# Source file src/pkg/math/big/int.go

```     1	// Copyright 2009 The Go Authors. All rights reserved.
2	// Use of this source code is governed by a BSD-style
4
5	// This file implements signed multi-precision integers.
6
7	package big
8
9	import (
10		"errors"
11		"fmt"
12		"io"
13		"math/rand"
14		"strings"
15	)
16
17	// An Int represents a signed multi-precision integer.
18	// The zero value for an Int represents the value 0.
19	type Int struct {
20		neg bool // sign
21		abs nat  // absolute value of the integer
22	}
23
24	var intOne = &Int{false, natOne}
25
26	// Sign returns:
27	//
28	//	-1 if x <  0
29	//	 0 if x == 0
30	//	+1 if x >  0
31	//
32	func (x *Int) Sign() int {
33		if len(x.abs) == 0 {
34			return 0
35		}
36		if x.neg {
37			return -1
38		}
39		return 1
40	}
41
42	// SetInt64 sets z to x and returns z.
43	func (z *Int) SetInt64(x int64) *Int {
44		neg := false
45		if x < 0 {
46			neg = true
47			x = -x
48		}
49		z.abs = z.abs.setUint64(uint64(x))
50		z.neg = neg
51		return z
52	}
53
54	// SetUint64 sets z to x and returns z.
55	func (z *Int) SetUint64(x uint64) *Int {
56		z.abs = z.abs.setUint64(x)
57		z.neg = false
58		return z
59	}
60
61	// NewInt allocates and returns a new Int set to x.
62	func NewInt(x int64) *Int {
63		return new(Int).SetInt64(x)
64	}
65
66	// Set sets z to x and returns z.
67	func (z *Int) Set(x *Int) *Int {
68		if z != x {
69			z.abs = z.abs.set(x.abs)
70			z.neg = x.neg
71		}
72		return z
73	}
74
75	// Bits provides raw (unchecked but fast) access to x by returning its
76	// absolute value as a little-endian Word slice. The result and x share
77	// the same underlying array.
78	// Bits is intended to support implementation of missing low-level Int
79	// functionality outside this package; it should be avoided otherwise.
80	func (x *Int) Bits() []Word {
81		return x.abs
82	}
83
84	// SetBits provides raw (unchecked but fast) access to z by setting its
85	// value to abs, interpreted as a little-endian Word slice, and returning
86	// z. The result and abs share the same underlying array.
87	// SetBits is intended to support implementation of missing low-level Int
88	// functionality outside this package; it should be avoided otherwise.
89	func (z *Int) SetBits(abs []Word) *Int {
90		z.abs = nat(abs).norm()
91		z.neg = false
92		return z
93	}
94
95	// Abs sets z to |x| (the absolute value of x) and returns z.
96	func (z *Int) Abs(x *Int) *Int {
97		z.Set(x)
98		z.neg = false
99		return z
100	}
101
102	// Neg sets z to -x and returns z.
103	func (z *Int) Neg(x *Int) *Int {
104		z.Set(x)
105		z.neg = len(z.abs) > 0 && !z.neg // 0 has no sign
106		return z
107	}
108
109	// Add sets z to the sum x+y and returns z.
110	func (z *Int) Add(x, y *Int) *Int {
111		neg := x.neg
112		if x.neg == y.neg {
113			// x + y == x + y
114			// (-x) + (-y) == -(x + y)
116		} else {
117			// x + (-y) == x - y == -(y - x)
118			// (-x) + y == y - x == -(x - y)
119			if x.abs.cmp(y.abs) >= 0 {
120				z.abs = z.abs.sub(x.abs, y.abs)
121			} else {
122				neg = !neg
123				z.abs = z.abs.sub(y.abs, x.abs)
124			}
125		}
126		z.neg = len(z.abs) > 0 && neg // 0 has no sign
127		return z
128	}
129
130	// Sub sets z to the difference x-y and returns z.
131	func (z *Int) Sub(x, y *Int) *Int {
132		neg := x.neg
133		if x.neg != y.neg {
134			// x - (-y) == x + y
135			// (-x) - y == -(x + y)
137		} else {
138			// x - y == x - y == -(y - x)
139			// (-x) - (-y) == y - x == -(x - y)
140			if x.abs.cmp(y.abs) >= 0 {
141				z.abs = z.abs.sub(x.abs, y.abs)
142			} else {
143				neg = !neg
144				z.abs = z.abs.sub(y.abs, x.abs)
145			}
146		}
147		z.neg = len(z.abs) > 0 && neg // 0 has no sign
148		return z
149	}
150
151	// Mul sets z to the product x*y and returns z.
152	func (z *Int) Mul(x, y *Int) *Int {
153		// x * y == x * y
154		// x * (-y) == -(x * y)
155		// (-x) * y == -(x * y)
156		// (-x) * (-y) == x * y
157		z.abs = z.abs.mul(x.abs, y.abs)
158		z.neg = len(z.abs) > 0 && x.neg != y.neg // 0 has no sign
159		return z
160	}
161
162	// MulRange sets z to the product of all integers
163	// in the range [a, b] inclusively and returns z.
164	// If a > b (empty range), the result is 1.
165	func (z *Int) MulRange(a, b int64) *Int {
166		switch {
167		case a > b:
168			return z.SetInt64(1) // empty range
169		case a <= 0 && b >= 0:
170			return z.SetInt64(0) // range includes 0
171		}
172		// a <= b && (b < 0 || a > 0)
173
174		neg := false
175		if a < 0 {
176			neg = (b-a)&1 == 0
177			a, b = -b, -a
178		}
179
180		z.abs = z.abs.mulRange(uint64(a), uint64(b))
181		z.neg = neg
182		return z
183	}
184
185	// Binomial sets z to the binomial coefficient of (n, k) and returns z.
186	func (z *Int) Binomial(n, k int64) *Int {
187		var a, b Int
188		a.MulRange(n-k+1, n)
189		b.MulRange(1, k)
190		return z.Quo(&a, &b)
191	}
192
193	// Quo sets z to the quotient x/y for y != 0 and returns z.
194	// If y == 0, a division-by-zero run-time panic occurs.
195	// Quo implements truncated division (like Go); see QuoRem for more details.
196	func (z *Int) Quo(x, y *Int) *Int {
197		z.abs, _ = z.abs.div(nil, x.abs, y.abs)
198		z.neg = len(z.abs) > 0 && x.neg != y.neg // 0 has no sign
199		return z
200	}
201
202	// Rem sets z to the remainder x%y for y != 0 and returns z.
203	// If y == 0, a division-by-zero run-time panic occurs.
204	// Rem implements truncated modulus (like Go); see QuoRem for more details.
205	func (z *Int) Rem(x, y *Int) *Int {
206		_, z.abs = nat(nil).div(z.abs, x.abs, y.abs)
207		z.neg = len(z.abs) > 0 && x.neg // 0 has no sign
208		return z
209	}
210
211	// QuoRem sets z to the quotient x/y and r to the remainder x%y
212	// and returns the pair (z, r) for y != 0.
213	// If y == 0, a division-by-zero run-time panic occurs.
214	//
215	// QuoRem implements T-division and modulus (like Go):
216	//
217	//	q = x/y      with the result truncated to zero
218	//	r = x - y*q
219	//
220	// (See Daan Leijen, ``Division and Modulus for Computer Scientists''.)
221	// See DivMod for Euclidean division and modulus (unlike Go).
222	//
223	func (z *Int) QuoRem(x, y, r *Int) (*Int, *Int) {
224		z.abs, r.abs = z.abs.div(r.abs, x.abs, y.abs)
225		z.neg, r.neg = len(z.abs) > 0 && x.neg != y.neg, len(r.abs) > 0 && x.neg // 0 has no sign
226		return z, r
227	}
228
229	// Div sets z to the quotient x/y for y != 0 and returns z.
230	// If y == 0, a division-by-zero run-time panic occurs.
231	// Div implements Euclidean division (unlike Go); see DivMod for more details.
232	func (z *Int) Div(x, y *Int) *Int {
233		y_neg := y.neg // z may be an alias for y
234		var r Int
235		z.QuoRem(x, y, &r)
236		if r.neg {
237			if y_neg {
239			} else {
240				z.Sub(z, intOne)
241			}
242		}
243		return z
244	}
245
246	// Mod sets z to the modulus x%y for y != 0 and returns z.
247	// If y == 0, a division-by-zero run-time panic occurs.
248	// Mod implements Euclidean modulus (unlike Go); see DivMod for more details.
249	func (z *Int) Mod(x, y *Int) *Int {
250		y0 := y // save y
251		if z == y || alias(z.abs, y.abs) {
252			y0 = new(Int).Set(y)
253		}
254		var q Int
255		q.QuoRem(x, y, z)
256		if z.neg {
257			if y0.neg {
258				z.Sub(z, y0)
259			} else {
261			}
262		}
263		return z
264	}
265
266	// DivMod sets z to the quotient x div y and m to the modulus x mod y
267	// and returns the pair (z, m) for y != 0.
268	// If y == 0, a division-by-zero run-time panic occurs.
269	//
270	// DivMod implements Euclidean division and modulus (unlike Go):
271	//
272	//	q = x div y  such that
273	//	m = x - y*q  with 0 <= m < |q|
274	//
275	// (See Raymond T. Boute, ``The Euclidean definition of the functions
276	// div and mod''. ACM Transactions on Programming Languages and
277	// Systems (TOPLAS), 14(2):127-144, New York, NY, USA, 4/1992.
278	// ACM press.)
279	// See QuoRem for T-division and modulus (like Go).
280	//
281	func (z *Int) DivMod(x, y, m *Int) (*Int, *Int) {
282		y0 := y // save y
283		if z == y || alias(z.abs, y.abs) {
284			y0 = new(Int).Set(y)
285		}
286		z.QuoRem(x, y, m)
287		if m.neg {
288			if y0.neg {
290				m.Sub(m, y0)
291			} else {
292				z.Sub(z, intOne)
294			}
295		}
296		return z, m
297	}
298
299	// Cmp compares x and y and returns:
300	//
301	//   -1 if x <  y
302	//    0 if x == y
303	//   +1 if x >  y
304	//
305	func (x *Int) Cmp(y *Int) (r int) {
306		// x cmp y == x cmp y
307		// x cmp (-y) == x
308		// (-x) cmp y == y
309		// (-x) cmp (-y) == -(x cmp y)
310		switch {
311		case x.neg == y.neg:
312			r = x.abs.cmp(y.abs)
313			if x.neg {
314				r = -r
315			}
316		case x.neg:
317			r = -1
318		default:
319			r = 1
320		}
321		return
322	}
323
324	func (x *Int) String() string {
325		switch {
326		case x == nil:
327			return "<nil>"
328		case x.neg:
329			return "-" + x.abs.decimalString()
330		}
331		return x.abs.decimalString()
332	}
333
334	func charset(ch rune) string {
335		switch ch {
336		case 'b':
337			return lowercaseDigits[0:2]
338		case 'o':
339			return lowercaseDigits[0:8]
340		case 'd', 's', 'v':
341			return lowercaseDigits[0:10]
342		case 'x':
343			return lowercaseDigits[0:16]
344		case 'X':
345			return uppercaseDigits[0:16]
346		}
347		return "" // unknown format
348	}
349
350	// write count copies of text to s
351	func writeMultiple(s fmt.State, text string, count int) {
352		if len(text) > 0 {
353			b := []byte(text)
354			for ; count > 0; count-- {
355				s.Write(b)
356			}
357		}
358	}
359
360	// Format is a support routine for fmt.Formatter. It accepts
361	// the formats 'b' (binary), 'o' (octal), 'd' (decimal), 'x'
363	// Also supported are the full suite of package fmt's format
364	// verbs for integral types, including '+', '-', and ' '
365	// for sign control, '#' for leading zero in octal and for
366	// hexadecimal, a leading "0x" or "0X" for "%#x" and "%#X"
367	// respectively, specification of minimum digits precision,
368	// output field width, space or zero padding, and left or
369	// right justification.
370	//
371	func (x *Int) Format(s fmt.State, ch rune) {
372		cs := charset(ch)
373
374		// special cases
375		switch {
376		case cs == "":
377			// unknown format
378			fmt.Fprintf(s, "%%!%c(big.Int=%s)", ch, x.String())
379			return
380		case x == nil:
381			fmt.Fprint(s, "<nil>")
382			return
383		}
384
385		// determine sign character
386		sign := ""
387		switch {
388		case x.neg:
389			sign = "-"
390		case s.Flag('+'): // supersedes ' ' when both specified
391			sign = "+"
392		case s.Flag(' '):
393			sign = " "
394		}
395
396		// determine prefix characters for indicating output base
397		prefix := ""
398		if s.Flag('#') {
399			switch ch {
400			case 'o': // octal
401				prefix = "0"
403				prefix = "0x"
404			case 'X':
405				prefix = "0X"
406			}
407		}
408
409		// determine digits with base set by len(cs) and digit characters from cs
410		digits := x.abs.string(cs)
411
412		// number of characters for the three classes of number padding
413		var left int   // space characters to left of digits for right justification ("%8d")
414		var zeroes int // zero characters (actually cs[0]) as left-most digits ("%.8d")
415		var right int  // space characters to right of digits for left justification ("%-8d")
416
417		// determine number padding from precision: the least number of digits to output
418		precision, precisionSet := s.Precision()
419		if precisionSet {
420			switch {
421			case len(digits) < precision:
422				zeroes = precision - len(digits) // count of zero padding
423			case digits == "0" && precision == 0:
424				return // print nothing if zero value (x == 0) and zero precision ("." or ".0")
425			}
426		}
427
428		// determine field pad from width: the least number of characters to output
429		length := len(sign) + len(prefix) + zeroes + len(digits)
430		if width, widthSet := s.Width(); widthSet && length < width { // pad as specified
431			switch d := width - length; {
432			case s.Flag('-'):
433				// pad on the right with spaces; supersedes '0' when both specified
434				right = d
435			case s.Flag('0') && !precisionSet:
436				// pad with zeroes unless precision also specified
437				zeroes = d
438			default:
439				// pad on the left with spaces
440				left = d
441			}
442		}
443
445		writeMultiple(s, " ", left)
446		writeMultiple(s, sign, 1)
447		writeMultiple(s, prefix, 1)
448		writeMultiple(s, "0", zeroes)
449		writeMultiple(s, digits, 1)
450		writeMultiple(s, " ", right)
451	}
452
453	// scan sets z to the integer value corresponding to the longest possible prefix
454	// read from r representing a signed integer number in a given conversion base.
455	// It returns z, the actual conversion base used, and an error, if any. In the
456	// error case, the value of z is undefined but the returned value is nil. The
457	// syntax follows the syntax of integer literals in Go.
458	//
459	// The base argument must be 0 or a value from 2 through MaxBase. If the base
460	// is 0, the string prefix determines the actual conversion base. A prefix of
461	// ``0x'' or ``0X'' selects base 16; the ``0'' prefix selects base 8, and a
462	// ``0b'' or ``0B'' prefix selects base 2. Otherwise the selected base is 10.
463	//
464	func (z *Int) scan(r io.RuneScanner, base int) (*Int, int, error) {
465		// determine sign
466		ch, _, err := r.ReadRune()
467		if err != nil {
468			return nil, 0, err
469		}
470		neg := false
471		switch ch {
472		case '-':
473			neg = true
474		case '+': // nothing to do
475		default:
477		}
478
479		// determine mantissa
480		z.abs, base, err = z.abs.scan(r, base)
481		if err != nil {
482			return nil, base, err
483		}
484		z.neg = len(z.abs) > 0 && neg // 0 has no sign
485
486		return z, base, nil
487	}
488
489	// Scan is a support routine for fmt.Scanner; it sets z to the value of
490	// the scanned number. It accepts the formats 'b' (binary), 'o' (octal),
492	func (z *Int) Scan(s fmt.ScanState, ch rune) error {
493		s.SkipSpace() // skip leading space characters
494		base := 0
495		switch ch {
496		case 'b':
497			base = 2
498		case 'o':
499			base = 8
500		case 'd':
501			base = 10
502		case 'x', 'X':
503			base = 16
504		case 's', 'v':
505			// let scan determine the base
506		default:
507			return errors.New("Int.Scan: invalid verb")
508		}
509		_, _, err := z.scan(s, base)
510		return err
511	}
512
513	// Int64 returns the int64 representation of x.
514	// If x cannot be represented in an int64, the result is undefined.
515	func (x *Int) Int64() int64 {
516		v := int64(x.Uint64())
517		if x.neg {
518			v = -v
519		}
520		return v
521	}
522
523	// Uint64 returns the uint64 representation of x.
524	// If x cannot be represented in a uint64, the result is undefined.
525	func (x *Int) Uint64() uint64 {
526		if len(x.abs) == 0 {
527			return 0
528		}
529		v := uint64(x.abs[0])
530		if _W == 32 && len(x.abs) > 1 {
531			v |= uint64(x.abs[1]) << 32
532		}
533		return v
534	}
535
536	// SetString sets z to the value of s, interpreted in the given base,
537	// and returns z and a boolean indicating success. If SetString fails,
538	// the value of z is undefined but the returned value is nil.
539	//
540	// The base argument must be 0 or a value from 2 through MaxBase. If the base
541	// is 0, the string prefix determines the actual conversion base. A prefix of
542	// ``0x'' or ``0X'' selects base 16; the ``0'' prefix selects base 8, and a
543	// ``0b'' or ``0B'' prefix selects base 2. Otherwise the selected base is 10.
544	//
545	func (z *Int) SetString(s string, base int) (*Int, bool) {
547		_, _, err := z.scan(r, base)
548		if err != nil {
549			return nil, false
550		}
551		_, _, err = r.ReadRune()
552		if err != io.EOF {
553			return nil, false
554		}
555		return z, true // err == io.EOF => scan consumed all of s
556	}
557
558	// SetBytes interprets buf as the bytes of a big-endian unsigned
559	// integer, sets z to that value, and returns z.
560	func (z *Int) SetBytes(buf []byte) *Int {
561		z.abs = z.abs.setBytes(buf)
562		z.neg = false
563		return z
564	}
565
566	// Bytes returns the absolute value of x as a big-endian byte slice.
567	func (x *Int) Bytes() []byte {
568		buf := make([]byte, len(x.abs)*_S)
569		return buf[x.abs.bytes(buf):]
570	}
571
572	// BitLen returns the length of the absolute value of x in bits.
573	// The bit length of 0 is 0.
574	func (x *Int) BitLen() int {
575		return x.abs.bitLen()
576	}
577
578	// Exp sets z = x**y mod |m| (i.e. the sign of m is ignored), and returns z.
579	// If y <= 0, the result is 1; if m == nil or m == 0, z = x**y.
580	// See Knuth, volume 2, section 4.6.3.
581	func (z *Int) Exp(x, y, m *Int) *Int {
582		if y.neg || len(y.abs) == 0 {
583			return z.SetInt64(1)
584		}
585		// y > 0
586
587		var mWords nat
588		if m != nil {
589			mWords = m.abs // m.abs may be nil for m == 0
590		}
591
592		z.abs = z.abs.expNN(x.abs, y.abs, mWords)
593		z.neg = len(z.abs) > 0 && x.neg && y.abs[0]&1 == 1 // 0 has no sign
594		return z
595	}
596
597	// GCD sets z to the greatest common divisor of a and b, which both must
598	// be > 0, and returns z.
599	// If x and y are not nil, GCD sets x and y such that z = a*x + b*y.
600	// If either a or b is <= 0, GCD sets z = x = y = 0.
601	func (z *Int) GCD(x, y, a, b *Int) *Int {
602		if a.Sign() <= 0 || b.Sign() <= 0 {
603			z.SetInt64(0)
604			if x != nil {
605				x.SetInt64(0)
606			}
607			if y != nil {
608				y.SetInt64(0)
609			}
610			return z
611		}
612		if x == nil && y == nil {
613			return z.binaryGCD(a, b)
614		}
615
616		A := new(Int).Set(a)
617		B := new(Int).Set(b)
618
619		X := new(Int)
620		Y := new(Int).SetInt64(1)
621
622		lastX := new(Int).SetInt64(1)
623		lastY := new(Int)
624
625		q := new(Int)
626		temp := new(Int)
627
628		for len(B.abs) > 0 {
629			r := new(Int)
630			q, r = q.QuoRem(A, B, r)
631
632			A, B = B, r
633
634			temp.Set(X)
635			X.Mul(X, q)
636			X.neg = !X.neg
638			lastX.Set(temp)
639
640			temp.Set(Y)
641			Y.Mul(Y, q)
642			Y.neg = !Y.neg
644			lastY.Set(temp)
645		}
646
647		if x != nil {
648			*x = *lastX
649		}
650
651		if y != nil {
652			*y = *lastY
653		}
654
655		*z = *A
656		return z
657	}
658
659	// binaryGCD sets z to the greatest common divisor of a and b, which both must
660	// be > 0, and returns z.
661	// See Knuth, The Art of Computer Programming, Vol. 2, Section 4.5.2, Algorithm B.
662	func (z *Int) binaryGCD(a, b *Int) *Int {
663		u := z
664		v := new(Int)
665
666		// use one Euclidean iteration to ensure that u and v are approx. the same size
667		switch {
668		case len(a.abs) > len(b.abs):
669			u.Set(b)
670			v.Rem(a, b)
671		case len(a.abs) < len(b.abs):
672			u.Set(a)
673			v.Rem(b, a)
674		default:
675			u.Set(a)
676			v.Set(b)
677		}
678
679		// v might be 0 now
680		if len(v.abs) == 0 {
681			return u
682		}
683		// u > 0 && v > 0
684
685		// determine largest k such that u = u' << k, v = v' << k
686		k := u.abs.trailingZeroBits()
687		if vk := v.abs.trailingZeroBits(); vk < k {
688			k = vk
689		}
690		u.Rsh(u, k)
691		v.Rsh(v, k)
692
693		// determine t (we know that u > 0)
694		t := new(Int)
695		if u.abs[0]&1 != 0 {
696			// u is odd
697			t.Neg(v)
698		} else {
699			t.Set(u)
700		}
701
702		for len(t.abs) > 0 {
703			// reduce t
704			t.Rsh(t, t.abs.trailingZeroBits())
705			if t.neg {
706				v, t = t, v
707				v.neg = len(v.abs) > 0 && !v.neg // 0 has no sign
708			} else {
709				u, t = t, u
710			}
711			t.Sub(u, v)
712		}
713
714		return z.Lsh(u, k)
715	}
716
717	// ProbablyPrime performs n Miller-Rabin tests to check whether x is prime.
718	// If it returns true, x is prime with probability 1 - 1/4^n.
719	// If it returns false, x is not prime.
720	func (x *Int) ProbablyPrime(n int) bool {
721		return !x.neg && x.abs.probablyPrime(n)
722	}
723
724	// Rand sets z to a pseudo-random number in [0, n) and returns z.
725	func (z *Int) Rand(rnd *rand.Rand, n *Int) *Int {
726		z.neg = false
727		if n.neg == true || len(n.abs) == 0 {
728			z.abs = nil
729			return z
730		}
731		z.abs = z.abs.random(rnd, n.abs, n.abs.bitLen())
732		return z
733	}
734
735	// ModInverse sets z to the multiplicative inverse of g in the group ℤ/pℤ (where
736	// p is a prime) and returns z.
737	func (z *Int) ModInverse(g, p *Int) *Int {
738		var d Int
739		d.GCD(z, nil, g, p)
740		// x and y are such that g*x + p*y = d. Since p is prime, d = 1. Taking
741		// that modulo p results in g*x = 1, therefore x is the inverse element.
742		if z.neg {
744		}
745		return z
746	}
747
748	// Lsh sets z = x << n and returns z.
749	func (z *Int) Lsh(x *Int, n uint) *Int {
750		z.abs = z.abs.shl(x.abs, n)
751		z.neg = x.neg
752		return z
753	}
754
755	// Rsh sets z = x >> n and returns z.
756	func (z *Int) Rsh(x *Int, n uint) *Int {
757		if x.neg {
758			// (-x) >> s == ^(x-1) >> s == ^((x-1) >> s) == -(((x-1) >> s) + 1)
759			t := z.abs.sub(x.abs, natOne) // no underflow because |x| > 0
760			t = t.shr(t, n)
762			z.neg = true // z cannot be zero if x is negative
763			return z
764		}
765
766		z.abs = z.abs.shr(x.abs, n)
767		z.neg = false
768		return z
769	}
770
771	// Bit returns the value of the i'th bit of x. That is, it
772	// returns (x>>i)&1. The bit index i must be >= 0.
773	func (x *Int) Bit(i int) uint {
774		if i == 0 {
775			// optimization for common case: odd/even test of x
776			if len(x.abs) > 0 {
777				return uint(x.abs[0] & 1) // bit 0 is same for -x
778			}
779			return 0
780		}
781		if i < 0 {
782			panic("negative bit index")
783		}
784		if x.neg {
785			t := nat(nil).sub(x.abs, natOne)
786			return t.bit(uint(i)) ^ 1
787		}
788
789		return x.abs.bit(uint(i))
790	}
791
792	// SetBit sets z to x, with x's i'th bit set to b (0 or 1).
793	// That is, if b is 1 SetBit sets z = x | (1 << i);
794	// if b is 0 SetBit sets z = x &^ (1 << i). If b is not 0 or 1,
795	// SetBit will panic.
796	func (z *Int) SetBit(x *Int, i int, b uint) *Int {
797		if i < 0 {
798			panic("negative bit index")
799		}
800		if x.neg {
801			t := z.abs.sub(x.abs, natOne)
802			t = t.setBit(t, uint(i), b^1)
804			z.neg = len(z.abs) > 0
805			return z
806		}
807		z.abs = z.abs.setBit(x.abs, uint(i), b)
808		z.neg = false
809		return z
810	}
811
812	// And sets z = x & y and returns z.
813	func (z *Int) And(x, y *Int) *Int {
814		if x.neg == y.neg {
815			if x.neg {
816				// (-x) & (-y) == ^(x-1) & ^(y-1) == ^((x-1) | (y-1)) == -(((x-1) | (y-1)) + 1)
817				x1 := nat(nil).sub(x.abs, natOne)
818				y1 := nat(nil).sub(y.abs, natOne)
819				z.abs = z.abs.add(z.abs.or(x1, y1), natOne)
820				z.neg = true // z cannot be zero if x and y are negative
821				return z
822			}
823
824			// x & y == x & y
825			z.abs = z.abs.and(x.abs, y.abs)
826			z.neg = false
827			return z
828		}
829
830		// x.neg != y.neg
831		if x.neg {
832			x, y = y, x // & is symmetric
833		}
834
835		// x & (-y) == x & ^(y-1) == x &^ (y-1)
836		y1 := nat(nil).sub(y.abs, natOne)
837		z.abs = z.abs.andNot(x.abs, y1)
838		z.neg = false
839		return z
840	}
841
842	// AndNot sets z = x &^ y and returns z.
843	func (z *Int) AndNot(x, y *Int) *Int {
844		if x.neg == y.neg {
845			if x.neg {
846				// (-x) &^ (-y) == ^(x-1) &^ ^(y-1) == ^(x-1) & (y-1) == (y-1) &^ (x-1)
847				x1 := nat(nil).sub(x.abs, natOne)
848				y1 := nat(nil).sub(y.abs, natOne)
849				z.abs = z.abs.andNot(y1, x1)
850				z.neg = false
851				return z
852			}
853
854			// x &^ y == x &^ y
855			z.abs = z.abs.andNot(x.abs, y.abs)
856			z.neg = false
857			return z
858		}
859
860		if x.neg {
861			// (-x) &^ y == ^(x-1) &^ y == ^(x-1) & ^y == ^((x-1) | y) == -(((x-1) | y) + 1)
862			x1 := nat(nil).sub(x.abs, natOne)
863			z.abs = z.abs.add(z.abs.or(x1, y.abs), natOne)
864			z.neg = true // z cannot be zero if x is negative and y is positive
865			return z
866		}
867
868		// x &^ (-y) == x &^ ^(y-1) == x & (y-1)
870		z.abs = z.abs.and(x.abs, y1)
871		z.neg = false
872		return z
873	}
874
875	// Or sets z = x | y and returns z.
876	func (z *Int) Or(x, y *Int) *Int {
877		if x.neg == y.neg {
878			if x.neg {
879				// (-x) | (-y) == ^(x-1) | ^(y-1) == ^((x-1) & (y-1)) == -(((x-1) & (y-1)) + 1)
880				x1 := nat(nil).sub(x.abs, natOne)
881				y1 := nat(nil).sub(y.abs, natOne)
882				z.abs = z.abs.add(z.abs.and(x1, y1), natOne)
883				z.neg = true // z cannot be zero if x and y are negative
884				return z
885			}
886
887			// x | y == x | y
888			z.abs = z.abs.or(x.abs, y.abs)
889			z.neg = false
890			return z
891		}
892
893		// x.neg != y.neg
894		if x.neg {
895			x, y = y, x // | is symmetric
896		}
897
898		// x | (-y) == x | ^(y-1) == ^((y-1) &^ x) == -(^((y-1) &^ x) + 1)
899		y1 := nat(nil).sub(y.abs, natOne)
900		z.abs = z.abs.add(z.abs.andNot(y1, x.abs), natOne)
901		z.neg = true // z cannot be zero if one of x or y is negative
902		return z
903	}
904
905	// Xor sets z = x ^ y and returns z.
906	func (z *Int) Xor(x, y *Int) *Int {
907		if x.neg == y.neg {
908			if x.neg {
909				// (-x) ^ (-y) == ^(x-1) ^ ^(y-1) == (x-1) ^ (y-1)
910				x1 := nat(nil).sub(x.abs, natOne)
911				y1 := nat(nil).sub(y.abs, natOne)
912				z.abs = z.abs.xor(x1, y1)
913				z.neg = false
914				return z
915			}
916
917			// x ^ y == x ^ y
918			z.abs = z.abs.xor(x.abs, y.abs)
919			z.neg = false
920			return z
921		}
922
923		// x.neg != y.neg
924		if x.neg {
925			x, y = y, x // ^ is symmetric
926		}
927
928		// x ^ (-y) == x ^ ^(y-1) == ^(x ^ (y-1)) == -((x ^ (y-1)) + 1)
929		y1 := nat(nil).sub(y.abs, natOne)
930		z.abs = z.abs.add(z.abs.xor(x.abs, y1), natOne)
931		z.neg = true // z cannot be zero if only one of x or y is negative
932		return z
933	}
934
935	// Not sets z = ^x and returns z.
936	func (z *Int) Not(x *Int) *Int {
937		if x.neg {
938			// ^(-x) == ^(^(x-1)) == x-1
939			z.abs = z.abs.sub(x.abs, natOne)
940			z.neg = false
941			return z
942		}
943
944		// ^x == -x-1 == -(x+1)
946		z.neg = true // z cannot be zero if x is positive
947		return z
948	}
949
950	// Gob codec version. Permits backward-compatible changes to the encoding.
951	const intGobVersion byte = 1
952
953	// GobEncode implements the gob.GobEncoder interface.
954	func (x *Int) GobEncode() ([]byte, error) {
955		if x == nil {
956			return nil, nil
957		}
958		buf := make([]byte, 1+len(x.abs)*_S) // extra byte for version and sign bit
959		i := x.abs.bytes(buf) - 1            // i >= 0
960		b := intGobVersion << 1              // make space for sign bit
961		if x.neg {
962			b |= 1
963		}
964		buf[i] = b
965		return buf[i:], nil
966	}
967
968	// GobDecode implements the gob.GobDecoder interface.
969	func (z *Int) GobDecode(buf []byte) error {
970		if len(buf) == 0 {
971			// Other side sent a nil or default value.
972			*z = Int{}
973			return nil
974		}
975		b := buf[0]
976		if b>>1 != intGobVersion {
977			return errors.New(fmt.Sprintf("Int.GobDecode: encoding version %d not supported", b>>1))
978		}
979		z.neg = b&1 != 0
980		z.abs = z.abs.setBytes(buf[1:])
981		return nil
982	}
983
984	// MarshalJSON implements the json.Marshaler interface.
985	func (x *Int) MarshalJSON() ([]byte, error) {
986		// TODO(gri): get rid of the []byte/string conversions
987		return []byte(x.String()), nil
988	}
989
990	// UnmarshalJSON implements the json.Unmarshaler interface.
991	func (z *Int) UnmarshalJSON(x []byte) error {
992		// TODO(gri): get rid of the []byte/string conversions
993		_, ok := z.SetString(string(x), 0)
994		if !ok {
995			return fmt.Errorf("math/big: cannot unmarshal %s into a *big.Int", x)
996		}
997		return nil
998	}
```

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