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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		"fmt"
11		"io"
12		"math/rand"
13		"strings"
14	)
15
16	// An Int represents a signed multi-precision integer.
17	// The zero value for an Int represents the value 0.
18	type Int struct {
19		neg bool // sign
20		abs nat  // absolute value of the integer
21	}
22
23	var intOne = &Int{false, natOne}
24
25	// Sign returns:
26	//
27	//	-1 if x <  0
28	//	 0 if x == 0
29	//	+1 if x >  0
30	//
31	func (x *Int) Sign() int {
32		if len(x.abs) == 0 {
33			return 0
34		}
35		if x.neg {
36			return -1
37		}
38		return 1
39	}
40
41	// SetInt64 sets z to x and returns z.
42	func (z *Int) SetInt64(x int64) *Int {
43		neg := false
44		if x < 0 {
45			neg = true
46			x = -x
47		}
48		z.abs = z.abs.setUint64(uint64(x))
49		z.neg = neg
50		return z
51	}
52
53	// SetUint64 sets z to x and returns z.
54	func (z *Int) SetUint64(x uint64) *Int {
55		z.abs = z.abs.setUint64(x)
56		z.neg = false
57		return z
58	}
59
60	// NewInt allocates and returns a new Int set to x.
61	func NewInt(x int64) *Int {
62		return new(Int).SetInt64(x)
63	}
64
65	// Set sets z to x and returns z.
66	func (z *Int) Set(x *Int) *Int {
67		if z != x {
68			z.abs = z.abs.set(x.abs)
69			z.neg = x.neg
70		}
71		return z
72	}
73
74	// Bits provides raw (unchecked but fast) access to x by returning its
75	// absolute value as a little-endian Word slice. The result and x share
76	// the same underlying array.
77	// Bits is intended to support implementation of missing low-level Int
78	// functionality outside this package; it should be avoided otherwise.
79	func (x *Int) Bits() []Word {
80		return x.abs
81	}
82
83	// SetBits provides raw (unchecked but fast) access to z by setting its
84	// value to abs, interpreted as a little-endian Word slice, and returning
85	// z. The result and abs share the same underlying array.
86	// SetBits is intended to support implementation of missing low-level Int
87	// functionality outside this package; it should be avoided otherwise.
88	func (z *Int) SetBits(abs []Word) *Int {
89		z.abs = nat(abs).norm()
90		z.neg = false
91		return z
92	}
93
94	// Abs sets z to |x| (the absolute value of x) and returns z.
95	func (z *Int) Abs(x *Int) *Int {
96		z.Set(x)
97		z.neg = false
98		return z
99	}
100
101	// Neg sets z to -x and returns z.
102	func (z *Int) Neg(x *Int) *Int {
103		z.Set(x)
104		z.neg = len(z.abs) > 0 && !z.neg // 0 has no sign
105		return z
106	}
107
108	// Add sets z to the sum x+y and returns z.
109	func (z *Int) Add(x, y *Int) *Int {
110		neg := x.neg
111		if x.neg == y.neg {
112			// x + y == x + y
113			// (-x) + (-y) == -(x + y)
115		} else {
116			// x + (-y) == x - y == -(y - x)
117			// (-x) + y == y - x == -(x - y)
118			if x.abs.cmp(y.abs) >= 0 {
119				z.abs = z.abs.sub(x.abs, y.abs)
120			} else {
121				neg = !neg
122				z.abs = z.abs.sub(y.abs, x.abs)
123			}
124		}
125		z.neg = len(z.abs) > 0 && neg // 0 has no sign
126		return z
127	}
128
129	// Sub sets z to the difference x-y and returns z.
130	func (z *Int) Sub(x, y *Int) *Int {
131		neg := x.neg
132		if x.neg != y.neg {
133			// x - (-y) == x + y
134			// (-x) - y == -(x + y)
136		} else {
137			// x - y == x - y == -(y - x)
138			// (-x) - (-y) == y - x == -(x - y)
139			if x.abs.cmp(y.abs) >= 0 {
140				z.abs = z.abs.sub(x.abs, y.abs)
141			} else {
142				neg = !neg
143				z.abs = z.abs.sub(y.abs, x.abs)
144			}
145		}
146		z.neg = len(z.abs) > 0 && neg // 0 has no sign
147		return z
148	}
149
150	// Mul sets z to the product x*y and returns z.
151	func (z *Int) Mul(x, y *Int) *Int {
152		// x * y == x * y
153		// x * (-y) == -(x * y)
154		// (-x) * y == -(x * y)
155		// (-x) * (-y) == x * y
156		z.abs = z.abs.mul(x.abs, y.abs)
157		z.neg = len(z.abs) > 0 && x.neg != y.neg // 0 has no sign
158		return z
159	}
160
161	// MulRange sets z to the product of all integers
162	// in the range [a, b] inclusively and returns z.
163	// If a > b (empty range), the result is 1.
164	func (z *Int) MulRange(a, b int64) *Int {
165		switch {
166		case a > b:
167			return z.SetInt64(1) // empty range
168		case a <= 0 && b >= 0:
169			return z.SetInt64(0) // range includes 0
170		}
171		// a <= b && (b < 0 || a > 0)
172
173		neg := false
174		if a < 0 {
175			neg = (b-a)&1 == 0
176			a, b = -b, -a
177		}
178
179		z.abs = z.abs.mulRange(uint64(a), uint64(b))
180		z.neg = neg
181		return z
182	}
183
184	// Binomial sets z to the binomial coefficient of (n, k) and returns z.
185	func (z *Int) Binomial(n, k int64) *Int {
186		// reduce the number of multiplications by reducing k
187		if n/2 < k && k <= n {
188			k = n - k // Binomial(n, k) == Binomial(n, n-k)
189		}
190		var a, b Int
191		a.MulRange(n-k+1, n)
192		b.MulRange(1, k)
193		return z.Quo(&a, &b)
194	}
195
196	// Quo sets z to the quotient x/y for y != 0 and returns z.
197	// If y == 0, a division-by-zero run-time panic occurs.
198	// Quo implements truncated division (like Go); see QuoRem for more details.
199	func (z *Int) Quo(x, y *Int) *Int {
200		z.abs, _ = z.abs.div(nil, x.abs, y.abs)
201		z.neg = len(z.abs) > 0 && x.neg != y.neg // 0 has no sign
202		return z
203	}
204
205	// Rem sets z to the remainder x%y for y != 0 and returns z.
206	// If y == 0, a division-by-zero run-time panic occurs.
207	// Rem implements truncated modulus (like Go); see QuoRem for more details.
208	func (z *Int) Rem(x, y *Int) *Int {
209		_, z.abs = nat(nil).div(z.abs, x.abs, y.abs)
210		z.neg = len(z.abs) > 0 && x.neg // 0 has no sign
211		return z
212	}
213
214	// QuoRem sets z to the quotient x/y and r to the remainder x%y
215	// and returns the pair (z, r) for y != 0.
216	// If y == 0, a division-by-zero run-time panic occurs.
217	//
218	// QuoRem implements T-division and modulus (like Go):
219	//
220	//	q = x/y      with the result truncated to zero
221	//	r = x - y*q
222	//
223	// (See Daan Leijen, ``Division and Modulus for Computer Scientists''.)
224	// See DivMod for Euclidean division and modulus (unlike Go).
225	//
226	func (z *Int) QuoRem(x, y, r *Int) (*Int, *Int) {
227		z.abs, r.abs = z.abs.div(r.abs, x.abs, y.abs)
228		z.neg, r.neg = len(z.abs) > 0 && x.neg != y.neg, len(r.abs) > 0 && x.neg // 0 has no sign
229		return z, r
230	}
231
232	// Div sets z to the quotient x/y for y != 0 and returns z.
233	// If y == 0, a division-by-zero run-time panic occurs.
234	// Div implements Euclidean division (unlike Go); see DivMod for more details.
235	func (z *Int) Div(x, y *Int) *Int {
236		y_neg := y.neg // z may be an alias for y
237		var r Int
238		z.QuoRem(x, y, &r)
239		if r.neg {
240			if y_neg {
242			} else {
243				z.Sub(z, intOne)
244			}
245		}
246		return z
247	}
248
249	// Mod sets z to the modulus x%y for y != 0 and returns z.
250	// If y == 0, a division-by-zero run-time panic occurs.
251	// Mod implements Euclidean modulus (unlike Go); see DivMod for more details.
252	func (z *Int) Mod(x, y *Int) *Int {
253		y0 := y // save y
254		if z == y || alias(z.abs, y.abs) {
255			y0 = new(Int).Set(y)
256		}
257		var q Int
258		q.QuoRem(x, y, z)
259		if z.neg {
260			if y0.neg {
261				z.Sub(z, y0)
262			} else {
264			}
265		}
266		return z
267	}
268
269	// DivMod sets z to the quotient x div y and m to the modulus x mod y
270	// and returns the pair (z, m) for y != 0.
271	// If y == 0, a division-by-zero run-time panic occurs.
272	//
273	// DivMod implements Euclidean division and modulus (unlike Go):
274	//
275	//	q = x div y  such that
276	//	m = x - y*q  with 0 <= m < |q|
277	//
278	// (See Raymond T. Boute, ``The Euclidean definition of the functions
279	// div and mod''. ACM Transactions on Programming Languages and
280	// Systems (TOPLAS), 14(2):127-144, New York, NY, USA, 4/1992.
281	// ACM press.)
282	// See QuoRem for T-division and modulus (like Go).
283	//
284	func (z *Int) DivMod(x, y, m *Int) (*Int, *Int) {
285		y0 := y // save y
286		if z == y || alias(z.abs, y.abs) {
287			y0 = new(Int).Set(y)
288		}
289		z.QuoRem(x, y, m)
290		if m.neg {
291			if y0.neg {
293				m.Sub(m, y0)
294			} else {
295				z.Sub(z, intOne)
297			}
298		}
299		return z, m
300	}
301
302	// Cmp compares x and y and returns:
303	//
304	//   -1 if x <  y
305	//    0 if x == y
306	//   +1 if x >  y
307	//
308	func (x *Int) Cmp(y *Int) (r int) {
309		// x cmp y == x cmp y
310		// x cmp (-y) == x
311		// (-x) cmp y == y
312		// (-x) cmp (-y) == -(x cmp y)
313		switch {
314		case x.neg == y.neg:
315			r = x.abs.cmp(y.abs)
316			if x.neg {
317				r = -r
318			}
319		case x.neg:
320			r = -1
321		default:
322			r = 1
323		}
324		return
325	}
326
327	// low32 returns the least significant 32 bits of z.
328	func low32(z nat) uint32 {
329		if len(z) == 0 {
330			return 0
331		}
332		return uint32(z[0])
333	}
334
335	// low64 returns the least significant 64 bits of z.
336	func low64(z nat) uint64 {
337		if len(z) == 0 {
338			return 0
339		}
340		v := uint64(z[0])
341		if _W == 32 && len(z) > 1 {
342			v |= uint64(z[1]) << 32
343		}
344		return v
345	}
346
347	// Int64 returns the int64 representation of x.
348	// If x cannot be represented in an int64, the result is undefined.
349	func (x *Int) Int64() int64 {
350		v := int64(low64(x.abs))
351		if x.neg {
352			v = -v
353		}
354		return v
355	}
356
357	// Uint64 returns the uint64 representation of x.
358	// If x cannot be represented in a uint64, the result is undefined.
359	func (x *Int) Uint64() uint64 {
360		return low64(x.abs)
361	}
362
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 but the returned value is nil.
366	//
367	// The base argument must be 0 or a value between 2 and MaxBase. If the base
368	// is 0, the string prefix determines the actual conversion base. A prefix of
369	// ``0x'' or ``0X'' selects base 16; the ``0'' prefix selects base 8, and a
370	// ``0b'' or ``0B'' prefix selects base 2. Otherwise the selected base is 10.
371	//
372	func (z *Int) SetString(s string, base int) (*Int, bool) {
374		_, _, err := z.scan(r, base)
375		if err != nil {
376			return nil, false
377		}
379		if err != io.EOF {
380			return nil, false
381		}
382		return z, true // err == io.EOF => scan consumed all of s
383	}
384
385	// SetBytes interprets buf as the bytes of a big-endian unsigned
386	// integer, sets z to that value, and returns z.
387	func (z *Int) SetBytes(buf []byte) *Int {
388		z.abs = z.abs.setBytes(buf)
389		z.neg = false
390		return z
391	}
392
393	// Bytes returns the absolute value of x as a big-endian byte slice.
394	func (x *Int) Bytes() []byte {
395		buf := make([]byte, len(x.abs)*_S)
396		return buf[x.abs.bytes(buf):]
397	}
398
399	// BitLen returns the length of the absolute value of x in bits.
400	// The bit length of 0 is 0.
401	func (x *Int) BitLen() int {
402		return x.abs.bitLen()
403	}
404
405	// Exp sets z = x**y mod |m| (i.e. the sign of m is ignored), and returns z.
406	// If y <= 0, the result is 1 mod |m|; if m == nil or m == 0, z = x**y.
407	// See Knuth, volume 2, section 4.6.3.
408	func (z *Int) Exp(x, y, m *Int) *Int {
409		var yWords nat
410		if !y.neg {
411			yWords = y.abs
412		}
413		// y >= 0
414
415		var mWords nat
416		if m != nil {
417			mWords = m.abs // m.abs may be nil for m == 0
418		}
419
420		z.abs = z.abs.expNN(x.abs, yWords, mWords)
421		z.neg = len(z.abs) > 0 && x.neg && len(yWords) > 0 && yWords[0]&1 == 1 // 0 has no sign
422		if z.neg && len(mWords) > 0 {
423			// make modulus result positive
424			z.abs = z.abs.sub(mWords, z.abs) // z == x**y mod |m| && 0 <= z < |m|
425			z.neg = false
426		}
427
428		return z
429	}
430
431	// GCD sets z to the greatest common divisor of a and b, which both must
432	// be > 0, and returns z.
433	// If x and y are not nil, GCD sets x and y such that z = a*x + b*y.
434	// If either a or b is <= 0, GCD sets z = x = y = 0.
435	func (z *Int) GCD(x, y, a, b *Int) *Int {
436		if a.Sign() <= 0 || b.Sign() <= 0 {
437			z.SetInt64(0)
438			if x != nil {
439				x.SetInt64(0)
440			}
441			if y != nil {
442				y.SetInt64(0)
443			}
444			return z
445		}
446		if x == nil && y == nil {
447			return z.binaryGCD(a, b)
448		}
449
450		A := new(Int).Set(a)
451		B := new(Int).Set(b)
452
453		X := new(Int)
454		Y := new(Int).SetInt64(1)
455
456		lastX := new(Int).SetInt64(1)
457		lastY := new(Int)
458
459		q := new(Int)
460		temp := new(Int)
461
462		for len(B.abs) > 0 {
463			r := new(Int)
464			q, r = q.QuoRem(A, B, r)
465
466			A, B = B, r
467
468			temp.Set(X)
469			X.Mul(X, q)
470			X.neg = !X.neg
472			lastX.Set(temp)
473
474			temp.Set(Y)
475			Y.Mul(Y, q)
476			Y.neg = !Y.neg
478			lastY.Set(temp)
479		}
480
481		if x != nil {
482			*x = *lastX
483		}
484
485		if y != nil {
486			*y = *lastY
487		}
488
489		*z = *A
490		return z
491	}
492
493	// binaryGCD sets z to the greatest common divisor of a and b, which both must
494	// be > 0, and returns z.
495	// See Knuth, The Art of Computer Programming, Vol. 2, Section 4.5.2, Algorithm B.
496	func (z *Int) binaryGCD(a, b *Int) *Int {
497		u := z
498		v := new(Int)
499
500		// use one Euclidean iteration to ensure that u and v are approx. the same size
501		switch {
502		case len(a.abs) > len(b.abs):
503			// must set v before u since u may be alias for a or b (was issue #11284)
504			v.Rem(a, b)
505			u.Set(b)
506		case len(a.abs) < len(b.abs):
507			v.Rem(b, a)
508			u.Set(a)
509		default:
510			v.Set(b)
511			u.Set(a)
512		}
513		// a, b must not be used anymore (may be aliases with u)
514
515		// v might be 0 now
516		if len(v.abs) == 0 {
517			return u
518		}
519		// u > 0 && v > 0
520
521		// determine largest k such that u = u' << k, v = v' << k
522		k := u.abs.trailingZeroBits()
523		if vk := v.abs.trailingZeroBits(); vk < k {
524			k = vk
525		}
526		u.Rsh(u, k)
527		v.Rsh(v, k)
528
529		// determine t (we know that u > 0)
530		t := new(Int)
531		if u.abs[0]&1 != 0 {
532			// u is odd
533			t.Neg(v)
534		} else {
535			t.Set(u)
536		}
537
538		for len(t.abs) > 0 {
539			// reduce t
540			t.Rsh(t, t.abs.trailingZeroBits())
541			if t.neg {
542				v, t = t, v
543				v.neg = len(v.abs) > 0 && !v.neg // 0 has no sign
544			} else {
545				u, t = t, u
546			}
547			t.Sub(u, v)
548		}
549
550		return z.Lsh(u, k)
551	}
552
553	// ProbablyPrime performs n Miller-Rabin tests to check whether x is prime.
554	// If it returns true, x is prime with probability 1 - 1/4^n.
555	// If it returns false, x is not prime. n must be > 0.
556	func (x *Int) ProbablyPrime(n int) bool {
557		if n <= 0 {
558			panic("non-positive n for ProbablyPrime")
559		}
560		return !x.neg && x.abs.probablyPrime(n)
561	}
562
563	// Rand sets z to a pseudo-random number in [0, n) and returns z.
564	func (z *Int) Rand(rnd *rand.Rand, n *Int) *Int {
565		z.neg = false
566		if n.neg == true || len(n.abs) == 0 {
567			z.abs = nil
568			return z
569		}
570		z.abs = z.abs.random(rnd, n.abs, n.abs.bitLen())
571		return z
572	}
573
574	// ModInverse sets z to the multiplicative inverse of g in the ring ℤ/nℤ
575	// and returns z. If g and n are not relatively prime, the result is undefined.
576	func (z *Int) ModInverse(g, n *Int) *Int {
577		var d Int
578		d.GCD(z, nil, g, n)
579		// x and y are such that g*x + n*y = d. Since g and n are
580		// relatively prime, d = 1. Taking that modulo n results in
581		// g*x = 1, therefore x is the inverse element.
582		if z.neg {
584		}
585		return z
586	}
587
588	// Jacobi returns the Jacobi symbol (x/y), either +1, -1, or 0.
589	// The y argument must be an odd integer.
590	func Jacobi(x, y *Int) int {
591		if len(y.abs) == 0 || y.abs[0]&1 == 0 {
592			panic(fmt.Sprintf("big: invalid 2nd argument to Int.Jacobi: need odd integer but got %s", y))
593		}
594
595		// We use the formulation described in chapter 2, section 2.4,
596		// "The Yacas Book of Algorithms":
597		// http://yacas.sourceforge.net/Algo.book.pdf
598
599		var a, b, c Int
600		a.Set(x)
601		b.Set(y)
602		j := 1
603
604		if b.neg {
605			if a.neg {
606				j = -1
607			}
608			b.neg = false
609		}
610
611		for {
612			if b.Cmp(intOne) == 0 {
613				return j
614			}
615			if len(a.abs) == 0 {
616				return 0
617			}
618			a.Mod(&a, &b)
619			if len(a.abs) == 0 {
620				return 0
621			}
622			// a > 0
623
624			// handle factors of 2 in 'a'
625			s := a.abs.trailingZeroBits()
626			if s&1 != 0 {
627				bmod8 := b.abs[0] & 7
628				if bmod8 == 3 || bmod8 == 5 {
629					j = -j
630				}
631			}
632			c.Rsh(&a, s) // a = 2^s*c
633
634			// swap numerator and denominator
635			if b.abs[0]&3 == 3 && c.abs[0]&3 == 3 {
636				j = -j
637			}
638			a.Set(&b)
639			b.Set(&c)
640		}
641	}
642
643	// ModSqrt sets z to a square root of x mod p if such a square root exists, and
644	// returns z. The modulus p must be an odd prime. If x is not a square mod p,
645	// ModSqrt leaves z unchanged and returns nil. This function panics if p is
646	// not an odd integer.
647	func (z *Int) ModSqrt(x, p *Int) *Int {
648		switch Jacobi(x, p) {
649		case -1:
650			return nil // x is not a square mod p
651		case 0:
652			return z.SetInt64(0) // sqrt(0) mod p = 0
653		case 1:
654			break
655		}
656		if x.neg || x.Cmp(p) >= 0 { // ensure 0 <= x < p
657			x = new(Int).Mod(x, p)
658		}
659
660		// Break p-1 into s*2^e such that s is odd.
661		var s Int
662		s.Sub(p, intOne)
663		e := s.abs.trailingZeroBits()
664		s.Rsh(&s, e)
665
666		// find some non-square n
667		var n Int
668		n.SetInt64(2)
669		for Jacobi(&n, p) != -1 {
671		}
672
673		// Core of the Tonelli-Shanks algorithm. Follows the description in
674		// section 6 of "Square roots from 1; 24, 51, 10 to Dan Shanks" by Ezra
675		// Brown:
677		var y, b, g, t Int
679		y.Rsh(&y, 1)
680		y.Exp(x, &y, p)  // y = x^((s+1)/2)
681		b.Exp(x, &s, p)  // b = x^s
682		g.Exp(&n, &s, p) // g = n^s
683		r := e
684		for {
685			// find the least m such that ord_p(b) = 2^m
686			var m uint
687			t.Set(&b)
688			for t.Cmp(intOne) != 0 {
689				t.Mul(&t, &t).Mod(&t, p)
690				m++
691			}
692
693			if m == 0 {
694				return z.Set(&y)
695			}
696
697			t.SetInt64(0).SetBit(&t, int(r-m-1), 1).Exp(&g, &t, p)
698			// t = g^(2^(r-m-1)) mod p
699			g.Mul(&t, &t).Mod(&g, p) // g = g^(2^(r-m)) mod p
700			y.Mul(&y, &t).Mod(&y, p)
701			b.Mul(&b, &g).Mod(&b, p)
702			r = m
703		}
704	}
705
706	// Lsh sets z = x << n and returns z.
707	func (z *Int) Lsh(x *Int, n uint) *Int {
708		z.abs = z.abs.shl(x.abs, n)
709		z.neg = x.neg
710		return z
711	}
712
713	// Rsh sets z = x >> n and returns z.
714	func (z *Int) Rsh(x *Int, n uint) *Int {
715		if x.neg {
716			// (-x) >> s == ^(x-1) >> s == ^((x-1) >> s) == -(((x-1) >> s) + 1)
717			t := z.abs.sub(x.abs, natOne) // no underflow because |x| > 0
718			t = t.shr(t, n)
720			z.neg = true // z cannot be zero if x is negative
721			return z
722		}
723
724		z.abs = z.abs.shr(x.abs, n)
725		z.neg = false
726		return z
727	}
728
729	// Bit returns the value of the i'th bit of x. That is, it
730	// returns (x>>i)&1. The bit index i must be >= 0.
731	func (x *Int) Bit(i int) uint {
732		if i == 0 {
733			// optimization for common case: odd/even test of x
734			if len(x.abs) > 0 {
735				return uint(x.abs[0] & 1) // bit 0 is same for -x
736			}
737			return 0
738		}
739		if i < 0 {
740			panic("negative bit index")
741		}
742		if x.neg {
743			t := nat(nil).sub(x.abs, natOne)
744			return t.bit(uint(i)) ^ 1
745		}
746
747		return x.abs.bit(uint(i))
748	}
749
750	// SetBit sets z to x, with x's i'th bit set to b (0 or 1).
751	// That is, if b is 1 SetBit sets z = x | (1 << i);
752	// if b is 0 SetBit sets z = x &^ (1 << i). If b is not 0 or 1,
753	// SetBit will panic.
754	func (z *Int) SetBit(x *Int, i int, b uint) *Int {
755		if i < 0 {
756			panic("negative bit index")
757		}
758		if x.neg {
759			t := z.abs.sub(x.abs, natOne)
760			t = t.setBit(t, uint(i), b^1)
762			z.neg = len(z.abs) > 0
763			return z
764		}
765		z.abs = z.abs.setBit(x.abs, uint(i), b)
766		z.neg = false
767		return z
768	}
769
770	// And sets z = x & y and returns z.
771	func (z *Int) And(x, y *Int) *Int {
772		if x.neg == y.neg {
773			if x.neg {
774				// (-x) & (-y) == ^(x-1) & ^(y-1) == ^((x-1) | (y-1)) == -(((x-1) | (y-1)) + 1)
775				x1 := nat(nil).sub(x.abs, natOne)
776				y1 := nat(nil).sub(y.abs, natOne)
777				z.abs = z.abs.add(z.abs.or(x1, y1), natOne)
778				z.neg = true // z cannot be zero if x and y are negative
779				return z
780			}
781
782			// x & y == x & y
783			z.abs = z.abs.and(x.abs, y.abs)
784			z.neg = false
785			return z
786		}
787
788		// x.neg != y.neg
789		if x.neg {
790			x, y = y, x // & is symmetric
791		}
792
793		// x & (-y) == x & ^(y-1) == x &^ (y-1)
794		y1 := nat(nil).sub(y.abs, natOne)
795		z.abs = z.abs.andNot(x.abs, y1)
796		z.neg = false
797		return z
798	}
799
800	// AndNot sets z = x &^ y and returns z.
801	func (z *Int) AndNot(x, y *Int) *Int {
802		if x.neg == y.neg {
803			if x.neg {
804				// (-x) &^ (-y) == ^(x-1) &^ ^(y-1) == ^(x-1) & (y-1) == (y-1) &^ (x-1)
805				x1 := nat(nil).sub(x.abs, natOne)
806				y1 := nat(nil).sub(y.abs, natOne)
807				z.abs = z.abs.andNot(y1, x1)
808				z.neg = false
809				return z
810			}
811
812			// x &^ y == x &^ y
813			z.abs = z.abs.andNot(x.abs, y.abs)
814			z.neg = false
815			return z
816		}
817
818		if x.neg {
819			// (-x) &^ y == ^(x-1) &^ y == ^(x-1) & ^y == ^((x-1) | y) == -(((x-1) | y) + 1)
820			x1 := nat(nil).sub(x.abs, natOne)
821			z.abs = z.abs.add(z.abs.or(x1, y.abs), natOne)
822			z.neg = true // z cannot be zero if x is negative and y is positive
823			return z
824		}
825
826		// x &^ (-y) == x &^ ^(y-1) == x & (y-1)
827		y1 := nat(nil).sub(y.abs, natOne)
828		z.abs = z.abs.and(x.abs, y1)
829		z.neg = false
830		return z
831	}
832
833	// Or sets z = x | y and returns z.
834	func (z *Int) Or(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.and(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.or(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) == ^((y-1) &^ x) == -(^((y-1) &^ x) + 1)
857		y1 := nat(nil).sub(y.abs, natOne)
858		z.abs = z.abs.add(z.abs.andNot(y1, x.abs), natOne)
859		z.neg = true // z cannot be zero if one of x or y is negative
860		return z
861	}
862
863	// Xor sets z = x ^ y and returns z.
864	func (z *Int) Xor(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)
868				x1 := nat(nil).sub(x.abs, natOne)
869				y1 := nat(nil).sub(y.abs, natOne)
870				z.abs = z.abs.xor(x1, y1)
871				z.neg = false
872				return z
873			}
874
875			// x ^ y == x ^ y
876			z.abs = z.abs.xor(x.abs, y.abs)
877			z.neg = false
878			return z
879		}
880
881		// x.neg != y.neg
882		if x.neg {
883			x, y = y, x // ^ is symmetric
884		}
885
886		// x ^ (-y) == x ^ ^(y-1) == ^(x ^ (y-1)) == -((x ^ (y-1)) + 1)
887		y1 := nat(nil).sub(y.abs, natOne)
888		z.abs = z.abs.add(z.abs.xor(x.abs, y1), natOne)
889		z.neg = true // z cannot be zero if only one of x or y is negative
890		return z
891	}
892
893	// Not sets z = ^x and returns z.
894	func (z *Int) Not(x *Int) *Int {
895		if x.neg {
896			// ^(-x) == ^(^(x-1)) == x-1
897			z.abs = z.abs.sub(x.abs, natOne)
898			z.neg = false
899			return z
900		}
901
902		// ^x == -x-1 == -(x+1)
904		z.neg = true // z cannot be zero if x is positive
905		return z
906	}
907
908	// Gob codec version. Permits backward-compatible changes to the encoding.
909	const intGobVersion byte = 1
910
911	// GobEncode implements the gob.GobEncoder interface.
912	func (x *Int) GobEncode() ([]byte, error) {
913		if x == nil {
914			return nil, nil
915		}
916		buf := make([]byte, 1+len(x.abs)*_S) // extra byte for version and sign bit
917		i := x.abs.bytes(buf) - 1            // i >= 0
918		b := intGobVersion << 1              // make space for sign bit
919		if x.neg {
920			b |= 1
921		}
922		buf[i] = b
923		return buf[i:], nil
924	}
925
926	// GobDecode implements the gob.GobDecoder interface.
927	func (z *Int) GobDecode(buf []byte) error {
928		if len(buf) == 0 {
929			// Other side sent a nil or default value.
930			*z = Int{}
931			return nil
932		}
933		b := buf[0]
934		if b>>1 != intGobVersion {
935			return fmt.Errorf("Int.GobDecode: encoding version %d not supported", b>>1)
936		}
937		z.neg = b&1 != 0
938		z.abs = z.abs.setBytes(buf[1:])
939		return nil
940	}
941
942	// MarshalJSON implements the json.Marshaler interface.
943	func (z *Int) MarshalJSON() ([]byte, error) {
944		// TODO(gri): get rid of the []byte/string conversions
945		return []byte(z.String()), nil
946	}
947
948	// UnmarshalJSON implements the json.Unmarshaler interface.
949	func (z *Int) UnmarshalJSON(text []byte) error {
950		// TODO(gri): get rid of the []byte/string conversions
951		if _, ok := z.SetString(string(text), 0); !ok {
952			return fmt.Errorf("math/big: cannot unmarshal %q into a *big.Int", text)
953		}
954		return nil
955	}
956
957	// MarshalText implements the encoding.TextMarshaler interface.
958	func (z *Int) MarshalText() (text []byte, err error) {
959		return []byte(z.String()), nil
960	}
961
962	// UnmarshalText implements the encoding.TextUnmarshaler interface.
963	func (z *Int) UnmarshalText(text []byte) error {
964		if _, ok := z.SetString(string(text), 0); !ok {
965			return fmt.Errorf("math/big: cannot unmarshal %q into a *big.Int", text)
966		}
967		return nil
968	}
969
```

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