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authorFilippo Valsorda <filippo@golang.org>2021-10-30 00:27:51 -0400
committerFilippo Valsorda <filippo@golang.org>2022-04-27 14:21:49 +0000
commit6796a7924c20d2c58b0cf78766b94543abfadc1b (patch)
treefab1c1c5956a94d6d631175b53df5fe38fba2ab9
parent24b570354caee33d4fb3934ce7ef1cc97fb403fd (diff)
downloadgo-6796a7924c20d2c58b0cf78766b94543abfadc1b.tar.gz
go-6796a7924c20d2c58b0cf78766b94543abfadc1b.zip
crypto/elliptic: refactor package structure
Not quite golang.org/wiki/TargetSpecific compliant, but almost. The only substantial code change is in randFieldElement: it used to use Params().BitSize instead of Params().N.BitLen(), which is semantically incorrect, even if the two values are the same for all named curves. For #52182 Change-Id: Ibc47450552afe23ea74fcf55d1d799d5d7e5487c Reviewed-on: https://go-review.googlesource.com/c/go/+/315273 Run-TryBot: Filippo Valsorda <filippo@golang.org> Reviewed-by: Than McIntosh <thanm@google.com> Reviewed-by: Roland Shoemaker <roland@golang.org> TryBot-Result: Gopher Robot <gobot@golang.org> Reviewed-by: Russ Cox <rsc@golang.org>
Diffstat
-rw-r--r--src/crypto/ecdsa/ecdsa.go16
-rw-r--r--src/crypto/elliptic/elliptic.go289
-rw-r--r--src/crypto/elliptic/p256.go1178
-rw-r--r--src/crypto/elliptic/p256_asm.go25
-rw-r--r--src/crypto/elliptic/p256_generic.go1171
-rw-r--r--src/crypto/elliptic/p256_noasm.go15
-rw-r--r--src/crypto/elliptic/p256_ppc64le.go7
-rw-r--r--src/crypto/elliptic/p256_s390x.go1
-rw-r--r--src/crypto/elliptic/params.go296
9 files changed, 1496 insertions, 1502 deletions
diff --git a/src/crypto/ecdsa/ecdsa.go b/src/crypto/ecdsa/ecdsa.go
index 9f9a09a884..c3f9459991 100644
--- a/src/crypto/ecdsa/ecdsa.go
+++ b/src/crypto/ecdsa/ecdsa.go
@@ -128,7 +128,7 @@ func randFieldElement(c elliptic.Curve, rand io.Reader) (k *big.Int, err error)
params := c.Params()
// Note that for P-521 this will actually be 63 bits more than the order, as
// division rounds down, but the extra bit is inconsequential.
- b := make([]byte, params.BitSize/8+8) // TODO: use params.N.BitLen()
+ b := make([]byte, params.N.BitLen()/8+8)
_, err = io.ReadFull(rand, b)
if err != nil {
return
@@ -228,13 +228,13 @@ func Sign(rand io.Reader, priv *PrivateKey, hash []byte) (r, s *big.Int, err err
// Create a CSPRNG that xors a stream of zeros with
// the output of the AES-CTR instance.
- csprng := cipher.StreamReader{
+ csprng := &cipher.StreamReader{
R: zeroReader,
S: cipher.NewCTR(block, []byte(aesIV)),
}
c := priv.PublicKey.Curve
- return sign(priv, &csprng, c, hash)
+ return sign(priv, csprng, c, hash)
}
func signGeneric(priv *PrivateKey, csprng *cipher.StreamReader, c elliptic.Curve, hash []byte) (r, s *big.Int, err error) {
@@ -353,16 +353,14 @@ func VerifyASN1(pub *PublicKey, hash, sig []byte) bool {
return Verify(pub, hash, r, s)
}
-type zr struct {
- io.Reader
-}
+type zr struct{}
-// Read replaces the contents of dst with zeros.
-func (z *zr) Read(dst []byte) (n int, err error) {
+// Read replaces the contents of dst with zeros. It is safe for concurrent use.
+func (zr) Read(dst []byte) (n int, err error) {
for i := range dst {
dst[i] = 0
}
return len(dst), nil
}
-var zeroReader = &zr{}
+var zeroReader = zr{}
diff --git a/src/crypto/elliptic/elliptic.go b/src/crypto/elliptic/elliptic.go
index 7ead09f8d3..522d7afbaf 100644
--- a/src/crypto/elliptic/elliptic.go
+++ b/src/crypto/elliptic/elliptic.go
@@ -36,295 +36,6 @@ type Curve interface {
ScalarBaseMult(k []byte) (x, y *big.Int)
}
-func matchesSpecificCurve(params *CurveParams, available ...Curve) (Curve, bool) {
- for _, c := range available {
- if params == c.Params() {
- return c, true
- }
- }
- return nil, false
-}
-
-// CurveParams contains the parameters of an elliptic curve and also provides
-// a generic, non-constant time implementation of Curve.
-type CurveParams struct {
- P *big.Int // the order of the underlying field
- N *big.Int // the order of the base point
- B *big.Int // the constant of the curve equation
- Gx, Gy *big.Int // (x,y) of the base point
- BitSize int // the size of the underlying field
- Name string // the canonical name of the curve
-}
-
-func (curve *CurveParams) Params() *CurveParams {
- return curve
-}
-
-// CurveParams operates, internally, on Jacobian coordinates. For a given
-// (x, y) position on the curve, the Jacobian coordinates are (x1, y1, z1)
-// where x = x1/z1² and y = y1/z1³. The greatest speedups come when the whole
-// calculation can be performed within the transform (as in ScalarMult and
-// ScalarBaseMult). But even for Add and Double, it's faster to apply and
-// reverse the transform than to operate in affine coordinates.
-
-// polynomial returns x³ - 3x + b.
-func (curve *CurveParams) polynomial(x *big.Int) *big.Int {
- x3 := new(big.Int).Mul(x, x)
- x3.Mul(x3, x)
-
- threeX := new(big.Int).Lsh(x, 1)
- threeX.Add(threeX, x)
-
- x3.Sub(x3, threeX)
- x3.Add(x3, curve.B)
- x3.Mod(x3, curve.P)
-
- return x3
-}
-
-func (curve *CurveParams) IsOnCurve(x, y *big.Int) bool {
- // If there is a dedicated constant-time implementation for this curve operation,
- // use that instead of the generic one.
- if specific, ok := matchesSpecificCurve(curve, p224, p384, p521); ok {
- return specific.IsOnCurve(x, y)
- }
-
- if x.Sign() < 0 || x.Cmp(curve.P) >= 0 ||
- y.Sign() < 0 || y.Cmp(curve.P) >= 0 {
- return false
- }
-
- // y² = x³ - 3x + b
- y2 := new(big.Int).Mul(y, y)
- y2.Mod(y2, curve.P)
-
- return curve.polynomial(x).Cmp(y2) == 0
-}
-
-// zForAffine returns a Jacobian Z value for the affine point (x, y). If x and
-// y are zero, it assumes that they represent the point at infinity because (0,
-// 0) is not on the any of the curves handled here.
-func zForAffine(x, y *big.Int) *big.Int {
- z := new(big.Int)
- if x.Sign() != 0 || y.Sign() != 0 {
- z.SetInt64(1)
- }
- return z
-}
-
-// affineFromJacobian reverses the Jacobian transform. See the comment at the
-// top of the file. If the point is ∞ it returns 0, 0.
-func (curve *CurveParams) affineFromJacobian(x, y, z *big.Int) (xOut, yOut *big.Int) {
- if z.Sign() == 0 {
- return new(big.Int), new(big.Int)
- }
-
- zinv := new(big.Int).ModInverse(z, curve.P)
- zinvsq := new(big.Int).Mul(zinv, zinv)
-
- xOut = new(big.Int).Mul(x, zinvsq)
- xOut.Mod(xOut, curve.P)
- zinvsq.Mul(zinvsq, zinv)
- yOut = new(big.Int).Mul(y, zinvsq)
- yOut.Mod(yOut, curve.P)
- return
-}
-
-func (curve *CurveParams) Add(x1, y1, x2, y2 *big.Int) (*big.Int, *big.Int) {
- // If there is a dedicated constant-time implementation for this curve operation,
- // use that instead of the generic one.
- if specific, ok := matchesSpecificCurve(curve, p224, p384, p521); ok {
- return specific.Add(x1, y1, x2, y2)
- }
-
- z1 := zForAffine(x1, y1)
- z2 := zForAffine(x2, y2)
- return curve.affineFromJacobian(curve.addJacobian(x1, y1, z1, x2, y2, z2))
-}
-
-// addJacobian takes two points in Jacobian coordinates, (x1, y1, z1) and
-// (x2, y2, z2) and returns their sum, also in Jacobian form.
-func (curve *CurveParams) addJacobian(x1, y1, z1, x2, y2, z2 *big.Int) (*big.Int, *big.Int, *big.Int) {
- // See https://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#addition-add-2007-bl
- x3, y3, z3 := new(big.Int), new(big.Int), new(big.Int)
- if z1.Sign() == 0 {
- x3.Set(x2)
- y3.Set(y2)
- z3.Set(z2)
- return x3, y3, z3
- }
- if z2.Sign() == 0 {
- x3.Set(x1)
- y3.Set(y1)
- z3.Set(z1)
- return x3, y3, z3
- }
-
- z1z1 := new(big.Int).Mul(z1, z1)
- z1z1.Mod(z1z1, curve.P)
- z2z2 := new(big.Int).Mul(z2, z2)
- z2z2.Mod(z2z2, curve.P)
-
- u1 := new(big.Int).Mul(x1, z2z2)
- u1.Mod(u1, curve.P)
- u2 := new(big.Int).Mul(x2, z1z1)
- u2.Mod(u2, curve.P)
- h := new(big.Int).Sub(u2, u1)
- xEqual := h.Sign() == 0
- if h.Sign() == -1 {
- h.Add(h, curve.P)
- }
- i := new(big.Int).Lsh(h, 1)
- i.Mul(i, i)
- j := new(big.Int).Mul(h, i)
-
- s1 := new(big.Int).Mul(y1, z2)
- s1.Mul(s1, z2z2)
- s1.Mod(s1, curve.P)
- s2 := new(big.Int).Mul(y2, z1)
- s2.Mul(s2, z1z1)
- s2.Mod(s2, curve.P)
- r := new(big.Int).Sub(s2, s1)
- if r.Sign() == -1 {
- r.Add(r, curve.P)
- }
- yEqual := r.Sign() == 0
- if xEqual && yEqual {
- return curve.doubleJacobian(x1, y1, z1)
- }
- r.Lsh(r, 1)
- v := new(big.Int).Mul(u1, i)
-
- x3.Set(r)
- x3.Mul(x3, x3)
- x3.Sub(x3, j)
- x3.Sub(x3, v)
- x3.Sub(x3, v)
- x3.Mod(x3, curve.P)
-
- y3.Set(r)
- v.Sub(v, x3)
- y3.Mul(y3, v)
- s1.Mul(s1, j)
- s1.Lsh(s1, 1)
- y3.Sub(y3, s1)
- y3.Mod(y3, curve.P)
-
- z3.Add(z1, z2)
- z3.Mul(z3, z3)
- z3.Sub(z3, z1z1)
- z3.Sub(z3, z2z2)
- z3.Mul(z3, h)
- z3.Mod(z3, curve.P)
-
- return x3, y3, z3
-}
-
-func (curve *CurveParams) Double(x1, y1 *big.Int) (*big.Int, *big.Int) {
- // If there is a dedicated constant-time implementation for this curve operation,
- // use that instead of the generic one.
- if specific, ok := matchesSpecificCurve(curve, p224, p384, p521); ok {
- return specific.Double(x1, y1)
- }
-
- z1 := zForAffine(x1, y1)
- return curve.affineFromJacobian(curve.doubleJacobian(x1, y1, z1))
-}
-
-// doubleJacobian takes a point in Jacobian coordinates, (x, y, z), and
-// returns its double, also in Jacobian form.
-func (curve *CurveParams) doubleJacobian(x, y, z *big.Int) (*big.Int, *big.Int, *big.Int) {
- // See https://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#doubling-dbl-2001-b
- delta := new(big.Int).Mul(z, z)
- delta.Mod(delta, curve.P)
- gamma := new(big.Int).Mul(y, y)
- gamma.Mod(gamma, curve.P)
- alpha := new(big.Int).Sub(x, delta)
- if alpha.Sign() == -1 {
- alpha.Add(alpha, curve.P)
- }
- alpha2 := new(big.Int).Add(x, delta)
- alpha.Mul(alpha, alpha2)
- alpha2.Set(alpha)
- alpha.Lsh(alpha, 1)
- alpha.Add(alpha, alpha2)
-
- beta := alpha2.Mul(x, gamma)
-
- x3 := new(big.Int).Mul(alpha, alpha)
- beta8 := new(big.Int).Lsh(beta, 3)
- beta8.Mod(beta8, curve.P)
- x3.Sub(x3, beta8)
- if x3.Sign() == -1 {
- x3.Add(x3, curve.P)
- }
- x3.Mod(x3, curve.P)
-
- z3 := new(big.Int).Add(y, z)
- z3.Mul(z3, z3)
- z3.Sub(z3, gamma)
- if z3.Sign() == -1 {
- z3.Add(z3, curve.P)
- }
- z3.Sub(z3, delta)
- if z3.Sign() == -1 {
- z3.Add(z3, curve.P)
- }
- z3.Mod(z3, curve.P)
-
- beta.Lsh(beta, 2)
- beta.Sub(beta, x3)
- if beta.Sign() == -1 {
- beta.Add(beta, curve.P)
- }
- y3 := alpha.Mul(alpha, beta)
-
- gamma.Mul(gamma, gamma)
- gamma.Lsh(gamma, 3)
- gamma.Mod(gamma, curve.P)
-
- y3.Sub(y3, gamma)
- if y3.Sign() == -1 {
- y3.Add(y3, curve.P)
- }
- y3.Mod(y3, curve.P)
-
- return x3, y3, z3
-}
-
-func (curve *CurveParams) ScalarMult(Bx, By *big.Int, k []byte) (*big.Int, *big.Int) {
- // If there is a dedicated constant-time implementation for this curve operation,
- // use that instead of the generic one.
- if specific, ok := matchesSpecificCurve(curve, p224, p256, p384, p521); ok {
- return specific.ScalarMult(Bx, By, k)
- }
-
- Bz := new(big.Int).SetInt64(1)
- x, y, z := new(big.Int), new(big.Int), new(big.Int)
-
- for _, byte := range k {
- for bitNum := 0; bitNum < 8; bitNum++ {
- x, y, z = curve.doubleJacobian(x, y, z)
- if byte&0x80 == 0x80 {
- x, y, z = curve.addJacobian(Bx, By, Bz, x, y, z)
- }
- byte <<= 1
- }
- }
-
- return curve.affineFromJacobian(x, y, z)
-}
-
-func (curve *CurveParams) ScalarBaseMult(k []byte) (*big.Int, *big.Int) {
- // If there is a dedicated constant-time implementation for this curve operation,
- // use that instead of the generic one.
- if specific, ok := matchesSpecificCurve(curve, p224, p256, p384, p521); ok {
- return specific.ScalarBaseMult(k)
- }
-
- return curve.ScalarMult(curve.Gx, curve.Gy, k)
-}
-
var mask = []byte{0xff, 0x1, 0x3, 0x7, 0xf, 0x1f, 0x3f, 0x7f}
// GenerateKey returns a public/private key pair. The private key is
diff --git a/src/crypto/elliptic/p256.go b/src/crypto/elliptic/p256.go
index 763b84283e..97ecda5a8e 100644
--- a/src/crypto/elliptic/p256.go
+++ b/src/crypto/elliptic/p256.go
@@ -1,28 +1,19 @@
-// Copyright 2013 The Go Authors. All rights reserved.
+// Copyright 2021 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.
-//go:build !amd64 && !arm64
-
package elliptic
// P-256 is implemented by various different backends, including a generic
-// 32-bit constant-time one in this file, which is used when assembly
+// 32-bit constant-time one in p256_generic.go, which is used when assembly
// implementations are not available, or not appropriate for the hardware.
import "math/big"
-type p256Curve struct {
- *CurveParams
-}
-
-var (
- p256Params *CurveParams
+var p256Params *CurveParams
- // RInverse contains 1/R mod p - the inverse of the Montgomery constant
- // (2**257).
- p256RInverse *big.Int
-)
+// RInverse contains 1/R mod p, the inverse of the Montgomery constant 2^257.
+var p256RInverse *big.Int
func initP256() {
// See FIPS 186-3, section D.2.3
@@ -39,1162 +30,3 @@ func initP256() {
// Arch-specific initialization, i.e. let a platform dynamically pick a P256 implementation
initP256Arch()
}
-
-func (curve p256Curve) Params() *CurveParams {
- return curve.CurveParams
-}
-
-// 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 *[32]byte, in []byte) {
- n := new(big.Int).SetBytes(in)
- var scalarBytes []byte
-
- if n.Cmp(p256Params.N) >= 0 || len(in) > len(out) {
- n.Mod(n, p256Params.N)
- scalarBytes = n.Bytes()
- } else {
- scalarBytes = in
- }
-
- for i, v := range scalarBytes {
- out[len(scalarBytes)-(1+i)] = v
- }
-}
-
-func (p256Curve) ScalarBaseMult(scalar []byte) (x, y *big.Int) {
- var scalarReversed [32]byte
- p256GetScalar(&scalarReversed, scalar)
-
- var x1, y1, z1 [p256Limbs]uint32
- p256ScalarBaseMult(&x1, &y1, &z1, &scalarReversed)
- return p256ToAffine(&x1, &y1, &z1)
-}
-
-func (p256Curve) ScalarMult(bigX, bigY *big.Int, scalar []byte) (x, y *big.Int) {
- var scalarReversed [32]byte
- p256GetScalar(&scalarReversed, scalar)
-
- var px, py, x1, y1, z1 [p256Limbs]uint32
- p256FromBig(&px, bigX)
- p256FromBig(&py, bigY)
- p256ScalarMult(&x1, &y1, &z1, &px, &py, &scalarReversed)
- return p256ToAffine(&x1, &y1, &z1)
-}
-
-// Field elements are represented as nine, unsigned 32-bit words.
-//
-// The value of a field element is:
-// x[0] + (x[1] * 2**29) + (x[2] * 2**57) + ... + (x[8] * 2**228)
-//
-// That is, each limb is alternately 29 or 28-bits wide in little-endian
-// order.
-//
-// This means that a field element hits 2**257, rather than 2**256 as we would
-// like. A 28, 29, ... pattern would cause us to hit 2**256, but that causes
-// problems when multiplying as terms end up one bit short of a limb which
-// would require much bit-shifting to correct.
-//
-// Finally, the values stored in a field element are in Montgomery form. So the
-// value |y| is stored as (y*R) mod p, where p is the P-256 prime and R is
-// 2**257.
-
-const (
- p256Limbs = 9
- bottom29Bits = 0x1fffffff
-)
-
-var (
- // p256One is the number 1 as a field element.
- p256One = [p256Limbs]uint32{2, 0, 0, 0xffff800, 0x1fffffff, 0xfffffff, 0x1fbfffff, 0x1ffffff, 0}
- p256Zero = [p256Limbs]uint32{0, 0, 0, 0, 0, 0, 0, 0, 0}
- // p256P is the prime modulus as a field element.
- p256P = [p256Limbs]uint32{0x1fffffff, 0xfffffff, 0x1fffffff, 0x3ff, 0, 0, 0x200000, 0xf000000, 0xfffffff}
- // p2562P is the twice prime modulus as a field element.
- p2562P = [p256Limbs]uint32{0x1ffffffe, 0xfffffff, 0x1fffffff, 0x7ff, 0, 0, 0x400000, 0xe000000, 0x1fffffff}
-)
-
-// p256Precomputed contains precomputed values to aid the calculation of scalar
-// multiples of the base point, G. It's actually two, equal length, tables
-// concatenated.
-//
-// The first table contains (x,y) field element pairs for 16 multiples of the
-// base point, G.
-//
-// Index | Index (binary) | Value
-// 0 | 0000 | 0G (all zeros, omitted)
-// 1 | 0001 | G
-// 2 | 0010 | 2**64G
-// 3 | 0011 | 2**64G + G
-// 4 | 0100 | 2**128G
-// 5 | 0101 | 2**128G + G
-// 6 | 0110 | 2**128G + 2**64G
-// 7 | 0111 | 2**128G + 2**64G + G
-// 8 | 1000 | 2**192G
-// 9 | 1001 | 2**192G + G
-// 10 | 1010 | 2**192G + 2**64G
-// 11 | 1011 | 2**192G + 2**64G + G
-// 12 | 1100 | 2**192G + 2**128G
-// 13 | 1101 | 2**192G + 2**128G + G
-// 14 | 1110 | 2**192G + 2**128G + 2**64G
-// 15 | 1111 | 2**192G + 2**128G + 2**64G + G
-//
-// The second table follows the same style, but the terms are 2**32G,
-// 2**96G, 2**160G, 2**224G.
-//
-// This is ~2KB of data.
-var p256Precomputed = [p256Limbs * 2 * 15 * 2]uint32{
- 0x11522878, 0xe730d41, 0xdb60179, 0x4afe2ff, 0x12883add, 0xcaddd88, 0x119e7edc, 0xd4a6eab, 0x3120bee,
- 0x1d2aac15, 0xf25357c, 0x19e45cdd, 0x5c721d0, 0x1992c5a5, 0xa237487, 0x154ba21, 0x14b10bb, 0xae3fe3,
- 0xd41a576, 0x922fc51, 0x234994f, 0x60b60d3, 0x164586ae, 0xce95f18, 0x1fe49073, 0x3fa36cc, 0x5ebcd2c,
- 0xb402f2f, 0x15c70bf, 0x1561925c, 0x5a26704, 0xda91e90, 0xcdc1c7f, 0x1ea12446, 0xe1ade1e, 0xec91f22,
- 0x26f7778, 0x566847e, 0xa0bec9e, 0x234f453, 0x1a31f21a, 0xd85e75c, 0x56c7109, 0xa267a00, 0xb57c050,
- 0x98fb57, 0xaa837cc, 0x60c0792, 0xcfa5e19, 0x61bab9e, 0x589e39b, 0xa324c5, 0x7d6dee7, 0x2976e4b,
- 0x1fc4124a, 0xa8c244b, 0x1ce86762, 0xcd61c7e, 0x1831c8e0, 0x75774e1, 0x1d96a5a9, 0x843a649, 0xc3ab0fa,
- 0x6e2e7d5, 0x7673a2a, 0x178b65e8, 0x4003e9b, 0x1a1f11c2, 0x7816ea, 0xf643e11, 0x58c43df, 0xf423fc2,
- 0x19633ffa, 0x891f2b2, 0x123c231c, 0x46add8c, 0x54700dd, 0x59e2b17, 0x172db40f, 0x83e277d, 0xb0dd609,
- 0xfd1da12, 0x35c6e52, 0x19ede20c, 0xd19e0c0, 0x97d0f40, 0xb015b19, 0x449e3f5, 0xe10c9e, 0x33ab581,
- 0x56a67ab, 0x577734d, 0x1dddc062, 0xc57b10d, 0x149b39d, 0x26a9e7b, 0xc35df9f, 0x48764cd, 0x76dbcca,
- 0xca4b366, 0xe9303ab, 0x1a7480e7, 0x57e9e81, 0x1e13eb50, 0xf466cf3, 0x6f16b20, 0x4ba3173, 0xc168c33,
- 0x15cb5439, 0x6a38e11, 0x73658bd, 0xb29564f, 0x3f6dc5b, 0x53b97e, 0x1322c4c0, 0x65dd7ff, 0x3a1e4f6,
- 0x14e614aa, 0x9246317, 0x1bc83aca, 0xad97eed, 0xd38ce4a, 0xf82b006, 0x341f077, 0xa6add89, 0x4894acd,
- 0x9f162d5, 0xf8410ef, 0x1b266a56, 0xd7f223, 0x3e0cb92, 0xe39b672, 0x6a2901a, 0x69a8556, 0x7e7c0,
- 0x9b7d8d3, 0x309a80, 0x1ad05f7f, 0xc2fb5dd, 0xcbfd41d, 0x9ceb638, 0x1051825c, 0xda0cf5b, 0x812e881,
- 0x6f35669, 0x6a56f2c, 0x1df8d184, 0x345820, 0x1477d477, 0x1645db1, 0xbe80c51, 0xc22be3e, 0xe35e65a,
- 0x1aeb7aa0, 0xc375315, 0xf67bc99, 0x7fdd7b9, 0x191fc1be, 0x61235d, 0x2c184e9, 0x1c5a839, 0x47a1e26,
- 0xb7cb456, 0x93e225d, 0x14f3c6ed, 0xccc1ac9, 0x17fe37f3, 0x4988989, 0x1a90c502, 0x2f32042, 0xa17769b,
- 0xafd8c7c, 0x8191c6e, 0x1dcdb237, 0x16200c0, 0x107b32a1, 0x66c08db, 0x10d06a02, 0x3fc93, 0x5620023,
- 0x16722b27, 0x68b5c59, 0x270fcfc, 0xfad0ecc, 0xe5de1c2, 0xeab466b, 0x2fc513c, 0x407f75c, 0xbaab133,
- 0x9705fe9, 0xb88b8e7, 0x734c993, 0x1e1ff8f, 0x19156970, 0xabd0f00, 0x10469ea7, 0x3293ac0, 0xcdc98aa,
- 0x1d843fd, 0xe14bfe8, 0x15be825f, 0x8b5212, 0xeb3fb67, 0x81cbd29, 0xbc62f16, 0x2b6fcc7, 0xf5a4e29,
- 0x13560b66, 0xc0b6ac2, 0x51ae690, 0xd41e271, 0xf3e9bd4, 0x1d70aab, 0x1029f72, 0x73e1c35, 0xee70fbc,
- 0xad81baf, 0x9ecc49a, 0x86c741e, 0xfe6be30, 0x176752e7, 0x23d416, 0x1f83de85, 0x27de188, 0x66f70b8,
- 0x181cd51f, 0x96b6e4c, 0x188f2335, 0xa5df759, 0x17a77eb6, 0xfeb0e73, 0x154ae914, 0x2f3ec51, 0x3826b59,
- 0xb91f17d, 0x1c72949, 0x1362bf0a, 0xe23fddf, 0xa5614b0, 0xf7d8f, 0x79061, 0x823d9d2, 0x8213f39,
- 0x1128ae0b, 0xd095d05, 0xb85c0c2, 0x1ecb2ef, 0x24ddc84, 0xe35e901, 0x18411a4a, 0xf5ddc3d, 0x3786689,
- 0x52260e8, 0x5ae3564, 0x542b10d, 0x8d93a45, 0x19952aa4, 0x996cc41, 0x1051a729, 0x4be3499, 0x52b23aa,
- 0x109f307e, 0x6f5b6bb, 0x1f84e1e7, 0x77a0cfa, 0x10c4df3f, 0x25a02ea, 0xb048035, 0xe31de66, 0xc6ecaa3,
- 0x28ea335, 0x2886024, 0x1372f020, 0xf55d35, 0x15e4684c, 0xf2a9e17, 0x1a4a7529, 0xcb7beb1, 0xb2a78a1,
- 0x1ab21f1f, 0x6361ccf, 0x6c9179d, 0xb135627, 0x1267b974, 0x4408bad, 0x1cbff658, 0xe3d6511, 0xc7d76f,
- 0x1cc7a69, 0xe7ee31b, 0x54fab4f, 0x2b914f, 0x1ad27a30, 0xcd3579e, 0xc50124c, 0x50daa90, 0xb13f72,
- 0xb06aa75, 0x70f5cc6, 0x1649e5aa, 0x84a5312, 0x329043c, 0x41c4011, 0x13d32411, 0xb04a838, 0xd760d2d,
- 0x1713b532, 0xbaa0c03, 0x84022ab, 0x6bcf5c1, 0x2f45379, 0x18ae070, 0x18c9e11e, 0x20bca9a, 0x66f496b,
- 0x3eef294, 0x67500d2, 0xd7f613c, 0x2dbbeb, 0xb741038, 0xe04133f, 0x1582968d, 0xbe985f7, 0x1acbc1a,
- 0x1a6a939f, 0x33e50f6, 0xd665ed4, 0xb4b7bd6, 0x1e5a3799, 0x6b33847, 0x17fa56ff, 0x65ef930, 0x21dc4a,
- 0x2b37659, 0x450fe17, 0xb357b65, 0xdf5efac, 0x15397bef, 0x9d35a7f, 0x112ac15f, 0x624e62e, 0xa90ae2f,
- 0x107eecd2, 0x1f69bbe, 0x77d6bce, 0x5741394, 0x13c684fc, 0x950c910, 0x725522b, 0xdc78583, 0x40eeabb,
- 0x1fde328a, 0xbd61d96, 0xd28c387, 0x9e77d89, 0x12550c40, 0x759cb7d, 0x367ef34, 0xae2a960, 0x91b8bdc,
- 0x93462a9, 0xf469ef, 0xb2e9aef, 0xd2ca771, 0x54e1f42, 0x7aaa49, 0x6316abb, 0x2413c8e, 0x5425bf9,
- 0x1bed3e3a, 0xf272274, 0x1f5e7326, 0x6416517, 0xea27072, 0x9cedea7, 0x6e7633, 0x7c91952, 0xd806dce,
- 0x8e2a7e1, 0xe421e1a, 0x418c9e1, 0x1dbc890, 0x1b395c36, 0xa1dc175, 0x1dc4ef73, 0x8956f34, 0xe4b5cf2,
- 0x1b0d3a18, 0x3194a36, 0x6c2641f, 0xe44124c, 0xa2f4eaa, 0xa8c25ba, 0xf927ed7, 0x627b614, 0x7371cca,
- 0xba16694, 0x417bc03, 0x7c0a7e3, 0x9c35c19, 0x1168a205, 0x8b6b00d, 0x10e3edc9, 0x9c19bf2, 0x5882229,
- 0x1b2b4162, 0xa5cef1a, 0x1543622b, 0x9bd433e, 0x364e04d, 0x7480792, 0x5c9b5b3, 0xe85ff25, 0x408ef57,
- 0x1814cfa4, 0x121b41b, 0xd248a0f, 0x3b05222, 0x39bb16a, 0xc75966d, 0xa038113, 0xa4a1769, 0x11fbc6c,
- 0x917e50e, 0xeec3da8, 0x169d6eac, 0x10c1699, 0xa416153, 0xf724912, 0x15cd60b7, 0x4acbad9, 0x5efc5fa,
- 0xf150ed7, 0x122b51, 0x1104b40a, 0xcb7f442, 0xfbb28ff, 0x6ac53ca, 0x196142cc, 0x7bf0fa9, 0x957651,
- 0x4e0f215, 0xed439f8, 0x3f46bd5, 0x5ace82f, 0x110916b6, 0x6db078, 0xffd7d57, 0xf2ecaac, 0xca86dec,
- 0x15d6b2da, 0x965ecc9, 0x1c92b4c2, 0x1f3811, 0x1cb080f5, 0x2d8b804, 0x19d1c12d, 0xf20bd46, 0x1951fa7,
- 0xa3656c3, 0x523a425, 0xfcd0692, 0xd44ddc8, 0x131f0f5b, 0xaf80e4a, 0xcd9fc74, 0x99bb618, 0x2db944c,
- 0xa673090, 0x1c210e1, 0x178c8d23, 0x1474383, 0x10b8743d, 0x985a55b, 0x2e74779, 0x576138, 0x9587927,
- 0x133130fa, 0xbe05516, 0x9f4d619, 0xbb62570, 0x99ec591, 0xd9468fe, 0x1d07782d, 0xfc72e0b, 0x701b298,
- 0x1863863b, 0x85954b8, 0x121a0c36, 0x9e7fedf, 0xf64b429, 0x9b9d71e, 0x14e2f5d8, 0xf858d3a, 0x942eea8,
- 0xda5b765, 0x6edafff, 0xa9d18cc, 0xc65e4ba, 0x1c747e86, 0xe4ea915, 0x1981d7a1, 0x8395659, 0x52ed4e2,
- 0x87d43b7, 0x37ab11b, 0x19d292ce, 0xf8d4692, 0x18c3053f, 0x8863e13, 0x4c146c0, 0x6bdf55a, 0x4e4457d,
- 0x16152289, 0xac78ec2, 0x1a59c5a2, 0x2028b97, 0x71c2d01, 0x295851f, 0x404747b, 0x878558d, 0x7d29aa4,
- 0x13d8341f, 0x8daefd7, 0x139c972d, 0x6b7ea75, 0xd4a9dde, 0xff163d8, 0x81d55d7, 0xa5bef68, 0xb7b30d8,
- 0xbe73d6f, 0xaa88141, 0xd976c81, 0x7e7a9cc, 0x18beb771, 0xd773cbd, 0x13f51951, 0x9d0c177, 0x1c49a78,
-}
-
-// Field element operations:
-
-const bottom28Bits = 0xfffffff
-
-// nonZeroToAllOnes returns:
-//
-// 0xffffffff for 0 < x <= 2**31
-// 0 for x == 0 or x > 2**31.
-func nonZeroToAllOnes(x uint32) uint32 {
- return ((x - 1) >> 31) - 1
-}
-
-// p256ReduceCarry adds a multiple of p in order to cancel |carry|,
-// which is a term at 2**257.
-//
-// On entry: carry < 2**3, inout[0,2,...] < 2**29, inout[1,3,...] < 2**28.
-// On exit: inout[0,2,..] < 2**30, inout[1,3,...] < 2**29.
-func p256ReduceCarry(inout *[p256Limbs]uint32, carry uint32) {
- carry_mask := nonZeroToAllOnes(carry)
-
- inout[0] += carry << 1
- inout[3] += 0x10000000 & carry_mask
- // carry < 2**3 thus (carry << 11) < 2**14 and we added 2**28 in the
- // previous line therefore this doesn't underflow.
- inout[3] -= carry << 11
- inout[4] += (0x20000000 - 1) & carry_mask
- inout[5] += (0x10000000 - 1) & carry_mask
- inout[6] += (0x20000000 - 1) & carry_mask
- inout[6] -= carry << 22
- // This may underflow if carry is non-zero but, if so, we'll fix it in the
- // next line.
- inout[7] -= 1 & carry_mask
- inout[7] += carry << 25
-}
-
-// p256Sum sets out = in+in2.
-//
-// On entry, in[i]+in2[i] must not overflow a 32-bit word.
-// On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29
-func p256Sum(out, in, in2 *[p256Limbs]uint32) {
- carry := uint32(0)
- for i := 0; ; i++ {
- out[i] = in[i] + in2[i]
- out[i] += carry
- carry = out[i] >> 29
- out[i] &= bottom29Bits
-
- i++
- if i == p256Limbs {
- break
- }
-
- out[i] = in[i] + in2[i]
- out[i] += carry
- carry = out[i] >> 28
- out[i] &= bottom28Bits
- }
-
- p256ReduceCarry(out, carry)
-}
-
-const (
- two30m2 = 1<<30 - 1<<2
- two30p13m2 = 1<<30 + 1<<13 - 1<<2
- two31m2 = 1<<31 - 1<<2
- two31m3 = 1<<31 - 1<<3
- two31p24m2 = 1<<31 + 1<<24 - 1<<2
- two30m27m2 = 1<<30 - 1<<27 - 1<<2
-)
-
-// p256Zero31 is 0 mod p.
-var p256Zero31 = [p256Limbs]uint32{two31m3, two30m2, two31m2, two30p13m2, two31m2, two30m2, two31p24m2, two30m27m2, two31m2}
-
-// p256Diff sets out = in-in2.
-//
-// On entry: in[0,2,...] < 2**30, in[1,3,...] < 2**29 and
-// in2[0,2,...] < 2**30, in2[1,3,...] < 2**29.
-//
-// On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
-func p256Diff(out, in, in2 *[p256Limbs]uint32) {
- var carry uint32
-
- for i := 0; ; i++ {
- out[i] = in[i] - in2[i]
- out[i] += p256Zero31[i]
- out[i] += carry
- carry = out[i] >> 29
- out[i] &= bottom29Bits
-
- i++
- if i == p256Limbs {
- break
- }
-
- out[i] = in[i] - in2[i]
- out[i] += p256Zero31[i]
- out[i] += carry
- carry = out[i] >> 28
- out[i] &= bottom28Bits
- }
-
- p256ReduceCarry(out, carry)
-}
-
-// p256ReduceDegree sets out = tmp/R mod p where tmp contains 64-bit words with
-// the same 29,28,... bit positions as a field element.
-//
-// The values in field elements are in Montgomery form: x*R mod p where R =
-// 2**257. Since we just multiplied two Montgomery values together, the result
-// is x*y*R*R mod p. We wish to divide by R in order for the result also to be
-// in Montgomery form.
-//
-// On entry: tmp[i] < 2**64
-// On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29
-func p256ReduceDegree(out *[p256Limbs]uint32, tmp [17]uint64) {
- // The following table may be helpful when reading this code:
- //
- // Limb number: 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10...
- // Width (bits): 29| 28| 29| 28| 29| 28| 29| 28| 29| 28| 29
- // Start bit: 0 | 29| 57| 86|114|143|171|200|228|257|285
- // (odd phase): 0 | 28| 57| 85|114|142|171|199|228|256|285
- var tmp2 [18]uint32
- var carry, x, xMask uint32
-
- // tmp contains 64-bit words with the same 29,28,29-bit positions as a
- // field element. So the top of an element of tmp might overlap with
- // another element two positions down. The following loop eliminates
- // this overlap.
- tmp2[0] = uint32(tmp[0]) & bottom29Bits
-
- tmp2[1] = uint32(tmp[0]) >> 29
- tmp2[1] |= (uint32(tmp[0]>>32) << 3) & bottom28Bits
- tmp2[1] += uint32(tmp[1]) & bottom28Bits
- carry = tmp2[1] >> 28
- tmp2[1] &= bottom28Bits
-
- for i := 2; i < 17; i++ {
- tmp2[i] = (uint32(tmp[i-2] >> 32)) >> 25
- tmp2[i] += (uint32(tmp[i-1])) >> 28
- tmp2[i] += (uint32(tmp[i-1]>>32) << 4) & bottom29Bits
- tmp2[i] += uint32(tmp[i]) & bottom29Bits
- tmp2[i] += carry
- carry = tmp2[i] >> 29
- tmp2[i] &= bottom29Bits
-
- i++
- if i == 17 {
- break
- }
- tmp2[i] = uint32(tmp[i-2]>>32) >> 25
- tmp2[i] += uint32(tmp[i-1]) >> 29
- tmp2[i] += ((uint32(tmp[i-1] >> 32)) << 3) & bottom28Bits
- tmp2[i] += uint32(tmp[i]) & bottom28Bits
- tmp2[i] += carry
- carry = tmp2[i] >> 28
- tmp2[i] &= bottom28Bits
- }
-
- tmp2[17] = uint32(tmp[15]>>32) >> 25
- tmp2[17] += uint32(tmp[16]) >> 29
- tmp2[17] += uint32(tmp[16]>>32) << 3
- tmp2[17] += carry
-
- // Montgomery elimination of terms:
- //
- // Since R is 2**257, we can divide by R with a bitwise shift if we can
- // ensure that the right-most 257 bits are all zero. We can make that true
- // by adding multiplies of p without affecting the value.
- //
- // So we eliminate limbs from right to left. Since the bottom 29 bits of p
- // are all ones, then by adding tmp2[0]*p to tmp2 we'll make tmp2[0] == 0.
- // We can do that for 8 further limbs and then right shift to eliminate the
- // extra factor of R.
- for i := 0; ; i += 2 {
- tmp2[i+1] += tmp2[i] >> 29
- x = tmp2[i] & bottom29Bits
- xMask = nonZeroToAllOnes(x)
- tmp2[i] = 0
-
- // The bounds calculations for this loop are tricky. Each iteration of
- // the loop eliminates two words by adding values to words to their
- // right.
- //
- // The following table contains the amounts added to each word (as an
- // offset from the value of i at the top of the loop). The amounts are
- // accounted for from the first and second half of the loop separately
- // and are written as, for example, 28 to mean a value <2**28.
- //
- // Word: 3 4 5 6 7 8 9 10
- // Added in top half: 28 11 29 21 29 28
- // 28 29
- // 29
- // Added in bottom half: 29 10 28 21 28 28
- // 29
- //
- // The value that is currently offset 7 will be offset 5 for the next
- // iteration and then offset 3 for the iteration after that. Therefore
- // the total value added will be the values added at 7, 5 and 3.
- //
- // The following table accumulates these values. The sums at the bottom
- // are written as, for example, 29+28, to mean a value < 2**29+2**28.
- //
- // Word: 3 4 5 6 7 8 9 10 11 12 13
- // 28 11 10 29 21 29 28 28 28 28 28
- // 29 28 11 28 29 28 29 28 29 28
- // 29 28 21 21 29 21 29 21
- // 10 29 28 21 28 21 28
- // 28 29 28 29 28 29 28
- // 11 10 29 10 29 10
- // 29 28 11 28 11
- // 29 29
- // --------------------------------------------
- // 30+ 31+ 30+ 31+ 30+
- // 28+ 29+ 28+ 29+ 21+
- // 21+ 28+ 21+ 28+ 10
- // 10 21+ 10 21+
- // 11 11
- //
- // So the greatest amount is added to tmp2[10] and tmp2[12]. If
- // tmp2[10/12] has an initial value of <2**29, then the maximum value
- // will be < 2**31 + 2**30 + 2**28 + 2**21 + 2**11, which is < 2**32,
- // as required.
- tmp2[i+3] += (x << 10) & bottom28Bits
- tmp2[i+4] += (x >> 18)
-
- tmp2[i+6] += (x << 21) & bottom29Bits
- tmp2[i+7] += x >> 8
-
- // At position 200, which is the starting bit position for word 7, we
- // have a factor of 0xf000000 = 2**28 - 2**24.
- tmp2[i+7] += 0x10000000 & xMask
- tmp2[i+8] += (x - 1) & xMask
- tmp2[i+7] -= (x << 24) & bottom28Bits
- tmp2[i+8] -= x >> 4
-
- tmp2[i+8] += 0x20000000 & xMask
- tmp2[i+8] -= x
- tmp2[i+8] += (x << 28) & bottom29Bits
- tmp2[i+9] += ((x >> 1) - 1) & xMask
-
- if i+1 == p256Limbs {
- break
- }
- tmp2[i+2] += tmp2[i+1] >> 28
- x = tmp2[i+1] & bottom28Bits
- xMask = nonZeroToAllOnes(x)
- tmp2[i+1] = 0
-
- tmp2[i+4] += (x << 11) & bottom29Bits
- tmp2[i+5] += (x >> 18)
-
- tmp2[i+7] += (x << 21) & bottom28Bits
- tmp2[i+8] += x >> 7
-
- // At position 199, which is the starting bit of the 8th word when
- // dealing with a context starting on an odd word, we have a factor of
- // 0x1e000000 = 2**29 - 2**25. Since we have not updated i, the 8th
- // word from i+1 is i+8.
- tmp2[i+8] += 0x20000000 & xMask
- tmp2[i+9] += (x - 1) & xMask
- tmp2[i+8] -= (x << 25) & bottom29Bits
- tmp2[i+9] -= x >> 4
-
- tmp2[i+9] += 0x10000000 & xMask
- tmp2[i+9] -= x
- tmp2[i+10] += (x - 1) & xMask
- }
-
- // We merge the right shift with a carry chain. The words above 2**257 have
- // widths of 28,29,... which we need to correct when copying them down.
- carry = 0
- for i := 0; i < 8; i++ {
- // The maximum value of tmp2[i + 9] occurs on the first iteration and
- // is < 2**30+2**29+2**28. Adding 2**29 (from tmp2[i + 10]) is
- // therefore safe.
- out[i] = tmp2[i+9]
- out[i] += carry
- out[i] += (tmp2[i+10] << 28) & bottom29Bits
- carry = out[i] >> 29
- out[i] &= bottom29Bits
-
- i++
- out[i] = tmp2[i+9] >> 1
- out[i] += carry
- carry = out[i] >> 28
- out[i] &= bottom28Bits
- }
-
- out[8] = tmp2[17]
- out[8] += carry
- carry = out[8] >> 29
- out[8] &= bottom29Bits
-
- p256ReduceCarry(out, carry)
-}
-
-// p256Square sets out=in*in.
-//
-// On entry: in[0,2,...] < 2**30, in[1,3,...] < 2**29.
-// On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
-func p256Square(out, in *[p256Limbs]uint32) {
- var tmp [17]uint64
-
- tmp[0] = uint64(in[0]) * uint64(in[0])
- tmp[1] = uint64(in[0]) * (uint64(in[1]) << 1)
- tmp[2] = uint64(in[0])*(uint64(in[2])<<1) +
- uint64(in[1])*(uint64(in[1])<<1)
- tmp[3] = uint64(in[0])*(uint64(in[3])<<1) +
- uint64(in[1])*(uint64(in[2])<<1)
- tmp[4] = uint64(in[0])*(uint64(in[4])<<1) +
- uint64(in[1])*(uint64(in[3])<<2) +
- uint64(in[2])*uint64(in[2])
- tmp[5] = uint64(in[0])*(uint64(in[5])<<1) +
- uint64(in[1])*(uint64(in[4])<<1) +
- uint64(in[2])*(uint64(in[3])<<1)
- tmp[6] = uint64(in[0])*(uint64(in[6])<<1) +
- uint64(in[1])*(uint64(in[5])<<2) +
- uint64(in[2])*(uint64(in[4])<<1) +
- uint64(in[3])*(uint64(in[3])<<1)
- tmp[7] = uint64(in[0])*(uint64(in[7])<<1) +
- uint64(in[1])*(uint64(in[6])<<1) +
- uint64(in[2])*(uint64(in[5])<<1) +
- uint64(in[3])*(uint64(in[4])<<1)
- // tmp[8] has the greatest value of 2**61 + 2**60 + 2**61 + 2**60 + 2**60,
- // which is < 2**64 as required.
- tmp[8] = uint64(in[0])*(uint64(in[8])<<1) +
- uint64(in[1])*(uint64(in[7])<<2) +
- uint64(in[2])*(uint64(in[6])<<1) +
- uint64(in[3])*(uint64(in[5])<<2) +
- uint64(in[4])*uint64(in[4])
- tmp[9] = uint64(in[1])*(uint64(in[8])<<1) +
- uint64(in[2])*(uint64(in[7])<<1) +
- uint64(in[3])*(uint64(in[6])<<1) +
- uint64(in[4])*(uint64(in[5])<<1)
- tmp[10] = uint64(in[2])*(uint64(in[8])<<1) +
- uint64(in[3])*(uint64(in[7])<<2) +
- uint64(in[4])*(uint64(in[6])<<1) +
- uint64(in[5])*(uint64(in[5])<<1)
- tmp[11] = uint64(in[3])*(uint64(in[8])<<1) +
- uint64(in[4])*(uint64(in[7])<<1) +
- uint64(in[5])*(uint64(in[6])<<1)
- tmp[12] = uint64(in[4])*(uint64(in[8])<<1) +
- uint64(in[5])*(uint64(in[7])<<2) +
- uint64(in[6])*uint64(in[6])
- tmp[13] = uint64(in[5])*(uint64(in[8])<<1) +
- uint64(in[6])*(uint64(in[7])<<1)
- tmp[14] = uint64(in[6])*(uint64(in[8])<<1) +
- uint64(in[7])*(uint64(in[7])<<1)
- tmp[15] = uint64(in[7]) * (uint64(in[8]) << 1)
- tmp[16] = uint64(in[8]) * uint64(in[8])
-
- p256ReduceDegree(out, tmp)
-}
-
-// p256Mul sets out=in*in2.
-//
-// On entry: in[0,2,...] < 2**30, in[1,3,...] < 2**29 and
-//
-// in2[0,2,...] < 2**30, in2[1,3,...] < 2**29.
-//
-// On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
-func p256Mul(out, in, in2 *[p256Limbs]uint32) {
- var tmp [17]uint64
-
- tmp[0] = uint64(in[0]) * uint64(in2[0])
- tmp[1] = uint64(in[0])*(uint64(in2[1])<<0) +
- uint64(in[1])*(uint64(in2[0])<<0)
- tmp[2] = uint64(in[0])*(uint64(in2[2])<<0) +
- uint64(in[1])*(uint64(in2[1])<<1) +
- uint64(in[2])*(uint64(in2[0])<<0)
- tmp[3] = uint64(in[0])*(uint64(in2[3])<<0) +
- uint64(in[1])*(uint64(in2[2])<<0) +
- uint64(in[2])*(uint64(in2[1])<<0) +
- uint64(in[3])*(uint64(in2[0])<<0)
- tmp[4] = uint64(in[0])*(uint64(in2[4])<<0) +
- uint64(in[1])*(uint64(in2[3])<<1) +
- uint64(in[2])*(uint64(in2[2])<<0) +
- uint64(in[3])*(uint64(in2[1])<<1) +
- uint64(in[4])*(uint64(in2[0])<<0)
- tmp[5] = uint64(in[0])*(uint64(in2[5])<<0) +
- uint64(in[1])*(uint64(in2[4])<<0) +
- uint64(in[2])*(uint64(in2[3])<<0) +
- uint64(in[3])*(uint64(in2[2])<<0) +
- uint64(in[4])*(uint64(in2[1])<<0) +
- uint64(in[5])*(uint64(in2[0])<<0)
- tmp[6] = uint64(in[0])*(uint64(in2[6])<<0) +
- uint64(in[1])*(uint64(in2[5])<<1) +
- uint64(in[2])*(uint64(in2[4])<<0) +
- uint64(in[3])*(uint64(in2[3])<<1) +
- uint64(in[4])*(uint64(in2[2])<<0) +
- uint64(in[5])*(uint64(in2[1])<<1) +
- uint64(in[6])*(uint64(in2[0])<<0)
- tmp[7] = uint64(in[0])*(uint64(in2[7])<<0) +
- uint64(in[1])*(uint64(in2[6])<<0) +
- uint64(in[2])*(uint64(in2[5])<<0) +
- uint64(in[3])*(uint64(in2[4])<<0) +
- uint64(in[4])*(uint64(in2[3])<<0) +
- uint64(in[5])*(uint64(in2[2])<<0) +
- uint64(in[6])*(uint64(in2[1])<<0) +
- uint64(in[7])*(uint64(in2[0])<<0)
- // tmp[8] has the greatest value but doesn't overflow. See logic in
- // p256Square.
- tmp[8] = uint64(in[0])*(uint64(in2[8])<<0) +
- uint64(in[1])*(uint64(in2[7])<<1) +
- uint64(in[2])*(uint64(in2[6])<<0) +
- uint64(in[3])*(uint64(in2[5])<<1) +
- uint64(in[4])*(uint64(in2[4])<<0) +
- uint64(in[5])*(uint64(in2[3])<<1) +
- uint64(in[6])*(uint64(in2[2])<<0) +
- uint64(in[7])*(uint64(in2[1])<<1) +
- uint64(in[8])*(uint64(in2[0])<<0)
- tmp[9] = uint64(in[1])*(uint64(in2[8])<<0) +
- uint64(in[2])*(uint64(in2[7])<<0) +
- uint64(in[3])*(uint64(in2[6])<<0) +
- uint64(in[4])*(uint64(in2[5])<<0) +
- uint64(in[5])*(uint64(in2[4])<<0) +
- uint64(in[6])*(uint64(in2[3])<<0) +
- uint64(in[7])*(uint64(in2[2])<<0) +
- uint64(in[8])*(uint64(in2[1])<<0)
- tmp[10] = uint64(in[2])*(uint64(in2[8])<<0) +
- uint64(in[3])*(uint64(in2[7])<<1) +
- uint64(in[4])*(uint64(in2[6])<<0) +
- uint64(in[5])*(uint64(in2[5])<<1) +
- uint64(in[6])*(uint64(in2[4])<<0) +
- uint64(in[7])*(uint64(in2[3])<<1) +
- uint64(in[8])*(uint64(in2[2])<<0)
- tmp[11] = uint64(in[3])*(uint64(in2[8])<<0) +
- uint64(in[4])*(uint64(in2[7])<<0) +
- uint64(in[5])*(uint64(in2[6])<<0) +
- uint64(in[6])*(uint64(in2[5])<<0) +
- uint64(in[7])*(uint64(in2[4])<<0) +
- uint64(in[8])*(uint64(in2[3])<<0)
- tmp[12] = uint64(in[4])*(uint64(in2[8])<<0) +
- uint64(in[5])*(uint64(in2[7])<<1) +
- uint64(in[6])*(uint64(in2[6])<<0) +
- uint64(in[7])*(uint64(in2[5])<<1) +
- uint64(in[8])*(uint64(in2[4])<<0)
- tmp[13] = uint64(in[5])*(uint64(in2[8])<<0) +
- uint64(in[6])*(uint64(in2[7])<<0) +
- uint64(in[7])*(uint64(in2[6])<<0) +
- uint64(in[8])*(uint64(in2[5])<<0)
- tmp[14] = uint64(in[6])*(uint64(in2[8])<<0) +
- uint64(in[7])*(uint64(in2[7])<<1) +
- uint64(in[8])*(uint64(in2[6])<<0)
- tmp[15] = uint64(in[7])*(uint64(in2[8])<<0) +
- uint64(in[8])*(uint64(in2[7])<<0)
- tmp[16] = uint64(in[8]) * (uint64(in2[8]) << 0)
-
- p256ReduceDegree(out, tmp)
-}
-
-func p256Assign(out, in *[p256Limbs]uint32) {
- *out = *in
-}
-
-// p256Invert calculates |out| = |in|^{-1}
-//
-// Based on Fermat's Little Theorem:
-//
-// a^p = a (mod p)
-// a^{p-1} = 1 (mod p)
-// a^{p-2} = a^{-1} (mod p)
-func p256Invert(out, in *[p256Limbs]uint32) {
- var ftmp, ftmp2 [p256Limbs]uint32
-
- // each e_I will hold |in|^{2^I - 1}
- var e2, e4, e8, e16, e32, e64 [p256Limbs]uint32
-
- p256Square(&ftmp, in) // 2^1
- p256Mul(&ftmp, in, &ftmp) // 2^2 - 2^0
- p256Assign(&e2, &ftmp)
- p256Square(&ftmp, &ftmp) // 2^3 - 2^1
- p256Square(&ftmp, &ftmp) // 2^4 - 2^2
- p256Mul(&ftmp, &ftmp, &e2) // 2^4 - 2^0
- p256Assign(&e4, &ftmp)
- p256Square(&ftmp, &ftmp) // 2^5 - 2^1
- p256Square(&ftmp, &ftmp) // 2^6 - 2^2
- p256Square(&ftmp, &ftmp) // 2^7 - 2^3
- p256Square(&ftmp, &ftmp) // 2^8 - 2^4
- p256Mul(&ftmp, &ftmp, &e4) // 2^8 - 2^0
- p256Assign(&e8, &ftmp)
- for i := 0; i < 8; i++ {
- p256Square(&ftmp, &ftmp)
- } // 2^16 - 2^8
- p256Mul(&ftmp, &ftmp, &e8) // 2^16 - 2^0
- p256Assign(&e16, &ftmp)
- for i := 0; i < 16; i++ {
- p256Square(&ftmp, &ftmp)
- } // 2^32 - 2^16
- p256Mul(&ftmp, &ftmp, &e16) // 2^32 - 2^0
- p256Assign(&e32, &ftmp)
- for i := 0; i < 32; i++ {
- p256Square(&ftmp, &ftmp)
- } // 2^64 - 2^32
- p256Assign(&e64, &ftmp)
- p256Mul(&ftmp, &ftmp, in) // 2^64 - 2^32 + 2^0
- for i := 0; i < 192; i++ {
- p256Square(&ftmp, &ftmp)
- } // 2^256 - 2^224 + 2^192
-
- p256Mul(&ftmp2, &e64, &e32) // 2^64 - 2^0
- for i := 0; i < 16; i++ {
- p256Square(&ftmp2, &ftmp2)
- } // 2^80 - 2^16
- p256Mul(&ftmp2, &ftmp2, &e16) // 2^80 - 2^0
- for i := 0; i < 8; i++ {
- p256Square(&ftmp2, &ftmp2)
- } // 2^88 - 2^8
- p256Mul(&ftmp2, &ftmp2, &e8) // 2^88 - 2^0
- for i := 0; i < 4; i++ {
- p256Square(&ftmp2, &ftmp2)
- } // 2^92 - 2^4
- p256Mul(&ftmp2, &ftmp2, &e4) // 2^92 - 2^0
- p256Square(&ftmp2, &ftmp2) // 2^93 - 2^1
- p256Square(&ftmp2, &ftmp2) // 2^94 - 2^2
- p256Mul(&ftmp2, &ftmp2, &e2) // 2^94 - 2^0
- p256Square(&ftmp2, &ftmp2) // 2^95 - 2^1
- p256Square(&ftmp2, &ftmp2) // 2^96 - 2^2
- p256Mul(&ftmp2, &ftmp2, in) // 2^96 - 3
-
- p256Mul(out, &ftmp2, &ftmp) // 2^256 - 2^224 + 2^192 + 2^96 - 3
-}
-
-// p256Scalar3 sets out=3*out.
-//
-// On entry: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
-// On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
-func p256Scalar3(out *[p256Limbs]uint32) {
- var carry uint32
-
- for i := 0; ; i++ {
- out[i] *= 3
- out[i] += carry
- carry = out[i] >> 29
- out[i] &= bottom29Bits
-
- i++
- if i == p256Limbs {
- break
- }
-
- out[i] *= 3
- out[i] += carry
- carry = out[i] >> 28
- out[i] &= bottom28Bits
- }
-
- p256ReduceCarry(out, carry)
-}
-
-// p256Scalar4 sets out=4*out.
-//
-// On entry: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
-// On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
-func p256Scalar4(out *[p256Limbs]uint32) {
- var carry, nextCarry uint32
-
- for i := 0; ; i++ {
- nextCarry = out[i] >> 27
- out[i] <<= 2
- out[i] &= bottom29Bits
- out[i] += carry
- carry = nextCarry + (out[i] >> 29)
- out[i] &= bottom29Bits
-
- i++
- if i == p256Limbs {
- break
- }
- nextCarry = out[i] >> 26
- out[i] <<= 2
- out[i] &= bottom28Bits
- out[i] += carry
- carry = nextCarry + (out[i] >> 28)
- out[i] &= bottom28Bits
- }
-
- p256ReduceCarry(out, carry)
-}
-
-// p256Scalar8 sets out=8*out.
-//
-// On entry: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
-// On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
-func p256Scalar8(out *[p256Limbs]uint32) {
- var carry, nextCarry uint32
-
- for i := 0; ; i++ {
- nextCarry = out[i] >> 26
- out[i] <<= 3
- out[i] &= bottom29Bits
- out[i] += carry
- carry = nextCarry + (out[i] >> 29)
- out[i] &= bottom29Bits
-
- i++
- if i == p256Limbs {
- break
- }
- nextCarry = out[i] >> 25
- out[i] <<= 3
- out[i] &= bottom28Bits
- out[i] += carry
- carry = nextCarry + (out[i] >> 28)
- out[i] &= bottom28Bits
- }
-
- p256ReduceCarry(out, carry)
-}
-
-// Group operations:
-//
-// Elements of the elliptic curve group are represented in Jacobian
-// coordinates: (x, y, z). An affine point (x', y') is x'=x/z**2, y'=y/z**3 in
-// Jacobian form.
-
-// p256PointDouble sets {xOut,yOut,zOut} = 2*{x,y,z}.
-//
-// See https://www.hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#doubling-dbl-2009-l
-func p256PointDouble(xOut, yOut, zOut, x, y, z *[p256Limbs]uint32) {
- var delta, gamma, alpha, beta, tmp, tmp2 [p256Limbs]uint32
-
- p256Square(&delta, z)
- p256Square(&gamma, y)
- p256Mul(&beta, x, &gamma)
-
- p256Sum(&tmp, x, &delta)
- p256Diff(&tmp2, x, &delta)
- p256Mul(&alpha, &tmp, &tmp2)
- p256Scalar3(&alpha)
-
- p256Sum(&tmp, y, z)
- p256Square(&tmp, &tmp)
- p256Diff(&tmp, &tmp, &gamma)
- p256Diff(zOut, &tmp, &delta)
-
- p256Scalar4(&beta)
- p256Square(xOut, &alpha)
- p256Diff(xOut, xOut, &beta)
- p256Diff(xOut, xOut, &beta)
-
- p256Diff(&tmp, &beta, xOut)
- p256Mul(&tmp, &alpha, &tmp)
- p256Square(&tmp2, &gamma)
- p256Scalar8(&tmp2)
- p256Diff(yOut, &tmp, &tmp2)
-}
-
-// p256PointAddMixed sets {xOut,yOut,zOut} = {x1,y1,z1} + {x2,y2,1}.
-// (i.e. the second point is affine.)
-//
-// See https://www.hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#addition-add-2007-bl
-//
-// Note that this function does not handle P+P, infinity+P nor P+infinity
-// correctly.
-func p256PointAddMixed(xOut, yOut, zOut, x1, y1, z1, x2, y2 *[p256Limbs]uint32) {
- var z1z1, z1z1z1, s2, u2, h, i, j, r, rr, v, tmp [p256Limbs]uint32
-
- p256Square(&z1z1, z1)
- p256Sum(&tmp, z1, z1)
-
- p256Mul(&u2, x2, &z1z1)
- p256Mul(&z1z1z1, z1, &z1z1)
- p256Mul(&s2, y2, &z1z1z1)
- p256Diff(&h, &u2, x1)
- p256Sum(&i, &h, &h)
- p256Square(&i, &i)
- p256Mul(&j, &h, &i)
- p256Diff(&r, &s2, y1)
- p256Sum(&r, &r, &r)
- p256Mul(&v, x1, &i)
-
- p256Mul(zOut, &tmp, &h)
- p256Square(&rr, &r)
- p256Diff(xOut, &rr, &j)
- p256Diff(xOut, xOut, &v)
- p256Diff(xOut, xOut, &v)
-
- p256Diff(&tmp, &v, xOut)
- p256Mul(yOut, &tmp, &r)
- p256Mul(&tmp, y1, &j)
- p256Diff(yOut, yOut, &tmp)
- p256Diff(yOut, yOut, &tmp)
-}
-
-// p256PointAdd sets {xOut,yOut,zOut} = {x1,y1,z1} + {x2,y2,z2}.
-//
-// See https://www.hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#addition-add-2007-bl
-//
-// Note that this function does not handle P+P, infinity+P nor P+infinity
-// correctly.
-func p256PointAdd(xOut, yOut, zOut, x1, y1, z1, x2, y2, z2 *[p256Limbs]uint32) {
- var z1z1, z1z1z1, z2z2, z2z2z2, s1, s2, u1, u2, h, i, j, r, rr, v, tmp [p256Limbs]uint32
-
- p256Square(&z1z1, z1)
- p256Square(&z2z2, z2)
- p256Mul(&u1, x1, &z2z2)
-
- p256Sum(&tmp, z1, z2)
- p256Square(&tmp, &tmp)
- p256Diff(&tmp, &tmp, &z1z1)
- p256Diff(&tmp, &tmp, &z2z2)
-
- p256Mul(&z2z2z2, z2, &z2z2)
- p256Mul(&s1, y1, &z2z2z2)
-
- p256Mul(&u2, x2, &z1z1)
- p256Mul(&z1z1z1, z1, &z1z1)
- p256Mul(&s2, y2, &z1z1z1)
- p256Diff(&h, &u2, &u1)
- p256Sum(&i, &h, &h)
- p256Square(&i, &i)
- p256Mul(&j, &h, &i)
- p256Diff(&r, &s2, &s1)
- p256Sum(&r, &r, &r)
- p256Mul(&v, &u1, &i)
-
- p256Mul(zOut, &tmp, &h)
- p256Square(&rr, &r)
- p256Diff(xOut, &rr, &j)
- p256Diff(xOut, xOut, &v)
- p256Diff(xOut, xOut, &v)
-
- p256Diff(&tmp, &v, xOut)
- p256Mul(yOut, &tmp, &r)
- p256Mul(&tmp, &s1, &j)
- p256Diff(yOut, yOut, &tmp)
- p256Diff(yOut, yOut, &tmp)
-}
-
-// p256CopyConditional sets out=in if mask = 0xffffffff in constant time.
-//
-// On entry: mask is either 0 or 0xffffffff.
-func p256CopyConditional(out, in *[p256Limbs]uint32, mask uint32) {
- for i := 0; i < p256Limbs; i++ {
- tmp := mask & (in[i] ^ out[i])
- out[i] ^= tmp
- }
-}
-
-// p256SelectAffinePoint sets {out_x,out_y} to the index'th entry of table.
-// On entry: index < 16, table[0] must be zero.
-func p256SelectAffinePoint(xOut, yOut *[p256Limbs]uint32, table []uint32, index uint32) {
- for i := range xOut {
- xOut[i] = 0
- }
- for i := range yOut {
- yOut[i] = 0
- }
-
- for i := uint32(1); i < 16; i++ {
- mask := i ^ index
- mask |= mask >> 2
- mask |= mask >> 1
- mask &= 1
- mask--
- for j := range xOut {
- xOut[j] |= table[0] & mask
- table = table[1:]
- }
- for j := range yOut {
- yOut[j] |= table[0] & mask
- table = table[1:]
- }
- }
-}
-
-// p256SelectJacobianPoint sets {out_x,out_y,out_z} to the index'th entry of
-// table.
-// On entry: index < 16, table[0] must be zero.
-func p256SelectJacobianPoint(xOut, yOut, zOut *[p256Limbs]uint32, table *[16][3][p256Limbs]uint32, index uint32) {
- for i := range xOut {
- xOut[i] = 0
- }
- for i := range yOut {
- yOut[i] = 0
- }
- for i := range zOut {
- zOut[i] = 0
- }
-
- // The implicit value at index 0 is all zero. We don't need to perform that
- // iteration of the loop because we already set out_* to zero.
- for i := uint32(1); i < 16; i++ {
- mask := i ^ index
- mask |= mask >> 2
- mask |= mask >> 1
- mask &= 1
- mask--
- for j := range xOut {
- xOut[j] |= table[i][0][j] & mask
- }
- for j := range yOut {
- yOut[j] |= table[i][1][j] & mask
- }
- for j := range zOut {
- zOut[j] |= table[i][2][j] & mask
- }
- }
-}
-
-// p256GetBit returns the bit'th bit of scalar.
-func p256GetBit(scalar *[32]uint8, bit uint) uint32 {
- return uint32(((scalar[bit>>3]) >> (bit & 7)) & 1)
-}
-
-// p256ScalarBaseMult sets {xOut,yOut,zOut} = scalar*G where scalar is a
-// little-endian number. Note that the value of scalar must be less than the
-// order of the group.
-func p256ScalarBaseMult(xOut, yOut, zOut *[p256Limbs]uint32, scalar *[32]uint8) {
- nIsInfinityMask := ^uint32(0)
- var pIsNoninfiniteMask, mask, tableOffset uint32
- var px, py, tx, ty, tz [p256Limbs]uint32
-
- for i := range xOut {
- xOut[i] = 0
- }
- for i := range yOut {
- yOut[i] = 0
- }
- for i := range zOut {
- zOut[i] = 0
- }
-
- // The loop adds bits at positions 0, 64, 128 and 192, followed by
- // positions 32,96,160 and 224 and does this 32 times.
- for i := uint(0); i < 32; i++ {
- if i != 0 {
- p256PointDouble(xOut, yOut, zOut, xOut, yOut, zOut)
- }
- tableOffset = 0
- for j := uint(0); j <= 32; j += 32 {
- bit0 := p256GetBit(scalar, 31-i+j)
- bit1 := p256GetBit(scalar, 95-i+j)
- bit2 := p256GetBit(scalar, 159-i+j)
- bit3 := p256GetBit(scalar, 223-i+j)
- index := bit0 | (bit1 << 1) | (bit2 << 2) | (bit3 << 3)
-
- p256SelectAffinePoint(&px, &py, p256Precomputed[tableOffset:], index)
- tableOffset += 30 * p256Limbs
-
- // Since scalar is less than the order of the group, we know that
- // {xOut,yOut,zOut} != {px,py,1}, unless both are zero, which we handle
- // below.
- p256PointAddMixed(&tx, &ty, &tz, xOut, yOut, zOut, &px, &py)
- // The result of pointAddMixed is incorrect if {xOut,yOut,zOut} is zero
- // (a.k.a. the point at infinity). We handle that situation by
- // copying the point from the table.
- p256CopyConditional(xOut, &px, nIsInfinityMask)
- p256CopyConditional(yOut, &py, nIsInfinityMask)
- p256CopyConditional(zOut, &p256One, nIsInfinityMask)
-
- // Equally, the result is also wrong if the point from the table is
- // zero, which happens when the index is zero. We handle that by
- // only copying from {tx,ty,tz} to {xOut,yOut,zOut} if index != 0.
- pIsNoninfiniteMask = nonZeroToAllOnes(index)
- mask = pIsNoninfiniteMask & ^nIsInfinityMask
- p256CopyConditional(xOut, &tx, mask)
- p256CopyConditional(yOut, &ty, mask)
- p256CopyConditional(zOut, &tz, mask)
- // If p was not zero, then n is now non-zero.
- nIsInfinityMask &^= pIsNoninfiniteMask
- }
- }
-}
-
-// p256PointToAffine converts a Jacobian point to an affine point. If the input
-// is the point at infinity then it returns (0, 0) in constant time.
-func p256PointToAffine(xOut, yOut, x, y, z *[p256Limbs]uint32) {
- var zInv, zInvSq [p256Limbs]uint32
-
- p256Invert(&zInv, z)
- p256Square(&zInvSq, &zInv)
- p256Mul(xOut, x, &zInvSq)
- p256Mul(&zInv, &zInv, &zInvSq)
- p256Mul(yOut, y, &zInv)
-}
-
-// p256ToAffine returns a pair of *big.Int containing the affine representation
-// of {x,y,z}.
-func p256ToAffine(x, y, z *[p256Limbs]uint32) (xOut, yOut *big.Int) {
- var xx, yy [p256Limbs]uint32
- p256PointToAffine(&xx, &yy, x, y, z)
- return p256ToBig(&xx), p256ToBig(&yy)
-}
-
-// p256ScalarMult sets {xOut,yOut,zOut} = scalar*{x,y}.
-func p256ScalarMult(xOut, yOut, zOut, x, y *[p256Limbs]uint32, scalar *[32]uint8) {
- var px, py, pz, tx, ty, tz [p256Limbs]uint32
- var precomp [16][3][p256Limbs]uint32
- var nIsInfinityMask, index, pIsNoninfiniteMask, mask uint32
-
- // We precompute 0,1,2,... times {x,y}.
- precomp[1][0] = *x
- precomp[1][1] = *y
- precomp[1][2] = p256One
-
- for i := 2; i < 16; i += 2 {
- p256PointDouble(&precomp[i][0], &precomp[i][1], &precomp[i][2], &precomp[i/2][0], &precomp[i/2][1], &precomp[i/2][2])
- p256PointAddMixed(&precomp[i+1][0], &precomp[i+1][1], &precomp[i+1][2], &precomp[i][0], &precomp[i][1], &precomp[i][2], x, y)
- }
-
- for i := range xOut {
- xOut[i] = 0
- }
- for i := range yOut {
- yOut[i] = 0
- }
- for i := range zOut {
- zOut[i] = 0
- }
- nIsInfinityMask = ^uint32(0)
-
- // We add in a window of four bits each iteration and do this 64 times.
- for i := 0; i < 64; i++ {
- if i != 0 {
- p256PointDouble(xOut, yOut, zOut, xOut, yOut, zOut)
- p256PointDouble(xOut, yOut, zOut, xOut, yOut, zOut)
- p256PointDouble(xOut, yOut, zOut, xOut, yOut, zOut)
- p256PointDouble(xOut, yOut, zOut, xOut, yOut, zOut)
- }
-
- index = uint32(scalar[31-i/2])
- if (i & 1) == 1 {
- index &= 15
- } else {
- index >>= 4
- }
-
- // See the comments in scalarBaseMult about handling infinities.
- p256SelectJacobianPoint(&px, &py, &pz, &precomp, index)
- p256PointAdd(&tx, &ty, &tz, xOut, yOut, zOut, &px, &py, &pz)
- p256CopyConditional(xOut, &px, nIsInfinityMask)
- p256CopyConditional(yOut, &py, nIsInfinityMask)
- p256CopyConditional(zOut, &pz, nIsInfinityMask)
-
- pIsNoninfiniteMask = nonZeroToAllOnes(index)
- mask = pIsNoninfiniteMask & ^nIsInfinityMask
- p256CopyConditional(xOut, &tx, mask)
- p256CopyConditional(yOut, &ty, mask)
- p256CopyConditional(zOut, &tz, mask)
- nIsInfinityMask &^= pIsNoninfiniteMask
- }
-}
-
-// p256FromBig sets out = R*in.
-func p256FromBig(out *[p256Limbs]uint32, in *big.Int) {
- tmp := new(big.Int).Lsh(in, 257)
- tmp.Mod(tmp, p256Params.P)
-
- for i := 0; i < p256Limbs; i++ {
- if bits := tmp.Bits(); len(bits) > 0 {
- out[i] = uint32(bits[0]) & bottom29Bits
- } else {
- out[i] = 0
- }
- tmp.Rsh(tmp, 29)
-
- i++
- if i == p256Limbs {
- break
- }
-
- if bits := tmp.Bits(); len(bits) > 0 {
- out[i] = uint32(bits[0]) & bottom28Bits
- } else {
- out[i] = 0
- }
- tmp.Rsh(tmp, 28)
- }
-}
-
-// p256ToBig returns a *big.Int containing the value of in.
-func p256ToBig(in *[p256Limbs]uint32) *big.Int {
- result, tmp := new(big.Int), new(big.Int)
-
- result.SetInt64(int64(in[p256Limbs-1]))
- for i := p256Limbs - 2; i >= 0; i-- {
- if (i & 1) == 0 {
- result.Lsh(result, 29)
- } else {
- result.Lsh(result, 28)
- }
- tmp.SetInt64(int64(in[i]))
- result.Add(result, tmp)
- }
-
- result.Mul(result, p256RInverse)
- result.Mod(result, p256Params.P)
- return result
-}
diff --git a/src/crypto/elliptic/p256_asm.go b/src/crypto/elliptic/p256_asm.go
index 93adaf9056..ce80282ed6 100644
--- a/src/crypto/elliptic/p256_asm.go
+++ b/src/crypto/elliptic/p256_asm.go
@@ -24,27 +24,18 @@ import (
//go:embed p256_asm_table.bin
var p256Precomputed string
-type (
- p256Curve struct {
- *CurveParams
- }
+type p256Curve struct {
+ *CurveParams
+}
- p256Point struct {
- xyz [12]uint64
- }
-)
+type p256Point struct {
+ xyz [12]uint64
+}
var p256 p256Curve
-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 initP256Arch() {
+ p256 = p256Curve{p256Params}
}
func (curve p256Curve) Params() *CurveParams {
diff --git a/src/crypto/elliptic/p256_generic.go b/src/crypto/elliptic/p256_generic.go
index 7f8fab5398..fc105c547c 100644
--- a/src/crypto/elliptic/p256_generic.go
+++ b/src/crypto/elliptic/p256_generic.go
@@ -1,14 +1,1173 @@
-// Copyright 2016 The Go Authors. All rights reserved.
+// Copyright 2013 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.
-//go:build !amd64 && !s390x && !arm64 && !ppc64le
+//go:build !amd64 && !arm64
package elliptic
-var p256 p256Curve
+// This file contains a constant-time, 32-bit implementation of P256.
-func initP256Arch() {
- // Use pure Go implementation.
- p256 = p256Curve{p256Params}
+import "math/big"
+
+type p256Curve struct {
+ *CurveParams
+}
+
+func (curve p256Curve) Params() *CurveParams {
+ return curve.CurveParams
+}
+
+// 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 *[32]byte, in []byte) {
+ n := new(big.Int).SetBytes(in)
+ var scalarBytes []byte
+
+ if n.Cmp(p256Params.N) >= 0 || len(in) > len(out) {
+ n.Mod(n, p256Params.N)
+ scalarBytes = n.Bytes()
+ } else {
+ scalarBytes = in
+ }
+
+ for i, v := range scalarBytes {
+ out[len(scalarBytes)-(1+i)] = v
+ }
+}
+
+func (p256Curve) ScalarBaseMult(scalar []byte) (x, y *big.Int) {
+ var scalarReversed [32]byte
+ p256GetScalar(&scalarReversed, scalar)
+
+ var x1, y1, z1 [p256Limbs]uint32
+ p256ScalarBaseMult(&x1, &y1, &z1, &scalarReversed)
+ return p256ToAffine(&x1, &y1, &z1)
+}
+
+func (p256Curve) ScalarMult(bigX, bigY *big.Int, scalar []byte) (x, y *big.Int) {
+ var scalarReversed [32]byte
+ p256GetScalar(&scalarReversed, scalar)
+
+ var px, py, x1, y1, z1 [p256Limbs]uint32
+ p256FromBig(&px, bigX)
+ p256FromBig(&py, bigY)
+ p256ScalarMult(&x1, &y1, &z1, &px, &py, &scalarReversed)
+ return p256ToAffine(&x1, &y1, &z1)
+}
+
+// Field elements are represented as nine, unsigned 32-bit words.
+//
+// The value of a field element is:
+// x[0] + (x[1] * 2**29) + (x[2] * 2**57) + ... + (x[8] * 2**228)
+//
+// That is, each limb is alternately 29 or 28-bits wide in little-endian
+// order.
+//
+// This means that a field element hits 2**257, rather than 2**256 as we would
+// like. A 28, 29, ... pattern would cause us to hit 2**256, but that causes
+// problems when multiplying as terms end up one bit short of a limb which
+// would require much bit-shifting to correct.
+//
+// Finally, the values stored in a field element are in Montgomery form. So the
+// value |y| is stored as (y*R) mod p, where p is the P-256 prime and R is
+// 2**257.
+
+const (
+ p256Limbs = 9
+ bottom29Bits = 0x1fffffff
+)
+
+var (
+ // p256One is the number 1 as a field element.
+ p256One = [p256Limbs]uint32{2, 0, 0, 0xffff800, 0x1fffffff, 0xfffffff, 0x1fbfffff, 0x1ffffff, 0}
+ p256Zero = [p256Limbs]uint32{0, 0, 0, 0, 0, 0, 0, 0, 0}
+ // p256P is the prime modulus as a field element.
+ p256P = [p256Limbs]uint32{0x1fffffff, 0xfffffff, 0x1fffffff, 0x3ff, 0, 0, 0x200000, 0xf000000, 0xfffffff}
+ // p2562P is the twice prime modulus as a field element.
+ p2562P = [p256Limbs]uint32{0x1ffffffe, 0xfffffff, 0x1fffffff, 0x7ff, 0, 0, 0x400000, 0xe000000, 0x1fffffff}
+)
+
+// p256Precomputed contains precomputed values to aid the calculation of scalar
+// multiples of the base point, G. It's actually two, equal length, tables
+// concatenated.
+//
+// The first table contains (x,y) field element pairs for 16 multiples of the
+// base point, G.
+//
+// Index | Index (binary) | Value
+// 0 | 0000 | 0G (all zeros, omitted)
+// 1 | 0001 | G
+// 2 | 0010 | 2**64G
+// 3 | 0011 | 2**64G + G
+// 4 | 0100 | 2**128G
+// 5 | 0101 | 2**128G + G
+// 6 | 0110 | 2**128G + 2**64G
+// 7 | 0111 | 2**128G + 2**64G + G
+// 8 | 1000 | 2**192G
+// 9 | 1001 | 2**192G + G
+// 10 | 1010 | 2**192G + 2**64G
+// 11 | 1011 | 2**192G + 2**64G + G
+// 12 | 1100 | 2**192G + 2**128G
+// 13 | 1101 | 2**192G + 2**128G + G
+// 14 | 1110 | 2**192G + 2**128G + 2**64G
+// 15 | 1111 | 2**192G + 2**128G + 2**64G + G
+//
+// The second table follows the same style, but the terms are 2**32G,
+// 2**96G, 2**160G, 2**224G.
+//
+// This is ~2KB of data.
+var p256Precomputed = [p256Limbs * 2 * 15 * 2]uint32{
+ 0x11522878, 0xe730d41, 0xdb60179, 0x4afe2ff, 0x12883add, 0xcaddd88, 0x119e7edc, 0xd4a6eab, 0x3120bee,
+ 0x1d2aac15, 0xf25357c, 0x19e45cdd, 0x5c721d0, 0x1992c5a5, 0xa237487, 0x154ba21, 0x14b10bb, 0xae3fe3,
+ 0xd41a576, 0x922fc51, 0x234994f, 0x60b60d3, 0x164586ae, 0xce95f18, 0x1fe49073, 0x3fa36cc, 0x5ebcd2c,
+ 0xb402f2f, 0x15c70bf, 0x1561925c, 0x5a26704, 0xda91e90, 0xcdc1c7f, 0x1ea12446, 0xe1ade1e, 0xec91f22,
+ 0x26f7778, 0x566847e, 0xa0bec9e, 0x234f453, 0x1a31f21a, 0xd85e75c, 0x56c7109, 0xa267a00, 0xb57c050,
+ 0x98fb57, 0xaa837cc, 0x60c0792, 0xcfa5e19, 0x61bab9e, 0x589e39b, 0xa324c5, 0x7d6dee7, 0x2976e4b,
+ 0x1fc4124a, 0xa8c244b, 0x1ce86762, 0xcd61c7e, 0x1831c8e0, 0x75774e1, 0x1d96a5a9, 0x843a649, 0xc3ab0fa,
+ 0x6e2e7d5, 0x7673a2a, 0x178b65e8, 0x4003e9b, 0x1a1f11c2, 0x7816ea, 0xf643e11, 0x58c43df, 0xf423fc2,
+ 0x19633ffa, 0x891f2b2, 0x123c231c, 0x46add8c, 0x54700dd, 0x59e2b17, 0x172db40f, 0x83e277d, 0xb0dd609,
+ 0xfd1da12, 0x35c6e52, 0x19ede20c, 0xd19e0c0, 0x97d0f40, 0xb015b19, 0x449e3f5, 0xe10c9e, 0x33ab581,
+ 0x56a67ab, 0x577734d, 0x1dddc062, 0xc57b10d, 0x149b39d, 0x26a9e7b, 0xc35df9f, 0x48764cd, 0x76dbcca,
+ 0xca4b366, 0xe9303ab, 0x1a7480e7, 0x57e9e81, 0x1e13eb50, 0xf466cf3, 0x6f16b20, 0x4ba3173, 0xc168c33,
+ 0x15cb5439, 0x6a38e11, 0x73658bd, 0xb29564f, 0x3f6dc5b, 0x53b97e, 0x1322c4c0, 0x65dd7ff, 0x3a1e4f6,
+ 0x14e614aa, 0x9246317, 0x1bc83aca, 0xad97eed, 0xd38ce4a, 0xf82b006, 0x341f077, 0xa6add89, 0x4894acd,
+ 0x9f162d5, 0xf8410ef, 0x1b266a56, 0xd7f223, 0x3e0cb92, 0xe39b672, 0x6a2901a, 0x69a8556, 0x7e7c0,
+ 0x9b7d8d3, 0x309a80, 0x1ad05f7f, 0xc2fb5dd, 0xcbfd41d, 0x9ceb638, 0x1051825c, 0xda0cf5b, 0x812e881,
+ 0x6f35669, 0x6a56f2c, 0x1df8d184, 0x345820, 0x1477d477, 0x1645db1, 0xbe80c51, 0xc22be3e, 0xe35e65a,
+ 0x1aeb7aa0, 0xc375315, 0xf67bc99, 0x7fdd7b9, 0x191fc1be, 0x61235d, 0x2c184e9, 0x1c5a839, 0x47a1e26,
+ 0xb7cb456, 0x93e225d, 0x14f3c6ed, 0xccc1ac9, 0x17fe37f3, 0x4988989, 0x1a90c502, 0x2f32042, 0xa17769b,
+ 0xafd8c7c, 0x8191c6e, 0x1dcdb237, 0x16200c0, 0x107b32a1, 0x66c08db, 0x10d06a02, 0x3fc93, 0x5620023,
+ 0x16722b27, 0x68b5c59, 0x270fcfc, 0xfad0ecc, 0xe5de1c2, 0xeab466b, 0x2fc513c, 0x407f75c, 0xbaab133,
+ 0x9705fe9, 0xb88b8e7, 0x734c993, 0x1e1ff8f, 0x19156970, 0xabd0f00, 0x10469ea7, 0x3293ac0, 0xcdc98aa,
+ 0x1d843fd, 0xe14bfe8, 0x15be825f, 0x8b5212, 0xeb3fb67, 0x81cbd29, 0xbc62f16, 0x2b6fcc7, 0xf5a4e29,
+ 0x13560b66, 0xc0b6ac2, 0x51ae690, 0xd41e271, 0xf3e9bd4, 0x1d70aab, 0x1029f72, 0x73e1c35, 0xee70fbc,
+ 0xad81baf, 0x9ecc49a, 0x86c741e, 0xfe6be30, 0x176752e7, 0x23d416, 0x1f83de85, 0x27de188, 0x66f70b8,
+ 0x181cd51f, 0x96b6e4c, 0x188f2335, 0xa5df759, 0x17a77eb6, 0xfeb0e73, 0x154ae914, 0x2f3ec51, 0x3826b59,
+ 0xb91f17d, 0x1c72949, 0x1362bf0a, 0xe23fddf, 0xa5614b0, 0xf7d8f, 0x79061, 0x823d9d2, 0x8213f39,
+ 0x1128ae0b, 0xd095d05, 0xb85c0c2, 0x1ecb2ef, 0x24ddc84, 0xe35e901, 0x18411a4a, 0xf5ddc3d, 0x3786689,
+ 0x52260e8, 0x5ae3564, 0x542b10d, 0x8d93a45, 0x19952aa4, 0x996cc41, 0x1051a729, 0x4be3499, 0x52b23aa,
+ 0x109f307e, 0x6f5b6bb, 0x1f84e1e7, 0x77a0cfa, 0x10c4df3f, 0x25a02ea, 0xb048035, 0xe31de66, 0xc6ecaa3,
+ 0x28ea335, 0x2886024, 0x1372f020, 0xf55d35, 0x15e4684c, 0xf2a9e17, 0x1a4a7529, 0xcb7beb1, 0xb2a78a1,
+ 0x1ab21f1f, 0x6361ccf, 0x6c9179d, 0xb135627, 0x1267b974, 0x4408bad, 0x1cbff658, 0xe3d6511, 0xc7d76f,
+ 0x1cc7a69, 0xe7ee31b, 0x54fab4f, 0x2b914f, 0x1ad27a30, 0xcd3579e, 0xc50124c, 0x50daa90, 0xb13f72,
+ 0xb06aa75, 0x70f5cc6, 0x1649e5aa, 0x84a5312, 0x329043c, 0x41c4011, 0x13d32411, 0xb04a838, 0xd760d2d,
+ 0x1713b532, 0xbaa0c03, 0x84022ab, 0x6bcf5c1, 0x2f45379, 0x18ae070, 0x18c9e11e, 0x20bca9a, 0x66f496b,
+ 0x3eef294, 0x67500d2, 0xd7f613c, 0x2dbbeb, 0xb741038, 0xe04133f, 0x1582968d, 0xbe985f7, 0x1acbc1a,
+ 0x1a6a939f, 0x33e50f6, 0xd665ed4, 0xb4b7bd6, 0x1e5a3799, 0x6b33847, 0x17fa56ff, 0x65ef930, 0x21dc4a,
+ 0x2b37659, 0x450fe17, 0xb357b65, 0xdf5efac, 0x15397bef, 0x9d35a7f, 0x112ac15f, 0x624e62e, 0xa90ae2f,
+ 0x107eecd2, 0x1f69bbe, 0x77d6bce, 0x5741394, 0x13c684fc, 0x950c910, 0x725522b, 0xdc78583, 0x40eeabb,
+ 0x1fde328a, 0xbd61d96, 0xd28c387, 0x9e77d89, 0x12550c40, 0x759cb7d, 0x367ef34, 0xae2a960, 0x91b8bdc,
+ 0x93462a9, 0xf469ef, 0xb2e9aef, 0xd2ca771, 0x54e1f42, 0x7aaa49, 0x6316abb, 0x2413c8e, 0x5425bf9,
+ 0x1bed3e3a, 0xf272274, 0x1f5e7326, 0x6416517, 0xea27072, 0x9cedea7, 0x6e7633, 0x7c91952, 0xd806dce,
+ 0x8e2a7e1, 0xe421e1a, 0x418c9e1, 0x1dbc890, 0x1b395c36, 0xa1dc175, 0x1dc4ef73, 0x8956f34, 0xe4b5cf2,
+ 0x1b0d3a18, 0x3194a36, 0x6c2641f, 0xe44124c, 0xa2f4eaa, 0xa8c25ba, 0xf927ed7, 0x627b614, 0x7371cca,
+ 0xba16694, 0x417bc03, 0x7c0a7e3, 0x9c35c19, 0x1168a205, 0x8b6b00d, 0x10e3edc9, 0x9c19bf2, 0x5882229,
+ 0x1b2b4162, 0xa5cef1a, 0x1543622b, 0x9bd433e, 0x364e04d, 0x7480792, 0x5c9b5b3, 0xe85ff25, 0x408ef57,
+ 0x1814cfa4, 0x121b41b, 0xd248a0f, 0x3b05222, 0x39bb16a, 0xc75966d, 0xa038113, 0xa4a1769, 0x11fbc6c,
+ 0x917e50e, 0xeec3da8, 0x169d6eac, 0x10c1699, 0xa416153, 0xf724912, 0x15cd60b7, 0x4acbad9, 0x5efc5fa,
+ 0xf150ed7, 0x122b51, 0x1104b40a, 0xcb7f442, 0xfbb28ff, 0x6ac53ca, 0x196142cc, 0x7bf0fa9, 0x957651,
+ 0x4e0f215, 0xed439f8, 0x3f46bd5, 0x5ace82f, 0x110916b6, 0x6db078, 0xffd7d57, 0xf2ecaac, 0xca86dec,
+ 0x15d6b2da, 0x965ecc9, 0x1c92b4c2, 0x1f3811, 0x1cb080f5, 0x2d8b804, 0x19d1c12d, 0xf20bd46, 0x1951fa7,
+ 0xa3656c3, 0x523a425, 0xfcd0692, 0xd44ddc8, 0x131f0f5b, 0xaf80e4a, 0xcd9fc74, 0x99bb618, 0x2db944c,
+ 0xa673090, 0x1c210e1, 0x178c8d23, 0x1474383, 0x10b8743d, 0x985a55b, 0x2e74779, 0x576138, 0x9587927,
+ 0x133130fa, 0xbe05516, 0x9f4d619, 0xbb62570, 0x99ec591, 0xd9468fe, 0x1d07782d, 0xfc72e0b, 0x701b298,
+ 0x1863863b, 0x85954b8, 0x121a0c36, 0x9e7fedf, 0xf64b429, 0x9b9d71e, 0x14e2f5d8, 0xf858d3a, 0x942eea8,
+ 0xda5b765, 0x6edafff, 0xa9d18cc, 0xc65e4ba, 0x1c747e86, 0xe4ea915, 0x1981d7a1, 0x8395659, 0x52ed4e2,
+ 0x87d43b7, 0x37ab11b, 0x19d292ce, 0xf8d4692, 0x18c3053f, 0x8863e13, 0x4c146c0, 0x6bdf55a, 0x4e4457d,
+ 0x16152289, 0xac78ec2, 0x1a59c5a2, 0x2028b97, 0x71c2d01, 0x295851f, 0x404747b, 0x878558d, 0x7d29aa4,
+ 0x13d8341f, 0x8daefd7, 0x139c972d, 0x6b7ea75, 0xd4a9dde, 0xff163d8, 0x81d55d7, 0xa5bef68, 0xb7b30d8,
+ 0xbe73d6f, 0xaa88141, 0xd976c81, 0x7e7a9cc, 0x18beb771, 0xd773cbd, 0x13f51951, 0x9d0c177, 0x1c49a78,
+}
+
+// Field element operations:
+
+const bottom28Bits = 0xfffffff
+
+// nonZeroToAllOnes returns:
+//
+// 0xffffffff for 0 < x <= 2**31
+// 0 for x == 0 or x > 2**31.
+func nonZeroToAllOnes(x uint32) uint32 {
+ return ((x - 1) >> 31) - 1
+}
+
+// p256ReduceCarry adds a multiple of p in order to cancel |carry|,
+// which is a term at 2**257.
+//
+// On entry: carry < 2**3, inout[0,2,...] < 2**29, inout[1,3,...] < 2**28.
+// On exit: inout[0,2,..] < 2**30, inout[1,3,...] < 2**29.
+func p256ReduceCarry(inout *[p256Limbs]uint32, carry uint32) {
+ carry_mask := nonZeroToAllOnes(carry)
+
+ inout[0] += carry << 1
+ inout[3] += 0x10000000 & carry_mask
+ // carry < 2**3 thus (carry << 11) < 2**14 and we added 2**28 in the
+ // previous line therefore this doesn't underflow.
+ inout[3] -= carry << 11
+ inout[4] += (0x20000000 - 1) & carry_mask
+ inout[5] += (0x10000000 - 1) & carry_mask
+ inout[6] += (0x20000000 - 1) & carry_mask
+ inout[6] -= carry << 22
+ // This may underflow if carry is non-zero but, if so, we'll fix it in the
+ // next line.
+ inout[7] -= 1 & carry_mask
+ inout[7] += carry << 25
+}
+
+// p256Sum sets out = in+in2.
+//
+// On entry: in[i]+in2[i] must not overflow a 32-bit word.
+// On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
+func p256Sum(out, in, in2 *[p256Limbs]uint32) {
+ carry := uint32(0)
+ for i := 0; ; i++ {
+ out[i] = in[i] + in2[i]
+ out[i] += carry
+ carry = out[i] >> 29
+ out[i] &= bottom29Bits
+
+ i++
+ if i == p256Limbs {
+ break
+ }
+
+ out[i] = in[i] + in2[i]
+ out[i] += carry
+ carry = out[i] >> 28
+ out[i] &= bottom28Bits
+ }
+
+ p256ReduceCarry(out, carry)
+}
+
+const (
+ two30m2 = 1<<30 - 1<<2
+ two30p13m2 = 1<<30 + 1<<13 - 1<<2
+ two31m2 = 1<<31 - 1<<2
+ two31m3 = 1<<31 - 1<<3
+ two31p24m2 = 1<<31 + 1<<24 - 1<<2
+ two30m27m2 = 1<<30 - 1<<27 - 1<<2
+)
+
+// p256Zero31 is 0 mod p.
+var p256Zero31 = [p256Limbs]uint32{two31m3, two30m2, two31m2, two30p13m2, two31m2, two30m2, two31p24m2, two30m27m2, two31m2}
+
+// p256Diff sets out = in-in2.
+//
+// On entry: in[0,2,...] < 2**30, in[1,3,...] < 2**29 and
+// in2[0,2,...] < 2**30, in2[1,3,...] < 2**29.
+// On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
+func p256Diff(out, in, in2 *[p256Limbs]uint32) {
+ var carry uint32
+
+ for i := 0; ; i++ {
+ out[i] = in[i] - in2[i]
+ out[i] += p256Zero31[i]
+ out[i] += carry
+ carry = out[i] >> 29
+ out[i] &= bottom29Bits
+
+ i++
+ if i == p256Limbs {
+ break
+ }
+
+ out[i] = in[i] - in2[i]
+ out[i] += p256Zero31[i]
+ out[i] += carry
+ carry = out[i] >> 28
+ out[i] &= bottom28Bits
+ }
+
+ p256ReduceCarry(out, carry)
+}
+
+// p256ReduceDegree sets out = tmp/R mod p where tmp contains 64-bit words with
+// the same 29,28,... bit positions as a field element.
+//
+// The values in field elements are in Montgomery form: x*R mod p where R =
+// 2**257. Since we just multiplied two Montgomery values together, the result
+// is x*y*R*R mod p. We wish to divide by R in order for the result also to be
+// in Montgomery form.
+//
+// On entry: tmp[i] < 2**64.
+// On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
+func p256ReduceDegree(out *[p256Limbs]uint32, tmp [17]uint64) {
+ // The following table may be helpful when reading this code:
+ //
+ // Limb number: 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10...
+ // Width (bits): 29| 28| 29| 28| 29| 28| 29| 28| 29| 28| 29
+ // Start bit: 0 | 29| 57| 86|114|143|171|200|228|257|285
+ // (odd phase): 0 | 28| 57| 85|114|142|171|199|228|256|285
+ var tmp2 [18]uint32
+ var carry, x, xMask uint32
+
+ // tmp contains 64-bit words with the same 29,28,29-bit positions as a
+ // field element. So the top of an element of tmp might overlap with
+ // another element two positions down. The following loop eliminates
+ // this overlap.
+ tmp2[0] = uint32(tmp[0]) & bottom29Bits
+
+ tmp2[1] = uint32(tmp[0]) >> 29
+ tmp2[1] |= (uint32(tmp[0]>>32) << 3) & bottom28Bits
+ tmp2[1] += uint32(tmp[1]) & bottom28Bits
+ carry = tmp2[1] >> 28
+ tmp2[1] &= bottom28Bits
+
+ for i := 2; i < 17; i++ {
+ tmp2[i] = (uint32(tmp[i-2] >> 32)) >> 25
+ tmp2[i] += (uint32(tmp[i-1])) >> 28
+ tmp2[i] += (uint32(tmp[i-1]>>32) << 4) & bottom29Bits
+ tmp2[i] += uint32(tmp[i]) & bottom29Bits
+ tmp2[i] += carry
+ carry = tmp2[i] >> 29
+ tmp2[i] &= bottom29Bits
+
+ i++
+ if i == 17 {
+ break
+ }
+ tmp2[i] = uint32(tmp[i-2]>>32) >> 25
+ tmp2[i] += uint32(tmp[i-1]) >> 29
+ tmp2[i] += ((uint32(tmp[i-1] >> 32)) << 3) & bottom28Bits
+ tmp2[i] += uint32(tmp[i]) & bottom28Bits
+ tmp2[i] += carry
+ carry = tmp2[i] >> 28
+ tmp2[i] &= bottom28Bits
+ }
+
+ tmp2[17] = uint32(tmp[15]>>32) >> 25
+ tmp2[17] += uint32(tmp[16]) >> 29
+ tmp2[17] += uint32(tmp[16]>>32) << 3
+ tmp2[17] += carry
+
+ // Montgomery elimination of terms:
+ //
+ // Since R is 2**257, we can divide by R with a bitwise shift if we can
+ // ensure that the right-most 257 bits are all zero. We can make that true
+ // by adding multiplies of p without affecting the value.
+ //
+ // So we eliminate limbs from right to left. Since the bottom 29 bits of p
+ // are all ones, then by adding tmp2[0]*p to tmp2 we'll make tmp2[0] == 0.
+ // We can do that for 8 further limbs and then right shift to eliminate the
+ // extra factor of R.
+ for i := 0; ; i += 2 {
+ tmp2[i+1] += tmp2[i] >> 29
+ x = tmp2[i] & bottom29Bits
+ xMask = nonZeroToAllOnes(x)
+ tmp2[i] = 0
+
+ // The bounds calculations for this loop are tricky. Each iteration of
+ // the loop eliminates two words by adding values to words to their
+ // right.
+ //
+ // The following table contains the amounts added to each word (as an
+ // offset from the value of i at the top of the loop). The amounts are
+ // accounted for from the first and second half of the loop separately
+ // and are written as, for example, 28 to mean a value <2**28.
+ //
+ // Word: 3 4 5 6 7 8 9 10
+ // Added in top half: 28 11 29 21 29 28
+ // 28 29
+ // 29
+ // Added in bottom half: 29 10 28 21 28 28
+ // 29
+ //
+ // The value that is currently offset 7 will be offset 5 for the next
+ // iteration and then offset 3 for the iteration after that. Therefore
+ // the total value added will be the values added at 7, 5 and 3.
+ //
+ // The following table accumulates these values. The sums at the bottom
+ // are written as, for example, 29+28, to mean a value < 2**29+2**28.
+ //
+ // Word: 3 4 5 6 7 8 9 10 11 12 13
+ // 28 11 10 29 21 29 28 28 28 28 28
+ // 29 28 11 28 29 28 29 28 29 28
+ // 29 28 21 21 29 21 29 21
+ // 10 29 28 21 28 21 28
+ // 28 29 28 29 28 29 28
+ // 11 10 29 10 29 10
+ // 29 28 11 28 11
+ // 29 29
+ // --------------------------------------------
+ // 30+ 31+ 30+ 31+ 30+
+ // 28+ 29+ 28+ 29+ 21+
+ // 21+ 28+ 21+ 28+ 10
+ // 10 21+ 10 21+
+ // 11 11
+ //
+ // So the greatest amount is added to tmp2[10] and tmp2[12]. If
+ // tmp2[10/12] has an initial value of <2**29, then the maximum value
+ // will be < 2**31 + 2**30 + 2**28 + 2**21 + 2**11, which is < 2**32,
+ // as required.
+ tmp2[i+3] += (x << 10) & bottom28Bits
+ tmp2[i+4] += (x >> 18)
+
+ tmp2[i+6] += (x << 21) & bottom29Bits
+ tmp2[i+7] += x >> 8
+
+ // At position 200, which is the starting bit position for word 7, we
+ // have a factor of 0xf000000 = 2**28 - 2**24.
+ tmp2[i+7] += 0x10000000 & xMask
+ tmp2[i+8] += (x - 1) & xMask
+ tmp2[i+7] -= (x << 24) & bottom28Bits
+ tmp2[i+8] -= x >> 4
+
+ tmp2[i+8] += 0x20000000 & xMask
+ tmp2[i+8] -= x
+ tmp2[i+8] += (x << 28) & bottom29Bits
+ tmp2[i+9] += ((x >> 1) - 1) & xMask
+
+ if i+1 == p256Limbs {
+ break
+ }
+ tmp2[i+2] += tmp2[i+1] >> 28
+ x = tmp2[i+1] & bottom28Bits
+ xMask = nonZeroToAllOnes(x)
+ tmp2[i+1] = 0
+
+ tmp2[i+4] += (x << 11) & bottom29Bits
+ tmp2[i+5] += (x >> 18)
+
+ tmp2[i+7] += (x << 21) & bottom28Bits
+ tmp2[i+8] += x >> 7
+
+ // At position 199, which is the starting bit of the 8th word when
+ // dealing with a context starting on an odd word, we have a factor of
+ // 0x1e000000 = 2**29 - 2**25. Since we have not updated i, the 8th
+ // word from i+1 is i+8.
+ tmp2[i+8] += 0x20000000 & xMask
+ tmp2[i+9] += (x - 1) & xMask
+ tmp2[i+8] -= (x << 25) & bottom29Bits
+ tmp2[i+9] -= x >> 4
+
+ tmp2[i+9] += 0x10000000 & xMask
+ tmp2[i+9] -= x
+ tmp2[i+10] += (x - 1) & xMask
+ }
+
+ // We merge the right shift with a carry chain. The words above 2**257 have
+ // widths of 28,29,... which we need to correct when copying them down.
+ carry = 0
+ for i := 0; i < 8; i++ {
+ // The maximum value of tmp2[i + 9] occurs on the first iteration and
+ // is < 2**30+2**29+2**28. Adding 2**29 (from tmp2[i + 10]) is
+ // therefore safe.
+ out[i] = tmp2[i+9]
+ out[i] += carry
+ out[i] += (tmp2[i+10] << 28) & bottom29Bits
+ carry = out[i] >> 29
+ out[i] &= bottom29Bits
+
+ i++
+ out[i] = tmp2[i+9] >> 1
+ out[i] += carry
+ carry = out[i] >> 28
+ out[i] &= bottom28Bits
+ }
+
+ out[8] = tmp2[17]
+ out[8] += carry
+ carry = out[8] >> 29
+ out[8] &= bottom29Bits
+
+ p256ReduceCarry(out, carry)
+}
+
+// p256Square sets out=in*in.
+//
+// On entry: in[0,2,...] < 2**30, in[1,3,...] < 2**29.
+// On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
+func p256Square(out, in *[p256Limbs]uint32) {
+ var tmp [17]uint64
+
+ tmp[0] = uint64(in[0]) * uint64(in[0])
+ tmp[1] = uint64(in[0]) * (uint64(in[1]) << 1)
+ tmp[2] = uint64(in[0])*(uint64(in[2])<<1) +
+ uint64(in[1])*(uint64(in[1])<<1)
+ tmp[3] = uint64(in[0])*(uint64(in[3])<<1) +
+ uint64(in[1])*(uint64(in[2])<<1)
+ tmp[4] = uint64(in[0])*(uint64(in[4])<<1) +
+ uint64(in[1])*(uint64(in[3])<<2) +
+ uint64(in[2])*uint64(in[2])
+ tmp[5] = uint64(in[0])*(uint64(in[5])<<1) +
+ uint64(in[1])*(uint64(in[4])<<1) +
+ uint64(in[2])*(uint64(in[3])<<1)
+ tmp[6] = uint64(in[0])*(uint64(in[6])<<1) +
+ uint64(in[1])*(uint64(in[5])<<2) +
+ uint64(in[2])*(uint64(in[4])<<1) +
+ uint64(in[3])*(uint64(in[3])<<1)
+ tmp[7] = uint64(in[0])*(uint64(in[7])<<1) +
+ uint64(in[1])*(uint64(in[6])<<1) +
+ uint64(in[2])*(uint64(in[5])<<1) +
+ uint64(in[3])*(uint64(in[4])<<1)
+ // tmp[8] has the greatest value of 2**61 + 2**60 + 2**61 + 2**60 + 2**60,
+ // which is < 2**64 as required.
+ tmp[8] = uint64(in[0])*(uint64(in[8])<<1) +
+ uint64(in[1])*(uint64(in[7])<<2) +
+ uint64(in[2])*(uint64(in[6])<<1) +
+ uint64(in[3])*(uint64(in[5])<<2) +
+ uint64(in[4])*uint64(in[4])
+ tmp[9] = uint64(in[1])*(uint64(in[8])<<1) +
+ uint64(in[2])*(uint64(in[7])<<1) +
+ uint64(in[3])*(uint64(in[6])<<1) +
+ uint64(in[4])*(uint64(in[5])<<1)
+ tmp[10] = uint64(in[2])*(uint64(in[8])<<1) +
+ uint64(in[3])*(uint64(in[7])<<2) +
+ uint64(in[4])*(uint64(in[6])<<1) +
+ uint64(in[5])*(uint64(in[5])<<1)
+ tmp[11] = uint64(in[3])*(uint64(in[8])<<1) +
+ uint64(in[4])*(uint64(in[7])<<1) +
+ uint64(in[5])*(uint64(in[6])<<1)
+ tmp[12] = uint64(in[4])*(uint64(in[8])<<1) +
+ uint64(in[5])*(uint64(in[7])<<2) +
+ uint64(in[6])*uint64(in[6])
+ tmp[13] = uint64(in[5])*(uint64(in[8])<<1) +
+ uint64(in[6])*(uint64(in[7])<<1)
+ tmp[14] = uint64(in[6])*(uint64(in[8])<<1) +
+ uint64(in[7])*(uint64(in[7])<<1)
+ tmp[15] = uint64(in[7]) * (uint64(in[8]) << 1)
+ tmp[16] = uint64(in[8]) * uint64(in[8])
+
+ p256ReduceDegree(out, tmp)
+}
+
+// p256Mul sets out=in*in2.
+//
+// On entry: in[0,2,...] < 2**30, in[1,3,...] < 2**29 and
+// in2[0,2,...] < 2**30, in2[1,3,...] < 2**29.
+// On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
+func p256Mul(out, in, in2 *[p256Limbs]uint32) {
+ var tmp [17]uint64
+
+ tmp[0] = uint64(in[0]) * uint64(in2[0])
+ tmp[1] = uint64(in[0])*(uint64(in2[1])<<0) +
+ uint64(in[1])*(uint64(in2[0])<<0)
+ tmp[2] = uint64(in[0])*(uint64(in2[2])<<0) +
+ uint64(in[1])*(uint64(in2[1])<<1) +
+ uint64(in[2])*(uint64(in2[0])<<0)
+ tmp[3] = uint64(in[0])*(uint64(in2[3])<<0) +
+ uint64(in[1])*(uint64(in2[2])<<0) +
+ uint64(in[2])*(uint64(in2[1])<<0) +
+ uint64(in[3])*(uint64(in2[0])<<0)
+ tmp[4] = uint64(in[0])*(uint64(in2[4])<<0) +
+ uint64(in[1])*(uint64(in2[3])<<1) +
+ uint64(in[2])*(uint64(in2[2])<<0) +
+ uint64(in[3])*(uint64(in2[1])<<1) +
+ uint64(in[4])*(uint64(in2[0])<<0)
+ tmp[5] = uint64(in[0])*(uint64(in2[5])<<0) +
+ uint64(in[1])*(uint64(in2[4])<<0) +
+ uint64(in[2])*(uint64(in2[3])<<0) +
+ uint64(in[3])*(uint64(in2[2])<<0) +
+ uint64(in[4])*(uint64(in2[1])<<0) +
+ uint64(in[5])*(uint64(in2[0])<<0)
+ tmp[6] = uint64(in[0])*(uint64(in2[6])<<0) +
+ uint64(in[1])*(uint64(in2[5])<<1) +
+ uint64(in[2])*(uint64(in2[4])<<0) +
+ uint64(in[3])*(uint64(in2[3])<<1) +
+ uint64(in[4])*(uint64(in2[2])<<0) +
+ uint64(in[5])*(uint64(in2[1])<<1) +
+ uint64(in[6])*(uint64(in2[0])<<0)
+ tmp[7] = uint64(in[0])*(uint64(in2[7])<<0) +
+ uint64(in[1])*(uint64(in2[6])<<0) +
+ uint64(in[2])*(uint64(in2[5])<<0) +
+ uint64(in[3])*(uint64(in2[4])<<0) +
+ uint64(in[4])*(uint64(in2[3])<<0) +
+ uint64(in[5])*(uint64(in2[2])<<0) +
+ uint64(in[6])*(uint64(in2[1])<<0) +
+ uint64(in[7])*(uint64(in2[0])<<0)
+ // tmp[8] has the greatest value but doesn't overflow. See logic in
+ // p256Square.
+ tmp[8] = uint64(in[0])*(uint64(in2[8])<<0) +
+ uint64(in[1])*(uint64(in2[7])<<1) +
+ uint64(in[2])*(uint64(in2[6])<<0) +
+ uint64(in[3])*(uint64(in2[5])<<1) +
+ uint64(in[4])*(uint64(in2[4])<<0) +
+ uint64(in[5])*(uint64(in2[3])<<1) +
+ uint64(in[6])*(uint64(in2[2])<<0) +
+ uint64(in[7])*(uint64(in2[1])<<1) +
+ uint64(in[8])*(uint64(in2[0])<<0)
+ tmp[9] = uint64(in[1])*(uint64(in2[8])<<0) +
+ uint64(in[2])*(uint64(in2[7])<<0) +
+ uint64(in[3])*(uint64(in2[6])<<0) +
+ uint64(in[4])*(uint64(in2[5])<<0) +
+ uint64(in[5])*(uint64(in2[4])<<0) +
+ uint64(in[6])*(uint64(in2[3])<<0) +
+ uint64(in[7])*(uint64(in2[2])<<0) +
+ uint64(in[8])*(uint64(in2[1])<<0)
+ tmp[10] = uint64(in[2])*(uint64(in2[8])<<0) +
+ uint64(in[3])*(uint64(in2[7])<<1) +
+ uint64(in[4])*(uint64(in2[6])<<0) +
+ uint64(in[5])*(uint64(in2[5])<<1) +
+ uint64(in[6])*(uint64(in2[4])<<0) +
+ uint64(in[7])*(uint64(in2[3])<<1) +
+ uint64(in[8])*(uint64(in2[2])<<0)
+ tmp[11] = uint64(in[3])*(uint64(in2[8])<<0) +
+ uint64(in[4])*(uint64(in2[7])<<0) +
+ uint64(in[5])*(uint64(in2[6])<<0) +
+ uint64(in[6])*(uint64(in2[5])<<0) +
+ uint64(in[7])*(uint64(in2[4])<<0) +
+ uint64(in[8])*(uint64(in2[3])<<0)
+ tmp[12] = uint64(in[4])*(uint64(in2[8])<<0) +
+ uint64(in[5])*(uint64(in2[7])<<1) +
+ uint64(in[6])*(uint64(in2[6])<<0) +
+ uint64(in[7])*(uint64(in2[5])<<1) +
+ uint64(in[8])*(uint64(in2[4])<<0)
+ tmp[13] = uint64(in[5])*(uint64(in2[8])<<0) +
+ uint64(in[6])*(uint64(in2[7])<<0) +
+ uint64(in[7])*(uint64(in2[6])<<0) +
+ uint64(in[8])*(uint64(in2[5])<<0)
+ tmp[14] = uint64(in[6])*(uint64(in2[8])<<0) +
+ uint64(in[7])*(uint64(in2[7])<<1) +
+ uint64(in[8])*(uint64(in2[6])<<0)
+ tmp[15] = uint64(in[7])*(uint64(in2[8])<<0) +
+ uint64(in[8])*(uint64(in2[7])<<0)
+ tmp[16] = uint64(in[8]) * (uint64(in2[8]) << 0)
+
+ p256ReduceDegree(out, tmp)
+}
+
+func p256Assign(out, in *[p256Limbs]uint32) {
+ *out = *in
+}
+
+// p256Invert calculates |out| = |in|^{-1}
+//
+// Based on Fermat's Little Theorem:
+//
+// a^p = a (mod p)
+// a^{p-1} = 1 (mod p)
+// a^{p-2} = a^{-1} (mod p)
+func p256Invert(out, in *[p256Limbs]uint32) {
+ var ftmp, ftmp2 [p256Limbs]uint32
+
+ // each e_I will hold |in|^{2^I - 1}
+ var e2, e4, e8, e16, e32, e64 [p256Limbs]uint32
+
+ p256Square(&ftmp, in) // 2^1
+ p256Mul(&ftmp, in, &ftmp) // 2^2 - 2^0
+ p256Assign(&e2, &ftmp)
+ p256Square(&ftmp, &ftmp) // 2^3 - 2^1
+ p256Square(&ftmp, &ftmp) // 2^4 - 2^2
+ p256Mul(&ftmp, &ftmp, &e2) // 2^4 - 2^0
+ p256Assign(&e4, &ftmp)
+ p256Square(&ftmp, &ftmp) // 2^5 - 2^1
+ p256Square(&ftmp, &ftmp) // 2^6 - 2^2
+ p256Square(&ftmp, &ftmp) // 2^7 - 2^3
+ p256Square(&ftmp, &ftmp) // 2^8 - 2^4
+ p256Mul(&ftmp, &ftmp, &e4) // 2^8 - 2^0
+ p256Assign(&e8, &ftmp)
+ for i := 0; i < 8; i++ {
+ p256Square(&ftmp, &ftmp)
+ } // 2^16 - 2^8
+ p256Mul(&ftmp, &ftmp, &e8) // 2^16 - 2^0
+ p256Assign(&e16, &ftmp)
+ for i := 0; i < 16; i++ {
+ p256Square(&ftmp, &ftmp)
+ } // 2^32 - 2^16
+ p256Mul(&ftmp, &ftmp, &e16) // 2^32 - 2^0
+ p256Assign(&e32, &ftmp)
+ for i := 0; i < 32; i++ {
+ p256Square(&ftmp, &ftmp)
+ } // 2^64 - 2^32
+ p256Assign(&e64, &ftmp)
+ p256Mul(&ftmp, &ftmp, in) // 2^64 - 2^32 + 2^0
+ for i := 0; i < 192; i++ {
+ p256Square(&ftmp, &ftmp)
+ } // 2^256 - 2^224 + 2^192
+
+ p256Mul(&ftmp2, &e64, &e32) // 2^64 - 2^0
+ for i := 0; i < 16; i++ {
+ p256Square(&ftmp2, &ftmp2)
+ } // 2^80 - 2^16
+ p256Mul(&ftmp2, &ftmp2, &e16) // 2^80 - 2^0
+ for i := 0; i < 8; i++ {
+ p256Square(&ftmp2, &ftmp2)
+ } // 2^88 - 2^8
+ p256Mul(&ftmp2, &ftmp2, &e8) // 2^88 - 2^0
+ for i := 0; i < 4; i++ {
+ p256Square(&ftmp2, &ftmp2)
+ } // 2^92 - 2^4
+ p256Mul(&ftmp2, &ftmp2, &e4) // 2^92 - 2^0
+ p256Square(&ftmp2, &ftmp2) // 2^93 - 2^1
+ p256Square(&ftmp2, &ftmp2) // 2^94 - 2^2
+ p256Mul(&ftmp2, &ftmp2, &e2) // 2^94 - 2^0
+ p256Square(&ftmp2, &ftmp2) // 2^95 - 2^1
+ p256Square(&ftmp2, &ftmp2) // 2^96 - 2^2
+ p256Mul(&ftmp2, &ftmp2, in) // 2^96 - 3
+
+ p256Mul(out, &ftmp2, &ftmp) // 2^256 - 2^224 + 2^192 + 2^96 - 3
+}
+
+// p256Scalar3 sets out=3*out.
+//
+// On entry: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
+// On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
+func p256Scalar3(out *[p256Limbs]uint32) {
+ var carry uint32
+
+ for i := 0; ; i++ {
+ out[i] *= 3
+ out[i] += carry
+ carry = out[i] >> 29
+ out[i] &= bottom29Bits
+
+ i++
+ if i == p256Limbs {
+ break
+ }
+
+ out[i] *= 3
+ out[i] += carry
+ carry = out[i] >> 28
+ out[i] &= bottom28Bits
+ }
+
+ p256ReduceCarry(out, carry)
+}
+
+// p256Scalar4 sets out=4*out.
+//
+// On entry: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
+// On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
+func p256Scalar4(out *[p256Limbs]uint32) {
+ var carry, nextCarry uint32
+
+ for i := 0; ; i++ {
+ nextCarry = out[i] >> 27
+ out[i] <<= 2
+ out[i] &= bottom29Bits
+ out[i] += carry
+ carry = nextCarry + (out[i] >> 29)
+ out[i] &= bottom29Bits
+
+ i++
+ if i == p256Limbs {
+ break
+ }
+ nextCarry = out[i] >> 26
+ out[i] <<= 2
+ out[i] &= bottom28Bits
+ out[i] += carry
+ carry = nextCarry + (out[i] >> 28)
+ out[i] &= bottom28Bits
+ }
+
+ p256ReduceCarry(out, carry)
+}
+
+// p256Scalar8 sets out=8*out.
+//
+// On entry: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
+// On exit: out[0,2,...] < 2**30, out[1,3,...] < 2**29.
+func p256Scalar8(out *[p256Limbs]uint32) {
+ var carry, nextCarry uint32
+
+ for i := 0; ; i++ {
+ nextCarry = out[i] >> 26
+ out[i] <<= 3
+ out[i] &= bottom29Bits
+ out[i] += carry
+ carry = nextCarry + (out[i] >> 29)
+ out[i] &= bottom29Bits
+
+ i++
+ if i == p256Limbs {
+ break
+ }
+ nextCarry = out[i] >> 25
+ out[i] <<= 3
+ out[i] &= bottom28Bits
+ out[i] += carry
+ carry = nextCarry + (out[i] >> 28)
+ out[i] &= bottom28Bits
+ }
+
+ p256ReduceCarry(out, carry)
+}
+
+// Group operations:
+//
+// Elements of the elliptic curve group are represented in Jacobian
+// coordinates: (x, y, z). An affine point (x', y') is x'=x/z**2, y'=y/z**3 in
+// Jacobian form.
+
+// p256PointDouble sets {xOut,yOut,zOut} = 2*{x,y,z}.
+//
+// See https://www.hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#doubling-dbl-2009-l
+func p256PointDouble(xOut, yOut, zOut, x, y, z *[p256Limbs]uint32) {
+ var delta, gamma, alpha, beta, tmp, tmp2 [p256Limbs]uint32
+
+ p256Square(&delta, z)
+ p256Square(&gamma, y)
+ p256Mul(&beta, x, &gamma)
+
+ p256Sum(&tmp, x, &delta)
+ p256Diff(&tmp2, x, &delta)
+ p256Mul(&alpha, &tmp, &tmp2)
+ p256Scalar3(&alpha)
+
+ p256Sum(&tmp, y, z)
+ p256Square(&tmp, &tmp)
+ p256Diff(&tmp, &tmp, &gamma)
+ p256Diff(zOut, &tmp, &delta)
+
+ p256Scalar4(&beta)
+ p256Square(xOut, &alpha)
+ p256Diff(xOut, xOut, &beta)
+ p256Diff(xOut, xOut, &beta)
+
+ p256Diff(&tmp, &beta, xOut)
+ p256Mul(&tmp, &alpha, &tmp)
+ p256Square(&tmp2, &gamma)
+ p256Scalar8(&tmp2)
+ p256Diff(yOut, &tmp, &tmp2)
+}
+
+// p256PointAddMixed sets {xOut,yOut,zOut} = {x1,y1,z1} + {x2,y2,1}.
+// (i.e. the second point is affine.)
+//
+// See https://www.hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#addition-add-2007-bl
+//
+// Note that this function does not handle P+P, infinity+P nor P+infinity
+// correctly.
+func p256PointAddMixed(xOut, yOut, zOut, x1, y1, z1, x2, y2 *[p256Limbs]uint32) {
+ var z1z1, z1z1z1, s2, u2, h, i, j, r, rr, v, tmp [p256Limbs]uint32
+
+ p256Square(&z1z1, z1)
+ p256Sum(&tmp, z1, z1)
+
+ p256Mul(&u2, x2, &z1z1)
+ p256Mul(&z1z1z1, z1, &z1z1)
+ p256Mul(&s2, y2, &z1z1z1)
+ p256Diff(&h, &u2, x1)
+ p256Sum(&i, &h, &h)
+ p256Square(&i, &i)
+ p256Mul(&j, &h, &i)
+ p256Diff(&r, &s2, y1)
+ p256Sum(&r, &r, &r)
+ p256Mul(&v, x1, &i)
+
+ p256Mul(zOut, &tmp, &h)
+ p256Square(&rr, &r)
+ p256Diff(xOut, &rr, &j)
+ p256Diff(xOut, xOut, &v)
+ p256Diff(xOut, xOut, &v)
+
+ p256Diff(&tmp, &v, xOut)
+ p256Mul(yOut, &tmp, &r)
+ p256Mul(&tmp, y1, &j)
+ p256Diff(yOut, yOut, &tmp)
+ p256Diff(yOut, yOut, &tmp)
+}
+
+// p256PointAdd sets {xOut,yOut,zOut} = {x1,y1,z1} + {x2,y2,z2}.
+//
+// See https://www.hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-0.html#addition-add-2007-bl
+//
+// Note that this function does not handle P+P, infinity+P nor P+infinity
+// correctly.
+func p256PointAdd(xOut, yOut, zOut, x1, y1, z1, x2, y2, z2 *[p256Limbs]uint32) {
+ var z1z1, z1z1z1, z2z2, z2z2z2, s1, s2, u1, u2, h, i, j, r, rr, v, tmp [p256Limbs]uint32
+
+ p256Square(&z1z1, z1)
+ p256Square(&z2z2, z2)
+ p256Mul(&u1, x1, &z2z2)
+
+ p256Sum(&tmp, z1, z2)
+ p256Square(&tmp, &tmp)
+ p256Diff(&tmp, &tmp, &z1z1)
+ p256Diff(&tmp, &tmp, &z2z2)
+
+ p256Mul(&z2z2z2, z2, &z2z2)
+ p256Mul(&s1, y1, &z2z2z2)
+
+ p256Mul(&u2, x2, &z1z1)
+ p256Mul(&z1z1z1, z1, &z1z1)
+ p256Mul(&s2, y2, &z1z1z1)
+ p256Diff(&h, &u2, &u1)
+ p256Sum(&i, &h, &h)
+ p256Square(&i, &i)
+ p256Mul(&j, &h, &i)
+ p256Diff(&r, &s2, &s1)
+ p256Sum(&r, &r, &r)
+ p256Mul(&v, &u1, &i)
+
+ p256Mul(zOut, &tmp, &h)
+ p256Square(&rr, &r)
+ p256Diff(xOut, &rr, &j)
+ p256Diff(xOut, xOut, &v)
+ p256Diff(xOut, xOut, &v)
+
+ p256Diff(&tmp, &v, xOut)
+ p256Mul(yOut, &tmp, &r)
+ p256Mul(&tmp, &s1, &j)
+ p256Diff(yOut, yOut, &tmp)
+ p256Diff(yOut, yOut, &tmp)
+}
+
+// p256CopyConditional sets out=in if mask = 0xffffffff in constant time.
+//
+// On entry: mask is either 0 or 0xffffffff.
+func p256CopyConditional(out, in *[p256Limbs]uint32, mask uint32) {
+ for i := 0; i < p256Limbs; i++ {
+ tmp := mask & (in[i] ^ out[i])
+ out[i] ^= tmp
+ }
+}
+
+// p256SelectAffinePoint sets {out_x,out_y} to the index'th entry of table.
+//
+// On entry: index < 16, table[0] must be zero.
+func p256SelectAffinePoint(xOut, yOut *[p256Limbs]uint32, table []uint32, index uint32) {
+ for i := range xOut {
+ xOut[i] = 0
+ }
+ for i := range yOut {
+ yOut[i] = 0
+ }
+
+ for i := uint32(1); i < 16; i++ {
+ mask := i ^ index
+ mask |= mask >> 2
+ mask |= mask >> 1
+ mask &= 1
+ mask--
+ for j := range xOut {
+ xOut[j] |= table[0] & mask
+ table = table[1:]
+ }
+ for j := range yOut {
+ yOut[j] |= table[0] & mask
+ table = table[1:]
+ }
+ }
+}
+
+// p256SelectJacobianPoint sets {out_x,out_y,out_z} to the index'th entry of
+// table.
+//
+// On entry: index < 16, table[0] must be zero.
+func p256SelectJacobianPoint(xOut, yOut, zOut *[p256Limbs]uint32, table *[16][3][p256Limbs]uint32, index uint32) {
+ for i := range xOut {
+ xOut[i] = 0
+ }
+ for i := range yOut {
+ yOut[i] = 0
+ }
+ for i := range zOut {
+ zOut[i] = 0
+ }
+
+ // The implicit value at index 0 is all zero. We don't need to perform that
+ // iteration of the loop because we already set out_* to zero.
+ for i := uint32(1); i < 16; i++ {
+ mask := i ^ index
+ mask |= mask >> 2
+ mask |= mask >> 1
+ mask &= 1
+ mask--
+ for j := range xOut {
+ xOut[j] |= table[i][0][j] & mask
+ }
+ for j := range yOut {
+ yOut[j] |= table[i][1][j] & mask
+ }
+ for j := range zOut {
+ zOut[j] |= table[i][2][j] & mask
+ }
+ }
+}
+
+// p256GetBit returns the bit'th bit of scalar.
+func p256GetBit(scalar *[32]uint8, bit uint) uint32 {
+ return uint32(((scalar[bit>>3]) >> (bit & 7)) & 1)
+}
+
+// p256ScalarBaseMult sets {xOut,yOut,zOut} = scalar*G where scalar is a
+// little-endian number. Note that the value of scalar must be less than the
+// order of the group.
+func p256ScalarBaseMult(xOut, yOut, zOut *[p256Limbs]uint32, scalar *[32]uint8) {
+ nIsInfinityMask := ^uint32(0)
+ var pIsNoninfiniteMask, mask, tableOffset uint32
+ var px, py, tx, ty, tz [p256Limbs]uint32
+
+ for i := range xOut {
+ xOut[i] = 0
+ }
+ for i := range yOut {
+ yOut[i] = 0
+ }
+ for i := range zOut {
+ zOut[i] = 0
+ }
+
+ // The loop adds bits at positions 0, 64, 128 and 192, followed by
+ // positions 32,96,160 and 224 and does this 32 times.
+ for i := uint(0); i < 32; i++ {
+ if i != 0 {
+ p256PointDouble(xOut, yOut, zOut, xOut, yOut, zOut)
+ }
+ tableOffset = 0
+ for j := uint(0); j <= 32; j += 32 {
+ bit0 := p256GetBit(scalar, 31-i+j)
+ bit1 := p256GetBit(scalar, 95-i+j)
+ bit2 := p256GetBit(scalar, 159-i+j)
+ bit3 := p256GetBit(scalar, 223-i+j)
+ index := bit0 | (bit1 << 1) | (bit2 << 2) | (bit3 << 3)
+
+ p256SelectAffinePoint(&px, &py, p256Precomputed[tableOffset:], index)
+ tableOffset += 30 * p256Limbs
+
+ // Since scalar is less than the order of the group, we know that
+ // {xOut,yOut,zOut} != {px,py,1}, unless both are zero, which we handle
+ // below.
+ p256PointAddMixed(&tx, &ty, &tz, xOut, yOut, zOut, &px, &py)
+ // The result of pointAddMixed is incorrect if {xOut,yOut,zOut} is zero
+ // (a.k.a. the point at infinity). We handle that situation by
+ // copying the point from the table.
+ p256CopyConditional(xOut, &px, nIsInfinityMask)
+ p256CopyConditional(yOut, &py, nIsInfinityMask)
+ p256CopyConditional(zOut, &p256One, nIsInfinityMask)
+
+ // Equally, the result is also wrong if the point from the table is
+ // zero, which happens when the index is zero. We handle that by
+ // only copying from {tx,ty,tz} to {xOut,yOut,zOut} if index != 0.
+ pIsNoninfiniteMask = nonZeroToAllOnes(index)
+ mask = pIsNoninfiniteMask & ^nIsInfinityMask
+ p256CopyConditional(xOut, &tx, mask)
+ p256CopyConditional(yOut, &ty, mask)
+ p256CopyConditional(zOut, &tz, mask)
+ // If p was not zero, then n is now non-zero.
+ nIsInfinityMask &^= pIsNoninfiniteMask
+ }
+ }
+}
+
+// p256PointToAffine converts a Jacobian point to an affine point. If the input
+// is the point at infinity then it returns (0, 0) in constant time.
+func p256PointToAffine(xOut, yOut, x, y, z *[p256Limbs]uint32) {
+ var zInv, zInvSq [p256Limbs]uint32
+
+ p256Invert(&zInv, z)
+ p256Square(&zInvSq, &zInv)
+ p256Mul(xOut, x, &zInvSq)
+ p256Mul(&zInv, &zInv, &zInvSq)
+ p256Mul(yOut, y, &zInv)
+}
+
+// p256ToAffine returns a pair of *big.Int containing the affine representation
+// of {x,y,z}.
+func p256ToAffine(x, y, z *[p256Limbs]uint32) (xOut, yOut *big.Int) {
+ var xx, yy [p256Limbs]uint32
+ p256PointToAffine(&xx, &yy, x, y, z)
+ return p256ToBig(&xx), p256ToBig(&yy)
+}
+
+// p256ScalarMult sets {xOut,yOut,zOut} = scalar*{x,y}.
+func p256ScalarMult(xOut, yOut, zOut, x, y *[p256Limbs]uint32, scalar *[32]uint8) {
+ var px, py, pz, tx, ty, tz [p256Limbs]uint32
+ var precomp [16][3][p256Limbs]uint32
+ var nIsInfinityMask, index, pIsNoninfiniteMask, mask uint32
+
+ // We precompute 0,1,2,... times {x,y}.
+ precomp[1][0] = *x
+ precomp[1][1] = *y
+ precomp[1][2] = p256One
+
+ for i := 2; i < 16; i += 2 {
+ p256PointDouble(&precomp[i][0], &precomp[i][1], &precomp[i][2], &precomp[i/2][0], &precomp[i/2][1], &precomp[i/2][2])
+ p256PointAddMixed(&precomp[i+1][0], &precomp[i+1][1], &precomp[i+1][2], &precomp[i][0], &precomp[i][1], &precomp[i][2], x, y)
+ }
+
+ for i := range xOut {
+ xOut[i] = 0
+ }
+ for i := range yOut {
+ yOut[i] = 0
+ }
+ for i := range zOut {
+ zOut[i] = 0
+ }
+ nIsInfinityMask = ^uint32(0)
+
+ // We add in a window of four bits each iteration and do this 64 times.
+ for i := 0; i < 64; i++ {
+ if i != 0 {
+ p256PointDouble(xOut, yOut, zOut, xOut, yOut, zOut)
+ p256PointDouble(xOut, yOut, zOut, xOut, yOut, zOut)
+ p256PointDouble(xOut, yOut, zOut, xOut, yOut, zOut)
+ p256PointDouble(xOut, yOut, zOut, xOut, yOut, zOut)
+ }
+
+ index = uint32(scalar[31-i/2])
+ if (i & 1) == 1 {
+ index &= 15
+ } else {
+ index >>= 4
+ }
+
+ // See the comments in scalarBaseMult about handling infinities.
+ p256SelectJacobianPoint(&px, &py, &pz, &precomp, index)
+ p256PointAdd(&tx, &ty, &tz, xOut, yOut, zOut, &px, &py, &pz)
+ p256CopyConditional(xOut, &px, nIsInfinityMask)
+ p256CopyConditional(yOut, &py, nIsInfinityMask)
+ p256CopyConditional(zOut, &pz, nIsInfinityMask)
+
+ pIsNoninfiniteMask = nonZeroToAllOnes(index)
+ mask = pIsNoninfiniteMask & ^nIsInfinityMask
+ p256CopyConditional(xOut, &tx, mask)
+ p256CopyConditional(yOut, &ty, mask)
+ p256CopyConditional(zOut, &tz, mask)
+ nIsInfinityMask &^= pIsNoninfiniteMask
+ }
+}
+
+// p256FromBig sets out = R*in.
+func p256FromBig(out *[p256Limbs]uint32, in *big.Int) {
+ tmp := new(big.Int).Lsh(in, 257)
+ tmp.Mod(tmp, p256Params.P)
+
+ for i := 0; i < p256Limbs; i++ {
+ if bits := tmp.Bits(); len(bits) > 0 {
+ out[i] = uint32(bits[0]) & bottom29Bits
+ } else {
+ out[i] = 0
+ }
+ tmp.Rsh(tmp, 29)
+
+ i++
+ if i == p256Limbs {
+ break
+ }
+
+ if bits := tmp.Bits(); len(bits) > 0 {
+ out[i] = uint32(bits[0]) & bottom28Bits
+ } else {
+ out[i] = 0
+ }
+ tmp.Rsh(tmp, 28)
+ }
+}
+
+// p256ToBig returns a *big.Int containing the value of in.
+func p256ToBig(in *[p256Limbs]uint32) *big.Int {
+ result, tmp := new(big.Int), new(big.Int)
+
+ result.SetInt64(int64(in[p256Limbs-1]))
+ for i := p256Limbs - 2; i >= 0; i-- {
+ if (i & 1) == 0 {
+ result.Lsh(result, 29)
+ } else {
+ result.Lsh(result, 28)
+ }
+ tmp.SetInt64(int64(in[i]))
+ result.Add(result, tmp)
+ }
+
+ result.Mul(result, p256RInverse)
+ result.Mod(result, p256Params.P)
+ return result
}
diff --git a/src/crypto/elliptic/p256_noasm.go b/src/crypto/elliptic/p256_noasm.go
new file mode 100644
index 0000000000..380ea66ac3
--- /dev/null
+++ b/src/crypto/elliptic/p256_noasm.go
@@ -0,0 +1,15 @@
+// Copyright 2016 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.
+
+//go:build !amd64 && !s390x && !arm64 && !ppc64le
+// +build !amd64,!s390x,!arm64,!ppc64le
+
+package elliptic
+
+var p256 p256Curve
+
+func initP256Arch() {
+ // Use pure Go constant-time implementation.
+ p256 = p256Curve{p256Params}
+}
diff --git a/src/crypto/elliptic/p256_ppc64le.go b/src/crypto/elliptic/p256_ppc64le.go
index dda1157564..3867a87e1f 100644
--- a/src/crypto/elliptic/p256_ppc64le.go
+++ b/src/crypto/elliptic/p256_ppc64le.go
@@ -35,7 +35,6 @@ var (
func initP256Arch() {
p256 = p256CurveFast{p256Params}
initTable()
- return
}
func (curve p256CurveFast) Params() *CurveParams {
@@ -73,7 +72,6 @@ func p256MovCond(res, a, b *p256Point, cond int)
//go:noescape
func p256Select(point *p256Point, table []p256Point, idx int)
-//
//go:noescape
func p256SelectBase(point *p256Point, table []p256Point, idx int)
@@ -85,12 +83,9 @@ func p256SelectBase(point *p256Point, table []p256Point, idx int)
//go:noescape
func p256PointAddAffineAsm(res, in1, in2 *p256Point, sign, sel, zero int)
-// Point add
-//
//go:noescape
func p256PointAddAsm(res, in1, in2 *p256Point) int
-//
//go:noescape
func p256PointDoubleAsm(res, in *p256Point)
@@ -340,7 +335,6 @@ func boothW7(in uint) (int, int) {
}
func initTable() {
-
p256PreFast = new([37][64]p256Point)
// TODO: For big endian, these slices should be in reverse byte order,
@@ -352,7 +346,6 @@ func initTable() {
0x25, 0xf3, 0x21, 0xdd, 0x88, 0x86, 0xe8, 0xd2, 0x85, 0x5d, 0x88, 0x25, 0x18, 0xff, 0x71, 0x85}, //(p256.y*2^256)%p
z: [32]byte{0x01, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0xff, 0xff, 0xff, 0xff,
0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xfe, 0xff, 0xff, 0xff, 0x00, 0x00, 0x00, 0x00}, //(p256.z*2^256)%p
-
}
t1 := new(p256Point)
diff --git a/src/crypto/elliptic/p256_s390x.go b/src/crypto/elliptic/p256_s390x.go
index 735e9f57f1..b7331ebbfd 100644
--- a/src/crypto/elliptic/p256_s390x.go
+++ b/src/crypto/elliptic/p256_s390x.go
@@ -60,7 +60,6 @@ func initP256Arch() {
// No vector support, use pure Go implementation.
p256 = p256Curve{p256Params}
- return
}
func (curve p256CurveFast) Params() *CurveParams {
diff --git a/src/crypto/elliptic/params.go b/src/crypto/elliptic/params.go
new file mode 100644
index 0000000000..586f2c0386
--- /dev/null
+++ b/src/crypto/elliptic/params.go
@@ -0,0 +1,296 @@
+// Copyright 2021 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 elliptic
+
+import "math/big"
+
+// CurveParams contains the parameters of an elliptic curve and also provides
+// a generic, non-constant time implementation of Curve.
+type CurveParams struct {
+ P *big.Int // the order of the underlying field
+ N *big.Int // the order of the base point
+ B *big.Int // the constant of the curve equation
+ Gx, Gy *big.Int // (x,y) of the base point
+ BitSize int // the size of the underlying field
+ Name string // the canonical name of the curve
+}
+
+func (curve *CurveParams) Params() *CurveParams {
+ return curve
+}
+
+// CurveParams operates, internally, on Jacobian coordinates. For a given
+// (x, y) position on the curve, the Jacobian coordinates are (x1, y1, z1)
+// where x = x1/z1² and y = y1/z1³. The greatest speedups come when the whole
+// calculation can be performed within the transform (as in ScalarMult and
+// ScalarBaseMult). But even for Add and Double, it's faster to apply and
+// reverse the transform than to operate in affine coordinates.
+
+// polynomial returns x³ - 3x + b.
+func (curve *CurveParams) polynomial(x *big.Int) *big.Int {
+ x3 := new(big.Int).Mul(x, x)
+ x3.Mul(x3, x)
+
+ threeX := new(big.Int).Lsh(x, 1)
+ threeX.Add(threeX, x)
+
+ x3.Sub(x3, threeX)
+ x3.Add(x3, curve.B)
+ x3.Mod(x3, curve.P)
+
+ return x3
+}
+
+func (curve *CurveParams) IsOnCurve(x, y *big.Int) bool {
+ // If there is a dedicated constant-time implementation for this curve operation,
+ // use that instead of the generic one.
+ if specific, ok := matchesSpecificCurve(curve, p224, p384, p521); ok {
+ return specific.IsOnCurve(x, y)
+ }
+
+ if x.Sign() < 0 || x.Cmp(curve.P) >= 0 ||
+ y.Sign() < 0 || y.Cmp(curve.P) >= 0 {
+ return false
+ }
+
+ // y² = x³ - 3x + b
+ y2 := new(big.Int).Mul(y, y)
+ y2.Mod(y2, curve.P)
+
+ return curve.polynomial(x).Cmp(y2) == 0
+}
+
+// zForAffine returns a Jacobian Z value for the affine point (x, y). If x and
+// y are zero, it assumes that they represent the point at infinity because (0,
+// 0) is not on the any of the curves handled here.
+func zForAffine(x, y *big.Int) *big.Int {
+ z := new(big.Int)
+ if x.Sign() != 0 || y.Sign() != 0 {
+ z.SetInt64(1)
+ }
+ return z
+}
+
+// affineFromJacobian reverses the Jacobian transform. See the comment at the
+// top of the file. If the point is ∞ it returns 0, 0.
+func (curve *CurveParams) affineFromJacobian(x, y, z *big.Int) (xOut, yOut *big.Int) {
+ if z.Sign() == 0 {
+ return new(big.Int), new(big.Int)
+ }
+
+ zinv := new(big.Int).ModInverse(z, curve.P)
+ zinvsq := new(big.Int).Mul(zinv, zinv)
+
+ xOut = new(big.Int).Mul(x, zinvsq)
+ xOut.Mod(xOut, curve.P)
+ zinvsq.Mul(zinvsq, zinv)
+ yOut = new(big.Int).Mul(y, zinvsq)
+ yOut.Mod(yOut, curve.P)
+ return
+}
+
+func (curve *CurveParams) Add(x1, y1, x2, y2 *big.Int) (*big.Int, *big.Int) {
+ // If there is a dedicated constant-time implementation for this curve operation,
+ // use that instead of the generic one.
+ if specific, ok := matchesSpecificCurve(curve, p224, p384, p521); ok {
+ return specific.Add(x1, y1, x2, y2)
+ }
+
+ z1 := zForAffine(x1, y1)
+ z2 := zForAffine(x2, y2)
+ return curve.affineFromJacobian(curve.addJacobian(x1, y1, z1, x2, y2, z2))
+}
+
+// addJacobian takes two points in Jacobian coordinates, (x1, y1, z1) and
+// (x2, y2, z2) and returns their sum, also in Jacobian form.
+func (curve *CurveParams) addJacobian(x1, y1, z1, x2, y2, z2 *big.Int) (*big.Int, *big.Int, *big.Int) {
+ // See https://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#addition-add-2007-bl
+ x3, y3, z3 := new(big.Int), new(big.Int), new(big.Int)
+ if z1.Sign() == 0 {
+ x3.Set(x2)
+ y3.Set(y2)
+ z3.Set(z2)
+ return x3, y3, z3
+ }
+ if z2.Sign() == 0 {
+ x3.Set(x1)
+ y3.Set(y1)
+ z3.Set(z1)
+ return x3, y3, z3
+ }
+
+ z1z1 := new(big.Int).Mul(z1, z1)
+ z1z1.Mod(z1z1, curve.P)
+ z2z2 := new(big.Int).Mul(z2, z2)
+ z2z2.Mod(z2z2, curve.P)
+
+ u1 := new(big.Int).Mul(x1, z2z2)
+ u1.Mod(u1, curve.P)
+ u2 := new(big.Int).Mul(x2, z1z1)
+ u2.Mod(u2, curve.P)
+ h := new(big.Int).Sub(u2, u1)
+ xEqual := h.Sign() == 0
+ if h.Sign() == -1 {
+ h.Add(h, curve.P)
+ }
+ i := new(big.Int).Lsh(h, 1)
+ i.Mul(i, i)
+ j := new(big.Int).Mul(h, i)
+
+ s1 := new(big.Int).Mul(y1, z2)
+ s1.Mul(s1, z2z2)
+ s1.Mod(s1, curve.P)
+ s2 := new(big.Int).Mul(y2, z1)
+ s2.Mul(s2, z1z1)
+ s2.Mod(s2, curve.P)
+ r := new(big.Int).Sub(s2, s1)
+ if r.Sign() == -1 {
+ r.Add(r, curve.P)
+ }
+ yEqual := r.Sign() == 0
+ if xEqual && yEqual {
+ return curve.doubleJacobian(x1, y1, z1)
+ }
+ r.Lsh(r, 1)
+ v := new(big.Int).Mul(u1, i)
+
+ x3.Set(r)
+ x3.Mul(x3, x3)
+ x3.Sub(x3, j)
+ x3.Sub(x3, v)
+ x3.Sub(x3, v)
+ x3.Mod(x3, curve.P)
+
+ y3.Set(r)
+ v.Sub(v, x3)
+ y3.Mul(y3, v)
+ s1.Mul(s1, j)
+ s1.Lsh(s1, 1)
+ y3.Sub(y3, s1)
+ y3.Mod(y3, curve.P)
+
+ z3.Add(z1, z2)
+ z3.Mul(z3, z3)
+ z3.Sub(z3, z1z1)
+ z3.Sub(z3, z2z2)
+ z3.Mul(z3, h)
+ z3.Mod(z3, curve.P)
+
+ return x3, y3, z3
+}
+
+func (curve *CurveParams) Double(x1, y1 *big.Int) (*big.Int, *big.Int) {
+ // If there is a dedicated constant-time implementation for this curve operation,
+ // use that instead of the generic one.
+ if specific, ok := matchesSpecificCurve(curve, p224, p384, p521); ok {
+ return specific.Double(x1, y1)
+ }
+
+ z1 := zForAffine(x1, y1)
+ return curve.affineFromJacobian(curve.doubleJacobian(x1, y1, z1))
+}
+
+// doubleJacobian takes a point in Jacobian coordinates, (x, y, z), and
+// returns its double, also in Jacobian form.
+func (curve *CurveParams) doubleJacobian(x, y, z *big.Int) (*big.Int, *big.Int, *big.Int) {
+ // See https://hyperelliptic.org/EFD/g1p/auto-shortw-jacobian-3.html#doubling-dbl-2001-b
+ delta := new(big.Int).Mul(z, z)
+ delta.Mod(delta, curve.P)
+ gamma := new(big.Int).Mul(y, y)
+ gamma.Mod(gamma, curve.P)
+ alpha := new(big.Int).Sub(x, delta)
+ if alpha.Sign() == -1 {
+ alpha.Add(alpha, curve.P)
+ }
+ alpha2 := new(big.Int).Add(x, delta)
+ alpha.Mul(alpha, alpha2)
+ alpha2.Set(alpha)
+ alpha.Lsh(alpha, 1)
+ alpha.Add(alpha, alpha2)
+
+ beta := alpha2.Mul(x, gamma)
+
+ x3 := new(big.Int).Mul(alpha, alpha)
+ beta8 := new(big.Int).Lsh(beta, 3)
+ beta8.Mod(beta8, curve.P)
+ x3.Sub(x3, beta8)
+ if x3.Sign() == -1 {
+ x3.Add(x3, curve.P)
+ }
+ x3.Mod(x3, curve.P)
+
+ z3 := new(big.Int).Add(y, z)
+ z3.Mul(z3, z3)
+ z3.Sub(z3, gamma)
+ if z3.Sign() == -1 {
+ z3.Add(z3, curve.P)
+ }
+ z3.Sub(z3, delta)
+ if z3.Sign() == -1 {
+ z3.Add(z3, curve.P)
+ }
+ z3.Mod(z3, curve.P)
+
+ beta.Lsh(beta, 2)
+ beta.Sub(beta, x3)
+ if beta.Sign() == -1 {
+ beta.Add(beta, curve.P)
+ }
+ y3 := alpha.Mul(alpha, beta)
+
+ gamma.Mul(gamma, gamma)
+ gamma.Lsh(gamma, 3)
+ gamma.Mod(gamma, curve.P)
+
+ y3.Sub(y3, gamma)
+ if y3.Sign() == -1 {
+ y3.Add(y3, curve.P)
+ }
+ y3.Mod(y3, curve.P)
+
+ return x3, y3, z3
+}
+
+func (curve *CurveParams) ScalarMult(Bx, By *big.Int, k []byte) (*big.Int, *big.Int) {
+ // If there is a dedicated constant-time implementation for this curve operation,
+ // use that instead of the generic one.
+ if specific, ok := matchesSpecificCurve(curve, p224, p256, p384, p521); ok {
+ return specific.ScalarMult(Bx, By, k)
+ }
+
+ Bz := new(big.Int).SetInt64(1)
+ x, y, z := new(big.Int), new(big.Int), new(big.Int)
+
+ for _, byte := range k {
+ for bitNum := 0; bitNum < 8; bitNum++ {
+ x, y, z = curve.doubleJacobian(x, y, z)
+ if byte&0x80 == 0x80 {
+ x, y, z = curve.addJacobian(Bx, By, Bz, x, y, z)
+ }
+ byte <<= 1
+ }
+ }
+
+ return curve.affineFromJacobian(x, y, z)
+}
+
+func (curve *CurveParams) ScalarBaseMult(k []byte) (*big.Int, *big.Int) {
+ // If there is a dedicated constant-time implementation for this curve operation,
+ // use that instead of the generic one.
+ if specific, ok := matchesSpecificCurve(curve, p224, p256, p384, p521); ok {
+ return specific.ScalarBaseMult(k)
+ }
+
+ return curve.ScalarMult(curve.Gx, curve.Gy, k)
+}
+
+func matchesSpecificCurve(params *CurveParams, available ...Curve) (Curve, bool) {
+ for _, c := range available {
+ if params == c.Params() {
+ return c, true
+ }
+ }
+ return nil, false
+}
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