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
5package big
6
7import (
8	"fmt"
9	"math"
10	"runtime"
11	"strings"
12	"testing"
13)
14
15var cmpTests = []struct {
16	x, y nat
17	r    int
18}{
19	{nil, nil, 0},
20	{nil, nat(nil), 0},
21	{nat(nil), nil, 0},
22	{nat(nil), nat(nil), 0},
23	{nat{0}, nat{0}, 0},
24	{nat{0}, nat{1}, -1},
25	{nat{1}, nat{0}, 1},
26	{nat{1}, nat{1}, 0},
27	{nat{0, _M}, nat{1}, 1},
28	{nat{1}, nat{0, _M}, -1},
29	{nat{1, _M}, nat{0, _M}, 1},
30	{nat{0, _M}, nat{1, _M}, -1},
31	{nat{16, 571956, 8794, 68}, nat{837, 9146, 1, 754489}, -1},
32	{nat{34986, 41, 105, 1957}, nat{56, 7458, 104, 1957}, 1},
33}
34
35func TestCmp(t *testing.T) {
36	for i, a := range cmpTests {
37		r := a.x.cmp(a.y)
38		if r != a.r {
39			t.Errorf("#%d got r = %v; want %v", i, r, a.r)
40		}
41	}
42}
43
44type funNN func(z, x, y nat) nat
45type argNN struct {
46	z, x, y nat
47}
48
49var sumNN = []argNN{
50	{},
51	{nat{1}, nil, nat{1}},
52	{nat{1111111110}, nat{123456789}, nat{987654321}},
53	{nat{0, 0, 0, 1}, nil, nat{0, 0, 0, 1}},
54	{nat{0, 0, 0, 1111111110}, nat{0, 0, 0, 123456789}, nat{0, 0, 0, 987654321}},
55	{nat{0, 0, 0, 1}, nat{0, 0, _M}, nat{0, 0, 1}},
56}
57
58var prodNN = []argNN{
59	{},
60	{nil, nil, nil},
61	{nil, nat{991}, nil},
62	{nat{991}, nat{991}, nat{1}},
63	{nat{991 * 991}, nat{991}, nat{991}},
64	{nat{0, 0, 991 * 991}, nat{0, 991}, nat{0, 991}},
65	{nat{1 * 991, 2 * 991, 3 * 991, 4 * 991}, nat{1, 2, 3, 4}, nat{991}},
66	{nat{4, 11, 20, 30, 20, 11, 4}, nat{1, 2, 3, 4}, nat{4, 3, 2, 1}},
67	// 3^100 * 3^28 = 3^128
68	{
69		natFromString("11790184577738583171520872861412518665678211592275841109096961"),
70		natFromString("515377520732011331036461129765621272702107522001"),
71		natFromString("22876792454961"),
72	},
73	// z = 111....1 (70000 digits)
74	// x = 10^(99*700) + ... + 10^1400 + 10^700 + 1
75	// y = 111....1 (700 digits, larger than Karatsuba threshold on 32-bit and 64-bit)
76	{
77		natFromString(strings.Repeat("1", 70000)),
78		natFromString("1" + strings.Repeat(strings.Repeat("0", 699)+"1", 99)),
79		natFromString(strings.Repeat("1", 700)),
80	},
81	// z = 111....1 (20000 digits)
82	// x = 10^10000 + 1
83	// y = 111....1 (10000 digits)
84	{
85		natFromString(strings.Repeat("1", 20000)),
86		natFromString("1" + strings.Repeat("0", 9999) + "1"),
87		natFromString(strings.Repeat("1", 10000)),
88	},
89}
90
91func natFromString(s string) nat {
92	x, _, _, err := nat(nil).scan(strings.NewReader(s), 0, false)
93	if err != nil {
94		panic(err)
95	}
96	return x
97}
98
99func TestSet(t *testing.T) {
100	for _, a := range sumNN {
101		z := nat(nil).set(a.z)
102		if z.cmp(a.z) != 0 {
103			t.Errorf("got z = %v; want %v", z, a.z)
104		}
105	}
106}
107
108func testFunNN(t *testing.T, msg string, f funNN, a argNN) {
109	z := f(nil, a.x, a.y)
110	if z.cmp(a.z) != 0 {
111		t.Errorf("%s%+v\n\tgot z = %v; want %v", msg, a, z, a.z)
112	}
113}
114
115func TestFunNN(t *testing.T) {
116	for _, a := range sumNN {
117		arg := a
118		testFunNN(t, "add", nat.add, arg)
119
120		arg = argNN{a.z, a.y, a.x}
121		testFunNN(t, "add symmetric", nat.add, arg)
122
123		arg = argNN{a.x, a.z, a.y}
124		testFunNN(t, "sub", nat.sub, arg)
125
126		arg = argNN{a.y, a.z, a.x}
127		testFunNN(t, "sub symmetric", nat.sub, arg)
128	}
129
130	for _, a := range prodNN {
131		arg := a
132		testFunNN(t, "mul", nat.mul, arg)
133
134		arg = argNN{a.z, a.y, a.x}
135		testFunNN(t, "mul symmetric", nat.mul, arg)
136	}
137}
138
139var mulRangesN = []struct {
140	a, b uint64
141	prod string
142}{
143	{0, 0, "0"},
144	{1, 1, "1"},
145	{1, 2, "2"},
146	{1, 3, "6"},
147	{10, 10, "10"},
148	{0, 100, "0"},
149	{0, 1e9, "0"},
150	{1, 0, "1"},                    // empty range
151	{100, 1, "1"},                  // empty range
152	{1, 10, "3628800"},             // 10!
153	{1, 20, "2432902008176640000"}, // 20!
154	{1, 100,
155		"933262154439441526816992388562667004907159682643816214685929" +
156			"638952175999932299156089414639761565182862536979208272237582" +
157			"51185210916864000000000000000000000000", // 100!
158	},
159	{math.MaxUint64 - 0, math.MaxUint64, "18446744073709551615"},
160	{math.MaxUint64 - 1, math.MaxUint64, "340282366920938463408034375210639556610"},
161	{math.MaxUint64 - 2, math.MaxUint64, "6277101735386680761794095221682035635525021984684230311930"},
162	{math.MaxUint64 - 3, math.MaxUint64, "115792089237316195360799967654821100226821973275796746098729803619699194331160"},
163}
164
165func TestMulRangeN(t *testing.T) {
166	for i, r := range mulRangesN {
167		prod := string(nat(nil).mulRange(r.a, r.b).utoa(10))
168		if prod != r.prod {
169			t.Errorf("#%d: got %s; want %s", i, prod, r.prod)
170		}
171	}
172}
173
174// allocBytes returns the number of bytes allocated by invoking f.
175func allocBytes(f func()) uint64 {
176	var stats runtime.MemStats
177	runtime.ReadMemStats(&stats)
178	t := stats.TotalAlloc
179	f()
180	runtime.ReadMemStats(&stats)
181	return stats.TotalAlloc - t
182}
183
184// TestMulUnbalanced tests that multiplying numbers of different lengths
185// does not cause deep recursion and in turn allocate too much memory.
186// Test case for issue 3807.
187func TestMulUnbalanced(t *testing.T) {
188	defer runtime.GOMAXPROCS(runtime.GOMAXPROCS(1))
189	x := rndNat(50000)
190	y := rndNat(40)
191	allocSize := allocBytes(func() {
192		nat(nil).mul(x, y)
193	})
194	inputSize := uint64(len(x)+len(y)) * _S
195	if ratio := allocSize / uint64(inputSize); ratio > 10 {
196		t.Errorf("multiplication uses too much memory (%d > %d times the size of inputs)", allocSize, ratio)
197	}
198}
199
200// rndNat returns a random nat value >= 0 of (usually) n words in length.
201// In extremely unlikely cases it may be smaller than n words if the top-
202// most words are 0.
203func rndNat(n int) nat {
204	return nat(rndV(n)).norm()
205}
206
207// rndNat1 is like rndNat but the result is guaranteed to be > 0.
208func rndNat1(n int) nat {
209	x := nat(rndV(n)).norm()
210	if len(x) == 0 {
211		x.setWord(1)
212	}
213	return x
214}
215
216func BenchmarkMul(b *testing.B) {
217	mulx := rndNat(1e4)
218	muly := rndNat(1e4)
219	b.ResetTimer()
220	for i := 0; i < b.N; i++ {
221		var z nat
222		z.mul(mulx, muly)
223	}
224}
225
226func benchmarkNatMul(b *testing.B, nwords int) {
227	x := rndNat(nwords)
228	y := rndNat(nwords)
229	var z nat
230	b.ResetTimer()
231	for i := 0; i < b.N; i++ {
232		z.mul(x, y)
233	}
234}
235
236var mulBenchSizes = []int{10, 100, 1000, 10000, 100000}
237
238func BenchmarkNatMul(b *testing.B) {
239	for _, n := range mulBenchSizes {
240		if isRaceBuilder && n > 1e3 {
241			continue
242		}
243		b.Run(fmt.Sprintf("%d", n), func(b *testing.B) {
244			benchmarkNatMul(b, n)
245		})
246	}
247}
248
249func TestNLZ(t *testing.T) {
250	var x Word = _B >> 1
251	for i := 0; i <= _W; i++ {
252		if int(nlz(x)) != i {
253			t.Errorf("failed at %x: got %d want %d", x, nlz(x), i)
254		}
255		x >>= 1
256	}
257}
258
259type shiftTest struct {
260	in    nat
261	shift uint
262	out   nat
263}
264
265var leftShiftTests = []shiftTest{
266	{nil, 0, nil},
267	{nil, 1, nil},
268	{natOne, 0, natOne},
269	{natOne, 1, natTwo},
270	{nat{1 << (_W - 1)}, 1, nat{0}},
271	{nat{1 << (_W - 1), 0}, 1, nat{0, 1}},
272}
273
274func TestShiftLeft(t *testing.T) {
275	for i, test := range leftShiftTests {
276		var z nat
277		z = z.shl(test.in, test.shift)
278		for j, d := range test.out {
279			if j >= len(z) || z[j] != d {
280				t.Errorf("#%d: got: %v want: %v", i, z, test.out)
281				break
282			}
283		}
284	}
285}
286
287var rightShiftTests = []shiftTest{
288	{nil, 0, nil},
289	{nil, 1, nil},
290	{natOne, 0, natOne},
291	{natOne, 1, nil},
292	{natTwo, 1, natOne},
293	{nat{0, 1}, 1, nat{1 << (_W - 1)}},
294	{nat{2, 1, 1}, 1, nat{1<<(_W-1) + 1, 1 << (_W - 1)}},
295}
296
297func TestShiftRight(t *testing.T) {
298	for i, test := range rightShiftTests {
299		var z nat
300		z = z.shr(test.in, test.shift)
301		for j, d := range test.out {
302			if j >= len(z) || z[j] != d {
303				t.Errorf("#%d: got: %v want: %v", i, z, test.out)
304				break
305			}
306		}
307	}
308}
309
310func BenchmarkZeroShifts(b *testing.B) {
311	x := rndNat(800)
312
313	b.Run("Shl", func(b *testing.B) {
314		for i := 0; i < b.N; i++ {
315			var z nat
316			z.shl(x, 0)
317		}
318	})
319	b.Run("ShlSame", func(b *testing.B) {
320		for i := 0; i < b.N; i++ {
321			x.shl(x, 0)
322		}
323	})
324
325	b.Run("Shr", func(b *testing.B) {
326		for i := 0; i < b.N; i++ {
327			var z nat
328			z.shr(x, 0)
329		}
330	})
331	b.Run("ShrSame", func(b *testing.B) {
332		for i := 0; i < b.N; i++ {
333			x.shr(x, 0)
334		}
335	})
336}
337
338type modWTest struct {
339	in       string
340	dividend string
341	out      string
342}
343
344var modWTests32 = []modWTest{
345	{"23492635982634928349238759823742", "252341", "220170"},
346}
347
348var modWTests64 = []modWTest{
349	{"6527895462947293856291561095690465243862946", "524326975699234", "375066989628668"},
350}
351
352func runModWTests(t *testing.T, tests []modWTest) {
353	for i, test := range tests {
354		in, _ := new(Int).SetString(test.in, 10)
355		d, _ := new(Int).SetString(test.dividend, 10)
356		out, _ := new(Int).SetString(test.out, 10)
357
358		r := in.abs.modW(d.abs[0])
359		if r != out.abs[0] {
360			t.Errorf("#%d failed: got %d want %s", i, r, out)
361		}
362	}
363}
364
365func TestModW(t *testing.T) {
366	if _W >= 32 {
367		runModWTests(t, modWTests32)
368	}
369	if _W >= 64 {
370		runModWTests(t, modWTests64)
371	}
372}
373
374var montgomeryTests = []struct {
375	x, y, m      string
376	k0           uint64
377	out32, out64 string
378}{
379	{
380		"0xffffffffffffffffffffffffffffffffffffffffffffffffe",
381		"0xffffffffffffffffffffffffffffffffffffffffffffffffe",
382		"0xfffffffffffffffffffffffffffffffffffffffffffffffff",
383		1,
384		"0x1000000000000000000000000000000000000000000",
385		"0x10000000000000000000000000000000000",
386	},
387	{
388		"0x000000000ffffff5",
389		"0x000000000ffffff0",
390		"0x0000000010000001",
391		0xff0000000fffffff,
392		"0x000000000bfffff4",
393		"0x0000000003400001",
394	},
395	{
396		"0x0000000080000000",
397		"0x00000000ffffffff",
398		"0x1000000000000001",
399		0xfffffffffffffff,
400		"0x0800000008000001",
401		"0x0800000008000001",
402	},
403	{
404		"0x0000000080000000",
405		"0x0000000080000000",
406		"0xffffffff00000001",
407		0xfffffffeffffffff,
408		"0xbfffffff40000001",
409		"0xbfffffff40000001",
410	},
411	{
412		"0x0000000080000000",
413		"0x0000000080000000",
414		"0x00ffffff00000001",
415		0xfffffeffffffff,
416		"0xbfffff40000001",
417		"0xbfffff40000001",
418	},
419	{
420		"0x0000000080000000",
421		"0x0000000080000000",
422		"0x0000ffff00000001",
423		0xfffeffffffff,
424		"0xbfff40000001",
425		"0xbfff40000001",
426	},
427	{
428		"0x3321ffffffffffffffffffffffffffff00000000000022222623333333332bbbb888c0",
429		"0x3321ffffffffffffffffffffffffffff00000000000022222623333333332bbbb888c0",
430		"0x33377fffffffffffffffffffffffffffffffffffffffffffff0000000000022222eee1",
431		0xdecc8f1249812adf,
432		"0x04eb0e11d72329dc0915f86784820fc403275bf2f6620a20e0dd344c5cd0875e50deb5",
433		"0x0d7144739a7d8e11d72329dc0915f86784820fc403275bf2f61ed96f35dd34dbb3d6a0",
434	},
435	{
436		"0x10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000ffffffffffffffffffffffffffffffff00000000000022222223333333333444444444",
437		"0x10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000ffffffffffffffffffffffffffffffff999999999999999aaabbbbbbbbcccccccccccc",
438		"0xffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff33377fffffffffffffffffffffffffffffffffffffffffffff0000000000022222eee1",
439		0xdecc8f1249812adf,
440		"0x5c0d52f451aec609b15da8e5e5626c4eaa88723bdeac9d25ca9b961269400410ca208a16af9c2fb07d7a11c7772cba02c22f9711078d51a3797eb18e691295293284d988e349fa6deba46b25a4ecd9f715",
441		"0x92fcad4b5c0d52f451aec609b15da8e5e5626c4eaa88723bdeac9d25ca9b961269400410ca208a16af9c2fb07d799c32fe2f3cc5422f9711078d51a3797eb18e691295293284d8f5e69caf6decddfe1df6",
442	},
443}
444
445func TestMontgomery(t *testing.T) {
446	one := NewInt(1)
447	_B := new(Int).Lsh(one, _W)
448	for i, test := range montgomeryTests {
449		x := natFromString(test.x)
450		y := natFromString(test.y)
451		m := natFromString(test.m)
452		for len(x) < len(m) {
453			x = append(x, 0)
454		}
455		for len(y) < len(m) {
456			y = append(y, 0)
457		}
458
459		if x.cmp(m) > 0 {
460			_, r := nat(nil).div(nil, x, m)
461			t.Errorf("#%d: x > m (0x%s > 0x%s; use 0x%s)", i, x.utoa(16), m.utoa(16), r.utoa(16))
462		}
463		if y.cmp(m) > 0 {
464			_, r := nat(nil).div(nil, x, m)
465			t.Errorf("#%d: y > m (0x%s > 0x%s; use 0x%s)", i, y.utoa(16), m.utoa(16), r.utoa(16))
466		}
467
468		var out nat
469		if _W == 32 {
470			out = natFromString(test.out32)
471		} else {
472			out = natFromString(test.out64)
473		}
474
475		// t.Logf("#%d: len=%d\n", i, len(m))
476
477		// check output in table
478		xi := &Int{abs: x}
479		yi := &Int{abs: y}
480		mi := &Int{abs: m}
481		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)
482		if out.cmp(p.abs.norm()) != 0 {
483			t.Errorf("#%d: out in table=0x%s, computed=0x%s", i, out.utoa(16), p.abs.norm().utoa(16))
484		}
485
486		// check k0 in table
487		k := new(Int).Mod(&Int{abs: m}, _B)
488		k = new(Int).Sub(_B, k)
489		k = new(Int).Mod(k, _B)
490		k0 := Word(new(Int).ModInverse(k, _B).Uint64())
491		if k0 != Word(test.k0) {
492			t.Errorf("#%d: k0 in table=%#x, computed=%#x\n", i, test.k0, k0)
493		}
494
495		// check montgomery with correct k0 produces correct output
496		z := nat(nil).montgomery(x, y, m, k0, len(m))
497		z = z.norm()
498		if z.cmp(out) != 0 {
499			t.Errorf("#%d: got 0x%s want 0x%s", i, z.utoa(16), out.utoa(16))
500		}
501	}
502}
503
504var expNNTests = []struct {
505	x, y, m string
506	out     string
507}{
508	{"0", "0", "0", "1"},
509	{"0", "0", "1", "0"},
510	{"1", "1", "1", "0"},
511	{"2", "1", "1", "0"},
512	{"2", "2", "1", "0"},
513	{"10", "100000000000", "1", "0"},
514	{"0x8000000000000000", "2", "", "0x40000000000000000000000000000000"},
515	{"0x8000000000000000", "2", "6719", "4944"},
516	{"0x8000000000000000", "3", "6719", "5447"},
517	{"0x8000000000000000", "1000", "6719", "1603"},
518	{"0x8000000000000000", "1000000", "6719", "3199"},
519	{
520		"2938462938472983472983659726349017249287491026512746239764525612965293865296239471239874193284792387498274256129746192347",
521		"298472983472983471903246121093472394872319615612417471234712061",
522		"29834729834729834729347290846729561262544958723956495615629569234729836259263598127342374289365912465901365498236492183464",
523		"23537740700184054162508175125554701713153216681790245129157191391322321508055833908509185839069455749219131480588829346291",
524	},
525	{
526		"11521922904531591643048817447554701904414021819823889996244743037378330903763518501116638828335352811871131385129455853417360623007349090150042001944696604737499160174391019030572483602867266711107136838523916077674888297896995042968746762200926853379",
527		"426343618817810911523",
528		"444747819283133684179",
529		"42",
530	},
531	{"375", "249", "388", "175"},
532	{"375", "18446744073709551801", "388", "175"},
533	{"0", "0x40000000000000", "0x200", "0"},
534	{"0xeffffff900002f00", "0x40000000000000", "0x200", "0"},
535	{"5", "1435700818", "72", "49"},
536	{"0xffff", "0x300030003000300030003000300030003000302a3000300030003000300030003000300030003000300030003000300030003030623066307f3030783062303430383064303630343036", "0x300000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000", "0xa3f94c08b0b90e87af637cacc9383f7ea032352b8961fc036a52b659b6c9b33491b335ffd74c927f64ddd62cfca0001"},
537}
538
539func TestExpNN(t *testing.T) {
540	for i, test := range expNNTests {
541		x := natFromString(test.x)
542		y := natFromString(test.y)
543		out := natFromString(test.out)
544
545		var m nat
546		if len(test.m) > 0 {
547			m = natFromString(test.m)
548		}
549
550		z := nat(nil).expNN(x, y, m, false)
551		if z.cmp(out) != 0 {
552			t.Errorf("#%d got %s want %s", i, z.utoa(10), out.utoa(10))
553		}
554	}
555}
556
557func FuzzExpMont(f *testing.F) {
558	f.Fuzz(func(t *testing.T, x1, x2, x3, y1, y2, y3, m1, m2, m3 uint) {
559		if m1 == 0 && m2 == 0 && m3 == 0 {
560			return
561		}
562		x := new(Int).SetBits([]Word{Word(x1), Word(x2), Word(x3)})
563		y := new(Int).SetBits([]Word{Word(y1), Word(y2), Word(y3)})
564		m := new(Int).SetBits([]Word{Word(m1), Word(m2), Word(m3)})
565		out := new(Int).Exp(x, y, m)
566		want := new(Int).expSlow(x, y, m)
567		if out.Cmp(want) != 0 {
568			t.Errorf("x = %#x\ny=%#x\nz=%#x\nout=%#x\nwant=%#x\ndc: 16o 16i %X %X %X |p", x, y, m, out, want, x, y, m)
569		}
570	})
571}
572
573func BenchmarkExp3Power(b *testing.B) {
574	const x = 3
575	for _, y := range []Word{
576		0x10, 0x40, 0x100, 0x400, 0x1000, 0x4000, 0x10000, 0x40000, 0x100000, 0x400000,
577	} {
578		b.Run(fmt.Sprintf("%#x", y), func(b *testing.B) {
579			var z nat
580			for i := 0; i < b.N; i++ {
581				z.expWW(x, y)
582			}
583		})
584	}
585}
586
587func fibo(n int) nat {
588	switch n {
589	case 0:
590		return nil
591	case 1:
592		return nat{1}
593	}
594	f0 := fibo(0)
595	f1 := fibo(1)
596	var f2 nat
597	for i := 1; i < n; i++ {
598		f2 = f2.add(f0, f1)
599		f0, f1, f2 = f1, f2, f0
600	}
601	return f1
602}
603
604var fiboNums = []string{
605	"0",
606	"55",
607	"6765",
608	"832040",
609	"102334155",
610	"12586269025",
611	"1548008755920",
612	"190392490709135",
613	"23416728348467685",
614	"2880067194370816120",
615	"354224848179261915075",
616}
617
618func TestFibo(t *testing.T) {
619	for i, want := range fiboNums {
620		n := i * 10
621		got := string(fibo(n).utoa(10))
622		if got != want {
623			t.Errorf("fibo(%d) failed: got %s want %s", n, got, want)
624		}
625	}
626}
627
628func BenchmarkFibo(b *testing.B) {
629	for i := 0; i < b.N; i++ {
630		fibo(1e0)
631		fibo(1e1)
632		fibo(1e2)
633		fibo(1e3)
634		fibo(1e4)
635		fibo(1e5)
636	}
637}
638
639var bitTests = []struct {
640	x    string
641	i    uint
642	want uint
643}{
644	{"0", 0, 0},
645	{"0", 1, 0},
646	{"0", 1000, 0},
647
648	{"0x1", 0, 1},
649	{"0x10", 0, 0},
650	{"0x10", 3, 0},
651	{"0x10", 4, 1},
652	{"0x10", 5, 0},
653
654	{"0x8000000000000000", 62, 0},
655	{"0x8000000000000000", 63, 1},
656	{"0x8000000000000000", 64, 0},
657
658	{"0x3" + strings.Repeat("0", 32), 127, 0},
659	{"0x3" + strings.Repeat("0", 32), 128, 1},
660	{"0x3" + strings.Repeat("0", 32), 129, 1},
661	{"0x3" + strings.Repeat("0", 32), 130, 0},
662}
663
664func TestBit(t *testing.T) {
665	for i, test := range bitTests {
666		x := natFromString(test.x)
667		if got := x.bit(test.i); got != test.want {
668			t.Errorf("#%d: %s.bit(%d) = %v; want %v", i, test.x, test.i, got, test.want)
669		}
670	}
671}
672
673var stickyTests = []struct {
674	x    string
675	i    uint
676	want uint
677}{
678	{"0", 0, 0},
679	{"0", 1, 0},
680	{"0", 1000, 0},
681
682	{"0x1", 0, 0},
683	{"0x1", 1, 1},
684
685	{"0x1350", 0, 0},
686	{"0x1350", 4, 0},
687	{"0x1350", 5, 1},
688
689	{"0x8000000000000000", 63, 0},
690	{"0x8000000000000000", 64, 1},
691
692	{"0x1" + strings.Repeat("0", 100), 400, 0},
693	{"0x1" + strings.Repeat("0", 100), 401, 1},
694}
695
696func TestSticky(t *testing.T) {
697	for i, test := range stickyTests {
698		x := natFromString(test.x)
699		if got := x.sticky(test.i); got != test.want {
700			t.Errorf("#%d: %s.sticky(%d) = %v; want %v", i, test.x, test.i, got, test.want)
701		}
702		if test.want == 1 {
703			// all subsequent i's should also return 1
704			for d := uint(1); d <= 3; d++ {
705				if got := x.sticky(test.i + d); got != 1 {
706					t.Errorf("#%d: %s.sticky(%d) = %v; want %v", i, test.x, test.i+d, got, 1)
707				}
708			}
709		}
710	}
711}
712
713func testSqr(t *testing.T, x nat) {
714	got := make(nat, 2*len(x))
715	want := make(nat, 2*len(x))
716	got = got.sqr(x)
717	want = want.mul(x, x)
718	if got.cmp(want) != 0 {
719		t.Errorf("basicSqr(%v), got %v, want %v", x, got, want)
720	}
721}
722
723func TestSqr(t *testing.T) {
724	for _, a := range prodNN {
725		if a.x != nil {
726			testSqr(t, a.x)
727		}
728		if a.y != nil {
729			testSqr(t, a.y)
730		}
731		if a.z != nil {
732			testSqr(t, a.z)
733		}
734	}
735}
736
737func benchmarkNatSqr(b *testing.B, nwords int) {
738	x := rndNat(nwords)
739	var z nat
740	b.ResetTimer()
741	for i := 0; i < b.N; i++ {
742		z.sqr(x)
743	}
744}
745
746var sqrBenchSizes = []int{
747	1, 2, 3, 5, 8, 10, 20, 30, 50, 80,
748	100, 200, 300, 500, 800,
749	1000, 10000, 100000,
750}
751
752func BenchmarkNatSqr(b *testing.B) {
753	for _, n := range sqrBenchSizes {
754		if isRaceBuilder && n > 1e3 {
755			continue
756		}
757		b.Run(fmt.Sprintf("%d", n), func(b *testing.B) {
758			benchmarkNatSqr(b, n)
759		})
760	}
761}
762
763var subMod2NTests = []struct {
764	x string
765	y string
766	n uint
767	z string
768}{
769	{"1", "2", 0, "0"},
770	{"1", "0", 1, "1"},
771	{"0", "1", 1, "1"},
772	{"3", "5", 3, "6"},
773	{"5", "3", 3, "2"},
774	// 2^65, 2^66-1, 2^65 - (2^66-1) + 2^67
775	{"36893488147419103232", "73786976294838206463", 67, "110680464442257309697"},
776	// 2^66-1, 2^65, 2^65-1
777	{"73786976294838206463", "36893488147419103232", 67, "36893488147419103231"},
778}
779
780func TestNatSubMod2N(t *testing.T) {
781	for _, mode := range []string{"noalias", "aliasX", "aliasY"} {
782		t.Run(mode, func(t *testing.T) {
783			for _, tt := range subMod2NTests {
784				x0 := natFromString(tt.x)
785				y0 := natFromString(tt.y)
786				want := natFromString(tt.z)
787				x := nat(nil).set(x0)
788				y := nat(nil).set(y0)
789				var z nat
790				switch mode {
791				case "aliasX":
792					z = x
793				case "aliasY":
794					z = y
795				}
796				z = z.subMod2N(x, y, tt.n)
797				if z.cmp(want) != 0 {
798					t.Fatalf("subMod2N(%d, %d, %d) = %d, want %d", x0, y0, tt.n, z, want)
799				}
800				if mode != "aliasX" && x.cmp(x0) != 0 {
801					t.Fatalf("subMod2N(%d, %d, %d) modified x", x0, y0, tt.n)
802				}
803				if mode != "aliasY" && y.cmp(y0) != 0 {
804					t.Fatalf("subMod2N(%d, %d, %d) modified y", x0, y0, tt.n)
805				}
806			}
807		})
808	}
809}
810
811func BenchmarkNatSetBytes(b *testing.B) {
812	const maxLength = 128
813	lengths := []int{
814		// No remainder:
815		8, 24, maxLength,
816		// With remainder:
817		7, 23, maxLength - 1,
818	}
819	n := make(nat, maxLength/_W) // ensure n doesn't need to grow during the test
820	buf := make([]byte, maxLength)
821	for _, l := range lengths {
822		b.Run(fmt.Sprint(l), func(b *testing.B) {
823			for i := 0; i < b.N; i++ {
824				n.setBytes(buf[:l])
825			}
826		})
827	}
828}
829
830func TestNatDiv(t *testing.T) {
831	sizes := []int{
832		1, 2, 5, 8, 15, 25, 40, 65, 100,
833		200, 500, 800, 1500, 2500, 4000, 6500, 10000,
834	}
835	for _, i := range sizes {
836		for _, j := range sizes {
837			a := rndNat1(i)
838			b := rndNat1(j)
839			// the test requires b >= 2
840			if len(b) == 1 && b[0] == 1 {
841				b[0] = 2
842			}
843			// choose a remainder c < b
844			c := rndNat1(len(b))
845			if len(c) == len(b) && c[len(c)-1] >= b[len(b)-1] {
846				c[len(c)-1] = 0
847				c = c.norm()
848			}
849			// compute x = a*b+c
850			x := nat(nil).mul(a, b)
851			x = x.add(x, c)
852
853			var q, r nat
854			q, r = q.div(r, x, b)
855			if q.cmp(a) != 0 {
856				t.Fatalf("wrong quotient: got %s; want %s for %s/%s", q.utoa(10), a.utoa(10), x.utoa(10), b.utoa(10))
857			}
858			if r.cmp(c) != 0 {
859				t.Fatalf("wrong remainder: got %s; want %s for %s/%s", r.utoa(10), c.utoa(10), x.utoa(10), b.utoa(10))
860			}
861		}
862	}
863}
864
865// TestIssue37499 triggers the edge case of divBasic where
866// the inaccurate estimate of the first word's quotient
867// happens at the very beginning of the loop.
868func TestIssue37499(t *testing.T) {
869	// Choose u and v such that v is slightly larger than u >> N.
870	// This tricks divBasic into choosing 1 as the first word
871	// of the quotient. This works in both 32-bit and 64-bit settings.
872	u := natFromString("0x2b6c385a05be027f5c22005b63c42a1165b79ff510e1706b39f8489c1d28e57bb5ba4ef9fd9387a3e344402c0a453381")
873	v := natFromString("0x2b6c385a05be027f5c22005b63c42a1165b79ff510e1706c")
874
875	q := nat(nil).make(8)
876	q.divBasic(u, v)
877	q = q.norm()
878	if s := string(q.utoa(16)); s != "fffffffffffffffffffffffffffffffffffffffffffffffb" {
879		t.Fatalf("incorrect quotient: %s", s)
880	}
881}
882
883// TestIssue42552 triggers an edge case of recursive division
884// where the first division loop is never entered, and correcting
885// the remainder takes exactly two iterations in the final loop.
886func TestIssue42552(t *testing.T) {
887	u := natFromString("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")
888	v := natFromString("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")
889	q := nat(nil).make(16)
890	q.div(q, u, v)
891}
892