libgo/go/math/big/nat_test.go

   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         "fmt"
   9         "runtime"
  10         "strings"
  11         "testing"
  12 )
  13
  14 var cmpTests = []struct {
  15         x, y nat
  16         r    int
  17 }{
  18         {nil, nil, 0},
  19         {nil, nat(nil), 0},
  20         {nat(nil), nil, 0},
  21         {nat(nil), nat(nil), 0},
  22         {nat{0}, nat{0}, 0},
  23         {nat{0}, nat{1}, -1},
  24         {nat{1}, nat{0}, 1},
  25         {nat{1}, nat{1}, 0},
  26         {nat{0, _M}, nat{1}, 1},
  27         {nat{1}, nat{0, _M}, -1},
  28         {nat{1, _M}, nat{0, _M}, 1},
  29         {nat{0, _M}, nat{1, _M}, -1},
  30         {nat{16, 571956, 8794, 68}, nat{837, 9146, 1, 754489}, -1},
  31         {nat{34986, 41, 105, 1957}, nat{56, 7458, 104, 1957}, 1},
  32 }
  33
  34 func TestCmp(t *testing.T) {
  35         for i, a := range cmpTests {
  36                 r := a.x.cmp(a.y)
  37                 if r != a.r {
  38                         t.Errorf("#%d got r = %v; want %v", i, r, a.r)
  39                 }
  40         }
  41 }
  42
  43 type funNN func(z, x, y nat) nat
  44 type argNN struct {
  45         z, x, y nat
  46 }
  47
  48 var sumNN = []argNN{
  49         {},
  50         {nat{1}, nil, nat{1}},
  51         {nat{1111111110}, nat{123456789}, nat{987654321}},
  52         {nat{0, 0, 0, 1}, nil, nat{0, 0, 0, 1}},
  53         {nat{0, 0, 0, 1111111110}, nat{0, 0, 0, 123456789}, nat{0, 0, 0, 987654321}},
  54         {nat{0, 0, 0, 1}, nat{0, 0, _M}, nat{0, 0, 1}},
  55 }
  56
  57 var prodNN = []argNN{
  58         {},
  59         {nil, nil, nil},
  60         {nil, nat{991}, nil},
  61         {nat{991}, nat{991}, nat{1}},
  62         {nat{991 * 991}, nat{991}, nat{991}},
  63         {nat{0, 0, 991 * 991}, nat{0, 991}, nat{0, 991}},
  64         {nat{1 * 991, 2 * 991, 3 * 991, 4 * 991}, nat{1, 2, 3, 4}, nat{991}},
  65         {nat{4, 11, 20, 30, 20, 11, 4}, nat{1, 2, 3, 4}, nat{4, 3, 2, 1}},
  66         // 3^100 * 3^28 = 3^128
  67         {
  68                 natFromString("11790184577738583171520872861412518665678211592275841109096961"),
  69                 natFromString("515377520732011331036461129765621272702107522001"),
  70                 natFromString("22876792454961"),
  71         },
  72         // z = 111....1 (70000 digits)
  73         // x = 10^(99*700) + ... + 10^1400 + 10^700 + 1
  74         // y = 111....1 (700 digits, larger than Karatsuba threshold on 32-bit and 64-bit)
  75         {
  76                 natFromString(strings.Repeat("1", 70000)),
  77                 natFromString("1" + strings.Repeat(strings.Repeat("0", 699)+"1", 99)),
  78                 natFromString(strings.Repeat("1", 700)),
  79         },
  80         // z = 111....1 (20000 digits)
  81         // x = 10^10000 + 1
  82         // y = 111....1 (10000 digits)
  83         {
  84                 natFromString(strings.Repeat("1", 20000)),
  85                 natFromString("1" + strings.Repeat("0", 9999) + "1"),
  86                 natFromString(strings.Repeat("1", 10000)),
  87         },
  88 }
  89
  90 func natFromString(s string) nat {
  91         x, _, _, err := nat(nil).scan(strings.NewReader(s), 0, false)
  92         if err != nil {
  93                 panic(err)
  94         }
  95         return x
  96 }
  97
  98 func TestSet(t *testing.T) {
  99         for _, a := range sumNN {
 100                 z := nat(nil).set(a.z)
 101                 if z.cmp(a.z) != 0 {
 102                         t.Errorf("got z = %v; want %v", z, a.z)
 103                 }
 104         }
 105 }
 106
 107 func testFunNN(t *testing.T, msg string, f funNN, a argNN) {
 108         z := f(nil, a.x, a.y)
 109         if z.cmp(a.z) != 0 {
 110                 t.Errorf("%s%+v\n\tgot z = %v; want %v", msg, a, z, a.z)
 111         }
 112 }
 113
 114 func TestFunNN(t *testing.T) {
 115         for _, a := range sumNN {
 116                 arg := a
 117                 testFunNN(t, "add", nat.add, arg)
 118
 119                 arg = argNN{a.z, a.y, a.x}
 120                 testFunNN(t, "add symmetric", nat.add, arg)
 121
 122                 arg = argNN{a.x, a.z, a.y}
 123                 testFunNN(t, "sub", nat.sub, arg)
 124
 125                 arg = argNN{a.y, a.z, a.x}
 126                 testFunNN(t, "sub symmetric", nat.sub, arg)
 127         }
 128
 129         for _, a := range prodNN {
 130                 arg := a
 131                 testFunNN(t, "mul", nat.mul, arg)
 132
 133                 arg = argNN{a.z, a.y, a.x}
 134                 testFunNN(t, "mul symmetric", nat.mul, arg)
 135         }
 136 }
 137
 138 var mulRangesN = []struct {
 139         a, b uint64
 140         prod string
 141 }{
 142         {0, 0, "0"},
 143         {1, 1, "1"},
 144         {1, 2, "2"},
 145         {1, 3, "6"},
 146         {10, 10, "10"},
 147         {0, 100, "0"},
 148         {0, 1e9, "0"},
 149         {1, 0, "1"},                    // empty range
 150         {100, 1, "1"},                  // empty range
 151         {1, 10, "3628800"},             // 10!
 152         {1, 20, "2432902008176640000"}, // 20!
 153         {1, 100,
 154                 "933262154439441526816992388562667004907159682643816214685929" +
 155                         "638952175999932299156089414639761565182862536979208272237582" +
 156                         "51185210916864000000000000000000000000", // 100!
 157         },
 158 }
 159
 160 func TestMulRangeN(t *testing.T) {
 161         for i, r := range mulRangesN {
 162                 prod := string(nat(nil).mulRange(r.a, r.b).utoa(10))
 163                 if prod != r.prod {
 164                         t.Errorf("#%d: got %s; want %s", i, prod, r.prod)
 165                 }
 166         }
 167 }
 168
 169 // allocBytes returns the number of bytes allocated by invoking f.
 170 func allocBytes(f func()) uint64 {
 171         var stats runtime.MemStats
 172         runtime.ReadMemStats(&stats)
 173         t := stats.TotalAlloc
 174         f()
 175         runtime.ReadMemStats(&stats)
 176         return stats.TotalAlloc - t
 177 }
 178
 179 // TestMulUnbalanced tests that multiplying numbers of different lengths
 180 // does not cause deep recursion and in turn allocate too much memory.
 181 // Test case for issue 3807.
 182 func TestMulUnbalanced(t *testing.T) {
 183         defer runtime.GOMAXPROCS(runtime.GOMAXPROCS(1))
 184         x := rndNat(50000)
 185         y := rndNat(40)
 186         allocSize := allocBytes(func() {
 187                 nat(nil).mul(x, y)
 188         })
 189         inputSize := uint64(len(x)+len(y)) * _S
 190         if ratio := allocSize / uint64(inputSize); ratio > 10 {
 191                 t.Errorf("multiplication uses too much memory (%d > %d times the size of inputs)", allocSize, ratio)
 192         }
 193 }
 194
 195 func rndNat(n int) nat {
 196         return nat(rndV(n)).norm()
 197 }
 198
 199 func BenchmarkMul(b *testing.B) {
 200         mulx := rndNat(1e4)
 201         muly := rndNat(1e4)
 202         b.ResetTimer()
 203         for i := 0; i < b.N; i++ {
 204                 var z nat
 205                 z.mul(mulx, muly)
 206         }
 207 }
 208
 209 func TestNLZ(t *testing.T) {
 210         var x Word = _B >> 1
 211         for i := 0; i <= _W; i++ {
 212                 if int(nlz(x)) != i {
 213                         t.Errorf("failed at %x: got %d want %d", x, nlz(x), i)
 214                 }
 215                 x >>= 1
 216         }
 217 }
 218
 219 type shiftTest struct {
 220         in    nat
 221         shift uint
 222         out   nat
 223 }
 224
 225 var leftShiftTests = []shiftTest{
 226         {nil, 0, nil},
 227         {nil, 1, nil},
 228         {natOne, 0, natOne},
 229         {natOne, 1, natTwo},
 230         {nat{1 << (_W - 1)}, 1, nat{0}},
 231         {nat{1 << (_W - 1), 0}, 1, nat{0, 1}},
 232 }
 233
 234 func TestShiftLeft(t *testing.T) {
 235         for i, test := range leftShiftTests {
 236                 var z nat
 237                 z = z.shl(test.in, test.shift)
 238                 for j, d := range test.out {
 239                         if j >= len(z) || z[j] != d {
 240                                 t.Errorf("#%d: got: %v want: %v", i, z, test.out)
 241                                 break
 242                         }
 243                 }
 244         }
 245 }
 246
 247 var rightShiftTests = []shiftTest{
 248         {nil, 0, nil},
 249         {nil, 1, nil},
 250         {natOne, 0, natOne},
 251         {natOne, 1, nil},
 252         {natTwo, 1, natOne},
 253         {nat{0, 1}, 1, nat{1 << (_W - 1)}},
 254         {nat{2, 1, 1}, 1, nat{1<<(_W-1) + 1, 1 << (_W - 1)}},
 255 }
 256
 257 func TestShiftRight(t *testing.T) {
 258         for i, test := range rightShiftTests {
 259                 var z nat
 260                 z = z.shr(test.in, test.shift)
 261                 for j, d := range test.out {
 262                         if j >= len(z) || z[j] != d {
 263                                 t.Errorf("#%d: got: %v want: %v", i, z, test.out)
 264                                 break
 265                         }
 266                 }
 267         }
 268 }
 269
 270 func BenchmarkZeroShifts(b *testing.B) {
 271         x := rndNat(800)
 272
 273         b.Run("Shl", func(b *testing.B) {
 274                 for i := 0; i < b.N; i++ {
 275                         var z nat
 276                         z.shl(x, 0)
 277                 }
 278         })
 279         b.Run("ShlSame", func(b *testing.B) {
 280                 for i := 0; i < b.N; i++ {
 281                         x.shl(x, 0)
 282                 }
 283         })
 284
 285         b.Run("Shr", func(b *testing.B) {
 286                 for i := 0; i < b.N; i++ {
 287                         var z nat
 288                         z.shr(x, 0)
 289                 }
 290         })
 291         b.Run("ShrSame", func(b *testing.B) {
 292                 for i := 0; i < b.N; i++ {
 293                         x.shr(x, 0)
 294                 }
 295         })
 296 }
 297
 298 type modWTest struct {
 299         in       string
 300         dividend string
 301         out      string
 302 }
 303
 304 var modWTests32 = []modWTest{
 305         {"23492635982634928349238759823742", "252341", "220170"},
 306 }
 307
 308 var modWTests64 = []modWTest{
 309         {"6527895462947293856291561095690465243862946", "524326975699234", "375066989628668"},
 310 }
 311
 312 func runModWTests(t *testing.T, tests []modWTest) {
 313         for i, test := range tests {
 314                 in, _ := new(Int).SetString(test.in, 10)
 315                 d, _ := new(Int).SetString(test.dividend, 10)
 316                 out, _ := new(Int).SetString(test.out, 10)
 317
 318                 r := in.abs.modW(d.abs[0])
 319                 if r != out.abs[0] {
 320                         t.Errorf("#%d failed: got %d want %s", i, r, out)
 321                 }
 322         }
 323 }
 324
 325 func TestModW(t *testing.T) {
 326         if _W >= 32 {
 327                 runModWTests(t, modWTests32)
 328         }
 329         if _W >= 64 {
 330                 runModWTests(t, modWTests64)
 331         }
 332 }
 333
 334 var montgomeryTests = []struct {
 335         x, y, m      string
 336         k0           uint64
 337         out32, out64 string
 338 }{
 339         {
 340                 "0xffffffffffffffffffffffffffffffffffffffffffffffffe",
 341                 "0xffffffffffffffffffffffffffffffffffffffffffffffffe",
 342                 "0xfffffffffffffffffffffffffffffffffffffffffffffffff",
 343                 1,
 344                 "0x1000000000000000000000000000000000000000000",
 345                 "0x10000000000000000000000000000000000",
 346         },
 347         {
 348                 "0x000000000ffffff5",
 349                 "0x000000000ffffff0",
 350                 "0x0000000010000001",
 351                 0xff0000000fffffff,
 352                 "0x000000000bfffff4",
 353                 "0x0000000003400001",
 354         },
 355         {
 356                 "0x0000000080000000",
 357                 "0x00000000ffffffff",
 358                 "0x1000000000000001",
 359                 0xfffffffffffffff,
 360                 "0x0800000008000001",
 361                 "0x0800000008000001",
 362         },
 363         {
 364                 "0x0000000080000000",
 365                 "0x0000000080000000",
 366                 "0xffffffff00000001",
 367                 0xfffffffeffffffff,
 368                 "0xbfffffff40000001",
 369                 "0xbfffffff40000001",
 370         },
 371         {
 372                 "0x0000000080000000",
 373                 "0x0000000080000000",
 374                 "0x00ffffff00000001",
 375                 0xfffffeffffffff,
 376                 "0xbfffff40000001",
 377                 "0xbfffff40000001",
 378         },
 379         {
 380                 "0x0000000080000000",
 381                 "0x0000000080000000",
 382                 "0x0000ffff00000001",
 383                 0xfffeffffffff,
 384                 "0xbfff40000001",
 385                 "0xbfff40000001",
 386         },
 387         {
 388                 "0x3321ffffffffffffffffffffffffffff00000000000022222623333333332bbbb888c0",
 389                 "0x3321ffffffffffffffffffffffffffff00000000000022222623333333332bbbb888c0",
 390                 "0x33377fffffffffffffffffffffffffffffffffffffffffffff0000000000022222eee1",
 391                 0xdecc8f1249812adf,
 392                 "0x04eb0e11d72329dc0915f86784820fc403275bf2f6620a20e0dd344c5cd0875e50deb5",
 393                 "0x0d7144739a7d8e11d72329dc0915f86784820fc403275bf2f61ed96f35dd34dbb3d6a0",
 394         },
 395         {
 396                 "0x10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000ffffffffffffffffffffffffffffffff00000000000022222223333333333444444444",
 397                 "0x10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000ffffffffffffffffffffffffffffffff999999999999999aaabbbbbbbbcccccccccccc",
 398                 "0xffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff33377fffffffffffffffffffffffffffffffffffffffffffff0000000000022222eee1",
 399                 0xdecc8f1249812adf,
 400                 "0x5c0d52f451aec609b15da8e5e5626c4eaa88723bdeac9d25ca9b961269400410ca208a16af9c2fb07d7a11c7772cba02c22f9711078d51a3797eb18e691295293284d988e349fa6deba46b25a4ecd9f715",
 401                 "0x92fcad4b5c0d52f451aec609b15da8e5e5626c4eaa88723bdeac9d25ca9b961269400410ca208a16af9c2fb07d799c32fe2f3cc5422f9711078d51a3797eb18e691295293284d8f5e69caf6decddfe1df6",
 402         },
 403 }
 404
 405 func TestMontgomery(t *testing.T) {
 406         one := NewInt(1)
 407         _B := new(Int).Lsh(one, _W)
 408         for i, test := range montgomeryTests {
 409                 x := natFromString(test.x)
 410                 y := natFromString(test.y)
 411                 m := natFromString(test.m)
 412                 for len(x) < len(m) {
 413                         x = append(x, 0)
 414                 }
 415                 for len(y) < len(m) {
 416                         y = append(y, 0)
 417                 }
 418
 419                 if x.cmp(m) > 0 {
 420                         _, r := nat(nil).div(nil, x, m)
 421                         t.Errorf("#%d: x > m (0x%s > 0x%s; use 0x%s)", i, x.utoa(16), m.utoa(16), r.utoa(16))
 422                 }
 423                 if y.cmp(m) > 0 {
 424                         _, r := nat(nil).div(nil, x, m)
 425                         t.Errorf("#%d: y > m (0x%s > 0x%s; use 0x%s)", i, y.utoa(16), m.utoa(16), r.utoa(16))
 426                 }
 427
 428                 var out nat
 429                 if _W == 32 {
 430                         out = natFromString(test.out32)
 431                 } else {
 432                         out = natFromString(test.out64)
 433                 }
 434
 435                 // t.Logf("#%d: len=%d\n", i, len(m))
 436
 437                 // check output in table
 438                 xi := &Int{abs: x}
 439                 yi := &Int{abs: y}
 440                 mi := &Int{abs: m}
 441                 p := new(Int).Mod(new(Int).Mul(xi, new(Int).Mul(yi, new(Int).ModInverse(new(Int).Lsh(one, uint(len(m))*_W), mi))), mi)
 442                 if out.cmp(p.abs.norm()) != 0 {
 443                         t.Errorf("#%d: out in table=0x%s, computed=0x%s", i, out.utoa(16), p.abs.norm().utoa(16))
 444                 }
 445
 446                 // check k0 in table
 447                 k := new(Int).Mod(&Int{abs: m}, _B)
 448                 k = new(Int).Sub(_B, k)
 449                 k = new(Int).Mod(k, _B)
 450                 k0 := Word(new(Int).ModInverse(k, _B).Uint64())
 451                 if k0 != Word(test.k0) {
 452                         t.Errorf("#%d: k0 in table=%#x, computed=%#x\n", i, test.k0, k0)
 453                 }
 454
 455                 // check montgomery with correct k0 produces correct output
 456                 z := nat(nil).montgomery(x, y, m, k0, len(m))
 457                 z = z.norm()
 458                 if z.cmp(out) != 0 {
 459                         t.Errorf("#%d: got 0x%s want 0x%s", i, z.utoa(16), out.utoa(16))
 460                 }
 461         }
 462 }
 463
 464 var expNNTests = []struct {
 465         x, y, m string
 466         out     string
 467 }{
 468         {"0", "0", "0", "1"},
 469         {"0", "0", "1", "0"},
 470         {"1", "1", "1", "0"},
 471         {"2", "1", "1", "0"},
 472         {"2", "2", "1", "0"},
 473         {"10", "100000000000", "1", "0"},
 474         {"0x8000000000000000", "2", "", "0x40000000000000000000000000000000"},
 475         {"0x8000000000000000", "2", "6719", "4944"},
 476         {"0x8000000000000000", "3", "6719", "5447"},
 477         {"0x8000000000000000", "1000", "6719", "1603"},
 478         {"0x8000000000000000", "1000000", "6719", "3199"},
 479         {
 480                 "2938462938472983472983659726349017249287491026512746239764525612965293865296239471239874193284792387498274256129746192347",
 481                 "298472983472983471903246121093472394872319615612417471234712061",
 482                 "29834729834729834729347290846729561262544958723956495615629569234729836259263598127342374289365912465901365498236492183464",
 483                 "23537740700184054162508175125554701713153216681790245129157191391322321508055833908509185839069455749219131480588829346291",
 484         },
 485         {
 486                 "11521922904531591643048817447554701904414021819823889996244743037378330903763518501116638828335352811871131385129455853417360623007349090150042001944696604737499160174391019030572483602867266711107136838523916077674888297896995042968746762200926853379",
 487                 "426343618817810911523",
 488                 "444747819283133684179",
 489                 "42",
 490         },
 491 }
 492
 493 func TestExpNN(t *testing.T) {
 494         for i, test := range expNNTests {
 495                 x := natFromString(test.x)
 496                 y := natFromString(test.y)
 497                 out := natFromString(test.out)
 498
 499                 var m nat
 500                 if len(test.m) > 0 {
 501                         m = natFromString(test.m)
 502                 }
 503
 504                 z := nat(nil).expNN(x, y, m)
 505                 if z.cmp(out) != 0 {
 506                         t.Errorf("#%d got %s want %s", i, z.utoa(10), out.utoa(10))
 507                 }
 508         }
 509 }
 510
 511 func BenchmarkExp3Power(b *testing.B) {
 512         const x = 3
 513         for _, y := range []Word{
 514                 0x10, 0x40, 0x100, 0x400, 0x1000, 0x4000, 0x10000, 0x40000, 0x100000, 0x400000,
 515         } {
 516                 b.Run(fmt.Sprintf("%#x", y), func(b *testing.B) {
 517                         var z nat
 518                         for i := 0; i < b.N; i++ {
 519                                 z.expWW(x, y)
 520                         }
 521                 })
 522         }
 523 }
 524
 525 func fibo(n int) nat {
 526         switch n {
 527         case 0:
 528                 return nil
 529         case 1:
 530                 return nat{1}
 531         }
 532         f0 := fibo(0)
 533         f1 := fibo(1)
 534         var f2 nat
 535         for i := 1; i < n; i++ {
 536                 f2 = f2.add(f0, f1)
 537                 f0, f1, f2 = f1, f2, f0
 538         }
 539         return f1
 540 }
 541
 542 var fiboNums = []string{
 543         "0",
 544         "55",
 545         "6765",
 546         "832040",
 547         "102334155",
 548         "12586269025",
 549         "1548008755920",
 550         "190392490709135",
 551         "23416728348467685",
 552         "2880067194370816120",
 553         "354224848179261915075",
 554 }
 555
 556 func TestFibo(t *testing.T) {
 557         for i, want := range fiboNums {
 558                 n := i * 10
 559                 got := string(fibo(n).utoa(10))
 560                 if got != want {
 561                         t.Errorf("fibo(%d) failed: got %s want %s", n, got, want)
 562                 }
 563         }
 564 }
 565
 566 func BenchmarkFibo(b *testing.B) {
 567         for i := 0; i < b.N; i++ {
 568                 fibo(1e0)
 569                 fibo(1e1)
 570                 fibo(1e2)
 571                 fibo(1e3)
 572                 fibo(1e4)
 573                 fibo(1e5)
 574         }
 575 }
 576
 577 var bitTests = []struct {
 578         x    string
 579         i    uint
 580         want uint
 581 }{
 582         {"0", 0, 0},
 583         {"0", 1, 0},
 584         {"0", 1000, 0},
 585
 586         {"0x1", 0, 1},
 587         {"0x10", 0, 0},
 588         {"0x10", 3, 0},
 589         {"0x10", 4, 1},
 590         {"0x10", 5, 0},
 591
 592         {"0x8000000000000000", 62, 0},
 593         {"0x8000000000000000", 63, 1},
 594         {"0x8000000000000000", 64, 0},
 595
 596         {"0x3" + strings.Repeat("0", 32), 127, 0},
 597         {"0x3" + strings.Repeat("0", 32), 128, 1},
 598         {"0x3" + strings.Repeat("0", 32), 129, 1},
 599         {"0x3" + strings.Repeat("0", 32), 130, 0},
 600 }
 601
 602 func TestBit(t *testing.T) {
 603         for i, test := range bitTests {
 604                 x := natFromString(test.x)
 605                 if got := x.bit(test.i); got != test.want {
 606                         t.Errorf("#%d: %s.bit(%d) = %v; want %v", i, test.x, test.i, got, test.want)
 607                 }
 608         }
 609 }
 610
 611 var stickyTests = []struct {
 612         x    string
 613         i    uint
 614         want uint
 615 }{
 616         {"0", 0, 0},
 617         {"0", 1, 0},
 618         {"0", 1000, 0},
 619
 620         {"0x1", 0, 0},
 621         {"0x1", 1, 1},
 622
 623         {"0x1350", 0, 0},
 624         {"0x1350", 4, 0},
 625         {"0x1350", 5, 1},
 626
 627         {"0x8000000000000000", 63, 0},
 628         {"0x8000000000000000", 64, 1},
 629
 630         {"0x1" + strings.Repeat("0", 100), 400, 0},
 631         {"0x1" + strings.Repeat("0", 100), 401, 1},
 632 }
 633
 634 func TestSticky(t *testing.T) {
 635         for i, test := range stickyTests {
 636                 x := natFromString(test.x)
 637                 if got := x.sticky(test.i); got != test.want {
 638                         t.Errorf("#%d: %s.sticky(%d) = %v; want %v", i, test.x, test.i, got, test.want)
 639                 }
 640                 if test.want == 1 {
 641                         // all subsequent i's should also return 1
 642                         for d := uint(1); d <= 3; d++ {
 643                                 if got := x.sticky(test.i + d); got != 1 {
 644                                         t.Errorf("#%d: %s.sticky(%d) = %v; want %v", i, test.x, test.i+d, got, 1)
 645                                 }
 646                         }
 647                 }
 648         }
 649 }
 650
 651 func testSqr(t *testing.T, x nat) {
 652         got := make(nat, 2*len(x))
 653         want := make(nat, 2*len(x))
 654         got = got.sqr(x)
 655         want = want.mul(x, x)
 656         if got.cmp(want) != 0 {
 657                 t.Errorf("basicSqr(%v), got %v, want %v", x, got, want)
 658         }
 659 }
 660
 661 func TestSqr(t *testing.T) {
 662         for _, a := range prodNN {
 663                 if a.x != nil {
 664                         testSqr(t, a.x)
 665                 }
 666                 if a.y != nil {
 667                         testSqr(t, a.y)
 668                 }
 669                 if a.z != nil {
 670                         testSqr(t, a.z)
 671                 }
 672         }
 673 }
 674
 675 func benchmarkNatSqr(b *testing.B, nwords int) {
 676         x := rndNat(nwords)
 677         var z nat
 678         b.ResetTimer()
 679         for i := 0; i < b.N; i++ {
 680                 z.sqr(x)
 681         }
 682 }
 683
 684 var sqrBenchSizes = []int{1, 2, 3, 5, 8, 10, 20, 30, 50, 80, 100, 200, 300, 500, 800, 1000}
 685
 686 func BenchmarkNatSqr(b *testing.B) {
 687         for _, n := range sqrBenchSizes {
 688                 if isRaceBuilder && n > 1e3 {
 689                         continue
 690                 }
 691                 b.Run(fmt.Sprintf("%d", n), func(b *testing.B) {
 692                         benchmarkNatSqr(b, n)
 693                 })
 694         }
 695 }
 696
 697 func BenchmarkNatSetBytes(b *testing.B) {
 698         const maxLength = 128
 699         lengths := []int{
 700                 // No remainder:
 701                 8, 24, maxLength,
 702                 // With remainder:
 703                 7, 23, maxLength - 1,
 704         }
 705         n := make(nat, maxLength/_W) // ensure n doesn't need to grow during the test
 706         buf := make([]byte, maxLength)
 707         for _, l := range lengths {
 708                 b.Run(fmt.Sprint(l), func(b *testing.B) {
 709                         for i := 0; i < b.N; i++ {
 710                                 n.setBytes(buf[:l])
 711                         }
 712                 })
 713         }
 714 }