mirror of
https://github.com/autc04/Retro68.git
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614 lines
16 KiB
Go
614 lines
16 KiB
Go
// Copyright 2015 The Go Authors. All rights reserved.
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// Use of this source code is governed by a BSD-style
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// license that can be found in the LICENSE file.
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// This file contains the Go wrapper for the constant-time, 64-bit assembly
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// implementation of P256. The optimizations performed here are described in
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// detail in:
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// S.Gueron and V.Krasnov, "Fast prime field elliptic-curve cryptography with
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// 256-bit primes"
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// http://link.springer.com/article/10.1007%2Fs13389-014-0090-x
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// https://eprint.iacr.org/2013/816.pdf
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// +build ignore
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// +build amd64
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package elliptic
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import (
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"math/big"
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"sync"
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)
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type (
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p256Curve struct {
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*CurveParams
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}
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p256Point struct {
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xyz [12]uint64
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}
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)
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var (
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p256 p256Curve
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p256Precomputed *[37][64 * 8]uint64
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precomputeOnce sync.Once
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)
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func initP256() {
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// See FIPS 186-3, section D.2.3
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p256.CurveParams = &CurveParams{Name: "P-256"}
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p256.P, _ = new(big.Int).SetString("115792089210356248762697446949407573530086143415290314195533631308867097853951", 10)
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p256.N, _ = new(big.Int).SetString("115792089210356248762697446949407573529996955224135760342422259061068512044369", 10)
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p256.B, _ = new(big.Int).SetString("5ac635d8aa3a93e7b3ebbd55769886bc651d06b0cc53b0f63bce3c3e27d2604b", 16)
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p256.Gx, _ = new(big.Int).SetString("6b17d1f2e12c4247f8bce6e563a440f277037d812deb33a0f4a13945d898c296", 16)
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p256.Gy, _ = new(big.Int).SetString("4fe342e2fe1a7f9b8ee7eb4a7c0f9e162bce33576b315ececbb6406837bf51f5", 16)
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p256.BitSize = 256
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}
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func (curve p256Curve) Params() *CurveParams {
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return curve.CurveParams
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}
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// Functions implemented in p256_asm_amd64.s
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// Montgomery multiplication modulo P256
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//go:noescape
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func p256Mul(res, in1, in2 []uint64)
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// Montgomery square modulo P256
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//go:noescape
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func p256Sqr(res, in []uint64)
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// Montgomery multiplication by 1
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//go:noescape
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func p256FromMont(res, in []uint64)
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// iff cond == 1 val <- -val
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//go:noescape
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func p256NegCond(val []uint64, cond int)
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// if cond == 0 res <- b; else res <- a
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//go:noescape
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func p256MovCond(res, a, b []uint64, cond int)
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// Endianness swap
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//go:noescape
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func p256BigToLittle(res []uint64, in []byte)
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//go:noescape
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func p256LittleToBig(res []byte, in []uint64)
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// Constant time table access
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//go:noescape
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func p256Select(point, table []uint64, idx int)
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//go:noescape
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func p256SelectBase(point, table []uint64, idx int)
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// Montgomery multiplication modulo Ord(G)
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//go:noescape
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func p256OrdMul(res, in1, in2 []uint64)
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// Montgomery square modulo Ord(G), repeated n times
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//go:noescape
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func p256OrdSqr(res, in []uint64, n int)
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// Point add with in2 being affine point
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// If sign == 1 -> in2 = -in2
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// If sel == 0 -> res = in1
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// if zero == 0 -> res = in2
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//go:noescape
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func p256PointAddAffineAsm(res, in1, in2 []uint64, sign, sel, zero int)
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// Point add. Returns one if the two input points were equal and zero
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// otherwise. (Note that, due to the way that the equations work out, some
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// representations of ∞ are considered equal to everything by this function.)
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//go:noescape
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func p256PointAddAsm(res, in1, in2 []uint64) int
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// Point double
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//go:noescape
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func p256PointDoubleAsm(res, in []uint64)
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func (curve p256Curve) Inverse(k *big.Int) *big.Int {
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if k.Sign() < 0 {
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// This should never happen.
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k = new(big.Int).Neg(k)
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}
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if k.Cmp(p256.N) >= 0 {
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// This should never happen.
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k = new(big.Int).Mod(k, p256.N)
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}
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// table will store precomputed powers of x. The four words at index
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// 4×i store x^(i+1).
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var table [4 * 15]uint64
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x := make([]uint64, 4)
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fromBig(x[:], k)
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// This code operates in the Montgomery domain where R = 2^256 mod n
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// and n is the order of the scalar field. (See initP256 for the
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// value.) Elements in the Montgomery domain take the form a×R and
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// multiplication of x and y in the calculates (x × y × R^-1) mod n. RR
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// is R×R mod n thus the Montgomery multiplication x and RR gives x×R,
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// i.e. converts x into the Montgomery domain.
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RR := []uint64{0x83244c95be79eea2, 0x4699799c49bd6fa6, 0x2845b2392b6bec59, 0x66e12d94f3d95620}
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p256OrdMul(table[:4], x, RR)
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// Prepare the table, no need in constant time access, because the
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// power is not a secret. (Entry 0 is never used.)
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for i := 2; i < 16; i += 2 {
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p256OrdSqr(table[4*(i-1):], table[4*((i/2)-1):], 1)
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p256OrdMul(table[4*i:], table[4*(i-1):], table[:4])
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}
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x[0] = table[4*14+0] // f
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x[1] = table[4*14+1]
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x[2] = table[4*14+2]
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x[3] = table[4*14+3]
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p256OrdSqr(x, x, 4)
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p256OrdMul(x, x, table[4*14:4*14+4]) // ff
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t := make([]uint64, 4, 4)
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t[0] = x[0]
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t[1] = x[1]
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t[2] = x[2]
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t[3] = x[3]
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p256OrdSqr(x, x, 8)
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p256OrdMul(x, x, t) // ffff
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t[0] = x[0]
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t[1] = x[1]
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t[2] = x[2]
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t[3] = x[3]
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p256OrdSqr(x, x, 16)
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p256OrdMul(x, x, t) // ffffffff
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t[0] = x[0]
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t[1] = x[1]
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t[2] = x[2]
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t[3] = x[3]
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p256OrdSqr(x, x, 64) // ffffffff0000000000000000
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p256OrdMul(x, x, t) // ffffffff00000000ffffffff
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p256OrdSqr(x, x, 32) // ffffffff00000000ffffffff00000000
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p256OrdMul(x, x, t) // ffffffff00000000ffffffffffffffff
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// Remaining 32 windows
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expLo := [32]byte{0xb, 0xc, 0xe, 0x6, 0xf, 0xa, 0xa, 0xd, 0xa, 0x7, 0x1, 0x7, 0x9, 0xe, 0x8, 0x4, 0xf, 0x3, 0xb, 0x9, 0xc, 0xa, 0xc, 0x2, 0xf, 0xc, 0x6, 0x3, 0x2, 0x5, 0x4, 0xf}
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for i := 0; i < 32; i++ {
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p256OrdSqr(x, x, 4)
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p256OrdMul(x, x, table[4*(expLo[i]-1):])
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}
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// Multiplying by one in the Montgomery domain converts a Montgomery
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// value out of the domain.
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one := []uint64{1, 0, 0, 0}
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p256OrdMul(x, x, one)
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xOut := make([]byte, 32)
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p256LittleToBig(xOut, x)
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return new(big.Int).SetBytes(xOut)
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}
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// fromBig converts a *big.Int into a format used by this code.
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func fromBig(out []uint64, big *big.Int) {
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for i := range out {
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out[i] = 0
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}
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for i, v := range big.Bits() {
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out[i] = uint64(v)
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}
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}
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// p256GetScalar endian-swaps the big-endian scalar value from in and writes it
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// to out. If the scalar is equal or greater than the order of the group, it's
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// reduced modulo that order.
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func p256GetScalar(out []uint64, in []byte) {
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n := new(big.Int).SetBytes(in)
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if n.Cmp(p256.N) >= 0 {
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n.Mod(n, p256.N)
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}
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fromBig(out, n)
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}
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// p256Mul operates in a Montgomery domain with R = 2^256 mod p, where p is the
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// underlying field of the curve. (See initP256 for the value.) Thus rr here is
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// R×R mod p. See comment in Inverse about how this is used.
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var rr = []uint64{0x0000000000000003, 0xfffffffbffffffff, 0xfffffffffffffffe, 0x00000004fffffffd}
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func maybeReduceModP(in *big.Int) *big.Int {
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if in.Cmp(p256.P) < 0 {
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return in
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}
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return new(big.Int).Mod(in, p256.P)
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}
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func (curve p256Curve) CombinedMult(bigX, bigY *big.Int, baseScalar, scalar []byte) (x, y *big.Int) {
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scalarReversed := make([]uint64, 4)
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var r1, r2 p256Point
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p256GetScalar(scalarReversed, baseScalar)
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r1IsInfinity := scalarIsZero(scalarReversed)
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r1.p256BaseMult(scalarReversed)
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p256GetScalar(scalarReversed, scalar)
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r2IsInfinity := scalarIsZero(scalarReversed)
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fromBig(r2.xyz[0:4], maybeReduceModP(bigX))
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fromBig(r2.xyz[4:8], maybeReduceModP(bigY))
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p256Mul(r2.xyz[0:4], r2.xyz[0:4], rr[:])
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p256Mul(r2.xyz[4:8], r2.xyz[4:8], rr[:])
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// This sets r2's Z value to 1, in the Montgomery domain.
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r2.xyz[8] = 0x0000000000000001
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r2.xyz[9] = 0xffffffff00000000
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r2.xyz[10] = 0xffffffffffffffff
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r2.xyz[11] = 0x00000000fffffffe
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r2.p256ScalarMult(scalarReversed)
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var sum, double p256Point
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pointsEqual := p256PointAddAsm(sum.xyz[:], r1.xyz[:], r2.xyz[:])
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p256PointDoubleAsm(double.xyz[:], r1.xyz[:])
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sum.CopyConditional(&double, pointsEqual)
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sum.CopyConditional(&r1, r2IsInfinity)
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sum.CopyConditional(&r2, r1IsInfinity)
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return sum.p256PointToAffine()
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}
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func (curve p256Curve) ScalarBaseMult(scalar []byte) (x, y *big.Int) {
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scalarReversed := make([]uint64, 4)
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p256GetScalar(scalarReversed, scalar)
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var r p256Point
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r.p256BaseMult(scalarReversed)
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return r.p256PointToAffine()
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}
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func (curve p256Curve) ScalarMult(bigX, bigY *big.Int, scalar []byte) (x, y *big.Int) {
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scalarReversed := make([]uint64, 4)
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p256GetScalar(scalarReversed, scalar)
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var r p256Point
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fromBig(r.xyz[0:4], maybeReduceModP(bigX))
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fromBig(r.xyz[4:8], maybeReduceModP(bigY))
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p256Mul(r.xyz[0:4], r.xyz[0:4], rr[:])
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p256Mul(r.xyz[4:8], r.xyz[4:8], rr[:])
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// This sets r2's Z value to 1, in the Montgomery domain.
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r.xyz[8] = 0x0000000000000001
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r.xyz[9] = 0xffffffff00000000
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r.xyz[10] = 0xffffffffffffffff
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r.xyz[11] = 0x00000000fffffffe
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r.p256ScalarMult(scalarReversed)
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return r.p256PointToAffine()
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}
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// uint64IsZero returns 1 if x is zero and zero otherwise.
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func uint64IsZero(x uint64) int {
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x = ^x
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x &= x >> 32
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x &= x >> 16
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x &= x >> 8
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x &= x >> 4
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x &= x >> 2
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x &= x >> 1
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return int(x & 1)
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}
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// scalarIsZero returns 1 if scalar represents the zero value, and zero
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// otherwise.
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func scalarIsZero(scalar []uint64) int {
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return uint64IsZero(scalar[0] | scalar[1] | scalar[2] | scalar[3])
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}
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func (p *p256Point) p256PointToAffine() (x, y *big.Int) {
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zInv := make([]uint64, 4)
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zInvSq := make([]uint64, 4)
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p256Inverse(zInv, p.xyz[8:12])
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p256Sqr(zInvSq, zInv)
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p256Mul(zInv, zInv, zInvSq)
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p256Mul(zInvSq, p.xyz[0:4], zInvSq)
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p256Mul(zInv, p.xyz[4:8], zInv)
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p256FromMont(zInvSq, zInvSq)
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p256FromMont(zInv, zInv)
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xOut := make([]byte, 32)
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yOut := make([]byte, 32)
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p256LittleToBig(xOut, zInvSq)
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p256LittleToBig(yOut, zInv)
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return new(big.Int).SetBytes(xOut), new(big.Int).SetBytes(yOut)
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}
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// CopyConditional copies overwrites p with src if v == 1, and leaves p
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// unchanged if v == 0.
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func (p *p256Point) CopyConditional(src *p256Point, v int) {
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pMask := uint64(v) - 1
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srcMask := ^pMask
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for i, n := range p.xyz {
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p.xyz[i] = (n & pMask) | (src.xyz[i] & srcMask)
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}
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}
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// p256Inverse sets out to in^-1 mod p.
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func p256Inverse(out, in []uint64) {
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var stack [6 * 4]uint64
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p2 := stack[4*0 : 4*0+4]
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p4 := stack[4*1 : 4*1+4]
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p8 := stack[4*2 : 4*2+4]
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p16 := stack[4*3 : 4*3+4]
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p32 := stack[4*4 : 4*4+4]
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p256Sqr(out, in)
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p256Mul(p2, out, in) // 3*p
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p256Sqr(out, p2)
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p256Sqr(out, out)
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p256Mul(p4, out, p2) // f*p
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p256Sqr(out, p4)
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p256Sqr(out, out)
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p256Sqr(out, out)
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p256Sqr(out, out)
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p256Mul(p8, out, p4) // ff*p
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p256Sqr(out, p8)
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for i := 0; i < 7; i++ {
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p256Sqr(out, out)
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}
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p256Mul(p16, out, p8) // ffff*p
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p256Sqr(out, p16)
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for i := 0; i < 15; i++ {
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p256Sqr(out, out)
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}
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p256Mul(p32, out, p16) // ffffffff*p
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p256Sqr(out, p32)
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for i := 0; i < 31; i++ {
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p256Sqr(out, out)
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}
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p256Mul(out, out, in)
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for i := 0; i < 32*4; i++ {
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p256Sqr(out, out)
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}
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p256Mul(out, out, p32)
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for i := 0; i < 32; i++ {
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p256Sqr(out, out)
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}
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p256Mul(out, out, p32)
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for i := 0; i < 16; i++ {
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p256Sqr(out, out)
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}
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p256Mul(out, out, p16)
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for i := 0; i < 8; i++ {
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p256Sqr(out, out)
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}
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p256Mul(out, out, p8)
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p256Sqr(out, out)
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p256Sqr(out, out)
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p256Sqr(out, out)
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p256Sqr(out, out)
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p256Mul(out, out, p4)
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p256Sqr(out, out)
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p256Sqr(out, out)
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p256Mul(out, out, p2)
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p256Sqr(out, out)
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p256Sqr(out, out)
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p256Mul(out, out, in)
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}
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func (p *p256Point) p256StorePoint(r *[16 * 4 * 3]uint64, index int) {
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copy(r[index*12:], p.xyz[:])
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}
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func boothW5(in uint) (int, int) {
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var s uint = ^((in >> 5) - 1)
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var d uint = (1 << 6) - in - 1
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d = (d & s) | (in & (^s))
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d = (d >> 1) + (d & 1)
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return int(d), int(s & 1)
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}
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func boothW7(in uint) (int, int) {
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var s uint = ^((in >> 7) - 1)
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var d uint = (1 << 8) - in - 1
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d = (d & s) | (in & (^s))
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d = (d >> 1) + (d & 1)
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return int(d), int(s & 1)
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}
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func initTable() {
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p256Precomputed = new([37][64 * 8]uint64)
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basePoint := []uint64{
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0x79e730d418a9143c, 0x75ba95fc5fedb601, 0x79fb732b77622510, 0x18905f76a53755c6,
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0xddf25357ce95560a, 0x8b4ab8e4ba19e45c, 0xd2e88688dd21f325, 0x8571ff1825885d85,
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0x0000000000000001, 0xffffffff00000000, 0xffffffffffffffff, 0x00000000fffffffe,
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}
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t1 := make([]uint64, 12)
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t2 := make([]uint64, 12)
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copy(t2, basePoint)
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zInv := make([]uint64, 4)
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zInvSq := make([]uint64, 4)
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for j := 0; j < 64; j++ {
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copy(t1, t2)
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for i := 0; i < 37; i++ {
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// The window size is 7 so we need to double 7 times.
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if i != 0 {
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for k := 0; k < 7; k++ {
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p256PointDoubleAsm(t1, t1)
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||
}
|
||
}
|
||
// Convert the point to affine form. (Its values are
|
||
// still in Montgomery form however.)
|
||
p256Inverse(zInv, t1[8:12])
|
||
p256Sqr(zInvSq, zInv)
|
||
p256Mul(zInv, zInv, zInvSq)
|
||
|
||
p256Mul(t1[:4], t1[:4], zInvSq)
|
||
p256Mul(t1[4:8], t1[4:8], zInv)
|
||
|
||
copy(t1[8:12], basePoint[8:12])
|
||
// Update the table entry
|
||
copy(p256Precomputed[i][j*8:], t1[:8])
|
||
}
|
||
if j == 0 {
|
||
p256PointDoubleAsm(t2, basePoint)
|
||
} else {
|
||
p256PointAddAsm(t2, t2, basePoint)
|
||
}
|
||
}
|
||
}
|
||
|
||
func (p *p256Point) p256BaseMult(scalar []uint64) {
|
||
precomputeOnce.Do(initTable)
|
||
|
||
wvalue := (scalar[0] << 1) & 0xff
|
||
sel, sign := boothW7(uint(wvalue))
|
||
p256SelectBase(p.xyz[0:8], p256Precomputed[0][0:], sel)
|
||
p256NegCond(p.xyz[4:8], sign)
|
||
|
||
// (This is one, in the Montgomery domain.)
|
||
p.xyz[8] = 0x0000000000000001
|
||
p.xyz[9] = 0xffffffff00000000
|
||
p.xyz[10] = 0xffffffffffffffff
|
||
p.xyz[11] = 0x00000000fffffffe
|
||
|
||
var t0 p256Point
|
||
// (This is one, in the Montgomery domain.)
|
||
t0.xyz[8] = 0x0000000000000001
|
||
t0.xyz[9] = 0xffffffff00000000
|
||
t0.xyz[10] = 0xffffffffffffffff
|
||
t0.xyz[11] = 0x00000000fffffffe
|
||
|
||
index := uint(6)
|
||
zero := sel
|
||
|
||
for i := 1; i < 37; i++ {
|
||
if index < 192 {
|
||
wvalue = ((scalar[index/64] >> (index % 64)) + (scalar[index/64+1] << (64 - (index % 64)))) & 0xff
|
||
} else {
|
||
wvalue = (scalar[index/64] >> (index % 64)) & 0xff
|
||
}
|
||
index += 7
|
||
sel, sign = boothW7(uint(wvalue))
|
||
p256SelectBase(t0.xyz[0:8], p256Precomputed[i][0:], sel)
|
||
p256PointAddAffineAsm(p.xyz[0:12], p.xyz[0:12], t0.xyz[0:8], sign, sel, zero)
|
||
zero |= sel
|
||
}
|
||
}
|
||
|
||
func (p *p256Point) p256ScalarMult(scalar []uint64) {
|
||
// precomp is a table of precomputed points that stores powers of p
|
||
// from p^1 to p^16.
|
||
var precomp [16 * 4 * 3]uint64
|
||
var t0, t1, t2, t3 p256Point
|
||
|
||
// Prepare the table
|
||
p.p256StorePoint(&precomp, 0) // 1
|
||
|
||
p256PointDoubleAsm(t0.xyz[:], p.xyz[:])
|
||
p256PointDoubleAsm(t1.xyz[:], t0.xyz[:])
|
||
p256PointDoubleAsm(t2.xyz[:], t1.xyz[:])
|
||
p256PointDoubleAsm(t3.xyz[:], t2.xyz[:])
|
||
t0.p256StorePoint(&precomp, 1) // 2
|
||
t1.p256StorePoint(&precomp, 3) // 4
|
||
t2.p256StorePoint(&precomp, 7) // 8
|
||
t3.p256StorePoint(&precomp, 15) // 16
|
||
|
||
p256PointAddAsm(t0.xyz[:], t0.xyz[:], p.xyz[:])
|
||
p256PointAddAsm(t1.xyz[:], t1.xyz[:], p.xyz[:])
|
||
p256PointAddAsm(t2.xyz[:], t2.xyz[:], p.xyz[:])
|
||
t0.p256StorePoint(&precomp, 2) // 3
|
||
t1.p256StorePoint(&precomp, 4) // 5
|
||
t2.p256StorePoint(&precomp, 8) // 9
|
||
|
||
p256PointDoubleAsm(t0.xyz[:], t0.xyz[:])
|
||
p256PointDoubleAsm(t1.xyz[:], t1.xyz[:])
|
||
t0.p256StorePoint(&precomp, 5) // 6
|
||
t1.p256StorePoint(&precomp, 9) // 10
|
||
|
||
p256PointAddAsm(t2.xyz[:], t0.xyz[:], p.xyz[:])
|
||
p256PointAddAsm(t1.xyz[:], t1.xyz[:], p.xyz[:])
|
||
t2.p256StorePoint(&precomp, 6) // 7
|
||
t1.p256StorePoint(&precomp, 10) // 11
|
||
|
||
p256PointDoubleAsm(t0.xyz[:], t0.xyz[:])
|
||
p256PointDoubleAsm(t2.xyz[:], t2.xyz[:])
|
||
t0.p256StorePoint(&precomp, 11) // 12
|
||
t2.p256StorePoint(&precomp, 13) // 14
|
||
|
||
p256PointAddAsm(t0.xyz[:], t0.xyz[:], p.xyz[:])
|
||
p256PointAddAsm(t2.xyz[:], t2.xyz[:], p.xyz[:])
|
||
t0.p256StorePoint(&precomp, 12) // 13
|
||
t2.p256StorePoint(&precomp, 14) // 15
|
||
|
||
// Start scanning the window from top bit
|
||
index := uint(254)
|
||
var sel, sign int
|
||
|
||
wvalue := (scalar[index/64] >> (index % 64)) & 0x3f
|
||
sel, _ = boothW5(uint(wvalue))
|
||
|
||
p256Select(p.xyz[0:12], precomp[0:], sel)
|
||
zero := sel
|
||
|
||
for index > 4 {
|
||
index -= 5
|
||
p256PointDoubleAsm(p.xyz[:], p.xyz[:])
|
||
p256PointDoubleAsm(p.xyz[:], p.xyz[:])
|
||
p256PointDoubleAsm(p.xyz[:], p.xyz[:])
|
||
p256PointDoubleAsm(p.xyz[:], p.xyz[:])
|
||
p256PointDoubleAsm(p.xyz[:], p.xyz[:])
|
||
|
||
if index < 192 {
|
||
wvalue = ((scalar[index/64] >> (index % 64)) + (scalar[index/64+1] << (64 - (index % 64)))) & 0x3f
|
||
} else {
|
||
wvalue = (scalar[index/64] >> (index % 64)) & 0x3f
|
||
}
|
||
|
||
sel, sign = boothW5(uint(wvalue))
|
||
|
||
p256Select(t0.xyz[0:], precomp[0:], sel)
|
||
p256NegCond(t0.xyz[4:8], sign)
|
||
p256PointAddAsm(t1.xyz[:], p.xyz[:], t0.xyz[:])
|
||
p256MovCond(t1.xyz[0:12], t1.xyz[0:12], p.xyz[0:12], sel)
|
||
p256MovCond(p.xyz[0:12], t1.xyz[0:12], t0.xyz[0:12], zero)
|
||
zero |= sel
|
||
}
|
||
|
||
p256PointDoubleAsm(p.xyz[:], p.xyz[:])
|
||
p256PointDoubleAsm(p.xyz[:], p.xyz[:])
|
||
p256PointDoubleAsm(p.xyz[:], p.xyz[:])
|
||
p256PointDoubleAsm(p.xyz[:], p.xyz[:])
|
||
p256PointDoubleAsm(p.xyz[:], p.xyz[:])
|
||
|
||
wvalue = (scalar[0] << 1) & 0x3f
|
||
sel, sign = boothW5(uint(wvalue))
|
||
|
||
p256Select(t0.xyz[0:], precomp[0:], sel)
|
||
p256NegCond(t0.xyz[4:8], sign)
|
||
p256PointAddAsm(t1.xyz[:], p.xyz[:], t0.xyz[:])
|
||
p256MovCond(t1.xyz[0:12], t1.xyz[0:12], p.xyz[0:12], sel)
|
||
p256MovCond(p.xyz[0:12], t1.xyz[0:12], t0.xyz[0:12], zero)
|
||
}
|