p224_sqrt.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  package nistec
     6  
     7  import (
     8  	"crypto/internal/nistec/fiat"
     9  	"sync"
    10  )
    11  
    12  var p224GG *[96]fiat.P224Element
    13  var p224GGOnce sync.Once
    14  
    15  // p224SqrtCandidate sets r to a square root candidate for x. r and x must not overlap.
    16  func p224SqrtCandidate(r, x *fiat.P224Element) {
    17  	// Since p = 1 mod 4, we can't use the exponentiation by (p + 1) / 4 like
    18  	// for the other primes. Instead, implement a variation of Tonelli–Shanks.
    19  	// The constant-time implementation is adapted from Thomas Pornin's ecGFp5.
    20  	//
    21  	// https://github.com/pornin/ecgfp5/blob/82325b965/rust/src/field.rs#L337-L385
    22  
    23  	// p = q*2^n + 1 with q odd -> q = 2^128 - 1 and n = 96
    24  	// g^(2^n) = 1 -> g = 11 ^ q (where 11 is the smallest non-square)
    25  	// GG[j] = g^(2^j) for j = 0 to n-1
    26  
    27  	p224GGOnce.Do(func() {
    28  		p224GG = new([96]fiat.P224Element)
    29  		for i := range p224GG {
    30  			if i == 0 {
    31  				p224GG[i].SetBytes([]byte{0x6a, 0x0f, 0xec, 0x67,
    32  					0x85, 0x98, 0xa7, 0x92, 0x0c, 0x55, 0xb2, 0xd4,
    33  					0x0b, 0x2d, 0x6f, 0xfb, 0xbe, 0xa3, 0xd8, 0xce,
    34  					0xf3, 0xfb, 0x36, 0x32, 0xdc, 0x69, 0x1b, 0x74})
    35  			} else {
    36  				p224GG[i].Square(&p224GG[i-1])
    37  			}
    38  		}
    39  	})
    40  
    41  	// r <- x^((q+1)/2) = x^(2^127)
    42  	// v <- x^q = x^(2^128-1)
    43  
    44  	// Compute x^(2^127-1) first.
    45  	//
    46  	// The sequence of 10 multiplications and 126 squarings is derived from the
    47  	// following addition chain generated with github.com/mmcloughlin/addchain v0.4.0.
    48  	//
    49  	//	_10      = 2*1
    50  	//	_11      = 1 + _10
    51  	//	_110     = 2*_11
    52  	//	_111     = 1 + _110
    53  	//	_111000  = _111 << 3
    54  	//	_111111  = _111 + _111000
    55  	//	_1111110 = 2*_111111
    56  	//	_1111111 = 1 + _1111110
    57  	//	x12      = _1111110 << 5 + _111111
    58  	//	x24      = x12 << 12 + x12
    59  	//	i36      = x24 << 7
    60  	//	x31      = _1111111 + i36
    61  	//	x48      = i36 << 17 + x24
    62  	//	x96      = x48 << 48 + x48
    63  	//	return     x96 << 31 + x31
    64  	//
    65  	var t0 = new(fiat.P224Element)
    66  	var t1 = new(fiat.P224Element)
    67  
    68  	r.Square(x)
    69  	r.Mul(x, r)
    70  	r.Square(r)
    71  	r.Mul(x, r)
    72  	t0.Square(r)
    73  	for s := 1; s < 3; s++ {
    74  		t0.Square(t0)
    75  	}
    76  	t0.Mul(r, t0)
    77  	t1.Square(t0)
    78  	r.Mul(x, t1)
    79  	for s := 0; s < 5; s++ {
    80  		t1.Square(t1)
    81  	}
    82  	t0.Mul(t0, t1)
    83  	t1.Square(t0)
    84  	for s := 1; s < 12; s++ {
    85  		t1.Square(t1)
    86  	}
    87  	t0.Mul(t0, t1)
    88  	t1.Square(t0)
    89  	for s := 1; s < 7; s++ {
    90  		t1.Square(t1)
    91  	}
    92  	r.Mul(r, t1)
    93  	for s := 0; s < 17; s++ {
    94  		t1.Square(t1)
    95  	}
    96  	t0.Mul(t0, t1)
    97  	t1.Square(t0)
    98  	for s := 1; s < 48; s++ {
    99  		t1.Square(t1)
   100  	}
   101  	t0.Mul(t0, t1)
   102  	for s := 0; s < 31; s++ {
   103  		t0.Square(t0)
   104  	}
   105  	r.Mul(r, t0)
   106  
   107  	// v = x^(2^127-1)^2 * x
   108  	v := new(fiat.P224Element).Square(r)
   109  	v.Mul(v, x)
   110  
   111  	// r = x^(2^127-1) * x
   112  	r.Mul(r, x)
   113  
   114  	// for i = n-1 down to 1:
   115  	//     w = v^(2^(i-1))
   116  	//     if w == -1 then:
   117  	//         v <- v*GG[n-i]
   118  	//         r <- r*GG[n-i-1]
   119  
   120  	var p224MinusOne = new(fiat.P224Element).Sub(
   121  		new(fiat.P224Element), new(fiat.P224Element).One())
   122  
   123  	for i := 96 - 1; i >= 1; i-- {
   124  		w := new(fiat.P224Element).Set(v)
   125  		for j := 0; j < i-1; j++ {
   126  			w.Square(w)
   127  		}
   128  		cond := w.Equal(p224MinusOne)
   129  		v.Select(t0.Mul(v, &p224GG[96-i]), v, cond)
   130  		r.Select(t0.Mul(r, &p224GG[96-i-1]), r, cond)
   131  	}
   132  }
   133
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