) {
  	carry_mask := nonZeroToAllOnes(carry)
  
  	inout[0] += carry << 1
  	inout[3] += 0x10000000 & carry_mask
  	// carry < 2**3 thus (carry << 11) < 2**14 and we added 2**28 in the
  	// previous line therefore this doesn't underflow.
  	inout[3] -= carry << 11
  	inout[4] += (0x20000000 - 1) & carry_mask
  	inout[5] += (0x10000000 - 1) & carry_mask
  	inout[6] += (0x20000000 - 1) & carry_mask
  	inout[6] -= carry << 22
  	// This may underflow if carry is non-zero but, if so, we'll fix it in the
  	// next line.
  	inout[7] -= 1 & carry_mask
  	inout[7] += carry << 25
  }
  
  // p256Sum sets out = in+in2.
  //
  // On entry, in[i]+in2[i] must not overflow a 32-bit word.
  // On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29
  func p256Sum(out, in, in2 *[p256Limbs]uint32) {
  	carry := uint32(0)
  	for i := 0; ; i++ {
  		out[i] = in[i] + in2[i]
  		out[i] += carry
  		carry = out[i] >> 29
  		out[i] &= bottom29Bits
  
  		i++
  		if i == p256Limbs {
  			break
  		}
  
  		out[i] = in[i] + in2[i]
  		out[i] += carry
  		carry = out[i] >> 28
  		out[i] &= bottom28Bits
  	}
  
  	p256ReduceCarry(out, carry)
  }
  
  const (
  	two30m2    = 1<<30 - 1<<2
  	two30p13m2 = 1<<30 + 1<<13 - 1<<2
  	two31m2    = 1<<31 - 1<<2
  	two31p24m2 = 1<<31 + 1<<24 - 1<<2
  	two30m27m2 = 1<<30 - 1<<27 - 1<<2
  )
  
  // p256Zero31 is 0 mod p.
  var p256Zero31 = [p256Limbs]uint32{two31m3, two30m2, two31m2, two30p13m2, two31m2, two30m2, two31p24m2, two30m27m2, two31m2}
  
  // p256Diff sets out = in-in2.
  //
  // On entry: in[0,2,...] < 2**30, in[1,3,...] < 2**29 and
  //           in2[0,2,...] < 2**30, in2[1,3,...] < 2**29.
  // On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
  func p256Diff(out, in, in2 *[p256Limbs]uint32) {
  	var carry uint32
  
  	for i := 0; ; i++ {
  		out[i] = in[i] - in2[i]
  		out[i] += p256Zero31[i]
  		out[i] += carry
  		carry = out[i] >> 29
  		out[i] &= bottom29Bits
  
  		i++
  		if i == p256Limbs {
  			break
  		}
  
  		out[i] = in[i] - in2[i]
  		out[i] += p256Zero31[i]
  		out[i] += carry
  		carry = out[i] >> 28
  		out[i] &= bottom28Bits
  	}
  
  	p256ReduceCarry(out, carry)
  }
  
  // p256ReduceDegree sets out = tmp/R mod p where tmp contains 64-bit words with
  // the same 29,28,... bit positions as an field element.
  //
  // The values in field elements are in Montgomery form: x*R mod p where R =
  // 2**257. Since we just multiplied two Montgomery values together, the result
  // is x*y*R*R mod p. We wish to divide by R in order for the result also to be
  // in Montgomery form.
  //
  // On entry: tmp[i] < 2**64
  // On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29
  func p256ReduceDegree(out *[p256Limbs]uint32, tmp [17]uint64) {
  	// The following table may be helpful when reading this code:
  	//
  	// Limb number:   0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10...
  	// Width (bits):  29| 28| 29| 28| 29| 28| 29| 28| 29| 28| 29
  	// Start bit:     0 | 29| 57| 86|114|143|171|200|228|257|285
  	//   (odd phase): 0 | 28| 57| 85|114|142|171|199|228|256|285
  	var tmp2 [18]uint32
  	var carry, x, xMask uint32
  
  	// tmp contains 64-bit words with the same 29,28,29-bit positions as an
  	// field element. So the top of an element of tmp might overlap with
  	// another element two positions down. The following loop eliminates
  	// this overlap.
  	tmp2[0] = uint32(tmp[0]) & bottom29Bits
  
  	tmp2[1] = uint32(tmp[0]) >> 29
  	tmp2[1] |= (uint32(tmp[0]>>32) << 3) & bottom28Bits
  	tmp2[1] += uint32(tmp[1]) & bottom28Bits
  	carry = tmp2[1] >> 28
  	tmp2[1] &= bottom28Bits
  
  	for i := 2; i < 17; i++ {
  		tmp2[i] = (uint32(tmp[i-2] >> 32)) >> 25
  		tmp2[i] += (uint32(tmp[i-1])) >> 28
  		tmp2[i] += (uint32(tmp[i-1]>>32) << 4) & bottom29Bits
  		tmp2[i] += uint32(tmp[i]) & bottom29Bits
  		tmp2[i] += carry
  		carry = tmp2[i] >> 29
  		tmp2[i] &= bottom29Bits
  
  		i++
  		if i == 17 {
  			break
  		}
  		tmp2[i] = uint32(tmp[i-2]>>32) >> 25
  		tmp2[i] += uint32(tmp[i-1]) >> 29
  		tmp2[i] += ((uint32(tmp[i-1] >> 32)) << 3) & bottom28Bits
  		tmp2[i] += uint32(tmp[i]) & bottom28Bits
  		tmp2[i] += carry
  		carry = tmp2[i] >> 28
  		tmp2[i] &= bottom28Bits
  	}
  
  	tmp2[17] = uint32(tmp[15]>>32) >> 25
  	tmp2[17] += uint32(tmp[16]) >> 29
  	tmp2[17] += uint32(tmp[16]>>32) << 3
  	tmp2[17] += carry
  
  	// Montgomery elimination of terms:
  	//
  	// Since R is 2**257, we can divide by R with a bitwise shift if we can
  	// ensure that the right-most 257 bits are all zero. We can make that true
  	// by adding multiplies of p without affecting the value.
  	//
  	// So we eliminate limbs from right to left. Since the bottom 29 bits of p
  	// are all ones, then by adding tmp2[0]*p to tmp2 we'll make tmp2[0] == 0.
  	// We can do that for 8 further limbs and then right shift to eliminate the
  	// extra factor of R.
  	for i := 0; ; i += 2 {
  		tmp2[i+1] += tmp2[i] >> 29
  		x = tmp2[i] & bottom29Bits
  		xMask = nonZeroToAllOnes(x)
  		tmp2[i] = 0
  
  		// The bounds calculations for this loop are tricky. Each iteration of
  		// the loop eliminates two words by adding values to words to their
  		// right.
  		//
  		// The following table contains the amounts added to each word (as an
  		// offset from the value of i at the top of the loop). The amounts are
  		// accounted for from the first and second half of the loop separately
  		// and are written as, for example, 28 to mean a value <2**28.
  		//
  		// Word:                   3   4   5   6   7   8   9   10
  		// Added in top half:     28  11      29  21  29  28
  		//                                        28  29
  		//                                            29
  		// Added in bottom half:      29  10      28  21  28   28
  		//                                            29
  		//
  		// The value that is currently offset 7 will be offset 5 for the next
  		// iteration and then offset 3 for the iteration after that. Therefore
  		// the total value added will be the values added at 7, 5 and 3.
  		//
  		// The following table accumulates these values. The sums at the bottom
  		// are written as, for example, 29+28, to mean a value < 2**29+2**28.
  		//
  		// Word:                   3   4   5   6   7   8   9  10  11  12  13
  		//                        28  11  10  29  21  29  28  28  28  28  28
  		//                            29  28  11  28  29  28  29  28  29  28
  		//                                    29  28  21  21  29  21  29  21
  		//                                        10  29  28  21  28  21  28
  		//                                        28  29  28  29  28  29  28
  		//                                            11  10  29  10  29  10
  		//                                            29  28  11  28  11
  		//                                                    29      29
  		//                        --------------------------------------------
  		//                                                30+ 31+ 30+ 31+ 30+
  		//                                                28+ 29+ 28+ 29+ 21+
  		//                                                21+ 28+ 21+ 28+ 10
  		//                                                10  21+ 10  21+
  		//                                                    11      11
  		//
  		// So the greatest amount is added to tmp2[10] and tmp2[12]. If
  		// tmp2[10/12] has an initial value of <2**29, then the maximum value
  		// will be < 2**31 + 2**30 + 2**28 + 2**21 + 2**11, which is < 2**32,
  		// as required.
  		tmp2[i+3] += (x << 10) & bottom28Bits
  		tmp2[i+4] += (x >> 18)
  
  		tmp2[i+6] += (x << 21) & bottom29Bits
  		tmp2[i+7] += x >> 8
  
  		// At position 200, which is the starting bit position for word 7, we
  		// have a factor of 0xf000000 = 2**28 - 2**24.
  		tmp2[i+7] += 0x10000000 & xMask
  		tmp2[i+8] += (x - 1) & xMask
  		tmp2[i+7] -= (x << 24) & bottom28Bits
  		tmp2[i+8] -= x >> 4
  
  		tmp2[i+8] += 0x20000000 & xMask
  		tmp2[i+8] -= x
  		tmp2[i+8] += (x << 28) & bottom29Bits
  		tmp2[i+9] += ((x >> 1) - 1) & xMask
  
  		if i+1 == p256Limbs {
  			break
  		}
  		tmp2[i+2] += tmp2[i+1] >> 28
  		x = tmp2[i+1] & bottom28Bits
  		xMask = nonZeroToAllOnes(x)
  		tmp2[i+1] = 0
  
  		tmp2[i+4] += (x << 11) & bottom29Bits
  		tmp2[i+5] += (x >> 18)
  
  		tmp2[i+7] += (x << 21) & bottom28Bits
  		tmp2[i+8] += x >> 7
  
  		// At position 199, which is the starting bit of the 8th word when
  		// dealing with a context starting on an odd word, we have a factor of
  		// 0x1e000000 = 2**29 - 2**25. Since we have not updated i, the 8th
  		// word from i+1 is i+8.
  		tmp2[i+8] += 0x20000000 & xMask
  		tmp2[i+9] += (x - 1) & xMask
  		tmp2[i+8] -= (x << 25) & bottom29Bits
  		tmp2[i+9] -= x >> 4
  
  		tmp2[i+9] += 0x10000000 & xMask
  		tmp2[i+9] -= x
  		tmp2[i+10] += (x - 1) & xMask
  	}
  
  	// We merge the right shift with a carry chain. The words above 2**257 have
  	// widths of 28,29,... which we need to correct when copying them down.
  	carry = 0
  	for i := 0; i < 8; i++ {
  		// The maximum value of tmp2[i + 9] occurs on the first iteration and
  		// is < 2**30+2**29+2**28. Adding 2**29 (from tmp2[i + 10]) is
  		// therefore safe.
  		out[i] = tmp2[i+9]
  		out[i] += carry
  		out[i] += (tmp2[i+10] << 28) & bottom29Bits
  		carry = out[i] >> 29
  		out[i] &= bottom29Bits
  
  		i++
  		out[i] = tmp2[i+9] >> 1
  		out[i] += carry
  		carry = out[i] >> 28
  		out[i] &= bottom28Bits
  	}
  
  	out[8] = tmp2[17]
  	out[8] += carry
  	carry = out[8] >> 29
  	out[8] &= bottom29Bits
  
  	p256ReduceCarry(out, carry)
  }
  
  // p256Square sets out=in*in.
  //
  // On entry: in[0,2,...] < 2**30, in[1,3,...] < 2**29.
  // On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
  func p256Square(out, in *[p256Limbs]uint32) {
  	var tmp [17]uint64
  
  	tmp[0] = uint64(in[0]) * uint64(in[0])
  	tmp[1] = uint64(in[0]) * (uint64(in[1]) << 1)
  	tmp[2] = uint64(in[0])*(uint64(in[2])<<1) +
  		uint64(in[1])*(uint64(in[1])<<1)
  	tmp[3] = uint64(in[0])*(uint64(in[3])<<1) +
  		uint64(in[1])*(uint64(in[2])<<1)
  	tmp[4] = uint64(in[0])*(uint64(in[4])<<1) +
  		uint64(in[1])*(uint64(in[3])<<2) +
  		uint64(in[2])*uint64(in[2])
  	tmp[5] = uint64(in[0])*(uint64(in[5])<<1) +
  		uint64(in[1])*(uint64(in[4])<<1) +
  		uint64(in[2])*(uint64(in[3])<<1)
  	tmp[6] = uint64(in[0])*(uint64(in[6])<<1) +
  		uint64(in[1])*(uint64(in[5])<<2) +
  		uint64(in[2])*(uint64(in[4])<<1) +
  		uint64(in[3])*(uint64(in[3])<<1)
  	tmp[7] = uint64(in[0])*(uint64(in[7])<<1) +
  		uint64(in[1])*(uint64(in[6])<<1) +
  		uint64(in[2])*(uint64(in[5])<<1) +
  		uint64(in[3])*(uint64(in[4])<<1)
  	// tmp[8] has the greatest value of 2**61 + 2**60 + 2**61 + 2**60 + 2**60,
  	// which is < 2**64 as required.
  	tmp[8] = uint64(in[0])*(uint64(in[8])<<1) +
  		uint64(in[1])*(uint64(in[7])<<2) +
  		uint64(in[2])*(uint64(in[6])<<1) +
  		uint64(in[3])*(uint64(in[5])<<2) +
  		uint64(in[4])*uint64(in[4])
  	tmp[9] = uint64(in[1])*(uint64(in[8])<<1) +
  		uint64(in[2])*(uint64(in[7])<<1) +
  		uint64(in[3])*(uint64(in[6])<<1) +
  		uint64(in[4])*(uint64(in[5])<<1)
  	tmp[10] = uint64(in[2])*(uint64(in[8])<<1) +
  		uint64(in[3])*(uint64(in[7])<<2) +
  		uint64(in[4])*(uint64(in[6])<<1) +
  		uint64(in[5])*(uint64(in[5])<<1)
  	tmp[11] = uint64(in[3])*(uint64(in[8])<<1) +
  		uint64(in[4])*(uint64(in[7])<<1) +
  		uint64(in[5])*(uint64(in[6])<<1)
  	tmp[12] = uint64(in[4])*(uint64(in[8])<<1) +
  		uint64(in[5])*(uint64(in[7])<<2) +
  		uint64(in[6])*uint64(in[6])
  	tmp[13] = uint64(in[5])*(uint64(in[8])<<1) +
  		uint64(in[6])*(uint64(in[7])<<1)
  	tmp[14] = uint64(in[6])*(uint64(in[8])<<1) +
  		uint64(in[7])*(uint64(in[7])<<1)
  	tmp[15] = uint64(in[7]) * (uint64(in[8]) << 1)
  	tmp[16] = uint64(in[8]) * uint64(in[8])
  
  	p256ReduceDegree(out, tmp)
  }
  
  // p256Mul sets out=in*in2.
  //
  // On entry: in[0,2,...] < 2**30, in[1,3,...] < 2**29 and
  //           in2[0,2,...] < 2**30, in2[1,3,...] < 2**29.
  // On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
  func p256Mul(out, in, in2 *[p256Limbs]uint32) {
  	var tmp [17]uint64
  
  	tmp[0] = uint64(in[0]) * uint64(in2[0])
  	tmp[1] = uint64(in[0])*(uint64(in2[1])<<0) +
  		uint64(in[1])*(uint64(in2[0])<<0)
  	tmp[2] = uint64(in[0])*(uint64(in2[2])<<0) +
  		uint64(in[1])*(uint64(in2[1])<<1) +
  		uint64(in[2])*(uint64(in2[0])<<0)
  	tmp[3] = uint64(in[0])*(uint64(in2[3])<<0) +
  		uint64(in[1])*(uint64(in2[2])<<0) +
  		uint64(in[2])*(uint64(in2[1])<<0) +
  		uint64(in[3])*(uint64(in2[0])<<0)
  	tmp[4] = uint64(in[0])*(uint64(in2[4])<<0) +
  		uint64(in[1])*(uint64(in2[3])<<1) +
  		uint64(in[2])*(uint64(in2[2])<<0) +
  		uint64(in[3])*(uint64(in2[1])<<1) +
  		uint64(in[4])*(uint64(in2[0])<<0)
  	tmp[5] = uint64(in[0])*(uint64(in2[5])<<0) +
  		uint64(in[1])*(uint64(in2[4])<<0) +
  		uint64(in[2])*(uint64(in2[3])<<0) +
  		uint64(in[3])*(uint64(in2[2])<<0) +
  		uint64(in[4])*(uint64(in2[1])<<0) +
  		uint64(in[5])*(uint64(in2[0])<<0)
  	tmp[6] = uint64(in[0])*(uint64(in2[6])<<0) +
  		uint64(in[1])*(uint64(in2[5])<<1) +
  		uint64(in[2])*(uint64(in2[4])<<0) +
  		uint64(in[3])*(uint64(in2[3])<<1) +
  		uint64(in[4])*(uint64(in2[2])<<0) +
  		uint64(in[5])*(uint64(in2[1])<<1) +
  		uint64(in[6])*(uint64(in2[0])<<0)
  	tmp[7] = uint64(in[0])*(uint64(in2[7])<<0) +
  		uint64(in[1])*(uint64(in2[6])<<0) +
  		uint64(in[2])*(uint64(in2[5])<<0) +
  		uint64(in[3])*(uint64(in2[4])<<0) +
  		uint64(in[4])*(uint64(in2[3])<<0) +
  		uint64(in[5])*(uint64(in2[2])<<0) +
  		uint64(in[6])*(uint64(in2[1])<<0) +
  		uint64(in[7])*(uint64(in2[0])<<0)
  	// tmp[8] has the greatest value but doesn't overflow. See logic in
  	// p256Square.
  	tmp[8] = uint64(in[0])*(uint64(in2[8])<<0) +
  		uint64(in[1])*(uint64(in2[7])<<1) +
  		uint64(in[2])*(uint64(in2[6])<<0) +
  		uint64(in[3])*(uint64(in2[5])<<1) +
  		uint64(in[4])*(uint64(in2[4])<<0) +
  		uint64(in[5])*(uint64(in2[3])<<1) +
  		uint64(in[6])*(uint64(in2[2])<<0) +
  		uint64(in[7])*(uint64(in2[1])<<1) +
  		uint64(in[8])*(uint64(in2[0])<<0)
  	tmp[9] = uint64(in[1])*(uint64(in2[8])<<0) +
  		uint64(in[2])*(uint64(in2[7])<<0) +
  		uint64(in[3])*(uint64(in2[6])<<0) +
  		uint64(in[4])*(uint64(in2[5])<<0) +
  		uint64(in[5])*(uint64(in2[4])<<0) +
  		uint64(in[6])*(uint64(in2[3])<<0) +
  		uint64(in[7])*(uint64(in2[2])<<0) +
  		uint64(in[8])*(uint64(in2[1])<<0)
  	tmp[10] = uint64(in[2])*(uint64(in2[8])<<0) +
  		uint64(in[3])*(uint64(in2[7])<<1) +
  		uint64(in[4])*(uint64(in2[6])<<0) +
  		uint64(in[5])*(uint64(in2[5])<<1) +
  		uint64(in[6])*(uint64(in2[4])<<0) +
  		uint64(in[7])*(uint64(in2[3])<<1) +
  		uint64(in[8])*(uint64(in2[2])<<0)
  	tmp[11] = uint64(in[3])*(uint64(in2[8])<<0) +
  		uint64(in[4])*(uint64(in2[7])<<0) +
  		uint64(in[5])*(uint64(in2[6])<<0) +
  		uint64(in[6])*(uint64(in2[5])<<0) +
  		uint64(in[7])*(uint64(in2[4])<<0) +
  		uint64(in[8])*(uint64(in2[3])<<0)
  	tmp[12] = uint64(in[4])*(uint64(in2[8])<<0) +
  		uint64(in[5])*(uint64(in2[7])<<1) +
  		uint64(in[6])*(uint64(in2[6])<<0) +
  		uint64(in[7])*(uint64(in2[5])<<1) +
  		uint64(in[8])*(uint64(in2[4])<<0)
  	tmp[13] = uint64(in[5])*(uint64(in2[8])<<0) +
  		uint64(in[6])*(uint64(in2[7])<<0) +
  		uint64(in[7])*(uint64(in2[6])<<0) +
  		uint64(in[8])*(uint64(in2[5])<<0)
  	tmp[14] = uint64(in[6])*(uint64(in2[8])<<0) +
  		uint64(in[7])*(uint64(in2[7])<<1) +
  		uint64(in[8])*(uint64(in2[6])<<0)
  	tmp[15] = uint64(in[7])*(uint64(in2[8])<<0) +
  		uint64(in[8])*(uint64(in2[7])<<0)
  	tmp[16] = uint64(in[8]) * (uint64(in2[8]) << 0)
  
  	p256ReduceDegree(out, tmp)
  }
  
  func p256Assign(out, in *[p256Limbs]uint32) {
  	*out = *in
  }
  
  // p256Invert 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)
  func p256Invert(out, in *[p256Limbs]uint32) {
  	var ftmp, ftmp2 [p256Limbs]uint32
  
  	// each e_I will hold |in|^{2^I - 1}
  	var e2, e4, e8, e16, e32, e64 [p256Limbs]uint32
  
  	p256Square(&ftmp, in)     // 2^1
  	p256Mul(&ftmp, in, &ftmp) // 2^2 - 2^0
  	p256Assign(&e2, &ftmp)
  	p256Square(&ftmp, &ftmp)   // 2^3 - 2^1
  	p256Square(&ftmp, &ftmp)   // 2^4 - 2^2
  	p256Mul(&ftmp, &ftmp, &e2) // 2^4 - 2^0
  	p256Assign(&e4, &ftmp)
  	p256Square(&ftmp, &ftmp)   // 2^5 - 2^1
  	p256Square(&ftmp, &ftmp)   // 2^6 - 2^2
  	p256Square(&ftmp, &ftmp)   // 2^7 - 2^3
  	p256Square(&ftmp, &ftmp)   // 2^8 - 2^4
  	p256Mul(&ftmp, &ftmp, &e4) // 2^8 - 2^0
  	p256Assign(&e8, &ftmp)
  	for i := 0; i < 8; i++ {
  		p256Square(&ftmp, &ftmp)
  	} // 2^16 - 2^8
  	p256Mul(&ftmp, &ftmp, &e8) // 2^16 - 2^0
  	p256Assign(&e16, &ftmp)
  	for i := 0; i < 16; i++ {
  		p256Square(&ftmp, &ftmp)
  	} // 2^32 - 2^16
  	p256Mul(&ftmp, &ftmp, &e16) // 2^32 - 2^0
  	p256Assign(&e32, &ftmp)
  	for i := 0; i < 32; i++ {
  		p256Square(&ftmp, &ftmp)
  	} // 2^64 - 2^32
  	p256Assign(&e64, &ftmp)
  	p256Mul(&ftmp, &ftmp, in) // 2^64 - 2^32 + 2^0
  	for i := 0; i < 192; i++ {
  		p256Square(&ftmp, &ftmp)
  	} // 2^256 - 2^224 + 2^192
  
  	p256Mul(&ftmp2, &e64, &e32) // 2^64 - 2^0
  	for i := 0; i < 16; i++ {
  		p256Square(&ftmp2, &ftmp2)
  	} // 2^80 - 2^16
  	p256Mul(&ftmp2, &ftmp2, &e16) // 2^80 - 2^0
  	for i := 0; i < 8; i++ {
  		p256Square(&ftmp2, &ftmp2)
  	} // 2^88 - 2^8
  	p256Mul(&ftmp2, &ftmp2, &e8) // 2^88 - 2^0
  	for i := 0; i < 4; i++ {
  		p256Square(&ftmp2, &ftmp2)
  	} // 2^92 - 2^4
  	p256Mul(&ftmp2, &ftmp2, &e4) // 2^92 - 2^0
  	p256Square(&ftmp2, &ftmp2)   // 2^93 - 2^1
  	p256Square(&ftmp2, &ftmp2)   // 2^94 - 2^2
  	p256Mul(&ftmp2, &ftmp2, &e2) // 2^94 - 2^0
  	p256Square(&ftmp2, &ftmp2)   // 2^95 - 2^1
  	p256Square(&ftmp2, &ftmp2)   // 2^96 - 2^2
  	p256Mul(&ftmp2, &ftmp2, in)  // 2^96 - 3
  
  	p256Mul(out, &ftmp2, &ftmp) // 2^256 - 2^224 + 2^192 + 2^96 - 3
  }
  
  // p256Scalar3 sets out=3*out.
  //
  // On entry: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
  // On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
  func p256Scalar3(out *[p256Limbs]uint32) {
  	var carry uint32
  
  	for i := 0; ; i++ {
  		out[i] *= 3
  		out[i] += carry
  		carry = out[i] >> 29
  		out[i] &= bottom29Bits
  
  		i++
  		if i == p256Limbs {
  			break
  		}
  
  		out[i] *= 3
  		out[i] += carry
  		carry = out[i] >> 28
  		out[i] &= bottom28Bits
  	}
  
  	p256ReduceCarry(out, carry)
  }
  
  // p256Scalar4 sets out=4*out.
  //
  // On entry: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
  // On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
  func p256Scalar4(out *[p256Limbs]uint32) {
  	var carry, nextCarry uint32
  
  	for i := 0; ; i++ {
  		nextCarry = out[i] >> 27
  		out[i] <<= 2
  		out[i] &= bottom29Bits
  		out[i] += carry
  		carry = nextCarry + (out[i] >> 29)
  		out[i] &= bottom29Bits
  
  		i++
  		if i == p256Limbs {
  			break
  		}
  		nextCarry = out[i] >> 26
  		out[i] <<= 2
  		out[i] &= bottom28Bits
  		out[i] += carry
  		carry = nextCarry + (out[i] >> 28)
  		out[i] &= bottom28Bits
  	}
  
  	p256ReduceCarry(out, carry)
  }
  
  // p256Scalar8 sets out=8*out.
  //
  // On entry: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
  // On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
  func p256Scalar8(out *[p256Limbs]uint32) {
  	var carry, nextCarry uint32
  
  	for i := 0; ; i++ {
  		nextCarry = out[i] >> 26
  		out[i] <<= 3
  		out[i] &= bottom29Bits
  		out[i] += carry
  		carry = nextCarry + (out[i] >> 29)
  		out[i] &= bottom29Bits
  
  		i++
  		if i == p256Limbs {
  			break
  		}
  		nextCarry = out[i] >> 25
  		out[i] <<= 3
  		out[i] &= bottom28Bits
  		out[i] += carry
  		carry = nextCarry + (out[i] >> 28)
  		out[i] &= bottom28Bits
  	}
  
  	p256ReduceCarry(out, carry)
  }
  
  // Group operations:
  //
  // Elements of the elliptic curve group are represented in Jacobian
  // coordinates: (x, y, z). An affine point (x', y') is x'=x/z**2, y'=y/z**3 in
  // Jacobian form.
  
  // p256PointDouble sets {xOut,yOut,zOut} = 2*{x,y,z}.
  //
  // See http://www.hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#doubling-dbl-2009-l
  func p256PointDouble(xOut, yOut, zOut, x, y, z *[p256Limbs]uint32) {
  	var delta, gamma, alpha, beta, tmp, tmp2 [p256Limbs]uint32
  
  	p256Square(&delta, z)
  	p256Square(&gamma, y)
  	p256Mul(&beta, x, &gamma)
  
  	p256Sum(&tmp, x, &delta)
  	p256Diff(&tmp2, x, &delta)
  	p256Mul(&alpha, &tmp, &tmp2)
  	p256Scalar3(&alpha)
  
  	p256Sum(&tmp, y, z)
  	p256Square(&tmp, &tmp)
  	p256Diff(&tmp, &tmp, &gamma)
  	p256Diff(zOut, &tmp, &delta)
  
  	p256Scalar4(&beta)
  	p256Square(xOut, &alpha)
  	p256Diff(xOut, xOut, &beta)
  	p256Diff(xOut, xOut, &beta)
  
  	p256Diff(&tmp, &beta, xOut)
  	p256Mul(&tmp, &alpha, &tmp)
  	p256Square(&tmp2, &gamma)
  	p256Scalar8(&tmp2)
  	p256Diff(yOut, &tmp, &tmp2)
  }
  
  // p256PointAddMixed sets {xOut,yOut,zOut} = {x1,y1,z1} + {x2,y2,1}.
  // (i.e. the second point is affine.)
  //
  // See http://www.hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#addition-add-2007-bl
  //
  // Note that this function does not handle P+P, infinity+P nor P+infinity
  // correctly.
  func p256PointAddMixed(xOut, yOut, zOut, x1, y1, z1, x2, y2 *[p256Limbs]uint32) {
  	var z1z1, z1z1z1, s2, u2, h, i, j, r, rr, v, tmp [p256Limbs]uint32
  
  	p256Square(&z1z1, z1)
  	p256Sum(&tmp, z1, z1)
  
  	p256Mul(&u2, x2, &z1z1)
  	p256Mul(&z1z1z1, z1, &z1z1)
  	p256Mul(&s2, y2, &z1z1z1)
  	p256Diff(&h, &u2, x1)
  	p256Sum(&i, &h, &h)
  	p256Square(&i, &i)
  	p256Mul(&j, &h, &i)
  	p256Diff(&r, &s2, y1)
  	p256Sum(&r, &r, &r)
  	p256Mul(&v, x1, &i)
  
  	p256Mul(zOut, &tmp, &h)
  	p256Square(&rr, &r)
  	p256Diff(xOut, &rr, &j)
  	p256Diff(xOut, xOut, &v)
  	p256Diff(xOut, xOut, &v)
  
  	p256Diff(&tmp, &v, xOut)
  	p256Mul(yOut, &tmp, &r)
  	p256Mul(&tmp, y1, &j)
  	p256Diff(yOut, yOut, &tmp)
  	p256Diff(yOut, yOut, &tmp)
  }
  
  // p256PointAdd sets {xOut,yOut,zOut} = {x1,y1,z1} + {x2,y2,z2}.
  //
  // See http://www.hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#addition-add-2007-bl
  //
  // Note that this function does not handle P+P, infinity+P nor P+infinity
  // correctly.
  func p256PointAdd(xOut, yOut, zOut, x1, y1, z1, x2, y2, z2 *[p256Limbs]uint32) {
  	var z1z1, z1z1z1, z2z2, z2z2z2, s1, s2, u1, u2, h, i, j, r, rr, v, tmp [p256Limbs]uint32
  
  	p256Square(&z1z1, z1)
  	p256Square(&z2z2, z2)
  	p256Mul(&u1, x1, &z2z2)
  
  	p256Sum(&tmp, z1, z2)
  	p256Square(&tmp, &tmp)
  	p256Diff(&tmp, &tmp, &z1z1)
  	p256Diff(&tmp, &tmp, &z2z2)
  
  	p256Mul(&z2z2z2, z2, &z2z2)
  	p256Mul(&s1, y1, &z2z2z2)
  
  	p256Mul(&u2, x2, &z1z1)
  	p256Mul(&z1z1z1, z1, &z1z1)
  	p256Mul(&s2, y2, &z1z1z1)
  	p256Diff(&h, &u2, &u1)
  	p256Sum(&i, &h, &h)
  	p256Square(&i, &i)
  	p256Mul(&j, &h, &i)
  	p256Diff(&r, &s2, &s1)
  	p256Sum(&r, &r, &r)
  	p256Mul(&v, &u1, &i)
  
  	p256Mul(zOut, &tmp, &h)
  	p256Square(&rr, &r)
  	p256Diff(xOut, &rr, &j)
  	p256Diff(xOut, xOut, &v)
  	p256Diff(xOut, xOut, &v)
  
  	p256Diff(&tmp, &v, xOut)
  	p256Mul(yOut, &tmp, &r)
  	p256Mul(&tmp, &s1, &j)
  	p256Diff(yOut, yOut, &tmp)
  	p256Diff(yOut, yOut, &tmp)
  }
  
  // p256CopyConditional sets out=in if mask = 0xffffffff in constant time.
  //
  // On entry: mask is either 0 or 0xffffffff.
  func p256CopyConditional(out, in *[p256Limbs]uint32, mask uint32) {
  	for i := 0; i < p256Limbs; i++ {
  		tmp := mask & (in[i] ^ out[i])
  		out[i] ^= tmp
  	}
  }
  
  // p256SelectAffinePoint sets {out_x,out_y} to the index'th entry of table.
  // On entry: index < 16, table[0] must be zero.
  func p256SelectAffinePoint(xOut, yOut *[p256Limbs]uint32, table []uint32, index uint32) {
  	for i := range xOut {
  		xOut[i] = 0
  	}
  	for i := range yOut {
  		yOut[i] = 0
  	}
  
  	for i := uint32(1); i < 16; i++ {
  		mask := i ^ index
  		mask |= mask >> 2
  		mask |= mask >> 1
  		mask &= 1
  		mask--
  		for j := range xOut {
  			xOut[j] |= table[0] & mask
  			table = table[1:]
  		}
  		for j := range yOut {
  			yOut[j] |= table[0] & mask
  			table = table[1:]
  		}
  	}
  }
  
  // p256SelectJacobianPoint sets {out_x,out_y,out_z} to the index'th entry of
  // table.
  // On entry: index < 16, table[0] must be zero.
  func p256SelectJacobianPoint(xOut, yOut, zOut *[p256Limbs]uint32, table *[16][3][p256Limbs]uint32, index uint32) {
  	for i := range xOut {
  		xOut[i] = 0
  	}
  	for i := range yOut {
  		yOut[i] = 0
  	}
  	for i := range zOut {
  		zOut[i] = 0
  	}
  
  	// The implicit value at index 0 is all zero. We don't need to perform that
  	// iteration of the loop because we already set out_* to zero.
  	for i := uint32(1); i < 16; i++ {
  		mask := i ^ index
  		mask |= mask >> 2
  		mask |= mask >> 1
  		mask &= 1
  		mask--
  		for j := range xOut {
  			xOut[j] |= table[i][0][j] & mask
  		}
  		for j := range yOut {
  			yOut[j] |= table[i][1][j] & mask
  		}
  		for j := range zOut {
  			zOut[j] |= table[i][2][j] & mask
  		}
  	}
  }
  
  // p256GetBit returns the bit'th bit of scalar.
  func p256GetBit(scalar *[32]uint8, bit uint) uint32 {
  	return uint32(((scalar[bit>>3]) >> (bit & 7)) & 1)
  }
  
  // p256ScalarBaseMult sets {xOut,yOut,zOut} = scalar*G where scalar is a
  // little-endian number. Note that the value of scalar must be less than the
  // order of the group.
  func p256ScalarBaseMult(xOut, yOut, zOut *[p256Limbs]uint32, scalar *[32]uint8) {
  	nIsInfinityMask := ^uint32(0)
  	var pIsNoninfiniteMask, mask, tableOffset uint32
  	var px, py, tx, ty, tz [p256Limbs]uint32
  
  	for i := range xOut {
  		xOut[i] = 0
  	}
  	for i := range yOut {
  		yOut[i] = 0
  	}
  	for i := range zOut {
  		zOut[i] = 0
  	}
  
  	// The loop adds bits at positions 0, 64, 128 and 192, followed by
  	// positions 32,96,160 and 224 and does this 32 times.
  	for i := uint(0); i < 32; i++ {
  		if i != 0 {
  			p256PointDouble(xOut, yOut, zOut, xOut, yOut, zOut)
  		}
  		tableOffset = 0
  		for j := uint(0); j <= 32; j += 32 {
  			bit0 := p256GetBit(scalar, 31-i+j)
  			bit1 := p256GetBit(scalar, 95-i+j)
  			bit2 := p256GetBit(scalar, 159-i+j)
  			bit3 := p256GetBit(scalar, 223-i+j)
  			index := bit0 | (bit1 << 1) | (bit2 << 2) | (bit3 << 3)
  
  			p256SelectAffinePoint(&px, &py, p256Precomputed[tableOffset:], index)
  			tableOffset += 30 * p256Limbs
  
  			// Since scalar is less than the order of the group, we know that
  			// {xOut,yOut,zOut} != {px,py,1}, unless both are zero, which we handle
  			// below.
  			p256PointAddMixed(&tx, &ty, &tz, xOut, yOut, zOut, &px, &py)
  			// The result of pointAddMixed is incorrect if {xOut,yOut,zOut} is zero
  			// (a.k.a.  the point at infinity). We handle that situation by
  			// copying the point from the table.
  			p256CopyConditional(xOut, &px, nIsInfinityMask)
  			p256CopyConditional(yOut, &py, nIsInfinityMask)
  			p256CopyConditional(zOut, &p256One, nIsInfinityMask)
  
  			// Equally, the result is also wrong if the point from the table is
  			// zero, which happens when the index is zero. We handle that by
  			// only copying from {tx,ty,tz} to {xOut,yOut,zOut} if index != 0.
  			pIsNoninfiniteMask = nonZeroToAllOnes(index)
  			mask = pIsNoninfiniteMask & ^nIsInfinityMask
  			p256CopyConditional(xOut, &tx, mask)
  			p256CopyConditional(yOut, &ty, mask)
  			p256CopyConditional(zOut, &tz, mask)
  			// If p was not zero, then n is now non-zero.
  			nIsInfinityMask &^= pIsNoninfiniteMask
  		}
  	}
  }
  
  // p256PointToAffine converts a Jacobian point to an affine point. If the input
  // is the point at infinity then it returns (0, 0) in constant time.
  func p256PointToAffine(xOut, yOut, x, y, z *[p256Limbs]uint32) {
  	var zInv, zInvSq [p256Limbs]uint32
  
  	p256Invert(&zInv, z)
  	p256Square(&zInvSq, &zInv)
  	p256Mul(xOut, x, &zInvSq)
  	p256Mul(&zInv, &zInv, &zInvSq)
  	p256Mul(yOut, y, &zInv)
  }
  
  // p256ToAffine returns a pair of *big.Int containing the affine representation
  // of {x,y,z}.
  func p256ToAffine(x, y, z *[p256Limbs]uint32) (xOut, yOut *big.Int) {
  	var xx, yy [p256Limbs]uint32
  	p256PointToAffine(&xx, &yy, x, y, z)
  	return p256ToBig(&xx), p256ToBig(&yy)
  }
  
  // p256ScalarMult sets {xOut,yOut,zOut} = scalar*{x,y}.
  func p256ScalarMult(xOut, yOut, zOut, x, y *[p256Limbs]uint32, scalar *[32]uint8) {
  	var px, py, pz, tx, ty, tz [p256Limbs]uint32
  	var precomp [16][3][p256Limbs]uint32
  	var nIsInfinityMask, index, pIsNoninfiniteMask, mask uint32
  
  	// We precompute 0,1,2,... times {x,y}.
  	precomp[1][0] = *x
  	precomp[1][1] = *y
  	precomp[1][2] = p256One
  
  	for i := 2; i < 16; i += 2 {
  		p256PointDouble(&precomp[i][0], &precomp[i][1], &precomp[i][2], &precomp[i/2][0], &precomp[i/2][1], &precomp[i/2][2])
  		p256PointAddMixed(&precomp[i+1][0], &precomp[i+1][1], &precomp[i+1][2], &precomp[i][0], &precomp[i][1], &precomp[i][2], x, y)
  	}
  
  	for i := range xOut {
  		xOut[i] = 0
  	}
  	for i := range yOut {
  		yOut[i] = 0
  	}
  	for i := range zOut {
  		zOut[i] = 0
  	}
  	nIsInfinityMask = ^uint32(0)
  
  	// We add in a window of four bits each iteration and do this 64 times.
  	for i := 0; i < 64; i++ {
  		if i != 0 {
  			p256PointDouble(xOut, yOut, zOut, xOut, yOut, zOut)
  			p256PointDouble(xOut, yOut, zOut, xOut, yOut, zOut)
  			p256PointDouble(xOut, yOut, zOut, xOut, yOut, zOut)
  			p256PointDouble(xOut, yOut, zOut, xOut, yOut, zOut)
  		}
  
  		index = uint32(scalar[31-i/2])
  		if (i & 1) == 1 {
  			index &= 15
  		} else {
  			index >>= 4
  		}
  
  		// See the comments in scalarBaseMult about handling infinities.
  		p256SelectJacobianPoint(&px, &py, &pz, &precomp, index)
  		p256PointAdd(&tx, &ty, &tz, xOut, yOut, zOut, &px, &py, &pz)
  		p256CopyConditional(xOut, &px, nIsInfinityMask)
  		p256CopyConditional(yOut, &py, nIsInfinityMask)
  		p256CopyConditional(zOut, &pz, nIsInfinityMask)
  
  		pIsNoninfiniteMask = nonZeroToAllOnes(index)
  		mask = pIsNoninfiniteMask & ^nIsInfinityMask
  		p256CopyConditional(xOut, &tx, mask)
  		p256CopyConditional(yOut, &ty, mask)
  		p256CopyConditional(zOut, &tz, mask)
  		nIsInfinityMask &^= pIsNoninfiniteMask
  	}
  }
  
  // p256FromBig sets out = R*in.
  func p256FromBig(out *[p256Limbs]uint32, in *big.Int) {
  	tmp := new(big.Int).Lsh(in, 257)
  	tmp.Mod(tmp, p256Params.P)
  
  	for i := 0; i < p256Limbs; i++ {
  		if bits := tmp.Bits(); len(bits) > 0 {
  			out[i] = uint32(bits[0]) & bottom29Bits
  		} else {
  			out[i] = 0
  		}
  		tmp.Rsh(tmp, 29)
  
  		i++
  		if i == p256Limbs {
  			break
  		}
  
  		if bits := tmp.Bits(); len(bits) > 0 {
  			out[i] = uint32(bits[0]) & bottom28Bits
  		} else {
  			out[i] = 0
  		}
  		tmp.Rsh(tmp, 28)
  	}
  }
  
  // p256ToBig returns a *big.Int containing the value of in.
  func p256ToBig(in *[p256Limbs]uint32) *big.Int {
  	result, tmp := new(big.Int), new(big.Int)
  
  	result.SetInt64(int64(in[p256Limbs-1]))
  	for i := p256Limbs - 2; i >= 0; i-- {
  		if (i & 1) == 0 {
  			result.Lsh(result, 29)
  		} else {
  			result.Lsh(result, 28)
  		}
  		tmp.SetInt64(int64(in[i]))
  		result.Add(result, tmp)
  	}
  
  	result.Mul(result, p256RInverse)
  	result.Mod(result, p256Params.P)
  	return result
  }
  

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