p521.go

     1  // Copyright 2022 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  // Code generated by generate.go. DO NOT EDIT.
     6  
     7  package nistec
     8  
     9  import (
    10  	"crypto/internal/fips140/nistec/fiat"
    11  	"crypto/internal/fips140/subtle"
    12  	"errors"
    13  	"sync"
    14  )
    15  
    16  // p521ElementLength is the length of an element of the base or scalar field,
    17  // which have the same bytes length for all NIST P curves.
    18  const p521ElementLength = 66
    19  
    20  // P521Point is a P521 point. The zero value is NOT valid.
    21  type P521Point struct {
    22  	// The point is represented in projective coordinates (X:Y:Z),
    23  	// where x = X/Z and y = Y/Z.
    24  	x, y, z *fiat.P521Element
    25  }
    26  
    27  // NewP521Point returns a new P521Point representing the point at infinity point.
    28  func NewP521Point() *P521Point {
    29  	return &P521Point{
    30  		x: new(fiat.P521Element),
    31  		y: new(fiat.P521Element).One(),
    32  		z: new(fiat.P521Element),
    33  	}
    34  }
    35  
    36  // SetGenerator sets p to the canonical generator and returns p.
    37  func (p *P521Point) SetGenerator() *P521Point {
    38  	p.x.SetBytes([]byte{0x0, 0xc6, 0x85, 0x8e, 0x6, 0xb7, 0x4, 0x4, 0xe9, 0xcd, 0x9e, 0x3e, 0xcb, 0x66, 0x23, 0x95, 0xb4, 0x42, 0x9c, 0x64, 0x81, 0x39, 0x5, 0x3f, 0xb5, 0x21, 0xf8, 0x28, 0xaf, 0x60, 0x6b, 0x4d, 0x3d, 0xba, 0xa1, 0x4b, 0x5e, 0x77, 0xef, 0xe7, 0x59, 0x28, 0xfe, 0x1d, 0xc1, 0x27, 0xa2, 0xff, 0xa8, 0xde, 0x33, 0x48, 0xb3, 0xc1, 0x85, 0x6a, 0x42, 0x9b, 0xf9, 0x7e, 0x7e, 0x31, 0xc2, 0xe5, 0xbd, 0x66})
    39  	p.y.SetBytes([]byte{0x1, 0x18, 0x39, 0x29, 0x6a, 0x78, 0x9a, 0x3b, 0xc0, 0x4, 0x5c, 0x8a, 0x5f, 0xb4, 0x2c, 0x7d, 0x1b, 0xd9, 0x98, 0xf5, 0x44, 0x49, 0x57, 0x9b, 0x44, 0x68, 0x17, 0xaf, 0xbd, 0x17, 0x27, 0x3e, 0x66, 0x2c, 0x97, 0xee, 0x72, 0x99, 0x5e, 0xf4, 0x26, 0x40, 0xc5, 0x50, 0xb9, 0x1, 0x3f, 0xad, 0x7, 0x61, 0x35, 0x3c, 0x70, 0x86, 0xa2, 0x72, 0xc2, 0x40, 0x88, 0xbe, 0x94, 0x76, 0x9f, 0xd1, 0x66, 0x50})
    40  	p.z.One()
    41  	return p
    42  }
    43  
    44  // Set sets p = q and returns p.
    45  func (p *P521Point) Set(q *P521Point) *P521Point {
    46  	p.x.Set(q.x)
    47  	p.y.Set(q.y)
    48  	p.z.Set(q.z)
    49  	return p
    50  }
    51  
    52  // SetBytes sets p to the compressed, uncompressed, or infinity value encoded in
    53  // b, as specified in SEC 1, Version 2.0, Section 2.3.4. If the point is not on
    54  // the curve, it returns nil and an error, and the receiver is unchanged.
    55  // Otherwise, it returns p.
    56  func (p *P521Point) SetBytes(b []byte) (*P521Point, error) {
    57  	switch {
    58  	// Point at infinity.
    59  	case len(b) == 1 && b[0] == 0:
    60  		return p.Set(NewP521Point()), nil
    61  
    62  	// Uncompressed form.
    63  	case len(b) == 1+2*p521ElementLength && b[0] == 4:
    64  		x, err := new(fiat.P521Element).SetBytes(b[1 : 1+p521ElementLength])
    65  		if err != nil {
    66  			return nil, err
    67  		}
    68  		y, err := new(fiat.P521Element).SetBytes(b[1+p521ElementLength:])
    69  		if err != nil {
    70  			return nil, err
    71  		}
    72  		if err := p521CheckOnCurve(x, y); err != nil {
    73  			return nil, err
    74  		}
    75  		p.x.Set(x)
    76  		p.y.Set(y)
    77  		p.z.One()
    78  		return p, nil
    79  
    80  	// Compressed form.
    81  	case len(b) == 1+p521ElementLength && (b[0] == 2 || b[0] == 3):
    82  		x, err := new(fiat.P521Element).SetBytes(b[1:])
    83  		if err != nil {
    84  			return nil, err
    85  		}
    86  
    87  		// y² = x³ - 3x + b
    88  		y := p521Polynomial(new(fiat.P521Element), x)
    89  		if !p521Sqrt(y, y) {
    90  			return nil, errors.New("invalid P521 compressed point encoding")
    91  		}
    92  
    93  		// Select the positive or negative root, as indicated by the least
    94  		// significant bit, based on the encoding type byte.
    95  		otherRoot := new(fiat.P521Element)
    96  		otherRoot.Sub(otherRoot, y)
    97  		cond := y.Bytes()[p521ElementLength-1]&1 ^ b[0]&1
    98  		y.Select(otherRoot, y, int(cond))
    99  
   100  		p.x.Set(x)
   101  		p.y.Set(y)
   102  		p.z.One()
   103  		return p, nil
   104  
   105  	default:
   106  		return nil, errors.New("invalid P521 point encoding")
   107  	}
   108  }
   109  
   110  var _p521B *fiat.P521Element
   111  var _p521BOnce sync.Once
   112  
   113  func p521B() *fiat.P521Element {
   114  	_p521BOnce.Do(func() {
   115  		_p521B, _ = new(fiat.P521Element).SetBytes([]byte{0x0, 0x51, 0x95, 0x3e, 0xb9, 0x61, 0x8e, 0x1c, 0x9a, 0x1f, 0x92, 0x9a, 0x21, 0xa0, 0xb6, 0x85, 0x40, 0xee, 0xa2, 0xda, 0x72, 0x5b, 0x99, 0xb3, 0x15, 0xf3, 0xb8, 0xb4, 0x89, 0x91, 0x8e, 0xf1, 0x9, 0xe1, 0x56, 0x19, 0x39, 0x51, 0xec, 0x7e, 0x93, 0x7b, 0x16, 0x52, 0xc0, 0xbd, 0x3b, 0xb1, 0xbf, 0x7, 0x35, 0x73, 0xdf, 0x88, 0x3d, 0x2c, 0x34, 0xf1, 0xef, 0x45, 0x1f, 0xd4, 0x6b, 0x50, 0x3f, 0x0})
   116  	})
   117  	return _p521B
   118  }
   119  
   120  // p521Polynomial sets y2 to x³ - 3x + b, and returns y2.
   121  func p521Polynomial(y2, x *fiat.P521Element) *fiat.P521Element {
   122  	y2.Square(x)
   123  	y2.Mul(y2, x)
   124  
   125  	threeX := new(fiat.P521Element).Add(x, x)
   126  	threeX.Add(threeX, x)
   127  	y2.Sub(y2, threeX)
   128  
   129  	return y2.Add(y2, p521B())
   130  }
   131  
   132  func p521CheckOnCurve(x, y *fiat.P521Element) error {
   133  	// y² = x³ - 3x + b
   134  	rhs := p521Polynomial(new(fiat.P521Element), x)
   135  	lhs := new(fiat.P521Element).Square(y)
   136  	if rhs.Equal(lhs) != 1 {
   137  		return errors.New("P521 point not on curve")
   138  	}
   139  	return nil
   140  }
   141  
   142  // Bytes returns the uncompressed or infinity encoding of p, as specified in
   143  // SEC 1, Version 2.0, Section 2.3.3. Note that the encoding of the point at
   144  // infinity is shorter than all other encodings.
   145  func (p *P521Point) Bytes() []byte {
   146  	// This function is outlined to make the allocations inline in the caller
   147  	// rather than happen on the heap.
   148  	var out [1 + 2*p521ElementLength]byte
   149  	return p.bytes(&out)
   150  }
   151  
   152  func (p *P521Point) bytes(out *[1 + 2*p521ElementLength]byte) []byte {
   153  	if p.z.IsZero() == 1 {
   154  		return append(out[:0], 0)
   155  	}
   156  
   157  	zinv := new(fiat.P521Element).Invert(p.z)
   158  	x := new(fiat.P521Element).Mul(p.x, zinv)
   159  	y := new(fiat.P521Element).Mul(p.y, zinv)
   160  
   161  	buf := append(out[:0], 4)
   162  	buf = append(buf, x.Bytes()...)
   163  	buf = append(buf, y.Bytes()...)
   164  	return buf
   165  }
   166  
   167  // BytesX returns the encoding of the x-coordinate of p, as specified in SEC 1,
   168  // Version 2.0, Section 2.3.5, or an error if p is the point at infinity.
   169  func (p *P521Point) BytesX() ([]byte, error) {
   170  	// This function is outlined to make the allocations inline in the caller
   171  	// rather than happen on the heap.
   172  	var out [p521ElementLength]byte
   173  	return p.bytesX(&out)
   174  }
   175  
   176  func (p *P521Point) bytesX(out *[p521ElementLength]byte) ([]byte, error) {
   177  	if p.z.IsZero() == 1 {
   178  		return nil, errors.New("P521 point is the point at infinity")
   179  	}
   180  
   181  	zinv := new(fiat.P521Element).Invert(p.z)
   182  	x := new(fiat.P521Element).Mul(p.x, zinv)
   183  
   184  	return append(out[:0], x.Bytes()...), nil
   185  }
   186  
   187  // BytesCompressed returns the compressed or infinity encoding of p, as
   188  // specified in SEC 1, Version 2.0, Section 2.3.3. Note that the encoding of the
   189  // point at infinity is shorter than all other encodings.
   190  func (p *P521Point) BytesCompressed() []byte {
   191  	// This function is outlined to make the allocations inline in the caller
   192  	// rather than happen on the heap.
   193  	var out [1 + p521ElementLength]byte
   194  	return p.bytesCompressed(&out)
   195  }
   196  
   197  func (p *P521Point) bytesCompressed(out *[1 + p521ElementLength]byte) []byte {
   198  	if p.z.IsZero() == 1 {
   199  		return append(out[:0], 0)
   200  	}
   201  
   202  	zinv := new(fiat.P521Element).Invert(p.z)
   203  	x := new(fiat.P521Element).Mul(p.x, zinv)
   204  	y := new(fiat.P521Element).Mul(p.y, zinv)
   205  
   206  	// Encode the sign of the y coordinate (indicated by the least significant
   207  	// bit) as the encoding type (2 or 3).
   208  	buf := append(out[:0], 2)
   209  	buf[0] |= y.Bytes()[p521ElementLength-1] & 1
   210  	buf = append(buf, x.Bytes()...)
   211  	return buf
   212  }
   213  
   214  // Add sets q = p1 + p2, and returns q. The points may overlap.
   215  func (q *P521Point) Add(p1, p2 *P521Point) *P521Point {
   216  	// Complete addition formula for a = -3 from "Complete addition formulas for
   217  	// prime order elliptic curves" (https://eprint.iacr.org/2015/1060), §A.2.
   218  
   219  	t0 := new(fiat.P521Element).Mul(p1.x, p2.x)  // t0 := X1 * X2
   220  	t1 := new(fiat.P521Element).Mul(p1.y, p2.y)  // t1 := Y1 * Y2
   221  	t2 := new(fiat.P521Element).Mul(p1.z, p2.z)  // t2 := Z1 * Z2
   222  	t3 := new(fiat.P521Element).Add(p1.x, p1.y)  // t3 := X1 + Y1
   223  	t4 := new(fiat.P521Element).Add(p2.x, p2.y)  // t4 := X2 + Y2
   224  	t3.Mul(t3, t4)                               // t3 := t3 * t4
   225  	t4.Add(t0, t1)                               // t4 := t0 + t1
   226  	t3.Sub(t3, t4)                               // t3 := t3 - t4
   227  	t4.Add(p1.y, p1.z)                           // t4 := Y1 + Z1
   228  	x3 := new(fiat.P521Element).Add(p2.y, p2.z)  // X3 := Y2 + Z2
   229  	t4.Mul(t4, x3)                               // t4 := t4 * X3
   230  	x3.Add(t1, t2)                               // X3 := t1 + t2
   231  	t4.Sub(t4, x3)                               // t4 := t4 - X3
   232  	x3.Add(p1.x, p1.z)                           // X3 := X1 + Z1
   233  	y3 := new(fiat.P521Element).Add(p2.x, p2.z)  // Y3 := X2 + Z2
   234  	x3.Mul(x3, y3)                               // X3 := X3 * Y3
   235  	y3.Add(t0, t2)                               // Y3 := t0 + t2
   236  	y3.Sub(x3, y3)                               // Y3 := X3 - Y3
   237  	z3 := new(fiat.P521Element).Mul(p521B(), t2) // Z3 := b * t2
   238  	x3.Sub(y3, z3)                               // X3 := Y3 - Z3
   239  	z3.Add(x3, x3)                               // Z3 := X3 + X3
   240  	x3.Add(x3, z3)                               // X3 := X3 + Z3
   241  	z3.Sub(t1, x3)                               // Z3 := t1 - X3
   242  	x3.Add(t1, x3)                               // X3 := t1 + X3
   243  	y3.Mul(p521B(), y3)                          // Y3 := b * Y3
   244  	t1.Add(t2, t2)                               // t1 := t2 + t2
   245  	t2.Add(t1, t2)                               // t2 := t1 + t2
   246  	y3.Sub(y3, t2)                               // Y3 := Y3 - t2
   247  	y3.Sub(y3, t0)                               // Y3 := Y3 - t0
   248  	t1.Add(y3, y3)                               // t1 := Y3 + Y3
   249  	y3.Add(t1, y3)                               // Y3 := t1 + Y3
   250  	t1.Add(t0, t0)                               // t1 := t0 + t0
   251  	t0.Add(t1, t0)                               // t0 := t1 + t0
   252  	t0.Sub(t0, t2)                               // t0 := t0 - t2
   253  	t1.Mul(t4, y3)                               // t1 := t4 * Y3
   254  	t2.Mul(t0, y3)                               // t2 := t0 * Y3
   255  	y3.Mul(x3, z3)                               // Y3 := X3 * Z3
   256  	y3.Add(y3, t2)                               // Y3 := Y3 + t2
   257  	x3.Mul(t3, x3)                               // X3 := t3 * X3
   258  	x3.Sub(x3, t1)                               // X3 := X3 - t1
   259  	z3.Mul(t4, z3)                               // Z3 := t4 * Z3
   260  	t1.Mul(t3, t0)                               // t1 := t3 * t0
   261  	z3.Add(z3, t1)                               // Z3 := Z3 + t1
   262  
   263  	q.x.Set(x3)
   264  	q.y.Set(y3)
   265  	q.z.Set(z3)
   266  	return q
   267  }
   268  
   269  // Double sets q = p + p, and returns q. The points may overlap.
   270  func (q *P521Point) Double(p *P521Point) *P521Point {
   271  	// Complete addition formula for a = -3 from "Complete addition formulas for
   272  	// prime order elliptic curves" (https://eprint.iacr.org/2015/1060), §A.2.
   273  
   274  	t0 := new(fiat.P521Element).Square(p.x)      // t0 := X ^ 2
   275  	t1 := new(fiat.P521Element).Square(p.y)      // t1 := Y ^ 2
   276  	t2 := new(fiat.P521Element).Square(p.z)      // t2 := Z ^ 2
   277  	t3 := new(fiat.P521Element).Mul(p.x, p.y)    // t3 := X * Y
   278  	t3.Add(t3, t3)                               // t3 := t3 + t3
   279  	z3 := new(fiat.P521Element).Mul(p.x, p.z)    // Z3 := X * Z
   280  	z3.Add(z3, z3)                               // Z3 := Z3 + Z3
   281  	y3 := new(fiat.P521Element).Mul(p521B(), t2) // Y3 := b * t2
   282  	y3.Sub(y3, z3)                               // Y3 := Y3 - Z3
   283  	x3 := new(fiat.P521Element).Add(y3, y3)      // X3 := Y3 + Y3
   284  	y3.Add(x3, y3)                               // Y3 := X3 + Y3
   285  	x3.Sub(t1, y3)                               // X3 := t1 - Y3
   286  	y3.Add(t1, y3)                               // Y3 := t1 + Y3
   287  	y3.Mul(x3, y3)                               // Y3 := X3 * Y3
   288  	x3.Mul(x3, t3)                               // X3 := X3 * t3
   289  	t3.Add(t2, t2)                               // t3 := t2 + t2
   290  	t2.Add(t2, t3)                               // t2 := t2 + t3
   291  	z3.Mul(p521B(), z3)                          // Z3 := b * Z3
   292  	z3.Sub(z3, t2)                               // Z3 := Z3 - t2
   293  	z3.Sub(z3, t0)                               // Z3 := Z3 - t0
   294  	t3.Add(z3, z3)                               // t3 := Z3 + Z3
   295  	z3.Add(z3, t3)                               // Z3 := Z3 + t3
   296  	t3.Add(t0, t0)                               // t3 := t0 + t0
   297  	t0.Add(t3, t0)                               // t0 := t3 + t0
   298  	t0.Sub(t0, t2)                               // t0 := t0 - t2
   299  	t0.Mul(t0, z3)                               // t0 := t0 * Z3
   300  	y3.Add(y3, t0)                               // Y3 := Y3 + t0
   301  	t0.Mul(p.y, p.z)                             // t0 := Y * Z
   302  	t0.Add(t0, t0)                               // t0 := t0 + t0
   303  	z3.Mul(t0, z3)                               // Z3 := t0 * Z3
   304  	x3.Sub(x3, z3)                               // X3 := X3 - Z3
   305  	z3.Mul(t0, t1)                               // Z3 := t0 * t1
   306  	z3.Add(z3, z3)                               // Z3 := Z3 + Z3
   307  	z3.Add(z3, z3)                               // Z3 := Z3 + Z3
   308  
   309  	q.x.Set(x3)
   310  	q.y.Set(y3)
   311  	q.z.Set(z3)
   312  	return q
   313  }
   314  
   315  // Select sets q to p1 if cond == 1, and to p2 if cond == 0.
   316  func (q *P521Point) Select(p1, p2 *P521Point, cond int) *P521Point {
   317  	q.x.Select(p1.x, p2.x, cond)
   318  	q.y.Select(p1.y, p2.y, cond)
   319  	q.z.Select(p1.z, p2.z, cond)
   320  	return q
   321  }
   322  
   323  // A p521Table holds the first 15 multiples of a point at offset -1, so [1]P
   324  // is at table[0], [15]P is at table[14], and [0]P is implicitly the identity
   325  // point.
   326  type p521Table [15]*P521Point
   327  
   328  // Select selects the n-th multiple of the table base point into p. It works in
   329  // constant time by iterating over every entry of the table. n must be in [0, 15].
   330  func (table *p521Table) Select(p *P521Point, n uint8) {
   331  	if n >= 16 {
   332  		panic("nistec: internal error: p521Table called with out-of-bounds value")
   333  	}
   334  	p.Set(NewP521Point())
   335  	for i := uint8(1); i < 16; i++ {
   336  		cond := subtle.ConstantTimeByteEq(i, n)
   337  		p.Select(table[i-1], p, cond)
   338  	}
   339  }
   340  
   341  // ScalarMult sets p = scalar * q, and returns p.
   342  func (p *P521Point) ScalarMult(q *P521Point, scalar []byte) (*P521Point, error) {
   343  	// Compute a p521Table for the base point q. The explicit NewP521Point
   344  	// calls get inlined, letting the allocations live on the stack.
   345  	var table = p521Table{NewP521Point(), NewP521Point(), NewP521Point(),
   346  		NewP521Point(), NewP521Point(), NewP521Point(), NewP521Point(),
   347  		NewP521Point(), NewP521Point(), NewP521Point(), NewP521Point(),
   348  		NewP521Point(), NewP521Point(), NewP521Point(), NewP521Point()}
   349  	table[0].Set(q)
   350  	for i := 1; i < 15; i += 2 {
   351  		table[i].Double(table[i/2])
   352  		table[i+1].Add(table[i], q)
   353  	}
   354  
   355  	// Instead of doing the classic double-and-add chain, we do it with a
   356  	// four-bit window: we double four times, and then add [0-15]P.
   357  	t := NewP521Point()
   358  	p.Set(NewP521Point())
   359  	for i, byte := range scalar {
   360  		// No need to double on the first iteration, as p is the identity at
   361  		// this point, and [N]∞ = ∞.
   362  		if i != 0 {
   363  			p.Double(p)
   364  			p.Double(p)
   365  			p.Double(p)
   366  			p.Double(p)
   367  		}
   368  
   369  		windowValue := byte >> 4
   370  		table.Select(t, windowValue)
   371  		p.Add(p, t)
   372  
   373  		p.Double(p)
   374  		p.Double(p)
   375  		p.Double(p)
   376  		p.Double(p)
   377  
   378  		windowValue = byte & 0b1111
   379  		table.Select(t, windowValue)
   380  		p.Add(p, t)
   381  	}
   382  
   383  	return p, nil
   384  }
   385  
   386  var p521GeneratorTable *[p521ElementLength * 2]p521Table
   387  var p521GeneratorTableOnce sync.Once
   388  
   389  // generatorTable returns a sequence of p521Tables. The first table contains
   390  // multiples of G. Each successive table is the previous table doubled four
   391  // times.
   392  func (p *P521Point) generatorTable() *[p521ElementLength * 2]p521Table {
   393  	p521GeneratorTableOnce.Do(func() {
   394  		p521GeneratorTable = new([p521ElementLength * 2]p521Table)
   395  		base := NewP521Point().SetGenerator()
   396  		for i := 0; i < p521ElementLength*2; i++ {
   397  			p521GeneratorTable[i][0] = NewP521Point().Set(base)
   398  			for j := 1; j < 15; j++ {
   399  				p521GeneratorTable[i][j] = NewP521Point().Add(p521GeneratorTable[i][j-1], base)
   400  			}
   401  			base.Double(base)
   402  			base.Double(base)
   403  			base.Double(base)
   404  			base.Double(base)
   405  		}
   406  	})
   407  	return p521GeneratorTable
   408  }
   409  
   410  // ScalarBaseMult sets p = scalar * B, where B is the canonical generator, and
   411  // returns p.
   412  func (p *P521Point) ScalarBaseMult(scalar []byte) (*P521Point, error) {
   413  	if len(scalar) != p521ElementLength {
   414  		return nil, errors.New("invalid scalar length")
   415  	}
   416  	tables := p.generatorTable()
   417  
   418  	// This is also a scalar multiplication with a four-bit window like in
   419  	// ScalarMult, but in this case the doublings are precomputed. The value
   420  	// [windowValue]G added at iteration k would normally get doubled
   421  	// (totIterations-k)×4 times, but with a larger precomputation we can
   422  	// instead add [2^((totIterations-k)×4)][windowValue]G and avoid the
   423  	// doublings between iterations.
   424  	t := NewP521Point()
   425  	p.Set(NewP521Point())
   426  	tableIndex := len(tables) - 1
   427  	for _, byte := range scalar {
   428  		windowValue := byte >> 4
   429  		tables[tableIndex].Select(t, windowValue)
   430  		p.Add(p, t)
   431  		tableIndex--
   432  
   433  		windowValue = byte & 0b1111
   434  		tables[tableIndex].Select(t, windowValue)
   435  		p.Add(p, t)
   436  		tableIndex--
   437  	}
   438  
   439  	return p, nil
   440  }
   441  
   442  // p521Sqrt sets e to a square root of x. If x is not a square, p521Sqrt returns
   443  // false and e is unchanged. e and x can overlap.
   444  func p521Sqrt(e, x *fiat.P521Element) (isSquare bool) {
   445  	candidate := new(fiat.P521Element)
   446  	p521SqrtCandidate(candidate, x)
   447  	square := new(fiat.P521Element).Square(candidate)
   448  	if square.Equal(x) != 1 {
   449  		return false
   450  	}
   451  	e.Set(candidate)
   452  	return true
   453  }
   454  
   455  // p521SqrtCandidate sets z to a square root candidate for x. z and x must not overlap.
   456  func p521SqrtCandidate(z, x *fiat.P521Element) {
   457  	// Since p = 3 mod 4, exponentiation by (p + 1) / 4 yields a square root candidate.
   458  	//
   459  	// The sequence of 0 multiplications and 519 squarings is derived from the
   460  	// following addition chain generated with github.com/mmcloughlin/addchain v0.4.0.
   461  	//
   462  	//	return  1 << 519
   463  	//
   464  
   465  	z.Square(x)
   466  	for s := 1; s < 519; s++ {
   467  		z.Square(z)
   468  	}
   469  }
   470
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