p256_ordinv.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  //go:build amd64 || arm64
     6  
     7  package nistec
     8  
     9  import "errors"
    10  
    11  // Montgomery multiplication modulo org(G). Sets res = in1 * in2 * R⁻¹.
    12  //
    13  //go:noescape
    14  func p256OrdMul(res, in1, in2 *p256OrdElement)
    15  
    16  // Montgomery square modulo org(G), repeated n times (n >= 1).
    17  //
    18  //go:noescape
    19  func p256OrdSqr(res, in *p256OrdElement, n int)
    20  
    21  func P256OrdInverse(k []byte) ([]byte, error) {
    22  	if len(k) != 32 {
    23  		return nil, errors.New("invalid scalar length")
    24  	}
    25  
    26  	x := new(p256OrdElement)
    27  	p256OrdBigToLittle(x, (*[32]byte)(k))
    28  	p256OrdReduce(x)
    29  
    30  	// Inversion is implemented as exponentiation by n - 2, per Fermat's little theorem.
    31  	//
    32  	// The sequence of 38 multiplications and 254 squarings is derived from
    33  	// https://briansmith.org/ecc-inversion-addition-chains-01#p256_scalar_inversion
    34  	_1 := new(p256OrdElement)
    35  	_11 := new(p256OrdElement)
    36  	_101 := new(p256OrdElement)
    37  	_111 := new(p256OrdElement)
    38  	_1111 := new(p256OrdElement)
    39  	_10101 := new(p256OrdElement)
    40  	_101111 := new(p256OrdElement)
    41  	t := new(p256OrdElement)
    42  
    43  	// This code operates in the Montgomery domain where R = 2²⁵⁶ mod n and n is
    44  	// the order of the scalar field. Elements in the Montgomery domain take the
    45  	// form a×R and p256OrdMul calculates (a × b × R⁻¹) mod n. RR is R in the
    46  	// domain, or R×R mod n, thus p256OrdMul(x, RR) gives x×R, i.e. converts x
    47  	// into the Montgomery domain.
    48  	RR := &p256OrdElement{0x83244c95be79eea2, 0x4699799c49bd6fa6,
    49  		0x2845b2392b6bec59, 0x66e12d94f3d95620}
    50  
    51  	p256OrdMul(_1, x, RR)      // _1
    52  	p256OrdSqr(x, _1, 1)       // _10
    53  	p256OrdMul(_11, x, _1)     // _11
    54  	p256OrdMul(_101, x, _11)   // _101
    55  	p256OrdMul(_111, x, _101)  // _111
    56  	p256OrdSqr(x, _101, 1)     // _1010
    57  	p256OrdMul(_1111, _101, x) // _1111
    58  
    59  	p256OrdSqr(t, x, 1)          // _10100
    60  	p256OrdMul(_10101, t, _1)    // _10101
    61  	p256OrdSqr(x, _10101, 1)     // _101010
    62  	p256OrdMul(_101111, _101, x) // _101111
    63  	p256OrdMul(x, _10101, x)     // _111111 = x6
    64  	p256OrdSqr(t, x, 2)          // _11111100
    65  	p256OrdMul(t, t, _11)        // _11111111 = x8
    66  	p256OrdSqr(x, t, 8)          // _ff00
    67  	p256OrdMul(x, x, t)          // _ffff = x16
    68  	p256OrdSqr(t, x, 16)         // _ffff0000
    69  	p256OrdMul(t, t, x)          // _ffffffff = x32
    70  
    71  	p256OrdSqr(x, t, 64)
    72  	p256OrdMul(x, x, t)
    73  	p256OrdSqr(x, x, 32)
    74  	p256OrdMul(x, x, t)
    75  
    76  	sqrs := []int{
    77  		6, 5, 4, 5, 5,
    78  		4, 3, 3, 5, 9,
    79  		6, 2, 5, 6, 5,
    80  		4, 5, 5, 3, 10,
    81  		2, 5, 5, 3, 7, 6}
    82  	muls := []*p256OrdElement{
    83  		_101111, _111, _11, _1111, _10101,
    84  		_101, _101, _101, _111, _101111,
    85  		_1111, _1, _1, _1111, _111,
    86  		_111, _111, _101, _11, _101111,
    87  		_11, _11, _11, _1, _10101, _1111}
    88  
    89  	for i, s := range sqrs {
    90  		p256OrdSqr(x, x, s)
    91  		p256OrdMul(x, x, muls[i])
    92  	}
    93  
    94  	// Montgomery multiplication by R⁻¹, or 1 outside the domain as R⁻¹×R = 1,
    95  	// converts a Montgomery value out of the domain.
    96  	one := &p256OrdElement{1}
    97  	p256OrdMul(x, x, one)
    98  
    99  	var xOut [32]byte
   100  	p256OrdLittleToBig(&xOut, x)
   101  	return xOut[:], nil
   102  }
   103
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