// Copyright 2022 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 ignore package main // Running this generator requires addchain v0.4.0, which can be installed with // // go install github.com/mmcloughlin/addchain/cmd/addchain@v0.4.0 // import ( "bytes" "crypto/elliptic" "fmt" "go/format" "io" "log" "math/big" "os" "os/exec" "strings" "text/template" ) var curves = []struct { P string Element string Params *elliptic.CurveParams }{ { P: "P224", Element: "fiat.P224Element", Params: elliptic.P224().Params(), }, { P: "P384", Element: "fiat.P384Element", Params: elliptic.P384().Params(), }, { P: "P521", Element: "fiat.P521Element", Params: elliptic.P521().Params(), }, } func main() { t := template.Must(template.New("tmplNISTEC").Parse(tmplNISTEC)) tmplAddchainFile, err := os.CreateTemp("", "addchain-template") if err != nil { log.Fatal(err) } defer os.Remove(tmplAddchainFile.Name()) if _, err := io.WriteString(tmplAddchainFile, tmplAddchain); err != nil { log.Fatal(err) } if err := tmplAddchainFile.Close(); err != nil { log.Fatal(err) } for _, c := range curves { p := strings.ToLower(c.P) elementLen := (c.Params.BitSize + 7) / 8 B := fmt.Sprintf("%#v", c.Params.B.FillBytes(make([]byte, elementLen))) Gx := fmt.Sprintf("%#v", c.Params.Gx.FillBytes(make([]byte, elementLen))) Gy := fmt.Sprintf("%#v", c.Params.Gy.FillBytes(make([]byte, elementLen))) log.Printf("Generating %s.go...", p) f, err := os.Create(p + ".go") if err != nil { log.Fatal(err) } defer f.Close() buf := &bytes.Buffer{} if err := t.Execute(buf, map[string]interface{}{ "P": c.P, "p": p, "B": B, "Gx": Gx, "Gy": Gy, "Element": c.Element, "ElementLen": elementLen, }); err != nil { log.Fatal(err) } out, err := format.Source(buf.Bytes()) if err != nil { log.Fatal(err) } if _, err := f.Write(out); err != nil { log.Fatal(err) } // If p = 3 mod 4, implement modular square root by exponentiation. mod4 := new(big.Int).Mod(c.Params.P, big.NewInt(4)) if mod4.Cmp(big.NewInt(3)) != 0 { continue } exp := new(big.Int).Add(c.Params.P, big.NewInt(1)) exp.Div(exp, big.NewInt(4)) tmp, err := os.CreateTemp("", "addchain-"+p) if err != nil { log.Fatal(err) } defer os.Remove(tmp.Name()) cmd := exec.Command("addchain", "search", fmt.Sprintf("%d", exp)) cmd.Stderr = os.Stderr cmd.Stdout = tmp if err := cmd.Run(); err != nil { log.Fatal(err) } if err := tmp.Close(); err != nil { log.Fatal(err) } cmd = exec.Command("addchain", "gen", "-tmpl", tmplAddchainFile.Name(), tmp.Name()) cmd.Stderr = os.Stderr out, err = cmd.Output() if err != nil { log.Fatal(err) } out = bytes.Replace(out, []byte("Element"), []byte(c.Element), -1) out = bytes.Replace(out, []byte("sqrtCandidate"), []byte(p+"SqrtCandidate"), -1) out, err = format.Source(out) if err != nil { log.Fatal(err) } if _, err := f.Write(out); err != nil { log.Fatal(err) } } } const tmplNISTEC = `// Copyright 2022 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. // Code generated by generate.go. DO NOT EDIT. package nistec import ( "crypto/internal/fips140/nistec/fiat" "crypto/internal/fips140/subtle" "errors" "sync" ) // {{.p}}ElementLength is the length of an element of the base or scalar field, // which have the same bytes length for all NIST P curves. const {{.p}}ElementLength = {{ .ElementLen }} // {{.P}}Point is a {{.P}} point. The zero value is NOT valid. type {{.P}}Point struct { // The point is represented in projective coordinates (X:Y:Z), // where x = X/Z and y = Y/Z. x, y, z *{{.Element}} } // New{{.P}}Point returns a new {{.P}}Point representing the point at infinity point. func New{{.P}}Point() *{{.P}}Point { return &{{.P}}Point{ x: new({{.Element}}), y: new({{.Element}}).One(), z: new({{.Element}}), } } // SetGenerator sets p to the canonical generator and returns p. func (p *{{.P}}Point) SetGenerator() *{{.P}}Point { p.x.SetBytes({{.Gx}}) p.y.SetBytes({{.Gy}}) p.z.One() return p } // Set sets p = q and returns p. func (p *{{.P}}Point) Set(q *{{.P}}Point) *{{.P}}Point { p.x.Set(q.x) p.y.Set(q.y) p.z.Set(q.z) return p } // SetBytes sets p to the compressed, uncompressed, or infinity value encoded in // b, as specified in SEC 1, Version 2.0, Section 2.3.4. If the point is not on // the curve, it returns nil and an error, and the receiver is unchanged. // Otherwise, it returns p. func (p *{{.P}}Point) SetBytes(b []byte) (*{{.P}}Point, error) { switch { // Point at infinity. case len(b) == 1 && b[0] == 0: return p.Set(New{{.P}}Point()), nil // Uncompressed form. case len(b) == 1+2*{{.p}}ElementLength && b[0] == 4: x, err := new({{.Element}}).SetBytes(b[1 : 1+{{.p}}ElementLength]) if err != nil { return nil, err } y, err := new({{.Element}}).SetBytes(b[1+{{.p}}ElementLength:]) if err != nil { return nil, err } if err := {{.p}}CheckOnCurve(x, y); err != nil { return nil, err } p.x.Set(x) p.y.Set(y) p.z.One() return p, nil // Compressed form. case len(b) == 1+{{.p}}ElementLength && (b[0] == 2 || b[0] == 3): x, err := new({{.Element}}).SetBytes(b[1:]) if err != nil { return nil, err } // y² = x³ - 3x + b y := {{.p}}Polynomial(new({{.Element}}), x) if !{{.p}}Sqrt(y, y) { return nil, errors.New("invalid {{.P}} compressed point encoding") } // Select the positive or negative root, as indicated by the least // significant bit, based on the encoding type byte. otherRoot := new({{.Element}}) otherRoot.Sub(otherRoot, y) cond := y.Bytes()[{{.p}}ElementLength-1]&1 ^ b[0]&1 y.Select(otherRoot, y, int(cond)) p.x.Set(x) p.y.Set(y) p.z.One() return p, nil default: return nil, errors.New("invalid {{.P}} point encoding") } } var _{{.p}}B *{{.Element}} var _{{.p}}BOnce sync.Once func {{.p}}B() *{{.Element}} { _{{.p}}BOnce.Do(func() { _{{.p}}B, _ = new({{.Element}}).SetBytes({{.B}}) }) return _{{.p}}B } // {{.p}}Polynomial sets y2 to x³ - 3x + b, and returns y2. func {{.p}}Polynomial(y2, x *{{.Element}}) *{{.Element}} { y2.Square(x) y2.Mul(y2, x) threeX := new({{.Element}}).Add(x, x) threeX.Add(threeX, x) y2.Sub(y2, threeX) return y2.Add(y2, {{.p}}B()) } func {{.p}}CheckOnCurve(x, y *{{.Element}}) error { // y² = x³ - 3x + b rhs := {{.p}}Polynomial(new({{.Element}}), x) lhs := new({{.Element}}).Square(y) if rhs.Equal(lhs) != 1 { return errors.New("{{.P}} point not on curve") } return nil } // Bytes returns the uncompressed or infinity encoding of p, as specified in // SEC 1, Version 2.0, Section 2.3.3. Note that the encoding of the point at // infinity is shorter than all other encodings. func (p *{{.P}}Point) Bytes() []byte { // This function is outlined to make the allocations inline in the caller // rather than happen on the heap. var out [1+2*{{.p}}ElementLength]byte return p.bytes(&out) } func (p *{{.P}}Point) bytes(out *[1+2*{{.p}}ElementLength]byte) []byte { if p.z.IsZero() == 1 { return append(out[:0], 0) } zinv := new({{.Element}}).Invert(p.z) x := new({{.Element}}).Mul(p.x, zinv) y := new({{.Element}}).Mul(p.y, zinv) buf := append(out[:0], 4) buf = append(buf, x.Bytes()...) buf = append(buf, y.Bytes()...) return buf } // BytesX returns the encoding of the x-coordinate of p, as specified in SEC 1, // Version 2.0, Section 2.3.5, or an error if p is the point at infinity. func (p *{{.P}}Point) BytesX() ([]byte, error) { // This function is outlined to make the allocations inline in the caller // rather than happen on the heap. var out [{{.p}}ElementLength]byte return p.bytesX(&out) } func (p *{{.P}}Point) bytesX(out *[{{.p}}ElementLength]byte) ([]byte, error) { if p.z.IsZero() == 1 { return nil, errors.New("{{.P}} point is the point at infinity") } zinv := new({{.Element}}).Invert(p.z) x := new({{.Element}}).Mul(p.x, zinv) return append(out[:0], x.Bytes()...), nil } // BytesCompressed returns the compressed or infinity encoding of p, as // specified in SEC 1, Version 2.0, Section 2.3.3. Note that the encoding of the // point at infinity is shorter than all other encodings. func (p *{{.P}}Point) BytesCompressed() []byte { // This function is outlined to make the allocations inline in the caller // rather than happen on the heap. var out [1 + {{.p}}ElementLength]byte return p.bytesCompressed(&out) } func (p *{{.P}}Point) bytesCompressed(out *[1 + {{.p}}ElementLength]byte) []byte { if p.z.IsZero() == 1 { return append(out[:0], 0) } zinv := new({{.Element}}).Invert(p.z) x := new({{.Element}}).Mul(p.x, zinv) y := new({{.Element}}).Mul(p.y, zinv) // Encode the sign of the y coordinate (indicated by the least significant // bit) as the encoding type (2 or 3). buf := append(out[:0], 2) buf[0] |= y.Bytes()[{{.p}}ElementLength-1] & 1 buf = append(buf, x.Bytes()...) return buf } // Add sets q = p1 + p2, and returns q. The points may overlap. func (q *{{.P}}Point) Add(p1, p2 *{{.P}}Point) *{{.P}}Point { // Complete addition formula for a = -3 from "Complete addition formulas for // prime order elliptic curves" (https://eprint.iacr.org/2015/1060), §A.2. t0 := new({{.Element}}).Mul(p1.x, p2.x) // t0 := X1 * X2 t1 := new({{.Element}}).Mul(p1.y, p2.y) // t1 := Y1 * Y2 t2 := new({{.Element}}).Mul(p1.z, p2.z) // t2 := Z1 * Z2 t3 := new({{.Element}}).Add(p1.x, p1.y) // t3 := X1 + Y1 t4 := new({{.Element}}).Add(p2.x, p2.y) // t4 := X2 + Y2 t3.Mul(t3, t4) // t3 := t3 * t4 t4.Add(t0, t1) // t4 := t0 + t1 t3.Sub(t3, t4) // t3 := t3 - t4 t4.Add(p1.y, p1.z) // t4 := Y1 + Z1 x3 := new({{.Element}}).Add(p2.y, p2.z) // X3 := Y2 + Z2 t4.Mul(t4, x3) // t4 := t4 * X3 x3.Add(t1, t2) // X3 := t1 + t2 t4.Sub(t4, x3) // t4 := t4 - X3 x3.Add(p1.x, p1.z) // X3 := X1 + Z1 y3 := new({{.Element}}).Add(p2.x, p2.z) // Y3 := X2 + Z2 x3.Mul(x3, y3) // X3 := X3 * Y3 y3.Add(t0, t2) // Y3 := t0 + t2 y3.Sub(x3, y3) // Y3 := X3 - Y3 z3 := new({{.Element}}).Mul({{.p}}B(), t2) // Z3 := b * t2 x3.Sub(y3, z3) // X3 := Y3 - Z3 z3.Add(x3, x3) // Z3 := X3 + X3 x3.Add(x3, z3) // X3 := X3 + Z3 z3.Sub(t1, x3) // Z3 := t1 - X3 x3.Add(t1, x3) // X3 := t1 + X3 y3.Mul({{.p}}B(), y3) // Y3 := b * Y3 t1.Add(t2, t2) // t1 := t2 + t2 t2.Add(t1, t2) // t2 := t1 + t2 y3.Sub(y3, t2) // Y3 := Y3 - t2 y3.Sub(y3, t0) // Y3 := Y3 - t0 t1.Add(y3, y3) // t1 := Y3 + Y3 y3.Add(t1, y3) // Y3 := t1 + Y3 t1.Add(t0, t0) // t1 := t0 + t0 t0.Add(t1, t0) // t0 := t1 + t0 t0.Sub(t0, t2) // t0 := t0 - t2 t1.Mul(t4, y3) // t1 := t4 * Y3 t2.Mul(t0, y3) // t2 := t0 * Y3 y3.Mul(x3, z3) // Y3 := X3 * Z3 y3.Add(y3, t2) // Y3 := Y3 + t2 x3.Mul(t3, x3) // X3 := t3 * X3 x3.Sub(x3, t1) // X3 := X3 - t1 z3.Mul(t4, z3) // Z3 := t4 * Z3 t1.Mul(t3, t0) // t1 := t3 * t0 z3.Add(z3, t1) // Z3 := Z3 + t1 q.x.Set(x3) q.y.Set(y3) q.z.Set(z3) return q } // Double sets q = p + p, and returns q. The points may overlap. func (q *{{.P}}Point) Double(p *{{.P}}Point) *{{.P}}Point { // Complete addition formula for a = -3 from "Complete addition formulas for // prime order elliptic curves" (https://eprint.iacr.org/2015/1060), §A.2. t0 := new({{.Element}}).Square(p.x) // t0 := X ^ 2 t1 := new({{.Element}}).Square(p.y) // t1 := Y ^ 2 t2 := new({{.Element}}).Square(p.z) // t2 := Z ^ 2 t3 := new({{.Element}}).Mul(p.x, p.y) // t3 := X * Y t3.Add(t3, t3) // t3 := t3 + t3 z3 := new({{.Element}}).Mul(p.x, p.z) // Z3 := X * Z z3.Add(z3, z3) // Z3 := Z3 + Z3 y3 := new({{.Element}}).Mul({{.p}}B(), t2) // Y3 := b * t2 y3.Sub(y3, z3) // Y3 := Y3 - Z3 x3 := new({{.Element}}).Add(y3, y3) // X3 := Y3 + Y3 y3.Add(x3, y3) // Y3 := X3 + Y3 x3.Sub(t1, y3) // X3 := t1 - Y3 y3.Add(t1, y3) // Y3 := t1 + Y3 y3.Mul(x3, y3) // Y3 := X3 * Y3 x3.Mul(x3, t3) // X3 := X3 * t3 t3.Add(t2, t2) // t3 := t2 + t2 t2.Add(t2, t3) // t2 := t2 + t3 z3.Mul({{.p}}B(), z3) // Z3 := b * Z3 z3.Sub(z3, t2) // Z3 := Z3 - t2 z3.Sub(z3, t0) // Z3 := Z3 - t0 t3.Add(z3, z3) // t3 := Z3 + Z3 z3.Add(z3, t3) // Z3 := Z3 + t3 t3.Add(t0, t0) // t3 := t0 + t0 t0.Add(t3, t0) // t0 := t3 + t0 t0.Sub(t0, t2) // t0 := t0 - t2 t0.Mul(t0, z3) // t0 := t0 * Z3 y3.Add(y3, t0) // Y3 := Y3 + t0 t0.Mul(p.y, p.z) // t0 := Y * Z t0.Add(t0, t0) // t0 := t0 + t0 z3.Mul(t0, z3) // Z3 := t0 * Z3 x3.Sub(x3, z3) // X3 := X3 - Z3 z3.Mul(t0, t1) // Z3 := t0 * t1 z3.Add(z3, z3) // Z3 := Z3 + Z3 z3.Add(z3, z3) // Z3 := Z3 + Z3 q.x.Set(x3) q.y.Set(y3) q.z.Set(z3) return q } // Select sets q to p1 if cond == 1, and to p2 if cond == 0. func (q *{{.P}}Point) Select(p1, p2 *{{.P}}Point, cond int) *{{.P}}Point { q.x.Select(p1.x, p2.x, cond) q.y.Select(p1.y, p2.y, cond) q.z.Select(p1.z, p2.z, cond) return q } // A {{.p}}Table holds the first 15 multiples of a point at offset -1, so [1]P // is at table[0], [15]P is at table[14], and [0]P is implicitly the identity // point. type {{.p}}Table [15]*{{.P}}Point // Select selects the n-th multiple of the table base point into p. It works in // constant time by iterating over every entry of the table. n must be in [0, 15]. func (table *{{.p}}Table) Select(p *{{.P}}Point, n uint8) { if n >= 16 { panic("nistec: internal error: {{.p}}Table called with out-of-bounds value") } p.Set(New{{.P}}Point()) for i := uint8(1); i < 16; i++ { cond := subtle.ConstantTimeByteEq(i, n) p.Select(table[i-1], p, cond) } } // ScalarMult sets p = scalar * q, and returns p. func (p *{{.P}}Point) ScalarMult(q *{{.P}}Point, scalar []byte) (*{{.P}}Point, error) { // Compute a {{.p}}Table for the base point q. The explicit New{{.P}}Point // calls get inlined, letting the allocations live on the stack. var table = {{.p}}Table{New{{.P}}Point(), New{{.P}}Point(), New{{.P}}Point(), New{{.P}}Point(), New{{.P}}Point(), New{{.P}}Point(), New{{.P}}Point(), New{{.P}}Point(), New{{.P}}Point(), New{{.P}}Point(), New{{.P}}Point(), New{{.P}}Point(), New{{.P}}Point(), New{{.P}}Point(), New{{.P}}Point()} table[0].Set(q) for i := 1; i < 15; i += 2 { table[i].Double(table[i/2]) table[i+1].Add(table[i], q) } // Instead of doing the classic double-and-add chain, we do it with a // four-bit window: we double four times, and then add [0-15]P. t := New{{.P}}Point() p.Set(New{{.P}}Point()) for i, byte := range scalar { // No need to double on the first iteration, as p is the identity at // this point, and [N]∞ = ∞. if i != 0 { p.Double(p) p.Double(p) p.Double(p) p.Double(p) } windowValue := byte >> 4 table.Select(t, windowValue) p.Add(p, t) p.Double(p) p.Double(p) p.Double(p) p.Double(p) windowValue = byte & 0b1111 table.Select(t, windowValue) p.Add(p, t) } return p, nil } var {{.p}}GeneratorTable *[{{.p}}ElementLength * 2]{{.p}}Table var {{.p}}GeneratorTableOnce sync.Once // generatorTable returns a sequence of {{.p}}Tables. The first table contains // multiples of G. Each successive table is the previous table doubled four // times. func (p *{{.P}}Point) generatorTable() *[{{.p}}ElementLength * 2]{{.p}}Table { {{.p}}GeneratorTableOnce.Do(func() { {{.p}}GeneratorTable = new([{{.p}}ElementLength * 2]{{.p}}Table) base := New{{.P}}Point().SetGenerator() for i := 0; i < {{.p}}ElementLength*2; i++ { {{.p}}GeneratorTable[i][0] = New{{.P}}Point().Set(base) for j := 1; j < 15; j++ { {{.p}}GeneratorTable[i][j] = New{{.P}}Point().Add({{.p}}GeneratorTable[i][j-1], base) } base.Double(base) base.Double(base) base.Double(base) base.Double(base) } }) return {{.p}}GeneratorTable } // ScalarBaseMult sets p = scalar * B, where B is the canonical generator, and // returns p. func (p *{{.P}}Point) ScalarBaseMult(scalar []byte) (*{{.P}}Point, error) { if len(scalar) != {{.p}}ElementLength { return nil, errors.New("invalid scalar length") } tables := p.generatorTable() // This is also a scalar multiplication with a four-bit window like in // ScalarMult, but in this case the doublings are precomputed. The value // [windowValue]G added at iteration k would normally get doubled // (totIterations-k)×4 times, but with a larger precomputation we can // instead add [2^((totIterations-k)×4)][windowValue]G and avoid the // doublings between iterations. t := New{{.P}}Point() p.Set(New{{.P}}Point()) tableIndex := len(tables) - 1 for _, byte := range scalar { windowValue := byte >> 4 tables[tableIndex].Select(t, windowValue) p.Add(p, t) tableIndex-- windowValue = byte & 0b1111 tables[tableIndex].Select(t, windowValue) p.Add(p, t) tableIndex-- } return p, nil } // {{.p}}Sqrt sets e to a square root of x. If x is not a square, {{.p}}Sqrt returns // false and e is unchanged. e and x can overlap. func {{.p}}Sqrt(e, x *{{ .Element }}) (isSquare bool) { candidate := new({{ .Element }}) {{.p}}SqrtCandidate(candidate, x) square := new({{ .Element }}).Square(candidate) if square.Equal(x) != 1 { return false } e.Set(candidate) return true } ` const tmplAddchain = ` // sqrtCandidate sets z to a square root candidate for x. z and x must not overlap. func sqrtCandidate(z, x *Element) { // Since p = 3 mod 4, exponentiation by (p + 1) / 4 yields a square root candidate. // // The sequence of {{ .Ops.Adds }} multiplications and {{ .Ops.Doubles }} squarings is derived from the // following addition chain generated with {{ .Meta.Module }} {{ .Meta.ReleaseTag }}. // {{- range lines (format .Script) }} // {{ . }} {{- end }} // {{- range .Program.Temporaries }} var {{ . }} = new(Element) {{- end }} {{ range $i := .Program.Instructions -}} {{- with add $i.Op }} {{ $i.Output }}.Mul({{ .X }}, {{ .Y }}) {{- end -}} {{- with double $i.Op }} {{ $i.Output }}.Square({{ .X }}) {{- end -}} {{- with shift $i.Op -}} {{- $first := 0 -}} {{- if ne $i.Output.Identifier .X.Identifier }} {{ $i.Output }}.Square({{ .X }}) {{- $first = 1 -}} {{- end }} for s := {{ $first }}; s < {{ .S }}; s++ { {{ $i.Output }}.Square({{ $i.Output }}) } {{- end -}} {{- end }} } `