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 // 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 Tonelliâ€“Shanks. // The contant-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) } }