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Source file src/math/big/int_test.go

Documentation: math/big

     1  // Copyright 2009 The Go Authors. All rights reserved.
     2  // Use of this source code is governed by a BSD-style
     3  // license that can be found in the LICENSE file.
     4  
     5  package big
     6  
     7  import (
     8  	"bytes"
     9  	"encoding/hex"
    10  	"fmt"
    11  	"math/rand"
    12  	"strconv"
    13  	"strings"
    14  	"testing"
    15  	"testing/quick"
    16  )
    17  
    18  func isNormalized(x *Int) bool {
    19  	if len(x.abs) == 0 {
    20  		return !x.neg
    21  	}
    22  	// len(x.abs) > 0
    23  	return x.abs[len(x.abs)-1] != 0
    24  }
    25  
    26  type funZZ func(z, x, y *Int) *Int
    27  type argZZ struct {
    28  	z, x, y *Int
    29  }
    30  
    31  var sumZZ = []argZZ{
    32  	{NewInt(0), NewInt(0), NewInt(0)},
    33  	{NewInt(1), NewInt(1), NewInt(0)},
    34  	{NewInt(1111111110), NewInt(123456789), NewInt(987654321)},
    35  	{NewInt(-1), NewInt(-1), NewInt(0)},
    36  	{NewInt(864197532), NewInt(-123456789), NewInt(987654321)},
    37  	{NewInt(-1111111110), NewInt(-123456789), NewInt(-987654321)},
    38  }
    39  
    40  var prodZZ = []argZZ{
    41  	{NewInt(0), NewInt(0), NewInt(0)},
    42  	{NewInt(0), NewInt(1), NewInt(0)},
    43  	{NewInt(1), NewInt(1), NewInt(1)},
    44  	{NewInt(-991 * 991), NewInt(991), NewInt(-991)},
    45  	// TODO(gri) add larger products
    46  }
    47  
    48  func TestSignZ(t *testing.T) {
    49  	var zero Int
    50  	for _, a := range sumZZ {
    51  		s := a.z.Sign()
    52  		e := a.z.Cmp(&zero)
    53  		if s != e {
    54  			t.Errorf("got %d; want %d for z = %v", s, e, a.z)
    55  		}
    56  	}
    57  }
    58  
    59  func TestSetZ(t *testing.T) {
    60  	for _, a := range sumZZ {
    61  		var z Int
    62  		z.Set(a.z)
    63  		if !isNormalized(&z) {
    64  			t.Errorf("%v is not normalized", z)
    65  		}
    66  		if (&z).Cmp(a.z) != 0 {
    67  			t.Errorf("got z = %v; want %v", z, a.z)
    68  		}
    69  	}
    70  }
    71  
    72  func TestAbsZ(t *testing.T) {
    73  	var zero Int
    74  	for _, a := range sumZZ {
    75  		var z Int
    76  		z.Abs(a.z)
    77  		var e Int
    78  		e.Set(a.z)
    79  		if e.Cmp(&zero) < 0 {
    80  			e.Sub(&zero, &e)
    81  		}
    82  		if z.Cmp(&e) != 0 {
    83  			t.Errorf("got z = %v; want %v", z, e)
    84  		}
    85  	}
    86  }
    87  
    88  func testFunZZ(t *testing.T, msg string, f funZZ, a argZZ) {
    89  	var z Int
    90  	f(&z, a.x, a.y)
    91  	if !isNormalized(&z) {
    92  		t.Errorf("%s%v is not normalized", msg, z)
    93  	}
    94  	if (&z).Cmp(a.z) != 0 {
    95  		t.Errorf("%s%+v\n\tgot z = %v; want %v", msg, a, &z, a.z)
    96  	}
    97  }
    98  
    99  func TestSumZZ(t *testing.T) {
   100  	AddZZ := func(z, x, y *Int) *Int { return z.Add(x, y) }
   101  	SubZZ := func(z, x, y *Int) *Int { return z.Sub(x, y) }
   102  	for _, a := range sumZZ {
   103  		arg := a
   104  		testFunZZ(t, "AddZZ", AddZZ, arg)
   105  
   106  		arg = argZZ{a.z, a.y, a.x}
   107  		testFunZZ(t, "AddZZ symmetric", AddZZ, arg)
   108  
   109  		arg = argZZ{a.x, a.z, a.y}
   110  		testFunZZ(t, "SubZZ", SubZZ, arg)
   111  
   112  		arg = argZZ{a.y, a.z, a.x}
   113  		testFunZZ(t, "SubZZ symmetric", SubZZ, arg)
   114  	}
   115  }
   116  
   117  func TestProdZZ(t *testing.T) {
   118  	MulZZ := func(z, x, y *Int) *Int { return z.Mul(x, y) }
   119  	for _, a := range prodZZ {
   120  		arg := a
   121  		testFunZZ(t, "MulZZ", MulZZ, arg)
   122  
   123  		arg = argZZ{a.z, a.y, a.x}
   124  		testFunZZ(t, "MulZZ symmetric", MulZZ, arg)
   125  	}
   126  }
   127  
   128  // mulBytes returns x*y via grade school multiplication. Both inputs
   129  // and the result are assumed to be in big-endian representation (to
   130  // match the semantics of Int.Bytes and Int.SetBytes).
   131  func mulBytes(x, y []byte) []byte {
   132  	z := make([]byte, len(x)+len(y))
   133  
   134  	// multiply
   135  	k0 := len(z) - 1
   136  	for j := len(y) - 1; j >= 0; j-- {
   137  		d := int(y[j])
   138  		if d != 0 {
   139  			k := k0
   140  			carry := 0
   141  			for i := len(x) - 1; i >= 0; i-- {
   142  				t := int(z[k]) + int(x[i])*d + carry
   143  				z[k], carry = byte(t), t>>8
   144  				k--
   145  			}
   146  			z[k] = byte(carry)
   147  		}
   148  		k0--
   149  	}
   150  
   151  	// normalize (remove leading 0's)
   152  	i := 0
   153  	for i < len(z) && z[i] == 0 {
   154  		i++
   155  	}
   156  
   157  	return z[i:]
   158  }
   159  
   160  func checkMul(a, b []byte) bool {
   161  	var x, y, z1 Int
   162  	x.SetBytes(a)
   163  	y.SetBytes(b)
   164  	z1.Mul(&x, &y)
   165  
   166  	var z2 Int
   167  	z2.SetBytes(mulBytes(a, b))
   168  
   169  	return z1.Cmp(&z2) == 0
   170  }
   171  
   172  func TestMul(t *testing.T) {
   173  	if err := quick.Check(checkMul, nil); err != nil {
   174  		t.Error(err)
   175  	}
   176  }
   177  
   178  var mulRangesZ = []struct {
   179  	a, b int64
   180  	prod string
   181  }{
   182  	// entirely positive ranges are covered by mulRangesN
   183  	{-1, 1, "0"},
   184  	{-2, -1, "2"},
   185  	{-3, -2, "6"},
   186  	{-3, -1, "-6"},
   187  	{1, 3, "6"},
   188  	{-10, -10, "-10"},
   189  	{0, -1, "1"},                      // empty range
   190  	{-1, -100, "1"},                   // empty range
   191  	{-1, 1, "0"},                      // range includes 0
   192  	{-1e9, 0, "0"},                    // range includes 0
   193  	{-1e9, 1e9, "0"},                  // range includes 0
   194  	{-10, -1, "3628800"},              // 10!
   195  	{-20, -2, "-2432902008176640000"}, // -20!
   196  	{-99, -1,
   197  		"-933262154439441526816992388562667004907159682643816214685929" +
   198  			"638952175999932299156089414639761565182862536979208272237582" +
   199  			"511852109168640000000000000000000000", // -99!
   200  	},
   201  }
   202  
   203  func TestMulRangeZ(t *testing.T) {
   204  	var tmp Int
   205  	// test entirely positive ranges
   206  	for i, r := range mulRangesN {
   207  		prod := tmp.MulRange(int64(r.a), int64(r.b)).String()
   208  		if prod != r.prod {
   209  			t.Errorf("#%da: got %s; want %s", i, prod, r.prod)
   210  		}
   211  	}
   212  	// test other ranges
   213  	for i, r := range mulRangesZ {
   214  		prod := tmp.MulRange(r.a, r.b).String()
   215  		if prod != r.prod {
   216  			t.Errorf("#%db: got %s; want %s", i, prod, r.prod)
   217  		}
   218  	}
   219  }
   220  
   221  func TestBinomial(t *testing.T) {
   222  	var z Int
   223  	for _, test := range []struct {
   224  		n, k int64
   225  		want string
   226  	}{
   227  		{0, 0, "1"},
   228  		{0, 1, "0"},
   229  		{1, 0, "1"},
   230  		{1, 1, "1"},
   231  		{1, 10, "0"},
   232  		{4, 0, "1"},
   233  		{4, 1, "4"},
   234  		{4, 2, "6"},
   235  		{4, 3, "4"},
   236  		{4, 4, "1"},
   237  		{10, 1, "10"},
   238  		{10, 9, "10"},
   239  		{10, 5, "252"},
   240  		{11, 5, "462"},
   241  		{11, 6, "462"},
   242  		{100, 10, "17310309456440"},
   243  		{100, 90, "17310309456440"},
   244  		{1000, 10, "263409560461970212832400"},
   245  		{1000, 990, "263409560461970212832400"},
   246  	} {
   247  		if got := z.Binomial(test.n, test.k).String(); got != test.want {
   248  			t.Errorf("Binomial(%d, %d) = %s; want %s", test.n, test.k, got, test.want)
   249  		}
   250  	}
   251  }
   252  
   253  func BenchmarkBinomial(b *testing.B) {
   254  	var z Int
   255  	for i := b.N - 1; i >= 0; i-- {
   256  		z.Binomial(1000, 990)
   257  	}
   258  }
   259  
   260  // Examples from the Go Language Spec, section "Arithmetic operators"
   261  var divisionSignsTests = []struct {
   262  	x, y int64
   263  	q, r int64 // T-division
   264  	d, m int64 // Euclidean division
   265  }{
   266  	{5, 3, 1, 2, 1, 2},
   267  	{-5, 3, -1, -2, -2, 1},
   268  	{5, -3, -1, 2, -1, 2},
   269  	{-5, -3, 1, -2, 2, 1},
   270  	{1, 2, 0, 1, 0, 1},
   271  	{8, 4, 2, 0, 2, 0},
   272  }
   273  
   274  func TestDivisionSigns(t *testing.T) {
   275  	for i, test := range divisionSignsTests {
   276  		x := NewInt(test.x)
   277  		y := NewInt(test.y)
   278  		q := NewInt(test.q)
   279  		r := NewInt(test.r)
   280  		d := NewInt(test.d)
   281  		m := NewInt(test.m)
   282  
   283  		q1 := new(Int).Quo(x, y)
   284  		r1 := new(Int).Rem(x, y)
   285  		if !isNormalized(q1) {
   286  			t.Errorf("#%d Quo: %v is not normalized", i, *q1)
   287  		}
   288  		if !isNormalized(r1) {
   289  			t.Errorf("#%d Rem: %v is not normalized", i, *r1)
   290  		}
   291  		if q1.Cmp(q) != 0 || r1.Cmp(r) != 0 {
   292  			t.Errorf("#%d QuoRem: got (%s, %s), want (%s, %s)", i, q1, r1, q, r)
   293  		}
   294  
   295  		q2, r2 := new(Int).QuoRem(x, y, new(Int))
   296  		if !isNormalized(q2) {
   297  			t.Errorf("#%d Quo: %v is not normalized", i, *q2)
   298  		}
   299  		if !isNormalized(r2) {
   300  			t.Errorf("#%d Rem: %v is not normalized", i, *r2)
   301  		}
   302  		if q2.Cmp(q) != 0 || r2.Cmp(r) != 0 {
   303  			t.Errorf("#%d QuoRem: got (%s, %s), want (%s, %s)", i, q2, r2, q, r)
   304  		}
   305  
   306  		d1 := new(Int).Div(x, y)
   307  		m1 := new(Int).Mod(x, y)
   308  		if !isNormalized(d1) {
   309  			t.Errorf("#%d Div: %v is not normalized", i, *d1)
   310  		}
   311  		if !isNormalized(m1) {
   312  			t.Errorf("#%d Mod: %v is not normalized", i, *m1)
   313  		}
   314  		if d1.Cmp(d) != 0 || m1.Cmp(m) != 0 {
   315  			t.Errorf("#%d DivMod: got (%s, %s), want (%s, %s)", i, d1, m1, d, m)
   316  		}
   317  
   318  		d2, m2 := new(Int).DivMod(x, y, new(Int))
   319  		if !isNormalized(d2) {
   320  			t.Errorf("#%d Div: %v is not normalized", i, *d2)
   321  		}
   322  		if !isNormalized(m2) {
   323  			t.Errorf("#%d Mod: %v is not normalized", i, *m2)
   324  		}
   325  		if d2.Cmp(d) != 0 || m2.Cmp(m) != 0 {
   326  			t.Errorf("#%d DivMod: got (%s, %s), want (%s, %s)", i, d2, m2, d, m)
   327  		}
   328  	}
   329  }
   330  
   331  func norm(x nat) nat {
   332  	i := len(x)
   333  	for i > 0 && x[i-1] == 0 {
   334  		i--
   335  	}
   336  	return x[:i]
   337  }
   338  
   339  func TestBits(t *testing.T) {
   340  	for _, test := range []nat{
   341  		nil,
   342  		{0},
   343  		{1},
   344  		{0, 1, 2, 3, 4},
   345  		{4, 3, 2, 1, 0},
   346  		{4, 3, 2, 1, 0, 0, 0, 0},
   347  	} {
   348  		var z Int
   349  		z.neg = true
   350  		got := z.SetBits(test)
   351  		want := norm(test)
   352  		if got.abs.cmp(want) != 0 {
   353  			t.Errorf("SetBits(%v) = %v; want %v", test, got.abs, want)
   354  		}
   355  
   356  		if got.neg {
   357  			t.Errorf("SetBits(%v): got negative result", test)
   358  		}
   359  
   360  		bits := nat(z.Bits())
   361  		if bits.cmp(want) != 0 {
   362  			t.Errorf("%v.Bits() = %v; want %v", z.abs, bits, want)
   363  		}
   364  	}
   365  }
   366  
   367  func checkSetBytes(b []byte) bool {
   368  	hex1 := hex.EncodeToString(new(Int).SetBytes(b).Bytes())
   369  	hex2 := hex.EncodeToString(b)
   370  
   371  	for len(hex1) < len(hex2) {
   372  		hex1 = "0" + hex1
   373  	}
   374  
   375  	for len(hex1) > len(hex2) {
   376  		hex2 = "0" + hex2
   377  	}
   378  
   379  	return hex1 == hex2
   380  }
   381  
   382  func TestSetBytes(t *testing.T) {
   383  	if err := quick.Check(checkSetBytes, nil); err != nil {
   384  		t.Error(err)
   385  	}
   386  }
   387  
   388  func checkBytes(b []byte) bool {
   389  	// trim leading zero bytes since Bytes() won't return them
   390  	// (was issue 12231)
   391  	for len(b) > 0 && b[0] == 0 {
   392  		b = b[1:]
   393  	}
   394  	b2 := new(Int).SetBytes(b).Bytes()
   395  	return bytes.Equal(b, b2)
   396  }
   397  
   398  func TestBytes(t *testing.T) {
   399  	if err := quick.Check(checkBytes, nil); err != nil {
   400  		t.Error(err)
   401  	}
   402  }
   403  
   404  func checkQuo(x, y []byte) bool {
   405  	u := new(Int).SetBytes(x)
   406  	v := new(Int).SetBytes(y)
   407  
   408  	if len(v.abs) == 0 {
   409  		return true
   410  	}
   411  
   412  	r := new(Int)
   413  	q, r := new(Int).QuoRem(u, v, r)
   414  
   415  	if r.Cmp(v) >= 0 {
   416  		return false
   417  	}
   418  
   419  	uprime := new(Int).Set(q)
   420  	uprime.Mul(uprime, v)
   421  	uprime.Add(uprime, r)
   422  
   423  	return uprime.Cmp(u) == 0
   424  }
   425  
   426  var quoTests = []struct {
   427  	x, y string
   428  	q, r string
   429  }{
   430  	{
   431  		"476217953993950760840509444250624797097991362735329973741718102894495832294430498335824897858659711275234906400899559094370964723884706254265559534144986498357",
   432  		"9353930466774385905609975137998169297361893554149986716853295022578535724979483772383667534691121982974895531435241089241440253066816724367338287092081996",
   433  		"50911",
   434  		"1",
   435  	},
   436  	{
   437  		"11510768301994997771168",
   438  		"1328165573307167369775",
   439  		"8",
   440  		"885443715537658812968",
   441  	},
   442  }
   443  
   444  func TestQuo(t *testing.T) {
   445  	if err := quick.Check(checkQuo, nil); err != nil {
   446  		t.Error(err)
   447  	}
   448  
   449  	for i, test := range quoTests {
   450  		x, _ := new(Int).SetString(test.x, 10)
   451  		y, _ := new(Int).SetString(test.y, 10)
   452  		expectedQ, _ := new(Int).SetString(test.q, 10)
   453  		expectedR, _ := new(Int).SetString(test.r, 10)
   454  
   455  		r := new(Int)
   456  		q, r := new(Int).QuoRem(x, y, r)
   457  
   458  		if q.Cmp(expectedQ) != 0 || r.Cmp(expectedR) != 0 {
   459  			t.Errorf("#%d got (%s, %s) want (%s, %s)", i, q, r, expectedQ, expectedR)
   460  		}
   461  	}
   462  }
   463  
   464  func TestQuoStepD6(t *testing.T) {
   465  	// See Knuth, Volume 2, section 4.3.1, exercise 21. This code exercises
   466  	// a code path which only triggers 1 in 10^{-19} cases.
   467  
   468  	u := &Int{false, nat{0, 0, 1 + 1<<(_W-1), _M ^ (1 << (_W - 1))}}
   469  	v := &Int{false, nat{5, 2 + 1<<(_W-1), 1 << (_W - 1)}}
   470  
   471  	r := new(Int)
   472  	q, r := new(Int).QuoRem(u, v, r)
   473  	const expectedQ64 = "18446744073709551613"
   474  	const expectedR64 = "3138550867693340382088035895064302439801311770021610913807"
   475  	const expectedQ32 = "4294967293"
   476  	const expectedR32 = "39614081266355540837921718287"
   477  	if q.String() != expectedQ64 && q.String() != expectedQ32 ||
   478  		r.String() != expectedR64 && r.String() != expectedR32 {
   479  		t.Errorf("got (%s, %s) want (%s, %s) or (%s, %s)", q, r, expectedQ64, expectedR64, expectedQ32, expectedR32)
   480  	}
   481  }
   482  
   483  func BenchmarkQuoRem(b *testing.B) {
   484  	x, _ := new(Int).SetString("153980389784927331788354528594524332344709972855165340650588877572729725338415474372475094155672066328274535240275856844648695200875763869073572078279316458648124537905600131008790701752441155668003033945258023841165089852359980273279085783159654751552359397986180318708491098942831252291841441726305535546071", 0)
   485  	y, _ := new(Int).SetString("7746362281539803897849273317883545285945243323447099728551653406505888775727297253384154743724750941556720663282745352402758568446486952008757638690735720782793164586481245379056001310087907017524411556680030339452580238411650898523599802732790857831596547515523593979861803187084910989428312522918414417263055355460715745539358014631136245887418412633787074173796862711588221766398229333338511838891484974940633857861775630560092874987828057333663969469797013996401149696897591265769095952887917296740109742927689053276850469671231961384715398038978492733178835452859452433234470997285516534065058887757272972533841547437247509415567206632827453524027585684464869520087576386907357207827931645864812453790560013100879070175244115566800303394525802384116508985235998027327908578315965475155235939798618031870849109894283125229184144172630553554607112725169432413343763989564437170644270643461665184965150423819594083121075825", 0)
   486  	q := new(Int)
   487  	r := new(Int)
   488  
   489  	b.ResetTimer()
   490  	for i := 0; i < b.N; i++ {
   491  		q.QuoRem(y, x, r)
   492  	}
   493  }
   494  
   495  var bitLenTests = []struct {
   496  	in  string
   497  	out int
   498  }{
   499  	{"-1", 1},
   500  	{"0", 0},
   501  	{"1", 1},
   502  	{"2", 2},
   503  	{"4", 3},
   504  	{"0xabc", 12},
   505  	{"0x8000", 16},
   506  	{"0x80000000", 32},
   507  	{"0x800000000000", 48},
   508  	{"0x8000000000000000", 64},
   509  	{"0x80000000000000000000", 80},
   510  	{"-0x4000000000000000000000", 87},
   511  }
   512  
   513  func TestBitLen(t *testing.T) {
   514  	for i, test := range bitLenTests {
   515  		x, ok := new(Int).SetString(test.in, 0)
   516  		if !ok {
   517  			t.Errorf("#%d test input invalid: %s", i, test.in)
   518  			continue
   519  		}
   520  
   521  		if n := x.BitLen(); n != test.out {
   522  			t.Errorf("#%d got %d want %d", i, n, test.out)
   523  		}
   524  	}
   525  }
   526  
   527  var expTests = []struct {
   528  	x, y, m string
   529  	out     string
   530  }{
   531  	// y <= 0
   532  	{"0", "0", "", "1"},
   533  	{"1", "0", "", "1"},
   534  	{"-10", "0", "", "1"},
   535  	{"1234", "-1", "", "1"},
   536  
   537  	// m == 1
   538  	{"0", "0", "1", "0"},
   539  	{"1", "0", "1", "0"},
   540  	{"-10", "0", "1", "0"},
   541  	{"1234", "-1", "1", "0"},
   542  
   543  	// misc
   544  	{"5", "1", "3", "2"},
   545  	{"5", "-7", "", "1"},
   546  	{"-5", "-7", "", "1"},
   547  	{"5", "0", "", "1"},
   548  	{"-5", "0", "", "1"},
   549  	{"5", "1", "", "5"},
   550  	{"-5", "1", "", "-5"},
   551  	{"-5", "1", "7", "2"},
   552  	{"-2", "3", "2", "0"},
   553  	{"5", "2", "", "25"},
   554  	{"1", "65537", "2", "1"},
   555  	{"0x8000000000000000", "2", "", "0x40000000000000000000000000000000"},
   556  	{"0x8000000000000000", "2", "6719", "4944"},
   557  	{"0x8000000000000000", "3", "6719", "5447"},
   558  	{"0x8000000000000000", "1000", "6719", "1603"},
   559  	{"0x8000000000000000", "1000000", "6719", "3199"},
   560  	{"0x8000000000000000", "-1000000", "6719", "3663"}, // 3663 = ModInverse(3199, 6719) Issue #25865
   561  
   562  	{"0xffffffffffffffffffffffffffffffff", "0x12345678123456781234567812345678123456789", "0x01112222333344445555666677778889", "0x36168FA1DB3AAE6C8CE647E137F97A"},
   563  
   564  	{
   565  		"2938462938472983472983659726349017249287491026512746239764525612965293865296239471239874193284792387498274256129746192347",
   566  		"298472983472983471903246121093472394872319615612417471234712061",
   567  		"29834729834729834729347290846729561262544958723956495615629569234729836259263598127342374289365912465901365498236492183464",
   568  		"23537740700184054162508175125554701713153216681790245129157191391322321508055833908509185839069455749219131480588829346291",
   569  	},
   570  	// test case for issue 8822
   571  	{
   572  		"11001289118363089646017359372117963499250546375269047542777928006103246876688756735760905680604646624353196869572752623285140408755420374049317646428185270079555372763503115646054602867593662923894140940837479507194934267532831694565516466765025434902348314525627418515646588160955862839022051353653052947073136084780742729727874803457643848197499548297570026926927502505634297079527299004267769780768565695459945235586892627059178884998772989397505061206395455591503771677500931269477503508150175717121828518985901959919560700853226255420793148986854391552859459511723547532575574664944815966793196961286234040892865",
   573  		"0xB08FFB20760FFED58FADA86DFEF71AD72AA0FA763219618FE022C197E54708BB1191C66470250FCE8879487507CEE41381CA4D932F81C2B3F1AB20B539D50DCD",
   574  		"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
   575  		"21484252197776302499639938883777710321993113097987201050501182909581359357618579566746556372589385361683610524730509041328855066514963385522570894839035884713051640171474186548713546686476761306436434146475140156284389181808675016576845833340494848283681088886584219750554408060556769486628029028720727393293111678826356480455433909233520504112074401376133077150471237549474149190242010469539006449596611576612573955754349042329130631128234637924786466585703488460540228477440853493392086251021228087076124706778899179648655221663765993962724699135217212118535057766739392069738618682722216712319320435674779146070442",
   576  	},
   577  	{
   578  		"-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
   579  		"0xB08FFB20760FFED58FADA86DFEF71AD72AA0FA763219618FE022C197E54708BB1191C66470250FCE8879487507CEE41381CA4D932F81C2B3F1AB20B539D50DCD",
   580  		"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
   581  		"21484252197776302499639938883777710321993113097987201050501182909581359357618579566746556372589385361683610524730509041328855066514963385522570894839035884713051640171474186548713546686476761306436434146475140156284389181808675016576845833340494848283681088886584219750554408060556769486628029028720727393293111678826356480455433909233520504112074401376133077150471237549474149190242010469539006449596611576612573955754349042329130631128234637924786466585703488460540228477440853493392086251021228087076124706778899179648655221663765993962724699135217212118535057766739392069738618682722216712319320435674779146070442",
   582  	},
   583  
   584  	// test cases for issue 13907
   585  	{"0xffffffff00000001", "0xffffffff00000001", "0xffffffff00000001", "0"},
   586  	{"0xffffffffffffffff00000001", "0xffffffffffffffff00000001", "0xffffffffffffffff00000001", "0"},
   587  	{"0xffffffffffffffffffffffff00000001", "0xffffffffffffffffffffffff00000001", "0xffffffffffffffffffffffff00000001", "0"},
   588  	{"0xffffffffffffffffffffffffffffffff00000001", "0xffffffffffffffffffffffffffffffff00000001", "0xffffffffffffffffffffffffffffffff00000001", "0"},
   589  
   590  	{
   591  		"2",
   592  		"0xB08FFB20760FFED58FADA86DFEF71AD72AA0FA763219618FE022C197E54708BB1191C66470250FCE8879487507CEE41381CA4D932F81C2B3F1AB20B539D50DCD",
   593  		"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odd
   594  		"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
   595  	},
   596  	{
   597  		"2",
   598  		"0xB08FFB20760FFED58FADA86DFEF71AD72AA0FA763219618FE022C197E54708BB1191C66470250FCE8879487507CEE41381CA4D932F81C2B3F1AB20B539D50DCD",
   599  		"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even
   600  		"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
   601  	},
   602  }
   603  
   604  func TestExp(t *testing.T) {
   605  	for i, test := range expTests {
   606  		x, ok1 := new(Int).SetString(test.x, 0)
   607  		y, ok2 := new(Int).SetString(test.y, 0)
   608  		out, ok3 := new(Int).SetString(test.out, 0)
   609  
   610  		var ok4 bool
   611  		var m *Int
   612  
   613  		if len(test.m) == 0 {
   614  			m, ok4 = nil, true
   615  		} else {
   616  			m, ok4 = new(Int).SetString(test.m, 0)
   617  		}
   618  
   619  		if !ok1 || !ok2 || !ok3 || !ok4 {
   620  			t.Errorf("#%d: error in input", i)
   621  			continue
   622  		}
   623  
   624  		z1 := new(Int).Exp(x, y, m)
   625  		if !isNormalized(z1) {
   626  			t.Errorf("#%d: %v is not normalized", i, *z1)
   627  		}
   628  		if z1.Cmp(out) != 0 {
   629  			t.Errorf("#%d: got %x want %x", i, z1, out)
   630  		}
   631  
   632  		if m == nil {
   633  			// The result should be the same as for m == 0;
   634  			// specifically, there should be no div-zero panic.
   635  			m = &Int{abs: nat{}} // m != nil && len(m.abs) == 0
   636  			z2 := new(Int).Exp(x, y, m)
   637  			if z2.Cmp(z1) != 0 {
   638  				t.Errorf("#%d: got %x want %x", i, z2, z1)
   639  			}
   640  		}
   641  	}
   642  }
   643  
   644  func BenchmarkExp(b *testing.B) {
   645  	x, _ := new(Int).SetString("11001289118363089646017359372117963499250546375269047542777928006103246876688756735760905680604646624353196869572752623285140408755420374049317646428185270079555372763503115646054602867593662923894140940837479507194934267532831694565516466765025434902348314525627418515646588160955862839022051353653052947073136084780742729727874803457643848197499548297570026926927502505634297079527299004267769780768565695459945235586892627059178884998772989397505061206395455591503771677500931269477503508150175717121828518985901959919560700853226255420793148986854391552859459511723547532575574664944815966793196961286234040892865", 0)
   646  	y, _ := new(Int).SetString("0xAC6BDB41324A9A9BF166DE5E1389582FAF72B6651987EE07FC3192943DB56050A37329CBB4A099ED8193E0757767A13DD52312AB4B03310DCD7F48A9DA04FD50E8083969EDB767B0CF6095179A163AB3661A05FBD5FAAAE82918A9962F0B93B855F97993EC975EEAA80D740ADBF4FF747359D041D5C33EA71D281E446B14773BCA97B43A23FB801676BD207A436C6481F1D2B9078717461A5B9D32E688F87748544523B524B0D57D5EA77A2775D2ECFA032CFBDBF52FB3786160279004E57AE6AF874E7303CE53299CCC041C7BC308D82A5698F3A8D0C38271AE35F8E9DBFBB694B5C803D89F7AE435DE236D525F54759B65E372FCD68EF20FA7111F9E4AFF72", 0)
   647  	n, _ := new(Int).SetString("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
   648  	out := new(Int)
   649  	for i := 0; i < b.N; i++ {
   650  		out.Exp(x, y, n)
   651  	}
   652  }
   653  
   654  func BenchmarkExp2(b *testing.B) {
   655  	x, _ := new(Int).SetString("2", 0)
   656  	y, _ := new(Int).SetString("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
   657  	n, _ := new(Int).SetString("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
   658  	out := new(Int)
   659  	for i := 0; i < b.N; i++ {
   660  		out.Exp(x, y, n)
   661  	}
   662  }
   663  
   664  func checkGcd(aBytes, bBytes []byte) bool {
   665  	x := new(Int)
   666  	y := new(Int)
   667  	a := new(Int).SetBytes(aBytes)
   668  	b := new(Int).SetBytes(bBytes)
   669  
   670  	d := new(Int).GCD(x, y, a, b)
   671  	x.Mul(x, a)
   672  	y.Mul(y, b)
   673  	x.Add(x, y)
   674  
   675  	return x.Cmp(d) == 0
   676  }
   677  
   678  // euclidExtGCD is a reference implementation of Euclid's
   679  // extended GCD algorithm for testing against optimized algorithms.
   680  // Requirements: a, b > 0
   681  func euclidExtGCD(a, b *Int) (g, x, y *Int) {
   682  	A := new(Int).Set(a)
   683  	B := new(Int).Set(b)
   684  
   685  	// A = Ua*a + Va*b
   686  	// B = Ub*a + Vb*b
   687  	Ua := new(Int).SetInt64(1)
   688  	Va := new(Int)
   689  
   690  	Ub := new(Int)
   691  	Vb := new(Int).SetInt64(1)
   692  
   693  	q := new(Int)
   694  	temp := new(Int)
   695  
   696  	r := new(Int)
   697  	for len(B.abs) > 0 {
   698  		q, r = q.QuoRem(A, B, r)
   699  
   700  		A, B, r = B, r, A
   701  
   702  		// Ua, Ub = Ub, Ua-q*Ub
   703  		temp.Set(Ub)
   704  		Ub.Mul(Ub, q)
   705  		Ub.Sub(Ua, Ub)
   706  		Ua.Set(temp)
   707  
   708  		// Va, Vb = Vb, Va-q*Vb
   709  		temp.Set(Vb)
   710  		Vb.Mul(Vb, q)
   711  		Vb.Sub(Va, Vb)
   712  		Va.Set(temp)
   713  	}
   714  	return A, Ua, Va
   715  }
   716  
   717  func checkLehmerGcd(aBytes, bBytes []byte) bool {
   718  	a := new(Int).SetBytes(aBytes)
   719  	b := new(Int).SetBytes(bBytes)
   720  
   721  	if a.Sign() <= 0 || b.Sign() <= 0 {
   722  		return true // can only test positive arguments
   723  	}
   724  
   725  	d := new(Int).lehmerGCD(nil, nil, a, b)
   726  	d0, _, _ := euclidExtGCD(a, b)
   727  
   728  	return d.Cmp(d0) == 0
   729  }
   730  
   731  func checkLehmerExtGcd(aBytes, bBytes []byte) bool {
   732  	a := new(Int).SetBytes(aBytes)
   733  	b := new(Int).SetBytes(bBytes)
   734  	x := new(Int)
   735  	y := new(Int)
   736  
   737  	if a.Sign() <= 0 || b.Sign() <= 0 {
   738  		return true // can only test positive arguments
   739  	}
   740  
   741  	d := new(Int).lehmerGCD(x, y, a, b)
   742  	d0, x0, y0 := euclidExtGCD(a, b)
   743  
   744  	return d.Cmp(d0) == 0 && x.Cmp(x0) == 0 && y.Cmp(y0) == 0
   745  }
   746  
   747  var gcdTests = []struct {
   748  	d, x, y, a, b string
   749  }{
   750  	// a <= 0 || b <= 0
   751  	{"0", "0", "0", "0", "0"},
   752  	{"0", "0", "0", "0", "7"},
   753  	{"0", "0", "0", "11", "0"},
   754  	{"0", "0", "0", "-77", "35"},
   755  	{"0", "0", "0", "64515", "-24310"},
   756  	{"0", "0", "0", "-64515", "-24310"},
   757  
   758  	{"1", "-9", "47", "120", "23"},
   759  	{"7", "1", "-2", "77", "35"},
   760  	{"935", "-3", "8", "64515", "24310"},
   761  	{"935000000000000000", "-3", "8", "64515000000000000000", "24310000000000000000"},
   762  	{"1", "-221", "22059940471369027483332068679400581064239780177629666810348940098015901108344", "98920366548084643601728869055592650835572950932266967461790948584315647051443", "991"},
   763  
   764  	// test early exit (after one Euclidean iteration) in binaryGCD
   765  	{"1", "", "", "1", "98920366548084643601728869055592650835572950932266967461790948584315647051443"},
   766  }
   767  
   768  func testGcd(t *testing.T, d, x, y, a, b *Int) {
   769  	var X *Int
   770  	if x != nil {
   771  		X = new(Int)
   772  	}
   773  	var Y *Int
   774  	if y != nil {
   775  		Y = new(Int)
   776  	}
   777  
   778  	D := new(Int).GCD(X, Y, a, b)
   779  	if D.Cmp(d) != 0 {
   780  		t.Errorf("GCD(%s, %s, %s, %s): got d = %s, want %s", x, y, a, b, D, d)
   781  	}
   782  	if x != nil && X.Cmp(x) != 0 {
   783  		t.Errorf("GCD(%s, %s, %s, %s): got x = %s, want %s", x, y, a, b, X, x)
   784  	}
   785  	if y != nil && Y.Cmp(y) != 0 {
   786  		t.Errorf("GCD(%s, %s, %s, %s): got y = %s, want %s", x, y, a, b, Y, y)
   787  	}
   788  
   789  	// check results in presence of aliasing (issue #11284)
   790  	a2 := new(Int).Set(a)
   791  	b2 := new(Int).Set(b)
   792  	a2.GCD(X, Y, a2, b2) // result is same as 1st argument
   793  	if a2.Cmp(d) != 0 {
   794  		t.Errorf("aliased z = a GCD(%s, %s, %s, %s): got d = %s, want %s", x, y, a, b, a2, d)
   795  	}
   796  
   797  	a2 = new(Int).Set(a)
   798  	b2 = new(Int).Set(b)
   799  	b2.GCD(X, Y, a2, b2) // result is same as 2nd argument
   800  	if b2.Cmp(d) != 0 {
   801  		t.Errorf("aliased z = b GCD(%s, %s, %s, %s): got d = %s, want %s", x, y, a, b, b2, d)
   802  	}
   803  }
   804  
   805  func TestGcd(t *testing.T) {
   806  	for _, test := range gcdTests {
   807  		d, _ := new(Int).SetString(test.d, 0)
   808  		x, _ := new(Int).SetString(test.x, 0)
   809  		y, _ := new(Int).SetString(test.y, 0)
   810  		a, _ := new(Int).SetString(test.a, 0)
   811  		b, _ := new(Int).SetString(test.b, 0)
   812  
   813  		testGcd(t, d, nil, nil, a, b)
   814  		testGcd(t, d, x, nil, a, b)
   815  		testGcd(t, d, nil, y, a, b)
   816  		testGcd(t, d, x, y, a, b)
   817  	}
   818  
   819  	if err := quick.Check(checkGcd, nil); err != nil {
   820  		t.Error(err)
   821  	}
   822  
   823  	if err := quick.Check(checkLehmerGcd, nil); err != nil {
   824  		t.Error(err)
   825  	}
   826  
   827  	if err := quick.Check(checkLehmerExtGcd, nil); err != nil {
   828  		t.Error(err)
   829  	}
   830  }
   831  
   832  type intShiftTest struct {
   833  	in    string
   834  	shift uint
   835  	out   string
   836  }
   837  
   838  var rshTests = []intShiftTest{
   839  	{"0", 0, "0"},
   840  	{"-0", 0, "0"},
   841  	{"0", 1, "0"},
   842  	{"0", 2, "0"},
   843  	{"1", 0, "1"},
   844  	{"1", 1, "0"},
   845  	{"1", 2, "0"},
   846  	{"2", 0, "2"},
   847  	{"2", 1, "1"},
   848  	{"-1", 0, "-1"},
   849  	{"-1", 1, "-1"},
   850  	{"-1", 10, "-1"},
   851  	{"-100", 2, "-25"},
   852  	{"-100", 3, "-13"},
   853  	{"-100", 100, "-1"},
   854  	{"4294967296", 0, "4294967296"},
   855  	{"4294967296", 1, "2147483648"},
   856  	{"4294967296", 2, "1073741824"},
   857  	{"18446744073709551616", 0, "18446744073709551616"},
   858  	{"18446744073709551616", 1, "9223372036854775808"},
   859  	{"18446744073709551616", 2, "4611686018427387904"},
   860  	{"18446744073709551616", 64, "1"},
   861  	{"340282366920938463463374607431768211456", 64, "18446744073709551616"},
   862  	{"340282366920938463463374607431768211456", 128, "1"},
   863  }
   864  
   865  func TestRsh(t *testing.T) {
   866  	for i, test := range rshTests {
   867  		in, _ := new(Int).SetString(test.in, 10)
   868  		expected, _ := new(Int).SetString(test.out, 10)
   869  		out := new(Int).Rsh(in, test.shift)
   870  
   871  		if !isNormalized(out) {
   872  			t.Errorf("#%d: %v is not normalized", i, *out)
   873  		}
   874  		if out.Cmp(expected) != 0 {
   875  			t.Errorf("#%d: got %s want %s", i, out, expected)
   876  		}
   877  	}
   878  }
   879  
   880  func TestRshSelf(t *testing.T) {
   881  	for i, test := range rshTests {
   882  		z, _ := new(Int).SetString(test.in, 10)
   883  		expected, _ := new(Int).SetString(test.out, 10)
   884  		z.Rsh(z, test.shift)
   885  
   886  		if !isNormalized(z) {
   887  			t.Errorf("#%d: %v is not normalized", i, *z)
   888  		}
   889  		if z.Cmp(expected) != 0 {
   890  			t.Errorf("#%d: got %s want %s", i, z, expected)
   891  		}
   892  	}
   893  }
   894  
   895  var lshTests = []intShiftTest{
   896  	{"0", 0, "0"},
   897  	{"0", 1, "0"},
   898  	{"0", 2, "0"},
   899  	{"1", 0, "1"},
   900  	{"1", 1, "2"},
   901  	{"1", 2, "4"},
   902  	{"2", 0, "2"},
   903  	{"2", 1, "4"},
   904  	{"2", 2, "8"},
   905  	{"-87", 1, "-174"},
   906  	{"4294967296", 0, "4294967296"},
   907  	{"4294967296", 1, "8589934592"},
   908  	{"4294967296", 2, "17179869184"},
   909  	{"18446744073709551616", 0, "18446744073709551616"},
   910  	{"9223372036854775808", 1, "18446744073709551616"},
   911  	{"4611686018427387904", 2, "18446744073709551616"},
   912  	{"1", 64, "18446744073709551616"},
   913  	{"18446744073709551616", 64, "340282366920938463463374607431768211456"},
   914  	{"1", 128, "340282366920938463463374607431768211456"},
   915  }
   916  
   917  func TestLsh(t *testing.T) {
   918  	for i, test := range lshTests {
   919  		in, _ := new(Int).SetString(test.in, 10)
   920  		expected, _ := new(Int).SetString(test.out, 10)
   921  		out := new(Int).Lsh(in, test.shift)
   922  
   923  		if !isNormalized(out) {
   924  			t.Errorf("#%d: %v is not normalized", i, *out)
   925  		}
   926  		if out.Cmp(expected) != 0 {
   927  			t.Errorf("#%d: got %s want %s", i, out, expected)
   928  		}
   929  	}
   930  }
   931  
   932  func TestLshSelf(t *testing.T) {
   933  	for i, test := range lshTests {
   934  		z, _ := new(Int).SetString(test.in, 10)
   935  		expected, _ := new(Int).SetString(test.out, 10)
   936  		z.Lsh(z, test.shift)
   937  
   938  		if !isNormalized(z) {
   939  			t.Errorf("#%d: %v is not normalized", i, *z)
   940  		}
   941  		if z.Cmp(expected) != 0 {
   942  			t.Errorf("#%d: got %s want %s", i, z, expected)
   943  		}
   944  	}
   945  }
   946  
   947  func TestLshRsh(t *testing.T) {
   948  	for i, test := range rshTests {
   949  		in, _ := new(Int).SetString(test.in, 10)
   950  		out := new(Int).Lsh(in, test.shift)
   951  		out = out.Rsh(out, test.shift)
   952  
   953  		if !isNormalized(out) {
   954  			t.Errorf("#%d: %v is not normalized", i, *out)
   955  		}
   956  		if in.Cmp(out) != 0 {
   957  			t.Errorf("#%d: got %s want %s", i, out, in)
   958  		}
   959  	}
   960  	for i, test := range lshTests {
   961  		in, _ := new(Int).SetString(test.in, 10)
   962  		out := new(Int).Lsh(in, test.shift)
   963  		out.Rsh(out, test.shift)
   964  
   965  		if !isNormalized(out) {
   966  			t.Errorf("#%d: %v is not normalized", i, *out)
   967  		}
   968  		if in.Cmp(out) != 0 {
   969  			t.Errorf("#%d: got %s want %s", i, out, in)
   970  		}
   971  	}
   972  }
   973  
   974  // Entries must be sorted by value in ascending order.
   975  var cmpAbsTests = []string{
   976  	"0",
   977  	"1",
   978  	"2",
   979  	"10",
   980  	"10000000",
   981  	"2783678367462374683678456387645876387564783686583485",
   982  	"2783678367462374683678456387645876387564783686583486",
   983  	"32957394867987420967976567076075976570670947609750670956097509670576075067076027578341538",
   984  }
   985  
   986  func TestCmpAbs(t *testing.T) {
   987  	values := make([]*Int, len(cmpAbsTests))
   988  	var prev *Int
   989  	for i, s := range cmpAbsTests {
   990  		x, ok := new(Int).SetString(s, 0)
   991  		if !ok {
   992  			t.Fatalf("SetString(%s, 0) failed", s)
   993  		}
   994  		if prev != nil && prev.Cmp(x) >= 0 {
   995  			t.Fatal("cmpAbsTests entries not sorted in ascending order")
   996  		}
   997  		values[i] = x
   998  		prev = x
   999  	}
  1000  
  1001  	for i, x := range values {
  1002  		for j, y := range values {
  1003  			// try all combinations of signs for x, y
  1004  			for k := 0; k < 4; k++ {
  1005  				var a, b Int
  1006  				a.Set(x)
  1007  				b.Set(y)
  1008  				if k&1 != 0 {
  1009  					a.Neg(&a)
  1010  				}
  1011  				if k&2 != 0 {
  1012  					b.Neg(&b)
  1013  				}
  1014  
  1015  				got := a.CmpAbs(&b)
  1016  				want := 0
  1017  				switch {
  1018  				case i > j:
  1019  					want = 1
  1020  				case i < j:
  1021  					want = -1
  1022  				}
  1023  				if got != want {
  1024  					t.Errorf("absCmp |%s|, |%s|: got %d; want %d", &a, &b, got, want)
  1025  				}
  1026  			}
  1027  		}
  1028  	}
  1029  }
  1030  
  1031  var int64Tests = []string{
  1032  	// int64
  1033  	"0",
  1034  	"1",
  1035  	"-1",
  1036  	"4294967295",
  1037  	"-4294967295",
  1038  	"4294967296",
  1039  	"-4294967296",
  1040  	"9223372036854775807",
  1041  	"-9223372036854775807",
  1042  	"-9223372036854775808",
  1043  
  1044  	// not int64
  1045  	"0x8000000000000000",
  1046  	"-0x8000000000000001",
  1047  	"38579843757496759476987459679745",
  1048  	"-38579843757496759476987459679745",
  1049  }
  1050  
  1051  func TestInt64(t *testing.T) {
  1052  	for _, s := range int64Tests {
  1053  		var x Int
  1054  		_, ok := x.SetString(s, 0)
  1055  		if !ok {
  1056  			t.Errorf("SetString(%s, 0) failed", s)
  1057  			continue
  1058  		}
  1059  
  1060  		want, err := strconv.ParseInt(s, 0, 64)
  1061  		if err != nil {
  1062  			if err.(*strconv.NumError).Err == strconv.ErrRange {
  1063  				if x.IsInt64() {
  1064  					t.Errorf("IsInt64(%s) succeeded unexpectedly", s)
  1065  				}
  1066  			} else {
  1067  				t.Errorf("ParseInt(%s) failed", s)
  1068  			}
  1069  			continue
  1070  		}
  1071  
  1072  		if !x.IsInt64() {
  1073  			t.Errorf("IsInt64(%s) failed unexpectedly", s)
  1074  		}
  1075  
  1076  		got := x.Int64()
  1077  		if got != want {
  1078  			t.Errorf("Int64(%s) = %d; want %d", s, got, want)
  1079  		}
  1080  	}
  1081  }
  1082  
  1083  var uint64Tests = []string{
  1084  	// uint64
  1085  	"0",
  1086  	"1",
  1087  	"4294967295",
  1088  	"4294967296",
  1089  	"8589934591",
  1090  	"8589934592",
  1091  	"9223372036854775807",
  1092  	"9223372036854775808",
  1093  	"0x08000000000000000",
  1094  
  1095  	// not uint64
  1096  	"0x10000000000000000",
  1097  	"-0x08000000000000000",
  1098  	"-1",
  1099  }
  1100  
  1101  func TestUint64(t *testing.T) {
  1102  	for _, s := range uint64Tests {
  1103  		var x Int
  1104  		_, ok := x.SetString(s, 0)
  1105  		if !ok {
  1106  			t.Errorf("SetString(%s, 0) failed", s)
  1107  			continue
  1108  		}
  1109  
  1110  		want, err := strconv.ParseUint(s, 0, 64)
  1111  		if err != nil {
  1112  			// check for sign explicitly (ErrRange doesn't cover signed input)
  1113  			if s[0] == '-' || err.(*strconv.NumError).Err == strconv.ErrRange {
  1114  				if x.IsUint64() {
  1115  					t.Errorf("IsUint64(%s) succeeded unexpectedly", s)
  1116  				}
  1117  			} else {
  1118  				t.Errorf("ParseUint(%s) failed", s)
  1119  			}
  1120  			continue
  1121  		}
  1122  
  1123  		if !x.IsUint64() {
  1124  			t.Errorf("IsUint64(%s) failed unexpectedly", s)
  1125  		}
  1126  
  1127  		got := x.Uint64()
  1128  		if got != want {
  1129  			t.Errorf("Uint64(%s) = %d; want %d", s, got, want)
  1130  		}
  1131  	}
  1132  }
  1133  
  1134  var bitwiseTests = []struct {
  1135  	x, y                 string
  1136  	and, or, xor, andNot string
  1137  }{
  1138  	{"0x00", "0x00", "0x00", "0x00", "0x00", "0x00"},
  1139  	{"0x00", "0x01", "0x00", "0x01", "0x01", "0x00"},
  1140  	{"0x01", "0x00", "0x00", "0x01", "0x01", "0x01"},
  1141  	{"-0x01", "0x00", "0x00", "-0x01", "-0x01", "-0x01"},
  1142  	{"-0xaf", "-0x50", "-0xf0", "-0x0f", "0xe1", "0x41"},
  1143  	{"0x00", "-0x01", "0x00", "-0x01", "-0x01", "0x00"},
  1144  	{"0x01", "0x01", "0x01", "0x01", "0x00", "0x00"},
  1145  	{"-0x01", "-0x01", "-0x01", "-0x01", "0x00", "0x00"},
  1146  	{"0x07", "0x08", "0x00", "0x0f", "0x0f", "0x07"},
  1147  	{"0x05", "0x0f", "0x05", "0x0f", "0x0a", "0x00"},
  1148  	{"0xff", "-0x0a", "0xf6", "-0x01", "-0xf7", "0x09"},
  1149  	{"0x013ff6", "0x9a4e", "0x1a46", "0x01bffe", "0x01a5b8", "0x0125b0"},
  1150  	{"-0x013ff6", "0x9a4e", "0x800a", "-0x0125b2", "-0x01a5bc", "-0x01c000"},
  1151  	{"-0x013ff6", "-0x9a4e", "-0x01bffe", "-0x1a46", "0x01a5b8", "0x8008"},
  1152  	{
  1153  		"0x1000009dc6e3d9822cba04129bcbe3401",
  1154  		"0xb9bd7d543685789d57cb918e833af352559021483cdb05cc21fd",
  1155  		"0x1000001186210100001000009048c2001",
  1156  		"0xb9bd7d543685789d57cb918e8bfeff7fddb2ebe87dfbbdfe35fd",
  1157  		"0xb9bd7d543685789d57ca918e8ae69d6fcdb2eae87df2b97215fc",
  1158  		"0x8c40c2d8822caa04120b8321400",
  1159  	},
  1160  	{
  1161  		"0x1000009dc6e3d9822cba04129bcbe3401",
  1162  		"-0xb9bd7d543685789d57cb918e833af352559021483cdb05cc21fd",
  1163  		"0x8c40c2d8822caa04120b8321401",
  1164  		"-0xb9bd7d543685789d57ca918e82229142459020483cd2014001fd",
  1165  		"-0xb9bd7d543685789d57ca918e8ae69d6fcdb2eae87df2b97215fe",
  1166  		"0x1000001186210100001000009048c2000",
  1167  	},
  1168  	{
  1169  		"-0x1000009dc6e3d9822cba04129bcbe3401",
  1170  		"-0xb9bd7d543685789d57cb918e833af352559021483cdb05cc21fd",
  1171  		"-0xb9bd7d543685789d57cb918e8bfeff7fddb2ebe87dfbbdfe35fd",
  1172  		"-0x1000001186210100001000009048c2001",
  1173  		"0xb9bd7d543685789d57ca918e8ae69d6fcdb2eae87df2b97215fc",
  1174  		"0xb9bd7d543685789d57ca918e82229142459020483cd2014001fc",
  1175  	},
  1176  }
  1177  
  1178  type bitFun func(z, x, y *Int) *Int
  1179  
  1180  func testBitFun(t *testing.T, msg string, f bitFun, x, y *Int, exp string) {
  1181  	expected := new(Int)
  1182  	expected.SetString(exp, 0)
  1183  
  1184  	out := f(new(Int), x, y)
  1185  	if out.Cmp(expected) != 0 {
  1186  		t.Errorf("%s: got %s want %s", msg, out, expected)
  1187  	}
  1188  }
  1189  
  1190  func testBitFunSelf(t *testing.T, msg string, f bitFun, x, y *Int, exp string) {
  1191  	self := new(Int)
  1192  	self.Set(x)
  1193  	expected := new(Int)
  1194  	expected.SetString(exp, 0)
  1195  
  1196  	self = f(self, self, y)
  1197  	if self.Cmp(expected) != 0 {
  1198  		t.Errorf("%s: got %s want %s", msg, self, expected)
  1199  	}
  1200  }
  1201  
  1202  func altBit(x *Int, i int) uint {
  1203  	z := new(Int).Rsh(x, uint(i))
  1204  	z = z.And(z, NewInt(1))
  1205  	if z.Cmp(new(Int)) != 0 {
  1206  		return 1
  1207  	}
  1208  	return 0
  1209  }
  1210  
  1211  func altSetBit(z *Int, x *Int, i int, b uint) *Int {
  1212  	one := NewInt(1)
  1213  	m := one.Lsh(one, uint(i))
  1214  	switch b {
  1215  	case 1:
  1216  		return z.Or(x, m)
  1217  	case 0:
  1218  		return z.AndNot(x, m)
  1219  	}
  1220  	panic("set bit is not 0 or 1")
  1221  }
  1222  
  1223  func testBitset(t *testing.T, x *Int) {
  1224  	n := x.BitLen()
  1225  	z := new(Int).Set(x)
  1226  	z1 := new(Int).Set(x)
  1227  	for i := 0; i < n+10; i++ {
  1228  		old := z.Bit(i)
  1229  		old1 := altBit(z1, i)
  1230  		if old != old1 {
  1231  			t.Errorf("bitset: inconsistent value for Bit(%s, %d), got %v want %v", z1, i, old, old1)
  1232  		}
  1233  		z := new(Int).SetBit(z, i, 1)
  1234  		z1 := altSetBit(new(Int), z1, i, 1)
  1235  		if z.Bit(i) == 0 {
  1236  			t.Errorf("bitset: bit %d of %s got 0 want 1", i, x)
  1237  		}
  1238  		if z.Cmp(z1) != 0 {
  1239  			t.Errorf("bitset: inconsistent value after SetBit 1, got %s want %s", z, z1)
  1240  		}
  1241  		z.SetBit(z, i, 0)
  1242  		altSetBit(z1, z1, i, 0)
  1243  		if z.Bit(i) != 0 {
  1244  			t.Errorf("bitset: bit %d of %s got 1 want 0", i, x)
  1245  		}
  1246  		if z.Cmp(z1) != 0 {
  1247  			t.Errorf("bitset: inconsistent value after SetBit 0, got %s want %s", z, z1)
  1248  		}
  1249  		altSetBit(z1, z1, i, old)
  1250  		z.SetBit(z, i, old)
  1251  		if z.Cmp(z1) != 0 {
  1252  			t.Errorf("bitset: inconsistent value after SetBit old, got %s want %s", z, z1)
  1253  		}
  1254  	}
  1255  	if z.Cmp(x) != 0 {
  1256  		t.Errorf("bitset: got %s want %s", z, x)
  1257  	}
  1258  }
  1259  
  1260  var bitsetTests = []struct {
  1261  	x string
  1262  	i int
  1263  	b uint
  1264  }{
  1265  	{"0", 0, 0},
  1266  	{"0", 200, 0},
  1267  	{"1", 0, 1},
  1268  	{"1", 1, 0},
  1269  	{"-1", 0, 1},
  1270  	{"-1", 200, 1},
  1271  	{"0x2000000000000000000000000000", 108, 0},
  1272  	{"0x2000000000000000000000000000", 109, 1},
  1273  	{"0x2000000000000000000000000000", 110, 0},
  1274  	{"-0x2000000000000000000000000001", 108, 1},
  1275  	{"-0x2000000000000000000000000001", 109, 0},
  1276  	{"-0x2000000000000000000000000001", 110, 1},
  1277  }
  1278  
  1279  func TestBitSet(t *testing.T) {
  1280  	for _, test := range bitwiseTests {
  1281  		x := new(Int)
  1282  		x.SetString(test.x, 0)
  1283  		testBitset(t, x)
  1284  		x = new(Int)
  1285  		x.SetString(test.y, 0)
  1286  		testBitset(t, x)
  1287  	}
  1288  	for i, test := range bitsetTests {
  1289  		x := new(Int)
  1290  		x.SetString(test.x, 0)
  1291  		b := x.Bit(test.i)
  1292  		if b != test.b {
  1293  			t.Errorf("#%d got %v want %v", i, b, test.b)
  1294  		}
  1295  	}
  1296  	z := NewInt(1)
  1297  	z.SetBit(NewInt(0), 2, 1)
  1298  	if z.Cmp(NewInt(4)) != 0 {
  1299  		t.Errorf("destination leaked into result; got %s want 4", z)
  1300  	}
  1301  }
  1302  
  1303  func BenchmarkBitset(b *testing.B) {
  1304  	z := new(Int)
  1305  	z.SetBit(z, 512, 1)
  1306  	b.ResetTimer()
  1307  	b.StartTimer()
  1308  	for i := b.N - 1; i >= 0; i-- {
  1309  		z.SetBit(z, i&512, 1)
  1310  	}
  1311  }
  1312  
  1313  func BenchmarkBitsetNeg(b *testing.B) {
  1314  	z := NewInt(-1)
  1315  	z.SetBit(z, 512, 0)
  1316  	b.ResetTimer()
  1317  	b.StartTimer()
  1318  	for i := b.N - 1; i >= 0; i-- {
  1319  		z.SetBit(z, i&512, 0)
  1320  	}
  1321  }
  1322  
  1323  func BenchmarkBitsetOrig(b *testing.B) {
  1324  	z := new(Int)
  1325  	altSetBit(z, z, 512, 1)
  1326  	b.ResetTimer()
  1327  	b.StartTimer()
  1328  	for i := b.N - 1; i >= 0; i-- {
  1329  		altSetBit(z, z, i&512, 1)
  1330  	}
  1331  }
  1332  
  1333  func BenchmarkBitsetNegOrig(b *testing.B) {
  1334  	z := NewInt(-1)
  1335  	altSetBit(z, z, 512, 0)
  1336  	b.ResetTimer()
  1337  	b.StartTimer()
  1338  	for i := b.N - 1; i >= 0; i-- {
  1339  		altSetBit(z, z, i&512, 0)
  1340  	}
  1341  }
  1342  
  1343  // tri generates the trinomial 2**(n*2) - 2**n - 1, which is always 3 mod 4 and
  1344  // 7 mod 8, so that 2 is always a quadratic residue.
  1345  func tri(n uint) *Int {
  1346  	x := NewInt(1)
  1347  	x.Lsh(x, n)
  1348  	x2 := new(Int).Lsh(x, n)
  1349  	x2.Sub(x2, x)
  1350  	x2.Sub(x2, intOne)
  1351  	return x2
  1352  }
  1353  
  1354  func BenchmarkModSqrt225_Tonelli(b *testing.B) {
  1355  	p := tri(225)
  1356  	x := NewInt(2)
  1357  	for i := 0; i < b.N; i++ {
  1358  		x.SetUint64(2)
  1359  		x.modSqrtTonelliShanks(x, p)
  1360  	}
  1361  }
  1362  
  1363  func BenchmarkModSqrt225_3Mod4(b *testing.B) {
  1364  	p := tri(225)
  1365  	x := new(Int).SetUint64(2)
  1366  	for i := 0; i < b.N; i++ {
  1367  		x.SetUint64(2)
  1368  		x.modSqrt3Mod4Prime(x, p)
  1369  	}
  1370  }
  1371  
  1372  func BenchmarkModSqrt231_Tonelli(b *testing.B) {
  1373  	p := tri(231)
  1374  	p.Sub(p, intOne)
  1375  	p.Sub(p, intOne) // tri(231) - 2 is a prime == 5 mod 8
  1376  	x := new(Int).SetUint64(7)
  1377  	for i := 0; i < b.N; i++ {
  1378  		x.SetUint64(7)
  1379  		x.modSqrtTonelliShanks(x, p)
  1380  	}
  1381  }
  1382  
  1383  func BenchmarkModSqrt231_5Mod8(b *testing.B) {
  1384  	p := tri(231)
  1385  	p.Sub(p, intOne)
  1386  	p.Sub(p, intOne) // tri(231) - 2 is a prime == 5 mod 8
  1387  	x := new(Int).SetUint64(7)
  1388  	for i := 0; i < b.N; i++ {
  1389  		x.SetUint64(7)
  1390  		x.modSqrt5Mod8Prime(x, p)
  1391  	}
  1392  }
  1393  
  1394  func TestBitwise(t *testing.T) {
  1395  	x := new(Int)
  1396  	y := new(Int)
  1397  	for _, test := range bitwiseTests {
  1398  		x.SetString(test.x, 0)
  1399  		y.SetString(test.y, 0)
  1400  
  1401  		testBitFun(t, "and", (*Int).And, x, y, test.and)
  1402  		testBitFunSelf(t, "and", (*Int).And, x, y, test.and)
  1403  		testBitFun(t, "andNot", (*Int).AndNot, x, y, test.andNot)
  1404  		testBitFunSelf(t, "andNot", (*Int).AndNot, x, y, test.andNot)
  1405  		testBitFun(t, "or", (*Int).Or, x, y, test.or)
  1406  		testBitFunSelf(t, "or", (*Int).Or, x, y, test.or)
  1407  		testBitFun(t, "xor", (*Int).Xor, x, y, test.xor)
  1408  		testBitFunSelf(t, "xor", (*Int).Xor, x, y, test.xor)
  1409  	}
  1410  }
  1411  
  1412  var notTests = []struct {
  1413  	in  string
  1414  	out string
  1415  }{
  1416  	{"0", "-1"},
  1417  	{"1", "-2"},
  1418  	{"7", "-8"},
  1419  	{"0", "-1"},
  1420  	{"-81910", "81909"},
  1421  	{
  1422  		"298472983472983471903246121093472394872319615612417471234712061",
  1423  		"-298472983472983471903246121093472394872319615612417471234712062",
  1424  	},
  1425  }
  1426  
  1427  func TestNot(t *testing.T) {
  1428  	in := new(Int)
  1429  	out := new(Int)
  1430  	expected := new(Int)
  1431  	for i, test := range notTests {
  1432  		in.SetString(test.in, 10)
  1433  		expected.SetString(test.out, 10)
  1434  		out = out.Not(in)
  1435  		if out.Cmp(expected) != 0 {
  1436  			t.Errorf("#%d: got %s want %s", i, out, expected)
  1437  		}
  1438  		out = out.Not(out)
  1439  		if out.Cmp(in) != 0 {
  1440  			t.Errorf("#%d: got %s want %s", i, out, in)
  1441  		}
  1442  	}
  1443  }
  1444  
  1445  var modInverseTests = []struct {
  1446  	element string
  1447  	modulus string
  1448  }{
  1449  	{"1234567", "458948883992"},
  1450  	{"239487239847", "2410312426921032588552076022197566074856950548502459942654116941958108831682612228890093858261341614673227141477904012196503648957050582631942730706805009223062734745341073406696246014589361659774041027169249453200378729434170325843778659198143763193776859869524088940195577346119843545301547043747207749969763750084308926339295559968882457872412993810129130294592999947926365264059284647209730384947211681434464714438488520940127459844288859336526896320919633919"},
  1451  	{"-10", "13"}, // issue #16984
  1452  	{"10", "-13"},
  1453  	{"-17", "-13"},
  1454  }
  1455  
  1456  func TestModInverse(t *testing.T) {
  1457  	var element, modulus, gcd, inverse Int
  1458  	one := NewInt(1)
  1459  	for _, test := range modInverseTests {
  1460  		(&element).SetString(test.element, 10)
  1461  		(&modulus).SetString(test.modulus, 10)
  1462  		(&inverse).ModInverse(&element, &modulus)
  1463  		(&inverse).Mul(&inverse, &element)
  1464  		(&inverse).Mod(&inverse, &modulus)
  1465  		if (&inverse).Cmp(one) != 0 {
  1466  			t.Errorf("ModInverse(%d,%d)*%d%%%d=%d, not 1", &element, &modulus, &element, &modulus, &inverse)
  1467  		}
  1468  	}
  1469  	// exhaustive test for small values
  1470  	for n := 2; n < 100; n++ {
  1471  		(&modulus).SetInt64(int64(n))
  1472  		for x := 1; x < n; x++ {
  1473  			(&element).SetInt64(int64(x))
  1474  			(&gcd).GCD(nil, nil, &element, &modulus)
  1475  			if (&gcd).Cmp(one) != 0 {
  1476  				continue
  1477  			}
  1478  			(&inverse).ModInverse(&element, &modulus)
  1479  			(&inverse).Mul(&inverse, &element)
  1480  			(&inverse).Mod(&inverse, &modulus)
  1481  			if (&inverse).Cmp(one) != 0 {
  1482  				t.Errorf("ModInverse(%d,%d)*%d%%%d=%d, not 1", &element, &modulus, &element, &modulus, &inverse)
  1483  			}
  1484  		}
  1485  	}
  1486  }
  1487  
  1488  func BenchmarkModInverse(b *testing.B) {
  1489  	p := new(Int).SetInt64(1) // Mersenne prime 2**1279 -1
  1490  	p.abs = p.abs.shl(p.abs, 1279)
  1491  	p.Sub(p, intOne)
  1492  	x := new(Int).Sub(p, intOne)
  1493  	z := new(Int)
  1494  	for i := 0; i < b.N; i++ {
  1495  		z.ModInverse(x, p)
  1496  	}
  1497  }
  1498  
  1499  // testModSqrt is a helper for TestModSqrt,
  1500  // which checks that ModSqrt can compute a square-root of elt^2.
  1501  func testModSqrt(t *testing.T, elt, mod, sq, sqrt *Int) bool {
  1502  	var sqChk, sqrtChk, sqrtsq Int
  1503  	sq.Mul(elt, elt)
  1504  	sq.Mod(sq, mod)
  1505  	z := sqrt.ModSqrt(sq, mod)
  1506  	if z != sqrt {
  1507  		t.Errorf("ModSqrt returned wrong value %s", z)
  1508  	}
  1509  
  1510  	// test ModSqrt arguments outside the range [0,mod)
  1511  	sqChk.Add(sq, mod)
  1512  	z = sqrtChk.ModSqrt(&sqChk, mod)
  1513  	if z != &sqrtChk || z.Cmp(sqrt) != 0 {
  1514  		t.Errorf("ModSqrt returned inconsistent value %s", z)
  1515  	}
  1516  	sqChk.Sub(sq, mod)
  1517  	z = sqrtChk.ModSqrt(&sqChk, mod)
  1518  	if z != &sqrtChk || z.Cmp(sqrt) != 0 {
  1519  		t.Errorf("ModSqrt returned inconsistent value %s", z)
  1520  	}
  1521  
  1522  	// test x aliasing z
  1523  	z = sqrtChk.ModSqrt(sqrtChk.Set(sq), mod)
  1524  	if z != &sqrtChk || z.Cmp(sqrt) != 0 {
  1525  		t.Errorf("ModSqrt returned inconsistent value %s", z)
  1526  	}
  1527  
  1528  	// make sure we actually got a square root
  1529  	if sqrt.Cmp(elt) == 0 {
  1530  		return true // we found the "desired" square root
  1531  	}
  1532  	sqrtsq.Mul(sqrt, sqrt) // make sure we found the "other" one
  1533  	sqrtsq.Mod(&sqrtsq, mod)
  1534  	return sq.Cmp(&sqrtsq) == 0
  1535  }
  1536  
  1537  func TestModSqrt(t *testing.T) {
  1538  	var elt, mod, modx4, sq, sqrt Int
  1539  	r := rand.New(rand.NewSource(9))
  1540  	for i, s := range primes[1:] { // skip 2, use only odd primes
  1541  		mod.SetString(s, 10)
  1542  		modx4.Lsh(&mod, 2)
  1543  
  1544  		// test a few random elements per prime
  1545  		for x := 1; x < 5; x++ {
  1546  			elt.Rand(r, &modx4)
  1547  			elt.Sub(&elt, &mod) // test range [-mod, 3*mod)
  1548  			if !testModSqrt(t, &elt, &mod, &sq, &sqrt) {
  1549  				t.Errorf("#%d: failed (sqrt(e) = %s)", i, &sqrt)
  1550  			}
  1551  		}
  1552  
  1553  		if testing.Short() && i > 2 {
  1554  			break
  1555  		}
  1556  	}
  1557  
  1558  	if testing.Short() {
  1559  		return
  1560  	}
  1561  
  1562  	// exhaustive test for small values
  1563  	for n := 3; n < 100; n++ {
  1564  		mod.SetInt64(int64(n))
  1565  		if !mod.ProbablyPrime(10) {
  1566  			continue
  1567  		}
  1568  		isSquare := make([]bool, n)
  1569  
  1570  		// test all the squares
  1571  		for x := 1; x < n; x++ {
  1572  			elt.SetInt64(int64(x))
  1573  			if !testModSqrt(t, &elt, &mod, &sq, &sqrt) {
  1574  				t.Errorf("#%d: failed (sqrt(%d,%d) = %s)", x, &elt, &mod, &sqrt)
  1575  			}
  1576  			isSquare[sq.Uint64()] = true
  1577  		}
  1578  
  1579  		// test all non-squares
  1580  		for x := 1; x < n; x++ {
  1581  			sq.SetInt64(int64(x))
  1582  			z := sqrt.ModSqrt(&sq, &mod)
  1583  			if !isSquare[x] && z != nil {
  1584  				t.Errorf("#%d: failed (sqrt(%d,%d) = nil)", x, &sqrt, &mod)
  1585  			}
  1586  		}
  1587  	}
  1588  }
  1589  
  1590  func TestJacobi(t *testing.T) {
  1591  	testCases := []struct {
  1592  		x, y   int64
  1593  		result int
  1594  	}{
  1595  		{0, 1, 1},
  1596  		{0, -1, 1},
  1597  		{1, 1, 1},
  1598  		{1, -1, 1},
  1599  		{0, 5, 0},
  1600  		{1, 5, 1},
  1601  		{2, 5, -1},
  1602  		{-2, 5, -1},
  1603  		{2, -5, -1},
  1604  		{-2, -5, 1},
  1605  		{3, 5, -1},
  1606  		{5, 5, 0},
  1607  		{-5, 5, 0},
  1608  		{6, 5, 1},
  1609  		{6, -5, 1},
  1610  		{-6, 5, 1},
  1611  		{-6, -5, -1},
  1612  	}
  1613  
  1614  	var x, y Int
  1615  
  1616  	for i, test := range testCases {
  1617  		x.SetInt64(test.x)
  1618  		y.SetInt64(test.y)
  1619  		expected := test.result
  1620  		actual := Jacobi(&x, &y)
  1621  		if actual != expected {
  1622  			t.Errorf("#%d: Jacobi(%d, %d) = %d, but expected %d", i, test.x, test.y, actual, expected)
  1623  		}
  1624  	}
  1625  }
  1626  
  1627  func TestJacobiPanic(t *testing.T) {
  1628  	const failureMsg = "test failure"
  1629  	defer func() {
  1630  		msg := recover()
  1631  		if msg == nil || msg == failureMsg {
  1632  			panic(msg)
  1633  		}
  1634  		t.Log(msg)
  1635  	}()
  1636  	x := NewInt(1)
  1637  	y := NewInt(2)
  1638  	// Jacobi should panic when the second argument is even.
  1639  	Jacobi(x, y)
  1640  	panic(failureMsg)
  1641  }
  1642  
  1643  func TestIssue2607(t *testing.T) {
  1644  	// This code sequence used to hang.
  1645  	n := NewInt(10)
  1646  	n.Rand(rand.New(rand.NewSource(9)), n)
  1647  }
  1648  
  1649  func TestSqrt(t *testing.T) {
  1650  	root := 0
  1651  	r := new(Int)
  1652  	for i := 0; i < 10000; i++ {
  1653  		if (root+1)*(root+1) <= i {
  1654  			root++
  1655  		}
  1656  		n := NewInt(int64(i))
  1657  		r.SetInt64(-2)
  1658  		r.Sqrt(n)
  1659  		if r.Cmp(NewInt(int64(root))) != 0 {
  1660  			t.Errorf("Sqrt(%v) = %v, want %v", n, r, root)
  1661  		}
  1662  	}
  1663  
  1664  	for i := 0; i < 1000; i += 10 {
  1665  		n, _ := new(Int).SetString("1"+strings.Repeat("0", i), 10)
  1666  		r := new(Int).Sqrt(n)
  1667  		root, _ := new(Int).SetString("1"+strings.Repeat("0", i/2), 10)
  1668  		if r.Cmp(root) != 0 {
  1669  			t.Errorf("Sqrt(1e%d) = %v, want 1e%d", i, r, i/2)
  1670  		}
  1671  	}
  1672  
  1673  	// Test aliasing.
  1674  	r.SetInt64(100)
  1675  	r.Sqrt(r)
  1676  	if r.Int64() != 10 {
  1677  		t.Errorf("Sqrt(100) = %v, want 10 (aliased output)", r.Int64())
  1678  	}
  1679  }
  1680  
  1681  // We can't test this together with the other Exp tests above because
  1682  // it requires a different receiver setup.
  1683  func TestIssue22830(t *testing.T) {
  1684  	one := new(Int).SetInt64(1)
  1685  	base, _ := new(Int).SetString("84555555300000000000", 10)
  1686  	mod, _ := new(Int).SetString("66666670001111111111", 10)
  1687  	want, _ := new(Int).SetString("17888885298888888889", 10)
  1688  
  1689  	var tests = []int64{
  1690  		0, 1, -1,
  1691  	}
  1692  
  1693  	for _, n := range tests {
  1694  		m := NewInt(n)
  1695  		if got := m.Exp(base, one, mod); got.Cmp(want) != 0 {
  1696  			t.Errorf("(%v).Exp(%s, 1, %s) = %s, want %s", n, base, mod, got, want)
  1697  		}
  1698  	}
  1699  }
  1700  
  1701  func BenchmarkSqrt(b *testing.B) {
  1702  	n, _ := new(Int).SetString("1"+strings.Repeat("0", 1001), 10)
  1703  	b.ResetTimer()
  1704  	t := new(Int)
  1705  	for i := 0; i < b.N; i++ {
  1706  		t.Sqrt(n)
  1707  	}
  1708  }
  1709  
  1710  func benchmarkIntSqr(b *testing.B, nwords int) {
  1711  	x := new(Int)
  1712  	x.abs = rndNat(nwords)
  1713  	t := new(Int)
  1714  	b.ResetTimer()
  1715  	for i := 0; i < b.N; i++ {
  1716  		t.Mul(x, x)
  1717  	}
  1718  }
  1719  
  1720  func BenchmarkIntSqr(b *testing.B) {
  1721  	for _, n := range sqrBenchSizes {
  1722  		if isRaceBuilder && n > 1e3 {
  1723  			continue
  1724  		}
  1725  		b.Run(fmt.Sprintf("%d", n), func(b *testing.B) {
  1726  			benchmarkIntSqr(b, n)
  1727  		})
  1728  	}
  1729  }
  1730  

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