Adding upstream version 1.16.10.upstream/1.16.10upstream

Signed-off-by: Daniel Baumann <daniel.baumann@progress-linux.org>

1 files changed, 1896 insertions, 0 deletions

diff --git a/src/math/big/int_test.go b/src/math/big/int_test.go
new file mode 100644
index 0000000..3c85573
--- /dev/null
+++ b/src/math/big/int_test.go
@@ -0,0 +1,1896 @@
+// Copyright 2009 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 big
+
+import (
+	"bytes"
+	"encoding/hex"
+	"fmt"
+	"math/rand"
+	"strconv"
+	"strings"
+	"testing"
+	"testing/quick"
+)
+
+func isNormalized(x *Int) bool {
+	if len(x.abs) == 0 {
+		return !x.neg
+	}
+	// len(x.abs) > 0
+	return x.abs[len(x.abs)-1] != 0
+}
+
+type funZZ func(z, x, y *Int) *Int
+type argZZ struct {
+	z, x, y *Int
+}
+
+var sumZZ = []argZZ{
+	{NewInt(0), NewInt(0), NewInt(0)},
+	{NewInt(1), NewInt(1), NewInt(0)},
+	{NewInt(1111111110), NewInt(123456789), NewInt(987654321)},
+	{NewInt(-1), NewInt(-1), NewInt(0)},
+	{NewInt(864197532), NewInt(-123456789), NewInt(987654321)},
+	{NewInt(-1111111110), NewInt(-123456789), NewInt(-987654321)},
+}
+
+var prodZZ = []argZZ{
+	{NewInt(0), NewInt(0), NewInt(0)},
+	{NewInt(0), NewInt(1), NewInt(0)},
+	{NewInt(1), NewInt(1), NewInt(1)},
+	{NewInt(-991 * 991), NewInt(991), NewInt(-991)},
+	// TODO(gri) add larger products
+}
+
+func TestSignZ(t *testing.T) {
+	var zero Int
+	for _, a := range sumZZ {
+		s := a.z.Sign()
+		e := a.z.Cmp(&zero)
+		if s != e {
+			t.Errorf("got %d; want %d for z = %v", s, e, a.z)
+		}
+	}
+}
+
+func TestSetZ(t *testing.T) {
+	for _, a := range sumZZ {
+		var z Int
+		z.Set(a.z)
+		if !isNormalized(&z) {
+			t.Errorf("%v is not normalized", z)
+		}
+		if (&z).Cmp(a.z) != 0 {
+			t.Errorf("got z = %v; want %v", z, a.z)
+		}
+	}
+}
+
+func TestAbsZ(t *testing.T) {
+	var zero Int
+	for _, a := range sumZZ {
+		var z Int
+		z.Abs(a.z)
+		var e Int
+		e.Set(a.z)
+		if e.Cmp(&zero) < 0 {
+			e.Sub(&zero, &e)
+		}
+		if z.Cmp(&e) != 0 {
+			t.Errorf("got z = %v; want %v", z, e)
+		}
+	}
+}
+
+func testFunZZ(t *testing.T, msg string, f funZZ, a argZZ) {
+	var z Int
+	f(&z, a.x, a.y)
+	if !isNormalized(&z) {
+		t.Errorf("%s%v is not normalized", msg, z)
+	}
+	if (&z).Cmp(a.z) != 0 {
+		t.Errorf("%s%+v\n\tgot z = %v; want %v", msg, a, &z, a.z)
+	}
+}
+
+func TestSumZZ(t *testing.T) {
+	AddZZ := func(z, x, y *Int) *Int { return z.Add(x, y) }
+	SubZZ := func(z, x, y *Int) *Int { return z.Sub(x, y) }
+	for _, a := range sumZZ {
+		arg := a
+		testFunZZ(t, "AddZZ", AddZZ, arg)
+
+		arg = argZZ{a.z, a.y, a.x}
+		testFunZZ(t, "AddZZ symmetric", AddZZ, arg)
+
+		arg = argZZ{a.x, a.z, a.y}
+		testFunZZ(t, "SubZZ", SubZZ, arg)
+
+		arg = argZZ{a.y, a.z, a.x}
+		testFunZZ(t, "SubZZ symmetric", SubZZ, arg)
+	}
+}
+
+func TestProdZZ(t *testing.T) {
+	MulZZ := func(z, x, y *Int) *Int { return z.Mul(x, y) }
+	for _, a := range prodZZ {
+		arg := a
+		testFunZZ(t, "MulZZ", MulZZ, arg)
+
+		arg = argZZ{a.z, a.y, a.x}
+		testFunZZ(t, "MulZZ symmetric", MulZZ, arg)
+	}
+}
+
+// mulBytes returns x*y via grade school multiplication. Both inputs
+// and the result are assumed to be in big-endian representation (to
+// match the semantics of Int.Bytes and Int.SetBytes).
+func mulBytes(x, y []byte) []byte {
+	z := make([]byte, len(x)+len(y))
+
+	// multiply
+	k0 := len(z) - 1
+	for j := len(y) - 1; j >= 0; j-- {
+		d := int(y[j])
+		if d != 0 {
+			k := k0
+			carry := 0
+			for i := len(x) - 1; i >= 0; i-- {
+				t := int(z[k]) + int(x[i])*d + carry
+				z[k], carry = byte(t), t>>8
+				k--
+			}
+			z[k] = byte(carry)
+		}
+		k0--
+	}
+
+	// normalize (remove leading 0's)
+	i := 0
+	for i < len(z) && z[i] == 0 {
+		i++
+	}
+
+	return z[i:]
+}
+
+func checkMul(a, b []byte) bool {
+	var x, y, z1 Int
+	x.SetBytes(a)
+	y.SetBytes(b)
+	z1.Mul(&x, &y)
+
+	var z2 Int
+	z2.SetBytes(mulBytes(a, b))
+
+	return z1.Cmp(&z2) == 0
+}
+
+func TestMul(t *testing.T) {
+	if err := quick.Check(checkMul, nil); err != nil {
+		t.Error(err)
+	}
+}
+
+var mulRangesZ = []struct {
+	a, b int64
+	prod string
+}{
+	// entirely positive ranges are covered by mulRangesN
+	{-1, 1, "0"},
+	{-2, -1, "2"},
+	{-3, -2, "6"},
+	{-3, -1, "-6"},
+	{1, 3, "6"},
+	{-10, -10, "-10"},
+	{0, -1, "1"},                      // empty range
+	{-1, -100, "1"},                   // empty range
+	{-1, 1, "0"},                      // range includes 0
+	{-1e9, 0, "0"},                    // range includes 0
+	{-1e9, 1e9, "0"},                  // range includes 0
+	{-10, -1, "3628800"},              // 10!
+	{-20, -2, "-2432902008176640000"}, // -20!
+	{-99, -1,
+		"-933262154439441526816992388562667004907159682643816214685929" +
+			"638952175999932299156089414639761565182862536979208272237582" +
+			"511852109168640000000000000000000000", // -99!
+	},
+}
+
+func TestMulRangeZ(t *testing.T) {
+	var tmp Int
+	// test entirely positive ranges
+	for i, r := range mulRangesN {
+		prod := tmp.MulRange(int64(r.a), int64(r.b)).String()
+		if prod != r.prod {
+			t.Errorf("#%da: got %s; want %s", i, prod, r.prod)
+		}
+	}
+	// test other ranges
+	for i, r := range mulRangesZ {
+		prod := tmp.MulRange(r.a, r.b).String()
+		if prod != r.prod {
+			t.Errorf("#%db: got %s; want %s", i, prod, r.prod)
+		}
+	}
+}
+
+func TestBinomial(t *testing.T) {
+	var z Int
+	for _, test := range []struct {
+		n, k int64
+		want string
+	}{
+		{0, 0, "1"},
+		{0, 1, "0"},
+		{1, 0, "1"},
+		{1, 1, "1"},
+		{1, 10, "0"},
+		{4, 0, "1"},
+		{4, 1, "4"},
+		{4, 2, "6"},
+		{4, 3, "4"},
+		{4, 4, "1"},
+		{10, 1, "10"},
+		{10, 9, "10"},
+		{10, 5, "252"},
+		{11, 5, "462"},
+		{11, 6, "462"},
+		{100, 10, "17310309456440"},
+		{100, 90, "17310309456440"},
+		{1000, 10, "263409560461970212832400"},
+		{1000, 990, "263409560461970212832400"},
+	} {
+		if got := z.Binomial(test.n, test.k).String(); got != test.want {
+			t.Errorf("Binomial(%d, %d) = %s; want %s", test.n, test.k, got, test.want)
+		}
+	}
+}
+
+func BenchmarkBinomial(b *testing.B) {
+	var z Int
+	for i := b.N - 1; i >= 0; i-- {
+		z.Binomial(1000, 990)
+	}
+}
+
+// Examples from the Go Language Spec, section "Arithmetic operators"
+var divisionSignsTests = []struct {
+	x, y int64
+	q, r int64 // T-division
+	d, m int64 // Euclidean division
+}{
+	{5, 3, 1, 2, 1, 2},
+	{-5, 3, -1, -2, -2, 1},
+	{5, -3, -1, 2, -1, 2},
+	{-5, -3, 1, -2, 2, 1},
+	{1, 2, 0, 1, 0, 1},
+	{8, 4, 2, 0, 2, 0},
+}
+
+func TestDivisionSigns(t *testing.T) {
+	for i, test := range divisionSignsTests {
+		x := NewInt(test.x)
+		y := NewInt(test.y)
+		q := NewInt(test.q)
+		r := NewInt(test.r)
+		d := NewInt(test.d)
+		m := NewInt(test.m)
+
+		q1 := new(Int).Quo(x, y)
+		r1 := new(Int).Rem(x, y)
+		if !isNormalized(q1) {
+			t.Errorf("#%d Quo: %v is not normalized", i, *q1)
+		}
+		if !isNormalized(r1) {
+			t.Errorf("#%d Rem: %v is not normalized", i, *r1)
+		}
+		if q1.Cmp(q) != 0 || r1.Cmp(r) != 0 {
+			t.Errorf("#%d QuoRem: got (%s, %s), want (%s, %s)", i, q1, r1, q, r)
+		}
+
+		q2, r2 := new(Int).QuoRem(x, y, new(Int))
+		if !isNormalized(q2) {
+			t.Errorf("#%d Quo: %v is not normalized", i, *q2)
+		}
+		if !isNormalized(r2) {
+			t.Errorf("#%d Rem: %v is not normalized", i, *r2)
+		}
+		if q2.Cmp(q) != 0 || r2.Cmp(r) != 0 {
+			t.Errorf("#%d QuoRem: got (%s, %s), want (%s, %s)", i, q2, r2, q, r)
+		}
+
+		d1 := new(Int).Div(x, y)
+		m1 := new(Int).Mod(x, y)
+		if !isNormalized(d1) {
+			t.Errorf("#%d Div: %v is not normalized", i, *d1)
+		}
+		if !isNormalized(m1) {
+			t.Errorf("#%d Mod: %v is not normalized", i, *m1)
+		}
+		if d1.Cmp(d) != 0 || m1.Cmp(m) != 0 {
+			t.Errorf("#%d DivMod: got (%s, %s), want (%s, %s)", i, d1, m1, d, m)
+		}
+
+		d2, m2 := new(Int).DivMod(x, y, new(Int))
+		if !isNormalized(d2) {
+			t.Errorf("#%d Div: %v is not normalized", i, *d2)
+		}
+		if !isNormalized(m2) {
+			t.Errorf("#%d Mod: %v is not normalized", i, *m2)
+		}
+		if d2.Cmp(d) != 0 || m2.Cmp(m) != 0 {
+			t.Errorf("#%d DivMod: got (%s, %s), want (%s, %s)", i, d2, m2, d, m)
+		}
+	}
+}
+
+func norm(x nat) nat {
+	i := len(x)
+	for i > 0 && x[i-1] == 0 {
+		i--
+	}
+	return x[:i]
+}
+
+func TestBits(t *testing.T) {
+	for _, test := range []nat{
+		nil,
+		{0},
+		{1},
+		{0, 1, 2, 3, 4},
+		{4, 3, 2, 1, 0},
+		{4, 3, 2, 1, 0, 0, 0, 0},
+	} {
+		var z Int
+		z.neg = true
+		got := z.SetBits(test)
+		want := norm(test)
+		if got.abs.cmp(want) != 0 {
+			t.Errorf("SetBits(%v) = %v; want %v", test, got.abs, want)
+		}
+
+		if got.neg {
+			t.Errorf("SetBits(%v): got negative result", test)
+		}
+
+		bits := nat(z.Bits())
+		if bits.cmp(want) != 0 {
+			t.Errorf("%v.Bits() = %v; want %v", z.abs, bits, want)
+		}
+	}
+}
+
+func checkSetBytes(b []byte) bool {
+	hex1 := hex.EncodeToString(new(Int).SetBytes(b).Bytes())
+	hex2 := hex.EncodeToString(b)
+
+	for len(hex1) < len(hex2) {
+		hex1 = "0" + hex1
+	}
+
+	for len(hex1) > len(hex2) {
+		hex2 = "0" + hex2
+	}
+
+	return hex1 == hex2
+}
+
+func TestSetBytes(t *testing.T) {
+	if err := quick.Check(checkSetBytes, nil); err != nil {
+		t.Error(err)
+	}
+}
+
+func checkBytes(b []byte) bool {
+	// trim leading zero bytes since Bytes() won't return them
+	// (was issue 12231)
+	for len(b) > 0 && b[0] == 0 {
+		b = b[1:]
+	}
+	b2 := new(Int).SetBytes(b).Bytes()
+	return bytes.Equal(b, b2)
+}
+
+func TestBytes(t *testing.T) {
+	if err := quick.Check(checkBytes, nil); err != nil {
+		t.Error(err)
+	}
+}
+
+func checkQuo(x, y []byte) bool {
+	u := new(Int).SetBytes(x)
+	v := new(Int).SetBytes(y)
+
+	if len(v.abs) == 0 {
+		return true
+	}
+
+	r := new(Int)
+	q, r := new(Int).QuoRem(u, v, r)
+
+	if r.Cmp(v) >= 0 {
+		return false
+	}
+
+	uprime := new(Int).Set(q)
+	uprime.Mul(uprime, v)
+	uprime.Add(uprime, r)
+
+	return uprime.Cmp(u) == 0
+}
+
+var quoTests = []struct {
+	x, y string
+	q, r string
+}{
+	{
+		"476217953993950760840509444250624797097991362735329973741718102894495832294430498335824897858659711275234906400899559094370964723884706254265559534144986498357",
+		"9353930466774385905609975137998169297361893554149986716853295022578535724979483772383667534691121982974895531435241089241440253066816724367338287092081996",
+		"50911",
+		"1",
+	},
+	{
+		"11510768301994997771168",
+		"1328165573307167369775",
+		"8",
+		"885443715537658812968",
+	},
+}
+
+func TestQuo(t *testing.T) {
+	if err := quick.Check(checkQuo, nil); err != nil {
+		t.Error(err)
+	}
+
+	for i, test := range quoTests {
+		x, _ := new(Int).SetString(test.x, 10)
+		y, _ := new(Int).SetString(test.y, 10)
+		expectedQ, _ := new(Int).SetString(test.q, 10)
+		expectedR, _ := new(Int).SetString(test.r, 10)
+
+		r := new(Int)
+		q, r := new(Int).QuoRem(x, y, r)
+
+		if q.Cmp(expectedQ) != 0 || r.Cmp(expectedR) != 0 {
+			t.Errorf("#%d got (%s, %s) want (%s, %s)", i, q, r, expectedQ, expectedR)
+		}
+	}
+}
+
+func TestQuoStepD6(t *testing.T) {
+	// See Knuth, Volume 2, section 4.3.1, exercise 21. This code exercises
+	// a code path which only triggers 1 in 10^{-19} cases.
+
+	u := &Int{false, nat{0, 0, 1 + 1<<(_W-1), _M ^ (1 << (_W - 1))}}
+	v := &Int{false, nat{5, 2 + 1<<(_W-1), 1 << (_W - 1)}}
+
+	r := new(Int)
+	q, r := new(Int).QuoRem(u, v, r)
+	const expectedQ64 = "18446744073709551613"
+	const expectedR64 = "3138550867693340382088035895064302439801311770021610913807"
+	const expectedQ32 = "4294967293"
+	const expectedR32 = "39614081266355540837921718287"
+	if q.String() != expectedQ64 && q.String() != expectedQ32 ||
+		r.String() != expectedR64 && r.String() != expectedR32 {
+		t.Errorf("got (%s, %s) want (%s, %s) or (%s, %s)", q, r, expectedQ64, expectedR64, expectedQ32, expectedR32)
+	}
+}
+
+func BenchmarkQuoRem(b *testing.B) {
+	x, _ := new(Int).SetString("153980389784927331788354528594524332344709972855165340650588877572729725338415474372475094155672066328274535240275856844648695200875763869073572078279316458648124537905600131008790701752441155668003033945258023841165089852359980273279085783159654751552359397986180318708491098942831252291841441726305535546071", 0)
+	y, _ := new(Int).SetString("7746362281539803897849273317883545285945243323447099728551653406505888775727297253384154743724750941556720663282745352402758568446486952008757638690735720782793164586481245379056001310087907017524411556680030339452580238411650898523599802732790857831596547515523593979861803187084910989428312522918414417263055355460715745539358014631136245887418412633787074173796862711588221766398229333338511838891484974940633857861775630560092874987828057333663969469797013996401149696897591265769095952887917296740109742927689053276850469671231961384715398038978492733178835452859452433234470997285516534065058887757272972533841547437247509415567206632827453524027585684464869520087576386907357207827931645864812453790560013100879070175244115566800303394525802384116508985235998027327908578315965475155235939798618031870849109894283125229184144172630553554607112725169432413343763989564437170644270643461665184965150423819594083121075825", 0)
+	q := new(Int)
+	r := new(Int)
+
+	b.ResetTimer()
+	for i := 0; i < b.N; i++ {
+		q.QuoRem(y, x, r)
+	}
+}
+
+var bitLenTests = []struct {
+	in  string
+	out int
+}{
+	{"-1", 1},
+	{"0", 0},
+	{"1", 1},
+	{"2", 2},
+	{"4", 3},
+	{"0xabc", 12},
+	{"0x8000", 16},
+	{"0x80000000", 32},
+	{"0x800000000000", 48},
+	{"0x8000000000000000", 64},
+	{"0x80000000000000000000", 80},
+	{"-0x4000000000000000000000", 87},
+}
+
+func TestBitLen(t *testing.T) {
+	for i, test := range bitLenTests {
+		x, ok := new(Int).SetString(test.in, 0)
+		if !ok {
+			t.Errorf("#%d test input invalid: %s", i, test.in)
+			continue
+		}
+
+		if n := x.BitLen(); n != test.out {
+			t.Errorf("#%d got %d want %d", i, n, test.out)
+		}
+	}
+}
+
+var expTests = []struct {
+	x, y, m string
+	out     string
+}{
+	// y <= 0
+	{"0", "0", "", "1"},
+	{"1", "0", "", "1"},
+	{"-10", "0", "", "1"},
+	{"1234", "-1", "", "1"},
+	{"1234", "-1", "0", "1"},
+	{"17", "-100", "1234", "865"},
+	{"2", "-100", "1234", ""},
+
+	// m == 1
+	{"0", "0", "1", "0"},
+	{"1", "0", "1", "0"},
+	{"-10", "0", "1", "0"},
+	{"1234", "-1", "1", "0"},
+
+	// misc
+	{"5", "1", "3", "2"},
+	{"5", "-7", "", "1"},
+	{"-5", "-7", "", "1"},
+	{"5", "0", "", "1"},
+	{"-5", "0", "", "1"},
+	{"5", "1", "", "5"},
+	{"-5", "1", "", "-5"},
+	{"-5", "1", "7", "2"},
+	{"-2", "3", "2", "0"},
+	{"5", "2", "", "25"},
+	{"1", "65537", "2", "1"},
+	{"0x8000000000000000", "2", "", "0x40000000000000000000000000000000"},
+	{"0x8000000000000000", "2", "6719", "4944"},
+	{"0x8000000000000000", "3", "6719", "5447"},
+	{"0x8000000000000000", "1000", "6719", "1603"},
+	{"0x8000000000000000", "1000000", "6719", "3199"},
+	{"0x8000000000000000", "-1000000", "6719", "3663"}, // 3663 = ModInverse(3199, 6719) Issue #25865
+
+	{"0xffffffffffffffffffffffffffffffff", "0x12345678123456781234567812345678123456789", "0x01112222333344445555666677778889", "0x36168FA1DB3AAE6C8CE647E137F97A"},
+
+	{
+		"2938462938472983472983659726349017249287491026512746239764525612965293865296239471239874193284792387498274256129746192347",
+		"298472983472983471903246121093472394872319615612417471234712061",
+		"29834729834729834729347290846729561262544958723956495615629569234729836259263598127342374289365912465901365498236492183464",
+		"23537740700184054162508175125554701713153216681790245129157191391322321508055833908509185839069455749219131480588829346291",
+	},
+	// test case for issue 8822
+	{
+		"11001289118363089646017359372117963499250546375269047542777928006103246876688756735760905680604646624353196869572752623285140408755420374049317646428185270079555372763503115646054602867593662923894140940837479507194934267532831694565516466765025434902348314525627418515646588160955862839022051353653052947073136084780742729727874803457643848197499548297570026926927502505634297079527299004267769780768565695459945235586892627059178884998772989397505061206395455591503771677500931269477503508150175717121828518985901959919560700853226255420793148986854391552859459511723547532575574664944815966793196961286234040892865",
+		"0xB08FFB20760FFED58FADA86DFEF71AD72AA0FA763219618FE022C197E54708BB1191C66470250FCE8879487507CEE41381CA4D932F81C2B3F1AB20B539D50DCD",
+		"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
+		"21484252197776302499639938883777710321993113097987201050501182909581359357618579566746556372589385361683610524730509041328855066514963385522570894839035884713051640171474186548713546686476761306436434146475140156284389181808675016576845833340494848283681088886584219750554408060556769486628029028720727393293111678826356480455433909233520504112074401376133077150471237549474149190242010469539006449596611576612573955754349042329130631128234637924786466585703488460540228477440853493392086251021228087076124706778899179648655221663765993962724699135217212118535057766739392069738618682722216712319320435674779146070442",
+	},
+	{
+		"-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
+		"0xB08FFB20760FFED58FADA86DFEF71AD72AA0FA763219618FE022C197E54708BB1191C66470250FCE8879487507CEE41381CA4D932F81C2B3F1AB20B539D50DCD",
+		"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
+		"21484252197776302499639938883777710321993113097987201050501182909581359357618579566746556372589385361683610524730509041328855066514963385522570894839035884713051640171474186548713546686476761306436434146475140156284389181808675016576845833340494848283681088886584219750554408060556769486628029028720727393293111678826356480455433909233520504112074401376133077150471237549474149190242010469539006449596611576612573955754349042329130631128234637924786466585703488460540228477440853493392086251021228087076124706778899179648655221663765993962724699135217212118535057766739392069738618682722216712319320435674779146070442",
+	},
+
+	// test cases for issue 13907
+	{"0xffffffff00000001", "0xffffffff00000001", "0xffffffff00000001", "0"},
+	{"0xffffffffffffffff00000001", "0xffffffffffffffff00000001", "0xffffffffffffffff00000001", "0"},
+	{"0xffffffffffffffffffffffff00000001", "0xffffffffffffffffffffffff00000001", "0xffffffffffffffffffffffff00000001", "0"},
+	{"0xffffffffffffffffffffffffffffffff00000001", "0xffffffffffffffffffffffffffffffff00000001", "0xffffffffffffffffffffffffffffffff00000001", "0"},
+
+	{
+		"2",
+		"0xB08FFB20760FFED58FADA86DFEF71AD72AA0FA763219618FE022C197E54708BB1191C66470250FCE8879487507CEE41381CA4D932F81C2B3F1AB20B539D50DCD",
+		"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odd
+		"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
+	},
+	{
+		"2",
+		"0xB08FFB20760FFED58FADA86DFEF71AD72AA0FA763219618FE022C197E54708BB1191C66470250FCE8879487507CEE41381CA4D932F81C2B3F1AB20B539D50DCD",
+		"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even
+		"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
+	},
+}
+
+func TestExp(t *testing.T) {
+	for i, test := range expTests {
+		x, ok1 := new(Int).SetString(test.x, 0)
+		y, ok2 := new(Int).SetString(test.y, 0)
+
+		var ok3, ok4 bool
+		var out, m *Int
+
+		if len(test.out) == 0 {
+			out, ok3 = nil, true
+		} else {
+			out, ok3 = new(Int).SetString(test.out, 0)
+		}
+
+		if len(test.m) == 0 {
+			m, ok4 = nil, true
+		} else {
+			m, ok4 = new(Int).SetString(test.m, 0)
+		}
+
+		if !ok1 || !ok2 || !ok3 || !ok4 {
+			t.Errorf("#%d: error in input", i)
+			continue
+		}
+
+		z1 := new(Int).Exp(x, y, m)
+		if z1 != nil && !isNormalized(z1) {
+			t.Errorf("#%d: %v is not normalized", i, *z1)
+		}
+		if !(z1 == nil && out == nil || z1.Cmp(out) == 0) {
+			t.Errorf("#%d: got %x want %x", i, z1, out)
+		}
+
+		if m == nil {
+			// The result should be the same as for m == 0;
+			// specifically, there should be no div-zero panic.
+			m = &Int{abs: nat{}} // m != nil && len(m.abs) == 0
+			z2 := new(Int).Exp(x, y, m)
+			if z2.Cmp(z1) != 0 {
+				t.Errorf("#%d: got %x want %x", i, z2, z1)
+			}
+		}
+	}
+}
+
+func BenchmarkExp(b *testing.B) {
+	x, _ := new(Int).SetString("11001289118363089646017359372117963499250546375269047542777928006103246876688756735760905680604646624353196869572752623285140408755420374049317646428185270079555372763503115646054602867593662923894140940837479507194934267532831694565516466765025434902348314525627418515646588160955862839022051353653052947073136084780742729727874803457643848197499548297570026926927502505634297079527299004267769780768565695459945235586892627059178884998772989397505061206395455591503771677500931269477503508150175717121828518985901959919560700853226255420793148986854391552859459511723547532575574664944815966793196961286234040892865", 0)
+	y, _ := new(Int).SetString("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
+	n, _ := new(Int).SetString("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
+	out := new(Int)
+	for i := 0; i < b.N; i++ {
+		out.Exp(x, y, n)
+	}
+}
+
+func BenchmarkExp2(b *testing.B) {
+	x, _ := new(Int).SetString("2", 0)
+	y, _ := new(Int).SetString("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
+	n, _ := new(Int).SetString("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
+	out := new(Int)
+	for i := 0; i < b.N; i++ {
+		out.Exp(x, y, n)
+	}
+}
+
+func checkGcd(aBytes, bBytes []byte) bool {
+	x := new(Int)
+	y := new(Int)
+	a := new(Int).SetBytes(aBytes)
+	b := new(Int).SetBytes(bBytes)
+
+	d := new(Int).GCD(x, y, a, b)
+	x.Mul(x, a)
+	y.Mul(y, b)
+	x.Add(x, y)
+
+	return x.Cmp(d) == 0
+}
+
+// euclidExtGCD is a reference implementation of Euclid's
+// extended GCD algorithm for testing against optimized algorithms.
+// Requirements: a, b > 0
+func euclidExtGCD(a, b *Int) (g, x, y *Int) {
+	A := new(Int).Set(a)
+	B := new(Int).Set(b)
+
+	// A = Ua*a + Va*b
+	// B = Ub*a + Vb*b
+	Ua := new(Int).SetInt64(1)
+	Va := new(Int)
+
+	Ub := new(Int)
+	Vb := new(Int).SetInt64(1)
+
+	q := new(Int)
+	temp := new(Int)
+
+	r := new(Int)
+	for len(B.abs) > 0 {
+		q, r = q.QuoRem(A, B, r)
+
+		A, B, r = B, r, A
+
+		// Ua, Ub = Ub, Ua-q*Ub
+		temp.Set(Ub)
+		Ub.Mul(Ub, q)
+		Ub.Sub(Ua, Ub)
+		Ua.Set(temp)
+
+		// Va, Vb = Vb, Va-q*Vb
+		temp.Set(Vb)
+		Vb.Mul(Vb, q)
+		Vb.Sub(Va, Vb)
+		Va.Set(temp)
+	}
+	return A, Ua, Va
+}
+
+func checkLehmerGcd(aBytes, bBytes []byte) bool {
+	a := new(Int).SetBytes(aBytes)
+	b := new(Int).SetBytes(bBytes)
+
+	if a.Sign() <= 0 || b.Sign() <= 0 {
+		return true // can only test positive arguments
+	}
+
+	d := new(Int).lehmerGCD(nil, nil, a, b)
+	d0, _, _ := euclidExtGCD(a, b)
+
+	return d.Cmp(d0) == 0
+}
+
+func checkLehmerExtGcd(aBytes, bBytes []byte) bool {
+	a := new(Int).SetBytes(aBytes)
+	b := new(Int).SetBytes(bBytes)
+	x := new(Int)
+	y := new(Int)
+
+	if a.Sign() <= 0 || b.Sign() <= 0 {
+		return true // can only test positive arguments
+	}
+
+	d := new(Int).lehmerGCD(x, y, a, b)
+	d0, x0, y0 := euclidExtGCD(a, b)
+
+	return d.Cmp(d0) == 0 && x.Cmp(x0) == 0 && y.Cmp(y0) == 0
+}
+
+var gcdTests = []struct {
+	d, x, y, a, b string
+}{
+	// a <= 0 || b <= 0
+	{"0", "0", "0", "0", "0"},
+	{"7", "0", "1", "0", "7"},
+	{"7", "0", "-1", "0", "-7"},
+	{"11", "1", "0", "11", "0"},
+	{"7", "-1", "-2", "-77", "35"},
+	{"935", "-3", "8", "64515", "24310"},
+	{"935", "-3", "-8", "64515", "-24310"},
+	{"935", "3", "-8", "-64515", "-24310"},
+
+	{"1", "-9", "47", "120", "23"},
+	{"7", "1", "-2", "77", "35"},
+	{"935", "-3", "8", "64515", "24310"},
+	{"935000000000000000", "-3", "8", "64515000000000000000", "24310000000000000000"},
+	{"1", "-221", "22059940471369027483332068679400581064239780177629666810348940098015901108344", "98920366548084643601728869055592650835572950932266967461790948584315647051443", "991"},
+}
+
+func testGcd(t *testing.T, d, x, y, a, b *Int) {
+	var X *Int
+	if x != nil {
+		X = new(Int)
+	}
+	var Y *Int
+	if y != nil {
+		Y = new(Int)
+	}
+
+	D := new(Int).GCD(X, Y, a, b)
+	if D.Cmp(d) != 0 {
+		t.Errorf("GCD(%s, %s, %s, %s): got d = %s, want %s", x, y, a, b, D, d)
+	}
+	if x != nil && X.Cmp(x) != 0 {
+		t.Errorf("GCD(%s, %s, %s, %s): got x = %s, want %s", x, y, a, b, X, x)
+	}
+	if y != nil && Y.Cmp(y) != 0 {
+		t.Errorf("GCD(%s, %s, %s, %s): got y = %s, want %s", x, y, a, b, Y, y)
+	}
+
+	// check results in presence of aliasing (issue #11284)
+	a2 := new(Int).Set(a)
+	b2 := new(Int).Set(b)
+	a2.GCD(X, Y, a2, b2) // result is same as 1st argument
+	if a2.Cmp(d) != 0 {
+		t.Errorf("aliased z = a GCD(%s, %s, %s, %s): got d = %s, want %s", x, y, a, b, a2, d)
+	}
+	if x != nil && X.Cmp(x) != 0 {
+		t.Errorf("aliased z = a GCD(%s, %s, %s, %s): got x = %s, want %s", x, y, a, b, X, x)
+	}
+	if y != nil && Y.Cmp(y) != 0 {
+		t.Errorf("aliased z = a GCD(%s, %s, %s, %s): got y = %s, want %s", x, y, a, b, Y, y)
+	}
+
+	a2 = new(Int).Set(a)
+	b2 = new(Int).Set(b)
+	b2.GCD(X, Y, a2, b2) // result is same as 2nd argument
+	if b2.Cmp(d) != 0 {
+		t.Errorf("aliased z = b GCD(%s, %s, %s, %s): got d = %s, want %s", x, y, a, b, b2, d)
+	}
+	if x != nil && X.Cmp(x) != 0 {
+		t.Errorf("aliased z = b GCD(%s, %s, %s, %s): got x = %s, want %s", x, y, a, b, X, x)
+	}
+	if y != nil && Y.Cmp(y) != 0 {
+		t.Errorf("aliased z = b GCD(%s, %s, %s, %s): got y = %s, want %s", x, y, a, b, Y, y)
+	}
+
+	a2 = new(Int).Set(a)
+	b2 = new(Int).Set(b)
+	D = new(Int).GCD(a2, b2, a2, b2) // x = a, y = b
+	if D.Cmp(d) != 0 {
+		t.Errorf("aliased x = a, y = b GCD(%s, %s, %s, %s): got d = %s, want %s", x, y, a, b, D, d)
+	}
+	if x != nil && a2.Cmp(x) != 0 {
+		t.Errorf("aliased x = a, y = b GCD(%s, %s, %s, %s): got x = %s, want %s", x, y, a, b, a2, x)
+	}
+	if y != nil && b2.Cmp(y) != 0 {
+		t.Errorf("aliased x = a, y = b GCD(%s, %s, %s, %s): got y = %s, want %s", x, y, a, b, b2, y)
+	}
+
+	a2 = new(Int).Set(a)
+	b2 = new(Int).Set(b)
+	D = new(Int).GCD(b2, a2, a2, b2) // x = b, y = a
+	if D.Cmp(d) != 0 {
+		t.Errorf("aliased x = b, y = a GCD(%s, %s, %s, %s): got d = %s, want %s", x, y, a, b, D, d)
+	}
+	if x != nil && b2.Cmp(x) != 0 {
+		t.Errorf("aliased x = b, y = a GCD(%s, %s, %s, %s): got x = %s, want %s", x, y, a, b, b2, x)
+	}
+	if y != nil && a2.Cmp(y) != 0 {
+		t.Errorf("aliased x = b, y = a GCD(%s, %s, %s, %s): got y = %s, want %s", x, y, a, b, a2, y)
+	}
+}
+
+func TestGcd(t *testing.T) {
+	for _, test := range gcdTests {
+		d, _ := new(Int).SetString(test.d, 0)
+		x, _ := new(Int).SetString(test.x, 0)
+		y, _ := new(Int).SetString(test.y, 0)
+		a, _ := new(Int).SetString(test.a, 0)
+		b, _ := new(Int).SetString(test.b, 0)
+
+		testGcd(t, d, nil, nil, a, b)
+		testGcd(t, d, x, nil, a, b)
+		testGcd(t, d, nil, y, a, b)
+		testGcd(t, d, x, y, a, b)
+	}
+
+	if err := quick.Check(checkGcd, nil); err != nil {
+		t.Error(err)
+	}
+
+	if err := quick.Check(checkLehmerGcd, nil); err != nil {
+		t.Error(err)
+	}
+
+	if err := quick.Check(checkLehmerExtGcd, nil); err != nil {
+		t.Error(err)
+	}
+}
+
+type intShiftTest struct {
+	in    string
+	shift uint
+	out   string
+}
+
+var rshTests = []intShiftTest{
+	{"0", 0, "0"},
+	{"-0", 0, "0"},
+	{"0", 1, "0"},
+	{"0", 2, "0"},
+	{"1", 0, "1"},
+	{"1", 1, "0"},
+	{"1", 2, "0"},
+	{"2", 0, "2"},
+	{"2", 1, "1"},
+	{"-1", 0, "-1"},
+	{"-1", 1, "-1"},
+	{"-1", 10, "-1"},
+	{"-100", 2, "-25"},
+	{"-100", 3, "-13"},
+	{"-100", 100, "-1"},
+	{"4294967296", 0, "4294967296"},
+	{"4294967296", 1, "2147483648"},
+	{"4294967296", 2, "1073741824"},
+	{"18446744073709551616", 0, "18446744073709551616"},
+	{"18446744073709551616", 1, "9223372036854775808"},
+	{"18446744073709551616", 2, "4611686018427387904"},
+	{"18446744073709551616", 64, "1"},
+	{"340282366920938463463374607431768211456", 64, "18446744073709551616"},
+	{"340282366920938463463374607431768211456", 128, "1"},
+}
+
+func TestRsh(t *testing.T) {
+	for i, test := range rshTests {
+		in, _ := new(Int).SetString(test.in, 10)
+		expected, _ := new(Int).SetString(test.out, 10)
+		out := new(Int).Rsh(in, test.shift)
+
+		if !isNormalized(out) {
+			t.Errorf("#%d: %v is not normalized", i, *out)
+		}
+		if out.Cmp(expected) != 0 {
+			t.Errorf("#%d: got %s want %s", i, out, expected)
+		}
+	}
+}
+
+func TestRshSelf(t *testing.T) {
+	for i, test := range rshTests {
+		z, _ := new(Int).SetString(test.in, 10)
+		expected, _ := new(Int).SetString(test.out, 10)
+		z.Rsh(z, test.shift)
+
+		if !isNormalized(z) {
+			t.Errorf("#%d: %v is not normalized", i, *z)
+		}
+		if z.Cmp(expected) != 0 {
+			t.Errorf("#%d: got %s want %s", i, z, expected)
+		}
+	}
+}
+
+var lshTests = []intShiftTest{
+	{"0", 0, "0"},
+	{"0", 1, "0"},
+	{"0", 2, "0"},
+	{"1", 0, "1"},
+	{"1", 1, "2"},
+	{"1", 2, "4"},
+	{"2", 0, "2"},
+	{"2", 1, "4"},
+	{"2", 2, "8"},
+	{"-87", 1, "-174"},
+	{"4294967296", 0, "4294967296"},
+	{"4294967296", 1, "8589934592"},
+	{"4294967296", 2, "17179869184"},
+	{"18446744073709551616", 0, "18446744073709551616"},
+	{"9223372036854775808", 1, "18446744073709551616"},
+	{"4611686018427387904", 2, "18446744073709551616"},
+	{"1", 64, "18446744073709551616"},
+	{"18446744073709551616", 64, "340282366920938463463374607431768211456"},
+	{"1", 128, "340282366920938463463374607431768211456"},
+}
+
+func TestLsh(t *testing.T) {
+	for i, test := range lshTests {
+		in, _ := new(Int).SetString(test.in, 10)
+		expected, _ := new(Int).SetString(test.out, 10)
+		out := new(Int).Lsh(in, test.shift)
+
+		if !isNormalized(out) {
+			t.Errorf("#%d: %v is not normalized", i, *out)
+		}
+		if out.Cmp(expected) != 0 {
+			t.Errorf("#%d: got %s want %s", i, out, expected)
+		}
+	}
+}
+
+func TestLshSelf(t *testing.T) {
+	for i, test := range lshTests {
+		z, _ := new(Int).SetString(test.in, 10)
+		expected, _ := new(Int).SetString(test.out, 10)
+		z.Lsh(z, test.shift)
+
+		if !isNormalized(z) {
+			t.Errorf("#%d: %v is not normalized", i, *z)
+		}
+		if z.Cmp(expected) != 0 {
+			t.Errorf("#%d: got %s want %s", i, z, expected)
+		}
+	}
+}
+
+func TestLshRsh(t *testing.T) {
+	for i, test := range rshTests {
+		in, _ := new(Int).SetString(test.in, 10)
+		out := new(Int).Lsh(in, test.shift)
+		out = out.Rsh(out, test.shift)
+
+		if !isNormalized(out) {
+			t.Errorf("#%d: %v is not normalized", i, *out)
+		}
+		if in.Cmp(out) != 0 {
+			t.Errorf("#%d: got %s want %s", i, out, in)
+		}
+	}
+	for i, test := range lshTests {
+		in, _ := new(Int).SetString(test.in, 10)
+		out := new(Int).Lsh(in, test.shift)
+		out.Rsh(out, test.shift)
+
+		if !isNormalized(out) {
+			t.Errorf("#%d: %v is not normalized", i, *out)
+		}
+		if in.Cmp(out) != 0 {
+			t.Errorf("#%d: got %s want %s", i, out, in)
+		}
+	}
+}
+
+// Entries must be sorted by value in ascending order.
+var cmpAbsTests = []string{
+	"0",
+	"1",
+	"2",
+	"10",
+	"10000000",
+	"2783678367462374683678456387645876387564783686583485",
+	"2783678367462374683678456387645876387564783686583486",
+	"32957394867987420967976567076075976570670947609750670956097509670576075067076027578341538",
+}
+
+func TestCmpAbs(t *testing.T) {
+	values := make([]*Int, len(cmpAbsTests))
+	var prev *Int
+	for i, s := range cmpAbsTests {
+		x, ok := new(Int).SetString(s, 0)
+		if !ok {
+			t.Fatalf("SetString(%s, 0) failed", s)
+		}
+		if prev != nil && prev.Cmp(x) >= 0 {
+			t.Fatal("cmpAbsTests entries not sorted in ascending order")
+		}
+		values[i] = x
+		prev = x
+	}
+
+	for i, x := range values {
+		for j, y := range values {
+			// try all combinations of signs for x, y
+			for k := 0; k < 4; k++ {
+				var a, b Int
+				a.Set(x)
+				b.Set(y)
+				if k&1 != 0 {
+					a.Neg(&a)
+				}
+				if k&2 != 0 {
+					b.Neg(&b)
+				}
+
+				got := a.CmpAbs(&b)
+				want := 0
+				switch {
+				case i > j:
+					want = 1
+				case i < j:
+					want = -1
+				}
+				if got != want {
+					t.Errorf("absCmp |%s|, |%s|: got %d; want %d", &a, &b, got, want)
+				}
+			}
+		}
+	}
+}
+
+func TestIntCmpSelf(t *testing.T) {
+	for _, s := range cmpAbsTests {
+		x, ok := new(Int).SetString(s, 0)
+		if !ok {
+			t.Fatalf("SetString(%s, 0) failed", s)
+		}
+		got := x.Cmp(x)
+		want := 0
+		if got != want {
+			t.Errorf("x = %s: x.Cmp(x): got %d; want %d", x, got, want)
+		}
+	}
+}
+
+var int64Tests = []string{
+	// int64
+	"0",
+	"1",
+	"-1",
+	"4294967295",
+	"-4294967295",
+	"4294967296",
+	"-4294967296",
+	"9223372036854775807",
+	"-9223372036854775807",
+	"-9223372036854775808",
+
+	// not int64
+	"0x8000000000000000",
+	"-0x8000000000000001",
+	"38579843757496759476987459679745",
+	"-38579843757496759476987459679745",
+}
+
+func TestInt64(t *testing.T) {
+	for _, s := range int64Tests {
+		var x Int
+		_, ok := x.SetString(s, 0)
+		if !ok {
+			t.Errorf("SetString(%s, 0) failed", s)
+			continue
+		}
+
+		want, err := strconv.ParseInt(s, 0, 64)
+		if err != nil {
+			if err.(*strconv.NumError).Err == strconv.ErrRange {
+				if x.IsInt64() {
+					t.Errorf("IsInt64(%s) succeeded unexpectedly", s)
+				}
+			} else {
+				t.Errorf("ParseInt(%s) failed", s)
+			}
+			continue
+		}
+
+		if !x.IsInt64() {
+			t.Errorf("IsInt64(%s) failed unexpectedly", s)
+		}
+
+		got := x.Int64()
+		if got != want {
+			t.Errorf("Int64(%s) = %d; want %d", s, got, want)
+		}
+	}
+}
+
+var uint64Tests = []string{
+	// uint64
+	"0",
+	"1",
+	"4294967295",
+	"4294967296",
+	"8589934591",
+	"8589934592",
+	"9223372036854775807",
+	"9223372036854775808",
+	"0x08000000000000000",
+
+	// not uint64
+	"0x10000000000000000",
+	"-0x08000000000000000",
+	"-1",
+}
+
+func TestUint64(t *testing.T) {
+	for _, s := range uint64Tests {
+		var x Int
+		_, ok := x.SetString(s, 0)
+		if !ok {
+			t.Errorf("SetString(%s, 0) failed", s)
+			continue
+		}
+
+		want, err := strconv.ParseUint(s, 0, 64)
+		if err != nil {
+			// check for sign explicitly (ErrRange doesn't cover signed input)
+			if s[0] == '-' || err.(*strconv.NumError).Err == strconv.ErrRange {
+				if x.IsUint64() {
+					t.Errorf("IsUint64(%s) succeeded unexpectedly", s)
+				}
+			} else {
+				t.Errorf("ParseUint(%s) failed", s)
+			}
+			continue
+		}
+
+		if !x.IsUint64() {
+			t.Errorf("IsUint64(%s) failed unexpectedly", s)
+		}
+
+		got := x.Uint64()
+		if got != want {
+			t.Errorf("Uint64(%s) = %d; want %d", s, got, want)
+		}
+	}
+}
+
+var bitwiseTests = []struct {
+	x, y                 string
+	and, or, xor, andNot string
+}{
+	{"0x00", "0x00", "0x00", "0x00", "0x00", "0x00"},
+	{"0x00", "0x01", "0x00", "0x01", "0x01", "0x00"},
+	{"0x01", "0x00", "0x00", "0x01", "0x01", "0x01"},
+	{"-0x01", "0x00", "0x00", "-0x01", "-0x01", "-0x01"},
+	{"-0xaf", "-0x50", "-0xf0", "-0x0f", "0xe1", "0x41"},
+	{"0x00", "-0x01", "0x00", "-0x01", "-0x01", "0x00"},
+	{"0x01", "0x01", "0x01", "0x01", "0x00", "0x00"},
+	{"-0x01", "-0x01", "-0x01", "-0x01", "0x00", "0x00"},
+	{"0x07", "0x08", "0x00", "0x0f", "0x0f", "0x07"},
+	{"0x05", "0x0f", "0x05", "0x0f", "0x0a", "0x00"},
+	{"0xff", "-0x0a", "0xf6", "-0x01", "-0xf7", "0x09"},
+	{"0x013ff6", "0x9a4e", "0x1a46", "0x01bffe", "0x01a5b8", "0x0125b0"},
+	{"-0x013ff6", "0x9a4e", "0x800a", "-0x0125b2", "-0x01a5bc", "-0x01c000"},
+	{"-0x013ff6", "-0x9a4e", "-0x01bffe", "-0x1a46", "0x01a5b8", "0x8008"},
+	{
+		"0x1000009dc6e3d9822cba04129bcbe3401",
+		"0xb9bd7d543685789d57cb918e833af352559021483cdb05cc21fd",
+		"0x1000001186210100001000009048c2001",
+		"0xb9bd7d543685789d57cb918e8bfeff7fddb2ebe87dfbbdfe35fd",
+		"0xb9bd7d543685789d57ca918e8ae69d6fcdb2eae87df2b97215fc",
+		"0x8c40c2d8822caa04120b8321400",
+	},
+	{
+		"0x1000009dc6e3d9822cba04129bcbe3401",
+		"-0xb9bd7d543685789d57cb918e833af352559021483cdb05cc21fd",
+		"0x8c40c2d8822caa04120b8321401",
+		"-0xb9bd7d543685789d57ca918e82229142459020483cd2014001fd",
+		"-0xb9bd7d543685789d57ca918e8ae69d6fcdb2eae87df2b97215fe",
+		"0x1000001186210100001000009048c2000",
+	},
+	{
+		"-0x1000009dc6e3d9822cba04129bcbe3401",
+		"-0xb9bd7d543685789d57cb918e833af352559021483cdb05cc21fd",
+		"-0xb9bd7d543685789d57cb918e8bfeff7fddb2ebe87dfbbdfe35fd",
+		"-0x1000001186210100001000009048c2001",
+		"0xb9bd7d543685789d57ca918e8ae69d6fcdb2eae87df2b97215fc",
+		"0xb9bd7d543685789d57ca918e82229142459020483cd2014001fc",
+	},
+}
+
+type bitFun func(z, x, y *Int) *Int
+
+func testBitFun(t *testing.T, msg string, f bitFun, x, y *Int, exp string) {
+	expected := new(Int)
+	expected.SetString(exp, 0)
+
+	out := f(new(Int), x, y)
+	if out.Cmp(expected) != 0 {
+		t.Errorf("%s: got %s want %s", msg, out, expected)
+	}
+}
+
+func testBitFunSelf(t *testing.T, msg string, f bitFun, x, y *Int, exp string) {
+	self := new(Int)
+	self.Set(x)
+	expected := new(Int)
+	expected.SetString(exp, 0)
+
+	self = f(self, self, y)
+	if self.Cmp(expected) != 0 {
+		t.Errorf("%s: got %s want %s", msg, self, expected)
+	}
+}
+
+func altBit(x *Int, i int) uint {
+	z := new(Int).Rsh(x, uint(i))
+	z = z.And(z, NewInt(1))
+	if z.Cmp(new(Int)) != 0 {
+		return 1
+	}
+	return 0
+}
+
+func altSetBit(z *Int, x *Int, i int, b uint) *Int {
+	one := NewInt(1)
+	m := one.Lsh(one, uint(i))
+	switch b {
+	case 1:
+		return z.Or(x, m)
+	case 0:
+		return z.AndNot(x, m)
+	}
+	panic("set bit is not 0 or 1")
+}
+
+func testBitset(t *testing.T, x *Int) {
+	n := x.BitLen()
+	z := new(Int).Set(x)
+	z1 := new(Int).Set(x)
+	for i := 0; i < n+10; i++ {
+		old := z.Bit(i)
+		old1 := altBit(z1, i)
+		if old != old1 {
+			t.Errorf("bitset: inconsistent value for Bit(%s, %d), got %v want %v", z1, i, old, old1)
+		}
+		z := new(Int).SetBit(z, i, 1)
+		z1 := altSetBit(new(Int), z1, i, 1)
+		if z.Bit(i) == 0 {
+			t.Errorf("bitset: bit %d of %s got 0 want 1", i, x)
+		}
+		if z.Cmp(z1) != 0 {
+			t.Errorf("bitset: inconsistent value after SetBit 1, got %s want %s", z, z1)
+		}
+		z.SetBit(z, i, 0)
+		altSetBit(z1, z1, i, 0)
+		if z.Bit(i) != 0 {
+			t.Errorf("bitset: bit %d of %s got 1 want 0", i, x)
+		}
+		if z.Cmp(z1) != 0 {
+			t.Errorf("bitset: inconsistent value after SetBit 0, got %s want %s", z, z1)
+		}
+		altSetBit(z1, z1, i, old)
+		z.SetBit(z, i, old)
+		if z.Cmp(z1) != 0 {
+			t.Errorf("bitset: inconsistent value after SetBit old, got %s want %s", z, z1)
+		}
+	}
+	if z.Cmp(x) != 0 {
+		t.Errorf("bitset: got %s want %s", z, x)
+	}
+}
+
+var bitsetTests = []struct {
+	x string
+	i int
+	b uint
+}{
+	{"0", 0, 0},
+	{"0", 200, 0},
+	{"1", 0, 1},
+	{"1", 1, 0},
+	{"-1", 0, 1},
+	{"-1", 200, 1},
+	{"0x2000000000000000000000000000", 108, 0},
+	{"0x2000000000000000000000000000", 109, 1},
+	{"0x2000000000000000000000000000", 110, 0},
+	{"-0x2000000000000000000000000001", 108, 1},
+	{"-0x2000000000000000000000000001", 109, 0},
+	{"-0x2000000000000000000000000001", 110, 1},
+}
+
+func TestBitSet(t *testing.T) {
+	for _, test := range bitwiseTests {
+		x := new(Int)
+		x.SetString(test.x, 0)
+		testBitset(t, x)
+		x = new(Int)
+		x.SetString(test.y, 0)
+		testBitset(t, x)
+	}
+	for i, test := range bitsetTests {
+		x := new(Int)
+		x.SetString(test.x, 0)
+		b := x.Bit(test.i)
+		if b != test.b {
+			t.Errorf("#%d got %v want %v", i, b, test.b)
+		}
+	}
+	z := NewInt(1)
+	z.SetBit(NewInt(0), 2, 1)
+	if z.Cmp(NewInt(4)) != 0 {
+		t.Errorf("destination leaked into result; got %s want 4", z)
+	}
+}
+
+var tzbTests = []struct {
+	in  string
+	out uint
+}{
+	{"0", 0},
+	{"1", 0},
+	{"-1", 0},
+	{"4", 2},
+	{"-8", 3},
+	{"0x4000000000000000000", 74},
+	{"-0x8000000000000000000", 75},
+}
+
+func TestTrailingZeroBits(t *testing.T) {
+	for i, test := range tzbTests {
+		in, _ := new(Int).SetString(test.in, 0)
+		want := test.out
+		got := in.TrailingZeroBits()
+
+		if got != want {
+			t.Errorf("#%d: got %v want %v", i, got, want)
+		}
+	}
+}
+
+func BenchmarkBitset(b *testing.B) {
+	z := new(Int)
+	z.SetBit(z, 512, 1)
+	b.ResetTimer()
+	b.StartTimer()
+	for i := b.N - 1; i >= 0; i-- {
+		z.SetBit(z, i&512, 1)
+	}
+}
+
+func BenchmarkBitsetNeg(b *testing.B) {
+	z := NewInt(-1)
+	z.SetBit(z, 512, 0)
+	b.ResetTimer()
+	b.StartTimer()
+	for i := b.N - 1; i >= 0; i-- {
+		z.SetBit(z, i&512, 0)
+	}
+}
+
+func BenchmarkBitsetOrig(b *testing.B) {
+	z := new(Int)
+	altSetBit(z, z, 512, 1)
+	b.ResetTimer()
+	b.StartTimer()
+	for i := b.N - 1; i >= 0; i-- {
+		altSetBit(z, z, i&512, 1)
+	}
+}
+
+func BenchmarkBitsetNegOrig(b *testing.B) {
+	z := NewInt(-1)
+	altSetBit(z, z, 512, 0)
+	b.ResetTimer()
+	b.StartTimer()
+	for i := b.N - 1; i >= 0; i-- {
+		altSetBit(z, z, i&512, 0)
+	}
+}
+
+// tri generates the trinomial 2**(n*2) - 2**n - 1, which is always 3 mod 4 and
+// 7 mod 8, so that 2 is always a quadratic residue.
+func tri(n uint) *Int {
+	x := NewInt(1)
+	x.Lsh(x, n)
+	x2 := new(Int).Lsh(x, n)
+	x2.Sub(x2, x)
+	x2.Sub(x2, intOne)
+	return x2
+}
+
+func BenchmarkModSqrt225_Tonelli(b *testing.B) {
+	p := tri(225)
+	x := NewInt(2)
+	for i := 0; i < b.N; i++ {
+		x.SetUint64(2)
+		x.modSqrtTonelliShanks(x, p)
+	}
+}
+
+func BenchmarkModSqrt225_3Mod4(b *testing.B) {
+	p := tri(225)
+	x := new(Int).SetUint64(2)
+	for i := 0; i < b.N; i++ {
+		x.SetUint64(2)
+		x.modSqrt3Mod4Prime(x, p)
+	}
+}
+
+func BenchmarkModSqrt231_Tonelli(b *testing.B) {
+	p := tri(231)
+	p.Sub(p, intOne)
+	p.Sub(p, intOne) // tri(231) - 2 is a prime == 5 mod 8
+	x := new(Int).SetUint64(7)
+	for i := 0; i < b.N; i++ {
+		x.SetUint64(7)
+		x.modSqrtTonelliShanks(x, p)
+	}
+}
+
+func BenchmarkModSqrt231_5Mod8(b *testing.B) {
+	p := tri(231)
+	p.Sub(p, intOne)
+	p.Sub(p, intOne) // tri(231) - 2 is a prime == 5 mod 8
+	x := new(Int).SetUint64(7)
+	for i := 0; i < b.N; i++ {
+		x.SetUint64(7)
+		x.modSqrt5Mod8Prime(x, p)
+	}
+}
+
+func TestBitwise(t *testing.T) {
+	x := new(Int)
+	y := new(Int)
+	for _, test := range bitwiseTests {
+		x.SetString(test.x, 0)
+		y.SetString(test.y, 0)
+
+		testBitFun(t, "and", (*Int).And, x, y, test.and)
+		testBitFunSelf(t, "and", (*Int).And, x, y, test.and)
+		testBitFun(t, "andNot", (*Int).AndNot, x, y, test.andNot)
+		testBitFunSelf(t, "andNot", (*Int).AndNot, x, y, test.andNot)
+		testBitFun(t, "or", (*Int).Or, x, y, test.or)
+		testBitFunSelf(t, "or", (*Int).Or, x, y, test.or)
+		testBitFun(t, "xor", (*Int).Xor, x, y, test.xor)
+		testBitFunSelf(t, "xor", (*Int).Xor, x, y, test.xor)
+	}
+}
+
+var notTests = []struct {
+	in  string
+	out string
+}{
+	{"0", "-1"},
+	{"1", "-2"},
+	{"7", "-8"},
+	{"0", "-1"},
+	{"-81910", "81909"},
+	{
+		"298472983472983471903246121093472394872319615612417471234712061",
+		"-298472983472983471903246121093472394872319615612417471234712062",
+	},
+}
+
+func TestNot(t *testing.T) {
+	in := new(Int)
+	out := new(Int)
+	expected := new(Int)
+	for i, test := range notTests {
+		in.SetString(test.in, 10)
+		expected.SetString(test.out, 10)
+		out = out.Not(in)
+		if out.Cmp(expected) != 0 {
+			t.Errorf("#%d: got %s want %s", i, out, expected)
+		}
+		out = out.Not(out)
+		if out.Cmp(in) != 0 {
+			t.Errorf("#%d: got %s want %s", i, out, in)
+		}
+	}
+}
+
+var modInverseTests = []struct {
+	element string
+	modulus string
+}{
+	{"1234567", "458948883992"},
+	{"239487239847", "2410312426921032588552076022197566074856950548502459942654116941958108831682612228890093858261341614673227141477904012196503648957050582631942730706805009223062734745341073406696246014589361659774041027169249453200378729434170325843778659198143763193776859869524088940195577346119843545301547043747207749969763750084308926339295559968882457872412993810129130294592999947926365264059284647209730384947211681434464714438488520940127459844288859336526896320919633919"},
+	{"-10", "13"}, // issue #16984
+	{"10", "-13"},
+	{"-17", "-13"},
+}
+
+func TestModInverse(t *testing.T) {
+	var element, modulus, gcd, inverse Int
+	one := NewInt(1)
+	for _, test := range modInverseTests {
+		(&element).SetString(test.element, 10)
+		(&modulus).SetString(test.modulus, 10)
+		(&inverse).ModInverse(&element, &modulus)
+		(&inverse).Mul(&inverse, &element)
+		(&inverse).Mod(&inverse, &modulus)
+		if (&inverse).Cmp(one) != 0 {
+			t.Errorf("ModInverse(%d,%d)*%d%%%d=%d, not 1", &element, &modulus, &element, &modulus, &inverse)
+		}
+	}
+	// exhaustive test for small values
+	for n := 2; n < 100; n++ {
+		(&modulus).SetInt64(int64(n))
+		for x := 1; x < n; x++ {
+			(&element).SetInt64(int64(x))
+			(&gcd).GCD(nil, nil, &element, &modulus)
+			if (&gcd).Cmp(one) != 0 {
+				continue
+			}
+			(&inverse).ModInverse(&element, &modulus)
+			(&inverse).Mul(&inverse, &element)
+			(&inverse).Mod(&inverse, &modulus)
+			if (&inverse).Cmp(one) != 0 {
+				t.Errorf("ModInverse(%d,%d)*%d%%%d=%d, not 1", &element, &modulus, &element, &modulus, &inverse)
+			}
+		}
+	}
+}
+
+func BenchmarkModInverse(b *testing.B) {
+	p := new(Int).SetInt64(1) // Mersenne prime 2**1279 -1
+	p.abs = p.abs.shl(p.abs, 1279)
+	p.Sub(p, intOne)
+	x := new(Int).Sub(p, intOne)
+	z := new(Int)
+	for i := 0; i < b.N; i++ {
+		z.ModInverse(x, p)
+	}
+}
+
+// testModSqrt is a helper for TestModSqrt,
+// which checks that ModSqrt can compute a square-root of elt^2.
+func testModSqrt(t *testing.T, elt, mod, sq, sqrt *Int) bool {
+	var sqChk, sqrtChk, sqrtsq Int
+	sq.Mul(elt, elt)
+	sq.Mod(sq, mod)
+	z := sqrt.ModSqrt(sq, mod)
+	if z != sqrt {
+		t.Errorf("ModSqrt returned wrong value %s", z)
+	}
+
+	// test ModSqrt arguments outside the range [0,mod)
+	sqChk.Add(sq, mod)
+	z = sqrtChk.ModSqrt(&sqChk, mod)
+	if z != &sqrtChk || z.Cmp(sqrt) != 0 {
+		t.Errorf("ModSqrt returned inconsistent value %s", z)
+	}
+	sqChk.Sub(sq, mod)
+	z = sqrtChk.ModSqrt(&sqChk, mod)
+	if z != &sqrtChk || z.Cmp(sqrt) != 0 {
+		t.Errorf("ModSqrt returned inconsistent value %s", z)
+	}
+
+	// test x aliasing z
+	z = sqrtChk.ModSqrt(sqrtChk.Set(sq), mod)
+	if z != &sqrtChk || z.Cmp(sqrt) != 0 {
+		t.Errorf("ModSqrt returned inconsistent value %s", z)
+	}
+
+	// make sure we actually got a square root
+	if sqrt.Cmp(elt) == 0 {
+		return true // we found the "desired" square root
+	}
+	sqrtsq.Mul(sqrt, sqrt) // make sure we found the "other" one
+	sqrtsq.Mod(&sqrtsq, mod)
+	return sq.Cmp(&sqrtsq) == 0
+}
+
+func TestModSqrt(t *testing.T) {
+	var elt, mod, modx4, sq, sqrt Int
+	r := rand.New(rand.NewSource(9))
+	for i, s := range primes[1:] { // skip 2, use only odd primes
+		mod.SetString(s, 10)
+		modx4.Lsh(&mod, 2)
+
+		// test a few random elements per prime
+		for x := 1; x < 5; x++ {
+			elt.Rand(r, &modx4)
+			elt.Sub(&elt, &mod) // test range [-mod, 3*mod)
+			if !testModSqrt(t, &elt, &mod, &sq, &sqrt) {
+				t.Errorf("#%d: failed (sqrt(e) = %s)", i, &sqrt)
+			}
+		}
+
+		if testing.Short() && i > 2 {
+			break
+		}
+	}
+
+	if testing.Short() {
+		return
+	}
+
+	// exhaustive test for small values
+	for n := 3; n < 100; n++ {
+		mod.SetInt64(int64(n))
+		if !mod.ProbablyPrime(10) {
+			continue
+		}
+		isSquare := make([]bool, n)
+
+		// test all the squares
+		for x := 1; x < n; x++ {
+			elt.SetInt64(int64(x))
+			if !testModSqrt(t, &elt, &mod, &sq, &sqrt) {
+				t.Errorf("#%d: failed (sqrt(%d,%d) = %s)", x, &elt, &mod, &sqrt)
+			}
+			isSquare[sq.Uint64()] = true
+		}
+
+		// test all non-squares
+		for x := 1; x < n; x++ {
+			sq.SetInt64(int64(x))
+			z := sqrt.ModSqrt(&sq, &mod)
+			if !isSquare[x] && z != nil {
+				t.Errorf("#%d: failed (sqrt(%d,%d) = nil)", x, &sqrt, &mod)
+			}
+		}
+	}
+}
+
+func TestJacobi(t *testing.T) {
+	testCases := []struct {
+		x, y   int64
+		result int
+	}{
+		{0, 1, 1},
+		{0, -1, 1},
+		{1, 1, 1},
+		{1, -1, 1},
+		{0, 5, 0},
+		{1, 5, 1},
+		{2, 5, -1},
+		{-2, 5, -1},
+		{2, -5, -1},
+		{-2, -5, 1},
+		{3, 5, -1},
+		{5, 5, 0},
+		{-5, 5, 0},
+		{6, 5, 1},
+		{6, -5, 1},
+		{-6, 5, 1},
+		{-6, -5, -1},
+	}
+
+	var x, y Int
+
+	for i, test := range testCases {
+		x.SetInt64(test.x)
+		y.SetInt64(test.y)
+		expected := test.result
+		actual := Jacobi(&x, &y)
+		if actual != expected {
+			t.Errorf("#%d: Jacobi(%d, %d) = %d, but expected %d", i, test.x, test.y, actual, expected)
+		}
+	}
+}
+
+func TestJacobiPanic(t *testing.T) {
+	const failureMsg = "test failure"
+	defer func() {
+		msg := recover()
+		if msg == nil || msg == failureMsg {
+			panic(msg)
+		}
+		t.Log(msg)
+	}()
+	x := NewInt(1)
+	y := NewInt(2)
+	// Jacobi should panic when the second argument is even.
+	Jacobi(x, y)
+	panic(failureMsg)
+}
+
+func TestIssue2607(t *testing.T) {
+	// This code sequence used to hang.
+	n := NewInt(10)
+	n.Rand(rand.New(rand.NewSource(9)), n)
+}
+
+func TestSqrt(t *testing.T) {
+	root := 0
+	r := new(Int)
+	for i := 0; i < 10000; i++ {
+		if (root+1)*(root+1) <= i {
+			root++
+		}
+		n := NewInt(int64(i))
+		r.SetInt64(-2)
+		r.Sqrt(n)
+		if r.Cmp(NewInt(int64(root))) != 0 {
+			t.Errorf("Sqrt(%v) = %v, want %v", n, r, root)
+		}
+	}
+
+	for i := 0; i < 1000; i += 10 {
+		n, _ := new(Int).SetString("1"+strings.Repeat("0", i), 10)
+		r := new(Int).Sqrt(n)
+		root, _ := new(Int).SetString("1"+strings.Repeat("0", i/2), 10)
+		if r.Cmp(root) != 0 {
+			t.Errorf("Sqrt(1e%d) = %v, want 1e%d", i, r, i/2)
+		}
+	}
+
+	// Test aliasing.
+	r.SetInt64(100)
+	r.Sqrt(r)
+	if r.Int64() != 10 {
+		t.Errorf("Sqrt(100) = %v, want 10 (aliased output)", r.Int64())
+	}
+}
+
+// We can't test this together with the other Exp tests above because
+// it requires a different receiver setup.
+func TestIssue22830(t *testing.T) {
+	one := new(Int).SetInt64(1)
+	base, _ := new(Int).SetString("84555555300000000000", 10)
+	mod, _ := new(Int).SetString("66666670001111111111", 10)
+	want, _ := new(Int).SetString("17888885298888888889", 10)
+
+	var tests = []int64{
+		0, 1, -1,
+	}
+
+	for _, n := range tests {
+		m := NewInt(n)
+		if got := m.Exp(base, one, mod); got.Cmp(want) != 0 {
+			t.Errorf("(%v).Exp(%s, 1, %s) = %s, want %s", n, base, mod, got, want)
+		}
+	}
+}
+
+func BenchmarkSqrt(b *testing.B) {
+	n, _ := new(Int).SetString("1"+strings.Repeat("0", 1001), 10)
+	b.ResetTimer()
+	t := new(Int)
+	for i := 0; i < b.N; i++ {
+		t.Sqrt(n)
+	}
+}
+
+func benchmarkIntSqr(b *testing.B, nwords int) {
+	x := new(Int)
+	x.abs = rndNat(nwords)
+	t := new(Int)
+	b.ResetTimer()
+	for i := 0; i < b.N; i++ {
+		t.Mul(x, x)
+	}
+}
+
+func BenchmarkIntSqr(b *testing.B) {
+	for _, n := range sqrBenchSizes {
+		if isRaceBuilder && n > 1e3 {
+			continue
+		}
+		b.Run(fmt.Sprintf("%d", n), func(b *testing.B) {
+			benchmarkIntSqr(b, n)
+		})
+	}
+}
+
+func benchmarkDiv(b *testing.B, aSize, bSize int) {
+	var r = rand.New(rand.NewSource(1234))
+	aa := randInt(r, uint(aSize))
+	bb := randInt(r, uint(bSize))
+	if aa.Cmp(bb) < 0 {
+		aa, bb = bb, aa
+	}
+	x := new(Int)
+	y := new(Int)
+
+	b.ResetTimer()
+	for i := 0; i < b.N; i++ {
+		x.DivMod(aa, bb, y)
+	}
+}
+
+func BenchmarkDiv(b *testing.B) {
+	sizes := []int{
+		10, 20, 50, 100, 200, 500, 1000,
+		1e4, 1e5, 1e6, 1e7,
+	}
+	for _, i := range sizes {
+		j := 2 * i
+		b.Run(fmt.Sprintf("%d/%d", j, i), func(b *testing.B) {
+			benchmarkDiv(b, j, i)
+		})
+	}
+}
+
+func TestFillBytes(t *testing.T) {
+	checkResult := func(t *testing.T, buf []byte, want *Int) {
+		t.Helper()
+		got := new(Int).SetBytes(buf)
+		if got.CmpAbs(want) != 0 {
+			t.Errorf("got 0x%x, want 0x%x: %x", got, want, buf)
+		}
+	}
+	panics := func(f func()) (panic bool) {
+		defer func() { panic = recover() != nil }()
+		f()
+		return
+	}
+
+	for _, n := range []string{
+		"0",
+		"1000",
+		"0xffffffff",
+		"-0xffffffff",
+		"0xffffffffffffffff",
+		"0x10000000000000000",
+		"0xabababababababababababababababababababababababababa",
+		"0xffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff",
+	} {
+		t.Run(n, func(t *testing.T) {
+			t.Logf(n)
+			x, ok := new(Int).SetString(n, 0)
+			if !ok {
+				panic("invalid test entry")
+			}
+
+			// Perfectly sized buffer.
+			byteLen := (x.BitLen() + 7) / 8
+			buf := make([]byte, byteLen)
+			checkResult(t, x.FillBytes(buf), x)
+
+			// Way larger, checking all bytes get zeroed.
+			buf = make([]byte, 100)
+			for i := range buf {
+				buf[i] = 0xff
+			}
+			checkResult(t, x.FillBytes(buf), x)
+
+			// Too small.
+			if byteLen > 0 {
+				buf = make([]byte, byteLen-1)
+				if !panics(func() { x.FillBytes(buf) }) {
+					t.Errorf("expected panic for small buffer and value %x", x)
+				}
+			}
+		})
+	}
+}