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

## Documentation: math/big

1  // 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  	"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  //
19  // Operations always take pointer arguments (*Int) rather
20  // than Int values, and each unique Int value requires
21  // its own unique *Int pointer. To "copy" an Int value,
22  // an existing (or newly allocated) Int must be set to
23  // a new value using the Int.Set method; shallow copies
24  // of Ints are not supported and may lead to errors.
25  type Int struct {
26  	neg bool // sign
27  	abs nat  // absolute value of the integer
28  }
29
30  var intOne = &Int{false, natOne}
31
32  // Sign returns:
33  //
34  //	-1 if x <  0
35  //	 0 if x == 0
36  //	+1 if x >  0
37  //
38  func (x *Int) Sign() int {
39  	if len(x.abs) == 0 {
40  		return 0
41  	}
42  	if x.neg {
43  		return -1
44  	}
45  	return 1
46  }
47
48  // SetInt64 sets z to x and returns z.
49  func (z *Int) SetInt64(x int64) *Int {
50  	neg := false
51  	if x < 0 {
52  		neg = true
53  		x = -x
54  	}
55  	z.abs = z.abs.setUint64(uint64(x))
56  	z.neg = neg
57  	return z
58  }
59
60  // SetUint64 sets z to x and returns z.
61  func (z *Int) SetUint64(x uint64) *Int {
62  	z.abs = z.abs.setUint64(x)
63  	z.neg = false
64  	return z
65  }
66
67  // NewInt allocates and returns a new Int set to x.
68  func NewInt(x int64) *Int {
69  	return new(Int).SetInt64(x)
70  }
71
72  // Set sets z to x and returns z.
73  func (z *Int) Set(x *Int) *Int {
74  	if z != x {
75  		z.abs = z.abs.set(x.abs)
76  		z.neg = x.neg
77  	}
78  	return z
79  }
80
81  // Bits provides raw (unchecked but fast) access to x by returning its
82  // absolute value as a little-endian Word slice. The result and x share
83  // the same underlying array.
84  // Bits is intended to support implementation of missing low-level Int
85  // functionality outside this package; it should be avoided otherwise.
86  func (x *Int) Bits() []Word {
87  	return x.abs
88  }
89
90  // SetBits provides raw (unchecked but fast) access to z by setting its
91  // value to abs, interpreted as a little-endian Word slice, and returning
92  // z. The result and abs share the same underlying array.
93  // SetBits is intended to support implementation of missing low-level Int
94  // functionality outside this package; it should be avoided otherwise.
95  func (z *Int) SetBits(abs []Word) *Int {
96  	z.abs = nat(abs).norm()
97  	z.neg = false
98  	return z
99  }
100
101  // Abs sets z to |x| (the absolute value of x) and returns z.
102  func (z *Int) Abs(x *Int) *Int {
103  	z.Set(x)
104  	z.neg = false
105  	return z
106  }
107
108  // Neg sets z to -x and returns z.
109  func (z *Int) Neg(x *Int) *Int {
110  	z.Set(x)
111  	z.neg = len(z.abs) > 0 && !z.neg // 0 has no sign
112  	return z
113  }
114
115  // Add sets z to the sum x+y and returns z.
116  func (z *Int) Add(x, y *Int) *Int {
117  	neg := x.neg
118  	if x.neg == y.neg {
119  		// x + y == x + y
120  		// (-x) + (-y) == -(x + y)
121  		z.abs = z.abs.add(x.abs, y.abs)
122  	} else {
123  		// x + (-y) == x - y == -(y - x)
124  		// (-x) + y == y - x == -(x - y)
125  		if x.abs.cmp(y.abs) >= 0 {
126  			z.abs = z.abs.sub(x.abs, y.abs)
127  		} else {
128  			neg = !neg
129  			z.abs = z.abs.sub(y.abs, x.abs)
130  		}
131  	}
132  	z.neg = len(z.abs) > 0 && neg // 0 has no sign
133  	return z
134  }
135
136  // Sub sets z to the difference x-y and returns z.
137  func (z *Int) Sub(x, y *Int) *Int {
138  	neg := x.neg
139  	if x.neg != y.neg {
140  		// x - (-y) == x + y
141  		// (-x) - y == -(x + y)
142  		z.abs = z.abs.add(x.abs, y.abs)
143  	} else {
144  		// x - y == x - y == -(y - x)
145  		// (-x) - (-y) == y - x == -(x - y)
146  		if x.abs.cmp(y.abs) >= 0 {
147  			z.abs = z.abs.sub(x.abs, y.abs)
148  		} else {
149  			neg = !neg
150  			z.abs = z.abs.sub(y.abs, x.abs)
151  		}
152  	}
153  	z.neg = len(z.abs) > 0 && neg // 0 has no sign
154  	return z
155  }
156
157  // Mul sets z to the product x*y and returns z.
158  func (z *Int) Mul(x, y *Int) *Int {
159  	// x * y == x * y
160  	// x * (-y) == -(x * y)
161  	// (-x) * y == -(x * y)
162  	// (-x) * (-y) == x * y
163  	if x == y {
164  		z.abs = z.abs.sqr(x.abs)
165  		z.neg = false
166  		return z
167  	}
168  	z.abs = z.abs.mul(x.abs, y.abs)
169  	z.neg = len(z.abs) > 0 && x.neg != y.neg // 0 has no sign
170  	return z
171  }
172
173  // MulRange sets z to the product of all integers
174  // in the range [a, b] inclusively and returns z.
175  // If a > b (empty range), the result is 1.
176  func (z *Int) MulRange(a, b int64) *Int {
177  	switch {
178  	case a > b:
179  		return z.SetInt64(1) // empty range
180  	case a <= 0 && b >= 0:
181  		return z.SetInt64(0) // range includes 0
182  	}
183  	// a <= b && (b < 0 || a > 0)
184
185  	neg := false
186  	if a < 0 {
187  		neg = (b-a)&1 == 0
188  		a, b = -b, -a
189  	}
190
191  	z.abs = z.abs.mulRange(uint64(a), uint64(b))
192  	z.neg = neg
193  	return z
194  }
195
196  // Binomial sets z to the binomial coefficient of (n, k) and returns z.
197  func (z *Int) Binomial(n, k int64) *Int {
198  	// reduce the number of multiplications by reducing k
199  	if n/2 < k && k <= n {
200  		k = n - k // Binomial(n, k) == Binomial(n, n-k)
201  	}
202  	var a, b Int
203  	a.MulRange(n-k+1, n)
204  	b.MulRange(1, k)
205  	return z.Quo(&a, &b)
206  }
207
208  // Quo sets z to the quotient x/y for y != 0 and returns z.
209  // If y == 0, a division-by-zero run-time panic occurs.
210  // Quo implements truncated division (like Go); see QuoRem for more details.
211  func (z *Int) Quo(x, y *Int) *Int {
212  	z.abs, _ = z.abs.div(nil, x.abs, y.abs)
213  	z.neg = len(z.abs) > 0 && x.neg != y.neg // 0 has no sign
214  	return z
215  }
216
217  // Rem sets z to the remainder x%y for y != 0 and returns z.
218  // If y == 0, a division-by-zero run-time panic occurs.
219  // Rem implements truncated modulus (like Go); see QuoRem for more details.
220  func (z *Int) Rem(x, y *Int) *Int {
221  	_, z.abs = nat(nil).div(z.abs, x.abs, y.abs)
222  	z.neg = len(z.abs) > 0 && x.neg // 0 has no sign
223  	return z
224  }
225
226  // QuoRem sets z to the quotient x/y and r to the remainder x%y
227  // and returns the pair (z, r) for y != 0.
228  // If y == 0, a division-by-zero run-time panic occurs.
229  //
230  // QuoRem implements T-division and modulus (like Go):
231  //
232  //	q = x/y      with the result truncated to zero
233  //	r = x - y*q
234  //
235  // (See Daan Leijen, ``Division and Modulus for Computer Scientists''.)
236  // See DivMod for Euclidean division and modulus (unlike Go).
237  //
238  func (z *Int) QuoRem(x, y, r *Int) (*Int, *Int) {
239  	z.abs, r.abs = z.abs.div(r.abs, x.abs, y.abs)
240  	z.neg, r.neg = len(z.abs) > 0 && x.neg != y.neg, len(r.abs) > 0 && x.neg // 0 has no sign
241  	return z, r
242  }
243
244  // Div sets z to the quotient x/y for y != 0 and returns z.
245  // If y == 0, a division-by-zero run-time panic occurs.
246  // Div implements Euclidean division (unlike Go); see DivMod for more details.
247  func (z *Int) Div(x, y *Int) *Int {
248  	y_neg := y.neg // z may be an alias for y
249  	var r Int
250  	z.QuoRem(x, y, &r)
251  	if r.neg {
252  		if y_neg {
253  			z.Add(z, intOne)
254  		} else {
255  			z.Sub(z, intOne)
256  		}
257  	}
258  	return z
259  }
260
261  // Mod sets z to the modulus x%y for y != 0 and returns z.
262  // If y == 0, a division-by-zero run-time panic occurs.
263  // Mod implements Euclidean modulus (unlike Go); see DivMod for more details.
264  func (z *Int) Mod(x, y *Int) *Int {
265  	y0 := y // save y
266  	if z == y || alias(z.abs, y.abs) {
267  		y0 = new(Int).Set(y)
268  	}
269  	var q Int
270  	q.QuoRem(x, y, z)
271  	if z.neg {
272  		if y0.neg {
273  			z.Sub(z, y0)
274  		} else {
275  			z.Add(z, y0)
276  		}
277  	}
278  	return z
279  }
280
281  // DivMod sets z to the quotient x div y and m to the modulus x mod y
282  // and returns the pair (z, m) for y != 0.
283  // If y == 0, a division-by-zero run-time panic occurs.
284  //
285  // DivMod implements Euclidean division and modulus (unlike Go):
286  //
287  //	q = x div y  such that
288  //	m = x - y*q  with 0 <= m < |y|
289  //
290  // (See Raymond T. Boute, ``The Euclidean definition of the functions
291  // div and mod''. ACM Transactions on Programming Languages and
292  // Systems (TOPLAS), 14(2):127-144, New York, NY, USA, 4/1992.
293  // ACM press.)
294  // See QuoRem for T-division and modulus (like Go).
295  //
296  func (z *Int) DivMod(x, y, m *Int) (*Int, *Int) {
297  	y0 := y // save y
298  	if z == y || alias(z.abs, y.abs) {
299  		y0 = new(Int).Set(y)
300  	}
301  	z.QuoRem(x, y, m)
302  	if m.neg {
303  		if y0.neg {
304  			z.Add(z, intOne)
305  			m.Sub(m, y0)
306  		} else {
307  			z.Sub(z, intOne)
308  			m.Add(m, y0)
309  		}
310  	}
311  	return z, m
312  }
313
314  // Cmp compares x and y and returns:
315  //
316  //   -1 if x <  y
317  //    0 if x == y
318  //   +1 if x >  y
319  //
320  func (x *Int) Cmp(y *Int) (r int) {
321  	// x cmp y == x cmp y
322  	// x cmp (-y) == x
323  	// (-x) cmp y == y
324  	// (-x) cmp (-y) == -(x cmp y)
325  	switch {
326  	case x.neg == y.neg:
327  		r = x.abs.cmp(y.abs)
328  		if x.neg {
329  			r = -r
330  		}
331  	case x.neg:
332  		r = -1
333  	default:
334  		r = 1
335  	}
336  	return
337  }
338
339  // CmpAbs compares the absolute values of x and y and returns:
340  //
341  //   -1 if |x| <  |y|
342  //    0 if |x| == |y|
343  //   +1 if |x| >  |y|
344  //
345  func (x *Int) CmpAbs(y *Int) int {
346  	return x.abs.cmp(y.abs)
347  }
348
349  // low32 returns the least significant 32 bits of x.
350  func low32(x nat) uint32 {
351  	if len(x) == 0 {
352  		return 0
353  	}
354  	return uint32(x[0])
355  }
356
357  // low64 returns the least significant 64 bits of x.
358  func low64(x nat) uint64 {
359  	if len(x) == 0 {
360  		return 0
361  	}
362  	v := uint64(x[0])
363  	if _W == 32 && len(x) > 1 {
364  		return uint64(x[1])<<32 | v
365  	}
366  	return v
367  }
368
369  // Int64 returns the int64 representation of x.
370  // If x cannot be represented in an int64, the result is undefined.
371  func (x *Int) Int64() int64 {
372  	v := int64(low64(x.abs))
373  	if x.neg {
374  		v = -v
375  	}
376  	return v
377  }
378
379  // Uint64 returns the uint64 representation of x.
380  // If x cannot be represented in a uint64, the result is undefined.
381  func (x *Int) Uint64() uint64 {
382  	return low64(x.abs)
383  }
384
385  // IsInt64 reports whether x can be represented as an int64.
386  func (x *Int) IsInt64() bool {
387  	if len(x.abs) <= 64/_W {
388  		w := int64(low64(x.abs))
389  		return w >= 0 || x.neg && w == -w
390  	}
391  	return false
392  }
393
394  // IsUint64 reports whether x can be represented as a uint64.
395  func (x *Int) IsUint64() bool {
396  	return !x.neg && len(x.abs) <= 64/_W
397  }
398
399  // SetString sets z to the value of s, interpreted in the given base,
400  // and returns z and a boolean indicating success. The entire string
401  // (not just a prefix) must be valid for success. If SetString fails,
402  // the value of z is undefined but the returned value is nil.
403  //
404  // The base argument must be 0 or a value between 2 and MaxBase. If the base
405  // is 0, the string prefix determines the actual conversion base. A prefix of
406  // ``0x'' or ``0X'' selects base 16; the ``0'' prefix selects base 8, and a
407  // ``0b'' or ``0B'' prefix selects base 2. Otherwise the selected base is 10.
408  //
409  // For bases <= 36, lower and upper case letters are considered the same:
410  // The letters 'a' to 'z' and 'A' to 'Z' represent digit values 10 to 35.
411  // For bases > 36, the upper case letters 'A' to 'Z' represent the digit
412  // values 36 to 61.
413  //
414  func (z *Int) SetString(s string, base int) (*Int, bool) {
415  	return z.setFromScanner(strings.NewReader(s), base)
416  }
417
418  // setFromScanner implements SetString given an io.BytesScanner.
419  // For documentation see comments of SetString.
420  func (z *Int) setFromScanner(r io.ByteScanner, base int) (*Int, bool) {
421  	if _, _, err := z.scan(r, base); err != nil {
422  		return nil, false
423  	}
424  	// entire content must have been consumed
425  	if _, err := r.ReadByte(); err != io.EOF {
426  		return nil, false
427  	}
428  	return z, true // err == io.EOF => scan consumed all content of r
429  }
430
431  // SetBytes interprets buf as the bytes of a big-endian unsigned
432  // integer, sets z to that value, and returns z.
433  func (z *Int) SetBytes(buf []byte) *Int {
434  	z.abs = z.abs.setBytes(buf)
435  	z.neg = false
436  	return z
437  }
438
439  // Bytes returns the absolute value of x as a big-endian byte slice.
440  func (x *Int) Bytes() []byte {
441  	buf := make([]byte, len(x.abs)*_S)
442  	return buf[x.abs.bytes(buf):]
443  }
444
445  // BitLen returns the length of the absolute value of x in bits.
446  // The bit length of 0 is 0.
447  func (x *Int) BitLen() int {
448  	return x.abs.bitLen()
449  }
450
451  // Exp sets z = x**y mod |m| (i.e. the sign of m is ignored), and returns z.
452  // If m == nil or m == 0, z = x**y unless y <= 0 then z = 1.
453  //
454  // Modular exponentation of inputs of a particular size is not a
455  // cryptographically constant-time operation.
456  func (z *Int) Exp(x, y, m *Int) *Int {
457  	// See Knuth, volume 2, section 4.6.3.
458  	xWords := x.abs
459  	if y.neg {
460  		if m == nil || len(m.abs) == 0 {
461  			return z.SetInt64(1)
462  		}
463  		// for y < 0: x**y mod m == (x**(-1))**|y| mod m
464  		xWords = new(Int).ModInverse(x, m).abs
465  	}
466  	yWords := y.abs
467
468  	var mWords nat
469  	if m != nil {
470  		mWords = m.abs // m.abs may be nil for m == 0
471  	}
472
473  	z.abs = z.abs.expNN(xWords, yWords, mWords)
474  	z.neg = len(z.abs) > 0 && x.neg && len(yWords) > 0 && yWords[0]&1 == 1 // 0 has no sign
475  	if z.neg && len(mWords) > 0 {
476  		// make modulus result positive
477  		z.abs = z.abs.sub(mWords, z.abs) // z == x**y mod |m| && 0 <= z < |m|
478  		z.neg = false
479  	}
480
481  	return z
482  }
483
484  // GCD sets z to the greatest common divisor of a and b, which both must
485  // be > 0, and returns z.
486  // If x or y are not nil, GCD sets their value such that z = a*x + b*y.
487  // If either a or b is <= 0, GCD sets z = x = y = 0.
488  func (z *Int) GCD(x, y, a, b *Int) *Int {
489  	if a.Sign() <= 0 || b.Sign() <= 0 {
490  		z.SetInt64(0)
491  		if x != nil {
492  			x.SetInt64(0)
493  		}
494  		if y != nil {
495  			y.SetInt64(0)
496  		}
497  		return z
498  	}
499
500  	return z.lehmerGCD(x, y, a, b)
501  }
502
503  // lehmerSimulate attempts to simulate several Euclidean update steps
504  // using the leading digits of A and B.  It returns u0, u1, v0, v1
505  // such that A and B can be updated as:
506  //		A = u0*A + v0*B
507  //		B = u1*A + v1*B
508  // Requirements: A >= B and len(B.abs) >= 2
509  // Since we are calculating with full words to avoid overflow,
510  // we use 'even' to track the sign of the cosequences.
511  // For even iterations: u0, v1 >= 0 && u1, v0 <= 0
512  // For odd  iterations: u0, v1 <= 0 && u1, v0 >= 0
513  func lehmerSimulate(A, B *Int) (u0, u1, v0, v1 Word, even bool) {
514  	// initialize the digits
515  	var a1, a2, u2, v2 Word
516
517  	m := len(B.abs) // m >= 2
518  	n := len(A.abs) // n >= m >= 2
519
520  	// extract the top Word of bits from A and B
521  	h := nlz(A.abs[n-1])
522  	a1 = A.abs[n-1]<<h | A.abs[n-2]>>(_W-h)
523  	// B may have implicit zero words in the high bits if the lengths differ
524  	switch {
525  	case n == m:
526  		a2 = B.abs[n-1]<<h | B.abs[n-2]>>(_W-h)
527  	case n == m+1:
528  		a2 = B.abs[n-2] >> (_W - h)
529  	default:
530  		a2 = 0
531  	}
532
533  	// Since we are calculating with full words to avoid overflow,
534  	// we use 'even' to track the sign of the cosequences.
535  	// For even iterations: u0, v1 >= 0 && u1, v0 <= 0
536  	// For odd  iterations: u0, v1 <= 0 && u1, v0 >= 0
537  	// The first iteration starts with k=1 (odd).
538  	even = false
539  	// variables to track the cosequences
540  	u0, u1, u2 = 0, 1, 0
541  	v0, v1, v2 = 0, 0, 1
542
543  	// Calculate the quotient and cosequences using Collins' stopping condition.
544  	// Note that overflow of a Word is not possible when computing the remainder
545  	// sequence and cosequences since the cosequence size is bounded by the input size.
546  	// See section 4.2 of Jebelean for details.
547  	for a2 >= v2 && a1-a2 >= v1+v2 {
548  		q, r := a1/a2, a1%a2
549  		a1, a2 = a2, r
550  		u0, u1, u2 = u1, u2, u1+q*u2
551  		v0, v1, v2 = v1, v2, v1+q*v2
552  		even = !even
553  	}
554  	return
555  }
556
557  // lehmerUpdate updates the inputs A and B such that:
558  //		A = u0*A + v0*B
559  //		B = u1*A + v1*B
560  // where the signs of u0, u1, v0, v1 are given by even
561  // For even == true: u0, v1 >= 0 && u1, v0 <= 0
562  // For even == false: u0, v1 <= 0 && u1, v0 >= 0
563  // q, r, s, t are temporary variables to avoid allocations in the multiplication
564  func lehmerUpdate(A, B, q, r, s, t *Int, u0, u1, v0, v1 Word, even bool) {
565
566  	t.abs = t.abs.setWord(u0)
567  	s.abs = s.abs.setWord(v0)
568  	t.neg = !even
569  	s.neg = even
570
571  	t.Mul(A, t)
572  	s.Mul(B, s)
573
574  	r.abs = r.abs.setWord(u1)
575  	q.abs = q.abs.setWord(v1)
576  	r.neg = even
577  	q.neg = !even
578
579  	r.Mul(A, r)
580  	q.Mul(B, q)
581
582  	A.Add(t, s)
583  	B.Add(r, q)
584  }
585
586  // euclidUpdate performs a single step of the Euclidean GCD algorithm
587  // if extended is true, it also updates the cosequence Ua, Ub
588  func euclidUpdate(A, B, Ua, Ub, q, r, s, t *Int, extended bool) {
589  	q, r = q.QuoRem(A, B, r)
590
591  	*A, *B, *r = *B, *r, *A
592
593  	if extended {
594  		// Ua, Ub = Ub, Ua - q*Ub
595  		t.Set(Ub)
596  		s.Mul(Ub, q)
597  		Ub.Sub(Ua, s)
598  		Ua.Set(t)
599  	}
600  }
601
602  // lehmerGCD sets z to the greatest common divisor of a and b,
603  // which both must be > 0, and returns z.
604  // If x or y are not nil, their values are set such that z = a*x + b*y.
605  // See Knuth, The Art of Computer Programming, Vol. 2, Section 4.5.2, Algorithm L.
606  // This implementation uses the improved condition by Collins requiring only one
607  // quotient and avoiding the possibility of single Word overflow.
608  // See Jebelean, "Improving the multiprecision Euclidean algorithm",
609  // Design and Implementation of Symbolic Computation Systems, pp 45-58.
610  // The cosequences are updated according to Algorithm 10.45 from
611  // Cohen et al. "Handbook of Elliptic and Hyperelliptic Curve Cryptography" pp 192.
612  func (z *Int) lehmerGCD(x, y, a, b *Int) *Int {
613  	var A, B, Ua, Ub *Int
614
615  	A = new(Int).Set(a)
616  	B = new(Int).Set(b)
617
618  	extended := x != nil || y != nil
619
620  	if extended {
621  		// Ua (Ub) tracks how many times input a has been accumulated into A (B).
622  		Ua = new(Int).SetInt64(1)
623  		Ub = new(Int)
624  	}
625
626  	// temp variables for multiprecision update
627  	q := new(Int)
628  	r := new(Int)
629  	s := new(Int)
630  	t := new(Int)
631
632  	// ensure A >= B
633  	if A.abs.cmp(B.abs) < 0 {
634  		A, B = B, A
635  		Ub, Ua = Ua, Ub
636  	}
637
638  	// loop invariant A >= B
639  	for len(B.abs) > 1 {
640  		// Attempt to calculate in single-precision using leading words of A and B.
641  		u0, u1, v0, v1, even := lehmerSimulate(A, B)
642
643  		// multiprecision Step
644  		if v0 != 0 {
645  			// Simulate the effect of the single-precision steps using the cosequences.
646  			// A = u0*A + v0*B
647  			// B = u1*A + v1*B
648  			lehmerUpdate(A, B, q, r, s, t, u0, u1, v0, v1, even)
649
650  			if extended {
651  				// Ua = u0*Ua + v0*Ub
652  				// Ub = u1*Ua + v1*Ub
653  				lehmerUpdate(Ua, Ub, q, r, s, t, u0, u1, v0, v1, even)
654  			}
655
656  		} else {
657  			// Single-digit calculations failed to simulate any quotients.
658  			// Do a standard Euclidean step.
659  			euclidUpdate(A, B, Ua, Ub, q, r, s, t, extended)
660  		}
661  	}
662
663  	if len(B.abs) > 0 {
664  		// extended Euclidean algorithm base case if B is a single Word
665  		if len(A.abs) > 1 {
666  			// A is longer than a single Word, so one update is needed.
667  			euclidUpdate(A, B, Ua, Ub, q, r, s, t, extended)
668  		}
669  		if len(B.abs) > 0 {
670  			// A and B are both a single Word.
671  			aWord, bWord := A.abs[0], B.abs[0]
672  			if extended {
673  				var ua, ub, va, vb Word
674  				ua, ub = 1, 0
675  				va, vb = 0, 1
676  				even := true
677  				for bWord != 0 {
678  					q, r := aWord/bWord, aWord%bWord
679  					aWord, bWord = bWord, r
680  					ua, ub = ub, ua+q*ub
681  					va, vb = vb, va+q*vb
682  					even = !even
683  				}
684
685  				t.abs = t.abs.setWord(ua)
686  				s.abs = s.abs.setWord(va)
687  				t.neg = !even
688  				s.neg = even
689
690  				t.Mul(Ua, t)
691  				s.Mul(Ub, s)
692
693  				Ua.Add(t, s)
694  			} else {
695  				for bWord != 0 {
696  					aWord, bWord = bWord, aWord%bWord
697  				}
698  			}
699  			A.abs[0] = aWord
700  		}
701  	}
702
703  	if x != nil {
704  		*x = *Ua
705  	}
706
707  	if y != nil {
708  		// y = (z - a*x)/b
709  		y.Mul(a, Ua)
710  		y.Sub(A, y)
711  		y.Div(y, b)
712  	}
713
714  	*z = *A
715
716  	return z
717  }
718
719  // Rand sets z to a pseudo-random number in [0, n) and returns z.
720  //
721  // As this uses the math/rand package, it must not be used for
722  // security-sensitive work. Use crypto/rand.Int instead.
723  func (z *Int) Rand(rnd *rand.Rand, n *Int) *Int {
724  	z.neg = false
725  	if n.neg || len(n.abs) == 0 {
726  		z.abs = nil
727  		return z
728  	}
729  	z.abs = z.abs.random(rnd, n.abs, n.abs.bitLen())
730  	return z
731  }
732
733  // ModInverse sets z to the multiplicative inverse of g in the ring ℤ/nℤ
734  // and returns z. If g and n are not relatively prime, g has no multiplicative
735  // inverse in the ring ℤ/nℤ.  In this case, z is unchanged and the return value
736  // is nil.
737  func (z *Int) ModInverse(g, n *Int) *Int {
738  	// GCD expects parameters a and b to be > 0.
739  	if n.neg {
740  		var n2 Int
741  		n = n2.Neg(n)
742  	}
743  	if g.neg {
744  		var g2 Int
745  		g = g2.Mod(g, n)
746  	}
747  	var d, x Int
748  	d.GCD(&x, nil, g, n)
749
750  	// if and only if d==1, g and n are relatively prime
751  	if d.Cmp(intOne) != 0 {
752  		return nil
753  	}
754
755  	// x and y are such that g*x + n*y = 1, therefore x is the inverse element,
756  	// but it may be negative, so convert to the range 0 <= z < |n|
757  	if x.neg {
758  		z.Add(&x, n)
759  	} else {
760  		z.Set(&x)
761  	}
762  	return z
763  }
764
765  // Jacobi returns the Jacobi symbol (x/y), either +1, -1, or 0.
766  // The y argument must be an odd integer.
767  func Jacobi(x, y *Int) int {
768  	if len(y.abs) == 0 || y.abs[0]&1 == 0 {
769  		panic(fmt.Sprintf("big: invalid 2nd argument to Int.Jacobi: need odd integer but got %s", y))
770  	}
771
772  	// We use the formulation described in chapter 2, section 2.4,
773  	// "The Yacas Book of Algorithms":
774  	// http://yacas.sourceforge.net/Algo.book.pdf
775
776  	var a, b, c Int
777  	a.Set(x)
778  	b.Set(y)
779  	j := 1
780
781  	if b.neg {
782  		if a.neg {
783  			j = -1
784  		}
785  		b.neg = false
786  	}
787
788  	for {
789  		if b.Cmp(intOne) == 0 {
790  			return j
791  		}
792  		if len(a.abs) == 0 {
793  			return 0
794  		}
795  		a.Mod(&a, &b)
796  		if len(a.abs) == 0 {
797  			return 0
798  		}
799  		// a > 0
800
801  		// handle factors of 2 in 'a'
802  		s := a.abs.trailingZeroBits()
803  		if s&1 != 0 {
804  			bmod8 := b.abs[0] & 7
805  			if bmod8 == 3 || bmod8 == 5 {
806  				j = -j
807  			}
808  		}
809  		c.Rsh(&a, s) // a = 2^s*c
810
811  		// swap numerator and denominator
812  		if b.abs[0]&3 == 3 && c.abs[0]&3 == 3 {
813  			j = -j
814  		}
815  		a.Set(&b)
816  		b.Set(&c)
817  	}
818  }
819
820  // modSqrt3Mod4 uses the identity
821  //      (a^((p+1)/4))^2  mod p
822  //   == u^(p+1)          mod p
823  //   == u^2              mod p
824  // to calculate the square root of any quadratic residue mod p quickly for 3
825  // mod 4 primes.
826  func (z *Int) modSqrt3Mod4Prime(x, p *Int) *Int {
827  	e := new(Int).Add(p, intOne) // e = p + 1
828  	e.Rsh(e, 2)                  // e = (p + 1) / 4
829  	z.Exp(x, e, p)               // z = x^e mod p
830  	return z
831  }
832
833  // modSqrt5Mod8 uses Atkin's observation that 2 is not a square mod p
834  //   alpha ==  (2*a)^((p-5)/8)    mod p
835  //   beta  ==  2*a*alpha^2        mod p  is a square root of -1
836  //   b     ==  a*alpha*(beta-1)   mod p  is a square root of a
837  // to calculate the square root of any quadratic residue mod p quickly for 5
838  // mod 8 primes.
839  func (z *Int) modSqrt5Mod8Prime(x, p *Int) *Int {
840  	// p == 5 mod 8 implies p = e*8 + 5
841  	// e is the quotient and 5 the remainder on division by 8
842  	e := new(Int).Rsh(p, 3)  // e = (p - 5) / 8
843  	tx := new(Int).Lsh(x, 1) // tx = 2*x
844  	alpha := new(Int).Exp(tx, e, p)
845  	beta := new(Int).Mul(alpha, alpha)
846  	beta.Mod(beta, p)
847  	beta.Mul(beta, tx)
848  	beta.Mod(beta, p)
849  	beta.Sub(beta, intOne)
850  	beta.Mul(beta, x)
851  	beta.Mod(beta, p)
852  	beta.Mul(beta, alpha)
853  	z.Mod(beta, p)
854  	return z
855  }
856
857  // modSqrtTonelliShanks uses the Tonelli-Shanks algorithm to find the square
858  // root of a quadratic residue modulo any prime.
859  func (z *Int) modSqrtTonelliShanks(x, p *Int) *Int {
860  	// Break p-1 into s*2^e such that s is odd.
861  	var s Int
862  	s.Sub(p, intOne)
863  	e := s.abs.trailingZeroBits()
864  	s.Rsh(&s, e)
865
866  	// find some non-square n
867  	var n Int
868  	n.SetInt64(2)
869  	for Jacobi(&n, p) != -1 {
870  		n.Add(&n, intOne)
871  	}
872
873  	// Core of the Tonelli-Shanks algorithm. Follows the description in
874  	// section 6 of "Square roots from 1; 24, 51, 10 to Dan Shanks" by Ezra
875  	// Brown:
876  	// https://www.maa.org/sites/default/files/pdf/upload_library/22/Polya/07468342.di020786.02p0470a.pdf
877  	var y, b, g, t Int
878  	y.Add(&s, intOne)
879  	y.Rsh(&y, 1)
880  	y.Exp(x, &y, p)  // y = x^((s+1)/2)
881  	b.Exp(x, &s, p)  // b = x^s
882  	g.Exp(&n, &s, p) // g = n^s
883  	r := e
884  	for {
885  		// find the least m such that ord_p(b) = 2^m
886  		var m uint
887  		t.Set(&b)
888  		for t.Cmp(intOne) != 0 {
889  			t.Mul(&t, &t).Mod(&t, p)
890  			m++
891  		}
892
893  		if m == 0 {
894  			return z.Set(&y)
895  		}
896
897  		t.SetInt64(0).SetBit(&t, int(r-m-1), 1).Exp(&g, &t, p)
898  		// t = g^(2^(r-m-1)) mod p
899  		g.Mul(&t, &t).Mod(&g, p) // g = g^(2^(r-m)) mod p
900  		y.Mul(&y, &t).Mod(&y, p)
901  		b.Mul(&b, &g).Mod(&b, p)
902  		r = m
903  	}
904  }
905
906  // ModSqrt sets z to a square root of x mod p if such a square root exists, and
907  // returns z. The modulus p must be an odd prime. If x is not a square mod p,
908  // ModSqrt leaves z unchanged and returns nil. This function panics if p is
909  // not an odd integer.
910  func (z *Int) ModSqrt(x, p *Int) *Int {
911  	switch Jacobi(x, p) {
912  	case -1:
913  		return nil // x is not a square mod p
914  	case 0:
915  		return z.SetInt64(0) // sqrt(0) mod p = 0
916  	case 1:
917  		break
918  	}
919  	if x.neg || x.Cmp(p) >= 0 { // ensure 0 <= x < p
920  		x = new(Int).Mod(x, p)
921  	}
922
923  	switch {
924  	case p.abs[0]%4 == 3:
925  		// Check whether p is 3 mod 4, and if so, use the faster algorithm.
926  		return z.modSqrt3Mod4Prime(x, p)
927  	case p.abs[0]%8 == 5:
928  		// Check whether p is 5 mod 8, use Atkin's algorithm.
929  		return z.modSqrt5Mod8Prime(x, p)
930  	default:
931  		// Otherwise, use Tonelli-Shanks.
932  		return z.modSqrtTonelliShanks(x, p)
933  	}
934  }
935
936  // Lsh sets z = x << n and returns z.
937  func (z *Int) Lsh(x *Int, n uint) *Int {
938  	z.abs = z.abs.shl(x.abs, n)
939  	z.neg = x.neg
940  	return z
941  }
942
943  // Rsh sets z = x >> n and returns z.
944  func (z *Int) Rsh(x *Int, n uint) *Int {
945  	if x.neg {
946  		// (-x) >> s == ^(x-1) >> s == ^((x-1) >> s) == -(((x-1) >> s) + 1)
947  		t := z.abs.sub(x.abs, natOne) // no underflow because |x| > 0
948  		t = t.shr(t, n)
949  		z.abs = t.add(t, natOne)
950  		z.neg = true // z cannot be zero if x is negative
951  		return z
952  	}
953
954  	z.abs = z.abs.shr(x.abs, n)
955  	z.neg = false
956  	return z
957  }
958
959  // Bit returns the value of the i'th bit of x. That is, it
960  // returns (x>>i)&1. The bit index i must be >= 0.
961  func (x *Int) Bit(i int) uint {
962  	if i == 0 {
963  		// optimization for common case: odd/even test of x
964  		if len(x.abs) > 0 {
965  			return uint(x.abs[0] & 1) // bit 0 is same for -x
966  		}
967  		return 0
968  	}
969  	if i < 0 {
970  		panic("negative bit index")
971  	}
972  	if x.neg {
973  		t := nat(nil).sub(x.abs, natOne)
974  		return t.bit(uint(i)) ^ 1
975  	}
976
977  	return x.abs.bit(uint(i))
978  }
979
980  // SetBit sets z to x, with x's i'th bit set to b (0 or 1).
981  // That is, if b is 1 SetBit sets z = x | (1 << i);
982  // if b is 0 SetBit sets z = x &^ (1 << i). If b is not 0 or 1,
983  // SetBit will panic.
984  func (z *Int) SetBit(x *Int, i int, b uint) *Int {
985  	if i < 0 {
986  		panic("negative bit index")
987  	}
988  	if x.neg {
989  		t := z.abs.sub(x.abs, natOne)
990  		t = t.setBit(t, uint(i), b^1)
991  		z.abs = t.add(t, natOne)
992  		z.neg = len(z.abs) > 0
993  		return z
994  	}
995  	z.abs = z.abs.setBit(x.abs, uint(i), b)
996  	z.neg = false
997  	return z
998  }
999
1000  // And sets z = x & y and returns z.
1001  func (z *Int) And(x, y *Int) *Int {
1002  	if x.neg == y.neg {
1003  		if x.neg {
1004  			// (-x) & (-y) == ^(x-1) & ^(y-1) == ^((x-1) | (y-1)) == -(((x-1) | (y-1)) + 1)
1005  			x1 := nat(nil).sub(x.abs, natOne)
1006  			y1 := nat(nil).sub(y.abs, natOne)
1007  			z.abs = z.abs.add(z.abs.or(x1, y1), natOne)
1008  			z.neg = true // z cannot be zero if x and y are negative
1009  			return z
1010  		}
1011
1012  		// x & y == x & y
1013  		z.abs = z.abs.and(x.abs, y.abs)
1014  		z.neg = false
1015  		return z
1016  	}
1017
1018  	// x.neg != y.neg
1019  	if x.neg {
1020  		x, y = y, x // & is symmetric
1021  	}
1022
1023  	// x & (-y) == x & ^(y-1) == x &^ (y-1)
1024  	y1 := nat(nil).sub(y.abs, natOne)
1025  	z.abs = z.abs.andNot(x.abs, y1)
1026  	z.neg = false
1027  	return z
1028  }
1029
1030  // AndNot sets z = x &^ y and returns z.
1031  func (z *Int) AndNot(x, y *Int) *Int {
1032  	if x.neg == y.neg {
1033  		if x.neg {
1034  			// (-x) &^ (-y) == ^(x-1) &^ ^(y-1) == ^(x-1) & (y-1) == (y-1) &^ (x-1)
1035  			x1 := nat(nil).sub(x.abs, natOne)
1036  			y1 := nat(nil).sub(y.abs, natOne)
1037  			z.abs = z.abs.andNot(y1, x1)
1038  			z.neg = false
1039  			return z
1040  		}
1041
1042  		// x &^ y == x &^ y
1043  		z.abs = z.abs.andNot(x.abs, y.abs)
1044  		z.neg = false
1045  		return z
1046  	}
1047
1048  	if x.neg {
1049  		// (-x) &^ y == ^(x-1) &^ y == ^(x-1) & ^y == ^((x-1) | y) == -(((x-1) | y) + 1)
1050  		x1 := nat(nil).sub(x.abs, natOne)
1051  		z.abs = z.abs.add(z.abs.or(x1, y.abs), natOne)
1052  		z.neg = true // z cannot be zero if x is negative and y is positive
1053  		return z
1054  	}
1055
1056  	// x &^ (-y) == x &^ ^(y-1) == x & (y-1)
1057  	y1 := nat(nil).sub(y.abs, natOne)
1058  	z.abs = z.abs.and(x.abs, y1)
1059  	z.neg = false
1060  	return z
1061  }
1062
1063  // Or sets z = x | y and returns z.
1064  func (z *Int) Or(x, y *Int) *Int {
1065  	if x.neg == y.neg {
1066  		if x.neg {
1067  			// (-x) | (-y) == ^(x-1) | ^(y-1) == ^((x-1) & (y-1)) == -(((x-1) & (y-1)) + 1)
1068  			x1 := nat(nil).sub(x.abs, natOne)
1069  			y1 := nat(nil).sub(y.abs, natOne)
1070  			z.abs = z.abs.add(z.abs.and(x1, y1), natOne)
1071  			z.neg = true // z cannot be zero if x and y are negative
1072  			return z
1073  		}
1074
1075  		// x | y == x | y
1076  		z.abs = z.abs.or(x.abs, y.abs)
1077  		z.neg = false
1078  		return z
1079  	}
1080
1081  	// x.neg != y.neg
1082  	if x.neg {
1083  		x, y = y, x // | is symmetric
1084  	}
1085
1086  	// x | (-y) == x | ^(y-1) == ^((y-1) &^ x) == -(^((y-1) &^ x) + 1)
1087  	y1 := nat(nil).sub(y.abs, natOne)
1088  	z.abs = z.abs.add(z.abs.andNot(y1, x.abs), natOne)
1089  	z.neg = true // z cannot be zero if one of x or y is negative
1090  	return z
1091  }
1092
1093  // Xor sets z = x ^ y and returns z.
1094  func (z *Int) Xor(x, y *Int) *Int {
1095  	if x.neg == y.neg {
1096  		if x.neg {
1097  			// (-x) ^ (-y) == ^(x-1) ^ ^(y-1) == (x-1) ^ (y-1)
1098  			x1 := nat(nil).sub(x.abs, natOne)
1099  			y1 := nat(nil).sub(y.abs, natOne)
1100  			z.abs = z.abs.xor(x1, y1)
1101  			z.neg = false
1102  			return z
1103  		}
1104
1105  		// x ^ y == x ^ y
1106  		z.abs = z.abs.xor(x.abs, y.abs)
1107  		z.neg = false
1108  		return z
1109  	}
1110
1111  	// x.neg != y.neg
1112  	if x.neg {
1113  		x, y = y, x // ^ is symmetric
1114  	}
1115
1116  	// x ^ (-y) == x ^ ^(y-1) == ^(x ^ (y-1)) == -((x ^ (y-1)) + 1)
1117  	y1 := nat(nil).sub(y.abs, natOne)
1118  	z.abs = z.abs.add(z.abs.xor(x.abs, y1), natOne)
1119  	z.neg = true // z cannot be zero if only one of x or y is negative
1120  	return z
1121  }
1122
1123  // Not sets z = ^x and returns z.
1124  func (z *Int) Not(x *Int) *Int {
1125  	if x.neg {
1126  		// ^(-x) == ^(^(x-1)) == x-1
1127  		z.abs = z.abs.sub(x.abs, natOne)
1128  		z.neg = false
1129  		return z
1130  	}
1131
1132  	// ^x == -x-1 == -(x+1)
1133  	z.abs = z.abs.add(x.abs, natOne)
1134  	z.neg = true // z cannot be zero if x is positive
1135  	return z
1136  }
1137
1138  // Sqrt sets z to ⌊√x⌋, the largest integer such that z² ≤ x, and returns z.
1139  // It panics if x is negative.
1140  func (z *Int) Sqrt(x *Int) *Int {
1141  	if x.neg {
1142  		panic("square root of negative number")
1143  	}
1144  	z.neg = false
1145  	z.abs = z.abs.sqrt(x.abs)
1146  	return z
1147  }
1148

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