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  	{"1234", "-1", "0", "1"},
   537  	{"17", "-100", "1234", "865"},
   538  	{"2", "-100", "1234", ""},
   539  
   540  	// m == 1
   541  	{"0", "0", "1", "0"},
   542  	{"1", "0", "1", "0"},
   543  	{"-10", "0", "1", "0"},
   544  	{"1234", "-1", "1", "0"},
   545  
   546  	// misc
   547  	{"5", "1", "3", "2"},
   548  	{"5", "-7", "", "1"},
   549  	{"-5", "-7", "", "1"},
   550  	{"5", "0", "", "1"},
   551  	{"-5", "0", "", "1"},
   552  	{"5", "1", "", "5"},
   553  	{"-5", "1", "", "-5"},
   554  	{"-5", "1", "7", "2"},
   555  	{"-2", "3", "2", "0"},
   556  	{"5", "2", "", "25"},
   557  	{"1", "65537", "2", "1"},
   558  	{"0x8000000000000000", "2", "", "0x40000000000000000000000000000000"},
   559  	{"0x8000000000000000", "2", "6719", "4944"},
   560  	{"0x8000000000000000", "3", "6719", "5447"},
   561  	{"0x8000000000000000", "1000", "6719", "1603"},
   562  	{"0x8000000000000000", "1000000", "6719", "3199"},
   563  	{"0x8000000000000000", "-1000000", "6719", "3663"}, // 3663 = ModInverse(3199, 6719) Issue #25865
   564  
   565  	{"0xffffffffffffffffffffffffffffffff", "0x12345678123456781234567812345678123456789", "0x01112222333344445555666677778889", "0x36168FA1DB3AAE6C8CE647E137F97A"},
   566  
   567  	{
   568  		"2938462938472983472983659726349017249287491026512746239764525612965293865296239471239874193284792387498274256129746192347",
   569  		"298472983472983471903246121093472394872319615612417471234712061",
   570  		"29834729834729834729347290846729561262544958723956495615629569234729836259263598127342374289365912465901365498236492183464",
   571  		"23537740700184054162508175125554701713153216681790245129157191391322321508055833908509185839069455749219131480588829346291",
   572  	},
   573  	// test case for issue 8822
   574  	{
   575  		"11001289118363089646017359372117963499250546375269047542777928006103246876688756735760905680604646624353196869572752623285140408755420374049317646428185270079555372763503115646054602867593662923894140940837479507194934267532831694565516466765025434902348314525627418515646588160955862839022051353653052947073136084780742729727874803457643848197499548297570026926927502505634297079527299004267769780768565695459945235586892627059178884998772989397505061206395455591503771677500931269477503508150175717121828518985901959919560700853226255420793148986854391552859459511723547532575574664944815966793196961286234040892865",
   576  		"0xB08FFB20760FFED58FADA86DFEF71AD72AA0FA763219618FE022C197E54708BB1191C66470250FCE8879487507CEE41381CA4D932F81C2B3F1AB20B539D50DCD",
   577  		"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
   578  		"21484252197776302499639938883777710321993113097987201050501182909581359357618579566746556372589385361683610524730509041328855066514963385522570894839035884713051640171474186548713546686476761306436434146475140156284389181808675016576845833340494848283681088886584219750554408060556769486628029028720727393293111678826356480455433909233520504112074401376133077150471237549474149190242010469539006449596611576612573955754349042329130631128234637924786466585703488460540228477440853493392086251021228087076124706778899179648655221663765993962724699135217212118535057766739392069738618682722216712319320435674779146070442",
   579  	},
   580  	{
   581  		"-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
   582  		"0xB08FFB20760FFED58FADA86DFEF71AD72AA0FA763219618FE022C197E54708BB1191C66470250FCE8879487507CEE41381CA4D932F81C2B3F1AB20B539D50DCD",
   583  		"0xAC6BDB41324A9A9BF166DE5E1389582FAF72B6651987EE07FC3192943DB56050A37329CBB4A099ED8193E0757767A13DD52312AB4B03310DCD7F48A9DA04FD50E8083969EDB767B0CF6095179A163AB3661A05FBD5FAAAE82918A9962F0B93B855F97993EC975EEAA80D740ADBF4FF747359D041D5C33EA71D281E446B14773BCA97B43A23FB801676BD207A436C6481F1D2B9078717461A5B9D32E688F87748544523B524B0D57D5EA77A2775D2ECFA032CFBDBF52FB3786160279004E57AE6AF874E7303CE53299CCC041C7BC308D82A5698F3A8D0C38271AE35F8E9DBFBB694B5C803D89F7AE435DE236D525F54759B65E372FCD68EF20FA7111F9E4AFF73",
   584  		"21484252197776302499639938883777710321993113097987201050501182909581359357618579566746556372589385361683610524730509041328855066514963385522570894839035884713051640171474186548713546686476761306436434146475140156284389181808675016576845833340494848283681088886584219750554408060556769486628029028720727393293111678826356480455433909233520504112074401376133077150471237549474149190242010469539006449596611576612573955754349042329130631128234637924786466585703488460540228477440853493392086251021228087076124706778899179648655221663765993962724699135217212118535057766739392069738618682722216712319320435674779146070442",
   585  	},
   586  
   587  	// test cases for issue 13907
   588  	{"0xffffffff00000001", "0xffffffff00000001", "0xffffffff00000001", "0"},
   589  	{"0xffffffffffffffff00000001", "0xffffffffffffffff00000001", "0xffffffffffffffff00000001", "0"},
   590  	{"0xffffffffffffffffffffffff00000001", "0xffffffffffffffffffffffff00000001", "0xffffffffffffffffffffffff00000001", "0"},
   591  	{"0xffffffffffffffffffffffffffffffff00000001", "0xffffffffffffffffffffffffffffffff00000001", "0xffffffffffffffffffffffffffffffff00000001", "0"},
   592  
   593  	{
   594  		"2",
   595  		"0xB08FFB20760FFED58FADA86DFEF71AD72AA0FA763219618FE022C197E54708BB1191C66470250FCE8879487507CEE41381CA4D932F81C2B3F1AB20B539D50DCD",
   596  		"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odd
   597  		"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
   598  	},
   599  	{
   600  		"2",
   601  		"0xB08FFB20760FFED58FADA86DFEF71AD72AA0FA763219618FE022C197E54708BB1191C66470250FCE8879487507CEE41381CA4D932F81C2B3F1AB20B539D50DCD",
   602  		"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even
   603  		"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
   604  	},
   605  }
   606  
   607  func TestExp(t *testing.T) {
   608  	for i, test := range expTests {
   609  		x, ok1 := new(Int).SetString(test.x, 0)
   610  		y, ok2 := new(Int).SetString(test.y, 0)
   611  
   612  		var ok3, ok4 bool
   613  		var out, m *Int
   614  
   615  		if len(test.out) == 0 {
   616  			out, ok3 = nil, true
   617  		} else {
   618  			out, ok3 = new(Int).SetString(test.out, 0)
   619  		}
   620  
   621  		if len(test.m) == 0 {
   622  			m, ok4 = nil, true
   623  		} else {
   624  			m, ok4 = new(Int).SetString(test.m, 0)
   625  		}
   626  
   627  		if !ok1 || !ok2 || !ok3 || !ok4 {
   628  			t.Errorf("#%d: error in input", i)
   629  			continue
   630  		}
   631  
   632  		z1 := new(Int).Exp(x, y, m)
   633  		if z1 != nil && !isNormalized(z1) {
   634  			t.Errorf("#%d: %v is not normalized", i, *z1)
   635  		}
   636  		if !(z1 == nil && out == nil || z1.Cmp(out) == 0) {
   637  			t.Errorf("#%d: got %x want %x", i, z1, out)
   638  		}
   639  
   640  		if m == nil {
   641  			// The result should be the same as for m == 0;
   642  			// specifically, there should be no div-zero panic.
   643  			m = &Int{abs: nat{}} // m != nil && len(m.abs) == 0
   644  			z2 := new(Int).Exp(x, y, m)
   645  			if z2.Cmp(z1) != 0 {
   646  				t.Errorf("#%d: got %x want %x", i, z2, z1)
   647  			}
   648  		}
   649  	}
   650  }
   651  
   652  func BenchmarkExp(b *testing.B) {
   653  	x, _ := new(Int).SetString("11001289118363089646017359372117963499250546375269047542777928006103246876688756735760905680604646624353196869572752623285140408755420374049317646428185270079555372763503115646054602867593662923894140940837479507194934267532831694565516466765025434902348314525627418515646588160955862839022051353653052947073136084780742729727874803457643848197499548297570026926927502505634297079527299004267769780768565695459945235586892627059178884998772989397505061206395455591503771677500931269477503508150175717121828518985901959919560700853226255420793148986854391552859459511723547532575574664944815966793196961286234040892865", 0)
   654  	y, _ := new(Int).SetString("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
   655  	n, _ := new(Int).SetString("0xAC6BDB41324A9A9BF166DE5E1389582FAF72B6651987EE07FC3192943DB56050A37329CBB4A099ED8193E0757767A13DD52312AB4B03310DCD7F48A9DA04FD50E8083969EDB767B0CF6095179A163AB3661A05FBD5FAAAE82918A9962F0B93B855F97993EC975EEAA80D740ADBF4FF747359D041D5C33EA71D281E446B14773BCA97B43A23FB801676BD207A436C6481F1D2B9078717461A5B9D32E688F87748544523B524B0D57D5EA77A2775D2ECFA032CFBDBF52FB3786160279004E57AE6AF874E7303CE53299CCC041C7BC308D82A5698F3A8D0C38271AE35F8E9DBFBB694B5C803D89F7AE435DE236D525F54759B65E372FCD68EF20FA7111F9E4AFF73", 0)
   656  	out := new(Int)
   657  	for i := 0; i < b.N; i++ {
   658  		out.Exp(x, y, n)
   659  	}
   660  }
   661  
   662  func BenchmarkExp2(b *testing.B) {
   663  	x, _ := new(Int).SetString("2", 0)
   664  	y, _ := new(Int).SetString("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
   665  	n, _ := new(Int).SetString("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
   666  	out := new(Int)
   667  	for i := 0; i < b.N; i++ {
   668  		out.Exp(x, y, n)
   669  	}
   670  }
   671  
   672  func checkGcd(aBytes, bBytes []byte) bool {
   673  	x := new(Int)
   674  	y := new(Int)
   675  	a := new(Int).SetBytes(aBytes)
   676  	b := new(Int).SetBytes(bBytes)
   677  
   678  	d := new(Int).GCD(x, y, a, b)
   679  	x.Mul(x, a)
   680  	y.Mul(y, b)
   681  	x.Add(x, y)
   682  
   683  	return x.Cmp(d) == 0
   684  }
   685  
   686  // euclidExtGCD is a reference implementation of Euclid's
   687  // extended GCD algorithm for testing against optimized algorithms.
   688  // Requirements: a, b > 0
   689  func euclidExtGCD(a, b *Int) (g, x, y *Int) {
   690  	A := new(Int).Set(a)
   691  	B := new(Int).Set(b)
   692  
   693  	// A = Ua*a + Va*b
   694  	// B = Ub*a + Vb*b
   695  	Ua := new(Int).SetInt64(1)
   696  	Va := new(Int)
   697  
   698  	Ub := new(Int)
   699  	Vb := new(Int).SetInt64(1)
   700  
   701  	q := new(Int)
   702  	temp := new(Int)
   703  
   704  	r := new(Int)
   705  	for len(B.abs) > 0 {
   706  		q, r = q.QuoRem(A, B, r)
   707  
   708  		A, B, r = B, r, A
   709  
   710  		// Ua, Ub = Ub, Ua-q*Ub
   711  		temp.Set(Ub)
   712  		Ub.Mul(Ub, q)
   713  		Ub.Sub(Ua, Ub)
   714  		Ua.Set(temp)
   715  
   716  		// Va, Vb = Vb, Va-q*Vb
   717  		temp.Set(Vb)
   718  		Vb.Mul(Vb, q)
   719  		Vb.Sub(Va, Vb)
   720  		Va.Set(temp)
   721  	}
   722  	return A, Ua, Va
   723  }
   724  
   725  func checkLehmerGcd(aBytes, bBytes []byte) bool {
   726  	a := new(Int).SetBytes(aBytes)
   727  	b := new(Int).SetBytes(bBytes)
   728  
   729  	if a.Sign() <= 0 || b.Sign() <= 0 {
   730  		return true // can only test positive arguments
   731  	}
   732  
   733  	d := new(Int).lehmerGCD(nil, nil, a, b)
   734  	d0, _, _ := euclidExtGCD(a, b)
   735  
   736  	return d.Cmp(d0) == 0
   737  }
   738  
   739  func checkLehmerExtGcd(aBytes, bBytes []byte) bool {
   740  	a := new(Int).SetBytes(aBytes)
   741  	b := new(Int).SetBytes(bBytes)
   742  	x := new(Int)
   743  	y := new(Int)
   744  
   745  	if a.Sign() <= 0 || b.Sign() <= 0 {
   746  		return true // can only test positive arguments
   747  	}
   748  
   749  	d := new(Int).lehmerGCD(x, y, a, b)
   750  	d0, x0, y0 := euclidExtGCD(a, b)
   751  
   752  	return d.Cmp(d0) == 0 && x.Cmp(x0) == 0 && y.Cmp(y0) == 0
   753  }
   754  
   755  var gcdTests = []struct {
   756  	d, x, y, a, b string
   757  }{
   758  	// a <= 0 || b <= 0
   759  	{"0", "0", "0", "0", "0"},
   760  	{"7", "0", "1", "0", "7"},
   761  	{"7", "0", "-1", "0", "-7"},
   762  	{"11", "1", "0", "11", "0"},
   763  	{"7", "-1", "-2", "-77", "35"},
   764  	{"935", "-3", "8", "64515", "24310"},
   765  	{"935", "-3", "-8", "64515", "-24310"},
   766  	{"935", "3", "-8", "-64515", "-24310"},
   767  
   768  	{"1", "-9", "47", "120", "23"},
   769  	{"7", "1", "-2", "77", "35"},
   770  	{"935", "-3", "8", "64515", "24310"},
   771  	{"935000000000000000", "-3", "8", "64515000000000000000", "24310000000000000000"},
   772  	{"1", "-221", "22059940471369027483332068679400581064239780177629666810348940098015901108344", "98920366548084643601728869055592650835572950932266967461790948584315647051443", "991"},
   773  }
   774  
   775  func testGcd(t *testing.T, d, x, y, a, b *Int) {
   776  	var X *Int
   777  	if x != nil {
   778  		X = new(Int)
   779  	}
   780  	var Y *Int
   781  	if y != nil {
   782  		Y = new(Int)
   783  	}
   784  
   785  	D := new(Int).GCD(X, Y, a, b)
   786  	if D.Cmp(d) != 0 {
   787  		t.Errorf("GCD(%s, %s, %s, %s): got d = %s, want %s", x, y, a, b, D, d)
   788  	}
   789  	if x != nil && X.Cmp(x) != 0 {
   790  		t.Errorf("GCD(%s, %s, %s, %s): got x = %s, want %s", x, y, a, b, X, x)
   791  	}
   792  	if y != nil && Y.Cmp(y) != 0 {
   793  		t.Errorf("GCD(%s, %s, %s, %s): got y = %s, want %s", x, y, a, b, Y, y)
   794  	}
   795  
   796  	// check results in presence of aliasing (issue #11284)
   797  	a2 := new(Int).Set(a)
   798  	b2 := new(Int).Set(b)
   799  	a2.GCD(X, Y, a2, b2) // result is same as 1st argument
   800  	if a2.Cmp(d) != 0 {
   801  		t.Errorf("aliased z = a GCD(%s, %s, %s, %s): got d = %s, want %s", x, y, a, b, a2, d)
   802  	}
   803  	if x != nil && X.Cmp(x) != 0 {
   804  		t.Errorf("aliased z = a GCD(%s, %s, %s, %s): got x = %s, want %s", x, y, a, b, X, x)
   805  	}
   806  	if y != nil && Y.Cmp(y) != 0 {
   807  		t.Errorf("aliased z = a GCD(%s, %s, %s, %s): got y = %s, want %s", x, y, a, b, Y, y)
   808  	}
   809  
   810  	a2 = new(Int).Set(a)
   811  	b2 = new(Int).Set(b)
   812  	b2.GCD(X, Y, a2, b2) // result is same as 2nd argument
   813  	if b2.Cmp(d) != 0 {
   814  		t.Errorf("aliased z = b GCD(%s, %s, %s, %s): got d = %s, want %s", x, y, a, b, b2, d)
   815  	}
   816  	if x != nil && X.Cmp(x) != 0 {
   817  		t.Errorf("aliased z = b GCD(%s, %s, %s, %s): got x = %s, want %s", x, y, a, b, X, x)
   818  	}
   819  	if y != nil && Y.Cmp(y) != 0 {
   820  		t.Errorf("aliased z = b GCD(%s, %s, %s, %s): got y = %s, want %s", x, y, a, b, Y, y)
   821  	}
   822  
   823  	a2 = new(Int).Set(a)
   824  	b2 = new(Int).Set(b)
   825  	D = new(Int).GCD(a2, b2, a2, b2) // x = a, y = b
   826  	if D.Cmp(d) != 0 {
   827  		t.Errorf("aliased x = a, y = b GCD(%s, %s, %s, %s): got d = %s, want %s", x, y, a, b, D, d)
   828  	}
   829  	if x != nil && a2.Cmp(x) != 0 {
   830  		t.Errorf("aliased x = a, y = b GCD(%s, %s, %s, %s): got x = %s, want %s", x, y, a, b, a2, x)
   831  	}
   832  	if y != nil && b2.Cmp(y) != 0 {
   833  		t.Errorf("aliased x = a, y = b GCD(%s, %s, %s, %s): got y = %s, want %s", x, y, a, b, b2, y)
   834  	}
   835  
   836  	a2 = new(Int).Set(a)
   837  	b2 = new(Int).Set(b)
   838  	D = new(Int).GCD(b2, a2, a2, b2) // x = b, y = a
   839  	if D.Cmp(d) != 0 {
   840  		t.Errorf("aliased x = b, y = a GCD(%s, %s, %s, %s): got d = %s, want %s", x, y, a, b, D, d)
   841  	}
   842  	if x != nil && b2.Cmp(x) != 0 {
   843  		t.Errorf("aliased x = b, y = a GCD(%s, %s, %s, %s): got x = %s, want %s", x, y, a, b, b2, x)
   844  	}
   845  	if y != nil && a2.Cmp(y) != 0 {
   846  		t.Errorf("aliased x = b, y = a GCD(%s, %s, %s, %s): got y = %s, want %s", x, y, a, b, a2, y)
   847  	}
   848  }
   849  
   850  func TestGcd(t *testing.T) {
   851  	for _, test := range gcdTests {
   852  		d, _ := new(Int).SetString(test.d, 0)
   853  		x, _ := new(Int).SetString(test.x, 0)
   854  		y, _ := new(Int).SetString(test.y, 0)
   855  		a, _ := new(Int).SetString(test.a, 0)
   856  		b, _ := new(Int).SetString(test.b, 0)
   857  
   858  		testGcd(t, d, nil, nil, a, b)
   859  		testGcd(t, d, x, nil, a, b)
   860  		testGcd(t, d, nil, y, a, b)
   861  		testGcd(t, d, x, y, a, b)
   862  	}
   863  
   864  	if err := quick.Check(checkGcd, nil); err != nil {
   865  		t.Error(err)
   866  	}
   867  
   868  	if err := quick.Check(checkLehmerGcd, nil); err != nil {
   869  		t.Error(err)
   870  	}
   871  
   872  	if err := quick.Check(checkLehmerExtGcd, nil); err != nil {
   873  		t.Error(err)
   874  	}
   875  }
   876  
   877  type intShiftTest struct {
   878  	in    string
   879  	shift uint
   880  	out   string
   881  }
   882  
   883  var rshTests = []intShiftTest{
   884  	{"0", 0, "0"},
   885  	{"-0", 0, "0"},
   886  	{"0", 1, "0"},
   887  	{"0", 2, "0"},
   888  	{"1", 0, "1"},
   889  	{"1", 1, "0"},
   890  	{"1", 2, "0"},
   891  	{"2", 0, "2"},
   892  	{"2", 1, "1"},
   893  	{"-1", 0, "-1"},
   894  	{"-1", 1, "-1"},
   895  	{"-1", 10, "-1"},
   896  	{"-100", 2, "-25"},
   897  	{"-100", 3, "-13"},
   898  	{"-100", 100, "-1"},
   899  	{"4294967296", 0, "4294967296"},
   900  	{"4294967296", 1, "2147483648"},
   901  	{"4294967296", 2, "1073741824"},
   902  	{"18446744073709551616", 0, "18446744073709551616"},
   903  	{"18446744073709551616", 1, "9223372036854775808"},
   904  	{"18446744073709551616", 2, "4611686018427387904"},
   905  	{"18446744073709551616", 64, "1"},
   906  	{"340282366920938463463374607431768211456", 64, "18446744073709551616"},
   907  	{"340282366920938463463374607431768211456", 128, "1"},
   908  }
   909  
   910  func TestRsh(t *testing.T) {
   911  	for i, test := range rshTests {
   912  		in, _ := new(Int).SetString(test.in, 10)
   913  		expected, _ := new(Int).SetString(test.out, 10)
   914  		out := new(Int).Rsh(in, test.shift)
   915  
   916  		if !isNormalized(out) {
   917  			t.Errorf("#%d: %v is not normalized", i, *out)
   918  		}
   919  		if out.Cmp(expected) != 0 {
   920  			t.Errorf("#%d: got %s want %s", i, out, expected)
   921  		}
   922  	}
   923  }
   924  
   925  func TestRshSelf(t *testing.T) {
   926  	for i, test := range rshTests {
   927  		z, _ := new(Int).SetString(test.in, 10)
   928  		expected, _ := new(Int).SetString(test.out, 10)
   929  		z.Rsh(z, test.shift)
   930  
   931  		if !isNormalized(z) {
   932  			t.Errorf("#%d: %v is not normalized", i, *z)
   933  		}
   934  		if z.Cmp(expected) != 0 {
   935  			t.Errorf("#%d: got %s want %s", i, z, expected)
   936  		}
   937  	}
   938  }
   939  
   940  var lshTests = []intShiftTest{
   941  	{"0", 0, "0"},
   942  	{"0", 1, "0"},
   943  	{"0", 2, "0"},
   944  	{"1", 0, "1"},
   945  	{"1", 1, "2"},
   946  	{"1", 2, "4"},
   947  	{"2", 0, "2"},
   948  	{"2", 1, "4"},
   949  	{"2", 2, "8"},
   950  	{"-87", 1, "-174"},
   951  	{"4294967296", 0, "4294967296"},
   952  	{"4294967296", 1, "8589934592"},
   953  	{"4294967296", 2, "17179869184"},
   954  	{"18446744073709551616", 0, "18446744073709551616"},
   955  	{"9223372036854775808", 1, "18446744073709551616"},
   956  	{"4611686018427387904", 2, "18446744073709551616"},
   957  	{"1", 64, "18446744073709551616"},
   958  	{"18446744073709551616", 64, "340282366920938463463374607431768211456"},
   959  	{"1", 128, "340282366920938463463374607431768211456"},
   960  }
   961  
   962  func TestLsh(t *testing.T) {
   963  	for i, test := range lshTests {
   964  		in, _ := new(Int).SetString(test.in, 10)
   965  		expected, _ := new(Int).SetString(test.out, 10)
   966  		out := new(Int).Lsh(in, test.shift)
   967  
   968  		if !isNormalized(out) {
   969  			t.Errorf("#%d: %v is not normalized", i, *out)
   970  		}
   971  		if out.Cmp(expected) != 0 {
   972  			t.Errorf("#%d: got %s want %s", i, out, expected)
   973  		}
   974  	}
   975  }
   976  
   977  func TestLshSelf(t *testing.T) {
   978  	for i, test := range lshTests {
   979  		z, _ := new(Int).SetString(test.in, 10)
   980  		expected, _ := new(Int).SetString(test.out, 10)
   981  		z.Lsh(z, test.shift)
   982  
   983  		if !isNormalized(z) {
   984  			t.Errorf("#%d: %v is not normalized", i, *z)
   985  		}
   986  		if z.Cmp(expected) != 0 {
   987  			t.Errorf("#%d: got %s want %s", i, z, expected)
   988  		}
   989  	}
   990  }
   991  
   992  func TestLshRsh(t *testing.T) {
   993  	for i, test := range rshTests {
   994  		in, _ := new(Int).SetString(test.in, 10)
   995  		out := new(Int).Lsh(in, test.shift)
   996  		out = out.Rsh(out, test.shift)
   997  
   998  		if !isNormalized(out) {
   999  			t.Errorf("#%d: %v is not normalized", i, *out)
  1000  		}
  1001  		if in.Cmp(out) != 0 {
  1002  			t.Errorf("#%d: got %s want %s", i, out, in)
  1003  		}
  1004  	}
  1005  	for i, test := range lshTests {
  1006  		in, _ := new(Int).SetString(test.in, 10)
  1007  		out := new(Int).Lsh(in, test.shift)
  1008  		out.Rsh(out, test.shift)
  1009  
  1010  		if !isNormalized(out) {
  1011  			t.Errorf("#%d: %v is not normalized", i, *out)
  1012  		}
  1013  		if in.Cmp(out) != 0 {
  1014  			t.Errorf("#%d: got %s want %s", i, out, in)
  1015  		}
  1016  	}
  1017  }
  1018  
  1019  // Entries must be sorted by value in ascending order.
  1020  var cmpAbsTests = []string{
  1021  	"0",
  1022  	"1",
  1023  	"2",
  1024  	"10",
  1025  	"10000000",
  1026  	"2783678367462374683678456387645876387564783686583485",
  1027  	"2783678367462374683678456387645876387564783686583486",
  1028  	"32957394867987420967976567076075976570670947609750670956097509670576075067076027578341538",
  1029  }
  1030  
  1031  func TestCmpAbs(t *testing.T) {
  1032  	values := make([]*Int, len(cmpAbsTests))
  1033  	var prev *Int
  1034  	for i, s := range cmpAbsTests {
  1035  		x, ok := new(Int).SetString(s, 0)
  1036  		if !ok {
  1037  			t.Fatalf("SetString(%s, 0) failed", s)
  1038  		}
  1039  		if prev != nil && prev.Cmp(x) >= 0 {
  1040  			t.Fatal("cmpAbsTests entries not sorted in ascending order")
  1041  		}
  1042  		values[i] = x
  1043  		prev = x
  1044  	}
  1045  
  1046  	for i, x := range values {
  1047  		for j, y := range values {
  1048  			// try all combinations of signs for x, y
  1049  			for k := 0; k < 4; k++ {
  1050  				var a, b Int
  1051  				a.Set(x)
  1052  				b.Set(y)
  1053  				if k&1 != 0 {
  1054  					a.Neg(&a)
  1055  				}
  1056  				if k&2 != 0 {
  1057  					b.Neg(&b)
  1058  				}
  1059  
  1060  				got := a.CmpAbs(&b)
  1061  				want := 0
  1062  				switch {
  1063  				case i > j:
  1064  					want = 1
  1065  				case i < j:
  1066  					want = -1
  1067  				}
  1068  				if got != want {
  1069  					t.Errorf("absCmp |%s|, |%s|: got %d; want %d", &a, &b, got, want)
  1070  				}
  1071  			}
  1072  		}
  1073  	}
  1074  }
  1075  
  1076  func TestIntCmpSelf(t *testing.T) {
  1077  	for _, s := range cmpAbsTests {
  1078  		x, ok := new(Int).SetString(s, 0)
  1079  		if !ok {
  1080  			t.Fatalf("SetString(%s, 0) failed", s)
  1081  		}
  1082  		got := x.Cmp(x)
  1083  		want := 0
  1084  		if got != want {
  1085  			t.Errorf("x = %s: x.Cmp(x): got %d; want %d", x, got, want)
  1086  		}
  1087  	}
  1088  }
  1089  
  1090  var int64Tests = []string{
  1091  	// int64
  1092  	"0",
  1093  	"1",
  1094  	"-1",
  1095  	"4294967295",
  1096  	"-4294967295",
  1097  	"4294967296",
  1098  	"-4294967296",
  1099  	"9223372036854775807",
  1100  	"-9223372036854775807",
  1101  	"-9223372036854775808",
  1102  
  1103  	// not int64
  1104  	"0x8000000000000000",
  1105  	"-0x8000000000000001",
  1106  	"38579843757496759476987459679745",
  1107  	"-38579843757496759476987459679745",
  1108  }
  1109  
  1110  func TestInt64(t *testing.T) {
  1111  	for _, s := range int64Tests {
  1112  		var x Int
  1113  		_, ok := x.SetString(s, 0)
  1114  		if !ok {
  1115  			t.Errorf("SetString(%s, 0) failed", s)
  1116  			continue
  1117  		}
  1118  
  1119  		want, err := strconv.ParseInt(s, 0, 64)
  1120  		if err != nil {
  1121  			if err.(*strconv.NumError).Err == strconv.ErrRange {
  1122  				if x.IsInt64() {
  1123  					t.Errorf("IsInt64(%s) succeeded unexpectedly", s)
  1124  				}
  1125  			} else {
  1126  				t.Errorf("ParseInt(%s) failed", s)
  1127  			}
  1128  			continue
  1129  		}
  1130  
  1131  		if !x.IsInt64() {
  1132  			t.Errorf("IsInt64(%s) failed unexpectedly", s)
  1133  		}
  1134  
  1135  		got := x.Int64()
  1136  		if got != want {
  1137  			t.Errorf("Int64(%s) = %d; want %d", s, got, want)
  1138  		}
  1139  	}
  1140  }
  1141  
  1142  var uint64Tests = []string{
  1143  	// uint64
  1144  	"0",
  1145  	"1",
  1146  	"4294967295",
  1147  	"4294967296",
  1148  	"8589934591",
  1149  	"8589934592",
  1150  	"9223372036854775807",
  1151  	"9223372036854775808",
  1152  	"0x08000000000000000",
  1153  
  1154  	// not uint64
  1155  	"0x10000000000000000",
  1156  	"-0x08000000000000000",
  1157  	"-1",
  1158  }
  1159  
  1160  func TestUint64(t *testing.T) {
  1161  	for _, s := range uint64Tests {
  1162  		var x Int
  1163  		_, ok := x.SetString(s, 0)
  1164  		if !ok {
  1165  			t.Errorf("SetString(%s, 0) failed", s)
  1166  			continue
  1167  		}
  1168  
  1169  		want, err := strconv.ParseUint(s, 0, 64)
  1170  		if err != nil {
  1171  			// check for sign explicitly (ErrRange doesn't cover signed input)
  1172  			if s[0] == '-' || err.(*strconv.NumError).Err == strconv.ErrRange {
  1173  				if x.IsUint64() {
  1174  					t.Errorf("IsUint64(%s) succeeded unexpectedly", s)
  1175  				}
  1176  			} else {
  1177  				t.Errorf("ParseUint(%s) failed", s)
  1178  			}
  1179  			continue
  1180  		}
  1181  
  1182  		if !x.IsUint64() {
  1183  			t.Errorf("IsUint64(%s) failed unexpectedly", s)
  1184  		}
  1185  
  1186  		got := x.Uint64()
  1187  		if got != want {
  1188  			t.Errorf("Uint64(%s) = %d; want %d", s, got, want)
  1189  		}
  1190  	}
  1191  }
  1192  
  1193  var bitwiseTests = []struct {
  1194  	x, y                 string
  1195  	and, or, xor, andNot string
  1196  }{
  1197  	{"0x00", "0x00", "0x00", "0x00", "0x00", "0x00"},
  1198  	{"0x00", "0x01", "0x00", "0x01", "0x01", "0x00"},
  1199  	{"0x01", "0x00", "0x00", "0x01", "0x01", "0x01"},
  1200  	{"-0x01", "0x00", "0x00", "-0x01", "-0x01", "-0x01"},
  1201  	{"-0xaf", "-0x50", "-0xf0", "-0x0f", "0xe1", "0x41"},
  1202  	{"0x00", "-0x01", "0x00", "-0x01", "-0x01", "0x00"},
  1203  	{"0x01", "0x01", "0x01", "0x01", "0x00", "0x00"},
  1204  	{"-0x01", "-0x01", "-0x01", "-0x01", "0x00", "0x00"},
  1205  	{"0x07", "0x08", "0x00", "0x0f", "0x0f", "0x07"},
  1206  	{"0x05", "0x0f", "0x05", "0x0f", "0x0a", "0x00"},
  1207  	{"0xff", "-0x0a", "0xf6", "-0x01", "-0xf7", "0x09"},
  1208  	{"0x013ff6", "0x9a4e", "0x1a46", "0x01bffe", "0x01a5b8", "0x0125b0"},
  1209  	{"-0x013ff6", "0x9a4e", "0x800a", "-0x0125b2", "-0x01a5bc", "-0x01c000"},
  1210  	{"-0x013ff6", "-0x9a4e", "-0x01bffe", "-0x1a46", "0x01a5b8", "0x8008"},
  1211  	{
  1212  		"0x1000009dc6e3d9822cba04129bcbe3401",
  1213  		"0xb9bd7d543685789d57cb918e833af352559021483cdb05cc21fd",
  1214  		"0x1000001186210100001000009048c2001",
  1215  		"0xb9bd7d543685789d57cb918e8bfeff7fddb2ebe87dfbbdfe35fd",
  1216  		"0xb9bd7d543685789d57ca918e8ae69d6fcdb2eae87df2b97215fc",
  1217  		"0x8c40c2d8822caa04120b8321400",
  1218  	},
  1219  	{
  1220  		"0x1000009dc6e3d9822cba04129bcbe3401",
  1221  		"-0xb9bd7d543685789d57cb918e833af352559021483cdb05cc21fd",
  1222  		"0x8c40c2d8822caa04120b8321401",
  1223  		"-0xb9bd7d543685789d57ca918e82229142459020483cd2014001fd",
  1224  		"-0xb9bd7d543685789d57ca918e8ae69d6fcdb2eae87df2b97215fe",
  1225  		"0x1000001186210100001000009048c2000",
  1226  	},
  1227  	{
  1228  		"-0x1000009dc6e3d9822cba04129bcbe3401",
  1229  		"-0xb9bd7d543685789d57cb918e833af352559021483cdb05cc21fd",
  1230  		"-0xb9bd7d543685789d57cb918e8bfeff7fddb2ebe87dfbbdfe35fd",
  1231  		"-0x1000001186210100001000009048c2001",
  1232  		"0xb9bd7d543685789d57ca918e8ae69d6fcdb2eae87df2b97215fc",
  1233  		"0xb9bd7d543685789d57ca918e82229142459020483cd2014001fc",
  1234  	},
  1235  }
  1236  
  1237  type bitFun func(z, x, y *Int) *Int
  1238  
  1239  func testBitFun(t *testing.T, msg string, f bitFun, x, y *Int, exp string) {
  1240  	expected := new(Int)
  1241  	expected.SetString(exp, 0)
  1242  
  1243  	out := f(new(Int), x, y)
  1244  	if out.Cmp(expected) != 0 {
  1245  		t.Errorf("%s: got %s want %s", msg, out, expected)
  1246  	}
  1247  }
  1248  
  1249  func testBitFunSelf(t *testing.T, msg string, f bitFun, x, y *Int, exp string) {
  1250  	self := new(Int)
  1251  	self.Set(x)
  1252  	expected := new(Int)
  1253  	expected.SetString(exp, 0)
  1254  
  1255  	self = f(self, self, y)
  1256  	if self.Cmp(expected) != 0 {
  1257  		t.Errorf("%s: got %s want %s", msg, self, expected)
  1258  	}
  1259  }
  1260  
  1261  func altBit(x *Int, i int) uint {
  1262  	z := new(Int).Rsh(x, uint(i))
  1263  	z = z.And(z, NewInt(1))
  1264  	if z.Cmp(new(Int)) != 0 {
  1265  		return 1
  1266  	}
  1267  	return 0
  1268  }
  1269  
  1270  func altSetBit(z *Int, x *Int, i int, b uint) *Int {
  1271  	one := NewInt(1)
  1272  	m := one.Lsh(one, uint(i))
  1273  	switch b {
  1274  	case 1:
  1275  		return z.Or(x, m)
  1276  	case 0:
  1277  		return z.AndNot(x, m)
  1278  	}
  1279  	panic("set bit is not 0 or 1")
  1280  }
  1281  
  1282  func testBitset(t *testing.T, x *Int) {
  1283  	n := x.BitLen()
  1284  	z := new(Int).Set(x)
  1285  	z1 := new(Int).Set(x)
  1286  	for i := 0; i < n+10; i++ {
  1287  		old := z.Bit(i)
  1288  		old1 := altBit(z1, i)
  1289  		if old != old1 {
  1290  			t.Errorf("bitset: inconsistent value for Bit(%s, %d), got %v want %v", z1, i, old, old1)
  1291  		}
  1292  		z := new(Int).SetBit(z, i, 1)
  1293  		z1 := altSetBit(new(Int), z1, i, 1)
  1294  		if z.Bit(i) == 0 {
  1295  			t.Errorf("bitset: bit %d of %s got 0 want 1", i, x)
  1296  		}
  1297  		if z.Cmp(z1) != 0 {
  1298  			t.Errorf("bitset: inconsistent value after SetBit 1, got %s want %s", z, z1)
  1299  		}
  1300  		z.SetBit(z, i, 0)
  1301  		altSetBit(z1, z1, i, 0)
  1302  		if z.Bit(i) != 0 {
  1303  			t.Errorf("bitset: bit %d of %s got 1 want 0", i, x)
  1304  		}
  1305  		if z.Cmp(z1) != 0 {
  1306  			t.Errorf("bitset: inconsistent value after SetBit 0, got %s want %s", z, z1)
  1307  		}
  1308  		altSetBit(z1, z1, i, old)
  1309  		z.SetBit(z, i, old)
  1310  		if z.Cmp(z1) != 0 {
  1311  			t.Errorf("bitset: inconsistent value after SetBit old, got %s want %s", z, z1)
  1312  		}
  1313  	}
  1314  	if z.Cmp(x) != 0 {
  1315  		t.Errorf("bitset: got %s want %s", z, x)
  1316  	}
  1317  }
  1318  
  1319  var bitsetTests = []struct {
  1320  	x string
  1321  	i int
  1322  	b uint
  1323  }{
  1324  	{"0", 0, 0},
  1325  	{"0", 200, 0},
  1326  	{"1", 0, 1},
  1327  	{"1", 1, 0},
  1328  	{"-1", 0, 1},
  1329  	{"-1", 200, 1},
  1330  	{"0x2000000000000000000000000000", 108, 0},
  1331  	{"0x2000000000000000000000000000", 109, 1},
  1332  	{"0x2000000000000000000000000000", 110, 0},
  1333  	{"-0x2000000000000000000000000001", 108, 1},
  1334  	{"-0x2000000000000000000000000001", 109, 0},
  1335  	{"-0x2000000000000000000000000001", 110, 1},
  1336  }
  1337  
  1338  func TestBitSet(t *testing.T) {
  1339  	for _, test := range bitwiseTests {
  1340  		x := new(Int)
  1341  		x.SetString(test.x, 0)
  1342  		testBitset(t, x)
  1343  		x = new(Int)
  1344  		x.SetString(test.y, 0)
  1345  		testBitset(t, x)
  1346  	}
  1347  	for i, test := range bitsetTests {
  1348  		x := new(Int)
  1349  		x.SetString(test.x, 0)
  1350  		b := x.Bit(test.i)
  1351  		if b != test.b {
  1352  			t.Errorf("#%d got %v want %v", i, b, test.b)
  1353  		}
  1354  	}
  1355  	z := NewInt(1)
  1356  	z.SetBit(NewInt(0), 2, 1)
  1357  	if z.Cmp(NewInt(4)) != 0 {
  1358  		t.Errorf("destination leaked into result; got %s want 4", z)
  1359  	}
  1360  }
  1361  
  1362  var tzbTests = []struct {
  1363  	in  string
  1364  	out uint
  1365  }{
  1366  	{"0", 0},
  1367  	{"1", 0},
  1368  	{"-1", 0},
  1369  	{"4", 2},
  1370  	{"-8", 3},
  1371  	{"0x4000000000000000000", 74},
  1372  	{"-0x8000000000000000000", 75},
  1373  }
  1374  
  1375  func TestTrailingZeroBits(t *testing.T) {
  1376  	for i, test := range tzbTests {
  1377  		in, _ := new(Int).SetString(test.in, 0)
  1378  		want := test.out
  1379  		got := in.TrailingZeroBits()
  1380  
  1381  		if got != want {
  1382  			t.Errorf("#%d: got %v want %v", i, got, want)
  1383  		}
  1384  	}
  1385  }
  1386  
  1387  func BenchmarkBitset(b *testing.B) {
  1388  	z := new(Int)
  1389  	z.SetBit(z, 512, 1)
  1390  	b.ResetTimer()
  1391  	b.StartTimer()
  1392  	for i := b.N - 1; i >= 0; i-- {
  1393  		z.SetBit(z, i&512, 1)
  1394  	}
  1395  }
  1396  
  1397  func BenchmarkBitsetNeg(b *testing.B) {
  1398  	z := NewInt(-1)
  1399  	z.SetBit(z, 512, 0)
  1400  	b.ResetTimer()
  1401  	b.StartTimer()
  1402  	for i := b.N - 1; i >= 0; i-- {
  1403  		z.SetBit(z, i&512, 0)
  1404  	}
  1405  }
  1406  
  1407  func BenchmarkBitsetOrig(b *testing.B) {
  1408  	z := new(Int)
  1409  	altSetBit(z, z, 512, 1)
  1410  	b.ResetTimer()
  1411  	b.StartTimer()
  1412  	for i := b.N - 1; i >= 0; i-- {
  1413  		altSetBit(z, z, i&512, 1)
  1414  	}
  1415  }
  1416  
  1417  func BenchmarkBitsetNegOrig(b *testing.B) {
  1418  	z := NewInt(-1)
  1419  	altSetBit(z, z, 512, 0)
  1420  	b.ResetTimer()
  1421  	b.StartTimer()
  1422  	for i := b.N - 1; i >= 0; i-- {
  1423  		altSetBit(z, z, i&512, 0)
  1424  	}
  1425  }
  1426  
  1427  // tri generates the trinomial 2**(n*2) - 2**n - 1, which is always 3 mod 4 and
  1428  // 7 mod 8, so that 2 is always a quadratic residue.
  1429  func tri(n uint) *Int {
  1430  	x := NewInt(1)
  1431  	x.Lsh(x, n)
  1432  	x2 := new(Int).Lsh(x, n)
  1433  	x2.Sub(x2, x)
  1434  	x2.Sub(x2, intOne)
  1435  	return x2
  1436  }
  1437  
  1438  func BenchmarkModSqrt225_Tonelli(b *testing.B) {
  1439  	p := tri(225)
  1440  	x := NewInt(2)
  1441  	for i := 0; i < b.N; i++ {
  1442  		x.SetUint64(2)
  1443  		x.modSqrtTonelliShanks(x, p)
  1444  	}
  1445  }
  1446  
  1447  func BenchmarkModSqrt225_3Mod4(b *testing.B) {
  1448  	p := tri(225)
  1449  	x := new(Int).SetUint64(2)
  1450  	for i := 0; i < b.N; i++ {
  1451  		x.SetUint64(2)
  1452  		x.modSqrt3Mod4Prime(x, p)
  1453  	}
  1454  }
  1455  
  1456  func BenchmarkModSqrt231_Tonelli(b *testing.B) {
  1457  	p := tri(231)
  1458  	p.Sub(p, intOne)
  1459  	p.Sub(p, intOne) // tri(231) - 2 is a prime == 5 mod 8
  1460  	x := new(Int).SetUint64(7)
  1461  	for i := 0; i < b.N; i++ {
  1462  		x.SetUint64(7)
  1463  		x.modSqrtTonelliShanks(x, p)
  1464  	}
  1465  }
  1466  
  1467  func BenchmarkModSqrt231_5Mod8(b *testing.B) {
  1468  	p := tri(231)
  1469  	p.Sub(p, intOne)
  1470  	p.Sub(p, intOne) // tri(231) - 2 is a prime == 5 mod 8
  1471  	x := new(Int).SetUint64(7)
  1472  	for i := 0; i < b.N; i++ {
  1473  		x.SetUint64(7)
  1474  		x.modSqrt5Mod8Prime(x, p)
  1475  	}
  1476  }
  1477  
  1478  func TestBitwise(t *testing.T) {
  1479  	x := new(Int)
  1480  	y := new(Int)
  1481  	for _, test := range bitwiseTests {
  1482  		x.SetString(test.x, 0)
  1483  		y.SetString(test.y, 0)
  1484  
  1485  		testBitFun(t, "and", (*Int).And, x, y, test.and)
  1486  		testBitFunSelf(t, "and", (*Int).And, x, y, test.and)
  1487  		testBitFun(t, "andNot", (*Int).AndNot, x, y, test.andNot)
  1488  		testBitFunSelf(t, "andNot", (*Int).AndNot, x, y, test.andNot)
  1489  		testBitFun(t, "or", (*Int).Or, x, y, test.or)
  1490  		testBitFunSelf(t, "or", (*Int).Or, x, y, test.or)
  1491  		testBitFun(t, "xor", (*Int).Xor, x, y, test.xor)
  1492  		testBitFunSelf(t, "xor", (*Int).Xor, x, y, test.xor)
  1493  	}
  1494  }
  1495  
  1496  var notTests = []struct {
  1497  	in  string
  1498  	out string
  1499  }{
  1500  	{"0", "-1"},
  1501  	{"1", "-2"},
  1502  	{"7", "-8"},
  1503  	{"0", "-1"},
  1504  	{"-81910", "81909"},
  1505  	{
  1506  		"298472983472983471903246121093472394872319615612417471234712061",
  1507  		"-298472983472983471903246121093472394872319615612417471234712062",
  1508  	},
  1509  }
  1510  
  1511  func TestNot(t *testing.T) {
  1512  	in := new(Int)
  1513  	out := new(Int)
  1514  	expected := new(Int)
  1515  	for i, test := range notTests {
  1516  		in.SetString(test.in, 10)
  1517  		expected.SetString(test.out, 10)
  1518  		out = out.Not(in)
  1519  		if out.Cmp(expected) != 0 {
  1520  			t.Errorf("#%d: got %s want %s", i, out, expected)
  1521  		}
  1522  		out = out.Not(out)
  1523  		if out.Cmp(in) != 0 {
  1524  			t.Errorf("#%d: got %s want %s", i, out, in)
  1525  		}
  1526  	}
  1527  }
  1528  
  1529  var modInverseTests = []struct {
  1530  	element string
  1531  	modulus string
  1532  }{
  1533  	{"1234567", "458948883992"},
  1534  	{"239487239847", "2410312426921032588552076022197566074856950548502459942654116941958108831682612228890093858261341614673227141477904012196503648957050582631942730706805009223062734745341073406696246014589361659774041027169249453200378729434170325843778659198143763193776859869524088940195577346119843545301547043747207749969763750084308926339295559968882457872412993810129130294592999947926365264059284647209730384947211681434464714438488520940127459844288859336526896320919633919"},
  1535  	{"-10", "13"}, // issue #16984
  1536  	{"10", "-13"},
  1537  	{"-17", "-13"},
  1538  }
  1539  
  1540  func TestModInverse(t *testing.T) {
  1541  	var element, modulus, gcd, inverse Int
  1542  	one := NewInt(1)
  1543  	for _, test := range modInverseTests {
  1544  		(&element).SetString(test.element, 10)
  1545  		(&modulus).SetString(test.modulus, 10)
  1546  		(&inverse).ModInverse(&element, &modulus)
  1547  		(&inverse).Mul(&inverse, &element)
  1548  		(&inverse).Mod(&inverse, &modulus)
  1549  		if (&inverse).Cmp(one) != 0 {
  1550  			t.Errorf("ModInverse(%d,%d)*%d%%%d=%d, not 1", &element, &modulus, &element, &modulus, &inverse)
  1551  		}
  1552  	}
  1553  	// exhaustive test for small values
  1554  	for n := 2; n < 100; n++ {
  1555  		(&modulus).SetInt64(int64(n))
  1556  		for x := 1; x < n; x++ {
  1557  			(&element).SetInt64(int64(x))
  1558  			(&gcd).GCD(nil, nil, &element, &modulus)
  1559  			if (&gcd).Cmp(one) != 0 {
  1560  				continue
  1561  			}
  1562  			(&inverse).ModInverse(&element, &modulus)
  1563  			(&inverse).Mul(&inverse, &element)
  1564  			(&inverse).Mod(&inverse, &modulus)
  1565  			if (&inverse).Cmp(one) != 0 {
  1566  				t.Errorf("ModInverse(%d,%d)*%d%%%d=%d, not 1", &element, &modulus, &element, &modulus, &inverse)
  1567  			}
  1568  		}
  1569  	}
  1570  }
  1571  
  1572  func BenchmarkModInverse(b *testing.B) {
  1573  	p := new(Int).SetInt64(1) // Mersenne prime 2**1279 -1
  1574  	p.abs = p.abs.shl(p.abs, 1279)
  1575  	p.Sub(p, intOne)
  1576  	x := new(Int).Sub(p, intOne)
  1577  	z := new(Int)
  1578  	for i := 0; i < b.N; i++ {
  1579  		z.ModInverse(x, p)
  1580  	}
  1581  }
  1582  
  1583  // testModSqrt is a helper for TestModSqrt,
  1584  // which checks that ModSqrt can compute a square-root of elt^2.
  1585  func testModSqrt(t *testing.T, elt, mod, sq, sqrt *Int) bool {
  1586  	var sqChk, sqrtChk, sqrtsq Int
  1587  	sq.Mul(elt, elt)
  1588  	sq.Mod(sq, mod)
  1589  	z := sqrt.ModSqrt(sq, mod)
  1590  	if z != sqrt {
  1591  		t.Errorf("ModSqrt returned wrong value %s", z)
  1592  	}
  1593  
  1594  	// test ModSqrt arguments outside the range [0,mod)
  1595  	sqChk.Add(sq, mod)
  1596  	z = sqrtChk.ModSqrt(&sqChk, mod)
  1597  	if z != &sqrtChk || z.Cmp(sqrt) != 0 {
  1598  		t.Errorf("ModSqrt returned inconsistent value %s", z)
  1599  	}
  1600  	sqChk.Sub(sq, mod)
  1601  	z = sqrtChk.ModSqrt(&sqChk, mod)
  1602  	if z != &sqrtChk || z.Cmp(sqrt) != 0 {
  1603  		t.Errorf("ModSqrt returned inconsistent value %s", z)
  1604  	}
  1605  
  1606  	// test x aliasing z
  1607  	z = sqrtChk.ModSqrt(sqrtChk.Set(sq), mod)
  1608  	if z != &sqrtChk || z.Cmp(sqrt) != 0 {
  1609  		t.Errorf("ModSqrt returned inconsistent value %s", z)
  1610  	}
  1611  
  1612  	// make sure we actually got a square root
  1613  	if sqrt.Cmp(elt) == 0 {
  1614  		return true // we found the "desired" square root
  1615  	}
  1616  	sqrtsq.Mul(sqrt, sqrt) // make sure we found the "other" one
  1617  	sqrtsq.Mod(&sqrtsq, mod)
  1618  	return sq.Cmp(&sqrtsq) == 0
  1619  }
  1620  
  1621  func TestModSqrt(t *testing.T) {
  1622  	var elt, mod, modx4, sq, sqrt Int
  1623  	r := rand.New(rand.NewSource(9))
  1624  	for i, s := range primes[1:] { // skip 2, use only odd primes
  1625  		mod.SetString(s, 10)
  1626  		modx4.Lsh(&mod, 2)
  1627  
  1628  		// test a few random elements per prime
  1629  		for x := 1; x < 5; x++ {
  1630  			elt.Rand(r, &modx4)
  1631  			elt.Sub(&elt, &mod) // test range [-mod, 3*mod)
  1632  			if !testModSqrt(t, &elt, &mod, &sq, &sqrt) {
  1633  				t.Errorf("#%d: failed (sqrt(e) = %s)", i, &sqrt)
  1634  			}
  1635  		}
  1636  
  1637  		if testing.Short() && i > 2 {
  1638  			break
  1639  		}
  1640  	}
  1641  
  1642  	if testing.Short() {
  1643  		return
  1644  	}
  1645  
  1646  	// exhaustive test for small values
  1647  	for n := 3; n < 100; n++ {
  1648  		mod.SetInt64(int64(n))
  1649  		if !mod.ProbablyPrime(10) {
  1650  			continue
  1651  		}
  1652  		isSquare := make([]bool, n)
  1653  
  1654  		// test all the squares
  1655  		for x := 1; x < n; x++ {
  1656  			elt.SetInt64(int64(x))
  1657  			if !testModSqrt(t, &elt, &mod, &sq, &sqrt) {
  1658  				t.Errorf("#%d: failed (sqrt(%d,%d) = %s)", x, &elt, &mod, &sqrt)
  1659  			}
  1660  			isSquare[sq.Uint64()] = true
  1661  		}
  1662  
  1663  		// test all non-squares
  1664  		for x := 1; x < n; x++ {
  1665  			sq.SetInt64(int64(x))
  1666  			z := sqrt.ModSqrt(&sq, &mod)
  1667  			if !isSquare[x] && z != nil {
  1668  				t.Errorf("#%d: failed (sqrt(%d,%d) = nil)", x, &sqrt, &mod)
  1669  			}
  1670  		}
  1671  	}
  1672  }
  1673  
  1674  func TestJacobi(t *testing.T) {
  1675  	testCases := []struct {
  1676  		x, y   int64
  1677  		result int
  1678  	}{
  1679  		{0, 1, 1},
  1680  		{0, -1, 1},
  1681  		{1, 1, 1},
  1682  		{1, -1, 1},
  1683  		{0, 5, 0},
  1684  		{1, 5, 1},
  1685  		{2, 5, -1},
  1686  		{-2, 5, -1},
  1687  		{2, -5, -1},
  1688  		{-2, -5, 1},
  1689  		{3, 5, -1},
  1690  		{5, 5, 0},
  1691  		{-5, 5, 0},
  1692  		{6, 5, 1},
  1693  		{6, -5, 1},
  1694  		{-6, 5, 1},
  1695  		{-6, -5, -1},
  1696  	}
  1697  
  1698  	var x, y Int
  1699  
  1700  	for i, test := range testCases {
  1701  		x.SetInt64(test.x)
  1702  		y.SetInt64(test.y)
  1703  		expected := test.result
  1704  		actual := Jacobi(&x, &y)
  1705  		if actual != expected {
  1706  			t.Errorf("#%d: Jacobi(%d, %d) = %d, but expected %d", i, test.x, test.y, actual, expected)
  1707  		}
  1708  	}
  1709  }
  1710  
  1711  func TestJacobiPanic(t *testing.T) {
  1712  	const failureMsg = "test failure"
  1713  	defer func() {
  1714  		msg := recover()
  1715  		if msg == nil || msg == failureMsg {
  1716  			panic(msg)
  1717  		}
  1718  		t.Log(msg)
  1719  	}()
  1720  	x := NewInt(1)
  1721  	y := NewInt(2)
  1722  	// Jacobi should panic when the second argument is even.
  1723  	Jacobi(x, y)
  1724  	panic(failureMsg)
  1725  }
  1726  
  1727  func TestIssue2607(t *testing.T) {
  1728  	// This code sequence used to hang.
  1729  	n := NewInt(10)
  1730  	n.Rand(rand.New(rand.NewSource(9)), n)
  1731  }
  1732  
  1733  func TestSqrt(t *testing.T) {
  1734  	root := 0
  1735  	r := new(Int)
  1736  	for i := 0; i < 10000; i++ {
  1737  		if (root+1)*(root+1) <= i {
  1738  			root++
  1739  		}
  1740  		n := NewInt(int64(i))
  1741  		r.SetInt64(-2)
  1742  		r.Sqrt(n)
  1743  		if r.Cmp(NewInt(int64(root))) != 0 {
  1744  			t.Errorf("Sqrt(%v) = %v, want %v", n, r, root)
  1745  		}
  1746  	}
  1747  
  1748  	for i := 0; i < 1000; i += 10 {
  1749  		n, _ := new(Int).SetString("1"+strings.Repeat("0", i), 10)
  1750  		r := new(Int).Sqrt(n)
  1751  		root, _ := new(Int).SetString("1"+strings.Repeat("0", i/2), 10)
  1752  		if r.Cmp(root) != 0 {
  1753  			t.Errorf("Sqrt(1e%d) = %v, want 1e%d", i, r, i/2)
  1754  		}
  1755  	}
  1756  
  1757  	// Test aliasing.
  1758  	r.SetInt64(100)
  1759  	r.Sqrt(r)
  1760  	if r.Int64() != 10 {
  1761  		t.Errorf("Sqrt(100) = %v, want 10 (aliased output)", r.Int64())
  1762  	}
  1763  }
  1764  
  1765  // We can't test this together with the other Exp tests above because
  1766  // it requires a different receiver setup.
  1767  func TestIssue22830(t *testing.T) {
  1768  	one := new(Int).SetInt64(1)
  1769  	base, _ := new(Int).SetString("84555555300000000000", 10)
  1770  	mod, _ := new(Int).SetString("66666670001111111111", 10)
  1771  	want, _ := new(Int).SetString("17888885298888888889", 10)
  1772  
  1773  	var tests = []int64{
  1774  		0, 1, -1,
  1775  	}
  1776  
  1777  	for _, n := range tests {
  1778  		m := NewInt(n)
  1779  		if got := m.Exp(base, one, mod); got.Cmp(want) != 0 {
  1780  			t.Errorf("(%v).Exp(%s, 1, %s) = %s, want %s", n, base, mod, got, want)
  1781  		}
  1782  	}
  1783  }
  1784  
  1785  func BenchmarkSqrt(b *testing.B) {
  1786  	n, _ := new(Int).SetString("1"+strings.Repeat("0", 1001), 10)
  1787  	b.ResetTimer()
  1788  	t := new(Int)
  1789  	for i := 0; i < b.N; i++ {
  1790  		t.Sqrt(n)
  1791  	}
  1792  }
  1793  
  1794  func benchmarkIntSqr(b *testing.B, nwords int) {
  1795  	x := new(Int)
  1796  	x.abs = rndNat(nwords)
  1797  	t := new(Int)
  1798  	b.ResetTimer()
  1799  	for i := 0; i < b.N; i++ {
  1800  		t.Mul(x, x)
  1801  	}
  1802  }
  1803  
  1804  func BenchmarkIntSqr(b *testing.B) {
  1805  	for _, n := range sqrBenchSizes {
  1806  		if isRaceBuilder && n > 1e3 {
  1807  			continue
  1808  		}
  1809  		b.Run(fmt.Sprintf("%d", n), func(b *testing.B) {
  1810  			benchmarkIntSqr(b, n)
  1811  		})
  1812  	}
  1813  }
  1814  
  1815  func benchmarkDiv(b *testing.B, aSize, bSize int) {
  1816  	var r = rand.New(rand.NewSource(1234))
  1817  	aa := randInt(r, uint(aSize))
  1818  	bb := randInt(r, uint(bSize))
  1819  	if aa.Cmp(bb) < 0 {
  1820  		aa, bb = bb, aa
  1821  	}
  1822  	x := new(Int)
  1823  	y := new(Int)
  1824  
  1825  	b.ResetTimer()
  1826  	for i := 0; i < b.N; i++ {
  1827  		x.DivMod(aa, bb, y)
  1828  	}
  1829  }
  1830  
  1831  func BenchmarkDiv(b *testing.B) {
  1832  	sizes := []int{
  1833  		10, 20, 50, 100, 200, 500, 1000,
  1834  		1e4, 1e5, 1e6, 1e7,
  1835  	}
  1836  	for _, i := range sizes {
  1837  		j := 2 * i
  1838  		b.Run(fmt.Sprintf("%d/%d", j, i), func(b *testing.B) {
  1839  			benchmarkDiv(b, j, i)
  1840  		})
  1841  	}
  1842  }
  1843  
  1844  func TestFillBytes(t *testing.T) {
  1845  	checkResult := func(t *testing.T, buf []byte, want *Int) {
  1846  		t.Helper()
  1847  		got := new(Int).SetBytes(buf)
  1848  		if got.CmpAbs(want) != 0 {
  1849  			t.Errorf("got 0x%x, want 0x%x: %x", got, want, buf)
  1850  		}
  1851  	}
  1852  	panics := func(f func()) (panic bool) {
  1853  		defer func() { panic = recover() != nil }()
  1854  		f()
  1855  		return
  1856  	}
  1857  
  1858  	for _, n := range []string{
  1859  		"0",
  1860  		"1000",
  1861  		"0xffffffff",
  1862  		"-0xffffffff",
  1863  		"0xffffffffffffffff",
  1864  		"0x10000000000000000",
  1865  		"0xabababababababababababababababababababababababababa",
  1866  		"0xffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff",
  1867  	} {
  1868  		t.Run(n, func(t *testing.T) {
  1869  			t.Logf(n)
  1870  			x, ok := new(Int).SetString(n, 0)
  1871  			if !ok {
  1872  				panic("invalid test entry")
  1873  			}
  1874  
  1875  			// Perfectly sized buffer.
  1876  			byteLen := (x.BitLen() + 7) / 8
  1877  			buf := make([]byte, byteLen)
  1878  			checkResult(t, x.FillBytes(buf), x)
  1879  
  1880  			// Way larger, checking all bytes get zeroed.
  1881  			buf = make([]byte, 100)
  1882  			for i := range buf {
  1883  				buf[i] = 0xff
  1884  			}
  1885  			checkResult(t, x.FillBytes(buf), x)
  1886  
  1887  			// Too small.
  1888  			if byteLen > 0 {
  1889  				buf = make([]byte, byteLen-1)
  1890  				if !panics(func() { x.FillBytes(buf) }) {
  1891  					t.Errorf("expected panic for small buffer and value %x", x)
  1892  				}
  1893  			}
  1894  		})
  1895  	}
  1896  }
  1897  

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