golang p224_sqrt 代码
文件路径：/src/crypto/internal/nistec/p224_sqrt.go
/ Copyright 2022 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.
package nistec
import (
"crypto/internal/nistec/fiat"
"sync"
)
var p224GG *[96]fiat.P224Element
var p224GGOnce sync.Once
var p224MinusOne = new(fiat.P224Element).Sub(
new(fiat.P224Element), new(fiat.P224Element).One())
/ p224SqrtCandidate sets r to a square root candidate for x. r and x must not overlap.
func p224SqrtCandidate(r, x *fiat.P224Element) {
/ Since p = 1 mod 4, we can't use the exponentiation by (p + 1) / 4 like
/ for the other primes. Instead, implement a variation of Tonelli–Shanks.
/ The constant-time implementation is adapted from Thomas Pornin's ecGFp5.
/
/ https://github.com/pornin/ecgfp5/blob/82325b965/rust/src/field.rs#L337-L385
/ p = q*2^n + 1 with q odd -> q = 2^128 - 1 and n = 96
/ g^(2^n) = 1 -> g = 11 ^ q (where 11 is the smallest non-square)
/ GG[j] = g^(2^j) for j = 0 to n-1
p224GGOnce.Do(func() {
p224GG = new([96]fiat.P224Element)
for i := range p224GG {
if i == 0 {
p224GG[i].SetBytes([]byte{0x6a, 0x0f, 0xec, 0x67,
0x85, 0x98, 0xa7, 0x92, 0x0c, 0x55, 0xb2, 0xd4,
0x0b, 0x2d, 0x6f, 0xfb, 0xbe, 0xa3, 0xd8, 0xce,
0xf3, 0xfb, 0x36, 0x32, 0xdc, 0x69, 0x1b, 0x74})
} else {
p224GG[i].Square(&p224GG[i-1])
}
}
})
/ r <- x^((q+1)/2) = x^(2^127)
/ v <- x^q = x^(2^128-1)
/ Compute x^(2^127-1) first.
/
/ The sequence of 10 multiplications and 126 squarings is derived from the
/ following addition chain generated with github.com/mmcloughlin/addchain v0.4.0.
/
/ _10 = 2*1
/ _11 = 1 + _10
/ _110 = 2*_11
/ _111 = 1 + _110
/ _111000 = _111 << 3
/ _111111 = _111 + _111000
/ _1111110 = 2*_111111
/ _1111111 = 1 + _1111110
/ x12 = _1111110 << 5 + _111111
/ x24 = x12 << 12 + x12
/ i36 = x24 << 7
/ x31 = _1111111 + i36
/ x48 = i36 << 17 + x24
/ x96 = x48 << 48 + x48
/ return x96 << 31 + x31
/
var t0 = new(fiat.P224Element)
var t1 = new(fiat.P224Element)
r.Square(x)
r.Mul(x, r)
r.Square(r)
r.Mul(x, r)
t0.Square(r)
for s := 1; s < 3; s++ {
t0.Square(t0)
}
t0.Mul(r, t0)
t1.Square(t0)
r.Mul(x, t1)
for s := 0; s < 5; s++ {
t1.Square(t1)
}
t0.Mul(t0, t1)
t1.Square(t0)
for s := 1; s < 12; s++ {
t1.Square(t1)
}
t0.Mul(t0, t1)
t1.Square(t0)
for s := 1; s < 7; s++ {
t1.Square(t1)
}
r.Mul(r, t1)
for s := 0; s < 17; s++ {
t1.Square(t1)
}
t0.Mul(t0, t1)
t1.Square(t0)
for s := 1; s < 48; s++ {
t1.Square(t1)
}
t0.Mul(t0, t1)
for s := 0; s < 31; s++ {
t0.Square(t0)
}
r.Mul(r, t0)
/ v = x^(2^127-1)^2 * x
v := new(fiat.P224Element).Square(r)
v.Mul(v, x)
/ r = x^(2^127-1) * x
r.Mul(r, x)
/ for i = n-1 down to 1:
/ w = v^(2^(i-1))
/ if w == -1 then:
/ v <- v*GG[n-i]
/ r <- r*GG[n-i-1]
for i := 96 - 1; i >= 1; i-- {
w := new(fiat.P224Element).Set(v)
for j := 0; j < i-1; j++ {
w.Square(w)
}
cond := w.Equal(p224MinusOne)
v.Select(t0.Mul(v, &p224GG[96-i]), v, cond)
r.Select(t0.Mul(r, &p224GG[96-i-1]), r, cond)
}
}