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1 747 jeremybenn
// Copyright 2009 The Go Authors. All rights reserved.
2
// Use of this source code is governed by a BSD-style
3
// license that can be found in the LICENSE file.
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
// NewInt allocates and returns a new Int set to x.
55
func NewInt(x int64) *Int {
56
        return new(Int).SetInt64(x)
57
}
58
 
59
// Set sets z to x and returns z.
60
func (z *Int) Set(x *Int) *Int {
61
        if z != x {
62
                z.abs = z.abs.set(x.abs)
63
                z.neg = x.neg
64
        }
65
        return z
66
}
67
 
68
// Bits provides raw (unchecked but fast) access to x by returning its
69
// absolute value as a little-endian Word slice. The result and x share
70
// the same underlying array.
71
// Bits is intended to support implementation of missing low-level Int
72
// functionality outside this package; it should be avoided otherwise.
73
func (x *Int) Bits() []Word {
74
        return x.abs
75
}
76
 
77
// SetBits provides raw (unchecked but fast) access to z by setting its
78
// value to abs, interpreted as a little-endian Word slice, and returning
79
// z. The result and abs share the same underlying array.
80
// SetBits is intended to support implementation of missing low-level Int
81
// functionality outside this package; it should be avoided otherwise.
82
func (z *Int) SetBits(abs []Word) *Int {
83
        z.abs = nat(abs).norm()
84
        z.neg = false
85
        return z
86
}
87
 
88
// Abs sets z to |x| (the absolute value of x) and returns z.
89
func (z *Int) Abs(x *Int) *Int {
90
        z.Set(x)
91
        z.neg = false
92
        return z
93
}
94
 
95
// Neg sets z to -x and returns z.
96
func (z *Int) Neg(x *Int) *Int {
97
        z.Set(x)
98
        z.neg = len(z.abs) > 0 && !z.neg // 0 has no sign
99
        return z
100
}
101
 
102
// Add sets z to the sum x+y and returns z.
103
func (z *Int) Add(x, y *Int) *Int {
104
        neg := x.neg
105
        if x.neg == y.neg {
106
                // x + y == x + y
107
                // (-x) + (-y) == -(x + y)
108
                z.abs = z.abs.add(x.abs, y.abs)
109
        } else {
110
                // x + (-y) == x - y == -(y - x)
111
                // (-x) + y == y - x == -(x - y)
112
                if x.abs.cmp(y.abs) >= 0 {
113
                        z.abs = z.abs.sub(x.abs, y.abs)
114
                } else {
115
                        neg = !neg
116
                        z.abs = z.abs.sub(y.abs, x.abs)
117
                }
118
        }
119
        z.neg = len(z.abs) > 0 && neg // 0 has no sign
120
        return z
121
}
122
 
123
// Sub sets z to the difference x-y and returns z.
124
func (z *Int) Sub(x, y *Int) *Int {
125
        neg := x.neg
126
        if x.neg != y.neg {
127
                // x - (-y) == x + y
128
                // (-x) - y == -(x + y)
129
                z.abs = z.abs.add(x.abs, y.abs)
130
        } else {
131
                // x - y == x - y == -(y - x)
132
                // (-x) - (-y) == y - x == -(x - y)
133
                if x.abs.cmp(y.abs) >= 0 {
134
                        z.abs = z.abs.sub(x.abs, y.abs)
135
                } else {
136
                        neg = !neg
137
                        z.abs = z.abs.sub(y.abs, x.abs)
138
                }
139
        }
140
        z.neg = len(z.abs) > 0 && neg // 0 has no sign
141
        return z
142
}
143
 
144
// Mul sets z to the product x*y and returns z.
145
func (z *Int) Mul(x, y *Int) *Int {
146
        // x * y == x * y
147
        // x * (-y) == -(x * y)
148
        // (-x) * y == -(x * y)
149
        // (-x) * (-y) == x * y
150
        z.abs = z.abs.mul(x.abs, y.abs)
151
        z.neg = len(z.abs) > 0 && x.neg != y.neg // 0 has no sign
152
        return z
153
}
154
 
155
// MulRange sets z to the product of all integers
156
// in the range [a, b] inclusively and returns z.
157
// If a > b (empty range), the result is 1.
158
func (z *Int) MulRange(a, b int64) *Int {
159
        switch {
160
        case a > b:
161
                return z.SetInt64(1) // empty range
162
        case a <= 0 && b >= 0:
163
                return z.SetInt64(0) // range includes 0
164
        }
165
        // a <= b && (b < 0 || a > 0)
166
 
167
        neg := false
168
        if a < 0 {
169
                neg = (b-a)&1 == 0
170
                a, b = -b, -a
171
        }
172
 
173
        z.abs = z.abs.mulRange(uint64(a), uint64(b))
174
        z.neg = neg
175
        return z
176
}
177
 
178
// Binomial sets z to the binomial coefficient of (n, k) and returns z.
179
func (z *Int) Binomial(n, k int64) *Int {
180
        var a, b Int
181
        a.MulRange(n-k+1, n)
182
        b.MulRange(1, k)
183
        return z.Quo(&a, &b)
184
}
185
 
186
// Quo sets z to the quotient x/y for y != 0 and returns z.
187
// If y == 0, a division-by-zero run-time panic occurs.
188
// Quo implements truncated division (like Go); see QuoRem for more details.
189
func (z *Int) Quo(x, y *Int) *Int {
190
        z.abs, _ = z.abs.div(nil, x.abs, y.abs)
191
        z.neg = len(z.abs) > 0 && x.neg != y.neg // 0 has no sign
192
        return z
193
}
194
 
195
// Rem sets z to the remainder x%y for y != 0 and returns z.
196
// If y == 0, a division-by-zero run-time panic occurs.
197
// Rem implements truncated modulus (like Go); see QuoRem for more details.
198
func (z *Int) Rem(x, y *Int) *Int {
199
        _, z.abs = nat(nil).div(z.abs, x.abs, y.abs)
200
        z.neg = len(z.abs) > 0 && x.neg // 0 has no sign
201
        return z
202
}
203
 
204
// QuoRem sets z to the quotient x/y and r to the remainder x%y
205
// and returns the pair (z, r) for y != 0.
206
// If y == 0, a division-by-zero run-time panic occurs.
207
//
208
// QuoRem implements T-division and modulus (like Go):
209
//
210
//      q = x/y      with the result truncated to zero
211
//      r = x - y*q
212
//
213
// (See Daan Leijen, ``Division and Modulus for Computer Scientists''.)
214
// See DivMod for Euclidean division and modulus (unlike Go).
215
//
216
func (z *Int) QuoRem(x, y, r *Int) (*Int, *Int) {
217
        z.abs, r.abs = z.abs.div(r.abs, x.abs, y.abs)
218
        z.neg, r.neg = len(z.abs) > 0 && x.neg != y.neg, len(r.abs) > 0 && x.neg // 0 has no sign
219
        return z, r
220
}
221
 
222
// Div sets z to the quotient x/y for y != 0 and returns z.
223
// If y == 0, a division-by-zero run-time panic occurs.
224
// Div implements Euclidean division (unlike Go); see DivMod for more details.
225
func (z *Int) Div(x, y *Int) *Int {
226
        y_neg := y.neg // z may be an alias for y
227
        var r Int
228
        z.QuoRem(x, y, &r)
229
        if r.neg {
230
                if y_neg {
231
                        z.Add(z, intOne)
232
                } else {
233
                        z.Sub(z, intOne)
234
                }
235
        }
236
        return z
237
}
238
 
239
// Mod sets z to the modulus x%y for y != 0 and returns z.
240
// If y == 0, a division-by-zero run-time panic occurs.
241
// Mod implements Euclidean modulus (unlike Go); see DivMod for more details.
242
func (z *Int) Mod(x, y *Int) *Int {
243
        y0 := y // save y
244
        if z == y || alias(z.abs, y.abs) {
245
                y0 = new(Int).Set(y)
246
        }
247
        var q Int
248
        q.QuoRem(x, y, z)
249
        if z.neg {
250
                if y0.neg {
251
                        z.Sub(z, y0)
252
                } else {
253
                        z.Add(z, y0)
254
                }
255
        }
256
        return z
257
}
258
 
259
// DivMod sets z to the quotient x div y and m to the modulus x mod y
260
// and returns the pair (z, m) for y != 0.
261
// If y == 0, a division-by-zero run-time panic occurs.
262
//
263
// DivMod implements Euclidean division and modulus (unlike Go):
264
//
265
//      q = x div y  such that
266
//      m = x - y*q  with 0 <= m < |q|
267
//
268
// (See Raymond T. Boute, ``The Euclidean definition of the functions
269
// div and mod''. ACM Transactions on Programming Languages and
270
// Systems (TOPLAS), 14(2):127-144, New York, NY, USA, 4/1992.
271
// ACM press.)
272
// See QuoRem for T-division and modulus (like Go).
273
//
274
func (z *Int) DivMod(x, y, m *Int) (*Int, *Int) {
275
        y0 := y // save y
276
        if z == y || alias(z.abs, y.abs) {
277
                y0 = new(Int).Set(y)
278
        }
279
        z.QuoRem(x, y, m)
280
        if m.neg {
281
                if y0.neg {
282
                        z.Add(z, intOne)
283
                        m.Sub(m, y0)
284
                } else {
285
                        z.Sub(z, intOne)
286
                        m.Add(m, y0)
287
                }
288
        }
289
        return z, m
290
}
291
 
292
// Cmp compares x and y and returns:
293
//
294
//   -1 if x <  y
295
//    0 if x == y
296
//   +1 if x >  y
297
//
298
func (x *Int) Cmp(y *Int) (r int) {
299
        // x cmp y == x cmp y
300
        // x cmp (-y) == x
301
        // (-x) cmp y == y
302
        // (-x) cmp (-y) == -(x cmp y)
303
        switch {
304
        case x.neg == y.neg:
305
                r = x.abs.cmp(y.abs)
306
                if x.neg {
307
                        r = -r
308
                }
309
        case x.neg:
310
                r = -1
311
        default:
312
                r = 1
313
        }
314
        return
315
}
316
 
317
func (x *Int) String() string {
318
        switch {
319
        case x == nil:
320
                return ""
321
        case x.neg:
322
                return "-" + x.abs.decimalString()
323
        }
324
        return x.abs.decimalString()
325
}
326
 
327
func charset(ch rune) string {
328
        switch ch {
329
        case 'b':
330
                return lowercaseDigits[0:2]
331
        case 'o':
332
                return lowercaseDigits[0:8]
333
        case 'd', 's', 'v':
334
                return lowercaseDigits[0:10]
335
        case 'x':
336
                return lowercaseDigits[0:16]
337
        case 'X':
338
                return uppercaseDigits[0:16]
339
        }
340
        return "" // unknown format
341
}
342
 
343
// write count copies of text to s
344
func writeMultiple(s fmt.State, text string, count int) {
345
        if len(text) > 0 {
346
                b := []byte(text)
347
                for ; count > 0; count-- {
348
                        s.Write(b)
349
                }
350
        }
351
}
352
 
353
// Format is a support routine for fmt.Formatter. It accepts
354
// the formats 'b' (binary), 'o' (octal), 'd' (decimal), 'x'
355
// (lowercase hexadecimal), and 'X' (uppercase hexadecimal).
356
// Also supported are the full suite of package fmt's format
357
// verbs for integral types, including '+', '-', and ' '
358
// for sign control, '#' for leading zero in octal and for
359
// hexadecimal, a leading "0x" or "0X" for "%#x" and "%#X"
360
// respectively, specification of minimum digits precision,
361
// output field width, space or zero padding, and left or
362
// right justification.
363
//
364
func (x *Int) Format(s fmt.State, ch rune) {
365
        cs := charset(ch)
366
 
367
        // special cases
368
        switch {
369
        case cs == "":
370
                // unknown format
371
                fmt.Fprintf(s, "%%!%c(big.Int=%s)", ch, x.String())
372
                return
373
        case x == nil:
374
                fmt.Fprint(s, "")
375
                return
376
        }
377
 
378
        // determine sign character
379
        sign := ""
380
        switch {
381
        case x.neg:
382
                sign = "-"
383
        case s.Flag('+'): // supersedes ' ' when both specified
384
                sign = "+"
385
        case s.Flag(' '):
386
                sign = " "
387
        }
388
 
389
        // determine prefix characters for indicating output base
390
        prefix := ""
391
        if s.Flag('#') {
392
                switch ch {
393
                case 'o': // octal
394
                        prefix = "0"
395
                case 'x': // hexadecimal
396
                        prefix = "0x"
397
                case 'X':
398
                        prefix = "0X"
399
                }
400
        }
401
 
402
        // determine digits with base set by len(cs) and digit characters from cs
403
        digits := x.abs.string(cs)
404
 
405
        // number of characters for the three classes of number padding
406
        var left int   // space characters to left of digits for right justification ("%8d")
407
        var zeroes int // zero characters (actually cs[0]) as left-most digits ("%.8d")
408
        var right int  // space characters to right of digits for left justification ("%-8d")
409
 
410
        // determine number padding from precision: the least number of digits to output
411
        precision, precisionSet := s.Precision()
412
        if precisionSet {
413
                switch {
414
                case len(digits) < precision:
415
                        zeroes = precision - len(digits) // count of zero padding
416
                case digits == "0" && precision == 0:
417
                        return // print nothing if zero value (x == 0) and zero precision ("." or ".0")
418
                }
419
        }
420
 
421
        // determine field pad from width: the least number of characters to output
422
        length := len(sign) + len(prefix) + zeroes + len(digits)
423
        if width, widthSet := s.Width(); widthSet && length < width { // pad as specified
424
                switch d := width - length; {
425
                case s.Flag('-'):
426
                        // pad on the right with spaces; supersedes '0' when both specified
427
                        right = d
428
                case s.Flag('0') && !precisionSet:
429
                        // pad with zeroes unless precision also specified
430
                        zeroes = d
431
                default:
432
                        // pad on the left with spaces
433
                        left = d
434
                }
435
        }
436
 
437
        // print number as [left pad][sign][prefix][zero pad][digits][right pad]
438
        writeMultiple(s, " ", left)
439
        writeMultiple(s, sign, 1)
440
        writeMultiple(s, prefix, 1)
441
        writeMultiple(s, "0", zeroes)
442
        writeMultiple(s, digits, 1)
443
        writeMultiple(s, " ", right)
444
}
445
 
446
// scan sets z to the integer value corresponding to the longest possible prefix
447
// read from r representing a signed integer number in a given conversion base.
448
// It returns z, the actual conversion base used, and an error, if any. In the
449
// error case, the value of z is undefined but the returned value is nil. The
450
// syntax follows the syntax of integer literals in Go.
451
//
452
// The base argument must be 0 or a value from 2 through MaxBase. If the base
453
// is 0, the string prefix determines the actual conversion base. A prefix of
454
// ``0x'' or ``0X'' selects base 16; the ``0'' prefix selects base 8, and a
455
// ``0b'' or ``0B'' prefix selects base 2. Otherwise the selected base is 10.
456
//
457
func (z *Int) scan(r io.RuneScanner, base int) (*Int, int, error) {
458
        // determine sign
459
        ch, _, err := r.ReadRune()
460
        if err != nil {
461
                return nil, 0, err
462
        }
463
        neg := false
464
        switch ch {
465
        case '-':
466
                neg = true
467
        case '+': // nothing to do
468
        default:
469
                r.UnreadRune()
470
        }
471
 
472
        // determine mantissa
473
        z.abs, base, err = z.abs.scan(r, base)
474
        if err != nil {
475
                return nil, base, err
476
        }
477
        z.neg = len(z.abs) > 0 && neg // 0 has no sign
478
 
479
        return z, base, nil
480
}
481
 
482
// Scan is a support routine for fmt.Scanner; it sets z to the value of
483
// the scanned number. It accepts the formats 'b' (binary), 'o' (octal),
484
// 'd' (decimal), 'x' (lowercase hexadecimal), and 'X' (uppercase hexadecimal).
485
func (z *Int) Scan(s fmt.ScanState, ch rune) error {
486
        s.SkipSpace() // skip leading space characters
487
        base := 0
488
        switch ch {
489
        case 'b':
490
                base = 2
491
        case 'o':
492
                base = 8
493
        case 'd':
494
                base = 10
495
        case 'x', 'X':
496
                base = 16
497
        case 's', 'v':
498
                // let scan determine the base
499
        default:
500
                return errors.New("Int.Scan: invalid verb")
501
        }
502
        _, _, err := z.scan(s, base)
503
        return err
504
}
505
 
506
// Int64 returns the int64 representation of x.
507
// If x cannot be represented in an int64, the result is undefined.
508
func (x *Int) Int64() int64 {
509
        if len(x.abs) == 0 {
510
                return 0
511
        }
512
        v := int64(x.abs[0])
513
        if _W == 32 && len(x.abs) > 1 {
514
                v |= int64(x.abs[1]) << 32
515
        }
516
        if x.neg {
517
                v = -v
518
        }
519
        return v
520
}
521
 
522
// SetString sets z to the value of s, interpreted in the given base,
523
// and returns z and a boolean indicating success. If SetString fails,
524
// the value of z is undefined but the returned value is nil.
525
//
526
// The base argument must be 0 or a value from 2 through MaxBase. If the base
527
// is 0, the string prefix determines the actual conversion base. A prefix of
528
// ``0x'' or ``0X'' selects base 16; the ``0'' prefix selects base 8, and a
529
// ``0b'' or ``0B'' prefix selects base 2. Otherwise the selected base is 10.
530
//
531
func (z *Int) SetString(s string, base int) (*Int, bool) {
532
        r := strings.NewReader(s)
533
        _, _, err := z.scan(r, base)
534
        if err != nil {
535
                return nil, false
536
        }
537
        _, _, err = r.ReadRune()
538
        if err != io.EOF {
539
                return nil, false
540
        }
541
        return z, true // err == io.EOF => scan consumed all of s
542
}
543
 
544
// SetBytes interprets buf as the bytes of a big-endian unsigned
545
// integer, sets z to that value, and returns z.
546
func (z *Int) SetBytes(buf []byte) *Int {
547
        z.abs = z.abs.setBytes(buf)
548
        z.neg = false
549
        return z
550
}
551
 
552
// Bytes returns the absolute value of z as a big-endian byte slice.
553
func (x *Int) Bytes() []byte {
554
        buf := make([]byte, len(x.abs)*_S)
555
        return buf[x.abs.bytes(buf):]
556
}
557
 
558
// BitLen returns the length of the absolute value of z in bits.
559
// The bit length of 0 is 0.
560
func (x *Int) BitLen() int {
561
        return x.abs.bitLen()
562
}
563
 
564
// Exp sets z = x**y mod m and returns z. If m is nil, z = x**y.
565
// See Knuth, volume 2, section 4.6.3.
566
func (z *Int) Exp(x, y, m *Int) *Int {
567
        if y.neg || len(y.abs) == 0 {
568
                neg := x.neg
569
                z.SetInt64(1)
570
                z.neg = neg
571
                return z
572
        }
573
 
574
        var mWords nat
575
        if m != nil {
576
                mWords = m.abs
577
        }
578
 
579
        z.abs = z.abs.expNN(x.abs, y.abs, mWords)
580
        z.neg = len(z.abs) > 0 && x.neg && y.abs[0]&1 == 1 // 0 has no sign
581
        return z
582
}
583
 
584
// GCD sets z to the greatest common divisor of a and b, which must be
585
// positive numbers, and returns z.
586
// If x and y are not nil, GCD sets x and y such that z = a*x + b*y.
587
// If either a or b is not positive, GCD sets z = x = y = 0.
588
func (z *Int) GCD(x, y, a, b *Int) *Int {
589
        if a.neg || b.neg {
590
                z.SetInt64(0)
591
                if x != nil {
592
                        x.SetInt64(0)
593
                }
594
                if y != nil {
595
                        y.SetInt64(0)
596
                }
597
                return z
598
        }
599
 
600
        A := new(Int).Set(a)
601
        B := new(Int).Set(b)
602
 
603
        X := new(Int)
604
        Y := new(Int).SetInt64(1)
605
 
606
        lastX := new(Int).SetInt64(1)
607
        lastY := new(Int)
608
 
609
        q := new(Int)
610
        temp := new(Int)
611
 
612
        for len(B.abs) > 0 {
613
                r := new(Int)
614
                q, r = q.QuoRem(A, B, r)
615
 
616
                A, B = B, r
617
 
618
                temp.Set(X)
619
                X.Mul(X, q)
620
                X.neg = !X.neg
621
                X.Add(X, lastX)
622
                lastX.Set(temp)
623
 
624
                temp.Set(Y)
625
                Y.Mul(Y, q)
626
                Y.neg = !Y.neg
627
                Y.Add(Y, lastY)
628
                lastY.Set(temp)
629
        }
630
 
631
        if x != nil {
632
                *x = *lastX
633
        }
634
 
635
        if y != nil {
636
                *y = *lastY
637
        }
638
 
639
        *z = *A
640
        return z
641
}
642
 
643
// ProbablyPrime performs n Miller-Rabin tests to check whether x is prime.
644
// If it returns true, x is prime with probability 1 - 1/4^n.
645
// If it returns false, x is not prime.
646
func (x *Int) ProbablyPrime(n int) bool {
647
        return !x.neg && x.abs.probablyPrime(n)
648
}
649
 
650
// Rand sets z to a pseudo-random number in [0, n) and returns z.
651
func (z *Int) Rand(rnd *rand.Rand, n *Int) *Int {
652
        z.neg = false
653
        if n.neg == true || len(n.abs) == 0 {
654
                z.abs = nil
655
                return z
656
        }
657
        z.abs = z.abs.random(rnd, n.abs, n.abs.bitLen())
658
        return z
659
}
660
 
661
// ModInverse sets z to the multiplicative inverse of g in the group ℤ/pℤ (where
662
// p is a prime) and returns z.
663
func (z *Int) ModInverse(g, p *Int) *Int {
664
        var d Int
665
        d.GCD(z, nil, g, p)
666
        // x and y are such that g*x + p*y = d. Since p is prime, d = 1. Taking
667
        // that modulo p results in g*x = 1, therefore x is the inverse element.
668
        if z.neg {
669
                z.Add(z, p)
670
        }
671
        return z
672
}
673
 
674
// Lsh sets z = x << n and returns z.
675
func (z *Int) Lsh(x *Int, n uint) *Int {
676
        z.abs = z.abs.shl(x.abs, n)
677
        z.neg = x.neg
678
        return z
679
}
680
 
681
// Rsh sets z = x >> n and returns z.
682
func (z *Int) Rsh(x *Int, n uint) *Int {
683
        if x.neg {
684
                // (-x) >> s == ^(x-1) >> s == ^((x-1) >> s) == -(((x-1) >> s) + 1)
685
                t := z.abs.sub(x.abs, natOne) // no underflow because |x| > 0
686
                t = t.shr(t, n)
687
                z.abs = t.add(t, natOne)
688
                z.neg = true // z cannot be zero if x is negative
689
                return z
690
        }
691
 
692
        z.abs = z.abs.shr(x.abs, n)
693
        z.neg = false
694
        return z
695
}
696
 
697
// Bit returns the value of the i'th bit of x. That is, it
698
// returns (x>>i)&1. The bit index i must be >= 0.
699
func (x *Int) Bit(i int) uint {
700
        if i < 0 {
701
                panic("negative bit index")
702
        }
703
        if x.neg {
704
                t := nat(nil).sub(x.abs, natOne)
705
                return t.bit(uint(i)) ^ 1
706
        }
707
 
708
        return x.abs.bit(uint(i))
709
}
710
 
711
// SetBit sets z to x, with x's i'th bit set to b (0 or 1).
712
// That is, if bit is 1 SetBit sets z = x | (1 << i);
713
// if bit is 0 it sets z = x &^ (1 << i). If bit is not 0 or 1,
714
// SetBit will panic.
715
func (z *Int) SetBit(x *Int, i int, b uint) *Int {
716
        if i < 0 {
717
                panic("negative bit index")
718
        }
719
        if x.neg {
720
                t := z.abs.sub(x.abs, natOne)
721
                t = t.setBit(t, uint(i), b^1)
722
                z.abs = t.add(t, natOne)
723
                z.neg = len(z.abs) > 0
724
                return z
725
        }
726
        z.abs = z.abs.setBit(x.abs, uint(i), b)
727
        z.neg = false
728
        return z
729
}
730
 
731
// And sets z = x & y and returns z.
732
func (z *Int) And(x, y *Int) *Int {
733
        if x.neg == y.neg {
734
                if x.neg {
735
                        // (-x) & (-y) == ^(x-1) & ^(y-1) == ^((x-1) | (y-1)) == -(((x-1) | (y-1)) + 1)
736
                        x1 := nat(nil).sub(x.abs, natOne)
737
                        y1 := nat(nil).sub(y.abs, natOne)
738
                        z.abs = z.abs.add(z.abs.or(x1, y1), natOne)
739
                        z.neg = true // z cannot be zero if x and y are negative
740
                        return z
741
                }
742
 
743
                // x & y == x & y
744
                z.abs = z.abs.and(x.abs, y.abs)
745
                z.neg = false
746
                return z
747
        }
748
 
749
        // x.neg != y.neg
750
        if x.neg {
751
                x, y = y, x // & is symmetric
752
        }
753
 
754
        // x & (-y) == x & ^(y-1) == x &^ (y-1)
755
        y1 := nat(nil).sub(y.abs, natOne)
756
        z.abs = z.abs.andNot(x.abs, y1)
757
        z.neg = false
758
        return z
759
}
760
 
761
// AndNot sets z = x &^ y and returns z.
762
func (z *Int) AndNot(x, y *Int) *Int {
763
        if x.neg == y.neg {
764
                if x.neg {
765
                        // (-x) &^ (-y) == ^(x-1) &^ ^(y-1) == ^(x-1) & (y-1) == (y-1) &^ (x-1)
766
                        x1 := nat(nil).sub(x.abs, natOne)
767
                        y1 := nat(nil).sub(y.abs, natOne)
768
                        z.abs = z.abs.andNot(y1, x1)
769
                        z.neg = false
770
                        return z
771
                }
772
 
773
                // x &^ y == x &^ y
774
                z.abs = z.abs.andNot(x.abs, y.abs)
775
                z.neg = false
776
                return z
777
        }
778
 
779
        if x.neg {
780
                // (-x) &^ y == ^(x-1) &^ y == ^(x-1) & ^y == ^((x-1) | y) == -(((x-1) | y) + 1)
781
                x1 := nat(nil).sub(x.abs, natOne)
782
                z.abs = z.abs.add(z.abs.or(x1, y.abs), natOne)
783
                z.neg = true // z cannot be zero if x is negative and y is positive
784
                return z
785
        }
786
 
787
        // x &^ (-y) == x &^ ^(y-1) == x & (y-1)
788
        y1 := nat(nil).add(y.abs, natOne)
789
        z.abs = z.abs.and(x.abs, y1)
790
        z.neg = false
791
        return z
792
}
793
 
794
// Or sets z = x | y and returns z.
795
func (z *Int) Or(x, y *Int) *Int {
796
        if x.neg == y.neg {
797
                if x.neg {
798
                        // (-x) | (-y) == ^(x-1) | ^(y-1) == ^((x-1) & (y-1)) == -(((x-1) & (y-1)) + 1)
799
                        x1 := nat(nil).sub(x.abs, natOne)
800
                        y1 := nat(nil).sub(y.abs, natOne)
801
                        z.abs = z.abs.add(z.abs.and(x1, y1), natOne)
802
                        z.neg = true // z cannot be zero if x and y are negative
803
                        return z
804
                }
805
 
806
                // x | y == x | y
807
                z.abs = z.abs.or(x.abs, y.abs)
808
                z.neg = false
809
                return z
810
        }
811
 
812
        // x.neg != y.neg
813
        if x.neg {
814
                x, y = y, x // | is symmetric
815
        }
816
 
817
        // x | (-y) == x | ^(y-1) == ^((y-1) &^ x) == -(^((y-1) &^ x) + 1)
818
        y1 := nat(nil).sub(y.abs, natOne)
819
        z.abs = z.abs.add(z.abs.andNot(y1, x.abs), natOne)
820
        z.neg = true // z cannot be zero if one of x or y is negative
821
        return z
822
}
823
 
824
// Xor sets z = x ^ y and returns z.
825
func (z *Int) Xor(x, y *Int) *Int {
826
        if x.neg == y.neg {
827
                if x.neg {
828
                        // (-x) ^ (-y) == ^(x-1) ^ ^(y-1) == (x-1) ^ (y-1)
829
                        x1 := nat(nil).sub(x.abs, natOne)
830
                        y1 := nat(nil).sub(y.abs, natOne)
831
                        z.abs = z.abs.xor(x1, y1)
832
                        z.neg = false
833
                        return z
834
                }
835
 
836
                // x ^ y == x ^ y
837
                z.abs = z.abs.xor(x.abs, y.abs)
838
                z.neg = false
839
                return z
840
        }
841
 
842
        // x.neg != y.neg
843
        if x.neg {
844
                x, y = y, x // ^ is symmetric
845
        }
846
 
847
        // x ^ (-y) == x ^ ^(y-1) == ^(x ^ (y-1)) == -((x ^ (y-1)) + 1)
848
        y1 := nat(nil).sub(y.abs, natOne)
849
        z.abs = z.abs.add(z.abs.xor(x.abs, y1), natOne)
850
        z.neg = true // z cannot be zero if only one of x or y is negative
851
        return z
852
}
853
 
854
// Not sets z = ^x and returns z.
855
func (z *Int) Not(x *Int) *Int {
856
        if x.neg {
857
                // ^(-x) == ^(^(x-1)) == x-1
858
                z.abs = z.abs.sub(x.abs, natOne)
859
                z.neg = false
860
                return z
861
        }
862
 
863
        // ^x == -x-1 == -(x+1)
864
        z.abs = z.abs.add(x.abs, natOne)
865
        z.neg = true // z cannot be zero if x is positive
866
        return z
867
}
868
 
869
// Gob codec version. Permits backward-compatible changes to the encoding.
870
const intGobVersion byte = 1
871
 
872
// GobEncode implements the gob.GobEncoder interface.
873
func (x *Int) GobEncode() ([]byte, error) {
874
        buf := make([]byte, 1+len(x.abs)*_S) // extra byte for version and sign bit
875
        i := x.abs.bytes(buf) - 1            // i >= 0
876
        b := intGobVersion << 1              // make space for sign bit
877
        if x.neg {
878
                b |= 1
879
        }
880
        buf[i] = b
881
        return buf[i:], nil
882
}
883
 
884
// GobDecode implements the gob.GobDecoder interface.
885
func (z *Int) GobDecode(buf []byte) error {
886
        if len(buf) == 0 {
887
                return errors.New("Int.GobDecode: no data")
888
        }
889
        b := buf[0]
890
        if b>>1 != intGobVersion {
891
                return errors.New(fmt.Sprintf("Int.GobDecode: encoding version %d not supported", b>>1))
892
        }
893
        z.neg = b&1 != 0
894
        z.abs = z.abs.setBytes(buf[1:])
895
        return nil
896
}

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