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# Source file src/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	// low32 returns the least significant 32 bits of z.
514	func low32(z nat) uint32 {
515		if len(z) == 0 {
516			return 0
517		}
518		return uint32(z[0])
519	}
520
521	// low64 returns the least significant 64 bits of z.
522	func low64(z nat) uint64 {
523		if len(z) == 0 {
524			return 0
525		}
526		v := uint64(z[0])
527		if _W == 32 && len(z) > 1 {
528			v |= uint64(z[1]) << 32
529		}
530		return v
531	}
532
533	// Int64 returns the int64 representation of x.
534	// If x cannot be represented in an int64, the result is undefined.
535	func (x *Int) Int64() int64 {
536		v := int64(low64(x.abs))
537		if x.neg {
538			v = -v
539		}
540		return v
541	}
542
543	// Uint64 returns the uint64 representation of x.
544	// If x cannot be represented in a uint64, the result is undefined.
545	func (x *Int) Uint64() uint64 {
546		return low64(x.abs)
547	}
548
549	// SetString sets z to the value of s, interpreted in the given base,
550	// and returns z and a boolean indicating success. If SetString fails,
551	// the value of z is undefined but the returned value is nil.
552	//
553	// The base argument must be 0 or a value from 2 through MaxBase. If the base
554	// is 0, the string prefix determines the actual conversion base. A prefix of
555	// ``0x'' or ``0X'' selects base 16; the ``0'' prefix selects base 8, and a
556	// ``0b'' or ``0B'' prefix selects base 2. Otherwise the selected base is 10.
557	//
558	func (z *Int) SetString(s string, base int) (*Int, bool) {
560		_, _, err := z.scan(r, base)
561		if err != nil {
562			return nil, false
563		}
564		_, _, err = r.ReadRune()
565		if err != io.EOF {
566			return nil, false
567		}
568		return z, true // err == io.EOF => scan consumed all of s
569	}
570
571	// SetBytes interprets buf as the bytes of a big-endian unsigned
572	// integer, sets z to that value, and returns z.
573	func (z *Int) SetBytes(buf []byte) *Int {
574		z.abs = z.abs.setBytes(buf)
575		z.neg = false
576		return z
577	}
578
579	// Bytes returns the absolute value of x as a big-endian byte slice.
580	func (x *Int) Bytes() []byte {
581		buf := make([]byte, len(x.abs)*_S)
582		return buf[x.abs.bytes(buf):]
583	}
584
585	// BitLen returns the length of the absolute value of x in bits.
586	// The bit length of 0 is 0.
587	func (x *Int) BitLen() int {
588		return x.abs.bitLen()
589	}
590
591	// Exp sets z = x**y mod |m| (i.e. the sign of m is ignored), and returns z.
592	// If y <= 0, the result is 1 mod |m|; if m == nil or m == 0, z = x**y.
593	// See Knuth, volume 2, section 4.6.3.
594	func (z *Int) Exp(x, y, m *Int) *Int {
595		var yWords nat
596		if !y.neg {
597			yWords = y.abs
598		}
599		// y >= 0
600
601		var mWords nat
602		if m != nil {
603			mWords = m.abs // m.abs may be nil for m == 0
604		}
605
606		z.abs = z.abs.expNN(x.abs, yWords, mWords)
607		z.neg = len(z.abs) > 0 && x.neg && len(yWords) > 0 && yWords[0]&1 == 1 // 0 has no sign
608		if z.neg && len(mWords) > 0 {
609			// make modulus result positive
610			z.abs = z.abs.sub(mWords, z.abs) // z == x**y mod |m| && 0 <= z < |m|
611			z.neg = false
612		}
613
614		return z
615	}
616
617	// GCD sets z to the greatest common divisor of a and b, which both must
618	// be > 0, and returns z.
619	// If x and y are not nil, GCD sets x and y such that z = a*x + b*y.
620	// If either a or b is <= 0, GCD sets z = x = y = 0.
621	func (z *Int) GCD(x, y, a, b *Int) *Int {
622		if a.Sign() <= 0 || b.Sign() <= 0 {
623			z.SetInt64(0)
624			if x != nil {
625				x.SetInt64(0)
626			}
627			if y != nil {
628				y.SetInt64(0)
629			}
630			return z
631		}
632		if x == nil && y == nil {
633			return z.binaryGCD(a, b)
634		}
635
636		A := new(Int).Set(a)
637		B := new(Int).Set(b)
638
639		X := new(Int)
640		Y := new(Int).SetInt64(1)
641
642		lastX := new(Int).SetInt64(1)
643		lastY := new(Int)
644
645		q := new(Int)
646		temp := new(Int)
647
648		for len(B.abs) > 0 {
649			r := new(Int)
650			q, r = q.QuoRem(A, B, r)
651
652			A, B = B, r
653
654			temp.Set(X)
655			X.Mul(X, q)
656			X.neg = !X.neg
658			lastX.Set(temp)
659
660			temp.Set(Y)
661			Y.Mul(Y, q)
662			Y.neg = !Y.neg
664			lastY.Set(temp)
665		}
666
667		if x != nil {
668			*x = *lastX
669		}
670
671		if y != nil {
672			*y = *lastY
673		}
674
675		*z = *A
676		return z
677	}
678
679	// binaryGCD sets z to the greatest common divisor of a and b, which both must
680	// be > 0, and returns z.
681	// See Knuth, The Art of Computer Programming, Vol. 2, Section 4.5.2, Algorithm B.
682	func (z *Int) binaryGCD(a, b *Int) *Int {
683		u := z
684		v := new(Int)
685
686		// use one Euclidean iteration to ensure that u and v are approx. the same size
687		switch {
688		case len(a.abs) > len(b.abs):
689			u.Set(b)
690			v.Rem(a, b)
691		case len(a.abs) < len(b.abs):
692			u.Set(a)
693			v.Rem(b, a)
694		default:
695			u.Set(a)
696			v.Set(b)
697		}
698
699		// v might be 0 now
700		if len(v.abs) == 0 {
701			return u
702		}
703		// u > 0 && v > 0
704
705		// determine largest k such that u = u' << k, v = v' << k
706		k := u.abs.trailingZeroBits()
707		if vk := v.abs.trailingZeroBits(); vk < k {
708			k = vk
709		}
710		u.Rsh(u, k)
711		v.Rsh(v, k)
712
713		// determine t (we know that u > 0)
714		t := new(Int)
715		if u.abs[0]&1 != 0 {
716			// u is odd
717			t.Neg(v)
718		} else {
719			t.Set(u)
720		}
721
722		for len(t.abs) > 0 {
723			// reduce t
724			t.Rsh(t, t.abs.trailingZeroBits())
725			if t.neg {
726				v, t = t, v
727				v.neg = len(v.abs) > 0 && !v.neg // 0 has no sign
728			} else {
729				u, t = t, u
730			}
731			t.Sub(u, v)
732		}
733
734		return z.Lsh(u, k)
735	}
736
737	// ProbablyPrime performs n Miller-Rabin tests to check whether x is prime.
738	// If it returns true, x is prime with probability 1 - 1/4^n.
739	// If it returns false, x is not prime.
740	func (x *Int) ProbablyPrime(n int) bool {
741		return !x.neg && x.abs.probablyPrime(n)
742	}
743
744	// Rand sets z to a pseudo-random number in [0, n) and returns z.
745	func (z *Int) Rand(rnd *rand.Rand, n *Int) *Int {
746		z.neg = false
747		if n.neg == true || len(n.abs) == 0 {
748			z.abs = nil
749			return z
750		}
751		z.abs = z.abs.random(rnd, n.abs, n.abs.bitLen())
752		return z
753	}
754
755	// ModInverse sets z to the multiplicative inverse of g in the ring ℤ/nℤ
756	// and returns z. If g and n are not relatively prime, the result is undefined.
757	func (z *Int) ModInverse(g, n *Int) *Int {
758		var d Int
759		d.GCD(z, nil, g, n)
760		// x and y are such that g*x + n*y = d. Since g and n are
761		// relatively prime, d = 1. Taking that modulo n results in
762		// g*x = 1, therefore x is the inverse element.
763		if z.neg {
765		}
766		return z
767	}
768
769	// Lsh sets z = x << n and returns z.
770	func (z *Int) Lsh(x *Int, n uint) *Int {
771		z.abs = z.abs.shl(x.abs, n)
772		z.neg = x.neg
773		return z
774	}
775
776	// Rsh sets z = x >> n and returns z.
777	func (z *Int) Rsh(x *Int, n uint) *Int {
778		if x.neg {
779			// (-x) >> s == ^(x-1) >> s == ^((x-1) >> s) == -(((x-1) >> s) + 1)
780			t := z.abs.sub(x.abs, natOne) // no underflow because |x| > 0
781			t = t.shr(t, n)
783			z.neg = true // z cannot be zero if x is negative
784			return z
785		}
786
787		z.abs = z.abs.shr(x.abs, n)
788		z.neg = false
789		return z
790	}
791
792	// Bit returns the value of the i'th bit of x. That is, it
793	// returns (x>>i)&1. The bit index i must be >= 0.
794	func (x *Int) Bit(i int) uint {
795		if i == 0 {
796			// optimization for common case: odd/even test of x
797			if len(x.abs) > 0 {
798				return uint(x.abs[0] & 1) // bit 0 is same for -x
799			}
800			return 0
801		}
802		if i < 0 {
803			panic("negative bit index")
804		}
805		if x.neg {
806			t := nat(nil).sub(x.abs, natOne)
807			return t.bit(uint(i)) ^ 1
808		}
809
810		return x.abs.bit(uint(i))
811	}
812
813	// SetBit sets z to x, with x's i'th bit set to b (0 or 1).
814	// That is, if b is 1 SetBit sets z = x | (1 << i);
815	// if b is 0 SetBit sets z = x &^ (1 << i). If b is not 0 or 1,
816	// SetBit will panic.
817	func (z *Int) SetBit(x *Int, i int, b uint) *Int {
818		if i < 0 {
819			panic("negative bit index")
820		}
821		if x.neg {
822			t := z.abs.sub(x.abs, natOne)
823			t = t.setBit(t, uint(i), b^1)
825			z.neg = len(z.abs) > 0
826			return z
827		}
828		z.abs = z.abs.setBit(x.abs, uint(i), b)
829		z.neg = false
830		return z
831	}
832
833	// And sets z = x & y and returns z.
834	func (z *Int) And(x, y *Int) *Int {
835		if x.neg == y.neg {
836			if x.neg {
837				// (-x) & (-y) == ^(x-1) & ^(y-1) == ^((x-1) | (y-1)) == -(((x-1) | (y-1)) + 1)
838				x1 := nat(nil).sub(x.abs, natOne)
839				y1 := nat(nil).sub(y.abs, natOne)
840				z.abs = z.abs.add(z.abs.or(x1, y1), natOne)
841				z.neg = true // z cannot be zero if x and y are negative
842				return z
843			}
844
845			// x & y == x & y
846			z.abs = z.abs.and(x.abs, y.abs)
847			z.neg = false
848			return z
849		}
850
851		// x.neg != y.neg
852		if x.neg {
853			x, y = y, x // & is symmetric
854		}
855
856		// x & (-y) == x & ^(y-1) == x &^ (y-1)
857		y1 := nat(nil).sub(y.abs, natOne)
858		z.abs = z.abs.andNot(x.abs, y1)
859		z.neg = false
860		return z
861	}
862
863	// AndNot sets z = x &^ y and returns z.
864	func (z *Int) AndNot(x, y *Int) *Int {
865		if x.neg == y.neg {
866			if x.neg {
867				// (-x) &^ (-y) == ^(x-1) &^ ^(y-1) == ^(x-1) & (y-1) == (y-1) &^ (x-1)
868				x1 := nat(nil).sub(x.abs, natOne)
869				y1 := nat(nil).sub(y.abs, natOne)
870				z.abs = z.abs.andNot(y1, x1)
871				z.neg = false
872				return z
873			}
874
875			// x &^ y == x &^ y
876			z.abs = z.abs.andNot(x.abs, y.abs)
877			z.neg = false
878			return z
879		}
880
881		if x.neg {
882			// (-x) &^ y == ^(x-1) &^ y == ^(x-1) & ^y == ^((x-1) | y) == -(((x-1) | y) + 1)
883			x1 := nat(nil).sub(x.abs, natOne)
884			z.abs = z.abs.add(z.abs.or(x1, y.abs), natOne)
885			z.neg = true // z cannot be zero if x is negative and y is positive
886			return z
887		}
888
889		// x &^ (-y) == x &^ ^(y-1) == x & (y-1)
890		y1 := nat(nil).sub(y.abs, natOne)
891		z.abs = z.abs.and(x.abs, y1)
892		z.neg = false
893		return z
894	}
895
896	// Or sets z = x | y and returns z.
897	func (z *Int) Or(x, y *Int) *Int {
898		if x.neg == y.neg {
899			if x.neg {
900				// (-x) | (-y) == ^(x-1) | ^(y-1) == ^((x-1) & (y-1)) == -(((x-1) & (y-1)) + 1)
901				x1 := nat(nil).sub(x.abs, natOne)
902				y1 := nat(nil).sub(y.abs, natOne)
903				z.abs = z.abs.add(z.abs.and(x1, y1), natOne)
904				z.neg = true // z cannot be zero if x and y are negative
905				return z
906			}
907
908			// x | y == x | y
909			z.abs = z.abs.or(x.abs, y.abs)
910			z.neg = false
911			return z
912		}
913
914		// x.neg != y.neg
915		if x.neg {
916			x, y = y, x // | is symmetric
917		}
918
919		// x | (-y) == x | ^(y-1) == ^((y-1) &^ x) == -(^((y-1) &^ x) + 1)
920		y1 := nat(nil).sub(y.abs, natOne)
921		z.abs = z.abs.add(z.abs.andNot(y1, x.abs), natOne)
922		z.neg = true // z cannot be zero if one of x or y is negative
923		return z
924	}
925
926	// Xor sets z = x ^ y and returns z.
927	func (z *Int) Xor(x, y *Int) *Int {
928		if x.neg == y.neg {
929			if x.neg {
930				// (-x) ^ (-y) == ^(x-1) ^ ^(y-1) == (x-1) ^ (y-1)
931				x1 := nat(nil).sub(x.abs, natOne)
932				y1 := nat(nil).sub(y.abs, natOne)
933				z.abs = z.abs.xor(x1, y1)
934				z.neg = false
935				return z
936			}
937
938			// x ^ y == x ^ y
939			z.abs = z.abs.xor(x.abs, y.abs)
940			z.neg = false
941			return z
942		}
943
944		// x.neg != y.neg
945		if x.neg {
946			x, y = y, x // ^ is symmetric
947		}
948
949		// x ^ (-y) == x ^ ^(y-1) == ^(x ^ (y-1)) == -((x ^ (y-1)) + 1)
950		y1 := nat(nil).sub(y.abs, natOne)
951		z.abs = z.abs.add(z.abs.xor(x.abs, y1), natOne)
952		z.neg = true // z cannot be zero if only one of x or y is negative
953		return z
954	}
955
956	// Not sets z = ^x and returns z.
957	func (z *Int) Not(x *Int) *Int {
958		if x.neg {
959			// ^(-x) == ^(^(x-1)) == x-1
960			z.abs = z.abs.sub(x.abs, natOne)
961			z.neg = false
962			return z
963		}
964
965		// ^x == -x-1 == -(x+1)
967		z.neg = true // z cannot be zero if x is positive
968		return z
969	}
970
971	// Gob codec version. Permits backward-compatible changes to the encoding.
972	const intGobVersion byte = 1
973
974	// GobEncode implements the gob.GobEncoder interface.
975	func (x *Int) GobEncode() ([]byte, error) {
976		if x == nil {
977			return nil, nil
978		}
979		buf := make([]byte, 1+len(x.abs)*_S) // extra byte for version and sign bit
980		i := x.abs.bytes(buf) - 1            // i >= 0
981		b := intGobVersion << 1              // make space for sign bit
982		if x.neg {
983			b |= 1
984		}
985		buf[i] = b
986		return buf[i:], nil
987	}
988
989	// GobDecode implements the gob.GobDecoder interface.
990	func (z *Int) GobDecode(buf []byte) error {
991		if len(buf) == 0 {
992			// Other side sent a nil or default value.
993			*z = Int{}
994			return nil
995		}
996		b := buf[0]
997		if b>>1 != intGobVersion {
998			return errors.New(fmt.Sprintf("Int.GobDecode: encoding version %d not supported", b>>1))
999		}
1000		z.neg = b&1 != 0
1001		z.abs = z.abs.setBytes(buf[1:])
1002		return nil
1003	}
1004
1005	// MarshalJSON implements the json.Marshaler interface.
1006	func (z *Int) MarshalJSON() ([]byte, error) {
1007		// TODO(gri): get rid of the []byte/string conversions
1008		return []byte(z.String()), nil
1009	}
1010
1011	// UnmarshalJSON implements the json.Unmarshaler interface.
1012	func (z *Int) UnmarshalJSON(text []byte) error {
1013		// TODO(gri): get rid of the []byte/string conversions
1014		if _, ok := z.SetString(string(text), 0); !ok {
1015			return fmt.Errorf("math/big: cannot unmarshal %q into a *big.Int", text)
1016		}
1017		return nil
1018	}
1019
1020	// MarshalText implements the encoding.TextMarshaler interface.
1021	func (z *Int) MarshalText() (text []byte, err error) {
1022		return []byte(z.String()), nil
1023	}
1024
1025	// UnmarshalText implements the encoding.TextUnmarshaler interface.
1026	func (z *Int) UnmarshalText(text []byte) error {
1027		if _, ok := z.SetString(string(text), 0); !ok {
1028			return fmt.Errorf("math/big: cannot unmarshal %q into a *big.Int", text)
1029		}
1030		return nil
1031	}
1032
```

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