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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
+func p256Mul(res, in1, in2 []uint64)
+
+// Montgomery square modulo P256
+func p256Sqr(res, in []uint64)
+
+// Montgomery multiplication by 1
+func p256FromMont(res, in []uint64)
+
+// iff cond == 1 val <- -val
+func p256NegCond(val []uint64, cond int)
+
+// if cond == 0 res <- b; else res <- a
+func p256MovCond(res, a, b []uint64, cond int)
+
+// Endianess swap
+func p256BigToLittle(res []uint64, in []byte)
+func p256LittleToBig(res []byte, in []uint64)
+
+// Constant time table access
+func p256Select(point, table []uint64, idx int)
+func p256SelectBase(point, table []uint64, idx int)
+
+// Montgomery multiplication modulo Ord(G)
+func p256OrdMul(res, in1, in2 []uint64)
+
+// Montgomery square modulo Ord(G), repeated n times
+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
+func p256PointAddAffineAsm(res, in1, in2 []uint64, sign, sel, zero int)
+
+// Point add
+func p256PointAddAsm(res, in1, in2 []uint64)
+
+// Point double
+func p256PointDoubleAsm(res, in []uint64)
+
+func (curve p256Curve) Inverse(k *big.Int) *big.Int {
+ if k.Cmp(p256.N) >= 0 {
+ // This should never happen.
+ reducedK := new(big.Int).Mod(k, p256.N)
+ k = reducedK
+ }
+
+ // 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)
+ r1.p256BaseMult(scalarReversed)
+
+ p256GetScalar(scalarReversed, scalar)
+ 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)
+ p256PointAddAsm(r1.xyz[:], r1.xyz[:], r2.xyz[:])
+ return r1.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()
+}
+
+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)
+}
+
+// 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)
+}
generated by cgit v1.1 at 2025-04-01 02:10:10 +0000