src/crypto/internal/bigmod/nat_test.go

// Copyright 2021 The Go Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.

package bigmod

import (
	"fmt"
	"math/big"
	"math/bits"
	"math/rand"
	"reflect"
	"strings"
	"testing"
	"testing/quick"
)

func (n *Nat) String() string {
	var limbs []string
	for i := range n.limbs {
		limbs = append(limbs, fmt.Sprintf("%016X", n.limbs[len(n.limbs)-1-i]))
	}
	return "{" + strings.Join(limbs, " ") + "}"
}

// Generate generates an even nat. It's used by testing/quick to produce random
// *nat values for quick.Check invocations.
func (*Nat) Generate(r *rand.Rand, size int) reflect.Value {
	limbs := make([]uint, size)
	for i := 0; i < size; i++ {
		limbs[i] = uint(r.Uint64()) & ((1 << _W) - 2)
	}
	return reflect.ValueOf(&Nat{limbs})
}

func testModAddCommutative(a *Nat, b *Nat) bool {
	m := maxModulus(uint(len(a.limbs)))
	aPlusB := new(Nat).set(a)
	aPlusB.Add(b, m)
	bPlusA := new(Nat).set(b)
	bPlusA.Add(a, m)
	return aPlusB.Equal(bPlusA) == 1
}

func TestModAddCommutative(t *testing.T) {
	err := quick.Check(testModAddCommutative, &quick.Config{})
	if err != nil {
		t.Error(err)
	}
}

func testModSubThenAddIdentity(a *Nat, b *Nat) bool {
	m := maxModulus(uint(len(a.limbs)))
	original := new(Nat).set(a)
	a.Sub(b, m)
	a.Add(b, m)
	return a.Equal(original) == 1
}

func TestModSubThenAddIdentity(t *testing.T) {
	err := quick.Check(testModSubThenAddIdentity, &quick.Config{})
	if err != nil {
		t.Error(err)
	}
}

func TestMontgomeryRoundtrip(t *testing.T) {
	err := quick.Check(func(a *Nat) bool {
		one := &Nat{make([]uint, len(a.limbs))}
		one.limbs[0] = 1
		aPlusOne := new(big.Int).SetBytes(natBytes(a))
		aPlusOne.Add(aPlusOne, big.NewInt(1))
		m, _ := NewModulusFromBig(aPlusOne)
		monty := new(Nat).set(a)
		monty.montgomeryRepresentation(m)
		aAgain := new(Nat).set(monty)
		aAgain.montgomeryMul(monty, one, m)
		if a.Equal(aAgain) != 1 {
			t.Errorf("%v != %v", a, aAgain)
			return false
		}
		return true
	}, &quick.Config{})
	if err != nil {
		t.Error(err)
	}
}

func TestShiftIn(t *testing.T) {
	if bits.UintSize != 64 {
		t.Skip("examples are only valid in 64 bit")
	}
	examples := []struct {
		m, x, expected []byte
		y              uint64
	}{{
		m:        []byte{13},
		x:        []byte{0},
		y:        0xFFFF_FFFF_FFFF_FFFF,
		expected: []byte{2},
	}, {
		m:        []byte{13},
		x:        []byte{7},
		y:        0xFFFF_FFFF_FFFF_FFFF,
		expected: []byte{10},
	}, {
		m:        []byte{0x06, 0x80, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x0d},
		x:        make([]byte, 9),
		y:        0xFFFF_FFFF_FFFF_FFFF,
		expected: []byte{0x00, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff},
	}, {
		m:        []byte{0x06, 0x80, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x0d},
		x:        []byte{0x00, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff},
		y:        0,
		expected: []byte{0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x06},
	}}

	for i, tt := range examples {
		m := modulusFromBytes(tt.m)
		got := natFromBytes(tt.x).ExpandFor(m).shiftIn(uint(tt.y), m)
		if exp := natFromBytes(tt.expected).ExpandFor(m); got.Equal(exp) != 1 {
			t.Errorf("%d: got %v, expected %v", i, got, exp)
		}
	}
}

func TestModulusAndNatSizes(t *testing.T) {
	// These are 126 bit (2 * _W on 64-bit architectures) values, serialized as
	// 128 bits worth of bytes. If leading zeroes are stripped, they fit in two
	// limbs, if they are not, they fit in three. This can be a problem because
	// modulus strips leading zeroes and nat does not.
	m := modulusFromBytes([]byte{
		0x3f, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
		0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff})
	xb := []byte{0x3f, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff,
		0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xfe}
	natFromBytes(xb).ExpandFor(m) // must not panic for shrinking
	NewNat().SetBytes(xb, m)
}

func TestSetBytes(t *testing.T) {
	tests := []struct {
		m, b []byte
		fail bool
	}{{
		m: []byte{0xff, 0xff},
		b: []byte{0x00, 0x01},
	}, {
		m:    []byte{0xff, 0xff},
		b:    []byte{0xff, 0xff},
		fail: true,
	}, {
		m: []byte{0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff},
		b: []byte{0x00, 0x01},
	}, {
		m: []byte{0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff},
		b: []byte{0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xfe},
	}, {
		m:    []byte{0xff, 0xff},
		b:    []byte{0xff, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00},
		fail: true,
	}, {
		m:    []byte{0xff, 0xff},
		b:    []byte{0xff, 0xff, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00},
		fail: true,
	}, {
		m: []byte{0x7f, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff},
		b: []byte{0x7f, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xfe},
	}, {
		m:    []byte{0x7f, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff},
		b:    []byte{0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xfe},
		fail: true,
	}, {
		m:    []byte{0x7f, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff},
		b:    []byte{0x7f, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff},
		fail: true,
	}, {
		m:    []byte{0x7f, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff},
		b:    []byte{0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xfe},
		fail: true,
	}, {
		m:    []byte{0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xfd},
		b:    []byte{0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff},
		fail: true,
	}}

	for i, tt := range tests {
		m := modulusFromBytes(tt.m)
		got, err := NewNat().SetBytes(tt.b, m)
		if err != nil {
			if !tt.fail {
				t.Errorf("%d: unexpected error: %v", i, err)
			}
			continue
		}
		if tt.fail {
			t.Errorf("%d: unexpected success", i)
			continue
		}
		if expected := natFromBytes(tt.b).ExpandFor(m); got.Equal(expected) != yes {
			t.Errorf("%d: got %v, expected %v", i, got, expected)
		}
	}

	f := func(xBytes []byte) bool {
		m := maxModulus(uint(len(xBytes)*8/_W + 1))
		got, err := NewNat().SetBytes(xBytes, m)
		if err != nil {
			return false
		}
		return got.Equal(natFromBytes(xBytes).ExpandFor(m)) == yes
	}

	err := quick.Check(f, &quick.Config{})
	if err != nil {
		t.Error(err)
	}
}

func TestExpand(t *testing.T) {
	sliced := []uint{1, 2, 3, 4}
	examples := []struct {
		in  []uint
		n   int
		out []uint
	}{{
		[]uint{1, 2},
		4,
		[]uint{1, 2, 0, 0},
	}, {
		sliced[:2],
		4,
		[]uint{1, 2, 0, 0},
	}, {
		[]uint{1, 2},
		2,
		[]uint{1, 2},
	}}

	for i, tt := range examples {
		got := (&Nat{tt.in}).expand(tt.n)
		if len(got.limbs) != len(tt.out) || got.Equal(&Nat{tt.out}) != 1 {
			t.Errorf("%d: got %v, expected %v", i, got, tt.out)
		}
	}
}

func TestMod(t *testing.T) {
	m := modulusFromBytes([]byte{0x06, 0x80, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x0d})
	x := natFromBytes([]byte{0x40, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x80, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x01})
	out := new(Nat)
	out.Mod(x, m)
	expected := natFromBytes([]byte{0x04, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x09})
	if out.Equal(expected) != 1 {
		t.Errorf("%+v != %+v", out, expected)
	}
}

func TestModSub(t *testing.T) {
	m := modulusFromBytes([]byte{13})
	x := &Nat{[]uint{6}}
	y := &Nat{[]uint{7}}
	x.Sub(y, m)
	expected := &Nat{[]uint{12}}
	if x.Equal(expected) != 1 {
		t.Errorf("%+v != %+v", x, expected)
	}
	x.Sub(y, m)
	expected = &Nat{[]uint{5}}
	if x.Equal(expected) != 1 {
		t.Errorf("%+v != %+v", x, expected)
	}
}

func TestModAdd(t *testing.T) {
	m := modulusFromBytes([]byte{13})
	x := &Nat{[]uint{6}}
	y := &Nat{[]uint{7}}
	x.Add(y, m)
	expected := &Nat{[]uint{0}}
	if x.Equal(expected) != 1 {
		t.Errorf("%+v != %+v", x, expected)
	}
	x.Add(y, m)
	expected = &Nat{[]uint{7}}
	if x.Equal(expected) != 1 {
		t.Errorf("%+v != %+v", x, expected)
	}
}

func TestExp(t *testing.T) {
	m := modulusFromBytes([]byte{13})
	x := &Nat{[]uint{3}}
	out := &Nat{[]uint{0}}
	out.Exp(x, []byte{12}, m)
	expected := &Nat{[]uint{1}}
	if out.Equal(expected) != 1 {
		t.Errorf("%+v != %+v", out, expected)
	}
}

func TestExpShort(t *testing.T) {
	m := modulusFromBytes([]byte{13})
	x := &Nat{[]uint{3}}
	out := &Nat{[]uint{0}}
	out.ExpShortVarTime(x, 12, m)
	expected := &Nat{[]uint{1}}
	if out.Equal(expected) != 1 {
		t.Errorf("%+v != %+v", out, expected)
	}
}

// TestMulReductions tests that Mul reduces results equal or slightly greater
// than the modulus. Some Montgomery algorithms don't and need extra care to
// return correct results. See https://go.dev/issue/13907.
func TestMulReductions(t *testing.T) {
	// Two short but multi-limb primes.
	a, _ := new(big.Int).SetString("773608962677651230850240281261679752031633236267106044359907", 10)
	b, _ := new(big.Int).SetString("180692823610368451951102211649591374573781973061758082626801", 10)
	n := new(big.Int).Mul(a, b)

	N, _ := NewModulusFromBig(n)
	A := NewNat().setBig(a).ExpandFor(N)
	B := NewNat().setBig(b).ExpandFor(N)

	if A.Mul(B, N).IsZero() != 1 {
		t.Error("a * b mod (a * b) != 0")
	}

	i := new(big.Int).ModInverse(a, b)
	N, _ = NewModulusFromBig(b)
	A = NewNat().setBig(a).ExpandFor(N)
	I := NewNat().setBig(i).ExpandFor(N)
	one := NewNat().setBig(big.NewInt(1)).ExpandFor(N)

	if A.Mul(I, N).Equal(one) != 1 {
		t.Error("a * inv(a) mod b != 1")
	}
}

func natBytes(n *Nat) []byte {
	return n.Bytes(maxModulus(uint(len(n.limbs))))
}

func natFromBytes(b []byte) *Nat {
	// Must not use Nat.SetBytes as it's used in TestSetBytes.
	bb := new(big.Int).SetBytes(b)
	return NewNat().setBig(bb)
}

func modulusFromBytes(b []byte) *Modulus {
	bb := new(big.Int).SetBytes(b)
	m, _ := NewModulusFromBig(bb)
	return m
}

// maxModulus returns the biggest modulus that can fit in n limbs.
func maxModulus(n uint) *Modulus {
	b := big.NewInt(1)
	b.Lsh(b, n*_W)
	b.Sub(b, big.NewInt(1))
	m, _ := NewModulusFromBig(b)
	return m
}

func makeBenchmarkModulus() *Modulus {
	return maxModulus(32)
}

func makeBenchmarkValue() *Nat {
	x := make([]uint, 32)
	for i := 0; i < 32; i++ {
		x[i]--
	}
	return &Nat{limbs: x}
}

func makeBenchmarkExponent() []byte {
	e := make([]byte, 256)
	for i := 0; i < 32; i++ {
		e[i] = 0xFF
	}
	return e
}

func BenchmarkModAdd(b *testing.B) {
	x := makeBenchmarkValue()
	y := makeBenchmarkValue()
	m := makeBenchmarkModulus()

	b.ResetTimer()
	for i := 0; i < b.N; i++ {
		x.Add(y, m)
	}
}

func BenchmarkModSub(b *testing.B) {
	x := makeBenchmarkValue()
	y := makeBenchmarkValue()
	m := makeBenchmarkModulus()

	b.ResetTimer()
	for i := 0; i < b.N; i++ {
		x.Sub(y, m)
	}
}

func BenchmarkMontgomeryRepr(b *testing.B) {
	x := makeBenchmarkValue()
	m := makeBenchmarkModulus()

	b.ResetTimer()
	for i := 0; i < b.N; i++ {
		x.montgomeryRepresentation(m)
	}
}

func BenchmarkMontgomeryMul(b *testing.B) {
	x := makeBenchmarkValue()
	y := makeBenchmarkValue()
	out := makeBenchmarkValue()
	m := makeBenchmarkModulus()

	b.ResetTimer()
	for i := 0; i < b.N; i++ {
		out.montgomeryMul(x, y, m)
	}
}

func BenchmarkModMul(b *testing.B) {
	x := makeBenchmarkValue()
	y := makeBenchmarkValue()
	m := makeBenchmarkModulus()

	b.ResetTimer()
	for i := 0; i < b.N; i++ {
		x.Mul(y, m)
	}
}

func BenchmarkExpBig(b *testing.B) {
	out := new(big.Int)
	exponentBytes := makeBenchmarkExponent()
	x := new(big.Int).SetBytes(exponentBytes)
	e := new(big.Int).SetBytes(exponentBytes)
	n := new(big.Int).SetBytes(exponentBytes)
	one := new(big.Int).SetUint64(1)
	n.Add(n, one)

	b.ResetTimer()
	for i := 0; i < b.N; i++ {
		out.Exp(x, e, n)
	}
}

func BenchmarkExp(b *testing.B) {
	x := makeBenchmarkValue()
	e := makeBenchmarkExponent()
	out := makeBenchmarkValue()
	m := makeBenchmarkModulus()

	b.ResetTimer()
	for i := 0; i < b.N; i++ {
		out.Exp(x, e, m)
	}
}

func TestNewModFromBigZero(t *testing.T) {
	expected := "modulus must be >= 0"
	_, err := NewModulusFromBig(big.NewInt(0))
	if err == nil || err.Error() != expected {
		t.Errorf("NewModulusFromBig(0) got %q, want %q", err, expected)
	}

	expected = "modulus must be odd"
	_, err = NewModulusFromBig(big.NewInt(2))
	if err == nil || err.Error() != expected {
		t.Errorf("NewModulusFromBig(2) got %q, want %q", err, expected)
	}
}