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| author | Daniel Baumann <daniel.baumann@progress-linux.org> | 2024-04-28 13:16:40 +0000 |
|---|---|---|
| committer | Daniel Baumann <daniel.baumann@progress-linux.org> | 2024-04-28 13:16:40 +0000 |
| commit | 47ab3d4a42e9ab51c465c4322d2ec233f6324e6b (patch) | |
| tree | a61a0ffd83f4a3def4b36e5c8e99630c559aa723 /src/crypto/ed25519/internal/edwards25519 | |
| parent | Initial commit. (diff) | |
| download | golang-1.18-47ab3d4a42e9ab51c465c4322d2ec233f6324e6b.tar.xz golang-1.18-47ab3d4a42e9ab51c465c4322d2ec233f6324e6b.zip | |
Adding upstream version 1.18.10.upstream/1.18.10upstream
Signed-off-by: Daniel Baumann <daniel.baumann@progress-linux.org>
Diffstat (limited to 'src/crypto/ed25519/internal/edwards25519')
24 files changed, 4975 insertions, 0 deletions
diff --git a/src/crypto/ed25519/internal/edwards25519/doc.go b/src/crypto/ed25519/internal/edwards25519/doc.go new file mode 100644 index 0000000..ff31cd2 --- /dev/null +++ b/src/crypto/ed25519/internal/edwards25519/doc.go @@ -0,0 +1,22 @@ +// Copyright (c) 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 edwards25519 implements group logic for the twisted Edwards curve +// +// -x^2 + y^2 = 1 + -(121665/121666)*x^2*y^2 +// +// This is better known as the Edwards curve equivalent to Curve25519, and is +// the curve used by the Ed25519 signature scheme. +// +// Most users don't need this package, and should instead use crypto/ed25519 for +// signatures, golang.org/x/crypto/curve25519 for Diffie-Hellman, or +// github.com/gtank/ristretto255 for prime order group logic. +// +// However, developers who do need to interact with low-level edwards25519 +// operations can use filippo.io/edwards25519, an extended version of this +// package repackaged as an importable module. +// +// (Note that filippo.io/edwards25519 and github.com/gtank/ristretto255 are not +// maintained by the Go team and are not covered by the Go 1 Compatibility Promise.) +package edwards25519 diff --git a/src/crypto/ed25519/internal/edwards25519/edwards25519.go b/src/crypto/ed25519/internal/edwards25519/edwards25519.go new file mode 100644 index 0000000..313e6c2 --- /dev/null +++ b/src/crypto/ed25519/internal/edwards25519/edwards25519.go @@ -0,0 +1,427 @@ +// Copyright (c) 2017 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 edwards25519 + +import ( + "crypto/ed25519/internal/edwards25519/field" + "errors" +) + +// Point types. + +type projP1xP1 struct { + X, Y, Z, T field.Element +} + +type projP2 struct { + X, Y, Z field.Element +} + +// Point represents a point on the edwards25519 curve. +// +// This type works similarly to math/big.Int, and all arguments and receivers +// are allowed to alias. +// +// The zero value is NOT valid, and it may be used only as a receiver. +type Point struct { + // The point is internally represented in extended coordinates (X, Y, Z, T) + // where x = X/Z, y = Y/Z, and xy = T/Z per https://eprint.iacr.org/2008/522. + x, y, z, t field.Element + + // Make the type not comparable (i.e. used with == or as a map key), as + // equivalent points can be represented by different Go values. + _ incomparable +} + +type incomparable [0]func() + +func checkInitialized(points ...*Point) { + for _, p := range points { + if p.x == (field.Element{}) && p.y == (field.Element{}) { + panic("edwards25519: use of uninitialized Point") + } + } +} + +type projCached struct { + YplusX, YminusX, Z, T2d field.Element +} + +type affineCached struct { + YplusX, YminusX, T2d field.Element +} + +// Constructors. + +func (v *projP2) Zero() *projP2 { + v.X.Zero() + v.Y.One() + v.Z.One() + return v +} + +// identity is the point at infinity. +var identity, _ = new(Point).SetBytes([]byte{ + 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, + 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}) + +// NewIdentityPoint returns a new Point set to the identity. +func NewIdentityPoint() *Point { + return new(Point).Set(identity) +} + +// generator is the canonical curve basepoint. See TestGenerator for the +// correspondence of this encoding with the values in RFC 8032. +var generator, _ = new(Point).SetBytes([]byte{ + 0x58, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, + 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, + 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, + 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66}) + +// NewGeneratorPoint returns a new Point set to the canonical generator. +func NewGeneratorPoint() *Point { + return new(Point).Set(generator) +} + +func (v *projCached) Zero() *projCached { + v.YplusX.One() + v.YminusX.One() + v.Z.One() + v.T2d.Zero() + return v +} + +func (v *affineCached) Zero() *affineCached { + v.YplusX.One() + v.YminusX.One() + v.T2d.Zero() + return v +} + +// Assignments. + +// Set sets v = u, and returns v. +func (v *Point) Set(u *Point) *Point { + *v = *u + return v +} + +// Encoding. + +// Bytes returns the canonical 32-byte encoding of v, according to RFC 8032, +// Section 5.1.2. +func (v *Point) Bytes() []byte { + // This function is outlined to make the allocations inline in the caller + // rather than happen on the heap. + var buf [32]byte + return v.bytes(&buf) +} + +func (v *Point) bytes(buf *[32]byte) []byte { + checkInitialized(v) + + var zInv, x, y field.Element + zInv.Invert(&v.z) // zInv = 1 / Z + x.Multiply(&v.x, &zInv) // x = X / Z + y.Multiply(&v.y, &zInv) // y = Y / Z + + out := copyFieldElement(buf, &y) + out[31] |= byte(x.IsNegative() << 7) + return out +} + +var feOne = new(field.Element).One() + +// SetBytes sets v = x, where x is a 32-byte encoding of v. If x does not +// represent a valid point on the curve, SetBytes returns nil and an error and +// the receiver is unchanged. Otherwise, SetBytes returns v. +// +// Note that SetBytes accepts all non-canonical encodings of valid points. +// That is, it follows decoding rules that match most implementations in +// the ecosystem rather than RFC 8032. +func (v *Point) SetBytes(x []byte) (*Point, error) { + // Specifically, the non-canonical encodings that are accepted are + // 1) the ones where the field element is not reduced (see the + // (*field.Element).SetBytes docs) and + // 2) the ones where the x-coordinate is zero and the sign bit is set. + // + // This is consistent with crypto/ed25519/internal/edwards25519. Read more + // at https://hdevalence.ca/blog/2020-10-04-its-25519am, specifically the + // "Canonical A, R" section. + + if len(x) != 32 { + return nil, errors.New("edwards25519: invalid point encoding length") + } + y := new(field.Element).SetBytes(x) + + // -x² + y² = 1 + dx²y² + // x² + dx²y² = x²(dy² + 1) = y² - 1 + // x² = (y² - 1) / (dy² + 1) + + // u = y² - 1 + y2 := new(field.Element).Square(y) + u := new(field.Element).Subtract(y2, feOne) + + // v = dy² + 1 + vv := new(field.Element).Multiply(y2, d) + vv = vv.Add(vv, feOne) + + // x = +√(u/v) + xx, wasSquare := new(field.Element).SqrtRatio(u, vv) + if wasSquare == 0 { + return nil, errors.New("edwards25519: invalid point encoding") + } + + // Select the negative square root if the sign bit is set. + xxNeg := new(field.Element).Negate(xx) + xx = xx.Select(xxNeg, xx, int(x[31]>>7)) + + v.x.Set(xx) + v.y.Set(y) + v.z.One() + v.t.Multiply(xx, y) // xy = T / Z + + return v, nil +} + +func copyFieldElement(buf *[32]byte, v *field.Element) []byte { + copy(buf[:], v.Bytes()) + return buf[:] +} + +// Conversions. + +func (v *projP2) FromP1xP1(p *projP1xP1) *projP2 { + v.X.Multiply(&p.X, &p.T) + v.Y.Multiply(&p.Y, &p.Z) + v.Z.Multiply(&p.Z, &p.T) + return v +} + +func (v *projP2) FromP3(p *Point) *projP2 { + v.X.Set(&p.x) + v.Y.Set(&p.y) + v.Z.Set(&p.z) + return v +} + +func (v *Point) fromP1xP1(p *projP1xP1) *Point { + v.x.Multiply(&p.X, &p.T) + v.y.Multiply(&p.Y, &p.Z) + v.z.Multiply(&p.Z, &p.T) + v.t.Multiply(&p.X, &p.Y) + return v +} + +func (v *Point) fromP2(p *projP2) *Point { + v.x.Multiply(&p.X, &p.Z) + v.y.Multiply(&p.Y, &p.Z) + v.z.Square(&p.Z) + v.t.Multiply(&p.X, &p.Y) + return v +} + +// d is a constant in the curve equation. +var d = new(field.Element).SetBytes([]byte{ + 0xa3, 0x78, 0x59, 0x13, 0xca, 0x4d, 0xeb, 0x75, + 0xab, 0xd8, 0x41, 0x41, 0x4d, 0x0a, 0x70, 0x00, + 0x98, 0xe8, 0x79, 0x77, 0x79, 0x40, 0xc7, 0x8c, + 0x73, 0xfe, 0x6f, 0x2b, 0xee, 0x6c, 0x03, 0x52}) +var d2 = new(field.Element).Add(d, d) + +func (v *projCached) FromP3(p *Point) *projCached { + v.YplusX.Add(&p.y, &p.x) + v.YminusX.Subtract(&p.y, &p.x) + v.Z.Set(&p.z) + v.T2d.Multiply(&p.t, d2) + return v +} + +func (v *affineCached) FromP3(p *Point) *affineCached { + v.YplusX.Add(&p.y, &p.x) + v.YminusX.Subtract(&p.y, &p.x) + v.T2d.Multiply(&p.t, d2) + + var invZ field.Element + invZ.Invert(&p.z) + v.YplusX.Multiply(&v.YplusX, &invZ) + v.YminusX.Multiply(&v.YminusX, &invZ) + v.T2d.Multiply(&v.T2d, &invZ) + return v +} + +// (Re)addition and subtraction. + +// Add sets v = p + q, and returns v. +func (v *Point) Add(p, q *Point) *Point { + checkInitialized(p, q) + qCached := new(projCached).FromP3(q) + result := new(projP1xP1).Add(p, qCached) + return v.fromP1xP1(result) +} + +// Subtract sets v = p - q, and returns v. +func (v *Point) Subtract(p, q *Point) *Point { + checkInitialized(p, q) + qCached := new(projCached).FromP3(q) + result := new(projP1xP1).Sub(p, qCached) + return v.fromP1xP1(result) +} + +func (v *projP1xP1) Add(p *Point, q *projCached) *projP1xP1 { + var YplusX, YminusX, PP, MM, TT2d, ZZ2 field.Element + + YplusX.Add(&p.y, &p.x) + YminusX.Subtract(&p.y, &p.x) + + PP.Multiply(&YplusX, &q.YplusX) + MM.Multiply(&YminusX, &q.YminusX) + TT2d.Multiply(&p.t, &q.T2d) + ZZ2.Multiply(&p.z, &q.Z) + + ZZ2.Add(&ZZ2, &ZZ2) + + v.X.Subtract(&PP, &MM) + v.Y.Add(&PP, &MM) + v.Z.Add(&ZZ2, &TT2d) + v.T.Subtract(&ZZ2, &TT2d) + return v +} + +func (v *projP1xP1) Sub(p *Point, q *projCached) *projP1xP1 { + var YplusX, YminusX, PP, MM, TT2d, ZZ2 field.Element + + YplusX.Add(&p.y, &p.x) + YminusX.Subtract(&p.y, &p.x) + + PP.Multiply(&YplusX, &q.YminusX) // flipped sign + MM.Multiply(&YminusX, &q.YplusX) // flipped sign + TT2d.Multiply(&p.t, &q.T2d) + ZZ2.Multiply(&p.z, &q.Z) + + ZZ2.Add(&ZZ2, &ZZ2) + + v.X.Subtract(&PP, &MM) + v.Y.Add(&PP, &MM) + v.Z.Subtract(&ZZ2, &TT2d) // flipped sign + v.T.Add(&ZZ2, &TT2d) // flipped sign + return v +} + +func (v *projP1xP1) AddAffine(p *Point, q *affineCached) *projP1xP1 { + var YplusX, YminusX, PP, MM, TT2d, Z2 field.Element + + YplusX.Add(&p.y, &p.x) + YminusX.Subtract(&p.y, &p.x) + + PP.Multiply(&YplusX, &q.YplusX) + MM.Multiply(&YminusX, &q.YminusX) + TT2d.Multiply(&p.t, &q.T2d) + + Z2.Add(&p.z, &p.z) + + v.X.Subtract(&PP, &MM) + v.Y.Add(&PP, &MM) + v.Z.Add(&Z2, &TT2d) + v.T.Subtract(&Z2, &TT2d) + return v +} + +func (v *projP1xP1) SubAffine(p *Point, q *affineCached) *projP1xP1 { + var YplusX, YminusX, PP, MM, TT2d, Z2 field.Element + + YplusX.Add(&p.y, &p.x) + YminusX.Subtract(&p.y, &p.x) + + PP.Multiply(&YplusX, &q.YminusX) // flipped sign + MM.Multiply(&YminusX, &q.YplusX) // flipped sign + TT2d.Multiply(&p.t, &q.T2d) + + Z2.Add(&p.z, &p.z) + + v.X.Subtract(&PP, &MM) + v.Y.Add(&PP, &MM) + v.Z.Subtract(&Z2, &TT2d) // flipped sign + v.T.Add(&Z2, &TT2d) // flipped sign + return v +} + +// Doubling. + +func (v *projP1xP1) Double(p *projP2) *projP1xP1 { + var XX, YY, ZZ2, XplusYsq field.Element + + XX.Square(&p.X) + YY.Square(&p.Y) + ZZ2.Square(&p.Z) + ZZ2.Add(&ZZ2, &ZZ2) + XplusYsq.Add(&p.X, &p.Y) + XplusYsq.Square(&XplusYsq) + + v.Y.Add(&YY, &XX) + v.Z.Subtract(&YY, &XX) + + v.X.Subtract(&XplusYsq, &v.Y) + v.T.Subtract(&ZZ2, &v.Z) + return v +} + +// Negation. + +// Negate sets v = -p, and returns v. +func (v *Point) Negate(p *Point) *Point { + checkInitialized(p) + v.x.Negate(&p.x) + v.y.Set(&p.y) + v.z.Set(&p.z) + v.t.Negate(&p.t) + return v +} + +// Equal returns 1 if v is equivalent to u, and 0 otherwise. +func (v *Point) Equal(u *Point) int { + checkInitialized(v, u) + + var t1, t2, t3, t4 field.Element + t1.Multiply(&v.x, &u.z) + t2.Multiply(&u.x, &v.z) + t3.Multiply(&v.y, &u.z) + t4.Multiply(&u.y, &v.z) + + return t1.Equal(&t2) & t3.Equal(&t4) +} + +// Constant-time operations + +// Select sets v to a if cond == 1 and to b if cond == 0. +func (v *projCached) Select(a, b *projCached, cond int) *projCached { + v.YplusX.Select(&a.YplusX, &b.YplusX, cond) + v.YminusX.Select(&a.YminusX, &b.YminusX, cond) + v.Z.Select(&a.Z, &b.Z, cond) + v.T2d.Select(&a.T2d, &b.T2d, cond) + return v +} + +// Select sets v to a if cond == 1 and to b if cond == 0. +func (v *affineCached) Select(a, b *affineCached, cond int) *affineCached { + v.YplusX.Select(&a.YplusX, &b.YplusX, cond) + v.YminusX.Select(&a.YminusX, &b.YminusX, cond) + v.T2d.Select(&a.T2d, &b.T2d, cond) + return v +} + +// CondNeg negates v if cond == 1 and leaves it unchanged if cond == 0. +func (v *projCached) CondNeg(cond int) *projCached { + v.YplusX.Swap(&v.YminusX, cond) + v.T2d.Select(new(field.Element).Negate(&v.T2d), &v.T2d, cond) + return v +} + +// CondNeg negates v if cond == 1 and leaves it unchanged if cond == 0. +func (v *affineCached) CondNeg(cond int) *affineCached { + v.YplusX.Swap(&v.YminusX, cond) + v.T2d.Select(new(field.Element).Negate(&v.T2d), &v.T2d, cond) + return v +} diff --git a/src/crypto/ed25519/internal/edwards25519/edwards25519_test.go b/src/crypto/ed25519/internal/edwards25519/edwards25519_test.go new file mode 100644 index 0000000..8031256 --- /dev/null +++ b/src/crypto/ed25519/internal/edwards25519/edwards25519_test.go @@ -0,0 +1,304 @@ +// Copyright (c) 2019 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 edwards25519 + +import ( + "crypto/ed25519/internal/edwards25519/field" + "encoding/hex" + "os" + "reflect" + "strings" + "testing" +) + +var B = NewGeneratorPoint() +var I = NewIdentityPoint() + +func checkOnCurve(t *testing.T, points ...*Point) { + t.Helper() + for i, p := range points { + var XX, YY, ZZ, ZZZZ field.Element + XX.Square(&p.x) + YY.Square(&p.y) + ZZ.Square(&p.z) + ZZZZ.Square(&ZZ) + // -x² + y² = 1 + dx²y² + // -(X/Z)² + (Y/Z)² = 1 + d(X/Z)²(Y/Z)² + // (-X² + Y²)/Z² = 1 + (dX²Y²)/Z⁴ + // (-X² + Y²)*Z² = Z⁴ + dX²Y² + var lhs, rhs field.Element + lhs.Subtract(&YY, &XX).Multiply(&lhs, &ZZ) + rhs.Multiply(d, &XX).Multiply(&rhs, &YY).Add(&rhs, &ZZZZ) + if lhs.Equal(&rhs) != 1 { + t.Errorf("X, Y, and Z do not specify a point on the curve\nX = %v\nY = %v\nZ = %v", p.x, p.y, p.z) + } + // xy = T/Z + lhs.Multiply(&p.x, &p.y) + rhs.Multiply(&p.z, &p.t) + if lhs.Equal(&rhs) != 1 { + t.Errorf("point %d is not valid\nX = %v\nY = %v\nZ = %v", i, p.x, p.y, p.z) + } + } +} + +func TestGenerator(t *testing.T) { + // These are the coordinates of B from RFC 8032, Section 5.1, converted to + // little endian hex. + x := "1ad5258f602d56c9b2a7259560c72c695cdcd6fd31e2a4c0fe536ecdd3366921" + y := "5866666666666666666666666666666666666666666666666666666666666666" + if got := hex.EncodeToString(B.x.Bytes()); got != x { + t.Errorf("wrong B.x: got %s, expected %s", got, x) + } + if got := hex.EncodeToString(B.y.Bytes()); got != y { + t.Errorf("wrong B.y: got %s, expected %s", got, y) + } + if B.z.Equal(feOne) != 1 { + t.Errorf("wrong B.z: got %v, expected 1", B.z) + } + // Check that t is correct. + checkOnCurve(t, B) +} + +func TestAddSubNegOnBasePoint(t *testing.T) { + checkLhs, checkRhs := &Point{}, &Point{} + + checkLhs.Add(B, B) + tmpP2 := new(projP2).FromP3(B) + tmpP1xP1 := new(projP1xP1).Double(tmpP2) + checkRhs.fromP1xP1(tmpP1xP1) + if checkLhs.Equal(checkRhs) != 1 { + t.Error("B + B != [2]B") + } + checkOnCurve(t, checkLhs, checkRhs) + + checkLhs.Subtract(B, B) + Bneg := new(Point).Negate(B) + checkRhs.Add(B, Bneg) + if checkLhs.Equal(checkRhs) != 1 { + t.Error("B - B != B + (-B)") + } + if I.Equal(checkLhs) != 1 { + t.Error("B - B != 0") + } + if I.Equal(checkRhs) != 1 { + t.Error("B + (-B) != 0") + } + checkOnCurve(t, checkLhs, checkRhs, Bneg) +} + +func TestComparable(t *testing.T) { + if reflect.TypeOf(Point{}).Comparable() { + t.Error("Point is unexpectedly comparable") + } +} + +func TestInvalidEncodings(t *testing.T) { + // An invalid point, that also happens to have y > p. + invalid := "efffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff7f" + p := NewGeneratorPoint() + if out, err := p.SetBytes(decodeHex(invalid)); err == nil { + t.Error("expected error for invalid point") + } else if out != nil { + t.Error("SetBytes did not return nil on an invalid encoding") + } else if p.Equal(B) != 1 { + t.Error("the Point was modified while decoding an invalid encoding") + } + checkOnCurve(t, p) +} + +func TestNonCanonicalPoints(t *testing.T) { + type test struct { + name string + encoding, canonical string + } + tests := []test{ + // Points with x = 0 and the sign bit set. With x = 0 the curve equation + // gives y² = 1, so y = ±1. 1 has two valid encodings. + { + "y=1,sign-", + "0100000000000000000000000000000000000000000000000000000000000080", + "0100000000000000000000000000000000000000000000000000000000000000", + }, + { + "y=p+1,sign-", + "eeffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff", + "0100000000000000000000000000000000000000000000000000000000000000", + }, + { + "y=p-1,sign-", + "ecffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff", + "ecffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff7f", + }, + + // Non-canonical y encodings with values 2²⁵⁵-19 (p) to 2²⁵⁵-1 (p+18). + { + "y=p,sign+", + "edffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff7f", + "0000000000000000000000000000000000000000000000000000000000000000", + }, + { + "y=p,sign-", + "edffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff", + "0000000000000000000000000000000000000000000000000000000000000080", + }, + { + "y=p+1,sign+", + "eeffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff7f", + "0100000000000000000000000000000000000000000000000000000000000000", + }, + // "y=p+1,sign-" is already tested above. + // p+2 is not a valid y-coordinate. + { + "y=p+3,sign+", + "f0ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff7f", + "0300000000000000000000000000000000000000000000000000000000000000", + }, + { + "y=p+3,sign-", + "f0ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff", + "0300000000000000000000000000000000000000000000000000000000000080", + }, + { + "y=p+4,sign+", + "f1ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff7f", + "0400000000000000000000000000000000000000000000000000000000000000", + }, + { + "y=p+4,sign-", + "f1ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff", + "0400000000000000000000000000000000000000000000000000000000000080", + }, + { + "y=p+5,sign+", + "f2ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff7f", + "0500000000000000000000000000000000000000000000000000000000000000", + }, + { + "y=p+5,sign-", + "f2ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff", + "0500000000000000000000000000000000000000000000000000000000000080", + }, + { + "y=p+6,sign+", + "f3ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff7f", + "0600000000000000000000000000000000000000000000000000000000000000", + }, + { + "y=p+6,sign-", + "f3ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff", + "0600000000000000000000000000000000000000000000000000000000000080", + }, + // p+7 is not a valid y-coordinate. + // p+8 is not a valid y-coordinate. + { + "y=p+9,sign+", + "f6ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff7f", + "0900000000000000000000000000000000000000000000000000000000000000", + }, + { + "y=p+9,sign-", + "f6ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff", + "0900000000000000000000000000000000000000000000000000000000000080", + }, + { + "y=p+10,sign+", + "f7ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff7f", + "0a00000000000000000000000000000000000000000000000000000000000000", + }, + { + "y=p+10,sign-", + "f7ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff", + "0a00000000000000000000000000000000000000000000000000000000000080", + }, + // p+11 is not a valid y-coordinate. + // p+12 is not a valid y-coordinate. + // p+13 is not a valid y-coordinate. + { + "y=p+14,sign+", + "fbffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff7f", + "0e00000000000000000000000000000000000000000000000000000000000000", + }, + { + "y=p+14,sign-", + "fbffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff", + "0e00000000000000000000000000000000000000000000000000000000000080", + }, + { + "y=p+15,sign+", + "fcffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff7f", + "0f00000000000000000000000000000000000000000000000000000000000000", + }, + { + "y=p+15,sign-", + "fcffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff", + "0f00000000000000000000000000000000000000000000000000000000000080", + }, + { + "y=p+16,sign+", + "fdffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff7f", + "1000000000000000000000000000000000000000000000000000000000000000", + }, + { + "y=p+16,sign-", + "fdffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff", + "1000000000000000000000000000000000000000000000000000000000000080", + }, + // p+17 is not a valid y-coordinate. + { + "y=p+18,sign+", + "ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff7f", + "1200000000000000000000000000000000000000000000000000000000000000", + }, + { + "y=p+18,sign-", + "ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff", + "1200000000000000000000000000000000000000000000000000000000000080", + }, + } + for _, tt := range tests { + t.Run(tt.name, func(t *testing.T) { + p1, err := new(Point).SetBytes(decodeHex(tt.encoding)) + if err != nil { + t.Fatalf("error decoding non-canonical point: %v", err) + } + p2, err := new(Point).SetBytes(decodeHex(tt.canonical)) + if err != nil { + t.Fatalf("error decoding canonical point: %v", err) + } + if p1.Equal(p2) != 1 { + t.Errorf("equivalent points are not equal: %v, %v", p1, p2) + } + if encoding := hex.EncodeToString(p1.Bytes()); encoding != tt.canonical { + t.Errorf("re-encoding does not match canonical; got %q, expected %q", encoding, tt.canonical) + } + checkOnCurve(t, p1, p2) + }) + } +} + +var testAllocationsSink byte + +func TestAllocations(t *testing.T) { + if strings.HasSuffix(os.Getenv("GO_BUILDER_NAME"), "-noopt") { + t.Skip("skipping allocations test without relevant optimizations") + } + if allocs := testing.AllocsPerRun(100, func() { + p := NewIdentityPoint() + p.Add(p, NewGeneratorPoint()) + s := NewScalar() + testAllocationsSink ^= s.Bytes()[0] + testAllocationsSink ^= p.Bytes()[0] + }); allocs > 0 { + t.Errorf("expected zero allocations, got %0.1v", allocs) + } +} + +func decodeHex(s string) []byte { + b, err := hex.DecodeString(s) + if err != nil { + panic(err) + } + return b +} diff --git a/src/crypto/ed25519/internal/edwards25519/field/_asm/fe_amd64_asm.go b/src/crypto/ed25519/internal/edwards25519/field/_asm/fe_amd64_asm.go new file mode 100644 index 0000000..fbc0cce --- /dev/null +++ b/src/crypto/ed25519/internal/edwards25519/field/_asm/fe_amd64_asm.go @@ -0,0 +1,294 @@ +// Copyright (c) 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 main + +import ( + "fmt" + + . "github.com/mmcloughlin/avo/build" + . "github.com/mmcloughlin/avo/gotypes" + . "github.com/mmcloughlin/avo/operand" + . "github.com/mmcloughlin/avo/reg" +) + +//go:generate go run . -out ../fe_amd64.s -stubs ../fe_amd64.go -pkg field + +func main() { + Package("crypto/ed25519/internal/edwards25519/field") + ConstraintExpr("amd64,gc,!purego") + feMul() + feSquare() + Generate() +} + +type namedComponent struct { + Component + name string +} + +func (c namedComponent) String() string { return c.name } + +type uint128 struct { + name string + hi, lo GPVirtual +} + +func (c uint128) String() string { return c.name } + +func feSquare() { + TEXT("feSquare", NOSPLIT, "func(out, a *Element)") + Doc("feSquare sets out = a * a. It works like feSquareGeneric.") + Pragma("noescape") + + a := Dereference(Param("a")) + l0 := namedComponent{a.Field("l0"), "l0"} + l1 := namedComponent{a.Field("l1"), "l1"} + l2 := namedComponent{a.Field("l2"), "l2"} + l3 := namedComponent{a.Field("l3"), "l3"} + l4 := namedComponent{a.Field("l4"), "l4"} + + // r0 = l0×l0 + 19×2×(l1×l4 + l2×l3) + r0 := uint128{"r0", GP64(), GP64()} + mul64(r0, 1, l0, l0) + addMul64(r0, 38, l1, l4) + addMul64(r0, 38, l2, l3) + + // r1 = 2×l0×l1 + 19×2×l2×l4 + 19×l3×l3 + r1 := uint128{"r1", GP64(), GP64()} + mul64(r1, 2, l0, l1) + addMul64(r1, 38, l2, l4) + addMul64(r1, 19, l3, l3) + + // r2 = = 2×l0×l2 + l1×l1 + 19×2×l3×l4 + r2 := uint128{"r2", GP64(), GP64()} + mul64(r2, 2, l0, l2) + addMul64(r2, 1, l1, l1) + addMul64(r2, 38, l3, l4) + + // r3 = = 2×l0×l3 + 2×l1×l2 + 19×l4×l4 + r3 := uint128{"r3", GP64(), GP64()} + mul64(r3, 2, l0, l3) + addMul64(r3, 2, l1, l2) + addMul64(r3, 19, l4, l4) + + // r4 = = 2×l0×l4 + 2×l1×l3 + l2×l2 + r4 := uint128{"r4", GP64(), GP64()} + mul64(r4, 2, l0, l4) + addMul64(r4, 2, l1, l3) + addMul64(r4, 1, l2, l2) + + Comment("First reduction chain") + maskLow51Bits := GP64() + MOVQ(Imm((1<<51)-1), maskLow51Bits) + c0, r0lo := shiftRightBy51(&r0) + c1, r1lo := shiftRightBy51(&r1) + c2, r2lo := shiftRightBy51(&r2) + c3, r3lo := shiftRightBy51(&r3) + c4, r4lo := shiftRightBy51(&r4) + maskAndAdd(r0lo, maskLow51Bits, c4, 19) + maskAndAdd(r1lo, maskLow51Bits, c0, 1) + maskAndAdd(r2lo, maskLow51Bits, c1, 1) + maskAndAdd(r3lo, maskLow51Bits, c2, 1) + maskAndAdd(r4lo, maskLow51Bits, c3, 1) + + Comment("Second reduction chain (carryPropagate)") + // c0 = r0 >> 51 + MOVQ(r0lo, c0) + SHRQ(Imm(51), c0) + // c1 = r1 >> 51 + MOVQ(r1lo, c1) + SHRQ(Imm(51), c1) + // c2 = r2 >> 51 + MOVQ(r2lo, c2) + SHRQ(Imm(51), c2) + // c3 = r3 >> 51 + MOVQ(r3lo, c3) + SHRQ(Imm(51), c3) + // c4 = r4 >> 51 + MOVQ(r4lo, c4) + SHRQ(Imm(51), c4) + maskAndAdd(r0lo, maskLow51Bits, c4, 19) + maskAndAdd(r1lo, maskLow51Bits, c0, 1) + maskAndAdd(r2lo, maskLow51Bits, c1, 1) + maskAndAdd(r3lo, maskLow51Bits, c2, 1) + maskAndAdd(r4lo, maskLow51Bits, c3, 1) + + Comment("Store output") + out := Dereference(Param("out")) + Store(r0lo, out.Field("l0")) + Store(r1lo, out.Field("l1")) + Store(r2lo, out.Field("l2")) + Store(r3lo, out.Field("l3")) + Store(r4lo, out.Field("l4")) + + RET() +} + +func feMul() { + TEXT("feMul", NOSPLIT, "func(out, a, b *Element)") + Doc("feMul sets out = a * b. It works like feMulGeneric.") + Pragma("noescape") + + a := Dereference(Param("a")) + a0 := namedComponent{a.Field("l0"), "a0"} + a1 := namedComponent{a.Field("l1"), "a1"} + a2 := namedComponent{a.Field("l2"), "a2"} + a3 := namedComponent{a.Field("l3"), "a3"} + a4 := namedComponent{a.Field("l4"), "a4"} + + b := Dereference(Param("b")) + b0 := namedComponent{b.Field("l0"), "b0"} + b1 := namedComponent{b.Field("l1"), "b1"} + b2 := namedComponent{b.Field("l2"), "b2"} + b3 := namedComponent{b.Field("l3"), "b3"} + b4 := namedComponent{b.Field("l4"), "b4"} + + // r0 = a0×b0 + 19×(a1×b4 + a2×b3 + a3×b2 + a4×b1) + r0 := uint128{"r0", GP64(), GP64()} + mul64(r0, 1, a0, b0) + addMul64(r0, 19, a1, b4) + addMul64(r0, 19, a2, b3) + addMul64(r0, 19, a3, b2) + addMul64(r0, 19, a4, b1) + + // r1 = a0×b1 + a1×b0 + 19×(a2×b4 + a3×b3 + a4×b2) + r1 := uint128{"r1", GP64(), GP64()} + mul64(r1, 1, a0, b1) + addMul64(r1, 1, a1, b0) + addMul64(r1, 19, a2, b4) + addMul64(r1, 19, a3, b3) + addMul64(r1, 19, a4, b2) + + // r2 = a0×b2 + a1×b1 + a2×b0 + 19×(a3×b4 + a4×b3) + r2 := uint128{"r2", GP64(), GP64()} + mul64(r2, 1, a0, b2) + addMul64(r2, 1, a1, b1) + addMul64(r2, 1, a2, b0) + addMul64(r2, 19, a3, b4) + addMul64(r2, 19, a4, b3) + + // r3 = a0×b3 + a1×b2 + a2×b1 + a3×b0 + 19×a4×b4 + r3 := uint128{"r3", GP64(), GP64()} + mul64(r3, 1, a0, b3) + addMul64(r3, 1, a1, b2) + addMul64(r3, 1, a2, b1) + addMul64(r3, 1, a3, b0) + addMul64(r3, 19, a4, b4) + + // r4 = a0×b4 + a1×b3 + a2×b2 + a3×b1 + a4×b0 + r4 := uint128{"r4", GP64(), GP64()} + mul64(r4, 1, a0, b4) + addMul64(r4, 1, a1, b3) + addMul64(r4, 1, a2, b2) + addMul64(r4, 1, a3, b1) + addMul64(r4, 1, a4, b0) + + Comment("First reduction chain") + maskLow51Bits := GP64() + MOVQ(Imm((1<<51)-1), maskLow51Bits) + c0, r0lo := shiftRightBy51(&r0) + c1, r1lo := shiftRightBy51(&r1) + c2, r2lo := shiftRightBy51(&r2) + c3, r3lo := shiftRightBy51(&r3) + c4, r4lo := shiftRightBy51(&r4) + maskAndAdd(r0lo, maskLow51Bits, c4, 19) + maskAndAdd(r1lo, maskLow51Bits, c0, 1) + maskAndAdd(r2lo, maskLow51Bits, c1, 1) + maskAndAdd(r3lo, maskLow51Bits, c2, 1) + maskAndAdd(r4lo, maskLow51Bits, c3, 1) + + Comment("Second reduction chain (carryPropagate)") + // c0 = r0 >> 51 + MOVQ(r0lo, c0) + SHRQ(Imm(51), c0) + // c1 = r1 >> 51 + MOVQ(r1lo, c1) + SHRQ(Imm(51), c1) + // c2 = r2 >> 51 + MOVQ(r2lo, c2) + SHRQ(Imm(51), c2) + // c3 = r3 >> 51 + MOVQ(r3lo, c3) + SHRQ(Imm(51), c3) + // c4 = r4 >> 51 + MOVQ(r4lo, c4) + SHRQ(Imm(51), c4) + maskAndAdd(r0lo, maskLow51Bits, c4, 19) + maskAndAdd(r1lo, maskLow51Bits, c0, 1) + maskAndAdd(r2lo, maskLow51Bits, c1, 1) + maskAndAdd(r3lo, maskLow51Bits, c2, 1) + maskAndAdd(r4lo, maskLow51Bits, c3, 1) + + Comment("Store output") + out := Dereference(Param("out")) + Store(r0lo, out.Field("l0")) + Store(r1lo, out.Field("l1")) + Store(r2lo, out.Field("l2")) + Store(r3lo, out.Field("l3")) + Store(r4lo, out.Field("l4")) + + RET() +} + +// mul64 sets r to i * aX * bX. +func mul64(r uint128, i int, aX, bX namedComponent) { + switch i { + case 1: + Comment(fmt.Sprintf("%s = %s×%s", r, aX, bX)) + Load(aX, RAX) + case 2: + Comment(fmt.Sprintf("%s = 2×%s×%s", r, aX, bX)) + Load(aX, RAX) + SHLQ(Imm(1), RAX) + default: + panic("unsupported i value") + } + MULQ(mustAddr(bX)) // RDX, RAX = RAX * bX + MOVQ(RAX, r.lo) + MOVQ(RDX, r.hi) +} + +// addMul64 sets r to r + i * aX * bX. +func addMul64(r uint128, i uint64, aX, bX namedComponent) { + switch i { + case 1: + Comment(fmt.Sprintf("%s += %s×%s", r, aX, bX)) + Load(aX, RAX) + default: + Comment(fmt.Sprintf("%s += %d×%s×%s", r, i, aX, bX)) + IMUL3Q(Imm(i), Load(aX, GP64()), RAX) + } + MULQ(mustAddr(bX)) // RDX, RAX = RAX * bX + ADDQ(RAX, r.lo) + ADCQ(RDX, r.hi) +} + +// shiftRightBy51 returns r >> 51 and r.lo. +// +// After this function is called, the uint128 may not be used anymore. +func shiftRightBy51(r *uint128) (out, lo GPVirtual) { + out = r.hi + lo = r.lo + SHLQ(Imm(64-51), r.lo, r.hi) + r.lo, r.hi = nil, nil // make sure the uint128 is unusable + return +} + +// maskAndAdd sets r = r&mask + c*i. +func maskAndAdd(r, mask, c GPVirtual, i uint64) { + ANDQ(mask, r) + if i != 1 { + IMUL3Q(Imm(i), c, c) + } + ADDQ(c, r) +} + +func mustAddr(c Component) Op { + b, err := c.Resolve() + if err != nil { + panic(err) + } + return b.Addr +} diff --git a/src/crypto/ed25519/internal/edwards25519/field/_asm/go.mod b/src/crypto/ed25519/internal/edwards25519/field/_asm/go.mod new file mode 100644 index 0000000..1127ade --- /dev/null +++ b/src/crypto/ed25519/internal/edwards25519/field/_asm/go.mod @@ -0,0 +1,5 @@ +module asm + +go 1.16 + +require github.com/mmcloughlin/avo v0.2.0 diff --git a/src/crypto/ed25519/internal/edwards25519/field/_asm/go.sum b/src/crypto/ed25519/internal/edwards25519/field/_asm/go.sum new file mode 100644 index 0000000..dae4777 --- /dev/null +++ b/src/crypto/ed25519/internal/edwards25519/field/_asm/go.sum @@ -0,0 +1,31 @@ +github.com/mmcloughlin/avo v0.2.0 h1:6vhoSaKtxb6f4RiH+LK2qL6GSMpFzhEwJYTTSZNy09w= +github.com/mmcloughlin/avo v0.2.0/go.mod h1:5tidO2Z9Z7N6X7UMcGg+1KTj51O8OxYDCMHxCZTVpEA= +github.com/yuin/goldmark v1.2.1/go.mod h1:3hX8gzYuyVAZsxl0MRgGTJEmQBFcNTphYh9decYSb74= +golang.org/x/arch v0.0.0-20210405154355-08b684f594a5/go.mod h1:flIaEI6LNU6xOCD5PaJvn9wGP0agmIOqjrtsKGRguv4= +golang.org/x/crypto v0.0.0-20190308221718-c2843e01d9a2/go.mod h1:djNgcEr1/C05ACkg1iLfiJU5Ep61QUkGW8qpdssI0+w= +golang.org/x/crypto v0.0.0-20191011191535-87dc89f01550/go.mod h1:yigFU9vqHzYiE8UmvKecakEJjdnWj3jj499lnFckfCI= +golang.org/x/crypto v0.0.0-20200622213623-75b288015ac9/go.mod h1:LzIPMQfyMNhhGPhUkYOs5KpL4U8rLKemX1yGLhDgUto= +golang.org/x/mod v0.3.0 h1:RM4zey1++hCTbCVQfnWeKs9/IEsaBLA8vTkd0WVtmH4= +golang.org/x/mod v0.3.0/go.mod h1:s0Qsj1ACt9ePp/hMypM3fl4fZqREWJwdYDEqhRiZZUA= +golang.org/x/net v0.0.0-20190404232315-eb5bcb51f2a3/go.mod h1:t9HGtf8HONx5eT2rtn7q6eTqICYqUVnKs3thJo3Qplg= +golang.org/x/net v0.0.0-20190620200207-3b0461eec859/go.mod h1:z5CRVTTTmAJ677TzLLGU+0bjPO0LkuOLi4/5GtJWs/s= +golang.org/x/net v0.0.0-20201021035429-f5854403a974/go.mod h1:sp8m0HH+o8qH0wwXwYZr8TS3Oi6o0r6Gce1SSxlDquU= +golang.org/x/sync v0.0.0-20190423024810-112230192c58/go.mod h1:RxMgew5VJxzue5/jJTE5uejpjVlOe/izrB70Jof72aM= +golang.org/x/sync v0.0.0-20201020160332-67f06af15bc9/go.mod h1:RxMgew5VJxzue5/jJTE5uejpjVlOe/izrB70Jof72aM= +golang.org/x/sys v0.0.0-20190215142949-d0b11bdaac8a/go.mod h1:STP8DvDyc/dI5b8T5hshtkjS+E42TnysNCUPdjciGhY= +golang.org/x/sys v0.0.0-20190412213103-97732733099d/go.mod h1:h1NjWce9XRLGQEsW7wpKNCjG9DtNlClVuFLEZdDNbEs= +golang.org/x/sys v0.0.0-20200930185726-fdedc70b468f/go.mod h1:h1NjWce9XRLGQEsW7wpKNCjG9DtNlClVuFLEZdDNbEs= +golang.org/x/sys v0.0.0-20210119212857-b64e53b001e4/go.mod h1:h1NjWce9XRLGQEsW7wpKNCjG9DtNlClVuFLEZdDNbEs= +golang.org/x/sys v0.0.0-20210403161142-5e06dd20ab57 h1:F5Gozwx4I1xtr/sr/8CFbb57iKi3297KFs0QDbGN60A= +golang.org/x/sys v0.0.0-20210403161142-5e06dd20ab57/go.mod h1:h1NjWce9XRLGQEsW7wpKNCjG9DtNlClVuFLEZdDNbEs= +golang.org/x/text v0.3.0/go.mod h1:NqM8EUOU14njkJ3fqMW+pc6Ldnwhi/IjpwHt7yyuwOQ= +golang.org/x/text v0.3.3/go.mod h1:5Zoc/QRtKVWzQhOtBMvqHzDpF6irO9z98xDceosuGiQ= +golang.org/x/tools v0.0.0-20180917221912-90fa682c2a6e/go.mod h1:n7NCudcB/nEzxVGmLbDWY5pfWTLqBcC2KZ6jyYvM4mQ= +golang.org/x/tools v0.0.0-20191119224855-298f0cb1881e/go.mod h1:b+2E5dAYhXwXZwtnZ6UAqBI28+e2cm9otk0dWdXHAEo= +golang.org/x/tools v0.1.0 h1:po9/4sTYwZU9lPhi1tOrb4hCv3qrhiQ77LZfGa2OjwY= +golang.org/x/tools v0.1.0/go.mod h1:xkSsbof2nBLbhDlRMhhhyNLN/zl3eTqcnHD5viDpcZ0= +golang.org/x/xerrors v0.0.0-20190717185122-a985d3407aa7/go.mod h1:I/5z698sn9Ka8TeJc9MKroUUfqBBauWjQqLJ2OPfmY0= +golang.org/x/xerrors v0.0.0-20191011141410-1b5146add898/go.mod h1:I/5z698sn9Ka8TeJc9MKroUUfqBBauWjQqLJ2OPfmY0= +golang.org/x/xerrors v0.0.0-20200804184101-5ec99f83aff1 h1:go1bK/D/BFZV2I8cIQd1NKEZ+0owSTG1fDTci4IqFcE= +golang.org/x/xerrors v0.0.0-20200804184101-5ec99f83aff1/go.mod h1:I/5z698sn9Ka8TeJc9MKroUUfqBBauWjQqLJ2OPfmY0= +rsc.io/pdf v0.1.1/go.mod h1:n8OzWcQ6Sp37PL01nO98y4iUCRdTGarVfzxY20ICaU4= diff --git a/src/crypto/ed25519/internal/edwards25519/field/fe.go b/src/crypto/ed25519/internal/edwards25519/field/fe.go new file mode 100644 index 0000000..dbe8659 --- /dev/null +++ b/src/crypto/ed25519/internal/edwards25519/field/fe.go @@ -0,0 +1,416 @@ +// Copyright (c) 2017 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 field implements fast arithmetic modulo 2^255-19. +package field + +import ( + "crypto/subtle" + "encoding/binary" + "math/bits" +) + +// Element represents an element of the field GF(2^255-19). Note that this +// is not a cryptographically secure group, and should only be used to interact +// with edwards25519.Point coordinates. +// +// This type works similarly to math/big.Int, and all arguments and receivers +// are allowed to alias. +// +// The zero value is a valid zero element. +type Element struct { + // An element t represents the integer + // t.l0 + t.l1*2^51 + t.l2*2^102 + t.l3*2^153 + t.l4*2^204 + // + // Between operations, all limbs are expected to be lower than 2^52. + l0 uint64 + l1 uint64 + l2 uint64 + l3 uint64 + l4 uint64 +} + +const maskLow51Bits uint64 = (1 << 51) - 1 + +var feZero = &Element{0, 0, 0, 0, 0} + +// Zero sets v = 0, and returns v. +func (v *Element) Zero() *Element { + *v = *feZero + return v +} + +var feOne = &Element{1, 0, 0, 0, 0} + +// One sets v = 1, and returns v. +func (v *Element) One() *Element { + *v = *feOne + return v +} + +// reduce reduces v modulo 2^255 - 19 and returns it. +func (v *Element) reduce() *Element { + v.carryPropagate() + + // After the light reduction we now have a field element representation + // v < 2^255 + 2^13 * 19, but need v < 2^255 - 19. + + // If v >= 2^255 - 19, then v + 19 >= 2^255, which would overflow 2^255 - 1, + // generating a carry. That is, c will be 0 if v < 2^255 - 19, and 1 otherwise. + c := (v.l0 + 19) >> 51 + c = (v.l1 + c) >> 51 + c = (v.l2 + c) >> 51 + c = (v.l3 + c) >> 51 + c = (v.l4 + c) >> 51 + + // If v < 2^255 - 19 and c = 0, this will be a no-op. Otherwise, it's + // effectively applying the reduction identity to the carry. + v.l0 += 19 * c + + v.l1 += v.l0 >> 51 + v.l0 = v.l0 & maskLow51Bits + v.l2 += v.l1 >> 51 + v.l1 = v.l1 & maskLow51Bits + v.l3 += v.l2 >> 51 + v.l2 = v.l2 & maskLow51Bits + v.l4 += v.l3 >> 51 + v.l3 = v.l3 & maskLow51Bits + // no additional carry + v.l4 = v.l4 & maskLow51Bits + + return v +} + +// Add sets v = a + b, and returns v. +func (v *Element) Add(a, b *Element) *Element { + v.l0 = a.l0 + b.l0 + v.l1 = a.l1 + b.l1 + v.l2 = a.l2 + b.l2 + v.l3 = a.l3 + b.l3 + v.l4 = a.l4 + b.l4 + // Using the generic implementation here is actually faster than the + // assembly. Probably because the body of this function is so simple that + // the compiler can figure out better optimizations by inlining the carry + // propagation. + return v.carryPropagateGeneric() +} + +// Subtract sets v = a - b, and returns v. +func (v *Element) Subtract(a, b *Element) *Element { + // We first add 2 * p, to guarantee the subtraction won't underflow, and + // then subtract b (which can be up to 2^255 + 2^13 * 19). + v.l0 = (a.l0 + 0xFFFFFFFFFFFDA) - b.l0 + v.l1 = (a.l1 + 0xFFFFFFFFFFFFE) - b.l1 + v.l2 = (a.l2 + 0xFFFFFFFFFFFFE) - b.l2 + v.l3 = (a.l3 + 0xFFFFFFFFFFFFE) - b.l3 + v.l4 = (a.l4 + 0xFFFFFFFFFFFFE) - b.l4 + return v.carryPropagate() +} + +// Negate sets v = -a, and returns v. +func (v *Element) Negate(a *Element) *Element { + return v.Subtract(feZero, a) +} + +// Invert sets v = 1/z mod p, and returns v. +// +// If z == 0, Invert returns v = 0. +func (v *Element) Invert(z *Element) *Element { + // Inversion is implemented as exponentiation with exponent p − 2. It uses the + // same sequence of 255 squarings and 11 multiplications as [Curve25519]. + var z2, z9, z11, z2_5_0, z2_10_0, z2_20_0, z2_50_0, z2_100_0, t Element + + z2.Square(z) // 2 + t.Square(&z2) // 4 + t.Square(&t) // 8 + z9.Multiply(&t, z) // 9 + z11.Multiply(&z9, &z2) // 11 + t.Square(&z11) // 22 + z2_5_0.Multiply(&t, &z9) // 31 = 2^5 - 2^0 + + t.Square(&z2_5_0) // 2^6 - 2^1 + for i := 0; i < 4; i++ { + t.Square(&t) // 2^10 - 2^5 + } + z2_10_0.Multiply(&t, &z2_5_0) // 2^10 - 2^0 + + t.Square(&z2_10_0) // 2^11 - 2^1 + for i := 0; i < 9; i++ { + t.Square(&t) // 2^20 - 2^10 + } + z2_20_0.Multiply(&t, &z2_10_0) // 2^20 - 2^0 + + t.Square(&z2_20_0) // 2^21 - 2^1 + for i := 0; i < 19; i++ { + t.Square(&t) // 2^40 - 2^20 + } + t.Multiply(&t, &z2_20_0) // 2^40 - 2^0 + + t.Square(&t) // 2^41 - 2^1 + for i := 0; i < 9; i++ { + t.Square(&t) // 2^50 - 2^10 + } + z2_50_0.Multiply(&t, &z2_10_0) // 2^50 - 2^0 + + t.Square(&z2_50_0) // 2^51 - 2^1 + for i := 0; i < 49; i++ { + t.Square(&t) // 2^100 - 2^50 + } + z2_100_0.Multiply(&t, &z2_50_0) // 2^100 - 2^0 + + t.Square(&z2_100_0) // 2^101 - 2^1 + for i := 0; i < 99; i++ { + t.Square(&t) // 2^200 - 2^100 + } + t.Multiply(&t, &z2_100_0) // 2^200 - 2^0 + + t.Square(&t) // 2^201 - 2^1 + for i := 0; i < 49; i++ { + t.Square(&t) // 2^250 - 2^50 + } + t.Multiply(&t, &z2_50_0) // 2^250 - 2^0 + + t.Square(&t) // 2^251 - 2^1 + t.Square(&t) // 2^252 - 2^2 + t.Square(&t) // 2^253 - 2^3 + t.Square(&t) // 2^254 - 2^4 + t.Square(&t) // 2^255 - 2^5 + + return v.Multiply(&t, &z11) // 2^255 - 21 +} + +// Set sets v = a, and returns v. +func (v *Element) Set(a *Element) *Element { + *v = *a + return v +} + +// SetBytes sets v to x, which must be a 32-byte little-endian encoding. +// +// Consistent with RFC 7748, the most significant bit (the high bit of the +// last byte) is ignored, and non-canonical values (2^255-19 through 2^255-1) +// are accepted. Note that this is laxer than specified by RFC 8032. +func (v *Element) SetBytes(x []byte) *Element { + if len(x) != 32 { + panic("edwards25519: invalid field element input size") + } + + // Bits 0:51 (bytes 0:8, bits 0:64, shift 0, mask 51). + v.l0 = binary.LittleEndian.Uint64(x[0:8]) + v.l0 &= maskLow51Bits + // Bits 51:102 (bytes 6:14, bits 48:112, shift 3, mask 51). + v.l1 = binary.LittleEndian.Uint64(x[6:14]) >> 3 + v.l1 &= maskLow51Bits + // Bits 102:153 (bytes 12:20, bits 96:160, shift 6, mask 51). + v.l2 = binary.LittleEndian.Uint64(x[12:20]) >> 6 + v.l2 &= maskLow51Bits + // Bits 153:204 (bytes 19:27, bits 152:216, shift 1, mask 51). + v.l3 = binary.LittleEndian.Uint64(x[19:27]) >> 1 + v.l3 &= maskLow51Bits + // Bits 204:251 (bytes 24:32, bits 192:256, shift 12, mask 51). + // Note: not bytes 25:33, shift 4, to avoid overread. + v.l4 = binary.LittleEndian.Uint64(x[24:32]) >> 12 + v.l4 &= maskLow51Bits + + return v +} + +// Bytes returns the canonical 32-byte little-endian encoding of v. +func (v *Element) Bytes() []byte { + // This function is outlined to make the allocations inline in the caller + // rather than happen on the heap. + var out [32]byte + return v.bytes(&out) +} + +func (v *Element) bytes(out *[32]byte) []byte { + t := *v + t.reduce() + + var buf [8]byte + for i, l := range [5]uint64{t.l0, t.l1, t.l2, t.l3, t.l4} { + bitsOffset := i * 51 + binary.LittleEndian.PutUint64(buf[:], l<<uint(bitsOffset%8)) + for i, bb := range buf { + off := bitsOffset/8 + i + if off >= len(out) { + break + } + out[off] |= bb + } + } + + return out[:] +} + +// Equal returns 1 if v and u are equal, and 0 otherwise. +func (v *Element) Equal(u *Element) int { + sa, sv := u.Bytes(), v.Bytes() + return subtle.ConstantTimeCompare(sa, sv) +} + +// mask64Bits returns 0xffffffff if cond is 1, and 0 otherwise. +func mask64Bits(cond int) uint64 { return ^(uint64(cond) - 1) } + +// Select sets v to a if cond == 1, and to b if cond == 0. +func (v *Element) Select(a, b *Element, cond int) *Element { + m := mask64Bits(cond) + v.l0 = (m & a.l0) | (^m & b.l0) + v.l1 = (m & a.l1) | (^m & b.l1) + v.l2 = (m & a.l2) | (^m & b.l2) + v.l3 = (m & a.l3) | (^m & b.l3) + v.l4 = (m & a.l4) | (^m & b.l4) + return v +} + +// Swap swaps v and u if cond == 1 or leaves them unchanged if cond == 0, and returns v. +func (v *Element) Swap(u *Element, cond int) { + m := mask64Bits(cond) + t := m & (v.l0 ^ u.l0) + v.l0 ^= t + u.l0 ^= t + t = m & (v.l1 ^ u.l1) + v.l1 ^= t + u.l1 ^= t + t = m & (v.l2 ^ u.l2) + v.l2 ^= t + u.l2 ^= t + t = m & (v.l3 ^ u.l3) + v.l3 ^= t + u.l3 ^= t + t = m & (v.l4 ^ u.l4) + v.l4 ^= t + u.l4 ^= t +} + +// IsNegative returns 1 if v is negative, and 0 otherwise. +func (v *Element) IsNegative() int { + return int(v.Bytes()[0] & 1) +} + +// Absolute sets v to |u|, and returns v. +func (v *Element) Absolute(u *Element) *Element { + return v.Select(new(Element).Negate(u), u, u.IsNegative()) +} + +// Multiply sets v = x * y, and returns v. +func (v *Element) Multiply(x, y *Element) *Element { + feMul(v, x, y) + return v +} + +// Square sets v = x * x, and returns v. +func (v *Element) Square(x *Element) *Element { + feSquare(v, x) + return v +} + +// Mult32 sets v = x * y, and returns v. +func (v *Element) Mult32(x *Element, y uint32) *Element { + x0lo, x0hi := mul51(x.l0, y) + x1lo, x1hi := mul51(x.l1, y) + x2lo, x2hi := mul51(x.l2, y) + x3lo, x3hi := mul51(x.l3, y) + x4lo, x4hi := mul51(x.l4, y) + v.l0 = x0lo + 19*x4hi // carried over per the reduction identity + v.l1 = x1lo + x0hi + v.l2 = x2lo + x1hi + v.l3 = x3lo + x2hi + v.l4 = x4lo + x3hi + // The hi portions are going to be only 32 bits, plus any previous excess, + // so we can skip the carry propagation. + return v +} + +// mul51 returns lo + hi * 2⁵¹ = a * b. +func mul51(a uint64, b uint32) (lo uint64, hi uint64) { + mh, ml := bits.Mul64(a, uint64(b)) + lo = ml & maskLow51Bits + hi = (mh << 13) | (ml >> 51) + return +} + +// Pow22523 set v = x^((p-5)/8), and returns v. (p-5)/8 is 2^252-3. +func (v *Element) Pow22523(x *Element) *Element { + var t0, t1, t2 Element + + t0.Square(x) // x^2 + t1.Square(&t0) // x^4 + t1.Square(&t1) // x^8 + t1.Multiply(x, &t1) // x^9 + t0.Multiply(&t0, &t1) // x^11 + t0.Square(&t0) // x^22 + t0.Multiply(&t1, &t0) // x^31 + t1.Square(&t0) // x^62 + for i := 1; i < 5; i++ { // x^992 + t1.Square(&t1) + } + t0.Multiply(&t1, &t0) // x^1023 -> 1023 = 2^10 - 1 + t1.Square(&t0) // 2^11 - 2 + for i := 1; i < 10; i++ { // 2^20 - 2^10 + t1.Square(&t1) + } + t1.Multiply(&t1, &t0) // 2^20 - 1 + t2.Square(&t1) // 2^21 - 2 + for i := 1; i < 20; i++ { // 2^40 - 2^20 + t2.Square(&t2) + } + t1.Multiply(&t2, &t1) // 2^40 - 1 + t1.Square(&t1) // 2^41 - 2 + for i := 1; i < 10; i++ { // 2^50 - 2^10 + t1.Square(&t1) + } + t0.Multiply(&t1, &t0) // 2^50 - 1 + t1.Square(&t0) // 2^51 - 2 + for i := 1; i < 50; i++ { // 2^100 - 2^50 + t1.Square(&t1) + } + t1.Multiply(&t1, &t0) // 2^100 - 1 + t2.Square(&t1) // 2^101 - 2 + for i := 1; i < 100; i++ { // 2^200 - 2^100 + t2.Square(&t2) + } + t1.Multiply(&t2, &t1) // 2^200 - 1 + t1.Square(&t1) // 2^201 - 2 + for i := 1; i < 50; i++ { // 2^250 - 2^50 + t1.Square(&t1) + } + t0.Multiply(&t1, &t0) // 2^250 - 1 + t0.Square(&t0) // 2^251 - 2 + t0.Square(&t0) // 2^252 - 4 + return v.Multiply(&t0, x) // 2^252 - 3 -> x^(2^252-3) +} + +// sqrtM1 is 2^((p-1)/4), which squared is equal to -1 by Euler's Criterion. +var sqrtM1 = &Element{1718705420411056, 234908883556509, + 2233514472574048, 2117202627021982, 765476049583133} + +// SqrtRatio sets r to the non-negative square root of the ratio of u and v. +// +// If u/v is square, SqrtRatio returns r and 1. If u/v is not square, SqrtRatio +// sets r according to Section 4.3 of draft-irtf-cfrg-ristretto255-decaf448-00, +// and returns r and 0. +func (r *Element) SqrtRatio(u, v *Element) (rr *Element, wasSquare int) { + var a, b Element + + // r = (u * v3) * (u * v7)^((p-5)/8) + v2 := a.Square(v) + uv3 := b.Multiply(u, b.Multiply(v2, v)) + uv7 := a.Multiply(uv3, a.Square(v2)) + r.Multiply(uv3, r.Pow22523(uv7)) + + check := a.Multiply(v, a.Square(r)) // check = v * r^2 + + uNeg := b.Negate(u) + correctSignSqrt := check.Equal(u) + flippedSignSqrt := check.Equal(uNeg) + flippedSignSqrtI := check.Equal(uNeg.Multiply(uNeg, sqrtM1)) + + rPrime := b.Multiply(r, sqrtM1) // r_prime = SQRT_M1 * r + // r = CT_SELECT(r_prime IF flipped_sign_sqrt | flipped_sign_sqrt_i ELSE r) + r.Select(rPrime, r, flippedSignSqrt|flippedSignSqrtI) + + r.Absolute(r) // Choose the nonnegative square root. + return r, correctSignSqrt | flippedSignSqrt +} diff --git a/src/crypto/ed25519/internal/edwards25519/field/fe_alias_test.go b/src/crypto/ed25519/internal/edwards25519/field/fe_alias_test.go new file mode 100644 index 0000000..5ad81df --- /dev/null +++ b/src/crypto/ed25519/internal/edwards25519/field/fe_alias_test.go @@ -0,0 +1,126 @@ +// Copyright (c) 2019 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 field + +import ( + "testing" + "testing/quick" +) + +func checkAliasingOneArg(f func(v, x *Element) *Element) func(v, x Element) bool { + return func(v, x Element) bool { + x1, v1 := x, x + + // Calculate a reference f(x) without aliasing. + if out := f(&v, &x); out != &v && isInBounds(out) { + return false + } + + // Test aliasing the argument and the receiver. + if out := f(&v1, &v1); out != &v1 || v1 != v { + return false + } + + // Ensure the arguments was not modified. + return x == x1 + } +} + +func checkAliasingTwoArgs(f func(v, x, y *Element) *Element) func(v, x, y Element) bool { + return func(v, x, y Element) bool { + x1, y1, v1 := x, y, Element{} + + // Calculate a reference f(x, y) without aliasing. + if out := f(&v, &x, &y); out != &v && isInBounds(out) { + return false + } + + // Test aliasing the first argument and the receiver. + v1 = x + if out := f(&v1, &v1, &y); out != &v1 || v1 != v { + return false + } + // Test aliasing the second argument and the receiver. + v1 = y + if out := f(&v1, &x, &v1); out != &v1 || v1 != v { + return false + } + + // Calculate a reference f(x, x) without aliasing. + if out := f(&v, &x, &x); out != &v { + return false + } + + // Test aliasing the first argument and the receiver. + v1 = x + if out := f(&v1, &v1, &x); out != &v1 || v1 != v { + return false + } + // Test aliasing the second argument and the receiver. + v1 = x + if out := f(&v1, &x, &v1); out != &v1 || v1 != v { + return false + } + // Test aliasing both arguments and the receiver. + v1 = x + if out := f(&v1, &v1, &v1); out != &v1 || v1 != v { + return false + } + + // Ensure the arguments were not modified. + return x == x1 && y == y1 + } +} + +// TestAliasing checks that receivers and arguments can alias each other without +// leading to incorrect results. That is, it ensures that it's safe to write +// +// v.Invert(v) +// +// or +// +// v.Add(v, v) +// +// without any of the inputs getting clobbered by the output being written. +func TestAliasing(t *testing.T) { + type target struct { + name string + oneArgF func(v, x *Element) *Element + twoArgsF func(v, x, y *Element) *Element + } + for _, tt := range []target{ + {name: "Absolute", oneArgF: (*Element).Absolute}, + {name: "Invert", oneArgF: (*Element).Invert}, + {name: "Negate", oneArgF: (*Element).Negate}, + {name: "Set", oneArgF: (*Element).Set}, + {name: "Square", oneArgF: (*Element).Square}, + {name: "Multiply", twoArgsF: (*Element).Multiply}, + {name: "Add", twoArgsF: (*Element).Add}, + {name: "Subtract", twoArgsF: (*Element).Subtract}, + { + name: "Select0", + twoArgsF: func(v, x, y *Element) *Element { + return (*Element).Select(v, x, y, 0) + }, + }, + { + name: "Select1", + twoArgsF: func(v, x, y *Element) *Element { + return (*Element).Select(v, x, y, 1) + }, + }, + } { + var err error + switch { + case tt.oneArgF != nil: + err = quick.Check(checkAliasingOneArg(tt.oneArgF), &quick.Config{MaxCountScale: 1 << 8}) + case tt.twoArgsF != nil: + err = quick.Check(checkAliasingTwoArgs(tt.twoArgsF), &quick.Config{MaxCountScale: 1 << 8}) + } + if err != nil { + t.Errorf("%v: %v", tt.name, err) + } + } +} diff --git a/src/crypto/ed25519/internal/edwards25519/field/fe_amd64.go b/src/crypto/ed25519/internal/edwards25519/field/fe_amd64.go new file mode 100644 index 0000000..363020b --- /dev/null +++ b/src/crypto/ed25519/internal/edwards25519/field/fe_amd64.go @@ -0,0 +1,13 @@ +// Code generated by command: go run fe_amd64_asm.go -out ../fe_amd64.s -stubs ../fe_amd64.go -pkg field. DO NOT EDIT. + +//go:build amd64 && gc && !purego + +package field + +// feMul sets out = a * b. It works like feMulGeneric. +//go:noescape +func feMul(out *Element, a *Element, b *Element) + +// feSquare sets out = a * a. It works like feSquareGeneric. +//go:noescape +func feSquare(out *Element, a *Element) diff --git a/src/crypto/ed25519/internal/edwards25519/field/fe_amd64.s b/src/crypto/ed25519/internal/edwards25519/field/fe_amd64.s new file mode 100644 index 0000000..0aa1e86 --- /dev/null +++ b/src/crypto/ed25519/internal/edwards25519/field/fe_amd64.s @@ -0,0 +1,378 @@ +// Code generated by command: go run fe_amd64_asm.go -out ../fe_amd64.s -stubs ../fe_amd64.go -pkg field. DO NOT EDIT. + +// +build amd64,gc,!purego + +#include "textflag.h" + +// func feMul(out *Element, a *Element, b *Element) +TEXT ·feMul(SB), NOSPLIT, $0-24 + MOVQ a+8(FP), CX + MOVQ b+16(FP), BX + + // r0 = a0×b0 + MOVQ (CX), AX + MULQ (BX) + MOVQ AX, DI + MOVQ DX, SI + + // r0 += 19×a1×b4 + MOVQ 8(CX), AX + IMUL3Q $0x13, AX, AX + MULQ 32(BX) + ADDQ AX, DI + ADCQ DX, SI + + // r0 += 19×a2×b3 + MOVQ 16(CX), AX + IMUL3Q $0x13, AX, AX + MULQ 24(BX) + ADDQ AX, DI + ADCQ DX, SI + + // r0 += 19×a3×b2 + MOVQ 24(CX), AX + IMUL3Q $0x13, AX, AX + MULQ 16(BX) + ADDQ AX, DI + ADCQ DX, SI + + // r0 += 19×a4×b1 + MOVQ 32(CX), AX + IMUL3Q $0x13, AX, AX + MULQ 8(BX) + ADDQ AX, DI + ADCQ DX, SI + + // r1 = a0×b1 + MOVQ (CX), AX + MULQ 8(BX) + MOVQ AX, R9 + MOVQ DX, R8 + + // r1 += a1×b0 + MOVQ 8(CX), AX + MULQ (BX) + ADDQ AX, R9 + ADCQ DX, R8 + + // r1 += 19×a2×b4 + MOVQ 16(CX), AX + IMUL3Q $0x13, AX, AX + MULQ 32(BX) + ADDQ AX, R9 + ADCQ DX, R8 + + // r1 += 19×a3×b3 + MOVQ 24(CX), AX + IMUL3Q $0x13, AX, AX + MULQ 24(BX) + ADDQ AX, R9 + ADCQ DX, R8 + + // r1 += 19×a4×b2 + MOVQ 32(CX), AX + IMUL3Q $0x13, AX, AX + MULQ 16(BX) + ADDQ AX, R9 + ADCQ DX, R8 + + // r2 = a0×b2 + MOVQ (CX), AX + MULQ 16(BX) + MOVQ AX, R11 + MOVQ DX, R10 + + // r2 += a1×b1 + MOVQ 8(CX), AX + MULQ 8(BX) + ADDQ AX, R11 + ADCQ DX, R10 + + // r2 += a2×b0 + MOVQ 16(CX), AX + MULQ (BX) + ADDQ AX, R11 + ADCQ DX, R10 + + // r2 += 19×a3×b4 + MOVQ 24(CX), AX + IMUL3Q $0x13, AX, AX + MULQ 32(BX) + ADDQ AX, R11 + ADCQ DX, R10 + + // r2 += 19×a4×b3 + MOVQ 32(CX), AX + IMUL3Q $0x13, AX, AX + MULQ 24(BX) + ADDQ AX, R11 + ADCQ DX, R10 + + // r3 = a0×b3 + MOVQ (CX), AX + MULQ 24(BX) + MOVQ AX, R13 + MOVQ DX, R12 + + // r3 += a1×b2 + MOVQ 8(CX), AX + MULQ 16(BX) + ADDQ AX, R13 + ADCQ DX, R12 + + // r3 += a2×b1 + MOVQ 16(CX), AX + MULQ 8(BX) + ADDQ AX, R13 + ADCQ DX, R12 + + // r3 += a3×b0 + MOVQ 24(CX), AX + MULQ (BX) + ADDQ AX, R13 + ADCQ DX, R12 + + // r3 += 19×a4×b4 + MOVQ 32(CX), AX + IMUL3Q $0x13, AX, AX + MULQ 32(BX) + ADDQ AX, R13 + ADCQ DX, R12 + + // r4 = a0×b4 + MOVQ (CX), AX + MULQ 32(BX) + MOVQ AX, R15 + MOVQ DX, R14 + + // r4 += a1×b3 + MOVQ 8(CX), AX + MULQ 24(BX) + ADDQ AX, R15 + ADCQ DX, R14 + + // r4 += a2×b2 + MOVQ 16(CX), AX + MULQ 16(BX) + ADDQ AX, R15 + ADCQ DX, R14 + + // r4 += a3×b1 + MOVQ 24(CX), AX + MULQ 8(BX) + ADDQ AX, R15 + ADCQ DX, R14 + + // r4 += a4×b0 + MOVQ 32(CX), AX + MULQ (BX) + ADDQ AX, R15 + ADCQ DX, R14 + + // First reduction chain + MOVQ $0x0007ffffffffffff, AX + SHLQ $0x0d, DI, SI + SHLQ $0x0d, R9, R8 + SHLQ $0x0d, R11, R10 + SHLQ $0x0d, R13, R12 + SHLQ $0x0d, R15, R14 + ANDQ AX, DI + IMUL3Q $0x13, R14, R14 + ADDQ R14, DI + ANDQ AX, R9 + ADDQ SI, R9 + ANDQ AX, R11 + ADDQ R8, R11 + ANDQ AX, R13 + ADDQ R10, R13 + ANDQ AX, R15 + ADDQ R12, R15 + + // Second reduction chain (carryPropagate) + MOVQ DI, SI + SHRQ $0x33, SI + MOVQ R9, R8 + SHRQ $0x33, R8 + MOVQ R11, R10 + SHRQ $0x33, R10 + MOVQ R13, R12 + SHRQ $0x33, R12 + MOVQ R15, R14 + SHRQ $0x33, R14 + ANDQ AX, DI + IMUL3Q $0x13, R14, R14 + ADDQ R14, DI + ANDQ AX, R9 + ADDQ SI, R9 + ANDQ AX, R11 + ADDQ R8, R11 + ANDQ AX, R13 + ADDQ R10, R13 + ANDQ AX, R15 + ADDQ R12, R15 + + // Store output + MOVQ out+0(FP), AX + MOVQ DI, (AX) + MOVQ R9, 8(AX) + MOVQ R11, 16(AX) + MOVQ R13, 24(AX) + MOVQ R15, 32(AX) + RET + +// func feSquare(out *Element, a *Element) +TEXT ·feSquare(SB), NOSPLIT, $0-16 + MOVQ a+8(FP), CX + + // r0 = l0×l0 + MOVQ (CX), AX + MULQ (CX) + MOVQ AX, SI + MOVQ DX, BX + + // r0 += 38×l1×l4 + MOVQ 8(CX), AX + IMUL3Q $0x26, AX, AX + MULQ 32(CX) + ADDQ AX, SI + ADCQ DX, BX + + // r0 += 38×l2×l3 + MOVQ 16(CX), AX + IMUL3Q $0x26, AX, AX + MULQ 24(CX) + ADDQ AX, SI + ADCQ DX, BX + + // r1 = 2×l0×l1 + MOVQ (CX), AX + SHLQ $0x01, AX + MULQ 8(CX) + MOVQ AX, R8 + MOVQ DX, DI + + // r1 += 38×l2×l4 + MOVQ 16(CX), AX + IMUL3Q $0x26, AX, AX + MULQ 32(CX) + ADDQ AX, R8 + ADCQ DX, DI + + // r1 += 19×l3×l3 + MOVQ 24(CX), AX + IMUL3Q $0x13, AX, AX + MULQ 24(CX) + ADDQ AX, R8 + ADCQ DX, DI + + // r2 = 2×l0×l2 + MOVQ (CX), AX + SHLQ $0x01, AX + MULQ 16(CX) + MOVQ AX, R10 + MOVQ DX, R9 + + // r2 += l1×l1 + MOVQ 8(CX), AX + MULQ 8(CX) + ADDQ AX, R10 + ADCQ DX, R9 + + // r2 += 38×l3×l4 + MOVQ 24(CX), AX + IMUL3Q $0x26, AX, AX + MULQ 32(CX) + ADDQ AX, R10 + ADCQ DX, R9 + + // r3 = 2×l0×l3 + MOVQ (CX), AX + SHLQ $0x01, AX + MULQ 24(CX) + MOVQ AX, R12 + MOVQ DX, R11 + + // r3 += 2×l1×l2 + MOVQ 8(CX), AX + IMUL3Q $0x02, AX, AX + MULQ 16(CX) + ADDQ AX, R12 + ADCQ DX, R11 + + // r3 += 19×l4×l4 + MOVQ 32(CX), AX + IMUL3Q $0x13, AX, AX + MULQ 32(CX) + ADDQ AX, R12 + ADCQ DX, R11 + + // r4 = 2×l0×l4 + MOVQ (CX), AX + SHLQ $0x01, AX + MULQ 32(CX) + MOVQ AX, R14 + MOVQ DX, R13 + + // r4 += 2×l1×l3 + MOVQ 8(CX), AX + IMUL3Q $0x02, AX, AX + MULQ 24(CX) + ADDQ AX, R14 + ADCQ DX, R13 + + // r4 += l2×l2 + MOVQ 16(CX), AX + MULQ 16(CX) + ADDQ AX, R14 + ADCQ DX, R13 + + // First reduction chain + MOVQ $0x0007ffffffffffff, AX + SHLQ $0x0d, SI, BX + SHLQ $0x0d, R8, DI + SHLQ $0x0d, R10, R9 + SHLQ $0x0d, R12, R11 + SHLQ $0x0d, R14, R13 + ANDQ AX, SI + IMUL3Q $0x13, R13, R13 + ADDQ R13, SI + ANDQ AX, R8 + ADDQ BX, R8 + ANDQ AX, R10 + ADDQ DI, R10 + ANDQ AX, R12 + ADDQ R9, R12 + ANDQ AX, R14 + ADDQ R11, R14 + + // Second reduction chain (carryPropagate) + MOVQ SI, BX + SHRQ $0x33, BX + MOVQ R8, DI + SHRQ $0x33, DI + MOVQ R10, R9 + SHRQ $0x33, R9 + MOVQ R12, R11 + SHRQ $0x33, R11 + MOVQ R14, R13 + SHRQ $0x33, R13 + ANDQ AX, SI + IMUL3Q $0x13, R13, R13 + ADDQ R13, SI + ANDQ AX, R8 + ADDQ BX, R8 + ANDQ AX, R10 + ADDQ DI, R10 + ANDQ AX, R12 + ADDQ R9, R12 + ANDQ AX, R14 + ADDQ R11, R14 + + // Store output + MOVQ out+0(FP), AX + MOVQ SI, (AX) + MOVQ R8, 8(AX) + MOVQ R10, 16(AX) + MOVQ R12, 24(AX) + MOVQ R14, 32(AX) + RET diff --git a/src/crypto/ed25519/internal/edwards25519/field/fe_amd64_noasm.go b/src/crypto/ed25519/internal/edwards25519/field/fe_amd64_noasm.go new file mode 100644 index 0000000..9da280d --- /dev/null +++ b/src/crypto/ed25519/internal/edwards25519/field/fe_amd64_noasm.go @@ -0,0 +1,11 @@ +// Copyright (c) 2019 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 || !gc || purego + +package field + +func feMul(v, x, y *Element) { feMulGeneric(v, x, y) } + +func feSquare(v, x *Element) { feSquareGeneric(v, x) } diff --git a/src/crypto/ed25519/internal/edwards25519/field/fe_arm64.go b/src/crypto/ed25519/internal/edwards25519/field/fe_arm64.go new file mode 100644 index 0000000..075fe9b --- /dev/null +++ b/src/crypto/ed25519/internal/edwards25519/field/fe_arm64.go @@ -0,0 +1,15 @@ +// Copyright (c) 2020 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 arm64 && gc && !purego + +package field + +//go:noescape +func carryPropagate(v *Element) + +func (v *Element) carryPropagate() *Element { + carryPropagate(v) + return v +} diff --git a/src/crypto/ed25519/internal/edwards25519/field/fe_arm64.s b/src/crypto/ed25519/internal/edwards25519/field/fe_arm64.s new file mode 100644 index 0000000..751ab2a --- /dev/null +++ b/src/crypto/ed25519/internal/edwards25519/field/fe_arm64.s @@ -0,0 +1,42 @@ +// Copyright (c) 2020 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. + +// +build arm64,gc,!purego + +#include "textflag.h" + +// carryPropagate works exactly like carryPropagateGeneric and uses the +// same AND, ADD, and LSR+MADD instructions emitted by the compiler, but +// avoids loading R0-R4 twice and uses LDP and STP. +// +// See https://golang.org/issues/43145 for the main compiler issue. +// +// func carryPropagate(v *Element) +TEXT ·carryPropagate(SB),NOFRAME|NOSPLIT,$0-8 + MOVD v+0(FP), R20 + + LDP 0(R20), (R0, R1) + LDP 16(R20), (R2, R3) + MOVD 32(R20), R4 + + AND $0x7ffffffffffff, R0, R10 + AND $0x7ffffffffffff, R1, R11 + AND $0x7ffffffffffff, R2, R12 + AND $0x7ffffffffffff, R3, R13 + AND $0x7ffffffffffff, R4, R14 + + ADD R0>>51, R11, R11 + ADD R1>>51, R12, R12 + ADD R2>>51, R13, R13 + ADD R3>>51, R14, R14 + // R4>>51 * 19 + R10 -> R10 + LSR $51, R4, R21 + MOVD $19, R22 + MADD R22, R10, R21, R10 + + STP (R10, R11), 0(R20) + STP (R12, R13), 16(R20) + MOVD R14, 32(R20) + + RET diff --git a/src/crypto/ed25519/internal/edwards25519/field/fe_arm64_noasm.go b/src/crypto/ed25519/internal/edwards25519/field/fe_arm64_noasm.go new file mode 100644 index 0000000..fc029ac --- /dev/null +++ b/src/crypto/ed25519/internal/edwards25519/field/fe_arm64_noasm.go @@ -0,0 +1,11 @@ +// Copyright (c) 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 !arm64 || !gc || purego + +package field + +func (v *Element) carryPropagate() *Element { + return v.carryPropagateGeneric() +} diff --git a/src/crypto/ed25519/internal/edwards25519/field/fe_bench_test.go b/src/crypto/ed25519/internal/edwards25519/field/fe_bench_test.go new file mode 100644 index 0000000..77dc06c --- /dev/null +++ b/src/crypto/ed25519/internal/edwards25519/field/fe_bench_test.go @@ -0,0 +1,36 @@ +// Copyright (c) 2019 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 field + +import "testing" + +func BenchmarkAdd(b *testing.B) { + var x, y Element + x.One() + y.Add(feOne, feOne) + b.ResetTimer() + for i := 0; i < b.N; i++ { + x.Add(&x, &y) + } +} + +func BenchmarkMultiply(b *testing.B) { + var x, y Element + x.One() + y.Add(feOne, feOne) + b.ResetTimer() + for i := 0; i < b.N; i++ { + x.Multiply(&x, &y) + } +} + +func BenchmarkMult32(b *testing.B) { + var x Element + x.One() + b.ResetTimer() + for i := 0; i < b.N; i++ { + x.Mult32(&x, 0xaa42aa42) + } +} diff --git a/src/crypto/ed25519/internal/edwards25519/field/fe_generic.go b/src/crypto/ed25519/internal/edwards25519/field/fe_generic.go new file mode 100644 index 0000000..bccf851 --- /dev/null +++ b/src/crypto/ed25519/internal/edwards25519/field/fe_generic.go @@ -0,0 +1,264 @@ +// Copyright (c) 2017 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 field + +import "math/bits" + +// uint128 holds a 128-bit number as two 64-bit limbs, for use with the +// bits.Mul64 and bits.Add64 intrinsics. +type uint128 struct { + lo, hi uint64 +} + +// mul64 returns a * b. +func mul64(a, b uint64) uint128 { + hi, lo := bits.Mul64(a, b) + return uint128{lo, hi} +} + +// addMul64 returns v + a * b. +func addMul64(v uint128, a, b uint64) uint128 { + hi, lo := bits.Mul64(a, b) + lo, c := bits.Add64(lo, v.lo, 0) + hi, _ = bits.Add64(hi, v.hi, c) + return uint128{lo, hi} +} + +// shiftRightBy51 returns a >> 51. a is assumed to be at most 115 bits. +func shiftRightBy51(a uint128) uint64 { + return (a.hi << (64 - 51)) | (a.lo >> 51) +} + +func feMulGeneric(v, a, b *Element) { + a0 := a.l0 + a1 := a.l1 + a2 := a.l2 + a3 := a.l3 + a4 := a.l4 + + b0 := b.l0 + b1 := b.l1 + b2 := b.l2 + b3 := b.l3 + b4 := b.l4 + + // Limb multiplication works like pen-and-paper columnar multiplication, but + // with 51-bit limbs instead of digits. + // + // a4 a3 a2 a1 a0 x + // b4 b3 b2 b1 b0 = + // ------------------------ + // a4b0 a3b0 a2b0 a1b0 a0b0 + + // a4b1 a3b1 a2b1 a1b1 a0b1 + + // a4b2 a3b2 a2b2 a1b2 a0b2 + + // a4b3 a3b3 a2b3 a1b3 a0b3 + + // a4b4 a3b4 a2b4 a1b4 a0b4 = + // ---------------------------------------------- + // r8 r7 r6 r5 r4 r3 r2 r1 r0 + // + // We can then use the reduction identity (a * 2²⁵⁵ + b = a * 19 + b) to + // reduce the limbs that would overflow 255 bits. r5 * 2²⁵⁵ becomes 19 * r5, + // r6 * 2³⁰⁶ becomes 19 * r6 * 2⁵¹, etc. + // + // Reduction can be carried out simultaneously to multiplication. For + // example, we do not compute r5: whenever the result of a multiplication + // belongs to r5, like a1b4, we multiply it by 19 and add the result to r0. + // + // a4b0 a3b0 a2b0 a1b0 a0b0 + + // a3b1 a2b1 a1b1 a0b1 19×a4b1 + + // a2b2 a1b2 a0b2 19×a4b2 19×a3b2 + + // a1b3 a0b3 19×a4b3 19×a3b3 19×a2b3 + + // a0b4 19×a4b4 19×a3b4 19×a2b4 19×a1b4 = + // -------------------------------------- + // r4 r3 r2 r1 r0 + // + // Finally we add up the columns into wide, overlapping limbs. + + a1_19 := a1 * 19 + a2_19 := a2 * 19 + a3_19 := a3 * 19 + a4_19 := a4 * 19 + + // r0 = a0×b0 + 19×(a1×b4 + a2×b3 + a3×b2 + a4×b1) + r0 := mul64(a0, b0) + r0 = addMul64(r0, a1_19, b4) + r0 = addMul64(r0, a2_19, b3) + r0 = addMul64(r0, a3_19, b2) + r0 = addMul64(r0, a4_19, b1) + + // r1 = a0×b1 + a1×b0 + 19×(a2×b4 + a3×b3 + a4×b2) + r1 := mul64(a0, b1) + r1 = addMul64(r1, a1, b0) + r1 = addMul64(r1, a2_19, b4) + r1 = addMul64(r1, a3_19, b3) + r1 = addMul64(r1, a4_19, b2) + + // r2 = a0×b2 + a1×b1 + a2×b0 + 19×(a3×b4 + a4×b3) + r2 := mul64(a0, b2) + r2 = addMul64(r2, a1, b1) + r2 = addMul64(r2, a2, b0) + r2 = addMul64(r2, a3_19, b4) + r2 = addMul64(r2, a4_19, b3) + + // r3 = a0×b3 + a1×b2 + a2×b1 + a3×b0 + 19×a4×b4 + r3 := mul64(a0, b3) + r3 = addMul64(r3, a1, b2) + r3 = addMul64(r3, a2, b1) + r3 = addMul64(r3, a3, b0) + r3 = addMul64(r3, a4_19, b4) + + // r4 = a0×b4 + a1×b3 + a2×b2 + a3×b1 + a4×b0 + r4 := mul64(a0, b4) + r4 = addMul64(r4, a1, b3) + r4 = addMul64(r4, a2, b2) + r4 = addMul64(r4, a3, b1) + r4 = addMul64(r4, a4, b0) + + // After the multiplication, we need to reduce (carry) the five coefficients + // to obtain a result with limbs that are at most slightly larger than 2⁵¹, + // to respect the Element invariant. + // + // Overall, the reduction works the same as carryPropagate, except with + // wider inputs: we take the carry for each coefficient by shifting it right + // by 51, and add it to the limb above it. The top carry is multiplied by 19 + // according to the reduction identity and added to the lowest limb. + // + // The largest coefficient (r0) will be at most 111 bits, which guarantees + // that all carries are at most 111 - 51 = 60 bits, which fits in a uint64. + // + // r0 = a0×b0 + 19×(a1×b4 + a2×b3 + a3×b2 + a4×b1) + // r0 < 2⁵²×2⁵² + 19×(2⁵²×2⁵² + 2⁵²×2⁵² + 2⁵²×2⁵² + 2⁵²×2⁵²) + // r0 < (1 + 19 × 4) × 2⁵² × 2⁵² + // r0 < 2⁷ × 2⁵² × 2⁵² + // r0 < 2¹¹¹ + // + // Moreover, the top coefficient (r4) is at most 107 bits, so c4 is at most + // 56 bits, and c4 * 19 is at most 61 bits, which again fits in a uint64 and + // allows us to easily apply the reduction identity. + // + // r4 = a0×b4 + a1×b3 + a2×b2 + a3×b1 + a4×b0 + // r4 < 5 × 2⁵² × 2⁵² + // r4 < 2¹⁰⁷ + // + + c0 := shiftRightBy51(r0) + c1 := shiftRightBy51(r1) + c2 := shiftRightBy51(r2) + c3 := shiftRightBy51(r3) + c4 := shiftRightBy51(r4) + + rr0 := r0.lo&maskLow51Bits + c4*19 + rr1 := r1.lo&maskLow51Bits + c0 + rr2 := r2.lo&maskLow51Bits + c1 + rr3 := r3.lo&maskLow51Bits + c2 + rr4 := r4.lo&maskLow51Bits + c3 + + // Now all coefficients fit into 64-bit registers but are still too large to + // be passed around as a Element. We therefore do one last carry chain, + // where the carries will be small enough to fit in the wiggle room above 2⁵¹. + *v = Element{rr0, rr1, rr2, rr3, rr4} + v.carryPropagate() +} + +func feSquareGeneric(v, a *Element) { + l0 := a.l0 + l1 := a.l1 + l2 := a.l2 + l3 := a.l3 + l4 := a.l4 + + // Squaring works precisely like multiplication above, but thanks to its + // symmetry we get to group a few terms together. + // + // l4 l3 l2 l1 l0 x + // l4 l3 l2 l1 l0 = + // ------------------------ + // l4l0 l3l0 l2l0 l1l0 l0l0 + + // l4l1 l3l1 l2l1 l1l1 l0l1 + + // l4l2 l3l2 l2l2 l1l2 l0l2 + + // l4l3 l3l3 l2l3 l1l3 l0l3 + + // l4l4 l3l4 l2l4 l1l4 l0l4 = + // ---------------------------------------------- + // r8 r7 r6 r5 r4 r3 r2 r1 r0 + // + // l4l0 l3l0 l2l0 l1l0 l0l0 + + // l3l1 l2l1 l1l1 l0l1 19×l4l1 + + // l2l2 l1l2 l0l2 19×l4l2 19×l3l2 + + // l1l3 l0l3 19×l4l3 19×l3l3 19×l2l3 + + // l0l4 19×l4l4 19×l3l4 19×l2l4 19×l1l4 = + // -------------------------------------- + // r4 r3 r2 r1 r0 + // + // With precomputed 2×, 19×, and 2×19× terms, we can compute each limb with + // only three Mul64 and four Add64, instead of five and eight. + + l0_2 := l0 * 2 + l1_2 := l1 * 2 + + l1_38 := l1 * 38 + l2_38 := l2 * 38 + l3_38 := l3 * 38 + + l3_19 := l3 * 19 + l4_19 := l4 * 19 + + // r0 = l0×l0 + 19×(l1×l4 + l2×l3 + l3×l2 + l4×l1) = l0×l0 + 19×2×(l1×l4 + l2×l3) + r0 := mul64(l0, l0) + r0 = addMul64(r0, l1_38, l4) + r0 = addMul64(r0, l2_38, l3) + + // r1 = l0×l1 + l1×l0 + 19×(l2×l4 + l3×l3 + l4×l2) = 2×l0×l1 + 19×2×l2×l4 + 19×l3×l3 + r1 := mul64(l0_2, l1) + r1 = addMul64(r1, l2_38, l4) + r1 = addMul64(r1, l3_19, l3) + + // r2 = l0×l2 + l1×l1 + l2×l0 + 19×(l3×l4 + l4×l3) = 2×l0×l2 + l1×l1 + 19×2×l3×l4 + r2 := mul64(l0_2, l2) + r2 = addMul64(r2, l1, l1) + r2 = addMul64(r2, l3_38, l4) + + // r3 = l0×l3 + l1×l2 + l2×l1 + l3×l0 + 19×l4×l4 = 2×l0×l3 + 2×l1×l2 + 19×l4×l4 + r3 := mul64(l0_2, l3) + r3 = addMul64(r3, l1_2, l2) + r3 = addMul64(r3, l4_19, l4) + + // r4 = l0×l4 + l1×l3 + l2×l2 + l3×l1 + l4×l0 = 2×l0×l4 + 2×l1×l3 + l2×l2 + r4 := mul64(l0_2, l4) + r4 = addMul64(r4, l1_2, l3) + r4 = addMul64(r4, l2, l2) + + c0 := shiftRightBy51(r0) + c1 := shiftRightBy51(r1) + c2 := shiftRightBy51(r2) + c3 := shiftRightBy51(r3) + c4 := shiftRightBy51(r4) + + rr0 := r0.lo&maskLow51Bits + c4*19 + rr1 := r1.lo&maskLow51Bits + c0 + rr2 := r2.lo&maskLow51Bits + c1 + rr3 := r3.lo&maskLow51Bits + c2 + rr4 := r4.lo&maskLow51Bits + c3 + + *v = Element{rr0, rr1, rr2, rr3, rr4} + v.carryPropagate() +} + +// carryPropagate brings the limbs below 52 bits by applying the reduction +// identity (a * 2²⁵⁵ + b = a * 19 + b) to the l4 carry. +func (v *Element) carryPropagateGeneric() *Element { + c0 := v.l0 >> 51 + c1 := v.l1 >> 51 + c2 := v.l2 >> 51 + c3 := v.l3 >> 51 + c4 := v.l4 >> 51 + + v.l0 = v.l0&maskLow51Bits + c4*19 + v.l1 = v.l1&maskLow51Bits + c0 + v.l2 = v.l2&maskLow51Bits + c1 + v.l3 = v.l3&maskLow51Bits + c2 + v.l4 = v.l4&maskLow51Bits + c3 + + return v +} diff --git a/src/crypto/ed25519/internal/edwards25519/field/fe_test.go b/src/crypto/ed25519/internal/edwards25519/field/fe_test.go new file mode 100644 index 0000000..b484459 --- /dev/null +++ b/src/crypto/ed25519/internal/edwards25519/field/fe_test.go @@ -0,0 +1,558 @@ +// Copyright (c) 2017 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 field + +import ( + "bytes" + "crypto/rand" + "encoding/hex" + "io" + "math/big" + "math/bits" + mathrand "math/rand" + "reflect" + "testing" + "testing/quick" +) + +func (v Element) String() string { + return hex.EncodeToString(v.Bytes()) +} + +// quickCheckConfig1024 will make each quickcheck test run (1024 * -quickchecks) +// times. The default value of -quickchecks is 100. +var quickCheckConfig1024 = &quick.Config{MaxCountScale: 1 << 10} + +func generateFieldElement(rand *mathrand.Rand) Element { + const maskLow52Bits = (1 << 52) - 1 + return Element{ + rand.Uint64() & maskLow52Bits, + rand.Uint64() & maskLow52Bits, + rand.Uint64() & maskLow52Bits, + rand.Uint64() & maskLow52Bits, + rand.Uint64() & maskLow52Bits, + } +} + +// weirdLimbs can be combined to generate a range of edge-case field elements. +// 0 and -1 are intentionally more weighted, as they combine well. +var ( + weirdLimbs51 = []uint64{ + 0, 0, 0, 0, + 1, + 19 - 1, + 19, + 0x2aaaaaaaaaaaa, + 0x5555555555555, + (1 << 51) - 20, + (1 << 51) - 19, + (1 << 51) - 1, (1 << 51) - 1, + (1 << 51) - 1, (1 << 51) - 1, + } + weirdLimbs52 = []uint64{ + 0, 0, 0, 0, 0, 0, + 1, + 19 - 1, + 19, + 0x2aaaaaaaaaaaa, + 0x5555555555555, + (1 << 51) - 20, + (1 << 51) - 19, + (1 << 51) - 1, (1 << 51) - 1, + (1 << 51) - 1, (1 << 51) - 1, + (1 << 51) - 1, (1 << 51) - 1, + 1 << 51, + (1 << 51) + 1, + (1 << 52) - 19, + (1 << 52) - 1, + } +) + +func generateWeirdFieldElement(rand *mathrand.Rand) Element { + return Element{ + weirdLimbs52[rand.Intn(len(weirdLimbs52))], + weirdLimbs51[rand.Intn(len(weirdLimbs51))], + weirdLimbs51[rand.Intn(len(weirdLimbs51))], + weirdLimbs51[rand.Intn(len(weirdLimbs51))], + weirdLimbs51[rand.Intn(len(weirdLimbs51))], + } +} + +func (Element) Generate(rand *mathrand.Rand, size int) reflect.Value { + if rand.Intn(2) == 0 { + return reflect.ValueOf(generateWeirdFieldElement(rand)) + } + return reflect.ValueOf(generateFieldElement(rand)) +} + +// isInBounds returns whether the element is within the expected bit size bounds +// after a light reduction. +func isInBounds(x *Element) bool { + return bits.Len64(x.l0) <= 52 && + bits.Len64(x.l1) <= 52 && + bits.Len64(x.l2) <= 52 && + bits.Len64(x.l3) <= 52 && + bits.Len64(x.l4) <= 52 +} + +func TestMultiplyDistributesOverAdd(t *testing.T) { + multiplyDistributesOverAdd := func(x, y, z Element) bool { + // Compute t1 = (x+y)*z + t1 := new(Element) + t1.Add(&x, &y) + t1.Multiply(t1, &z) + + // Compute t2 = x*z + y*z + t2 := new(Element) + t3 := new(Element) + t2.Multiply(&x, &z) + t3.Multiply(&y, &z) + t2.Add(t2, t3) + + return t1.Equal(t2) == 1 && isInBounds(t1) && isInBounds(t2) + } + + if err := quick.Check(multiplyDistributesOverAdd, quickCheckConfig1024); err != nil { + t.Error(err) + } +} + +func TestMul64to128(t *testing.T) { + a := uint64(5) + b := uint64(5) + r := mul64(a, b) + if r.lo != 0x19 || r.hi != 0 { + t.Errorf("lo-range wide mult failed, got %d + %d*(2**64)", r.lo, r.hi) + } + + a = uint64(18014398509481983) // 2^54 - 1 + b = uint64(18014398509481983) // 2^54 - 1 + r = mul64(a, b) + if r.lo != 0xff80000000000001 || r.hi != 0xfffffffffff { + t.Errorf("hi-range wide mult failed, got %d + %d*(2**64)", r.lo, r.hi) + } + + a = uint64(1125899906842661) + b = uint64(2097155) + r = mul64(a, b) + r = addMul64(r, a, b) + r = addMul64(r, a, b) + r = addMul64(r, a, b) + r = addMul64(r, a, b) + if r.lo != 16888498990613035 || r.hi != 640 { + t.Errorf("wrong answer: %d + %d*(2**64)", r.lo, r.hi) + } +} + +func TestSetBytesRoundTrip(t *testing.T) { + f1 := func(in [32]byte, fe Element) bool { + fe.SetBytes(in[:]) + + // Mask the most significant bit as it's ignored by SetBytes. (Now + // instead of earlier so we check the masking in SetBytes is working.) + in[len(in)-1] &= (1 << 7) - 1 + + return bytes.Equal(in[:], fe.Bytes()) && isInBounds(&fe) + } + if err := quick.Check(f1, nil); err != nil { + t.Errorf("failed bytes->FE->bytes round-trip: %v", err) + } + + f2 := func(fe, r Element) bool { + r.SetBytes(fe.Bytes()) + + // Intentionally not using Equal not to go through Bytes again. + // Calling reduce because both Generate and SetBytes can produce + // non-canonical representations. + fe.reduce() + r.reduce() + return fe == r + } + if err := quick.Check(f2, nil); err != nil { + t.Errorf("failed FE->bytes->FE round-trip: %v", err) + } + + // Check some fixed vectors from dalek + type feRTTest struct { + fe Element + b []byte + } + var tests = []feRTTest{ + { + fe: Element{358744748052810, 1691584618240980, 977650209285361, 1429865912637724, 560044844278676}, + b: []byte{74, 209, 69, 197, 70, 70, 161, 222, 56, 226, 229, 19, 112, 60, 25, 92, 187, 74, 222, 56, 50, 153, 51, 233, 40, 74, 57, 6, 160, 185, 213, 31}, + }, + { + fe: Element{84926274344903, 473620666599931, 365590438845504, 1028470286882429, 2146499180330972}, + b: []byte{199, 23, 106, 112, 61, 77, 216, 79, 186, 60, 11, 118, 13, 16, 103, 15, 42, 32, 83, 250, 44, 57, 204, 198, 78, 199, 253, 119, 146, 172, 3, 122}, + }, + } + + for _, tt := range tests { + b := tt.fe.Bytes() + if !bytes.Equal(b, tt.b) || new(Element).SetBytes(tt.b).Equal(&tt.fe) != 1 { + t.Errorf("Failed fixed roundtrip: %v", tt) + } + } +} + +func swapEndianness(buf []byte) []byte { + for i := 0; i < len(buf)/2; i++ { + buf[i], buf[len(buf)-i-1] = buf[len(buf)-i-1], buf[i] + } + return buf +} + +func TestBytesBigEquivalence(t *testing.T) { + f1 := func(in [32]byte, fe, fe1 Element) bool { + fe.SetBytes(in[:]) + + in[len(in)-1] &= (1 << 7) - 1 // mask the most significant bit + b := new(big.Int).SetBytes(swapEndianness(in[:])) + fe1.fromBig(b) + + if fe != fe1 { + return false + } + + buf := make([]byte, 32) // pad with zeroes + copy(buf, swapEndianness(fe1.toBig().Bytes())) + + return bytes.Equal(fe.Bytes(), buf) && isInBounds(&fe) && isInBounds(&fe1) + } + if err := quick.Check(f1, nil); err != nil { + t.Error(err) + } +} + +// fromBig sets v = n, and returns v. The bit length of n must not exceed 256. +func (v *Element) fromBig(n *big.Int) *Element { + if n.BitLen() > 32*8 { + panic("edwards25519: invalid field element input size") + } + + buf := make([]byte, 0, 32) + for _, word := range n.Bits() { + for i := 0; i < bits.UintSize; i += 8 { + if len(buf) >= cap(buf) { + break + } + buf = append(buf, byte(word)) + word >>= 8 + } + } + + return v.SetBytes(buf[:32]) +} + +func (v *Element) fromDecimal(s string) *Element { + n, ok := new(big.Int).SetString(s, 10) + if !ok { + panic("not a valid decimal: " + s) + } + return v.fromBig(n) +} + +// toBig returns v as a big.Int. +func (v *Element) toBig() *big.Int { + buf := v.Bytes() + + words := make([]big.Word, 32*8/bits.UintSize) + for n := range words { + for i := 0; i < bits.UintSize; i += 8 { + if len(buf) == 0 { + break + } + words[n] |= big.Word(buf[0]) << big.Word(i) + buf = buf[1:] + } + } + + return new(big.Int).SetBits(words) +} + +func TestDecimalConstants(t *testing.T) { + sqrtM1String := "19681161376707505956807079304988542015446066515923890162744021073123829784752" + if exp := new(Element).fromDecimal(sqrtM1String); sqrtM1.Equal(exp) != 1 { + t.Errorf("sqrtM1 is %v, expected %v", sqrtM1, exp) + } + // d is in the parent package, and we don't want to expose d or fromDecimal. + // dString := "37095705934669439343138083508754565189542113879843219016388785533085940283555" + // if exp := new(Element).fromDecimal(dString); d.Equal(exp) != 1 { + // t.Errorf("d is %v, expected %v", d, exp) + // } +} + +func TestSetBytesRoundTripEdgeCases(t *testing.T) { + // TODO: values close to 0, close to 2^255-19, between 2^255-19 and 2^255-1, + // and between 2^255 and 2^256-1. Test both the documented SetBytes + // behavior, and that Bytes reduces them. +} + +// Tests self-consistency between Multiply and Square. +func TestConsistency(t *testing.T) { + var x Element + var x2, x2sq Element + + x = Element{1, 1, 1, 1, 1} + x2.Multiply(&x, &x) + x2sq.Square(&x) + + if x2 != x2sq { + t.Fatalf("all ones failed\nmul: %x\nsqr: %x\n", x2, x2sq) + } + + var bytes [32]byte + + _, err := io.ReadFull(rand.Reader, bytes[:]) + if err != nil { + t.Fatal(err) + } + x.SetBytes(bytes[:]) + + x2.Multiply(&x, &x) + x2sq.Square(&x) + + if x2 != x2sq { + t.Fatalf("all ones failed\nmul: %x\nsqr: %x\n", x2, x2sq) + } +} + +func TestEqual(t *testing.T) { + x := Element{1, 1, 1, 1, 1} + y := Element{5, 4, 3, 2, 1} + + eq := x.Equal(&x) + if eq != 1 { + t.Errorf("wrong about equality") + } + + eq = x.Equal(&y) + if eq != 0 { + t.Errorf("wrong about inequality") + } +} + +func TestInvert(t *testing.T) { + x := Element{1, 1, 1, 1, 1} + one := Element{1, 0, 0, 0, 0} + var xinv, r Element + + xinv.Invert(&x) + r.Multiply(&x, &xinv) + r.reduce() + + if one != r { + t.Errorf("inversion identity failed, got: %x", r) + } + + var bytes [32]byte + + _, err := io.ReadFull(rand.Reader, bytes[:]) + if err != nil { + t.Fatal(err) + } + x.SetBytes(bytes[:]) + + xinv.Invert(&x) + r.Multiply(&x, &xinv) + r.reduce() + + if one != r { + t.Errorf("random inversion identity failed, got: %x for field element %x", r, x) + } + + zero := Element{} + x.Set(&zero) + if xx := xinv.Invert(&x); xx != &xinv { + t.Errorf("inverting zero did not return the receiver") + } else if xinv.Equal(&zero) != 1 { + t.Errorf("inverting zero did not return zero") + } +} + +func TestSelectSwap(t *testing.T) { + a := Element{358744748052810, 1691584618240980, 977650209285361, 1429865912637724, 560044844278676} + b := Element{84926274344903, 473620666599931, 365590438845504, 1028470286882429, 2146499180330972} + + var c, d Element + + c.Select(&a, &b, 1) + d.Select(&a, &b, 0) + + if c.Equal(&a) != 1 || d.Equal(&b) != 1 { + t.Errorf("Select failed") + } + + c.Swap(&d, 0) + + if c.Equal(&a) != 1 || d.Equal(&b) != 1 { + t.Errorf("Swap failed") + } + + c.Swap(&d, 1) + + if c.Equal(&b) != 1 || d.Equal(&a) != 1 { + t.Errorf("Swap failed") + } +} + +func TestMult32(t *testing.T) { + mult32EquivalentToMul := func(x Element, y uint32) bool { + t1 := new(Element) + for i := 0; i < 100; i++ { + t1.Mult32(&x, y) + } + + ty := new(Element) + ty.l0 = uint64(y) + + t2 := new(Element) + for i := 0; i < 100; i++ { + t2.Multiply(&x, ty) + } + + return t1.Equal(t2) == 1 && isInBounds(t1) && isInBounds(t2) + } + + if err := quick.Check(mult32EquivalentToMul, quickCheckConfig1024); err != nil { + t.Error(err) + } +} + +func TestSqrtRatio(t *testing.T) { + // From draft-irtf-cfrg-ristretto255-decaf448-00, Appendix A.4. + type test struct { + u, v string + wasSquare int + r string + } + var tests = []test{ + // If u is 0, the function is defined to return (0, TRUE), even if v + // is zero. Note that where used in this package, the denominator v + // is never zero. + { + "0000000000000000000000000000000000000000000000000000000000000000", + "0000000000000000000000000000000000000000000000000000000000000000", + 1, "0000000000000000000000000000000000000000000000000000000000000000", + }, + // 0/1 == 0² + { + "0000000000000000000000000000000000000000000000000000000000000000", + "0100000000000000000000000000000000000000000000000000000000000000", + 1, "0000000000000000000000000000000000000000000000000000000000000000", + }, + // If u is non-zero and v is zero, defined to return (0, FALSE). + { + "0100000000000000000000000000000000000000000000000000000000000000", + "0000000000000000000000000000000000000000000000000000000000000000", + 0, "0000000000000000000000000000000000000000000000000000000000000000", + }, + // 2/1 is not square in this field. + { + "0200000000000000000000000000000000000000000000000000000000000000", + "0100000000000000000000000000000000000000000000000000000000000000", + 0, "3c5ff1b5d8e4113b871bd052f9e7bcd0582804c266ffb2d4f4203eb07fdb7c54", + }, + // 4/1 == 2² + { + "0400000000000000000000000000000000000000000000000000000000000000", + "0100000000000000000000000000000000000000000000000000000000000000", + 1, "0200000000000000000000000000000000000000000000000000000000000000", + }, + // 1/4 == (2⁻¹)² == (2^(p-2))² per Euler's theorem + { + "0100000000000000000000000000000000000000000000000000000000000000", + "0400000000000000000000000000000000000000000000000000000000000000", + 1, "f6ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff3f", + }, + } + + for i, tt := range tests { + u := new(Element).SetBytes(decodeHex(tt.u)) + v := new(Element).SetBytes(decodeHex(tt.v)) + want := new(Element).SetBytes(decodeHex(tt.r)) + got, wasSquare := new(Element).SqrtRatio(u, v) + if got.Equal(want) == 0 || wasSquare != tt.wasSquare { + t.Errorf("%d: got (%v, %v), want (%v, %v)", i, got, wasSquare, want, tt.wasSquare) + } + } +} + +func TestCarryPropagate(t *testing.T) { + asmLikeGeneric := func(a [5]uint64) bool { + t1 := &Element{a[0], a[1], a[2], a[3], a[4]} + t2 := &Element{a[0], a[1], a[2], a[3], a[4]} + + t1.carryPropagate() + t2.carryPropagateGeneric() + + if *t1 != *t2 { + t.Logf("got: %#v,\nexpected: %#v", t1, t2) + } + + return *t1 == *t2 && isInBounds(t2) + } + + if err := quick.Check(asmLikeGeneric, quickCheckConfig1024); err != nil { + t.Error(err) + } + + if !asmLikeGeneric([5]uint64{0xffffffffffffffff, 0xffffffffffffffff, 0xffffffffffffffff, 0xffffffffffffffff, 0xffffffffffffffff}) { + t.Errorf("failed for {0xffffffffffffffff, 0xffffffffffffffff, 0xffffffffffffffff, 0xffffffffffffffff, 0xffffffffffffffff}") + } +} + +func TestFeSquare(t *testing.T) { + asmLikeGeneric := func(a Element) bool { + t1 := a + t2 := a + + feSquareGeneric(&t1, &t1) + feSquare(&t2, &t2) + + if t1 != t2 { + t.Logf("got: %#v,\nexpected: %#v", t1, t2) + } + + return t1 == t2 && isInBounds(&t2) + } + + if err := quick.Check(asmLikeGeneric, quickCheckConfig1024); err != nil { + t.Error(err) + } +} + +func TestFeMul(t *testing.T) { + asmLikeGeneric := func(a, b Element) bool { + a1 := a + a2 := a + b1 := b + b2 := b + + feMulGeneric(&a1, &a1, &b1) + feMul(&a2, &a2, &b2) + + if a1 != a2 || b1 != b2 { + t.Logf("got: %#v,\nexpected: %#v", a1, a2) + t.Logf("got: %#v,\nexpected: %#v", b1, b2) + } + + return a1 == a2 && isInBounds(&a2) && + b1 == b2 && isInBounds(&b2) + } + + if err := quick.Check(asmLikeGeneric, quickCheckConfig1024); err != nil { + t.Error(err) + } +} + +func decodeHex(s string) []byte { + b, err := hex.DecodeString(s) + if err != nil { + panic(err) + } + return b +} diff --git a/src/crypto/ed25519/internal/edwards25519/scalar.go b/src/crypto/ed25519/internal/edwards25519/scalar.go new file mode 100644 index 0000000..889acaa --- /dev/null +++ b/src/crypto/ed25519/internal/edwards25519/scalar.go @@ -0,0 +1,1025 @@ +// Copyright (c) 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. + +package edwards25519 + +import ( + "crypto/subtle" + "encoding/binary" + "errors" +) + +// A Scalar is an integer modulo +// +// l = 2^252 + 27742317777372353535851937790883648493 +// +// which is the prime order of the edwards25519 group. +// +// This type works similarly to math/big.Int, and all arguments and +// receivers are allowed to alias. +// +// The zero value is a valid zero element. +type Scalar struct { + // s is the Scalar value in little-endian. The value is always reduced + // between operations. + s [32]byte +} + +var ( + scZero = Scalar{[32]byte{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}} + + scOne = Scalar{[32]byte{1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}} + + scMinusOne = Scalar{[32]byte{236, 211, 245, 92, 26, 99, 18, 88, 214, 156, 247, 162, 222, 249, 222, 20, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16}} +) + +// NewScalar returns a new zero Scalar. +func NewScalar() *Scalar { + return &Scalar{} +} + +// MultiplyAdd sets s = x * y + z mod l, and returns s. +func (s *Scalar) MultiplyAdd(x, y, z *Scalar) *Scalar { + scMulAdd(&s.s, &x.s, &y.s, &z.s) + return s +} + +// Add sets s = x + y mod l, and returns s. +func (s *Scalar) Add(x, y *Scalar) *Scalar { + // s = 1 * x + y mod l + scMulAdd(&s.s, &scOne.s, &x.s, &y.s) + return s +} + +// Subtract sets s = x - y mod l, and returns s. +func (s *Scalar) Subtract(x, y *Scalar) *Scalar { + // s = -1 * y + x mod l + scMulAdd(&s.s, &scMinusOne.s, &y.s, &x.s) + return s +} + +// Negate sets s = -x mod l, and returns s. +func (s *Scalar) Negate(x *Scalar) *Scalar { + // s = -1 * x + 0 mod l + scMulAdd(&s.s, &scMinusOne.s, &x.s, &scZero.s) + return s +} + +// Multiply sets s = x * y mod l, and returns s. +func (s *Scalar) Multiply(x, y *Scalar) *Scalar { + // s = x * y + 0 mod l + scMulAdd(&s.s, &x.s, &y.s, &scZero.s) + return s +} + +// Set sets s = x, and returns s. +func (s *Scalar) Set(x *Scalar) *Scalar { + *s = *x + return s +} + +// SetUniformBytes sets s to an uniformly distributed value given 64 uniformly +// distributed random bytes. +func (s *Scalar) SetUniformBytes(x []byte) *Scalar { + if len(x) != 64 { + panic("edwards25519: invalid SetUniformBytes input length") + } + var wideBytes [64]byte + copy(wideBytes[:], x[:]) + scReduce(&s.s, &wideBytes) + return s +} + +// SetCanonicalBytes sets s = x, where x is a 32-byte little-endian encoding of +// s, and returns s. If x is not a canonical encoding of s, SetCanonicalBytes +// returns nil and an error, and the receiver is unchanged. +func (s *Scalar) SetCanonicalBytes(x []byte) (*Scalar, error) { + if len(x) != 32 { + return nil, errors.New("invalid scalar length") + } + ss := &Scalar{} + copy(ss.s[:], x) + if !isReduced(ss) { + return nil, errors.New("invalid scalar encoding") + } + s.s = ss.s + return s, nil +} + +// isReduced returns whether the given scalar is reduced modulo l. +func isReduced(s *Scalar) bool { + for i := len(s.s) - 1; i >= 0; i-- { + switch { + case s.s[i] > scMinusOne.s[i]: + return false + case s.s[i] < scMinusOne.s[i]: + return true + } + } + return true +} + +// SetBytesWithClamping applies the buffer pruning described in RFC 8032, +// Section 5.1.5 (also known as clamping) and sets s to the result. The input +// must be 32 bytes, and it is not modified. +// +// Note that since Scalar values are always reduced modulo the prime order of +// the curve, the resulting value will not preserve any of the cofactor-clearing +// properties that clamping is meant to provide. It will however work as +// expected as long as it is applied to points on the prime order subgroup, like +// in Ed25519. In fact, it is lost to history why RFC 8032 adopted the +// irrelevant RFC 7748 clamping, but it is now required for compatibility. +func (s *Scalar) SetBytesWithClamping(x []byte) *Scalar { + // The description above omits the purpose of the high bits of the clamping + // for brevity, but those are also lost to reductions, and are also + // irrelevant to edwards25519 as they protect against a specific + // implementation bug that was once observed in a generic Montgomery ladder. + if len(x) != 32 { + panic("edwards25519: invalid SetBytesWithClamping input length") + } + var wideBytes [64]byte + copy(wideBytes[:], x[:]) + wideBytes[0] &= 248 + wideBytes[31] &= 63 + wideBytes[31] |= 64 + scReduce(&s.s, &wideBytes) + return s +} + +// Bytes returns the canonical 32-byte little-endian encoding of s. +func (s *Scalar) Bytes() []byte { + buf := make([]byte, 32) + copy(buf, s.s[:]) + return buf +} + +// Equal returns 1 if s and t are equal, and 0 otherwise. +func (s *Scalar) Equal(t *Scalar) int { + return subtle.ConstantTimeCompare(s.s[:], t.s[:]) +} + +// scMulAdd and scReduce are ported from the public domain, “ref10” +// implementation of ed25519 from SUPERCOP. + +func load3(in []byte) int64 { + r := int64(in[0]) + r |= int64(in[1]) << 8 + r |= int64(in[2]) << 16 + return r +} + +func load4(in []byte) int64 { + r := int64(in[0]) + r |= int64(in[1]) << 8 + r |= int64(in[2]) << 16 + r |= int64(in[3]) << 24 + return r +} + +// Input: +// a[0]+256*a[1]+...+256^31*a[31] = a +// b[0]+256*b[1]+...+256^31*b[31] = b +// c[0]+256*c[1]+...+256^31*c[31] = c +// +// Output: +// s[0]+256*s[1]+...+256^31*s[31] = (ab+c) mod l +// where l = 2^252 + 27742317777372353535851937790883648493. +func scMulAdd(s, a, b, c *[32]byte) { + a0 := 2097151 & load3(a[:]) + a1 := 2097151 & (load4(a[2:]) >> 5) + a2 := 2097151 & (load3(a[5:]) >> 2) + a3 := 2097151 & (load4(a[7:]) >> 7) + a4 := 2097151 & (load4(a[10:]) >> 4) + a5 := 2097151 & (load3(a[13:]) >> 1) + a6 := 2097151 & (load4(a[15:]) >> 6) + a7 := 2097151 & (load3(a[18:]) >> 3) + a8 := 2097151 & load3(a[21:]) + a9 := 2097151 & (load4(a[23:]) >> 5) + a10 := 2097151 & (load3(a[26:]) >> 2) + a11 := (load4(a[28:]) >> 7) + b0 := 2097151 & load3(b[:]) + b1 := 2097151 & (load4(b[2:]) >> 5) + b2 := 2097151 & (load3(b[5:]) >> 2) + b3 := 2097151 & (load4(b[7:]) >> 7) + b4 := 2097151 & (load4(b[10:]) >> 4) + b5 := 2097151 & (load3(b[13:]) >> 1) + b6 := 2097151 & (load4(b[15:]) >> 6) + b7 := 2097151 & (load3(b[18:]) >> 3) + b8 := 2097151 & load3(b[21:]) + b9 := 2097151 & (load4(b[23:]) >> 5) + b10 := 2097151 & (load3(b[26:]) >> 2) + b11 := (load4(b[28:]) >> 7) + c0 := 2097151 & load3(c[:]) + c1 := 2097151 & (load4(c[2:]) >> 5) + c2 := 2097151 & (load3(c[5:]) >> 2) + c3 := 2097151 & (load4(c[7:]) >> 7) + c4 := 2097151 & (load4(c[10:]) >> 4) + c5 := 2097151 & (load3(c[13:]) >> 1) + c6 := 2097151 & (load4(c[15:]) >> 6) + c7 := 2097151 & (load3(c[18:]) >> 3) + c8 := 2097151 & load3(c[21:]) + c9 := 2097151 & (load4(c[23:]) >> 5) + c10 := 2097151 & (load3(c[26:]) >> 2) + c11 := (load4(c[28:]) >> 7) + var carry [23]int64 + + s0 := c0 + a0*b0 + s1 := c1 + a0*b1 + a1*b0 + s2 := c2 + a0*b2 + a1*b1 + a2*b0 + s3 := c3 + a0*b3 + a1*b2 + a2*b1 + a3*b0 + s4 := c4 + a0*b4 + a1*b3 + a2*b2 + a3*b1 + a4*b0 + s5 := c5 + a0*b5 + a1*b4 + a2*b3 + a3*b2 + a4*b1 + a5*b0 + s6 := c6 + a0*b6 + a1*b5 + a2*b4 + a3*b3 + a4*b2 + a5*b1 + a6*b0 + s7 := c7 + a0*b7 + a1*b6 + a2*b5 + a3*b4 + a4*b3 + a5*b2 + a6*b1 + a7*b0 + s8 := c8 + a0*b8 + a1*b7 + a2*b6 + a3*b5 + a4*b4 + a5*b3 + a6*b2 + a7*b1 + a8*b0 + s9 := c9 + a0*b9 + a1*b8 + a2*b7 + a3*b6 + a4*b5 + a5*b4 + a6*b3 + a7*b2 + a8*b1 + a9*b0 + s10 := c10 + a0*b10 + a1*b9 + a2*b8 + a3*b7 + a4*b6 + a5*b5 + a6*b4 + a7*b3 + a8*b2 + a9*b1 + a10*b0 + s11 := c11 + a0*b11 + a1*b10 + a2*b9 + a3*b8 + a4*b7 + a5*b6 + a6*b5 + a7*b4 + a8*b3 + a9*b2 + a10*b1 + a11*b0 + s12 := a1*b11 + a2*b10 + a3*b9 + a4*b8 + a5*b7 + a6*b6 + a7*b5 + a8*b4 + a9*b3 + a10*b2 + a11*b1 + s13 := a2*b11 + a3*b10 + a4*b9 + a5*b8 + a6*b7 + a7*b6 + a8*b5 + a9*b4 + a10*b3 + a11*b2 + s14 := a3*b11 + a4*b10 + a5*b9 + a6*b8 + a7*b7 + a8*b6 + a9*b5 + a10*b4 + a11*b3 + s15 := a4*b11 + a5*b10 + a6*b9 + a7*b8 + a8*b7 + a9*b6 + a10*b5 + a11*b4 + s16 := a5*b11 + a6*b10 + a7*b9 + a8*b8 + a9*b7 + a10*b6 + a11*b5 + s17 := a6*b11 + a7*b10 + a8*b9 + a9*b8 + a10*b7 + a11*b6 + s18 := a7*b11 + a8*b10 + a9*b9 + a10*b8 + a11*b7 + s19 := a8*b11 + a9*b10 + a10*b9 + a11*b8 + s20 := a9*b11 + a10*b10 + a11*b9 + s21 := a10*b11 + a11*b10 + s22 := a11 * b11 + s23 := int64(0) + + carry[0] = (s0 + (1 << 20)) >> 21 + s1 += carry[0] + s0 -= carry[0] << 21 + carry[2] = (s2 + (1 << 20)) >> 21 + s3 += carry[2] + s2 -= carry[2] << 21 + carry[4] = (s4 + (1 << 20)) >> 21 + s5 += carry[4] + s4 -= carry[4] << 21 + carry[6] = (s6 + (1 << 20)) >> 21 + s7 += carry[6] + s6 -= carry[6] << 21 + carry[8] = (s8 + (1 << 20)) >> 21 + s9 += carry[8] + s8 -= carry[8] << 21 + carry[10] = (s10 + (1 << 20)) >> 21 + s11 += carry[10] + s10 -= carry[10] << 21 + carry[12] = (s12 + (1 << 20)) >> 21 + s13 += carry[12] + s12 -= carry[12] << 21 + carry[14] = (s14 + (1 << 20)) >> 21 + s15 += carry[14] + s14 -= carry[14] << 21 + carry[16] = (s16 + (1 << 20)) >> 21 + s17 += carry[16] + s16 -= carry[16] << 21 + carry[18] = (s18 + (1 << 20)) >> 21 + s19 += carry[18] + s18 -= carry[18] << 21 + carry[20] = (s20 + (1 << 20)) >> 21 + s21 += carry[20] + s20 -= carry[20] << 21 + carry[22] = (s22 + (1 << 20)) >> 21 + s23 += carry[22] + s22 -= carry[22] << 21 + + carry[1] = (s1 + (1 << 20)) >> 21 + s2 += carry[1] + s1 -= carry[1] << 21 + carry[3] = (s3 + (1 << 20)) >> 21 + s4 += carry[3] + s3 -= carry[3] << 21 + carry[5] = (s5 + (1 << 20)) >> 21 + s6 += carry[5] + s5 -= carry[5] << 21 + carry[7] = (s7 + (1 << 20)) >> 21 + s8 += carry[7] + s7 -= carry[7] << 21 + carry[9] = (s9 + (1 << 20)) >> 21 + s10 += carry[9] + s9 -= carry[9] << 21 + carry[11] = (s11 + (1 << 20)) >> 21 + s12 += carry[11] + s11 -= carry[11] << 21 + carry[13] = (s13 + (1 << 20)) >> 21 + s14 += carry[13] + s13 -= carry[13] << 21 + carry[15] = (s15 + (1 << 20)) >> 21 + s16 += carry[15] + s15 -= carry[15] << 21 + carry[17] = (s17 + (1 << 20)) >> 21 + s18 += carry[17] + s17 -= carry[17] << 21 + carry[19] = (s19 + (1 << 20)) >> 21 + s20 += carry[19] + s19 -= carry[19] << 21 + carry[21] = (s21 + (1 << 20)) >> 21 + s22 += carry[21] + s21 -= carry[21] << 21 + + s11 += s23 * 666643 + s12 += s23 * 470296 + s13 += s23 * 654183 + s14 -= s23 * 997805 + s15 += s23 * 136657 + s16 -= s23 * 683901 + s23 = 0 + + s10 += s22 * 666643 + s11 += s22 * 470296 + s12 += s22 * 654183 + s13 -= s22 * 997805 + s14 += s22 * 136657 + s15 -= s22 * 683901 + s22 = 0 + + s9 += s21 * 666643 + s10 += s21 * 470296 + s11 += s21 * 654183 + s12 -= s21 * 997805 + s13 += s21 * 136657 + s14 -= s21 * 683901 + s21 = 0 + + s8 += s20 * 666643 + s9 += s20 * 470296 + s10 += s20 * 654183 + s11 -= s20 * 997805 + s12 += s20 * 136657 + s13 -= s20 * 683901 + s20 = 0 + + s7 += s19 * 666643 + s8 += s19 * 470296 + s9 += s19 * 654183 + s10 -= s19 * 997805 + s11 += s19 * 136657 + s12 -= s19 * 683901 + s19 = 0 + + s6 += s18 * 666643 + s7 += s18 * 470296 + s8 += s18 * 654183 + s9 -= s18 * 997805 + s10 += s18 * 136657 + s11 -= s18 * 683901 + s18 = 0 + + carry[6] = (s6 + (1 << 20)) >> 21 + s7 += carry[6] + s6 -= carry[6] << 21 + carry[8] = (s8 + (1 << 20)) >> 21 + s9 += carry[8] + s8 -= carry[8] << 21 + carry[10] = (s10 + (1 << 20)) >> 21 + s11 += carry[10] + s10 -= carry[10] << 21 + carry[12] = (s12 + (1 << 20)) >> 21 + s13 += carry[12] + s12 -= carry[12] << 21 + carry[14] = (s14 + (1 << 20)) >> 21 + s15 += carry[14] + s14 -= carry[14] << 21 + carry[16] = (s16 + (1 << 20)) >> 21 + s17 += carry[16] + s16 -= carry[16] << 21 + + carry[7] = (s7 + (1 << 20)) >> 21 + s8 += carry[7] + s7 -= carry[7] << 21 + carry[9] = (s9 + (1 << 20)) >> 21 + s10 += carry[9] + s9 -= carry[9] << 21 + carry[11] = (s11 + (1 << 20)) >> 21 + s12 += carry[11] + s11 -= carry[11] << 21 + carry[13] = (s13 + (1 << 20)) >> 21 + s14 += carry[13] + s13 -= carry[13] << 21 + carry[15] = (s15 + (1 << 20)) >> 21 + s16 += carry[15] + s15 -= carry[15] << 21 + + s5 += s17 * 666643 + s6 += s17 * 470296 + s7 += s17 * 654183 + s8 -= s17 * 997805 + s9 += s17 * 136657 + s10 -= s17 * 683901 + s17 = 0 + + s4 += s16 * 666643 + s5 += s16 * 470296 + s6 += s16 * 654183 + s7 -= s16 * 997805 + s8 += s16 * 136657 + s9 -= s16 * 683901 + s16 = 0 + + s3 += s15 * 666643 + s4 += s15 * 470296 + s5 += s15 * 654183 + s6 -= s15 * 997805 + s7 += s15 * 136657 + s8 -= s15 * 683901 + s15 = 0 + + s2 += s14 * 666643 + s3 += s14 * 470296 + s4 += s14 * 654183 + s5 -= s14 * 997805 + s6 += s14 * 136657 + s7 -= s14 * 683901 + s14 = 0 + + s1 += s13 * 666643 + s2 += s13 * 470296 + s3 += s13 * 654183 + s4 -= s13 * 997805 + s5 += s13 * 136657 + s6 -= s13 * 683901 + s13 = 0 + + s0 += s12 * 666643 + s1 += s12 * 470296 + s2 += s12 * 654183 + s3 -= s12 * 997805 + s4 += s12 * 136657 + s5 -= s12 * 683901 + s12 = 0 + + carry[0] = (s0 + (1 << 20)) >> 21 + s1 += carry[0] + s0 -= carry[0] << 21 + carry[2] = (s2 + (1 << 20)) >> 21 + s3 += carry[2] + s2 -= carry[2] << 21 + carry[4] = (s4 + (1 << 20)) >> 21 + s5 += carry[4] + s4 -= carry[4] << 21 + carry[6] = (s6 + (1 << 20)) >> 21 + s7 += carry[6] + s6 -= carry[6] << 21 + carry[8] = (s8 + (1 << 20)) >> 21 + s9 += carry[8] + s8 -= carry[8] << 21 + carry[10] = (s10 + (1 << 20)) >> 21 + s11 += carry[10] + s10 -= carry[10] << 21 + + carry[1] = (s1 + (1 << 20)) >> 21 + s2 += carry[1] + s1 -= carry[1] << 21 + carry[3] = (s3 + (1 << 20)) >> 21 + s4 += carry[3] + s3 -= carry[3] << 21 + carry[5] = (s5 + (1 << 20)) >> 21 + s6 += carry[5] + s5 -= carry[5] << 21 + carry[7] = (s7 + (1 << 20)) >> 21 + s8 += carry[7] + s7 -= carry[7] << 21 + carry[9] = (s9 + (1 << 20)) >> 21 + s10 += carry[9] + s9 -= carry[9] << 21 + carry[11] = (s11 + (1 << 20)) >> 21 + s12 += carry[11] + s11 -= carry[11] << 21 + + s0 += s12 * 666643 + s1 += s12 * 470296 + s2 += s12 * 654183 + s3 -= s12 * 997805 + s4 += s12 * 136657 + s5 -= s12 * 683901 + s12 = 0 + + carry[0] = s0 >> 21 + s1 += carry[0] + s0 -= carry[0] << 21 + carry[1] = s1 >> 21 + s2 += carry[1] + s1 -= carry[1] << 21 + carry[2] = s2 >> 21 + s3 += carry[2] + s2 -= carry[2] << 21 + carry[3] = s3 >> 21 + s4 += carry[3] + s3 -= carry[3] << 21 + carry[4] = s4 >> 21 + s5 += carry[4] + s4 -= carry[4] << 21 + carry[5] = s5 >> 21 + s6 += carry[5] + s5 -= carry[5] << 21 + carry[6] = s6 >> 21 + s7 += carry[6] + s6 -= carry[6] << 21 + carry[7] = s7 >> 21 + s8 += carry[7] + s7 -= carry[7] << 21 + carry[8] = s8 >> 21 + s9 += carry[8] + s8 -= carry[8] << 21 + carry[9] = s9 >> 21 + s10 += carry[9] + s9 -= carry[9] << 21 + carry[10] = s10 >> 21 + s11 += carry[10] + s10 -= carry[10] << 21 + carry[11] = s11 >> 21 + s12 += carry[11] + s11 -= carry[11] << 21 + + s0 += s12 * 666643 + s1 += s12 * 470296 + s2 += s12 * 654183 + s3 -= s12 * 997805 + s4 += s12 * 136657 + s5 -= s12 * 683901 + s12 = 0 + + carry[0] = s0 >> 21 + s1 += carry[0] + s0 -= carry[0] << 21 + carry[1] = s1 >> 21 + s2 += carry[1] + s1 -= carry[1] << 21 + carry[2] = s2 >> 21 + s3 += carry[2] + s2 -= carry[2] << 21 + carry[3] = s3 >> 21 + s4 += carry[3] + s3 -= carry[3] << 21 + carry[4] = s4 >> 21 + s5 += carry[4] + s4 -= carry[4] << 21 + carry[5] = s5 >> 21 + s6 += carry[5] + s5 -= carry[5] << 21 + carry[6] = s6 >> 21 + s7 += carry[6] + s6 -= carry[6] << 21 + carry[7] = s7 >> 21 + s8 += carry[7] + s7 -= carry[7] << 21 + carry[8] = s8 >> 21 + s9 += carry[8] + s8 -= carry[8] << 21 + carry[9] = s9 >> 21 + s10 += carry[9] + s9 -= carry[9] << 21 + carry[10] = s10 >> 21 + s11 += carry[10] + s10 -= carry[10] << 21 + + s[0] = byte(s0 >> 0) + s[1] = byte(s0 >> 8) + s[2] = byte((s0 >> 16) | (s1 << 5)) + s[3] = byte(s1 >> 3) + s[4] = byte(s1 >> 11) + s[5] = byte((s1 >> 19) | (s2 << 2)) + s[6] = byte(s2 >> 6) + s[7] = byte((s2 >> 14) | (s3 << 7)) + s[8] = byte(s3 >> 1) + s[9] = byte(s3 >> 9) + s[10] = byte((s3 >> 17) | (s4 << 4)) + s[11] = byte(s4 >> 4) + s[12] = byte(s4 >> 12) + s[13] = byte((s4 >> 20) | (s5 << 1)) + s[14] = byte(s5 >> 7) + s[15] = byte((s5 >> 15) | (s6 << 6)) + s[16] = byte(s6 >> 2) + s[17] = byte(s6 >> 10) + s[18] = byte((s6 >> 18) | (s7 << 3)) + s[19] = byte(s7 >> 5) + s[20] = byte(s7 >> 13) + s[21] = byte(s8 >> 0) + s[22] = byte(s8 >> 8) + s[23] = byte((s8 >> 16) | (s9 << 5)) + s[24] = byte(s9 >> 3) + s[25] = byte(s9 >> 11) + s[26] = byte((s9 >> 19) | (s10 << 2)) + s[27] = byte(s10 >> 6) + s[28] = byte((s10 >> 14) | (s11 << 7)) + s[29] = byte(s11 >> 1) + s[30] = byte(s11 >> 9) + s[31] = byte(s11 >> 17) +} + +// Input: +// s[0]+256*s[1]+...+256^63*s[63] = s +// +// Output: +// s[0]+256*s[1]+...+256^31*s[31] = s mod l +// where l = 2^252 + 27742317777372353535851937790883648493. +func scReduce(out *[32]byte, s *[64]byte) { + s0 := 2097151 & load3(s[:]) + s1 := 2097151 & (load4(s[2:]) >> 5) + s2 := 2097151 & (load3(s[5:]) >> 2) + s3 := 2097151 & (load4(s[7:]) >> 7) + s4 := 2097151 & (load4(s[10:]) >> 4) + s5 := 2097151 & (load3(s[13:]) >> 1) + s6 := 2097151 & (load4(s[15:]) >> 6) + s7 := 2097151 & (load3(s[18:]) >> 3) + s8 := 2097151 & load3(s[21:]) + s9 := 2097151 & (load4(s[23:]) >> 5) + s10 := 2097151 & (load3(s[26:]) >> 2) + s11 := 2097151 & (load4(s[28:]) >> 7) + s12 := 2097151 & (load4(s[31:]) >> 4) + s13 := 2097151 & (load3(s[34:]) >> 1) + s14 := 2097151 & (load4(s[36:]) >> 6) + s15 := 2097151 & (load3(s[39:]) >> 3) + s16 := 2097151 & load3(s[42:]) + s17 := 2097151 & (load4(s[44:]) >> 5) + s18 := 2097151 & (load3(s[47:]) >> 2) + s19 := 2097151 & (load4(s[49:]) >> 7) + s20 := 2097151 & (load4(s[52:]) >> 4) + s21 := 2097151 & (load3(s[55:]) >> 1) + s22 := 2097151 & (load4(s[57:]) >> 6) + s23 := (load4(s[60:]) >> 3) + + s11 += s23 * 666643 + s12 += s23 * 470296 + s13 += s23 * 654183 + s14 -= s23 * 997805 + s15 += s23 * 136657 + s16 -= s23 * 683901 + s23 = 0 + + s10 += s22 * 666643 + s11 += s22 * 470296 + s12 += s22 * 654183 + s13 -= s22 * 997805 + s14 += s22 * 136657 + s15 -= s22 * 683901 + s22 = 0 + + s9 += s21 * 666643 + s10 += s21 * 470296 + s11 += s21 * 654183 + s12 -= s21 * 997805 + s13 += s21 * 136657 + s14 -= s21 * 683901 + s21 = 0 + + s8 += s20 * 666643 + s9 += s20 * 470296 + s10 += s20 * 654183 + s11 -= s20 * 997805 + s12 += s20 * 136657 + s13 -= s20 * 683901 + s20 = 0 + + s7 += s19 * 666643 + s8 += s19 * 470296 + s9 += s19 * 654183 + s10 -= s19 * 997805 + s11 += s19 * 136657 + s12 -= s19 * 683901 + s19 = 0 + + s6 += s18 * 666643 + s7 += s18 * 470296 + s8 += s18 * 654183 + s9 -= s18 * 997805 + s10 += s18 * 136657 + s11 -= s18 * 683901 + s18 = 0 + + var carry [17]int64 + + carry[6] = (s6 + (1 << 20)) >> 21 + s7 += carry[6] + s6 -= carry[6] << 21 + carry[8] = (s8 + (1 << 20)) >> 21 + s9 += carry[8] + s8 -= carry[8] << 21 + carry[10] = (s10 + (1 << 20)) >> 21 + s11 += carry[10] + s10 -= carry[10] << 21 + carry[12] = (s12 + (1 << 20)) >> 21 + s13 += carry[12] + s12 -= carry[12] << 21 + carry[14] = (s14 + (1 << 20)) >> 21 + s15 += carry[14] + s14 -= carry[14] << 21 + carry[16] = (s16 + (1 << 20)) >> 21 + s17 += carry[16] + s16 -= carry[16] << 21 + + carry[7] = (s7 + (1 << 20)) >> 21 + s8 += carry[7] + s7 -= carry[7] << 21 + carry[9] = (s9 + (1 << 20)) >> 21 + s10 += carry[9] + s9 -= carry[9] << 21 + carry[11] = (s11 + (1 << 20)) >> 21 + s12 += carry[11] + s11 -= carry[11] << 21 + carry[13] = (s13 + (1 << 20)) >> 21 + s14 += carry[13] + s13 -= carry[13] << 21 + carry[15] = (s15 + (1 << 20)) >> 21 + s16 += carry[15] + s15 -= carry[15] << 21 + + s5 += s17 * 666643 + s6 += s17 * 470296 + s7 += s17 * 654183 + s8 -= s17 * 997805 + s9 += s17 * 136657 + s10 -= s17 * 683901 + s17 = 0 + + s4 += s16 * 666643 + s5 += s16 * 470296 + s6 += s16 * 654183 + s7 -= s16 * 997805 + s8 += s16 * 136657 + s9 -= s16 * 683901 + s16 = 0 + + s3 += s15 * 666643 + s4 += s15 * 470296 + s5 += s15 * 654183 + s6 -= s15 * 997805 + s7 += s15 * 136657 + s8 -= s15 * 683901 + s15 = 0 + + s2 += s14 * 666643 + s3 += s14 * 470296 + s4 += s14 * 654183 + s5 -= s14 * 997805 + s6 += s14 * 136657 + s7 -= s14 * 683901 + s14 = 0 + + s1 += s13 * 666643 + s2 += s13 * 470296 + s3 += s13 * 654183 + s4 -= s13 * 997805 + s5 += s13 * 136657 + s6 -= s13 * 683901 + s13 = 0 + + s0 += s12 * 666643 + s1 += s12 * 470296 + s2 += s12 * 654183 + s3 -= s12 * 997805 + s4 += s12 * 136657 + s5 -= s12 * 683901 + s12 = 0 + + carry[0] = (s0 + (1 << 20)) >> 21 + s1 += carry[0] + s0 -= carry[0] << 21 + carry[2] = (s2 + (1 << 20)) >> 21 + s3 += carry[2] + s2 -= carry[2] << 21 + carry[4] = (s4 + (1 << 20)) >> 21 + s5 += carry[4] + s4 -= carry[4] << 21 + carry[6] = (s6 + (1 << 20)) >> 21 + s7 += carry[6] + s6 -= carry[6] << 21 + carry[8] = (s8 + (1 << 20)) >> 21 + s9 += carry[8] + s8 -= carry[8] << 21 + carry[10] = (s10 + (1 << 20)) >> 21 + s11 += carry[10] + s10 -= carry[10] << 21 + + carry[1] = (s1 + (1 << 20)) >> 21 + s2 += carry[1] + s1 -= carry[1] << 21 + carry[3] = (s3 + (1 << 20)) >> 21 + s4 += carry[3] + s3 -= carry[3] << 21 + carry[5] = (s5 + (1 << 20)) >> 21 + s6 += carry[5] + s5 -= carry[5] << 21 + carry[7] = (s7 + (1 << 20)) >> 21 + s8 += carry[7] + s7 -= carry[7] << 21 + carry[9] = (s9 + (1 << 20)) >> 21 + s10 += carry[9] + s9 -= carry[9] << 21 + carry[11] = (s11 + (1 << 20)) >> 21 + s12 += carry[11] + s11 -= carry[11] << 21 + + s0 += s12 * 666643 + s1 += s12 * 470296 + s2 += s12 * 654183 + s3 -= s12 * 997805 + s4 += s12 * 136657 + s5 -= s12 * 683901 + s12 = 0 + + carry[0] = s0 >> 21 + s1 += carry[0] + s0 -= carry[0] << 21 + carry[1] = s1 >> 21 + s2 += carry[1] + s1 -= carry[1] << 21 + carry[2] = s2 >> 21 + s3 += carry[2] + s2 -= carry[2] << 21 + carry[3] = s3 >> 21 + s4 += carry[3] + s3 -= carry[3] << 21 + carry[4] = s4 >> 21 + s5 += carry[4] + s4 -= carry[4] << 21 + carry[5] = s5 >> 21 + s6 += carry[5] + s5 -= carry[5] << 21 + carry[6] = s6 >> 21 + s7 += carry[6] + s6 -= carry[6] << 21 + carry[7] = s7 >> 21 + s8 += carry[7] + s7 -= carry[7] << 21 + carry[8] = s8 >> 21 + s9 += carry[8] + s8 -= carry[8] << 21 + carry[9] = s9 >> 21 + s10 += carry[9] + s9 -= carry[9] << 21 + carry[10] = s10 >> 21 + s11 += carry[10] + s10 -= carry[10] << 21 + carry[11] = s11 >> 21 + s12 += carry[11] + s11 -= carry[11] << 21 + + s0 += s12 * 666643 + s1 += s12 * 470296 + s2 += s12 * 654183 + s3 -= s12 * 997805 + s4 += s12 * 136657 + s5 -= s12 * 683901 + s12 = 0 + + carry[0] = s0 >> 21 + s1 += carry[0] + s0 -= carry[0] << 21 + carry[1] = s1 >> 21 + s2 += carry[1] + s1 -= carry[1] << 21 + carry[2] = s2 >> 21 + s3 += carry[2] + s2 -= carry[2] << 21 + carry[3] = s3 >> 21 + s4 += carry[3] + s3 -= carry[3] << 21 + carry[4] = s4 >> 21 + s5 += carry[4] + s4 -= carry[4] << 21 + carry[5] = s5 >> 21 + s6 += carry[5] + s5 -= carry[5] << 21 + carry[6] = s6 >> 21 + s7 += carry[6] + s6 -= carry[6] << 21 + carry[7] = s7 >> 21 + s8 += carry[7] + s7 -= carry[7] << 21 + carry[8] = s8 >> 21 + s9 += carry[8] + s8 -= carry[8] << 21 + carry[9] = s9 >> 21 + s10 += carry[9] + s9 -= carry[9] << 21 + carry[10] = s10 >> 21 + s11 += carry[10] + s10 -= carry[10] << 21 + + out[0] = byte(s0 >> 0) + out[1] = byte(s0 >> 8) + out[2] = byte((s0 >> 16) | (s1 << 5)) + out[3] = byte(s1 >> 3) + out[4] = byte(s1 >> 11) + out[5] = byte((s1 >> 19) | (s2 << 2)) + out[6] = byte(s2 >> 6) + out[7] = byte((s2 >> 14) | (s3 << 7)) + out[8] = byte(s3 >> 1) + out[9] = byte(s3 >> 9) + out[10] = byte((s3 >> 17) | (s4 << 4)) + out[11] = byte(s4 >> 4) + out[12] = byte(s4 >> 12) + out[13] = byte((s4 >> 20) | (s5 << 1)) + out[14] = byte(s5 >> 7) + out[15] = byte((s5 >> 15) | (s6 << 6)) + out[16] = byte(s6 >> 2) + out[17] = byte(s6 >> 10) + out[18] = byte((s6 >> 18) | (s7 << 3)) + out[19] = byte(s7 >> 5) + out[20] = byte(s7 >> 13) + out[21] = byte(s8 >> 0) + out[22] = byte(s8 >> 8) + out[23] = byte((s8 >> 16) | (s9 << 5)) + out[24] = byte(s9 >> 3) + out[25] = byte(s9 >> 11) + out[26] = byte((s9 >> 19) | (s10 << 2)) + out[27] = byte(s10 >> 6) + out[28] = byte((s10 >> 14) | (s11 << 7)) + out[29] = byte(s11 >> 1) + out[30] = byte(s11 >> 9) + out[31] = byte(s11 >> 17) +} + +// nonAdjacentForm computes a width-w non-adjacent form for this scalar. +// +// w must be between 2 and 8, or nonAdjacentForm will panic. +func (s *Scalar) nonAdjacentForm(w uint) [256]int8 { + // This implementation is adapted from the one + // in curve25519-dalek and is documented there: + // https://github.com/dalek-cryptography/curve25519-dalek/blob/f630041af28e9a405255f98a8a93adca18e4315b/src/scalar.rs#L800-L871 + if s.s[31] > 127 { + panic("scalar has high bit set illegally") + } + if w < 2 { + panic("w must be at least 2 by the definition of NAF") + } else if w > 8 { + panic("NAF digits must fit in int8") + } + + var naf [256]int8 + var digits [5]uint64 + + for i := 0; i < 4; i++ { + digits[i] = binary.LittleEndian.Uint64(s.s[i*8:]) + } + + width := uint64(1 << w) + windowMask := uint64(width - 1) + + pos := uint(0) + carry := uint64(0) + for pos < 256 { + indexU64 := pos / 64 + indexBit := pos % 64 + var bitBuf uint64 + if indexBit < 64-w { + // This window's bits are contained in a single u64 + bitBuf = digits[indexU64] >> indexBit + } else { + // Combine the current 64 bits with bits from the next 64 + bitBuf = (digits[indexU64] >> indexBit) | (digits[1+indexU64] << (64 - indexBit)) + } + + // Add carry into the current window + window := carry + (bitBuf & windowMask) + + if window&1 == 0 { + // If the window value is even, preserve the carry and continue. + // Why is the carry preserved? + // If carry == 0 and window & 1 == 0, + // then the next carry should be 0 + // If carry == 1 and window & 1 == 0, + // then bit_buf & 1 == 1 so the next carry should be 1 + pos += 1 + continue + } + + if window < width/2 { + carry = 0 + naf[pos] = int8(window) + } else { + carry = 1 + naf[pos] = int8(window) - int8(width) + } + + pos += w + } + return naf +} + +func (s *Scalar) signedRadix16() [64]int8 { + if s.s[31] > 127 { + panic("scalar has high bit set illegally") + } + + var digits [64]int8 + + // Compute unsigned radix-16 digits: + for i := 0; i < 32; i++ { + digits[2*i] = int8(s.s[i] & 15) + digits[2*i+1] = int8((s.s[i] >> 4) & 15) + } + + // Recenter coefficients: + for i := 0; i < 63; i++ { + carry := (digits[i] + 8) >> 4 + digits[i] -= carry << 4 + digits[i+1] += carry + } + + return digits +} diff --git a/src/crypto/ed25519/internal/edwards25519/scalar_alias_test.go b/src/crypto/ed25519/internal/edwards25519/scalar_alias_test.go new file mode 100644 index 0000000..827153b --- /dev/null +++ b/src/crypto/ed25519/internal/edwards25519/scalar_alias_test.go @@ -0,0 +1,93 @@ +// Copyright (c) 2019 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 edwards25519 + +import ( + "testing" + "testing/quick" +) + +func TestScalarAliasing(t *testing.T) { + checkAliasingOneArg := func(f func(v, x *Scalar) *Scalar, v, x Scalar) bool { + x1, v1 := x, x + + // Calculate a reference f(x) without aliasing. + if out := f(&v, &x); out != &v || !isReduced(out) { + return false + } + + // Test aliasing the argument and the receiver. + if out := f(&v1, &v1); out != &v1 || v1 != v || !isReduced(out) { + return false + } + + // Ensure the arguments was not modified. + return x == x1 + } + + checkAliasingTwoArgs := func(f func(v, x, y *Scalar) *Scalar, v, x, y Scalar) bool { + x1, y1, v1 := x, y, Scalar{} + + // Calculate a reference f(x, y) without aliasing. + if out := f(&v, &x, &y); out != &v || !isReduced(out) { + return false + } + + // Test aliasing the first argument and the receiver. + v1 = x + if out := f(&v1, &v1, &y); out != &v1 || v1 != v || !isReduced(out) { + return false + } + // Test aliasing the second argument and the receiver. + v1 = y + if out := f(&v1, &x, &v1); out != &v1 || v1 != v || !isReduced(out) { + return false + } + + // Calculate a reference f(x, x) without aliasing. + if out := f(&v, &x, &x); out != &v || !isReduced(out) { + return false + } + + // Test aliasing the first argument and the receiver. + v1 = x + if out := f(&v1, &v1, &x); out != &v1 || v1 != v || !isReduced(out) { + return false + } + // Test aliasing the second argument and the receiver. + v1 = x + if out := f(&v1, &x, &v1); out != &v1 || v1 != v || !isReduced(out) { + return false + } + // Test aliasing both arguments and the receiver. + v1 = x + if out := f(&v1, &v1, &v1); out != &v1 || v1 != v || !isReduced(out) { + return false + } + + // Ensure the arguments were not modified. + return x == x1 && y == y1 + } + + for name, f := range map[string]any{ + "Negate": func(v, x Scalar) bool { + return checkAliasingOneArg((*Scalar).Negate, v, x) + }, + "Multiply": func(v, x, y Scalar) bool { + return checkAliasingTwoArgs((*Scalar).Multiply, v, x, y) + }, + "Add": func(v, x, y Scalar) bool { + return checkAliasingTwoArgs((*Scalar).Add, v, x, y) + }, + "Subtract": func(v, x, y Scalar) bool { + return checkAliasingTwoArgs((*Scalar).Subtract, v, x, y) + }, + } { + err := quick.Check(f, &quick.Config{MaxCountScale: 1 << 5}) + if err != nil { + t.Errorf("%v: %v", name, err) + } + } +} diff --git a/src/crypto/ed25519/internal/edwards25519/scalar_test.go b/src/crypto/ed25519/internal/edwards25519/scalar_test.go new file mode 100644 index 0000000..704caff --- /dev/null +++ b/src/crypto/ed25519/internal/edwards25519/scalar_test.go @@ -0,0 +1,233 @@ +// Copyright (c) 2019 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 edwards25519 + +import ( + "bytes" + "encoding/hex" + "math/big" + mathrand "math/rand" + "reflect" + "testing" + "testing/quick" +) + +// Generate returns a valid (reduced modulo l) Scalar with a distribution +// weighted towards high, low, and edge values. +func (Scalar) Generate(rand *mathrand.Rand, size int) reflect.Value { + s := scZero + diceRoll := rand.Intn(100) + switch { + case diceRoll == 0: + case diceRoll == 1: + s = scOne + case diceRoll == 2: + s = scMinusOne + case diceRoll < 5: + // Generate a low scalar in [0, 2^125). + rand.Read(s.s[:16]) + s.s[15] &= (1 << 5) - 1 + case diceRoll < 10: + // Generate a high scalar in [2^252, 2^252 + 2^124). + s.s[31] = 1 << 4 + rand.Read(s.s[:16]) + s.s[15] &= (1 << 4) - 1 + default: + // Generate a valid scalar in [0, l) by returning [0, 2^252) which has a + // negligibly different distribution (the former has a 2^-127.6 chance + // of being out of the latter range). + rand.Read(s.s[:]) + s.s[31] &= (1 << 4) - 1 + } + return reflect.ValueOf(s) +} + +// quickCheckConfig1024 will make each quickcheck test run (1024 * -quickchecks) +// times. The default value of -quickchecks is 100. +var quickCheckConfig1024 = &quick.Config{MaxCountScale: 1 << 10} + +func TestScalarGenerate(t *testing.T) { + f := func(sc Scalar) bool { + return isReduced(&sc) + } + if err := quick.Check(f, quickCheckConfig1024); err != nil { + t.Errorf("generated unreduced scalar: %v", err) + } +} + +func TestScalarSetCanonicalBytes(t *testing.T) { + f1 := func(in [32]byte, sc Scalar) bool { + // Mask out top 4 bits to guarantee value falls in [0, l). + in[len(in)-1] &= (1 << 4) - 1 + if _, err := sc.SetCanonicalBytes(in[:]); err != nil { + return false + } + return bytes.Equal(in[:], sc.Bytes()) && isReduced(&sc) + } + if err := quick.Check(f1, quickCheckConfig1024); err != nil { + t.Errorf("failed bytes->scalar->bytes round-trip: %v", err) + } + + f2 := func(sc1, sc2 Scalar) bool { + if _, err := sc2.SetCanonicalBytes(sc1.Bytes()); err != nil { + return false + } + return sc1 == sc2 + } + if err := quick.Check(f2, quickCheckConfig1024); err != nil { + t.Errorf("failed scalar->bytes->scalar round-trip: %v", err) + } + + b := scMinusOne.s + b[31] += 1 + s := scOne + if out, err := s.SetCanonicalBytes(b[:]); err == nil { + t.Errorf("SetCanonicalBytes worked on a non-canonical value") + } else if s != scOne { + t.Errorf("SetCanonicalBytes modified its receiver") + } else if out != nil { + t.Errorf("SetCanonicalBytes did not return nil with an error") + } +} + +func TestScalarSetUniformBytes(t *testing.T) { + mod, _ := new(big.Int).SetString("27742317777372353535851937790883648493", 10) + mod.Add(mod, new(big.Int).Lsh(big.NewInt(1), 252)) + f := func(in [64]byte, sc Scalar) bool { + sc.SetUniformBytes(in[:]) + if !isReduced(&sc) { + return false + } + scBig := bigIntFromLittleEndianBytes(sc.s[:]) + inBig := bigIntFromLittleEndianBytes(in[:]) + return inBig.Mod(inBig, mod).Cmp(scBig) == 0 + } + if err := quick.Check(f, quickCheckConfig1024); err != nil { + t.Error(err) + } +} + +func TestScalarSetBytesWithClamping(t *testing.T) { + // Generated with libsodium.js 1.0.18 crypto_scalarmult_ed25519_base. + + random := "633d368491364dc9cd4c1bf891b1d59460face1644813240a313e61f2c88216e" + s := new(Scalar).SetBytesWithClamping(decodeHex(random)) + p := new(Point).ScalarBaseMult(s) + want := "1d87a9026fd0126a5736fe1628c95dd419172b5b618457e041c9c861b2494a94" + if got := hex.EncodeToString(p.Bytes()); got != want { + t.Errorf("random: got %q, want %q", got, want) + } + + zero := "0000000000000000000000000000000000000000000000000000000000000000" + s = new(Scalar).SetBytesWithClamping(decodeHex(zero)) + p = new(Point).ScalarBaseMult(s) + want = "693e47972caf527c7883ad1b39822f026f47db2ab0e1919955b8993aa04411d1" + if got := hex.EncodeToString(p.Bytes()); got != want { + t.Errorf("zero: got %q, want %q", got, want) + } + + one := "ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff" + s = new(Scalar).SetBytesWithClamping(decodeHex(one)) + p = new(Point).ScalarBaseMult(s) + want = "12e9a68b73fd5aacdbcaf3e88c46fea6ebedb1aa84eed1842f07f8edab65e3a7" + if got := hex.EncodeToString(p.Bytes()); got != want { + t.Errorf("one: got %q, want %q", got, want) + } +} + +func bigIntFromLittleEndianBytes(b []byte) *big.Int { + bb := make([]byte, len(b)) + for i := range b { + bb[i] = b[len(b)-i-1] + } + return new(big.Int).SetBytes(bb) +} + +func TestScalarMultiplyDistributesOverAdd(t *testing.T) { + multiplyDistributesOverAdd := func(x, y, z Scalar) bool { + // Compute t1 = (x+y)*z + var t1 Scalar + t1.Add(&x, &y) + t1.Multiply(&t1, &z) + + // Compute t2 = x*z + y*z + var t2 Scalar + var t3 Scalar + t2.Multiply(&x, &z) + t3.Multiply(&y, &z) + t2.Add(&t2, &t3) + + return t1 == t2 && isReduced(&t1) && isReduced(&t3) + } + + if err := quick.Check(multiplyDistributesOverAdd, quickCheckConfig1024); err != nil { + t.Error(err) + } +} + +func TestScalarAddLikeSubNeg(t *testing.T) { + addLikeSubNeg := func(x, y Scalar) bool { + // Compute t1 = x - y + var t1 Scalar + t1.Subtract(&x, &y) + + // Compute t2 = -y + x + var t2 Scalar + t2.Negate(&y) + t2.Add(&t2, &x) + + return t1 == t2 && isReduced(&t1) + } + + if err := quick.Check(addLikeSubNeg, quickCheckConfig1024); err != nil { + t.Error(err) + } +} + +func TestScalarNonAdjacentForm(t *testing.T) { + s := Scalar{[32]byte{ + 0x1a, 0x0e, 0x97, 0x8a, 0x90, 0xf6, 0x62, 0x2d, + 0x37, 0x47, 0x02, 0x3f, 0x8a, 0xd8, 0x26, 0x4d, + 0xa7, 0x58, 0xaa, 0x1b, 0x88, 0xe0, 0x40, 0xd1, + 0x58, 0x9e, 0x7b, 0x7f, 0x23, 0x76, 0xef, 0x09, + }} + expectedNaf := [256]int8{ + 0, 13, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, -9, 0, 0, 0, 0, -11, 0, 0, 0, 0, 3, 0, 0, 0, 0, 1, + 0, 0, 0, 0, 9, 0, 0, 0, 0, -5, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 11, 0, 0, 0, 0, 11, 0, 0, 0, 0, 0, + -9, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 9, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 9, 0, + 0, 0, 0, -15, 0, 0, 0, 0, -7, 0, 0, 0, 0, -9, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 13, 0, 0, 0, 0, 0, -3, 0, + 0, 0, 0, -11, 0, 0, 0, 0, -7, 0, 0, 0, 0, -13, 0, 0, 0, 0, 11, 0, 0, 0, 0, -9, 0, 0, 0, 0, 0, 1, 0, 0, + 0, 0, 0, -15, 0, 0, 0, 0, 1, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 13, 0, 0, 0, + 0, 0, 0, 11, 0, 0, 0, 0, 0, 15, 0, 0, 0, 0, 0, -9, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 7, + 0, 0, 0, 0, 0, -15, 0, 0, 0, 0, 0, 15, 0, 0, 0, 0, 15, 0, 0, 0, 0, 15, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, + } + + sNaf := s.nonAdjacentForm(5) + + for i := 0; i < 256; i++ { + if expectedNaf[i] != sNaf[i] { + t.Errorf("Wrong digit at position %d, got %d, expected %d", i, sNaf[i], expectedNaf[i]) + } + } +} + +type notZeroScalar Scalar + +func (notZeroScalar) Generate(rand *mathrand.Rand, size int) reflect.Value { + var s Scalar + for s == scZero { + s = Scalar{}.Generate(rand, size).Interface().(Scalar) + } + return reflect.ValueOf(notZeroScalar(s)) +} + +func TestScalarEqual(t *testing.T) { + if scOne.Equal(&scMinusOne) == 1 { + t.Errorf("scOne.Equal(&scMinusOne) is true") + } + if scMinusOne.Equal(&scMinusOne) == 0 { + t.Errorf("scMinusOne.Equal(&scMinusOne) is false") + } +} diff --git a/src/crypto/ed25519/internal/edwards25519/scalarmult.go b/src/crypto/ed25519/internal/edwards25519/scalarmult.go new file mode 100644 index 0000000..f7ca3ce --- /dev/null +++ b/src/crypto/ed25519/internal/edwards25519/scalarmult.go @@ -0,0 +1,214 @@ +// Copyright (c) 2019 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 edwards25519 + +import "sync" + +// basepointTable is a set of 32 affineLookupTables, where table i is generated +// from 256i * basepoint. It is precomputed the first time it's used. +func basepointTable() *[32]affineLookupTable { + basepointTablePrecomp.initOnce.Do(func() { + p := NewGeneratorPoint() + for i := 0; i < 32; i++ { + basepointTablePrecomp.table[i].FromP3(p) + for j := 0; j < 8; j++ { + p.Add(p, p) + } + } + }) + return &basepointTablePrecomp.table +} + +var basepointTablePrecomp struct { + table [32]affineLookupTable + initOnce sync.Once +} + +// ScalarBaseMult sets v = x * B, where B is the canonical generator, and +// returns v. +// +// The scalar multiplication is done in constant time. +func (v *Point) ScalarBaseMult(x *Scalar) *Point { + basepointTable := basepointTable() + + // Write x = sum(x_i * 16^i) so x*B = sum( B*x_i*16^i ) + // as described in the Ed25519 paper + // + // Group even and odd coefficients + // x*B = x_0*16^0*B + x_2*16^2*B + ... + x_62*16^62*B + // + x_1*16^1*B + x_3*16^3*B + ... + x_63*16^63*B + // x*B = x_0*16^0*B + x_2*16^2*B + ... + x_62*16^62*B + // + 16*( x_1*16^0*B + x_3*16^2*B + ... + x_63*16^62*B) + // + // We use a lookup table for each i to get x_i*16^(2*i)*B + // and do four doublings to multiply by 16. + digits := x.signedRadix16() + + multiple := &affineCached{} + tmp1 := &projP1xP1{} + tmp2 := &projP2{} + + // Accumulate the odd components first + v.Set(NewIdentityPoint()) + for i := 1; i < 64; i += 2 { + basepointTable[i/2].SelectInto(multiple, digits[i]) + tmp1.AddAffine(v, multiple) + v.fromP1xP1(tmp1) + } + + // Multiply by 16 + tmp2.FromP3(v) // tmp2 = v in P2 coords + tmp1.Double(tmp2) // tmp1 = 2*v in P1xP1 coords + tmp2.FromP1xP1(tmp1) // tmp2 = 2*v in P2 coords + tmp1.Double(tmp2) // tmp1 = 4*v in P1xP1 coords + tmp2.FromP1xP1(tmp1) // tmp2 = 4*v in P2 coords + tmp1.Double(tmp2) // tmp1 = 8*v in P1xP1 coords + tmp2.FromP1xP1(tmp1) // tmp2 = 8*v in P2 coords + tmp1.Double(tmp2) // tmp1 = 16*v in P1xP1 coords + v.fromP1xP1(tmp1) // now v = 16*(odd components) + + // Accumulate the even components + for i := 0; i < 64; i += 2 { + basepointTable[i/2].SelectInto(multiple, digits[i]) + tmp1.AddAffine(v, multiple) + v.fromP1xP1(tmp1) + } + + return v +} + +// ScalarMult sets v = x * q, and returns v. +// +// The scalar multiplication is done in constant time. +func (v *Point) ScalarMult(x *Scalar, q *Point) *Point { + checkInitialized(q) + + var table projLookupTable + table.FromP3(q) + + // Write x = sum(x_i * 16^i) + // so x*Q = sum( Q*x_i*16^i ) + // = Q*x_0 + 16*(Q*x_1 + 16*( ... + Q*x_63) ... ) + // <------compute inside out--------- + // + // We use the lookup table to get the x_i*Q values + // and do four doublings to compute 16*Q + digits := x.signedRadix16() + + // Unwrap first loop iteration to save computing 16*identity + multiple := &projCached{} + tmp1 := &projP1xP1{} + tmp2 := &projP2{} + table.SelectInto(multiple, digits[63]) + + v.Set(NewIdentityPoint()) + tmp1.Add(v, multiple) // tmp1 = x_63*Q in P1xP1 coords + for i := 62; i >= 0; i-- { + tmp2.FromP1xP1(tmp1) // tmp2 = (prev) in P2 coords + tmp1.Double(tmp2) // tmp1 = 2*(prev) in P1xP1 coords + tmp2.FromP1xP1(tmp1) // tmp2 = 2*(prev) in P2 coords + tmp1.Double(tmp2) // tmp1 = 4*(prev) in P1xP1 coords + tmp2.FromP1xP1(tmp1) // tmp2 = 4*(prev) in P2 coords + tmp1.Double(tmp2) // tmp1 = 8*(prev) in P1xP1 coords + tmp2.FromP1xP1(tmp1) // tmp2 = 8*(prev) in P2 coords + tmp1.Double(tmp2) // tmp1 = 16*(prev) in P1xP1 coords + v.fromP1xP1(tmp1) // v = 16*(prev) in P3 coords + table.SelectInto(multiple, digits[i]) + tmp1.Add(v, multiple) // tmp1 = x_i*Q + 16*(prev) in P1xP1 coords + } + v.fromP1xP1(tmp1) + return v +} + +// basepointNafTable is the nafLookupTable8 for the basepoint. +// It is precomputed the first time it's used. +func basepointNafTable() *nafLookupTable8 { + basepointNafTablePrecomp.initOnce.Do(func() { + basepointNafTablePrecomp.table.FromP3(NewGeneratorPoint()) + }) + return &basepointNafTablePrecomp.table +} + +var basepointNafTablePrecomp struct { + table nafLookupTable8 + initOnce sync.Once +} + +// VarTimeDoubleScalarBaseMult sets v = a * A + b * B, where B is the canonical +// generator, and returns v. +// +// Execution time depends on the inputs. +func (v *Point) VarTimeDoubleScalarBaseMult(a *Scalar, A *Point, b *Scalar) *Point { + checkInitialized(A) + + // Similarly to the single variable-base approach, we compute + // digits and use them with a lookup table. However, because + // we are allowed to do variable-time operations, we don't + // need constant-time lookups or constant-time digit + // computations. + // + // So we use a non-adjacent form of some width w instead of + // radix 16. This is like a binary representation (one digit + // for each binary place) but we allow the digits to grow in + // magnitude up to 2^{w-1} so that the nonzero digits are as + // sparse as possible. Intuitively, this "condenses" the + // "mass" of the scalar onto sparse coefficients (meaning + // fewer additions). + + basepointNafTable := basepointNafTable() + var aTable nafLookupTable5 + aTable.FromP3(A) + // Because the basepoint is fixed, we can use a wider NAF + // corresponding to a bigger table. + aNaf := a.nonAdjacentForm(5) + bNaf := b.nonAdjacentForm(8) + + // Find the first nonzero coefficient. + i := 255 + for j := i; j >= 0; j-- { + if aNaf[j] != 0 || bNaf[j] != 0 { + break + } + } + + multA := &projCached{} + multB := &affineCached{} + tmp1 := &projP1xP1{} + tmp2 := &projP2{} + tmp2.Zero() + + // Move from high to low bits, doubling the accumulator + // at each iteration and checking whether there is a nonzero + // coefficient to look up a multiple of. + for ; i >= 0; i-- { + tmp1.Double(tmp2) + + // Only update v if we have a nonzero coeff to add in. + if aNaf[i] > 0 { + v.fromP1xP1(tmp1) + aTable.SelectInto(multA, aNaf[i]) + tmp1.Add(v, multA) + } else if aNaf[i] < 0 { + v.fromP1xP1(tmp1) + aTable.SelectInto(multA, -aNaf[i]) + tmp1.Sub(v, multA) + } + + if bNaf[i] > 0 { + v.fromP1xP1(tmp1) + basepointNafTable.SelectInto(multB, bNaf[i]) + tmp1.AddAffine(v, multB) + } else if bNaf[i] < 0 { + v.fromP1xP1(tmp1) + basepointNafTable.SelectInto(multB, -bNaf[i]) + tmp1.SubAffine(v, multB) + } + + tmp2.FromP1xP1(tmp1) + } + + v.fromP2(tmp2) + return v +} diff --git a/src/crypto/ed25519/internal/edwards25519/scalarmult_test.go b/src/crypto/ed25519/internal/edwards25519/scalarmult_test.go new file mode 100644 index 0000000..c2027f5 --- /dev/null +++ b/src/crypto/ed25519/internal/edwards25519/scalarmult_test.go @@ -0,0 +1,209 @@ +// Copyright (c) 2019 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 edwards25519 + +import ( + "testing" + "testing/quick" +) + +var ( + // quickCheckConfig32 will make each quickcheck test run (32 * -quickchecks) + // times. The default value of -quickchecks is 100. + quickCheckConfig32 = &quick.Config{MaxCountScale: 1 << 5} + + // a random scalar generated using dalek. + dalekScalar = Scalar{[32]byte{219, 106, 114, 9, 174, 249, 155, 89, 69, 203, 201, 93, 92, 116, 234, 187, 78, 115, 103, 172, 182, 98, 62, 103, 187, 136, 13, 100, 248, 110, 12, 4}} + // the above, times the edwards25519 basepoint. + dalekScalarBasepoint, _ = new(Point).SetBytes([]byte{0xf4, 0xef, 0x7c, 0xa, 0x34, 0x55, 0x7b, 0x9f, 0x72, 0x3b, 0xb6, 0x1e, 0xf9, 0x46, 0x9, 0x91, 0x1c, 0xb9, 0xc0, 0x6c, 0x17, 0x28, 0x2d, 0x8b, 0x43, 0x2b, 0x5, 0x18, 0x6a, 0x54, 0x3e, 0x48}) +) + +func TestScalarMultSmallScalars(t *testing.T) { + var z Scalar + var p Point + p.ScalarMult(&z, B) + if I.Equal(&p) != 1 { + t.Error("0*B != 0") + } + checkOnCurve(t, &p) + + z = Scalar{[32]byte{1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}} + p.ScalarMult(&z, B) + if B.Equal(&p) != 1 { + t.Error("1*B != 1") + } + checkOnCurve(t, &p) +} + +func TestScalarMultVsDalek(t *testing.T) { + var p Point + p.ScalarMult(&dalekScalar, B) + if dalekScalarBasepoint.Equal(&p) != 1 { + t.Error("Scalar mul does not match dalek") + } + checkOnCurve(t, &p) +} + +func TestBaseMultVsDalek(t *testing.T) { + var p Point + p.ScalarBaseMult(&dalekScalar) + if dalekScalarBasepoint.Equal(&p) != 1 { + t.Error("Scalar mul does not match dalek") + } + checkOnCurve(t, &p) +} + +func TestVarTimeDoubleBaseMultVsDalek(t *testing.T) { + var p Point + var z Scalar + p.VarTimeDoubleScalarBaseMult(&dalekScalar, B, &z) + if dalekScalarBasepoint.Equal(&p) != 1 { + t.Error("VarTimeDoubleScalarBaseMult fails with b=0") + } + checkOnCurve(t, &p) + p.VarTimeDoubleScalarBaseMult(&z, B, &dalekScalar) + if dalekScalarBasepoint.Equal(&p) != 1 { + t.Error("VarTimeDoubleScalarBaseMult fails with a=0") + } + checkOnCurve(t, &p) +} + +func TestScalarMultDistributesOverAdd(t *testing.T) { + scalarMultDistributesOverAdd := func(x, y Scalar) bool { + var z Scalar + z.Add(&x, &y) + var p, q, r, check Point + p.ScalarMult(&x, B) + q.ScalarMult(&y, B) + r.ScalarMult(&z, B) + check.Add(&p, &q) + checkOnCurve(t, &p, &q, &r, &check) + return check.Equal(&r) == 1 + } + + if err := quick.Check(scalarMultDistributesOverAdd, quickCheckConfig32); err != nil { + t.Error(err) + } +} + +func TestScalarMultNonIdentityPoint(t *testing.T) { + // Check whether p.ScalarMult and q.ScalaBaseMult give the same, + // when p and q are originally set to the base point. + + scalarMultNonIdentityPoint := func(x Scalar) bool { + var p, q Point + p.Set(B) + q.Set(B) + + p.ScalarMult(&x, B) + q.ScalarBaseMult(&x) + + checkOnCurve(t, &p, &q) + + return p.Equal(&q) == 1 + } + + if err := quick.Check(scalarMultNonIdentityPoint, quickCheckConfig32); err != nil { + t.Error(err) + } +} + +func TestBasepointTableGeneration(t *testing.T) { + // The basepoint table is 32 affineLookupTables, + // corresponding to (16^2i)*B for table i. + basepointTable := basepointTable() + + tmp1 := &projP1xP1{} + tmp2 := &projP2{} + tmp3 := &Point{} + tmp3.Set(B) + table := make([]affineLookupTable, 32) + for i := 0; i < 32; i++ { + // Build the table + table[i].FromP3(tmp3) + // Assert equality with the hardcoded one + if table[i] != basepointTable[i] { + t.Errorf("Basepoint table %d does not match", i) + } + + // Set p = (16^2)*p = 256*p = 2^8*p + tmp2.FromP3(tmp3) + for j := 0; j < 7; j++ { + tmp1.Double(tmp2) + tmp2.FromP1xP1(tmp1) + } + tmp1.Double(tmp2) + tmp3.fromP1xP1(tmp1) + checkOnCurve(t, tmp3) + } +} + +func TestScalarMultMatchesBaseMult(t *testing.T) { + scalarMultMatchesBaseMult := func(x Scalar) bool { + var p, q Point + p.ScalarMult(&x, B) + q.ScalarBaseMult(&x) + checkOnCurve(t, &p, &q) + return p.Equal(&q) == 1 + } + + if err := quick.Check(scalarMultMatchesBaseMult, quickCheckConfig32); err != nil { + t.Error(err) + } +} + +func TestBasepointNafTableGeneration(t *testing.T) { + var table nafLookupTable8 + table.FromP3(B) + + if table != *basepointNafTable() { + t.Error("BasepointNafTable does not match") + } +} + +func TestVarTimeDoubleBaseMultMatchesBaseMult(t *testing.T) { + varTimeDoubleBaseMultMatchesBaseMult := func(x, y Scalar) bool { + var p, q1, q2, check Point + + p.VarTimeDoubleScalarBaseMult(&x, B, &y) + + q1.ScalarBaseMult(&x) + q2.ScalarBaseMult(&y) + check.Add(&q1, &q2) + + checkOnCurve(t, &p, &check, &q1, &q2) + return p.Equal(&check) == 1 + } + + if err := quick.Check(varTimeDoubleBaseMultMatchesBaseMult, quickCheckConfig32); err != nil { + t.Error(err) + } +} + +// Benchmarks. + +func BenchmarkScalarBaseMult(t *testing.B) { + var p Point + + for i := 0; i < t.N; i++ { + p.ScalarBaseMult(&dalekScalar) + } +} + +func BenchmarkScalarMult(t *testing.B) { + var p Point + + for i := 0; i < t.N; i++ { + p.ScalarMult(&dalekScalar, B) + } +} + +func BenchmarkVarTimeDoubleScalarBaseMult(t *testing.B) { + var p Point + + for i := 0; i < t.N; i++ { + p.VarTimeDoubleScalarBaseMult(&dalekScalar, B, &dalekScalar) + } +} diff --git a/src/crypto/ed25519/internal/edwards25519/tables.go b/src/crypto/ed25519/internal/edwards25519/tables.go new file mode 100644 index 0000000..5ca40f7 --- /dev/null +++ b/src/crypto/ed25519/internal/edwards25519/tables.go @@ -0,0 +1,129 @@ +// Copyright (c) 2019 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 edwards25519 + +import ( + "crypto/subtle" +) + +// A dynamic lookup table for variable-base, constant-time scalar muls. +type projLookupTable struct { + points [8]projCached +} + +// A precomputed lookup table for fixed-base, constant-time scalar muls. +type affineLookupTable struct { + points [8]affineCached +} + +// A dynamic lookup table for variable-base, variable-time scalar muls. +type nafLookupTable5 struct { + points [8]projCached +} + +// A precomputed lookup table for fixed-base, variable-time scalar muls. +type nafLookupTable8 struct { + points [64]affineCached +} + +// Constructors. + +// Builds a lookup table at runtime. Fast. +func (v *projLookupTable) FromP3(q *Point) { + // Goal: v.points[i] = (i+1)*Q, i.e., Q, 2Q, ..., 8Q + // This allows lookup of -8Q, ..., -Q, 0, Q, ..., 8Q + v.points[0].FromP3(q) + tmpP3 := Point{} + tmpP1xP1 := projP1xP1{} + for i := 0; i < 7; i++ { + // Compute (i+1)*Q as Q + i*Q and convert to a ProjCached + // This is needlessly complicated because the API has explicit + // receivers instead of creating stack objects and relying on RVO + v.points[i+1].FromP3(tmpP3.fromP1xP1(tmpP1xP1.Add(q, &v.points[i]))) + } +} + +// This is not optimised for speed; fixed-base tables should be precomputed. +func (v *affineLookupTable) FromP3(q *Point) { + // Goal: v.points[i] = (i+1)*Q, i.e., Q, 2Q, ..., 8Q + // This allows lookup of -8Q, ..., -Q, 0, Q, ..., 8Q + v.points[0].FromP3(q) + tmpP3 := Point{} + tmpP1xP1 := projP1xP1{} + for i := 0; i < 7; i++ { + // Compute (i+1)*Q as Q + i*Q and convert to AffineCached + v.points[i+1].FromP3(tmpP3.fromP1xP1(tmpP1xP1.AddAffine(q, &v.points[i]))) + } +} + +// Builds a lookup table at runtime. Fast. +func (v *nafLookupTable5) FromP3(q *Point) { + // Goal: v.points[i] = (2*i+1)*Q, i.e., Q, 3Q, 5Q, ..., 15Q + // This allows lookup of -15Q, ..., -3Q, -Q, 0, Q, 3Q, ..., 15Q + v.points[0].FromP3(q) + q2 := Point{} + q2.Add(q, q) + tmpP3 := Point{} + tmpP1xP1 := projP1xP1{} + for i := 0; i < 7; i++ { + v.points[i+1].FromP3(tmpP3.fromP1xP1(tmpP1xP1.Add(&q2, &v.points[i]))) + } +} + +// This is not optimised for speed; fixed-base tables should be precomputed. +func (v *nafLookupTable8) FromP3(q *Point) { + v.points[0].FromP3(q) + q2 := Point{} + q2.Add(q, q) + tmpP3 := Point{} + tmpP1xP1 := projP1xP1{} + for i := 0; i < 63; i++ { + v.points[i+1].FromP3(tmpP3.fromP1xP1(tmpP1xP1.AddAffine(&q2, &v.points[i]))) + } +} + +// Selectors. + +// Set dest to x*Q, where -8 <= x <= 8, in constant time. +func (v *projLookupTable) SelectInto(dest *projCached, x int8) { + // Compute xabs = |x| + xmask := x >> 7 + xabs := uint8((x + xmask) ^ xmask) + + dest.Zero() + for j := 1; j <= 8; j++ { + // Set dest = j*Q if |x| = j + cond := subtle.ConstantTimeByteEq(xabs, uint8(j)) + dest.Select(&v.points[j-1], dest, cond) + } + // Now dest = |x|*Q, conditionally negate to get x*Q + dest.CondNeg(int(xmask & 1)) +} + +// Set dest to x*Q, where -8 <= x <= 8, in constant time. +func (v *affineLookupTable) SelectInto(dest *affineCached, x int8) { + // Compute xabs = |x| + xmask := x >> 7 + xabs := uint8((x + xmask) ^ xmask) + + dest.Zero() + for j := 1; j <= 8; j++ { + // Set dest = j*Q if |x| = j + cond := subtle.ConstantTimeByteEq(xabs, uint8(j)) + dest.Select(&v.points[j-1], dest, cond) + } + // Now dest = |x|*Q, conditionally negate to get x*Q + dest.CondNeg(int(xmask & 1)) +} + +// Given odd x with 0 < x < 2^4, return x*Q (in variable time). +func (v *nafLookupTable5) SelectInto(dest *projCached, x int8) { + *dest = v.points[x/2] +} + +// Given odd x with 0 < x < 2^7, return x*Q (in variable time). +func (v *nafLookupTable8) SelectInto(dest *affineCached, x int8) { + *dest = v.points[x/2] +} diff --git a/src/crypto/ed25519/internal/edwards25519/tables_test.go b/src/crypto/ed25519/internal/edwards25519/tables_test.go new file mode 100644 index 0000000..b5d161a --- /dev/null +++ b/src/crypto/ed25519/internal/edwards25519/tables_test.go @@ -0,0 +1,119 @@ +// Copyright (c) 2019 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 edwards25519 + +import ( + "testing" +) + +func TestProjLookupTable(t *testing.T) { + var table projLookupTable + table.FromP3(B) + + var tmp1, tmp2, tmp3 projCached + table.SelectInto(&tmp1, 6) + table.SelectInto(&tmp2, -2) + table.SelectInto(&tmp3, -4) + // Expect T1 + T2 + T3 = identity + + var accP1xP1 projP1xP1 + accP3 := NewIdentityPoint() + + accP1xP1.Add(accP3, &tmp1) + accP3.fromP1xP1(&accP1xP1) + accP1xP1.Add(accP3, &tmp2) + accP3.fromP1xP1(&accP1xP1) + accP1xP1.Add(accP3, &tmp3) + accP3.fromP1xP1(&accP1xP1) + + if accP3.Equal(I) != 1 { + t.Errorf("Consistency check on ProjLookupTable.SelectInto failed! %x %x %x", tmp1, tmp2, tmp3) + } +} + +func TestAffineLookupTable(t *testing.T) { + var table affineLookupTable + table.FromP3(B) + + var tmp1, tmp2, tmp3 affineCached + table.SelectInto(&tmp1, 3) + table.SelectInto(&tmp2, -7) + table.SelectInto(&tmp3, 4) + // Expect T1 + T2 + T3 = identity + + var accP1xP1 projP1xP1 + accP3 := NewIdentityPoint() + + accP1xP1.AddAffine(accP3, &tmp1) + accP3.fromP1xP1(&accP1xP1) + accP1xP1.AddAffine(accP3, &tmp2) + accP3.fromP1xP1(&accP1xP1) + accP1xP1.AddAffine(accP3, &tmp3) + accP3.fromP1xP1(&accP1xP1) + + if accP3.Equal(I) != 1 { + t.Errorf("Consistency check on ProjLookupTable.SelectInto failed! %x %x %x", tmp1, tmp2, tmp3) + } +} + +func TestNafLookupTable5(t *testing.T) { + var table nafLookupTable5 + table.FromP3(B) + + var tmp1, tmp2, tmp3, tmp4 projCached + table.SelectInto(&tmp1, 9) + table.SelectInto(&tmp2, 11) + table.SelectInto(&tmp3, 7) + table.SelectInto(&tmp4, 13) + // Expect T1 + T2 = T3 + T4 + + var accP1xP1 projP1xP1 + lhs := NewIdentityPoint() + rhs := NewIdentityPoint() + + accP1xP1.Add(lhs, &tmp1) + lhs.fromP1xP1(&accP1xP1) + accP1xP1.Add(lhs, &tmp2) + lhs.fromP1xP1(&accP1xP1) + + accP1xP1.Add(rhs, &tmp3) + rhs.fromP1xP1(&accP1xP1) + accP1xP1.Add(rhs, &tmp4) + rhs.fromP1xP1(&accP1xP1) + + if lhs.Equal(rhs) != 1 { + t.Errorf("Consistency check on nafLookupTable5 failed") + } +} + +func TestNafLookupTable8(t *testing.T) { + var table nafLookupTable8 + table.FromP3(B) + + var tmp1, tmp2, tmp3, tmp4 affineCached + table.SelectInto(&tmp1, 49) + table.SelectInto(&tmp2, 11) + table.SelectInto(&tmp3, 35) + table.SelectInto(&tmp4, 25) + // Expect T1 + T2 = T3 + T4 + + var accP1xP1 projP1xP1 + lhs := NewIdentityPoint() + rhs := NewIdentityPoint() + + accP1xP1.AddAffine(lhs, &tmp1) + lhs.fromP1xP1(&accP1xP1) + accP1xP1.AddAffine(lhs, &tmp2) + lhs.fromP1xP1(&accP1xP1) + + accP1xP1.AddAffine(rhs, &tmp3) + rhs.fromP1xP1(&accP1xP1) + accP1xP1.AddAffine(rhs, &tmp4) + rhs.fromP1xP1(&accP1xP1) + + if lhs.Equal(rhs) != 1 { + t.Errorf("Consistency check on nafLookupTable8 failed") + } +} |