Source file src/crypto/elliptic/p224.go

     1	// Copyright 2012 The Go Authors. All rights reserved.
     2	// Use of this source code is governed by a BSD-style
     3	// license that can be found in the LICENSE file.
     4	
     5	package elliptic
     6	
     7	// This is a constant-time, 32-bit implementation of P224. See FIPS 186-3,
     8	// section D.2.2.
     9	//
    10	// See https://www.imperialviolet.org/2010/12/04/ecc.html ([1]) for background.
    11	
    12	import (
    13		"math/big"
    14	)
    15	
    16	var p224 p224Curve
    17	
    18	type p224Curve struct {
    19		*CurveParams
    20		gx, gy, b p224FieldElement
    21	}
    22	
    23	func initP224() {
    24		// See FIPS 186-3, section D.2.2
    25		p224.CurveParams = &CurveParams{Name: "P-224"}
    26		p224.P, _ = new(big.Int).SetString("26959946667150639794667015087019630673557916260026308143510066298881", 10)
    27		p224.N, _ = new(big.Int).SetString("26959946667150639794667015087019625940457807714424391721682722368061", 10)
    28		p224.B, _ = new(big.Int).SetString("b4050a850c04b3abf54132565044b0b7d7bfd8ba270b39432355ffb4", 16)
    29		p224.Gx, _ = new(big.Int).SetString("b70e0cbd6bb4bf7f321390b94a03c1d356c21122343280d6115c1d21", 16)
    30		p224.Gy, _ = new(big.Int).SetString("bd376388b5f723fb4c22dfe6cd4375a05a07476444d5819985007e34", 16)
    31		p224.BitSize = 224
    32	
    33		p224FromBig(&p224.gx, p224.Gx)
    34		p224FromBig(&p224.gy, p224.Gy)
    35		p224FromBig(&p224.b, p224.B)
    36	}
    37	
    38	// P224 returns a Curve which implements P-224 (see FIPS 186-3, section D.2.2).
    39	//
    40	// The cryptographic operations are implemented using constant-time algorithms.
    41	func P224() Curve {
    42		initonce.Do(initAll)
    43		return p224
    44	}
    45	
    46	func (curve p224Curve) Params() *CurveParams {
    47		return curve.CurveParams
    48	}
    49	
    50	func (curve p224Curve) IsOnCurve(bigX, bigY *big.Int) bool {
    51		var x, y p224FieldElement
    52		p224FromBig(&x, bigX)
    53		p224FromBig(&y, bigY)
    54	
    55		// y² = x³ - 3x + b
    56		var tmp p224LargeFieldElement
    57		var x3 p224FieldElement
    58		p224Square(&x3, &x, &tmp)
    59		p224Mul(&x3, &x3, &x, &tmp)
    60	
    61		for i := 0; i < 8; i++ {
    62			x[i] *= 3
    63		}
    64		p224Sub(&x3, &x3, &x)
    65		p224Reduce(&x3)
    66		p224Add(&x3, &x3, &curve.b)
    67		p224Contract(&x3, &x3)
    68	
    69		p224Square(&y, &y, &tmp)
    70		p224Contract(&y, &y)
    71	
    72		for i := 0; i < 8; i++ {
    73			if y[i] != x3[i] {
    74				return false
    75			}
    76		}
    77		return true
    78	}
    79	
    80	func (p224Curve) Add(bigX1, bigY1, bigX2, bigY2 *big.Int) (x, y *big.Int) {
    81		var x1, y1, z1, x2, y2, z2, x3, y3, z3 p224FieldElement
    82	
    83		p224FromBig(&x1, bigX1)
    84		p224FromBig(&y1, bigY1)
    85		if bigX1.Sign() != 0 || bigY1.Sign() != 0 {
    86			z1[0] = 1
    87		}
    88		p224FromBig(&x2, bigX2)
    89		p224FromBig(&y2, bigY2)
    90		if bigX2.Sign() != 0 || bigY2.Sign() != 0 {
    91			z2[0] = 1
    92		}
    93	
    94		p224AddJacobian(&x3, &y3, &z3, &x1, &y1, &z1, &x2, &y2, &z2)
    95		return p224ToAffine(&x3, &y3, &z3)
    96	}
    97	
    98	func (p224Curve) Double(bigX1, bigY1 *big.Int) (x, y *big.Int) {
    99		var x1, y1, z1, x2, y2, z2 p224FieldElement
   100	
   101		p224FromBig(&x1, bigX1)
   102		p224FromBig(&y1, bigY1)
   103		z1[0] = 1
   104	
   105		p224DoubleJacobian(&x2, &y2, &z2, &x1, &y1, &z1)
   106		return p224ToAffine(&x2, &y2, &z2)
   107	}
   108	
   109	func (p224Curve) ScalarMult(bigX1, bigY1 *big.Int, scalar []byte) (x, y *big.Int) {
   110		var x1, y1, z1, x2, y2, z2 p224FieldElement
   111	
   112		p224FromBig(&x1, bigX1)
   113		p224FromBig(&y1, bigY1)
   114		z1[0] = 1
   115	
   116		p224ScalarMult(&x2, &y2, &z2, &x1, &y1, &z1, scalar)
   117		return p224ToAffine(&x2, &y2, &z2)
   118	}
   119	
   120	func (curve p224Curve) ScalarBaseMult(scalar []byte) (x, y *big.Int) {
   121		var z1, x2, y2, z2 p224FieldElement
   122	
   123		z1[0] = 1
   124		p224ScalarMult(&x2, &y2, &z2, &curve.gx, &curve.gy, &z1, scalar)
   125		return p224ToAffine(&x2, &y2, &z2)
   126	}
   127	
   128	// Field element functions.
   129	//
   130	// The field that we're dealing with is ℤ/pℤ where p = 2**224 - 2**96 + 1.
   131	//
   132	// Field elements are represented by a FieldElement, which is a typedef to an
   133	// array of 8 uint32's. The value of a FieldElement, a, is:
   134	//   a[0] + 2**28·a[1] + 2**56·a[1] + ... + 2**196·a[7]
   135	//
   136	// Using 28-bit limbs means that there's only 4 bits of headroom, which is less
   137	// than we would really like. But it has the useful feature that we hit 2**224
   138	// exactly, making the reflections during a reduce much nicer.
   139	type p224FieldElement [8]uint32
   140	
   141	// p224P is the order of the field, represented as a p224FieldElement.
   142	var p224P = [8]uint32{1, 0, 0, 0xffff000, 0xfffffff, 0xfffffff, 0xfffffff, 0xfffffff}
   143	
   144	// p224IsZero returns 1 if a == 0 mod p and 0 otherwise.
   145	//
   146	// a[i] < 2**29
   147	func p224IsZero(a *p224FieldElement) uint32 {
   148		// Since a p224FieldElement contains 224 bits there are two possible
   149		// representations of 0: 0 and p.
   150		var minimal p224FieldElement
   151		p224Contract(&minimal, a)
   152	
   153		var isZero, isP uint32
   154		for i, v := range minimal {
   155			isZero |= v
   156			isP |= v - p224P[i]
   157		}
   158	
   159		// If either isZero or isP is 0, then we should return 1.
   160		isZero |= isZero >> 16
   161		isZero |= isZero >> 8
   162		isZero |= isZero >> 4
   163		isZero |= isZero >> 2
   164		isZero |= isZero >> 1
   165	
   166		isP |= isP >> 16
   167		isP |= isP >> 8
   168		isP |= isP >> 4
   169		isP |= isP >> 2
   170		isP |= isP >> 1
   171	
   172		// For isZero and isP, the LSB is 0 iff all the bits are zero.
   173		result := isZero & isP
   174		result = (^result) & 1
   175	
   176		return result
   177	}
   178	
   179	// p224Add computes *out = a+b
   180	//
   181	// a[i] + b[i] < 2**32
   182	func p224Add(out, a, b *p224FieldElement) {
   183		for i := 0; i < 8; i++ {
   184			out[i] = a[i] + b[i]
   185		}
   186	}
   187	
   188	const two31p3 = 1<<31 + 1<<3
   189	const two31m3 = 1<<31 - 1<<3
   190	const two31m15m3 = 1<<31 - 1<<15 - 1<<3
   191	
   192	// p224ZeroModP31 is 0 mod p where bit 31 is set in all limbs so that we can
   193	// subtract smaller amounts without underflow. See the section "Subtraction" in
   194	// [1] for reasoning.
   195	var p224ZeroModP31 = []uint32{two31p3, two31m3, two31m3, two31m15m3, two31m3, two31m3, two31m3, two31m3}
   196	
   197	// p224Sub computes *out = a-b
   198	//
   199	// a[i], b[i] < 2**30
   200	// out[i] < 2**32
   201	func p224Sub(out, a, b *p224FieldElement) {
   202		for i := 0; i < 8; i++ {
   203			out[i] = a[i] + p224ZeroModP31[i] - b[i]
   204		}
   205	}
   206	
   207	// LargeFieldElement also represents an element of the field. The limbs are
   208	// still spaced 28-bits apart and in little-endian order. So the limbs are at
   209	// 0, 28, 56, ..., 392 bits, each 64-bits wide.
   210	type p224LargeFieldElement [15]uint64
   211	
   212	const two63p35 = 1<<63 + 1<<35
   213	const two63m35 = 1<<63 - 1<<35
   214	const two63m35m19 = 1<<63 - 1<<35 - 1<<19
   215	
   216	// p224ZeroModP63 is 0 mod p where bit 63 is set in all limbs. See the section
   217	// "Subtraction" in [1] for why.
   218	var p224ZeroModP63 = [8]uint64{two63p35, two63m35, two63m35, two63m35, two63m35m19, two63m35, two63m35, two63m35}
   219	
   220	const bottom12Bits = 0xfff
   221	const bottom28Bits = 0xfffffff
   222	
   223	// p224Mul computes *out = a*b
   224	//
   225	// a[i] < 2**29, b[i] < 2**30 (or vice versa)
   226	// out[i] < 2**29
   227	func p224Mul(out, a, b *p224FieldElement, tmp *p224LargeFieldElement) {
   228		for i := 0; i < 15; i++ {
   229			tmp[i] = 0
   230		}
   231	
   232		for i := 0; i < 8; i++ {
   233			for j := 0; j < 8; j++ {
   234				tmp[i+j] += uint64(a[i]) * uint64(b[j])
   235			}
   236		}
   237	
   238		p224ReduceLarge(out, tmp)
   239	}
   240	
   241	// Square computes *out = a*a
   242	//
   243	// a[i] < 2**29
   244	// out[i] < 2**29
   245	func p224Square(out, a *p224FieldElement, tmp *p224LargeFieldElement) {
   246		for i := 0; i < 15; i++ {
   247			tmp[i] = 0
   248		}
   249	
   250		for i := 0; i < 8; i++ {
   251			for j := 0; j <= i; j++ {
   252				r := uint64(a[i]) * uint64(a[j])
   253				if i == j {
   254					tmp[i+j] += r
   255				} else {
   256					tmp[i+j] += r << 1
   257				}
   258			}
   259		}
   260	
   261		p224ReduceLarge(out, tmp)
   262	}
   263	
   264	// ReduceLarge converts a p224LargeFieldElement to a p224FieldElement.
   265	//
   266	// in[i] < 2**62
   267	func p224ReduceLarge(out *p224FieldElement, in *p224LargeFieldElement) {
   268		for i := 0; i < 8; i++ {
   269			in[i] += p224ZeroModP63[i]
   270		}
   271	
   272		// Eliminate the coefficients at 2**224 and greater.
   273		for i := 14; i >= 8; i-- {
   274			in[i-8] -= in[i]
   275			in[i-5] += (in[i] & 0xffff) << 12
   276			in[i-4] += in[i] >> 16
   277		}
   278		in[8] = 0
   279		// in[0..8] < 2**64
   280	
   281		// As the values become small enough, we start to store them in |out|
   282		// and use 32-bit operations.
   283		for i := 1; i < 8; i++ {
   284			in[i+1] += in[i] >> 28
   285			out[i] = uint32(in[i] & bottom28Bits)
   286		}
   287		in[0] -= in[8]
   288		out[3] += uint32(in[8]&0xffff) << 12
   289		out[4] += uint32(in[8] >> 16)
   290		// in[0] < 2**64
   291		// out[3] < 2**29
   292		// out[4] < 2**29
   293		// out[1,2,5..7] < 2**28
   294	
   295		out[0] = uint32(in[0] & bottom28Bits)
   296		out[1] += uint32((in[0] >> 28) & bottom28Bits)
   297		out[2] += uint32(in[0] >> 56)
   298		// out[0] < 2**28
   299		// out[1..4] < 2**29
   300		// out[5..7] < 2**28
   301	}
   302	
   303	// Reduce reduces the coefficients of a to smaller bounds.
   304	//
   305	// On entry: a[i] < 2**31 + 2**30
   306	// On exit: a[i] < 2**29
   307	func p224Reduce(a *p224FieldElement) {
   308		for i := 0; i < 7; i++ {
   309			a[i+1] += a[i] >> 28
   310			a[i] &= bottom28Bits
   311		}
   312		top := a[7] >> 28
   313		a[7] &= bottom28Bits
   314	
   315		// top < 2**4
   316		mask := top
   317		mask |= mask >> 2
   318		mask |= mask >> 1
   319		mask <<= 31
   320		mask = uint32(int32(mask) >> 31)
   321		// Mask is all ones if top != 0, all zero otherwise
   322	
   323		a[0] -= top
   324		a[3] += top << 12
   325	
   326		// We may have just made a[0] negative but, if we did, then we must
   327		// have added something to a[3], this it's > 2**12. Therefore we can
   328		// carry down to a[0].
   329		a[3] -= 1 & mask
   330		a[2] += mask & (1<<28 - 1)
   331		a[1] += mask & (1<<28 - 1)
   332		a[0] += mask & (1 << 28)
   333	}
   334	
   335	// p224Invert calculates *out = in**-1 by computing in**(2**224 - 2**96 - 1),
   336	// i.e. Fermat's little theorem.
   337	func p224Invert(out, in *p224FieldElement) {
   338		var f1, f2, f3, f4 p224FieldElement
   339		var c p224LargeFieldElement
   340	
   341		p224Square(&f1, in, &c)    // 2
   342		p224Mul(&f1, &f1, in, &c)  // 2**2 - 1
   343		p224Square(&f1, &f1, &c)   // 2**3 - 2
   344		p224Mul(&f1, &f1, in, &c)  // 2**3 - 1
   345		p224Square(&f2, &f1, &c)   // 2**4 - 2
   346		p224Square(&f2, &f2, &c)   // 2**5 - 4
   347		p224Square(&f2, &f2, &c)   // 2**6 - 8
   348		p224Mul(&f1, &f1, &f2, &c) // 2**6 - 1
   349		p224Square(&f2, &f1, &c)   // 2**7 - 2
   350		for i := 0; i < 5; i++ {   // 2**12 - 2**6
   351			p224Square(&f2, &f2, &c)
   352		}
   353		p224Mul(&f2, &f2, &f1, &c) // 2**12 - 1
   354		p224Square(&f3, &f2, &c)   // 2**13 - 2
   355		for i := 0; i < 11; i++ {  // 2**24 - 2**12
   356			p224Square(&f3, &f3, &c)
   357		}
   358		p224Mul(&f2, &f3, &f2, &c) // 2**24 - 1
   359		p224Square(&f3, &f2, &c)   // 2**25 - 2
   360		for i := 0; i < 23; i++ {  // 2**48 - 2**24
   361			p224Square(&f3, &f3, &c)
   362		}
   363		p224Mul(&f3, &f3, &f2, &c) // 2**48 - 1
   364		p224Square(&f4, &f3, &c)   // 2**49 - 2
   365		for i := 0; i < 47; i++ {  // 2**96 - 2**48
   366			p224Square(&f4, &f4, &c)
   367		}
   368		p224Mul(&f3, &f3, &f4, &c) // 2**96 - 1
   369		p224Square(&f4, &f3, &c)   // 2**97 - 2
   370		for i := 0; i < 23; i++ {  // 2**120 - 2**24
   371			p224Square(&f4, &f4, &c)
   372		}
   373		p224Mul(&f2, &f4, &f2, &c) // 2**120 - 1
   374		for i := 0; i < 6; i++ {   // 2**126 - 2**6
   375			p224Square(&f2, &f2, &c)
   376		}
   377		p224Mul(&f1, &f1, &f2, &c) // 2**126 - 1
   378		p224Square(&f1, &f1, &c)   // 2**127 - 2
   379		p224Mul(&f1, &f1, in, &c)  // 2**127 - 1
   380		for i := 0; i < 97; i++ {  // 2**224 - 2**97
   381			p224Square(&f1, &f1, &c)
   382		}
   383		p224Mul(out, &f1, &f3, &c) // 2**224 - 2**96 - 1
   384	}
   385	
   386	// p224Contract converts a FieldElement to its unique, minimal form.
   387	//
   388	// On entry, in[i] < 2**29
   389	// On exit, in[i] < 2**28
   390	func p224Contract(out, in *p224FieldElement) {
   391		copy(out[:], in[:])
   392	
   393		for i := 0; i < 7; i++ {
   394			out[i+1] += out[i] >> 28
   395			out[i] &= bottom28Bits
   396		}
   397		top := out[7] >> 28
   398		out[7] &= bottom28Bits
   399	
   400		out[0] -= top
   401		out[3] += top << 12
   402	
   403		// We may just have made out[i] negative. So we carry down. If we made
   404		// out[0] negative then we know that out[3] is sufficiently positive
   405		// because we just added to it.
   406		for i := 0; i < 3; i++ {
   407			mask := uint32(int32(out[i]) >> 31)
   408			out[i] += (1 << 28) & mask
   409			out[i+1] -= 1 & mask
   410		}
   411	
   412		// We might have pushed out[3] over 2**28 so we perform another, partial,
   413		// carry chain.
   414		for i := 3; i < 7; i++ {
   415			out[i+1] += out[i] >> 28
   416			out[i] &= bottom28Bits
   417		}
   418		top = out[7] >> 28
   419		out[7] &= bottom28Bits
   420	
   421		// Eliminate top while maintaining the same value mod p.
   422		out[0] -= top
   423		out[3] += top << 12
   424	
   425		// There are two cases to consider for out[3]:
   426		//   1) The first time that we eliminated top, we didn't push out[3] over
   427		//      2**28. In this case, the partial carry chain didn't change any values
   428		//      and top is zero.
   429		//   2) We did push out[3] over 2**28 the first time that we eliminated top.
   430		//      The first value of top was in [0..16), therefore, prior to eliminating
   431		//      the first top, 0xfff1000 <= out[3] <= 0xfffffff. Therefore, after
   432		//      overflowing and being reduced by the second carry chain, out[3] <=
   433		//      0xf000. Thus it cannot have overflowed when we eliminated top for the
   434		//      second time.
   435	
   436		// Again, we may just have made out[0] negative, so do the same carry down.
   437		// As before, if we made out[0] negative then we know that out[3] is
   438		// sufficiently positive.
   439		for i := 0; i < 3; i++ {
   440			mask := uint32(int32(out[i]) >> 31)
   441			out[i] += (1 << 28) & mask
   442			out[i+1] -= 1 & mask
   443		}
   444	
   445		// Now we see if the value is >= p and, if so, subtract p.
   446	
   447		// First we build a mask from the top four limbs, which must all be
   448		// equal to bottom28Bits if the whole value is >= p. If top4AllOnes
   449		// ends up with any zero bits in the bottom 28 bits, then this wasn't
   450		// true.
   451		top4AllOnes := uint32(0xffffffff)
   452		for i := 4; i < 8; i++ {
   453			top4AllOnes &= out[i]
   454		}
   455		top4AllOnes |= 0xf0000000
   456		// Now we replicate any zero bits to all the bits in top4AllOnes.
   457		top4AllOnes &= top4AllOnes >> 16
   458		top4AllOnes &= top4AllOnes >> 8
   459		top4AllOnes &= top4AllOnes >> 4
   460		top4AllOnes &= top4AllOnes >> 2
   461		top4AllOnes &= top4AllOnes >> 1
   462		top4AllOnes = uint32(int32(top4AllOnes<<31) >> 31)
   463	
   464		// Now we test whether the bottom three limbs are non-zero.
   465		bottom3NonZero := out[0] | out[1] | out[2]
   466		bottom3NonZero |= bottom3NonZero >> 16
   467		bottom3NonZero |= bottom3NonZero >> 8
   468		bottom3NonZero |= bottom3NonZero >> 4
   469		bottom3NonZero |= bottom3NonZero >> 2
   470		bottom3NonZero |= bottom3NonZero >> 1
   471		bottom3NonZero = uint32(int32(bottom3NonZero<<31) >> 31)
   472	
   473		// Everything depends on the value of out[3].
   474		//    If it's > 0xffff000 and top4AllOnes != 0 then the whole value is >= p
   475		//    If it's = 0xffff000 and top4AllOnes != 0 and bottom3NonZero != 0,
   476		//      then the whole value is >= p
   477		//    If it's < 0xffff000, then the whole value is < p
   478		n := out[3] - 0xffff000
   479		out3Equal := n
   480		out3Equal |= out3Equal >> 16
   481		out3Equal |= out3Equal >> 8
   482		out3Equal |= out3Equal >> 4
   483		out3Equal |= out3Equal >> 2
   484		out3Equal |= out3Equal >> 1
   485		out3Equal = ^uint32(int32(out3Equal<<31) >> 31)
   486	
   487		// If out[3] > 0xffff000 then n's MSB will be zero.
   488		out3GT := ^uint32(int32(n) >> 31)
   489	
   490		mask := top4AllOnes & ((out3Equal & bottom3NonZero) | out3GT)
   491		out[0] -= 1 & mask
   492		out[3] -= 0xffff000 & mask
   493		out[4] -= 0xfffffff & mask
   494		out[5] -= 0xfffffff & mask
   495		out[6] -= 0xfffffff & mask
   496		out[7] -= 0xfffffff & mask
   497	}
   498	
   499	// Group element functions.
   500	//
   501	// These functions deal with group elements. The group is an elliptic curve
   502	// group with a = -3 defined in FIPS 186-3, section D.2.2.
   503	
   504	// p224AddJacobian computes *out = a+b where a != b.
   505	func p224AddJacobian(x3, y3, z3, x1, y1, z1, x2, y2, z2 *p224FieldElement) {
   506		// See https://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#addition-p224Add-2007-bl
   507		var z1z1, z2z2, u1, u2, s1, s2, h, i, j, r, v p224FieldElement
   508		var c p224LargeFieldElement
   509	
   510		z1IsZero := p224IsZero(z1)
   511		z2IsZero := p224IsZero(z2)
   512	
   513		// Z1Z1 = Z1²
   514		p224Square(&z1z1, z1, &c)
   515		// Z2Z2 = Z2²
   516		p224Square(&z2z2, z2, &c)
   517		// U1 = X1*Z2Z2
   518		p224Mul(&u1, x1, &z2z2, &c)
   519		// U2 = X2*Z1Z1
   520		p224Mul(&u2, x2, &z1z1, &c)
   521		// S1 = Y1*Z2*Z2Z2
   522		p224Mul(&s1, z2, &z2z2, &c)
   523		p224Mul(&s1, y1, &s1, &c)
   524		// S2 = Y2*Z1*Z1Z1
   525		p224Mul(&s2, z1, &z1z1, &c)
   526		p224Mul(&s2, y2, &s2, &c)
   527		// H = U2-U1
   528		p224Sub(&h, &u2, &u1)
   529		p224Reduce(&h)
   530		xEqual := p224IsZero(&h)
   531		// I = (2*H)²
   532		for j := 0; j < 8; j++ {
   533			i[j] = h[j] << 1
   534		}
   535		p224Reduce(&i)
   536		p224Square(&i, &i, &c)
   537		// J = H*I
   538		p224Mul(&j, &h, &i, &c)
   539		// r = 2*(S2-S1)
   540		p224Sub(&r, &s2, &s1)
   541		p224Reduce(&r)
   542		yEqual := p224IsZero(&r)
   543		if xEqual == 1 && yEqual == 1 && z1IsZero == 0 && z2IsZero == 0 {
   544			p224DoubleJacobian(x3, y3, z3, x1, y1, z1)
   545			return
   546		}
   547		for i := 0; i < 8; i++ {
   548			r[i] <<= 1
   549		}
   550		p224Reduce(&r)
   551		// V = U1*I
   552		p224Mul(&v, &u1, &i, &c)
   553		// Z3 = ((Z1+Z2)²-Z1Z1-Z2Z2)*H
   554		p224Add(&z1z1, &z1z1, &z2z2)
   555		p224Add(&z2z2, z1, z2)
   556		p224Reduce(&z2z2)
   557		p224Square(&z2z2, &z2z2, &c)
   558		p224Sub(z3, &z2z2, &z1z1)
   559		p224Reduce(z3)
   560		p224Mul(z3, z3, &h, &c)
   561		// X3 = r²-J-2*V
   562		for i := 0; i < 8; i++ {
   563			z1z1[i] = v[i] << 1
   564		}
   565		p224Add(&z1z1, &j, &z1z1)
   566		p224Reduce(&z1z1)
   567		p224Square(x3, &r, &c)
   568		p224Sub(x3, x3, &z1z1)
   569		p224Reduce(x3)
   570		// Y3 = r*(V-X3)-2*S1*J
   571		for i := 0; i < 8; i++ {
   572			s1[i] <<= 1
   573		}
   574		p224Mul(&s1, &s1, &j, &c)
   575		p224Sub(&z1z1, &v, x3)
   576		p224Reduce(&z1z1)
   577		p224Mul(&z1z1, &z1z1, &r, &c)
   578		p224Sub(y3, &z1z1, &s1)
   579		p224Reduce(y3)
   580	
   581		p224CopyConditional(x3, x2, z1IsZero)
   582		p224CopyConditional(x3, x1, z2IsZero)
   583		p224CopyConditional(y3, y2, z1IsZero)
   584		p224CopyConditional(y3, y1, z2IsZero)
   585		p224CopyConditional(z3, z2, z1IsZero)
   586		p224CopyConditional(z3, z1, z2IsZero)
   587	}
   588	
   589	// p224DoubleJacobian computes *out = a+a.
   590	func p224DoubleJacobian(x3, y3, z3, x1, y1, z1 *p224FieldElement) {
   591		var delta, gamma, beta, alpha, t p224FieldElement
   592		var c p224LargeFieldElement
   593	
   594		p224Square(&delta, z1, &c)
   595		p224Square(&gamma, y1, &c)
   596		p224Mul(&beta, x1, &gamma, &c)
   597	
   598		// alpha = 3*(X1-delta)*(X1+delta)
   599		p224Add(&t, x1, &delta)
   600		for i := 0; i < 8; i++ {
   601			t[i] += t[i] << 1
   602		}
   603		p224Reduce(&t)
   604		p224Sub(&alpha, x1, &delta)
   605		p224Reduce(&alpha)
   606		p224Mul(&alpha, &alpha, &t, &c)
   607	
   608		// Z3 = (Y1+Z1)²-gamma-delta
   609		p224Add(z3, y1, z1)
   610		p224Reduce(z3)
   611		p224Square(z3, z3, &c)
   612		p224Sub(z3, z3, &gamma)
   613		p224Reduce(z3)
   614		p224Sub(z3, z3, &delta)
   615		p224Reduce(z3)
   616	
   617		// X3 = alpha²-8*beta
   618		for i := 0; i < 8; i++ {
   619			delta[i] = beta[i] << 3
   620		}
   621		p224Reduce(&delta)
   622		p224Square(x3, &alpha, &c)
   623		p224Sub(x3, x3, &delta)
   624		p224Reduce(x3)
   625	
   626		// Y3 = alpha*(4*beta-X3)-8*gamma²
   627		for i := 0; i < 8; i++ {
   628			beta[i] <<= 2
   629		}
   630		p224Sub(&beta, &beta, x3)
   631		p224Reduce(&beta)
   632		p224Square(&gamma, &gamma, &c)
   633		for i := 0; i < 8; i++ {
   634			gamma[i] <<= 3
   635		}
   636		p224Reduce(&gamma)
   637		p224Mul(y3, &alpha, &beta, &c)
   638		p224Sub(y3, y3, &gamma)
   639		p224Reduce(y3)
   640	}
   641	
   642	// p224CopyConditional sets *out = *in iff the least-significant-bit of control
   643	// is true, and it runs in constant time.
   644	func p224CopyConditional(out, in *p224FieldElement, control uint32) {
   645		control <<= 31
   646		control = uint32(int32(control) >> 31)
   647	
   648		for i := 0; i < 8; i++ {
   649			out[i] ^= (out[i] ^ in[i]) & control
   650		}
   651	}
   652	
   653	func p224ScalarMult(outX, outY, outZ, inX, inY, inZ *p224FieldElement, scalar []byte) {
   654		var xx, yy, zz p224FieldElement
   655		for i := 0; i < 8; i++ {
   656			outX[i] = 0
   657			outY[i] = 0
   658			outZ[i] = 0
   659		}
   660	
   661		for _, byte := range scalar {
   662			for bitNum := uint(0); bitNum < 8; bitNum++ {
   663				p224DoubleJacobian(outX, outY, outZ, outX, outY, outZ)
   664				bit := uint32((byte >> (7 - bitNum)) & 1)
   665				p224AddJacobian(&xx, &yy, &zz, inX, inY, inZ, outX, outY, outZ)
   666				p224CopyConditional(outX, &xx, bit)
   667				p224CopyConditional(outY, &yy, bit)
   668				p224CopyConditional(outZ, &zz, bit)
   669			}
   670		}
   671	}
   672	
   673	// p224ToAffine converts from Jacobian to affine form.
   674	func p224ToAffine(x, y, z *p224FieldElement) (*big.Int, *big.Int) {
   675		var zinv, zinvsq, outx, outy p224FieldElement
   676		var tmp p224LargeFieldElement
   677	
   678		if isPointAtInfinity := p224IsZero(z); isPointAtInfinity == 1 {
   679			return new(big.Int), new(big.Int)
   680		}
   681	
   682		p224Invert(&zinv, z)
   683		p224Square(&zinvsq, &zinv, &tmp)
   684		p224Mul(x, x, &zinvsq, &tmp)
   685		p224Mul(&zinvsq, &zinvsq, &zinv, &tmp)
   686		p224Mul(y, y, &zinvsq, &tmp)
   687	
   688		p224Contract(&outx, x)
   689		p224Contract(&outy, y)
   690		return p224ToBig(&outx), p224ToBig(&outy)
   691	}
   692	
   693	// get28BitsFromEnd returns the least-significant 28 bits from buf>>shift,
   694	// where buf is interpreted as a big-endian number.
   695	func get28BitsFromEnd(buf []byte, shift uint) (uint32, []byte) {
   696		var ret uint32
   697	
   698		for i := uint(0); i < 4; i++ {
   699			var b byte
   700			if l := len(buf); l > 0 {
   701				b = buf[l-1]
   702				// We don't remove the byte if we're about to return and we're not
   703				// reading all of it.
   704				if i != 3 || shift == 4 {
   705					buf = buf[:l-1]
   706				}
   707			}
   708			ret |= uint32(b) << (8 * i) >> shift
   709		}
   710		ret &= bottom28Bits
   711		return ret, buf
   712	}
   713	
   714	// p224FromBig sets *out = *in.
   715	func p224FromBig(out *p224FieldElement, in *big.Int) {
   716		bytes := in.Bytes()
   717		out[0], bytes = get28BitsFromEnd(bytes, 0)
   718		out[1], bytes = get28BitsFromEnd(bytes, 4)
   719		out[2], bytes = get28BitsFromEnd(bytes, 0)
   720		out[3], bytes = get28BitsFromEnd(bytes, 4)
   721		out[4], bytes = get28BitsFromEnd(bytes, 0)
   722		out[5], bytes = get28BitsFromEnd(bytes, 4)
   723		out[6], bytes = get28BitsFromEnd(bytes, 0)
   724		out[7], bytes = get28BitsFromEnd(bytes, 4)
   725	}
   726	
   727	// p224ToBig returns in as a big.Int.
   728	func p224ToBig(in *p224FieldElement) *big.Int {
   729		var buf [28]byte
   730		buf[27] = byte(in[0])
   731		buf[26] = byte(in[0] >> 8)
   732		buf[25] = byte(in[0] >> 16)
   733		buf[24] = byte(((in[0] >> 24) & 0x0f) | (in[1]<<4)&0xf0)
   734	
   735		buf[23] = byte(in[1] >> 4)
   736		buf[22] = byte(in[1] >> 12)
   737		buf[21] = byte(in[1] >> 20)
   738	
   739		buf[20] = byte(in[2])
   740		buf[19] = byte(in[2] >> 8)
   741		buf[18] = byte(in[2] >> 16)
   742		buf[17] = byte(((in[2] >> 24) & 0x0f) | (in[3]<<4)&0xf0)
   743	
   744		buf[16] = byte(in[3] >> 4)
   745		buf[15] = byte(in[3] >> 12)
   746		buf[14] = byte(in[3] >> 20)
   747	
   748		buf[13] = byte(in[4])
   749		buf[12] = byte(in[4] >> 8)
   750		buf[11] = byte(in[4] >> 16)
   751		buf[10] = byte(((in[4] >> 24) & 0x0f) | (in[5]<<4)&0xf0)
   752	
   753		buf[9] = byte(in[5] >> 4)
   754		buf[8] = byte(in[5] >> 12)
   755		buf[7] = byte(in[5] >> 20)
   756	
   757		buf[6] = byte(in[6])
   758		buf[5] = byte(in[6] >> 8)
   759		buf[4] = byte(in[6] >> 16)
   760		buf[3] = byte(((in[6] >> 24) & 0x0f) | (in[7]<<4)&0xf0)
   761	
   762		buf[2] = byte(in[7] >> 4)
   763		buf[1] = byte(in[7] >> 12)
   764		buf[0] = byte(in[7] >> 20)
   765	
   766		return new(big.Int).SetBytes(buf[:])
   767	}
   768	

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