Source: p256_asm.go in package crypto/elliptic

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" https://link.springer.com/article/10.1007%2Fs13389-014-0090-x https://eprint.iacr.org/2013/816.pdf

+build amd64 arm64


package elliptic

import (
	"math/big"
	"sync"
)

type (
	p256Curve struct {
		*CurveParams
	}

	p256Point struct {
		xyz [12]uint64
	}
)

var (
	p256            p256Curve
	p256Precomputed *[43][32 * 8]uint64
	precomputeOnce  sync.Once
)

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_*64.s Montgomery multiplication modulo P256go:noescape

func p256Mul(res, in1, in2 []uint64)

Montgomery square modulo P256, repeated n times (n >= 1)go:noescape

func p256Sqr(res, in []uint64, n int)

Montgomery multiplication by 1go:noescape

func p256FromMont(res, in []uint64)

iff cond == 1 val <- -valgo:noescape

func p256NegCond(val []uint64, cond int)

if cond == 0 res <- b; else res <- ago:noescape

func p256MovCond(res, a, b []uint64, cond int)

Endianness swapgo:noescape

func p256BigToLittle(res []uint64, in []byte)

go:noescape

func p256LittleToBig(res []byte, in []uint64)

Constant time table accessgo: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 timesgo: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 = in2go: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 doublego:noescape

func p256PointDoubleAsm(res, in []uint64)

func (curve p256Curve) Inverse(k *big.Int) *big.Int {

This should never happen.

		k = new(big.Int).Neg(k)
	}

This should never happen.

		k = new(big.Int).Mod(k, p256.N)
	}

table will store precomputed powers of x.

	var table [4 * 9]uint64
	var (
		_1      = table[4*0 : 4*1]
		_11     = table[4*1 : 4*2]
		_101    = table[4*2 : 4*3]
		_111    = table[4*3 : 4*4]
		_1111   = table[4*4 : 4*5]
		_10101  = table[4*5 : 4*6]
		_101111 = table[4*6 : 4*7]
		x       = table[4*7 : 4*8]
		t       = table[4*8 : 4*9]
	)

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. Window values borrowed from https://briansmith.org/ecc-inversion-addition-chains-01#p256_scalar_inversion

	RR := []uint64{0x83244c95be79eea2, 0x4699799c49bd6fa6, 0x2845b2392b6bec59, 0x66e12d94f3d95620}
	p256OrdMul(_1, x, RR)      // _1
	p256OrdSqr(x, _1, 1)       // _10
	p256OrdMul(_11, x, _1)     // _11
	p256OrdMul(_101, x, _11)   // _101
	p256OrdMul(_111, x, _101)  // _111
	p256OrdSqr(x, _101, 1)     // _1010
	p256OrdMul(_1111, _101, x) // _1111

	p256OrdSqr(t, x, 1)          // _10100
	p256OrdMul(_10101, t, _1)    // _10101
	p256OrdSqr(x, _10101, 1)     // _101010
	p256OrdMul(_101111, _101, x) // _101111
	p256OrdMul(x, _10101, x)     // _111111 = x6
	p256OrdSqr(t, x, 2)          // _11111100
	p256OrdMul(t, t, _11)        // _11111111 = x8
	p256OrdSqr(x, t, 8)          // _ff00
	p256OrdMul(x, x, t)          // _ffff = x16
	p256OrdSqr(t, x, 16)         // _ffff0000
	p256OrdMul(t, t, x)          // _ffffffff = x32

	p256OrdSqr(x, t, 64)
	p256OrdMul(x, x, t)
	p256OrdSqr(x, x, 32)
	p256OrdMul(x, x, t)

	sqrs := []uint8{
		6, 5, 4, 5, 5,
		4, 3, 3, 5, 9,
		6, 2, 5, 6, 5,
		4, 5, 5, 3, 10,
		2, 5, 5, 3, 7, 6}
	muls := [][]uint64{
		_101111, _111, _11, _1111, _10101,
		_101, _101, _101, _111, _101111,
		_1111, _1, _1, _1111, _111,
		_111, _111, _101, _11, _101111,
		_11, _11, _11, _1, _10101, _1111}

	for i, s := range sqrs {
		p256OrdSqr(x, x, int(s))
		p256OrdMul(x, x, muls[i])
	}

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[:])

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, 1)
	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, 1)
	p256Mul(p2, out, in) // 3*p

	p256Sqr(out, p2, 2)
	p256Mul(p4, out, p2) // f*p

	p256Sqr(out, p4, 4)
	p256Mul(p8, out, p4) // ff*p

	p256Sqr(out, p8, 8)
	p256Mul(p16, out, p8) // ffff*p

	p256Sqr(out, p16, 16)
	p256Mul(p32, out, p16) // ffffffff*p

	p256Sqr(out, p32, 32)
	p256Mul(out, out, in)

	p256Sqr(out, out, 128)
	p256Mul(out, out, p32)

	p256Sqr(out, out, 32)
	p256Mul(out, out, p32)

	p256Sqr(out, out, 16)
	p256Mul(out, out, p16)

	p256Sqr(out, out, 8)
	p256Mul(out, out, p8)

	p256Sqr(out, out, 4)
	p256Mul(out, out, p4)

	p256Sqr(out, out, 2)
	p256Mul(out, out, p2)

	p256Sqr(out, out, 2)
	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 boothW6(in uint) (int, int) {
	var s uint = ^((in >> 6) - 1)
	var d uint = (1 << 7) - in - 1
	d = (d & s) | (in & (^s))
	d = (d >> 1) + (d & 1)
	return int(d), int(s & 1)
}

func initTable() {
	p256Precomputed = new([43][32 * 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 < 32; j++ {
		copy(t1, t2)

The window size is 6 so we need to double 6 times.

			if i != 0 {
				for k := 0; k < 6; 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, 1)
			p256Mul(zInv, zInv, zInvSq)

			p256Mul(t1[:4], t1[:4], zInvSq)
			p256Mul(t1[4:8], t1[4:8], zInv)

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) & 0x7f
	sel, sign := boothW6(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

(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(5)
	zero := sel

	for i := 1; i < 43; i++ {
		if index < 192 {
			wvalue = ((scalar[index/64] >> (index % 64)) + (scalar[index/64+1] << (64 - (index % 64)))) & 0x7f
		} else {
			wvalue = (scalar[index/64] >> (index % 64)) & 0x7f
		}
		index += 6
		sel, sign = boothW6(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
	}
}

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)