Adding upstream version 1.16.10.upstream/1.16.10upstream

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

1 files changed, 816 insertions, 0 deletions

diff --git a/src/math/big/nat_test.go b/src/math/big/nat_test.go
new file mode 100644
index 0000000..0850818
--- /dev/null
+++ b/src/math/big/nat_test.go
@@ -0,0 +1,816 @@
+// Copyright 2009 The Go Authors. All rights reserved.
+// Use of this source code is governed by a BSD-style
+// license that can be found in the LICENSE file.
+
+package big
+
+import (
+	"fmt"
+	"runtime"
+	"strings"
+	"testing"
+)
+
+var cmpTests = []struct {
+	x, y nat
+	r    int
+}{
+	{nil, nil, 0},
+	{nil, nat(nil), 0},
+	{nat(nil), nil, 0},
+	{nat(nil), nat(nil), 0},
+	{nat{0}, nat{0}, 0},
+	{nat{0}, nat{1}, -1},
+	{nat{1}, nat{0}, 1},
+	{nat{1}, nat{1}, 0},
+	{nat{0, _M}, nat{1}, 1},
+	{nat{1}, nat{0, _M}, -1},
+	{nat{1, _M}, nat{0, _M}, 1},
+	{nat{0, _M}, nat{1, _M}, -1},
+	{nat{16, 571956, 8794, 68}, nat{837, 9146, 1, 754489}, -1},
+	{nat{34986, 41, 105, 1957}, nat{56, 7458, 104, 1957}, 1},
+}
+
+func TestCmp(t *testing.T) {
+	for i, a := range cmpTests {
+		r := a.x.cmp(a.y)
+		if r != a.r {
+			t.Errorf("#%d got r = %v; want %v", i, r, a.r)
+		}
+	}
+}
+
+type funNN func(z, x, y nat) nat
+type argNN struct {
+	z, x, y nat
+}
+
+var sumNN = []argNN{
+	{},
+	{nat{1}, nil, nat{1}},
+	{nat{1111111110}, nat{123456789}, nat{987654321}},
+	{nat{0, 0, 0, 1}, nil, nat{0, 0, 0, 1}},
+	{nat{0, 0, 0, 1111111110}, nat{0, 0, 0, 123456789}, nat{0, 0, 0, 987654321}},
+	{nat{0, 0, 0, 1}, nat{0, 0, _M}, nat{0, 0, 1}},
+}
+
+var prodNN = []argNN{
+	{},
+	{nil, nil, nil},
+	{nil, nat{991}, nil},
+	{nat{991}, nat{991}, nat{1}},
+	{nat{991 * 991}, nat{991}, nat{991}},
+	{nat{0, 0, 991 * 991}, nat{0, 991}, nat{0, 991}},
+	{nat{1 * 991, 2 * 991, 3 * 991, 4 * 991}, nat{1, 2, 3, 4}, nat{991}},
+	{nat{4, 11, 20, 30, 20, 11, 4}, nat{1, 2, 3, 4}, nat{4, 3, 2, 1}},
+	// 3^100 * 3^28 = 3^128
+	{
+		natFromString("11790184577738583171520872861412518665678211592275841109096961"),
+		natFromString("515377520732011331036461129765621272702107522001"),
+		natFromString("22876792454961"),
+	},
+	// z = 111....1 (70000 digits)
+	// x = 10^(99*700) + ... + 10^1400 + 10^700 + 1
+	// y = 111....1 (700 digits, larger than Karatsuba threshold on 32-bit and 64-bit)
+	{
+		natFromString(strings.Repeat("1", 70000)),
+		natFromString("1" + strings.Repeat(strings.Repeat("0", 699)+"1", 99)),
+		natFromString(strings.Repeat("1", 700)),
+	},
+	// z = 111....1 (20000 digits)
+	// x = 10^10000 + 1
+	// y = 111....1 (10000 digits)
+	{
+		natFromString(strings.Repeat("1", 20000)),
+		natFromString("1" + strings.Repeat("0", 9999) + "1"),
+		natFromString(strings.Repeat("1", 10000)),
+	},
+}
+
+func natFromString(s string) nat {
+	x, _, _, err := nat(nil).scan(strings.NewReader(s), 0, false)
+	if err != nil {
+		panic(err)
+	}
+	return x
+}
+
+func TestSet(t *testing.T) {
+	for _, a := range sumNN {
+		z := nat(nil).set(a.z)
+		if z.cmp(a.z) != 0 {
+			t.Errorf("got z = %v; want %v", z, a.z)
+		}
+	}
+}
+
+func testFunNN(t *testing.T, msg string, f funNN, a argNN) {
+	z := f(nil, a.x, a.y)
+	if z.cmp(a.z) != 0 {
+		t.Errorf("%s%+v\n\tgot z = %v; want %v", msg, a, z, a.z)
+	}
+}
+
+func TestFunNN(t *testing.T) {
+	for _, a := range sumNN {
+		arg := a
+		testFunNN(t, "add", nat.add, arg)
+
+		arg = argNN{a.z, a.y, a.x}
+		testFunNN(t, "add symmetric", nat.add, arg)
+
+		arg = argNN{a.x, a.z, a.y}
+		testFunNN(t, "sub", nat.sub, arg)
+
+		arg = argNN{a.y, a.z, a.x}
+		testFunNN(t, "sub symmetric", nat.sub, arg)
+	}
+
+	for _, a := range prodNN {
+		arg := a
+		testFunNN(t, "mul", nat.mul, arg)
+
+		arg = argNN{a.z, a.y, a.x}
+		testFunNN(t, "mul symmetric", nat.mul, arg)
+	}
+}
+
+var mulRangesN = []struct {
+	a, b uint64
+	prod string
+}{
+	{0, 0, "0"},
+	{1, 1, "1"},
+	{1, 2, "2"},
+	{1, 3, "6"},
+	{10, 10, "10"},
+	{0, 100, "0"},
+	{0, 1e9, "0"},
+	{1, 0, "1"},                    // empty range
+	{100, 1, "1"},                  // empty range
+	{1, 10, "3628800"},             // 10!
+	{1, 20, "2432902008176640000"}, // 20!
+	{1, 100,
+		"933262154439441526816992388562667004907159682643816214685929" +
+			"638952175999932299156089414639761565182862536979208272237582" +
+			"51185210916864000000000000000000000000", // 100!
+	},
+}
+
+func TestMulRangeN(t *testing.T) {
+	for i, r := range mulRangesN {
+		prod := string(nat(nil).mulRange(r.a, r.b).utoa(10))
+		if prod != r.prod {
+			t.Errorf("#%d: got %s; want %s", i, prod, r.prod)
+		}
+	}
+}
+
+// allocBytes returns the number of bytes allocated by invoking f.
+func allocBytes(f func()) uint64 {
+	var stats runtime.MemStats
+	runtime.ReadMemStats(&stats)
+	t := stats.TotalAlloc
+	f()
+	runtime.ReadMemStats(&stats)
+	return stats.TotalAlloc - t
+}
+
+// TestMulUnbalanced tests that multiplying numbers of different lengths
+// does not cause deep recursion and in turn allocate too much memory.
+// Test case for issue 3807.
+func TestMulUnbalanced(t *testing.T) {
+	defer runtime.GOMAXPROCS(runtime.GOMAXPROCS(1))
+	x := rndNat(50000)
+	y := rndNat(40)
+	allocSize := allocBytes(func() {
+		nat(nil).mul(x, y)
+	})
+	inputSize := uint64(len(x)+len(y)) * _S
+	if ratio := allocSize / uint64(inputSize); ratio > 10 {
+		t.Errorf("multiplication uses too much memory (%d > %d times the size of inputs)", allocSize, ratio)
+	}
+}
+
+// rndNat returns a random nat value >= 0 of (usually) n words in length.
+// In extremely unlikely cases it may be smaller than n words if the top-
+// most words are 0.
+func rndNat(n int) nat {
+	return nat(rndV(n)).norm()
+}
+
+// rndNat1 is like rndNat but the result is guaranteed to be > 0.
+func rndNat1(n int) nat {
+	x := nat(rndV(n)).norm()
+	if len(x) == 0 {
+		x.setWord(1)
+	}
+	return x
+}
+
+func BenchmarkMul(b *testing.B) {
+	mulx := rndNat(1e4)
+	muly := rndNat(1e4)
+	b.ResetTimer()
+	for i := 0; i < b.N; i++ {
+		var z nat
+		z.mul(mulx, muly)
+	}
+}
+
+func benchmarkNatMul(b *testing.B, nwords int) {
+	x := rndNat(nwords)
+	y := rndNat(nwords)
+	var z nat
+	b.ResetTimer()
+	for i := 0; i < b.N; i++ {
+		z.mul(x, y)
+	}
+}
+
+var mulBenchSizes = []int{10, 100, 1000, 10000, 100000}
+
+func BenchmarkNatMul(b *testing.B) {
+	for _, n := range mulBenchSizes {
+		if isRaceBuilder && n > 1e3 {
+			continue
+		}
+		b.Run(fmt.Sprintf("%d", n), func(b *testing.B) {
+			benchmarkNatMul(b, n)
+		})
+	}
+}
+
+func TestNLZ(t *testing.T) {
+	var x Word = _B >> 1
+	for i := 0; i <= _W; i++ {
+		if int(nlz(x)) != i {
+			t.Errorf("failed at %x: got %d want %d", x, nlz(x), i)
+		}
+		x >>= 1
+	}
+}
+
+type shiftTest struct {
+	in    nat
+	shift uint
+	out   nat
+}
+
+var leftShiftTests = []shiftTest{
+	{nil, 0, nil},
+	{nil, 1, nil},
+	{natOne, 0, natOne},
+	{natOne, 1, natTwo},
+	{nat{1 << (_W - 1)}, 1, nat{0}},
+	{nat{1 << (_W - 1), 0}, 1, nat{0, 1}},
+}
+
+func TestShiftLeft(t *testing.T) {
+	for i, test := range leftShiftTests {
+		var z nat
+		z = z.shl(test.in, test.shift)
+		for j, d := range test.out {
+			if j >= len(z) || z[j] != d {
+				t.Errorf("#%d: got: %v want: %v", i, z, test.out)
+				break
+			}
+		}
+	}
+}
+
+var rightShiftTests = []shiftTest{
+	{nil, 0, nil},
+	{nil, 1, nil},
+	{natOne, 0, natOne},
+	{natOne, 1, nil},
+	{natTwo, 1, natOne},
+	{nat{0, 1}, 1, nat{1 << (_W - 1)}},
+	{nat{2, 1, 1}, 1, nat{1<<(_W-1) + 1, 1 << (_W - 1)}},
+}
+
+func TestShiftRight(t *testing.T) {
+	for i, test := range rightShiftTests {
+		var z nat
+		z = z.shr(test.in, test.shift)
+		for j, d := range test.out {
+			if j >= len(z) || z[j] != d {
+				t.Errorf("#%d: got: %v want: %v", i, z, test.out)
+				break
+			}
+		}
+	}
+}
+
+func BenchmarkZeroShifts(b *testing.B) {
+	x := rndNat(800)
+
+	b.Run("Shl", func(b *testing.B) {
+		for i := 0; i < b.N; i++ {
+			var z nat
+			z.shl(x, 0)
+		}
+	})
+	b.Run("ShlSame", func(b *testing.B) {
+		for i := 0; i < b.N; i++ {
+			x.shl(x, 0)
+		}
+	})
+
+	b.Run("Shr", func(b *testing.B) {
+		for i := 0; i < b.N; i++ {
+			var z nat
+			z.shr(x, 0)
+		}
+	})
+	b.Run("ShrSame", func(b *testing.B) {
+		for i := 0; i < b.N; i++ {
+			x.shr(x, 0)
+		}
+	})
+}
+
+type modWTest struct {
+	in       string
+	dividend string
+	out      string
+}
+
+var modWTests32 = []modWTest{
+	{"23492635982634928349238759823742", "252341", "220170"},
+}
+
+var modWTests64 = []modWTest{
+	{"6527895462947293856291561095690465243862946", "524326975699234", "375066989628668"},
+}
+
+func runModWTests(t *testing.T, tests []modWTest) {
+	for i, test := range tests {
+		in, _ := new(Int).SetString(test.in, 10)
+		d, _ := new(Int).SetString(test.dividend, 10)
+		out, _ := new(Int).SetString(test.out, 10)
+
+		r := in.abs.modW(d.abs[0])
+		if r != out.abs[0] {
+			t.Errorf("#%d failed: got %d want %s", i, r, out)
+		}
+	}
+}
+
+func TestModW(t *testing.T) {
+	if _W >= 32 {
+		runModWTests(t, modWTests32)
+	}
+	if _W >= 64 {
+		runModWTests(t, modWTests64)
+	}
+}
+
+var montgomeryTests = []struct {
+	x, y, m      string
+	k0           uint64
+	out32, out64 string
+}{
+	{
+		"0xffffffffffffffffffffffffffffffffffffffffffffffffe",
+		"0xffffffffffffffffffffffffffffffffffffffffffffffffe",
+		"0xfffffffffffffffffffffffffffffffffffffffffffffffff",
+		1,
+		"0x1000000000000000000000000000000000000000000",
+		"0x10000000000000000000000000000000000",
+	},
+	{
+		"0x000000000ffffff5",
+		"0x000000000ffffff0",
+		"0x0000000010000001",
+		0xff0000000fffffff,
+		"0x000000000bfffff4",
+		"0x0000000003400001",
+	},
+	{
+		"0x0000000080000000",
+		"0x00000000ffffffff",
+		"0x1000000000000001",
+		0xfffffffffffffff,
+		"0x0800000008000001",
+		"0x0800000008000001",
+	},
+	{
+		"0x0000000080000000",
+		"0x0000000080000000",
+		"0xffffffff00000001",
+		0xfffffffeffffffff,
+		"0xbfffffff40000001",
+		"0xbfffffff40000001",
+	},
+	{
+		"0x0000000080000000",
+		"0x0000000080000000",
+		"0x00ffffff00000001",
+		0xfffffeffffffff,
+		"0xbfffff40000001",
+		"0xbfffff40000001",
+	},
+	{
+		"0x0000000080000000",
+		"0x0000000080000000",
+		"0x0000ffff00000001",
+		0xfffeffffffff,
+		"0xbfff40000001",
+		"0xbfff40000001",
+	},
+	{
+		"0x3321ffffffffffffffffffffffffffff00000000000022222623333333332bbbb888c0",
+		"0x3321ffffffffffffffffffffffffffff00000000000022222623333333332bbbb888c0",
+		"0x33377fffffffffffffffffffffffffffffffffffffffffffff0000000000022222eee1",
+		0xdecc8f1249812adf,
+		"0x04eb0e11d72329dc0915f86784820fc403275bf2f6620a20e0dd344c5cd0875e50deb5",
+		"0x0d7144739a7d8e11d72329dc0915f86784820fc403275bf2f61ed96f35dd34dbb3d6a0",
+	},
+	{
+		"0x10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000ffffffffffffffffffffffffffffffff00000000000022222223333333333444444444",
+		"0x10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000ffffffffffffffffffffffffffffffff999999999999999aaabbbbbbbbcccccccccccc",
+		"0xffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff33377fffffffffffffffffffffffffffffffffffffffffffff0000000000022222eee1",
+		0xdecc8f1249812adf,
+		"0x5c0d52f451aec609b15da8e5e5626c4eaa88723bdeac9d25ca9b961269400410ca208a16af9c2fb07d7a11c7772cba02c22f9711078d51a3797eb18e691295293284d988e349fa6deba46b25a4ecd9f715",
+		"0x92fcad4b5c0d52f451aec609b15da8e5e5626c4eaa88723bdeac9d25ca9b961269400410ca208a16af9c2fb07d799c32fe2f3cc5422f9711078d51a3797eb18e691295293284d8f5e69caf6decddfe1df6",
+	},
+}
+
+func TestMontgomery(t *testing.T) {
+	one := NewInt(1)
+	_B := new(Int).Lsh(one, _W)
+	for i, test := range montgomeryTests {
+		x := natFromString(test.x)
+		y := natFromString(test.y)
+		m := natFromString(test.m)
+		for len(x) < len(m) {
+			x = append(x, 0)
+		}
+		for len(y) < len(m) {
+			y = append(y, 0)
+		}
+
+		if x.cmp(m) > 0 {
+			_, r := nat(nil).div(nil, x, m)
+			t.Errorf("#%d: x > m (0x%s > 0x%s; use 0x%s)", i, x.utoa(16), m.utoa(16), r.utoa(16))
+		}
+		if y.cmp(m) > 0 {
+			_, r := nat(nil).div(nil, x, m)
+			t.Errorf("#%d: y > m (0x%s > 0x%s; use 0x%s)", i, y.utoa(16), m.utoa(16), r.utoa(16))
+		}
+
+		var out nat
+		if _W == 32 {
+			out = natFromString(test.out32)
+		} else {
+			out = natFromString(test.out64)
+		}
+
+		// t.Logf("#%d: len=%d\n", i, len(m))
+
+		// check output in table
+		xi := &Int{abs: x}
+		yi := &Int{abs: y}
+		mi := &Int{abs: m}
+		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)
+		if out.cmp(p.abs.norm()) != 0 {
+			t.Errorf("#%d: out in table=0x%s, computed=0x%s", i, out.utoa(16), p.abs.norm().utoa(16))
+		}
+
+		// check k0 in table
+		k := new(Int).Mod(&Int{abs: m}, _B)
+		k = new(Int).Sub(_B, k)
+		k = new(Int).Mod(k, _B)
+		k0 := Word(new(Int).ModInverse(k, _B).Uint64())
+		if k0 != Word(test.k0) {
+			t.Errorf("#%d: k0 in table=%#x, computed=%#x\n", i, test.k0, k0)
+		}
+
+		// check montgomery with correct k0 produces correct output
+		z := nat(nil).montgomery(x, y, m, k0, len(m))
+		z = z.norm()
+		if z.cmp(out) != 0 {
+			t.Errorf("#%d: got 0x%s want 0x%s", i, z.utoa(16), out.utoa(16))
+		}
+	}
+}
+
+var expNNTests = []struct {
+	x, y, m string
+	out     string
+}{
+	{"0", "0", "0", "1"},
+	{"0", "0", "1", "0"},
+	{"1", "1", "1", "0"},
+	{"2", "1", "1", "0"},
+	{"2", "2", "1", "0"},
+	{"10", "100000000000", "1", "0"},
+	{"0x8000000000000000", "2", "", "0x40000000000000000000000000000000"},
+	{"0x8000000000000000", "2", "6719", "4944"},
+	{"0x8000000000000000", "3", "6719", "5447"},
+	{"0x8000000000000000", "1000", "6719", "1603"},
+	{"0x8000000000000000", "1000000", "6719", "3199"},
+	{
+		"2938462938472983472983659726349017249287491026512746239764525612965293865296239471239874193284792387498274256129746192347",
+		"298472983472983471903246121093472394872319615612417471234712061",
+		"29834729834729834729347290846729561262544958723956495615629569234729836259263598127342374289365912465901365498236492183464",
+		"23537740700184054162508175125554701713153216681790245129157191391322321508055833908509185839069455749219131480588829346291",
+	},
+	{
+		"11521922904531591643048817447554701904414021819823889996244743037378330903763518501116638828335352811871131385129455853417360623007349090150042001944696604737499160174391019030572483602867266711107136838523916077674888297896995042968746762200926853379",
+		"426343618817810911523",
+		"444747819283133684179",
+		"42",
+	},
+}
+
+func TestExpNN(t *testing.T) {
+	for i, test := range expNNTests {
+		x := natFromString(test.x)
+		y := natFromString(test.y)
+		out := natFromString(test.out)
+
+		var m nat
+		if len(test.m) > 0 {
+			m = natFromString(test.m)
+		}
+
+		z := nat(nil).expNN(x, y, m)
+		if z.cmp(out) != 0 {
+			t.Errorf("#%d got %s want %s", i, z.utoa(10), out.utoa(10))
+		}
+	}
+}
+
+func BenchmarkExp3Power(b *testing.B) {
+	const x = 3
+	for _, y := range []Word{
+		0x10, 0x40, 0x100, 0x400, 0x1000, 0x4000, 0x10000, 0x40000, 0x100000, 0x400000,
+	} {
+		b.Run(fmt.Sprintf("%#x", y), func(b *testing.B) {
+			var z nat
+			for i := 0; i < b.N; i++ {
+				z.expWW(x, y)
+			}
+		})
+	}
+}
+
+func fibo(n int) nat {
+	switch n {
+	case 0:
+		return nil
+	case 1:
+		return nat{1}
+	}
+	f0 := fibo(0)
+	f1 := fibo(1)
+	var f2 nat
+	for i := 1; i < n; i++ {
+		f2 = f2.add(f0, f1)
+		f0, f1, f2 = f1, f2, f0
+	}
+	return f1
+}
+
+var fiboNums = []string{
+	"0",
+	"55",
+	"6765",
+	"832040",
+	"102334155",
+	"12586269025",
+	"1548008755920",
+	"190392490709135",
+	"23416728348467685",
+	"2880067194370816120",
+	"354224848179261915075",
+}
+
+func TestFibo(t *testing.T) {
+	for i, want := range fiboNums {
+		n := i * 10
+		got := string(fibo(n).utoa(10))
+		if got != want {
+			t.Errorf("fibo(%d) failed: got %s want %s", n, got, want)
+		}
+	}
+}
+
+func BenchmarkFibo(b *testing.B) {
+	for i := 0; i < b.N; i++ {
+		fibo(1e0)
+		fibo(1e1)
+		fibo(1e2)
+		fibo(1e3)
+		fibo(1e4)
+		fibo(1e5)
+	}
+}
+
+var bitTests = []struct {
+	x    string
+	i    uint
+	want uint
+}{
+	{"0", 0, 0},
+	{"0", 1, 0},
+	{"0", 1000, 0},
+
+	{"0x1", 0, 1},
+	{"0x10", 0, 0},
+	{"0x10", 3, 0},
+	{"0x10", 4, 1},
+	{"0x10", 5, 0},
+
+	{"0x8000000000000000", 62, 0},
+	{"0x8000000000000000", 63, 1},
+	{"0x8000000000000000", 64, 0},
+
+	{"0x3" + strings.Repeat("0", 32), 127, 0},
+	{"0x3" + strings.Repeat("0", 32), 128, 1},
+	{"0x3" + strings.Repeat("0", 32), 129, 1},
+	{"0x3" + strings.Repeat("0", 32), 130, 0},
+}
+
+func TestBit(t *testing.T) {
+	for i, test := range bitTests {
+		x := natFromString(test.x)
+		if got := x.bit(test.i); got != test.want {
+			t.Errorf("#%d: %s.bit(%d) = %v; want %v", i, test.x, test.i, got, test.want)
+		}
+	}
+}
+
+var stickyTests = []struct {
+	x    string
+	i    uint
+	want uint
+}{
+	{"0", 0, 0},
+	{"0", 1, 0},
+	{"0", 1000, 0},
+
+	{"0x1", 0, 0},
+	{"0x1", 1, 1},
+
+	{"0x1350", 0, 0},
+	{"0x1350", 4, 0},
+	{"0x1350", 5, 1},
+
+	{"0x8000000000000000", 63, 0},
+	{"0x8000000000000000", 64, 1},
+
+	{"0x1" + strings.Repeat("0", 100), 400, 0},
+	{"0x1" + strings.Repeat("0", 100), 401, 1},
+}
+
+func TestSticky(t *testing.T) {
+	for i, test := range stickyTests {
+		x := natFromString(test.x)
+		if got := x.sticky(test.i); got != test.want {
+			t.Errorf("#%d: %s.sticky(%d) = %v; want %v", i, test.x, test.i, got, test.want)
+		}
+		if test.want == 1 {
+			// all subsequent i's should also return 1
+			for d := uint(1); d <= 3; d++ {
+				if got := x.sticky(test.i + d); got != 1 {
+					t.Errorf("#%d: %s.sticky(%d) = %v; want %v", i, test.x, test.i+d, got, 1)
+				}
+			}
+		}
+	}
+}
+
+func testSqr(t *testing.T, x nat) {
+	got := make(nat, 2*len(x))
+	want := make(nat, 2*len(x))
+	got = got.sqr(x)
+	want = want.mul(x, x)
+	if got.cmp(want) != 0 {
+		t.Errorf("basicSqr(%v), got %v, want %v", x, got, want)
+	}
+}
+
+func TestSqr(t *testing.T) {
+	for _, a := range prodNN {
+		if a.x != nil {
+			testSqr(t, a.x)
+		}
+		if a.y != nil {
+			testSqr(t, a.y)
+		}
+		if a.z != nil {
+			testSqr(t, a.z)
+		}
+	}
+}
+
+func benchmarkNatSqr(b *testing.B, nwords int) {
+	x := rndNat(nwords)
+	var z nat
+	b.ResetTimer()
+	for i := 0; i < b.N; i++ {
+		z.sqr(x)
+	}
+}
+
+var sqrBenchSizes = []int{
+	1, 2, 3, 5, 8, 10, 20, 30, 50, 80,
+	100, 200, 300, 500, 800,
+	1000, 10000, 100000,
+}
+
+func BenchmarkNatSqr(b *testing.B) {
+	for _, n := range sqrBenchSizes {
+		if isRaceBuilder && n > 1e3 {
+			continue
+		}
+		b.Run(fmt.Sprintf("%d", n), func(b *testing.B) {
+			benchmarkNatSqr(b, n)
+		})
+	}
+}
+
+func BenchmarkNatSetBytes(b *testing.B) {
+	const maxLength = 128
+	lengths := []int{
+		// No remainder:
+		8, 24, maxLength,
+		// With remainder:
+		7, 23, maxLength - 1,
+	}
+	n := make(nat, maxLength/_W) // ensure n doesn't need to grow during the test
+	buf := make([]byte, maxLength)
+	for _, l := range lengths {
+		b.Run(fmt.Sprint(l), func(b *testing.B) {
+			for i := 0; i < b.N; i++ {
+				n.setBytes(buf[:l])
+			}
+		})
+	}
+}
+
+func TestNatDiv(t *testing.T) {
+	sizes := []int{
+		1, 2, 5, 8, 15, 25, 40, 65, 100,
+		200, 500, 800, 1500, 2500, 4000, 6500, 10000,
+	}
+	for _, i := range sizes {
+		for _, j := range sizes {
+			a := rndNat1(i)
+			b := rndNat1(j)
+			// the test requires b >= 2
+			if len(b) == 1 && b[0] == 1 {
+				b[0] = 2
+			}
+			// choose a remainder c < b
+			c := rndNat1(len(b))
+			if len(c) == len(b) && c[len(c)-1] >= b[len(b)-1] {
+				c[len(c)-1] = 0
+				c = c.norm()
+			}
+			// compute x = a*b+c
+			x := nat(nil).mul(a, b)
+			x = x.add(x, c)
+
+			var q, r nat
+			q, r = q.div(r, x, b)
+			if q.cmp(a) != 0 {
+				t.Fatalf("wrong quotient: got %s; want %s for %s/%s", q.utoa(10), a.utoa(10), x.utoa(10), b.utoa(10))
+			}
+			if r.cmp(c) != 0 {
+				t.Fatalf("wrong remainder: got %s; want %s for %s/%s", r.utoa(10), c.utoa(10), x.utoa(10), b.utoa(10))
+			}
+		}
+	}
+}
+
+// TestIssue37499 triggers the edge case of divBasic where
+// the inaccurate estimate of the first word's quotient
+// happens at the very beginning of the loop.
+func TestIssue37499(t *testing.T) {
+	// Choose u and v such that v is slightly larger than u >> N.
+	// This tricks divBasic into choosing 1 as the first word
+	// of the quotient. This works in both 32-bit and 64-bit settings.
+	u := natFromString("0x2b6c385a05be027f5c22005b63c42a1165b79ff510e1706b39f8489c1d28e57bb5ba4ef9fd9387a3e344402c0a453381")
+	v := natFromString("0x2b6c385a05be027f5c22005b63c42a1165b79ff510e1706c")
+
+	q := nat(nil).make(8)
+	q.divBasic(u, v)
+	q = q.norm()
+	if s := string(q.utoa(16)); s != "fffffffffffffffffffffffffffffffffffffffffffffffb" {
+		t.Fatalf("incorrect quotient: %s", s)
+	}
+}
+
+// TestIssue42552 triggers an edge case of recursive division
+// where the first division loop is never entered, and correcting
+// the remainder takes exactly two iterations in the final loop.
+func TestIssue42552(t *testing.T) {
+	u := natFromString("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")
+	v := natFromString("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")
+	q := nat(nil).make(16)
+	q.div(q, u, v)
+}