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