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.
5
// This file implements signed multi-precision integers.
14
// An Int represents a signed multi-precision integer.
15
// The zero value for an Int represents the value 0.
18
abs nat // absolute value of the integer
22
var intOne = &Int{false, natOne}
31
func (x *Int) Sign() int {
42
// SetInt64 sets z to x and returns z.
43
func (z *Int) SetInt64(x int64) *Int {
49
z.abs = z.abs.setUint64(uint64(x))
55
// NewInt allocates and returns a new Int set to x.
56
func NewInt(x int64) *Int {
57
return new(Int).SetInt64(x)
61
// Set sets z to x and returns z.
62
func (z *Int) Set(x *Int) *Int {
63
z.abs = z.abs.set(x.abs)
69
// Abs sets z to |x| (the absolute value of x) and returns z.
70
func (z *Int) Abs(x *Int) *Int {
71
z.abs = z.abs.set(x.abs)
77
// Neg sets z to -x and returns z.
78
func (z *Int) Neg(x *Int) *Int {
79
z.abs = z.abs.set(x.abs)
80
z.neg = len(z.abs) > 0 && !x.neg // 0 has no sign
85
// Add sets z to the sum x+y and returns z.
86
func (z *Int) Add(x, y *Int) *Int {
90
// (-x) + (-y) == -(x + y)
91
z.abs = z.abs.add(x.abs, y.abs)
93
// x + (-y) == x - y == -(y - x)
94
// (-x) + y == y - x == -(x - y)
95
if x.abs.cmp(y.abs) >= 0 {
96
z.abs = z.abs.sub(x.abs, y.abs)
99
z.abs = z.abs.sub(y.abs, x.abs)
102
z.neg = len(z.abs) > 0 && neg // 0 has no sign
107
// Sub sets z to the difference x-y and returns z.
108
func (z *Int) Sub(x, y *Int) *Int {
112
// (-x) - y == -(x + y)
113
z.abs = z.abs.add(x.abs, y.abs)
115
// x - y == x - y == -(y - x)
116
// (-x) - (-y) == y - x == -(x - y)
117
if x.abs.cmp(y.abs) >= 0 {
118
z.abs = z.abs.sub(x.abs, y.abs)
121
z.abs = z.abs.sub(y.abs, x.abs)
124
z.neg = len(z.abs) > 0 && neg // 0 has no sign
129
// Mul sets z to the product x*y and returns z.
130
func (z *Int) Mul(x, y *Int) *Int {
132
// x * (-y) == -(x * y)
133
// (-x) * y == -(x * y)
134
// (-x) * (-y) == x * y
135
z.abs = z.abs.mul(x.abs, y.abs)
136
z.neg = len(z.abs) > 0 && x.neg != y.neg // 0 has no sign
141
// MulRange sets z to the product of all integers
142
// in the range [a, b] inclusively and returns z.
143
// If a > b (empty range), the result is 1.
144
func (z *Int) MulRange(a, b int64) *Int {
147
return z.SetInt64(1) // empty range
148
case a <= 0 && b >= 0:
149
return z.SetInt64(0) // range includes 0
151
// a <= b && (b < 0 || a > 0)
159
z.abs = z.abs.mulRange(uint64(a), uint64(b))
165
// Binomial sets z to the binomial coefficient of (n, k) and returns z.
166
func (z *Int) Binomial(n, k int64) *Int {
174
// Quo sets z to the quotient x/y for y != 0 and returns z.
175
// If y == 0, a division-by-zero run-time panic occurs.
176
// See QuoRem for more details.
177
func (z *Int) Quo(x, y *Int) *Int {
178
z.abs, _ = z.abs.div(nil, x.abs, y.abs)
179
z.neg = len(z.abs) > 0 && x.neg != y.neg // 0 has no sign
184
// Rem sets z to the remainder x%y for y != 0 and returns z.
185
// If y == 0, a division-by-zero run-time panic occurs.
186
// See QuoRem for more details.
187
func (z *Int) Rem(x, y *Int) *Int {
188
_, z.abs = nat(nil).div(z.abs, x.abs, y.abs)
189
z.neg = len(z.abs) > 0 && x.neg // 0 has no sign
194
// QuoRem sets z to the quotient x/y and r to the remainder x%y
195
// and returns the pair (z, r) for y != 0.
196
// If y == 0, a division-by-zero run-time panic occurs.
198
// QuoRem implements T-division and modulus (like Go):
200
// q = x/y with the result truncated to zero
203
// (See Daan Leijen, ``Division and Modulus for Computer Scientists''.)
205
func (z *Int) QuoRem(x, y, r *Int) (*Int, *Int) {
206
z.abs, r.abs = z.abs.div(r.abs, x.abs, y.abs)
207
z.neg, r.neg = len(z.abs) > 0 && x.neg != y.neg, len(r.abs) > 0 && x.neg // 0 has no sign
212
// Div sets z to the quotient x/y for y != 0 and returns z.
213
// If y == 0, a division-by-zero run-time panic occurs.
214
// See DivMod for more details.
215
func (z *Int) Div(x, y *Int) *Int {
216
y_neg := y.neg // z may be an alias for y
230
// Mod sets z to the modulus x%y for y != 0 and returns z.
231
// If y == 0, a division-by-zero run-time panic occurs.
232
// See DivMod for more details.
233
func (z *Int) Mod(x, y *Int) *Int {
235
if z == y || alias(z.abs, y.abs) {
251
// DivMod sets z to the quotient x div y and m to the modulus x mod y
252
// and returns the pair (z, m) for y != 0.
253
// If y == 0, a division-by-zero run-time panic occurs.
255
// DivMod implements Euclidean division and modulus (unlike Go):
257
// q = x div y such that
258
// m = x - y*q with 0 <= m < |q|
260
// (See Raymond T. Boute, ``The Euclidean definition of the functions
261
// div and mod''. ACM Transactions on Programming Languages and
262
// Systems (TOPLAS), 14(2):127-144, New York, NY, USA, 4/1992.
265
func (z *Int) DivMod(x, y, m *Int) (*Int, *Int) {
267
if z == y || alias(z.abs, y.abs) {
284
// Cmp compares x and y and returns:
290
func (x *Int) Cmp(y *Int) (r int) {
291
// x cmp y == x cmp y
294
// (-x) cmp (-y) == -(x cmp y)
310
func (x *Int) String() string {
315
return s + x.abs.string(10)
319
func fmtbase(ch int) int {
334
// Format is a support routine for fmt.Formatter. It accepts
335
// the formats 'b' (binary), 'o' (octal), 'd' (decimal) and
336
// 'x' (hexadecimal).
338
func (x *Int) Format(s fmt.State, ch int) {
342
fmt.Fprint(s, x.abs.string(fmtbase(ch)))
346
// Int64 returns the int64 representation of z.
347
// If z cannot be represented in an int64, the result is undefined.
348
func (x *Int) Int64() int64 {
353
if _W == 32 && len(x.abs) > 1 {
354
v |= int64(x.abs[1]) << 32
363
// SetString sets z to the value of s, interpreted in the given base,
364
// and returns z and a boolean indicating success. If SetString fails,
365
// the value of z is undefined.
367
// If the base argument is 0, the string prefix determines the actual
368
// conversion base. A prefix of ``0x'' or ``0X'' selects base 16; the
369
// ``0'' prefix selects base 8, and a ``0b'' or ``0B'' prefix selects
370
// base 2. Otherwise the selected base is 10.
372
func (z *Int) SetString(s string, base int) (*Int, bool) {
373
if len(s) == 0 || base < 0 || base == 1 || 16 < base {
378
if neg || s[0] == '+' {
386
z.abs, _, scanned = z.abs.scan(s, base)
387
if scanned != len(s) {
390
z.neg = len(z.abs) > 0 && neg // 0 has no sign
396
// SetBytes interprets b as the bytes of a big-endian, unsigned integer and
397
// sets z to that value.
398
func (z *Int) SetBytes(b []byte) *Int {
400
z.abs = z.abs.make((len(b) + s - 1) / s)
406
for i := s; i > 0; i-- {
408
w |= Word(b[len(b)-i])
419
for i := len(b); i > 0; i-- {
421
w |= Word(b[len(b)-i])
433
// Bytes returns the absolute value of x as a big-endian byte array.
434
func (z *Int) Bytes() []byte {
436
b := make([]byte, len(z.abs)*s)
438
for i, w := range z.abs {
439
wordBytes := b[(len(z.abs)-i-1)*s : (len(z.abs)-i)*s]
440
for j := s - 1; j >= 0; j-- {
441
wordBytes[j] = byte(w)
447
for i < len(b) && b[i] == 0 {
455
// BitLen returns the length of the absolute value of z in bits.
456
// The bit length of 0 is 0.
457
func (z *Int) BitLen() int {
458
return z.abs.bitLen()
462
// Exp sets z = x**y mod m. If m is nil, z = x**y.
463
// See Knuth, volume 2, section 4.6.3.
464
func (z *Int) Exp(x, y, m *Int) *Int {
465
if y.neg || len(y.abs) == 0 {
477
z.abs = z.abs.expNN(x.abs, y.abs, mWords)
478
z.neg = len(z.abs) > 0 && x.neg && y.abs[0]&1 == 1 // 0 has no sign
483
// GcdInt sets d to the greatest common divisor of a and b, which must be
485
// If x and y are not nil, GcdInt sets x and y such that d = a*x + b*y.
486
// If either a or b is not positive, GcdInt sets d = x = y = 0.
487
func GcdInt(d, x, y, a, b *Int) {
503
Y := new(Int).SetInt64(1)
505
lastX := new(Int).SetInt64(1)
513
q, r = q.QuoRem(A, B, r)
542
// ProbablyPrime performs n Miller-Rabin tests to check whether z is prime.
543
// If it returns true, z is prime with probability 1 - 1/4^n.
544
// If it returns false, z is not prime.
545
func ProbablyPrime(z *Int, n int) bool {
546
return !z.neg && z.abs.probablyPrime(n)
550
// Rand sets z to a pseudo-random number in [0, n) and returns z.
551
func (z *Int) Rand(rnd *rand.Rand, n *Int) *Int {
553
if n.neg == true || len(n.abs) == 0 {
557
z.abs = z.abs.random(rnd, n.abs, n.abs.bitLen())
562
// ModInverse sets z to the multiplicative inverse of g in the group ℤ/pℤ (where
563
// p is a prime) and returns z.
564
func (z *Int) ModInverse(g, p *Int) *Int {
566
GcdInt(&d, z, nil, g, p)
567
// x and y are such that g*x + p*y = d. Since p is prime, d = 1. Taking
568
// that modulo p results in g*x = 1, therefore x is the inverse element.
576
// Lsh sets z = x << n and returns z.
577
func (z *Int) Lsh(x *Int, n uint) *Int {
578
z.abs = z.abs.shl(x.abs, n)
584
// Rsh sets z = x >> n and returns z.
585
func (z *Int) Rsh(x *Int, n uint) *Int {
587
// (-x) >> s == ^(x-1) >> s == ^((x-1) >> s) == -(((x-1) >> s) + 1)
588
t := z.abs.sub(x.abs, natOne) // no underflow because |x| > 0
590
z.abs = t.add(t, natOne)
591
z.neg = true // z cannot be zero if x is negative
595
z.abs = z.abs.shr(x.abs, n)
601
// And sets z = x & y and returns z.
602
func (z *Int) And(x, y *Int) *Int {
605
// (-x) & (-y) == ^(x-1) & ^(y-1) == ^((x-1) | (y-1)) == -(((x-1) | (y-1)) + 1)
606
x1 := nat{}.sub(x.abs, natOne)
607
y1 := nat{}.sub(y.abs, natOne)
608
z.abs = z.abs.add(z.abs.or(x1, y1), natOne)
609
z.neg = true // z cannot be zero if x and y are negative
614
z.abs = z.abs.and(x.abs, y.abs)
621
x, y = y, x // & is symmetric
624
// x & (-y) == x & ^(y-1) == x &^ (y-1)
625
y1 := nat{}.sub(y.abs, natOne)
626
z.abs = z.abs.andNot(x.abs, y1)
632
// AndNot sets z = x &^ y and returns z.
633
func (z *Int) AndNot(x, y *Int) *Int {
636
// (-x) &^ (-y) == ^(x-1) &^ ^(y-1) == ^(x-1) & (y-1) == (y-1) &^ (x-1)
637
x1 := nat{}.sub(x.abs, natOne)
638
y1 := nat{}.sub(y.abs, natOne)
639
z.abs = z.abs.andNot(y1, x1)
645
z.abs = z.abs.andNot(x.abs, y.abs)
651
// (-x) &^ y == ^(x-1) &^ y == ^(x-1) & ^y == ^((x-1) | y) == -(((x-1) | y) + 1)
652
x1 := nat{}.sub(x.abs, natOne)
653
z.abs = z.abs.add(z.abs.or(x1, y.abs), natOne)
654
z.neg = true // z cannot be zero if x is negative and y is positive
658
// x &^ (-y) == x &^ ^(y-1) == x & (y-1)
659
y1 := nat{}.add(y.abs, natOne)
660
z.abs = z.abs.and(x.abs, y1)
666
// Or sets z = x | y and returns z.
667
func (z *Int) Or(x, y *Int) *Int {
670
// (-x) | (-y) == ^(x-1) | ^(y-1) == ^((x-1) & (y-1)) == -(((x-1) & (y-1)) + 1)
671
x1 := nat{}.sub(x.abs, natOne)
672
y1 := nat{}.sub(y.abs, natOne)
673
z.abs = z.abs.add(z.abs.and(x1, y1), natOne)
674
z.neg = true // z cannot be zero if x and y are negative
679
z.abs = z.abs.or(x.abs, y.abs)
686
x, y = y, x // | is symmetric
689
// x | (-y) == x | ^(y-1) == ^((y-1) &^ x) == -(^((y-1) &^ x) + 1)
690
y1 := nat{}.sub(y.abs, natOne)
691
z.abs = z.abs.add(z.abs.andNot(y1, x.abs), natOne)
692
z.neg = true // z cannot be zero if one of x or y is negative
697
// Xor sets z = x ^ y and returns z.
698
func (z *Int) Xor(x, y *Int) *Int {
701
// (-x) ^ (-y) == ^(x-1) ^ ^(y-1) == (x-1) ^ (y-1)
702
x1 := nat{}.sub(x.abs, natOne)
703
y1 := nat{}.sub(y.abs, natOne)
704
z.abs = z.abs.xor(x1, y1)
710
z.abs = z.abs.xor(x.abs, y.abs)
717
x, y = y, x // ^ is symmetric
720
// x ^ (-y) == x ^ ^(y-1) == ^(x ^ (y-1)) == -((x ^ (y-1)) + 1)
721
y1 := nat{}.sub(y.abs, natOne)
722
z.abs = z.abs.add(z.abs.xor(x.abs, y1), natOne)
723
z.neg = true // z cannot be zero if only one of x or y is negative
728
// Not sets z = ^x and returns z.
729
func (z *Int) Not(x *Int) *Int {
731
// ^(-x) == ^(^(x-1)) == x-1
732
z.abs = z.abs.sub(x.abs, natOne)
737
// ^x == -x-1 == -(x+1)
738
z.abs = z.abs.add(x.abs, natOne)
739
z.neg = true // z cannot be zero if x is positive
added
removed
src/pkg/big/int.go
