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

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 < |y|
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. The entire string
365	// (not just a prefix) must be valid for success. If SetString fails,
366	// the value of z is undefined but the returned value is nil.
367	//
368	// The base argument must be 0 or a value between 2 and MaxBase. If the base
369	// is 0, the string prefix determines the actual conversion base. A prefix of
370	// ``0x'' or ``0X'' selects base 16; the ``0'' prefix selects base 8, and a
371	// ``0b'' or ``0B'' prefix selects base 2. Otherwise the selected base is 10.
372	//
373	func (z *Int) SetString(s string, base int) (*Int, bool) {
375		if _, _, err := z.scan(r, base); err != nil {
376			return nil, false
377		}
378		// entire string must have been consumed
379		if _, err := r.ReadByte(); 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	//
408	// Modular exponentation of inputs of a particular size is not a
409	// cryptographically constant-time operation.
410	func (z *Int) Exp(x, y, m *Int) *Int {
411		// See Knuth, volume 2, section 4.6.3.
412		var yWords nat
413		if !y.neg {
414			yWords = y.abs
415		}
416		// y >= 0
417
418		var mWords nat
419		if m != nil {
420			mWords = m.abs // m.abs may be nil for m == 0
421		}
422
423		z.abs = z.abs.expNN(x.abs, yWords, mWords)
424		z.neg = len(z.abs) > 0 && x.neg && len(yWords) > 0 && yWords[0]&1 == 1 // 0 has no sign
425		if z.neg && len(mWords) > 0 {
426			// make modulus result positive
427			z.abs = z.abs.sub(mWords, z.abs) // z == x**y mod |m| && 0 <= z < |m|
428			z.neg = false
429		}
430
431		return z
432	}
433
434	// GCD sets z to the greatest common divisor of a and b, which both must
435	// be > 0, and returns z.
436	// If x and y are not nil, GCD sets x and y such that z = a*x + b*y.
437	// If either a or b is <= 0, GCD sets z = x = y = 0.
438	func (z *Int) GCD(x, y, a, b *Int) *Int {
439		if a.Sign() <= 0 || b.Sign() <= 0 {
440			z.SetInt64(0)
441			if x != nil {
442				x.SetInt64(0)
443			}
444			if y != nil {
445				y.SetInt64(0)
446			}
447			return z
448		}
449		if x == nil && y == nil {
450			return z.binaryGCD(a, b)
451		}
452
453		A := new(Int).Set(a)
454		B := new(Int).Set(b)
455
456		X := new(Int)
457		Y := new(Int).SetInt64(1)
458
459		lastX := new(Int).SetInt64(1)
460		lastY := new(Int)
461
462		q := new(Int)
463		temp := new(Int)
464
465		r := new(Int)
466		for len(B.abs) > 0 {
467			q, r = q.QuoRem(A, B, r)
468
469			A, B, r = B, r, A
470
471			temp.Set(X)
472			X.Mul(X, q)
473			X.neg = !X.neg
475			lastX.Set(temp)
476
477			temp.Set(Y)
478			Y.Mul(Y, q)
479			Y.neg = !Y.neg
481			lastY.Set(temp)
482		}
483
484		if x != nil {
485			*x = *lastX
486		}
487
488		if y != nil {
489			*y = *lastY
490		}
491
492		*z = *A
493		return z
494	}
495
496	// binaryGCD sets z to the greatest common divisor of a and b, which both must
497	// be > 0, and returns z.
498	// See Knuth, The Art of Computer Programming, Vol. 2, Section 4.5.2, Algorithm B.
499	func (z *Int) binaryGCD(a, b *Int) *Int {
500		u := z
501		v := new(Int)
502
503		// use one Euclidean iteration to ensure that u and v are approx. the same size
504		switch {
505		case len(a.abs) > len(b.abs):
506			// must set v before u since u may be alias for a or b (was issue #11284)
507			v.Rem(a, b)
508			u.Set(b)
509		case len(a.abs) < len(b.abs):
510			v.Rem(b, a)
511			u.Set(a)
512		default:
513			v.Set(b)
514			u.Set(a)
515		}
516		// a, b must not be used anymore (may be aliases with u)
517
518		// v might be 0 now
519		if len(v.abs) == 0 {
520			return u
521		}
522		// u > 0 && v > 0
523
524		// determine largest k such that u = u' << k, v = v' << k
525		k := u.abs.trailingZeroBits()
526		if vk := v.abs.trailingZeroBits(); vk < k {
527			k = vk
528		}
529		u.Rsh(u, k)
530		v.Rsh(v, k)
531
532		// determine t (we know that u > 0)
533		t := new(Int)
534		if u.abs[0]&1 != 0 {
535			// u is odd
536			t.Neg(v)
537		} else {
538			t.Set(u)
539		}
540
541		for len(t.abs) > 0 {
542			// reduce t
543			t.Rsh(t, t.abs.trailingZeroBits())
544			if t.neg {
545				v, t = t, v
546				v.neg = len(v.abs) > 0 && !v.neg // 0 has no sign
547			} else {
548				u, t = t, u
549			}
550			t.Sub(u, v)
551		}
552
553		return z.Lsh(u, k)
554	}
555
556	// Rand sets z to a pseudo-random number in [0, n) and returns z.
557	func (z *Int) Rand(rnd *rand.Rand, n *Int) *Int {
558		z.neg = false
559		if n.neg == true || len(n.abs) == 0 {
560			z.abs = nil
561			return z
562		}
563		z.abs = z.abs.random(rnd, n.abs, n.abs.bitLen())
564		return z
565	}
566
567	// ModInverse sets z to the multiplicative inverse of g in the ring ℤ/nℤ
568	// and returns z. If g and n are not relatively prime, the result is undefined.
569	func (z *Int) ModInverse(g, n *Int) *Int {
570		if g.neg {
571			// GCD expects parameters a and b to be > 0.
572			var g2 Int
573			g = g2.Mod(g, n)
574		}
575		var d Int
576		d.GCD(z, nil, g, n)
577		// x and y are such that g*x + n*y = d. Since g and n are
578		// relatively prime, d = 1. Taking that modulo n results in
579		// g*x = 1, therefore x is the inverse element.
580		if z.neg {
582		}
583		return z
584	}
585
586	// Jacobi returns the Jacobi symbol (x/y), either +1, -1, or 0.
587	// The y argument must be an odd integer.
588	func Jacobi(x, y *Int) int {
589		if len(y.abs) == 0 || y.abs[0]&1 == 0 {
590			panic(fmt.Sprintf("big: invalid 2nd argument to Int.Jacobi: need odd integer but got %s", y))
591		}
592
593		// We use the formulation described in chapter 2, section 2.4,
594		// "The Yacas Book of Algorithms":
595		// http://yacas.sourceforge.net/Algo.book.pdf
596
597		var a, b, c Int
598		a.Set(x)
599		b.Set(y)
600		j := 1
601
602		if b.neg {
603			if a.neg {
604				j = -1
605			}
606			b.neg = false
607		}
608
609		for {
610			if b.Cmp(intOne) == 0 {
611				return j
612			}
613			if len(a.abs) == 0 {
614				return 0
615			}
616			a.Mod(&a, &b)
617			if len(a.abs) == 0 {
618				return 0
619			}
620			// a > 0
621
622			// handle factors of 2 in 'a'
623			s := a.abs.trailingZeroBits()
624			if s&1 != 0 {
625				bmod8 := b.abs[0] & 7
626				if bmod8 == 3 || bmod8 == 5 {
627					j = -j
628				}
629			}
630			c.Rsh(&a, s) // a = 2^s*c
631
632			// swap numerator and denominator
633			if b.abs[0]&3 == 3 && c.abs[0]&3 == 3 {
634				j = -j
635			}
636			a.Set(&b)
637			b.Set(&c)
638		}
639	}
640
641	// modSqrt3Mod4 uses the identity
642	//      (a^((p+1)/4))^2  mod p
643	//   == u^(p+1)          mod p
644	//   == u^2              mod p
645	// to calculate the square root of any quadratic residue mod p quickly for 3
646	// mod 4 primes.
647	func (z *Int) modSqrt3Mod4Prime(x, p *Int) *Int {
648		z.Set(p)         // z = p
649		z.Add(z, intOne) // z = p + 1
650		z.Rsh(z, 2)      // z = (p + 1) / 4
651		z.Exp(x, z, p)   // z = x^z mod p
652		return z
653	}
654
655	// modSqrtTonelliShanks uses the Tonelli-Shanks algorithm to find the square
656	// root of a quadratic residue modulo any prime.
657	func (z *Int) modSqrtTonelliShanks(x, p *Int) *Int {
658		// Break p-1 into s*2^e such that s is odd.
659		var s Int
660		s.Sub(p, intOne)
661		e := s.abs.trailingZeroBits()
662		s.Rsh(&s, e)
663
664		// find some non-square n
665		var n Int
666		n.SetInt64(2)
667		for Jacobi(&n, p) != -1 {
669		}
670
671		// Core of the Tonelli-Shanks algorithm. Follows the description in
672		// section 6 of "Square roots from 1; 24, 51, 10 to Dan Shanks" by Ezra
673		// Brown:
675		var y, b, g, t Int
677		y.Rsh(&y, 1)
678		y.Exp(x, &y, p)  // y = x^((s+1)/2)
679		b.Exp(x, &s, p)  // b = x^s
680		g.Exp(&n, &s, p) // g = n^s
681		r := e
682		for {
683			// find the least m such that ord_p(b) = 2^m
684			var m uint
685			t.Set(&b)
686			for t.Cmp(intOne) != 0 {
687				t.Mul(&t, &t).Mod(&t, p)
688				m++
689			}
690
691			if m == 0 {
692				return z.Set(&y)
693			}
694
695			t.SetInt64(0).SetBit(&t, int(r-m-1), 1).Exp(&g, &t, p)
696			// t = g^(2^(r-m-1)) mod p
697			g.Mul(&t, &t).Mod(&g, p) // g = g^(2^(r-m)) mod p
698			y.Mul(&y, &t).Mod(&y, p)
699			b.Mul(&b, &g).Mod(&b, p)
700			r = m
701		}
702	}
703
704	// ModSqrt sets z to a square root of x mod p if such a square root exists, and
705	// returns z. The modulus p must be an odd prime. If x is not a square mod p,
706	// ModSqrt leaves z unchanged and returns nil. This function panics if p is
707	// not an odd integer.
708	func (z *Int) ModSqrt(x, p *Int) *Int {
709		switch Jacobi(x, p) {
710		case -1:
711			return nil // x is not a square mod p
712		case 0:
713			return z.SetInt64(0) // sqrt(0) mod p = 0
714		case 1:
715			break
716		}
717		if x.neg || x.Cmp(p) >= 0 { // ensure 0 <= x < p
718			x = new(Int).Mod(x, p)
719		}
720
721		// Check whether p is 3 mod 4, and if so, use the faster algorithm.
722		if len(p.abs) > 0 && p.abs[0]%4 == 3 {
723			return z.modSqrt3Mod4Prime(x, p)
724		}
725		// Otherwise, use Tonelli-Shanks.
726		return z.modSqrtTonelliShanks(x, p)
727	}
728
729	// Lsh sets z = x << n and returns z.
730	func (z *Int) Lsh(x *Int, n uint) *Int {
731		z.abs = z.abs.shl(x.abs, n)
732		z.neg = x.neg
733		return z
734	}
735
736	// Rsh sets z = x >> n and returns z.
737	func (z *Int) Rsh(x *Int, n uint) *Int {
738		if x.neg {
739			// (-x) >> s == ^(x-1) >> s == ^((x-1) >> s) == -(((x-1) >> s) + 1)
740			t := z.abs.sub(x.abs, natOne) // no underflow because |x| > 0
741			t = t.shr(t, n)
743			z.neg = true // z cannot be zero if x is negative
744			return z
745		}
746
747		z.abs = z.abs.shr(x.abs, n)
748		z.neg = false
749		return z
750	}
751
752	// Bit returns the value of the i'th bit of x. That is, it
753	// returns (x>>i)&1. The bit index i must be >= 0.
754	func (x *Int) Bit(i int) uint {
755		if i == 0 {
756			// optimization for common case: odd/even test of x
757			if len(x.abs) > 0 {
758				return uint(x.abs[0] & 1) // bit 0 is same for -x
759			}
760			return 0
761		}
762		if i < 0 {
763			panic("negative bit index")
764		}
765		if x.neg {
766			t := nat(nil).sub(x.abs, natOne)
767			return t.bit(uint(i)) ^ 1
768		}
769
770		return x.abs.bit(uint(i))
771	}
772
773	// SetBit sets z to x, with x's i'th bit set to b (0 or 1).
774	// That is, if b is 1 SetBit sets z = x | (1 << i);
775	// if b is 0 SetBit sets z = x &^ (1 << i). If b is not 0 or 1,
776	// SetBit will panic.
777	func (z *Int) SetBit(x *Int, i int, b uint) *Int {
778		if i < 0 {
779			panic("negative bit index")
780		}
781		if x.neg {
782			t := z.abs.sub(x.abs, natOne)
783			t = t.setBit(t, uint(i), b^1)
785			z.neg = len(z.abs) > 0
786			return z
787		}
788		z.abs = z.abs.setBit(x.abs, uint(i), b)
789		z.neg = false
790		return z
791	}
792
793	// And sets z = x & y and returns z.
794	func (z *Int) And(x, y *Int) *Int {
795		if x.neg == y.neg {
796			if x.neg {
797				// (-x) & (-y) == ^(x-1) & ^(y-1) == ^((x-1) | (y-1)) == -(((x-1) | (y-1)) + 1)
798				x1 := nat(nil).sub(x.abs, natOne)
799				y1 := nat(nil).sub(y.abs, natOne)
800				z.abs = z.abs.add(z.abs.or(x1, y1), natOne)
801				z.neg = true // z cannot be zero if x and y are negative
802				return z
803			}
804
805			// x & y == x & y
806			z.abs = z.abs.and(x.abs, y.abs)
807			z.neg = false
808			return z
809		}
810
811		// x.neg != y.neg
812		if x.neg {
813			x, y = y, x // & is symmetric
814		}
815
816		// x & (-y) == x & ^(y-1) == x &^ (y-1)
817		y1 := nat(nil).sub(y.abs, natOne)
818		z.abs = z.abs.andNot(x.abs, y1)
819		z.neg = false
820		return z
821	}
822
823	// AndNot sets z = x &^ y and returns z.
824	func (z *Int) AndNot(x, y *Int) *Int {
825		if x.neg == y.neg {
826			if x.neg {
827				// (-x) &^ (-y) == ^(x-1) &^ ^(y-1) == ^(x-1) & (y-1) == (y-1) &^ (x-1)
828				x1 := nat(nil).sub(x.abs, natOne)
829				y1 := nat(nil).sub(y.abs, natOne)
830				z.abs = z.abs.andNot(y1, x1)
831				z.neg = false
832				return z
833			}
834
835			// x &^ y == x &^ y
836			z.abs = z.abs.andNot(x.abs, y.abs)
837			z.neg = false
838			return z
839		}
840
841		if x.neg {
842			// (-x) &^ y == ^(x-1) &^ y == ^(x-1) & ^y == ^((x-1) | y) == -(((x-1) | y) + 1)
843			x1 := nat(nil).sub(x.abs, natOne)
844			z.abs = z.abs.add(z.abs.or(x1, y.abs), natOne)
845			z.neg = true // z cannot be zero if x is negative and y is positive
846			return z
847		}
848
849		// x &^ (-y) == x &^ ^(y-1) == x & (y-1)
850		y1 := nat(nil).sub(y.abs, natOne)
851		z.abs = z.abs.and(x.abs, y1)
852		z.neg = false
853		return z
854	}
855
856	// Or sets z = x | y and returns z.
857	func (z *Int) Or(x, y *Int) *Int {
858		if x.neg == y.neg {
859			if x.neg {
860				// (-x) | (-y) == ^(x-1) | ^(y-1) == ^((x-1) & (y-1)) == -(((x-1) & (y-1)) + 1)
861				x1 := nat(nil).sub(x.abs, natOne)
862				y1 := nat(nil).sub(y.abs, natOne)
863				z.abs = z.abs.add(z.abs.and(x1, y1), natOne)
864				z.neg = true // z cannot be zero if x and y are negative
865				return z
866			}
867
868			// x | y == x | y
869			z.abs = z.abs.or(x.abs, y.abs)
870			z.neg = false
871			return z
872		}
873
874		// x.neg != y.neg
875		if x.neg {
876			x, y = y, x // | is symmetric
877		}
878
879		// x | (-y) == x | ^(y-1) == ^((y-1) &^ x) == -(^((y-1) &^ x) + 1)
880		y1 := nat(nil).sub(y.abs, natOne)
881		z.abs = z.abs.add(z.abs.andNot(y1, x.abs), natOne)
882		z.neg = true // z cannot be zero if one of x or y is negative
883		return z
884	}
885
886	// Xor sets z = x ^ y and returns z.
887	func (z *Int) Xor(x, y *Int) *Int {
888		if x.neg == y.neg {
889			if x.neg {
890				// (-x) ^ (-y) == ^(x-1) ^ ^(y-1) == (x-1) ^ (y-1)
891				x1 := nat(nil).sub(x.abs, natOne)
892				y1 := nat(nil).sub(y.abs, natOne)
893				z.abs = z.abs.xor(x1, y1)
894				z.neg = false
895				return z
896			}
897
898			// x ^ y == x ^ y
899			z.abs = z.abs.xor(x.abs, y.abs)
900			z.neg = false
901			return z
902		}
903
904		// x.neg != y.neg
905		if x.neg {
906			x, y = y, x // ^ is symmetric
907		}
908
909		// x ^ (-y) == x ^ ^(y-1) == ^(x ^ (y-1)) == -((x ^ (y-1)) + 1)
910		y1 := nat(nil).sub(y.abs, natOne)
911		z.abs = z.abs.add(z.abs.xor(x.abs, y1), natOne)
912		z.neg = true // z cannot be zero if only one of x or y is negative
913		return z
914	}
915
916	// Not sets z = ^x and returns z.
917	func (z *Int) Not(x *Int) *Int {
918		if x.neg {
919			// ^(-x) == ^(^(x-1)) == x-1
920			z.abs = z.abs.sub(x.abs, natOne)
921			z.neg = false
922			return z
923		}
924
925		// ^x == -x-1 == -(x+1)
927		z.neg = true // z cannot be zero if x is positive
928		return z
929	}
930
931	// Sqrt sets z to ⌊√x⌋, the largest integer such that z² ≤ x, and returns z.
932	// It panics if x is negative.
933	func (z *Int) Sqrt(x *Int) *Int {
934		if x.neg {
935			panic("square root of negative number")
936		}
937		z.neg = false
938		z.abs = z.abs.sqrt(x.abs)
939		return z
940	}
941

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