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# Source file src/math/big/nat.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
4
5  // This file implements unsigned multi-precision integers (natural
6  // numbers). They are the building blocks for the implementation
7  // of signed integers, rationals, and floating-point numbers.
8  //
9  // Caution: This implementation relies on the function "alias"
10  //          which assumes that (nat) slice capacities are never
11  //          changed (no 3-operand slice expressions). If that
12  //          changes, alias needs to be updated for correctness.
13
14  package big
15
16  import (
17  	"encoding/binary"
18  	"math/bits"
19  	"math/rand"
20  	"sync"
21  )
22
23  // An unsigned integer x of the form
24  //
25  //   x = x[n-1]*_B^(n-1) + x[n-2]*_B^(n-2) + ... + x[1]*_B + x[0]
26  //
27  // with 0 <= x[i] < _B and 0 <= i < n is stored in a slice of length n,
28  // with the digits x[i] as the slice elements.
29  //
30  // A number is normalized if the slice contains no leading 0 digits.
31  // During arithmetic operations, denormalized values may occur but are
32  // always normalized before returning the final result. The normalized
33  // representation of 0 is the empty or nil slice (length = 0).
34  //
35  type nat []Word
36
37  var (
38  	natOne  = nat{1}
39  	natTwo  = nat{2}
40  	natFive = nat{5}
41  	natTen  = nat{10}
42  )
43
44  func (z nat) clear() {
45  	for i := range z {
46  		z[i] = 0
47  	}
48  }
49
50  func (z nat) norm() nat {
51  	i := len(z)
52  	for i > 0 && z[i-1] == 0 {
53  		i--
54  	}
55  	return z[0:i]
56  }
57
58  func (z nat) make(n int) nat {
59  	if n <= cap(z) {
60  		return z[:n] // reuse z
61  	}
62  	if n == 1 {
63  		// Most nats start small and stay that way; don't over-allocate.
64  		return make(nat, 1)
65  	}
66  	// Choosing a good value for e has significant performance impact
67  	// because it increases the chance that a value can be reused.
68  	const e = 4 // extra capacity
69  	return make(nat, n, n+e)
70  }
71
72  func (z nat) setWord(x Word) nat {
73  	if x == 0 {
74  		return z[:0]
75  	}
76  	z = z.make(1)
77  	z[0] = x
78  	return z
79  }
80
81  func (z nat) setUint64(x uint64) nat {
82  	// single-word value
83  	if w := Word(x); uint64(w) == x {
84  		return z.setWord(w)
85  	}
86  	// 2-word value
87  	z = z.make(2)
88  	z[1] = Word(x >> 32)
89  	z[0] = Word(x)
90  	return z
91  }
92
93  func (z nat) set(x nat) nat {
94  	z = z.make(len(x))
95  	copy(z, x)
96  	return z
97  }
98
99  func (z nat) add(x, y nat) nat {
100  	m := len(x)
101  	n := len(y)
102
103  	switch {
104  	case m < n:
106  	case m == 0:
107  		// n == 0 because m >= n; result is 0
108  		return z[:0]
109  	case n == 0:
110  		// result is x
111  		return z.set(x)
112  	}
113  	// m > 0
114
115  	z = z.make(m + 1)
116  	c := addVV(z[0:n], x, y)
117  	if m > n {
118  		c = addVW(z[n:m], x[n:], c)
119  	}
120  	z[m] = c
121
122  	return z.norm()
123  }
124
125  func (z nat) sub(x, y nat) nat {
126  	m := len(x)
127  	n := len(y)
128
129  	switch {
130  	case m < n:
131  		panic("underflow")
132  	case m == 0:
133  		// n == 0 because m >= n; result is 0
134  		return z[:0]
135  	case n == 0:
136  		// result is x
137  		return z.set(x)
138  	}
139  	// m > 0
140
141  	z = z.make(m)
142  	c := subVV(z[0:n], x, y)
143  	if m > n {
144  		c = subVW(z[n:], x[n:], c)
145  	}
146  	if c != 0 {
147  		panic("underflow")
148  	}
149
150  	return z.norm()
151  }
152
153  func (x nat) cmp(y nat) (r int) {
154  	m := len(x)
155  	n := len(y)
156  	if m != n || m == 0 {
157  		switch {
158  		case m < n:
159  			r = -1
160  		case m > n:
161  			r = 1
162  		}
163  		return
164  	}
165
166  	i := m - 1
167  	for i > 0 && x[i] == y[i] {
168  		i--
169  	}
170
171  	switch {
172  	case x[i] < y[i]:
173  		r = -1
174  	case x[i] > y[i]:
175  		r = 1
176  	}
177  	return
178  }
179
180  func (z nat) mulAddWW(x nat, y, r Word) nat {
181  	m := len(x)
182  	if m == 0 || y == 0 {
183  		return z.setWord(r) // result is r
184  	}
185  	// m > 0
186
187  	z = z.make(m + 1)
188  	z[m] = mulAddVWW(z[0:m], x, y, r)
189
190  	return z.norm()
191  }
192
193  // basicMul multiplies x and y and leaves the result in z.
194  // The (non-normalized) result is placed in z[0 : len(x) + len(y)].
195  func basicMul(z, x, y nat) {
196  	z[0 : len(x)+len(y)].clear() // initialize z
197  	for i, d := range y {
198  		if d != 0 {
199  			z[len(x)+i] = addMulVVW(z[i:i+len(x)], x, d)
200  		}
201  	}
202  }
203
204  // montgomery computes z mod m = x*y*2**(-n*_W) mod m,
205  // assuming k = -1/m mod 2**_W.
206  // z is used for storing the result which is returned;
207  // z must not alias x, y or m.
208  // See Gueron, "Efficient Software Implementations of Modular Exponentiation".
209  // https://eprint.iacr.org/2011/239.pdf
210  // In the terminology of that paper, this is an "Almost Montgomery Multiplication":
211  // x and y are required to satisfy 0 <= z < 2**(n*_W) and then the result
212  // z is guaranteed to satisfy 0 <= z < 2**(n*_W), but it may not be < m.
213  func (z nat) montgomery(x, y, m nat, k Word, n int) nat {
214  	// This code assumes x, y, m are all the same length, n.
215  	// (required by addMulVVW and the for loop).
216  	// It also assumes that x, y are already reduced mod m,
217  	// or else the result will not be properly reduced.
218  	if len(x) != n || len(y) != n || len(m) != n {
219  		panic("math/big: mismatched montgomery number lengths")
220  	}
221  	z = z.make(n * 2)
222  	z.clear()
223  	var c Word
224  	for i := 0; i < n; i++ {
225  		d := y[i]
226  		c2 := addMulVVW(z[i:n+i], x, d)
227  		t := z[i] * k
228  		c3 := addMulVVW(z[i:n+i], m, t)
229  		cx := c + c2
230  		cy := cx + c3
231  		z[n+i] = cy
232  		if cx < c2 || cy < c3 {
233  			c = 1
234  		} else {
235  			c = 0
236  		}
237  	}
238  	if c != 0 {
239  		subVV(z[:n], z[n:], m)
240  	} else {
241  		copy(z[:n], z[n:])
242  	}
243  	return z[:n]
244  }
245
246  // Fast version of z[0:n+n>>1].add(z[0:n+n>>1], x[0:n]) w/o bounds checks.
247  // Factored out for readability - do not use outside karatsuba.
248  func karatsubaAdd(z, x nat, n int) {
249  	if c := addVV(z[0:n], z, x); c != 0 {
251  	}
252  }
253
254  // Like karatsubaAdd, but does subtract.
255  func karatsubaSub(z, x nat, n int) {
256  	if c := subVV(z[0:n], z, x); c != 0 {
257  		subVW(z[n:n+n>>1], z[n:], c)
258  	}
259  }
260
261  // Operands that are shorter than karatsubaThreshold are multiplied using
262  // "grade school" multiplication; for longer operands the Karatsuba algorithm
263  // is used.
264  var karatsubaThreshold = 40 // computed by calibrate_test.go
265
266  // karatsuba multiplies x and y and leaves the result in z.
267  // Both x and y must have the same length n and n must be a
268  // power of 2. The result vector z must have len(z) >= 6*n.
269  // The (non-normalized) result is placed in z[0 : 2*n].
270  func karatsuba(z, x, y nat) {
271  	n := len(y)
272
273  	// Switch to basic multiplication if numbers are odd or small.
274  	// (n is always even if karatsubaThreshold is even, but be
275  	// conservative)
276  	if n&1 != 0 || n < karatsubaThreshold || n < 2 {
277  		basicMul(z, x, y)
278  		return
279  	}
280  	// n&1 == 0 && n >= karatsubaThreshold && n >= 2
281
282  	// Karatsuba multiplication is based on the observation that
283  	// for two numbers x and y with:
284  	//
285  	//   x = x1*b + x0
286  	//   y = y1*b + y0
287  	//
288  	// the product x*y can be obtained with 3 products z2, z1, z0
290  	//
291  	//   x*y = x1*y1*b*b + (x1*y0 + x0*y1)*b + x0*y0
292  	//       =    z2*b*b +              z1*b +    z0
293  	//
294  	// with:
295  	//
296  	//   xd = x1 - x0
297  	//   yd = y0 - y1
298  	//
299  	//   z1 =      xd*yd                    + z2 + z0
300  	//      = (x1-x0)*(y0 - y1)             + z2 + z0
301  	//      = x1*y0 - x1*y1 - x0*y0 + x0*y1 + z2 + z0
302  	//      = x1*y0 -    z2 -    z0 + x0*y1 + z2 + z0
303  	//      = x1*y0                 + x0*y1
304
305  	// split x, y into "digits"
306  	n2 := n >> 1              // n2 >= 1
307  	x1, x0 := x[n2:], x[0:n2] // x = x1*b + y0
308  	y1, y0 := y[n2:], y[0:n2] // y = y1*b + y0
309
310  	// z is used for the result and temporary storage:
311  	//
312  	//   6*n     5*n     4*n     3*n     2*n     1*n     0*n
313  	// z = [z2 copy|z0 copy| xd*yd | yd:xd | x1*y1 | x0*y0 ]
314  	//
315  	// For each recursive call of karatsuba, an unused slice of
316  	// z is passed in that has (at least) half the length of the
317  	// caller's z.
318
319  	// compute z0 and z2 with the result "in place" in z
320  	karatsuba(z, x0, y0)     // z0 = x0*y0
321  	karatsuba(z[n:], x1, y1) // z2 = x1*y1
322
323  	// compute xd (or the negative value if underflow occurs)
324  	s := 1 // sign of product xd*yd
325  	xd := z[2*n : 2*n+n2]
326  	if subVV(xd, x1, x0) != 0 { // x1-x0
327  		s = -s
328  		subVV(xd, x0, x1) // x0-x1
329  	}
330
331  	// compute yd (or the negative value if underflow occurs)
332  	yd := z[2*n+n2 : 3*n]
333  	if subVV(yd, y0, y1) != 0 { // y0-y1
334  		s = -s
335  		subVV(yd, y1, y0) // y1-y0
336  	}
337
338  	// p = (x1-x0)*(y0-y1) == x1*y0 - x1*y1 - x0*y0 + x0*y1 for s > 0
339  	// p = (x0-x1)*(y0-y1) == x0*y0 - x0*y1 - x1*y0 + x1*y1 for s < 0
340  	p := z[n*3:]
341  	karatsuba(p, xd, yd)
342
343  	// save original z2:z0
344  	// (ok to use upper half of z since we're done recursing)
345  	r := z[n*4:]
346  	copy(r, z[:n*2])
347
348  	// add up all partial products
349  	//
350  	//   2*n     n     0
351  	// z = [ z2  | z0  ]
352  	//   +    [ z0  ]
353  	//   +    [ z2  ]
354  	//   +    [  p  ]
355  	//
358  	if s > 0 {
360  	} else {
361  		karatsubaSub(z[n2:], p, n)
362  	}
363  }
364
365  // alias reports whether x and y share the same base array.
366  // Note: alias assumes that the capacity of underlying arrays
367  //       is never changed for nat values; i.e. that there are
368  //       no 3-operand slice expressions in this code (or worse,
369  //       reflect-based operations to the same effect).
370  func alias(x, y nat) bool {
371  	return cap(x) > 0 && cap(y) > 0 && &x[0:cap(x)][cap(x)-1] == &y[0:cap(y)][cap(y)-1]
372  }
373
374  // addAt implements z += x<<(_W*i); z must be long enough.
375  // (we don't use nat.add because we need z to stay the same
376  // slice, and we don't need to normalize z after each addition)
377  func addAt(z, x nat, i int) {
378  	if n := len(x); n > 0 {
379  		if c := addVV(z[i:i+n], z[i:], x); c != 0 {
380  			j := i + n
381  			if j < len(z) {
383  			}
384  		}
385  	}
386  }
387
388  func max(x, y int) int {
389  	if x > y {
390  		return x
391  	}
392  	return y
393  }
394
395  // karatsubaLen computes an approximation to the maximum k <= n such that
396  // k = p<<i for a number p <= threshold and an i >= 0. Thus, the
397  // result is the largest number that can be divided repeatedly by 2 before
398  // becoming about the value of threshold.
399  func karatsubaLen(n, threshold int) int {
400  	i := uint(0)
401  	for n > threshold {
402  		n >>= 1
403  		i++
404  	}
405  	return n << i
406  }
407
408  func (z nat) mul(x, y nat) nat {
409  	m := len(x)
410  	n := len(y)
411
412  	switch {
413  	case m < n:
414  		return z.mul(y, x)
415  	case m == 0 || n == 0:
416  		return z[:0]
417  	case n == 1:
419  	}
420  	// m >= n > 1
421
422  	// determine if z can be reused
423  	if alias(z, x) || alias(z, y) {
424  		z = nil // z is an alias for x or y - cannot reuse
425  	}
426
427  	// use basic multiplication if the numbers are small
428  	if n < karatsubaThreshold {
429  		z = z.make(m + n)
430  		basicMul(z, x, y)
431  		return z.norm()
432  	}
433  	// m >= n && n >= karatsubaThreshold && n >= 2
434
435  	// determine Karatsuba length k such that
436  	//
437  	//   x = xh*b + x0  (0 <= x0 < b)
438  	//   y = yh*b + y0  (0 <= y0 < b)
439  	//   b = 1<<(_W*k)  ("base" of digits xi, yi)
440  	//
441  	k := karatsubaLen(n, karatsubaThreshold)
442  	// k <= n
443
444  	// multiply x0 and y0 via Karatsuba
445  	x0 := x[0:k]              // x0 is not normalized
446  	y0 := y[0:k]              // y0 is not normalized
447  	z = z.make(max(6*k, m+n)) // enough space for karatsuba of x0*y0 and full result of x*y
448  	karatsuba(z, x0, y0)
449  	z = z[0 : m+n]  // z has final length but may be incomplete
450  	z[2*k:].clear() // upper portion of z is garbage (and 2*k <= m+n since k <= n <= m)
451
452  	// If xh != 0 or yh != 0, add the missing terms to z. For
453  	//
454  	//   xh = xi*b^i + ... + x2*b^2 + x1*b (0 <= xi < b)
455  	//   yh =                         y1*b (0 <= y1 < b)
456  	//
457  	// the missing terms are
458  	//
459  	//   x0*y1*b and xi*y0*b^i, xi*y1*b^(i+1) for i > 0
460  	//
461  	// since all the yi for i > 1 are 0 by choice of k: If any of them
462  	// were > 0, then yh >= b^2 and thus y >= b^2. Then k' = k*2 would
463  	// be a larger valid threshold contradicting the assumption about k.
464  	//
465  	if k < n || m != n {
466  		tp := getNat(3 * k)
467  		t := *tp
468
470  		x0 := x0.norm()
471  		y1 := y[k:]       // y1 is normalized because y is
472  		t = t.mul(x0, y1) // update t so we don't lose t's underlying array
474
476  		y0 := y0.norm()
477  		for i := k; i < len(x); i += k {
478  			xi := x[i:]
479  			if len(xi) > k {
480  				xi = xi[:k]
481  			}
482  			xi = xi.norm()
483  			t = t.mul(xi, y0)
485  			t = t.mul(xi, y1)
487  		}
488
489  		putNat(tp)
490  	}
491
492  	return z.norm()
493  }
494
495  // basicSqr sets z = x*x and is asymptotically faster than basicMul
496  // by about a factor of 2, but slower for small arguments due to overhead.
497  // Requirements: len(x) > 0, len(z) == 2*len(x)
498  // The (non-normalized) result is placed in z.
499  func basicSqr(z, x nat) {
500  	n := len(x)
501  	tp := getNat(2 * n)
502  	t := *tp // temporary variable to hold the products
503  	t.clear()
504  	z[1], z[0] = mulWW(x[0], x[0]) // the initial square
505  	for i := 1; i < n; i++ {
506  		d := x[i]
507  		// z collects the squares x[i] * x[i]
508  		z[2*i+1], z[2*i] = mulWW(d, d)
509  		// t collects the products x[i] * x[j] where j < i
510  		t[2*i] = addMulVVW(t[i:2*i], x[0:i], d)
511  	}
512  	t[2*n-1] = shlVU(t[1:2*n-1], t[1:2*n-1], 1) // double the j < i products
513  	addVV(z, z, t)                              // combine the result
514  	putNat(tp)
515  }
516
517  // karatsubaSqr squares x and leaves the result in z.
518  // len(x) must be a power of 2 and len(z) >= 6*len(x).
519  // The (non-normalized) result is placed in z[0 : 2*len(x)].
520  //
521  // The algorithm and the layout of z are the same as for karatsuba.
522  func karatsubaSqr(z, x nat) {
523  	n := len(x)
524
525  	if n&1 != 0 || n < karatsubaSqrThreshold || n < 2 {
526  		basicSqr(z[:2*n], x)
527  		return
528  	}
529
530  	n2 := n >> 1
531  	x1, x0 := x[n2:], x[0:n2]
532
533  	karatsubaSqr(z, x0)
534  	karatsubaSqr(z[n:], x1)
535
536  	// s = sign(xd*yd) == -1 for xd != 0; s == 1 for xd == 0
537  	xd := z[2*n : 2*n+n2]
538  	if subVV(xd, x1, x0) != 0 {
539  		subVV(xd, x0, x1)
540  	}
541
542  	p := z[n*3:]
543  	karatsubaSqr(p, xd)
544
545  	r := z[n*4:]
546  	copy(r, z[:n*2])
547
550  	karatsubaSub(z[n2:], p, n) // s == -1 for p != 0; s == 1 for p == 0
551  }
552
553  // Operands that are shorter than basicSqrThreshold are squared using
554  // "grade school" multiplication; for operands longer than karatsubaSqrThreshold
555  // we use the Karatsuba algorithm optimized for x == y.
556  var basicSqrThreshold = 20      // computed by calibrate_test.go
557  var karatsubaSqrThreshold = 260 // computed by calibrate_test.go
558
559  // z = x*x
560  func (z nat) sqr(x nat) nat {
561  	n := len(x)
562  	switch {
563  	case n == 0:
564  		return z[:0]
565  	case n == 1:
566  		d := x[0]
567  		z = z.make(2)
568  		z[1], z[0] = mulWW(d, d)
569  		return z.norm()
570  	}
571
572  	if alias(z, x) {
573  		z = nil // z is an alias for x - cannot reuse
574  	}
575
576  	if n < basicSqrThreshold {
577  		z = z.make(2 * n)
578  		basicMul(z, x, x)
579  		return z.norm()
580  	}
581  	if n < karatsubaSqrThreshold {
582  		z = z.make(2 * n)
583  		basicSqr(z, x)
584  		return z.norm()
585  	}
586
587  	// Use Karatsuba multiplication optimized for x == y.
588  	// The algorithm and layout of z are the same as for mul.
589
590  	// z = (x1*b + x0)^2 = x1^2*b^2 + 2*x1*x0*b + x0^2
591
592  	k := karatsubaLen(n, karatsubaSqrThreshold)
593
594  	x0 := x[0:k]
595  	z = z.make(max(6*k, 2*n))
596  	karatsubaSqr(z, x0) // z = x0^2
597  	z = z[0 : 2*n]
598  	z[2*k:].clear()
599
600  	if k < n {
601  		tp := getNat(2 * k)
602  		t := *tp
603  		x0 := x0.norm()
604  		x1 := x[k:]
605  		t = t.mul(x0, x1)
607  		addAt(z, t, k) // z = 2*x1*x0*b + x0^2
608  		t = t.sqr(x1)
609  		addAt(z, t, 2*k) // z = x1^2*b^2 + 2*x1*x0*b + x0^2
610  		putNat(tp)
611  	}
612
613  	return z.norm()
614  }
615
616  // mulRange computes the product of all the unsigned integers in the
617  // range [a, b] inclusively. If a > b (empty range), the result is 1.
618  func (z nat) mulRange(a, b uint64) nat {
619  	switch {
620  	case a == 0:
621  		// cut long ranges short (optimization)
622  		return z.setUint64(0)
623  	case a > b:
624  		return z.setUint64(1)
625  	case a == b:
626  		return z.setUint64(a)
627  	case a+1 == b:
628  		return z.mul(nat(nil).setUint64(a), nat(nil).setUint64(b))
629  	}
630  	m := (a + b) / 2
631  	return z.mul(nat(nil).mulRange(a, m), nat(nil).mulRange(m+1, b))
632  }
633
634  // getNat returns a *nat of len n. The contents may not be zero.
635  // The pool holds *nat to avoid allocation when converting to interface{}.
636  func getNat(n int) *nat {
637  	var z *nat
638  	if v := natPool.Get(); v != nil {
639  		z = v.(*nat)
640  	}
641  	if z == nil {
642  		z = new(nat)
643  	}
644  	*z = z.make(n)
645  	return z
646  }
647
648  func putNat(x *nat) {
649  	natPool.Put(x)
650  }
651
652  var natPool sync.Pool
653
654  // Length of x in bits. x must be normalized.
655  func (x nat) bitLen() int {
656  	if i := len(x) - 1; i >= 0 {
657  		return i*_W + bits.Len(uint(x[i]))
658  	}
659  	return 0
660  }
661
662  // trailingZeroBits returns the number of consecutive least significant zero
663  // bits of x.
664  func (x nat) trailingZeroBits() uint {
665  	if len(x) == 0 {
666  		return 0
667  	}
668  	var i uint
669  	for x[i] == 0 {
670  		i++
671  	}
672  	// x[i] != 0
673  	return i*_W + uint(bits.TrailingZeros(uint(x[i])))
674  }
675
676  func same(x, y nat) bool {
677  	return len(x) == len(y) && len(x) > 0 && &x[0] == &y[0]
678  }
679
680  // z = x << s
681  func (z nat) shl(x nat, s uint) nat {
682  	if s == 0 {
683  		if same(z, x) {
684  			return z
685  		}
686  		if !alias(z, x) {
687  			return z.set(x)
688  		}
689  	}
690
691  	m := len(x)
692  	if m == 0 {
693  		return z[:0]
694  	}
695  	// m > 0
696
697  	n := m + int(s/_W)
698  	z = z.make(n + 1)
699  	z[n] = shlVU(z[n-m:n], x, s%_W)
700  	z[0 : n-m].clear()
701
702  	return z.norm()
703  }
704
705  // z = x >> s
706  func (z nat) shr(x nat, s uint) nat {
707  	if s == 0 {
708  		if same(z, x) {
709  			return z
710  		}
711  		if !alias(z, x) {
712  			return z.set(x)
713  		}
714  	}
715
716  	m := len(x)
717  	n := m - int(s/_W)
718  	if n <= 0 {
719  		return z[:0]
720  	}
721  	// n > 0
722
723  	z = z.make(n)
724  	shrVU(z, x[m-n:], s%_W)
725
726  	return z.norm()
727  }
728
729  func (z nat) setBit(x nat, i uint, b uint) nat {
730  	j := int(i / _W)
731  	m := Word(1) << (i % _W)
732  	n := len(x)
733  	switch b {
734  	case 0:
735  		z = z.make(n)
736  		copy(z, x)
737  		if j >= n {
738  			// no need to grow
739  			return z
740  		}
741  		z[j] &^= m
742  		return z.norm()
743  	case 1:
744  		if j >= n {
745  			z = z.make(j + 1)
746  			z[n:].clear()
747  		} else {
748  			z = z.make(n)
749  		}
750  		copy(z, x)
751  		z[j] |= m
752  		// no need to normalize
753  		return z
754  	}
755  	panic("set bit is not 0 or 1")
756  }
757
758  // bit returns the value of the i'th bit, with lsb == bit 0.
759  func (x nat) bit(i uint) uint {
760  	j := i / _W
761  	if j >= uint(len(x)) {
762  		return 0
763  	}
764  	// 0 <= j < len(x)
765  	return uint(x[j] >> (i % _W) & 1)
766  }
767
768  // sticky returns 1 if there's a 1 bit within the
769  // i least significant bits, otherwise it returns 0.
770  func (x nat) sticky(i uint) uint {
771  	j := i / _W
772  	if j >= uint(len(x)) {
773  		if len(x) == 0 {
774  			return 0
775  		}
776  		return 1
777  	}
778  	// 0 <= j < len(x)
779  	for _, x := range x[:j] {
780  		if x != 0 {
781  			return 1
782  		}
783  	}
784  	if x[j]<<(_W-i%_W) != 0 {
785  		return 1
786  	}
787  	return 0
788  }
789
790  func (z nat) and(x, y nat) nat {
791  	m := len(x)
792  	n := len(y)
793  	if m > n {
794  		m = n
795  	}
796  	// m <= n
797
798  	z = z.make(m)
799  	for i := 0; i < m; i++ {
800  		z[i] = x[i] & y[i]
801  	}
802
803  	return z.norm()
804  }
805
806  func (z nat) andNot(x, y nat) nat {
807  	m := len(x)
808  	n := len(y)
809  	if n > m {
810  		n = m
811  	}
812  	// m >= n
813
814  	z = z.make(m)
815  	for i := 0; i < n; i++ {
816  		z[i] = x[i] &^ y[i]
817  	}
818  	copy(z[n:m], x[n:m])
819
820  	return z.norm()
821  }
822
823  func (z nat) or(x, y nat) nat {
824  	m := len(x)
825  	n := len(y)
826  	s := x
827  	if m < n {
828  		n, m = m, n
829  		s = y
830  	}
831  	// m >= n
832
833  	z = z.make(m)
834  	for i := 0; i < n; i++ {
835  		z[i] = x[i] | y[i]
836  	}
837  	copy(z[n:m], s[n:m])
838
839  	return z.norm()
840  }
841
842  func (z nat) xor(x, y nat) nat {
843  	m := len(x)
844  	n := len(y)
845  	s := x
846  	if m < n {
847  		n, m = m, n
848  		s = y
849  	}
850  	// m >= n
851
852  	z = z.make(m)
853  	for i := 0; i < n; i++ {
854  		z[i] = x[i] ^ y[i]
855  	}
856  	copy(z[n:m], s[n:m])
857
858  	return z.norm()
859  }
860
861  // random creates a random integer in [0..limit), using the space in z if
862  // possible. n is the bit length of limit.
863  func (z nat) random(rand *rand.Rand, limit nat, n int) nat {
864  	if alias(z, limit) {
865  		z = nil // z is an alias for limit - cannot reuse
866  	}
867  	z = z.make(len(limit))
868
869  	bitLengthOfMSW := uint(n % _W)
870  	if bitLengthOfMSW == 0 {
871  		bitLengthOfMSW = _W
872  	}
873  	mask := Word((1 << bitLengthOfMSW) - 1)
874
875  	for {
876  		switch _W {
877  		case 32:
878  			for i := range z {
879  				z[i] = Word(rand.Uint32())
880  			}
881  		case 64:
882  			for i := range z {
883  				z[i] = Word(rand.Uint32()) | Word(rand.Uint32())<<32
884  			}
885  		default:
886  			panic("unknown word size")
887  		}
889  		if z.cmp(limit) < 0 {
890  			break
891  		}
892  	}
893
894  	return z.norm()
895  }
896
897  // If m != 0 (i.e., len(m) != 0), expNN sets z to x**y mod m;
898  // otherwise it sets z to x**y. The result is the value of z.
899  func (z nat) expNN(x, y, m nat) nat {
900  	if alias(z, x) || alias(z, y) {
901  		// We cannot allow in-place modification of x or y.
902  		z = nil
903  	}
904
905  	// x**y mod 1 == 0
906  	if len(m) == 1 && m[0] == 1 {
907  		return z.setWord(0)
908  	}
909  	// m == 0 || m > 1
910
911  	// x**0 == 1
912  	if len(y) == 0 {
913  		return z.setWord(1)
914  	}
915  	// y > 0
916
917  	// x**1 mod m == x mod m
918  	if len(y) == 1 && y[0] == 1 && len(m) != 0 {
919  		_, z = nat(nil).div(z, x, m)
920  		return z
921  	}
922  	// y > 1
923
924  	if len(m) != 0 {
925  		// We likely end up being as long as the modulus.
926  		z = z.make(len(m))
927  	}
928  	z = z.set(x)
929
930  	// If the base is non-trivial and the exponent is large, we use
931  	// 4-bit, windowed exponentiation. This involves precomputing 14 values
932  	// (x^2...x^15) but then reduces the number of multiply-reduces by a
933  	// third. Even for a 32-bit exponent, this reduces the number of
934  	// operations. Uses Montgomery method for odd moduli.
935  	if x.cmp(natOne) > 0 && len(y) > 1 && len(m) > 0 {
936  		if m[0]&1 == 1 {
937  			return z.expNNMontgomery(x, y, m)
938  		}
939  		return z.expNNWindowed(x, y, m)
940  	}
941
942  	v := y[len(y)-1] // v > 0 because y is normalized and y > 0
943  	shift := nlz(v) + 1
944  	v <<= shift
945  	var q nat
946
947  	const mask = 1 << (_W - 1)
948
949  	// We walk through the bits of the exponent one by one. Each time we
950  	// see a bit, we square, thus doubling the power. If the bit is a one,
951  	// we also multiply by x, thus adding one to the power.
952
953  	w := _W - int(shift)
954  	// zz and r are used to avoid allocating in mul and div as
955  	// otherwise the arguments would alias.
956  	var zz, r nat
957  	for j := 0; j < w; j++ {
958  		zz = zz.sqr(z)
959  		zz, z = z, zz
960
961  		if v&mask != 0 {
962  			zz = zz.mul(z, x)
963  			zz, z = z, zz
964  		}
965
966  		if len(m) != 0 {
967  			zz, r = zz.div(r, z, m)
968  			zz, r, q, z = q, z, zz, r
969  		}
970
971  		v <<= 1
972  	}
973
974  	for i := len(y) - 2; i >= 0; i-- {
975  		v = y[i]
976
977  		for j := 0; j < _W; j++ {
978  			zz = zz.sqr(z)
979  			zz, z = z, zz
980
981  			if v&mask != 0 {
982  				zz = zz.mul(z, x)
983  				zz, z = z, zz
984  			}
985
986  			if len(m) != 0 {
987  				zz, r = zz.div(r, z, m)
988  				zz, r, q, z = q, z, zz, r
989  			}
990
991  			v <<= 1
992  		}
993  	}
994
995  	return z.norm()
996  }
997
998  // expNNWindowed calculates x**y mod m using a fixed, 4-bit window.
999  func (z nat) expNNWindowed(x, y, m nat) nat {
1000  	// zz and r are used to avoid allocating in mul and div as otherwise
1001  	// the arguments would alias.
1002  	var zz, r nat
1003
1004  	const n = 4
1005  	// powers[i] contains x^i.
1006  	var powers [1 << n]nat
1007  	powers[0] = natOne
1008  	powers[1] = x
1009  	for i := 2; i < 1<<n; i += 2 {
1010  		p2, p, p1 := &powers[i/2], &powers[i], &powers[i+1]
1011  		*p = p.sqr(*p2)
1012  		zz, r = zz.div(r, *p, m)
1013  		*p, r = r, *p
1014  		*p1 = p1.mul(*p, x)
1015  		zz, r = zz.div(r, *p1, m)
1016  		*p1, r = r, *p1
1017  	}
1018
1019  	z = z.setWord(1)
1020
1021  	for i := len(y) - 1; i >= 0; i-- {
1022  		yi := y[i]
1023  		for j := 0; j < _W; j += n {
1024  			if i != len(y)-1 || j != 0 {
1025  				// Unrolled loop for significant performance
1026  				// gain. Use go test -bench=".*" in crypto/rsa
1027  				// to check performance before making changes.
1028  				zz = zz.sqr(z)
1029  				zz, z = z, zz
1030  				zz, r = zz.div(r, z, m)
1031  				z, r = r, z
1032
1033  				zz = zz.sqr(z)
1034  				zz, z = z, zz
1035  				zz, r = zz.div(r, z, m)
1036  				z, r = r, z
1037
1038  				zz = zz.sqr(z)
1039  				zz, z = z, zz
1040  				zz, r = zz.div(r, z, m)
1041  				z, r = r, z
1042
1043  				zz = zz.sqr(z)
1044  				zz, z = z, zz
1045  				zz, r = zz.div(r, z, m)
1046  				z, r = r, z
1047  			}
1048
1049  			zz = zz.mul(z, powers[yi>>(_W-n)])
1050  			zz, z = z, zz
1051  			zz, r = zz.div(r, z, m)
1052  			z, r = r, z
1053
1054  			yi <<= n
1055  		}
1056  	}
1057
1058  	return z.norm()
1059  }
1060
1061  // expNNMontgomery calculates x**y mod m using a fixed, 4-bit window.
1062  // Uses Montgomery representation.
1063  func (z nat) expNNMontgomery(x, y, m nat) nat {
1064  	numWords := len(m)
1065
1066  	// We want the lengths of x and m to be equal.
1067  	// It is OK if x >= m as long as len(x) == len(m).
1068  	if len(x) > numWords {
1069  		_, x = nat(nil).div(nil, x, m)
1070  		// Note: now len(x) <= numWords, not guaranteed ==.
1071  	}
1072  	if len(x) < numWords {
1073  		rr := make(nat, numWords)
1074  		copy(rr, x)
1075  		x = rr
1076  	}
1077
1078  	// Ideally the precomputations would be performed outside, and reused
1079  	// k0 = -m**-1 mod 2**_W. Algorithm from: Dumas, J.G. "On Newton–Raphson
1080  	// Iteration for Multiplicative Inverses Modulo Prime Powers".
1081  	k0 := 2 - m[0]
1082  	t := m[0] - 1
1083  	for i := 1; i < _W; i <<= 1 {
1084  		t *= t
1085  		k0 *= (t + 1)
1086  	}
1087  	k0 = -k0
1088
1089  	// RR = 2**(2*_W*len(m)) mod m
1090  	RR := nat(nil).setWord(1)
1091  	zz := nat(nil).shl(RR, uint(2*numWords*_W))
1092  	_, RR = nat(nil).div(RR, zz, m)
1093  	if len(RR) < numWords {
1094  		zz = zz.make(numWords)
1095  		copy(zz, RR)
1096  		RR = zz
1097  	}
1098  	// one = 1, with equal length to that of m
1099  	one := make(nat, numWords)
1100  	one[0] = 1
1101
1102  	const n = 4
1103  	// powers[i] contains x^i
1104  	var powers [1 << n]nat
1105  	powers[0] = powers[0].montgomery(one, RR, m, k0, numWords)
1106  	powers[1] = powers[1].montgomery(x, RR, m, k0, numWords)
1107  	for i := 2; i < 1<<n; i++ {
1108  		powers[i] = powers[i].montgomery(powers[i-1], powers[1], m, k0, numWords)
1109  	}
1110
1111  	// initialize z = 1 (Montgomery 1)
1112  	z = z.make(numWords)
1113  	copy(z, powers[0])
1114
1115  	zz = zz.make(numWords)
1116
1117  	// same windowed exponent, but with Montgomery multiplications
1118  	for i := len(y) - 1; i >= 0; i-- {
1119  		yi := y[i]
1120  		for j := 0; j < _W; j += n {
1121  			if i != len(y)-1 || j != 0 {
1122  				zz = zz.montgomery(z, z, m, k0, numWords)
1123  				z = z.montgomery(zz, zz, m, k0, numWords)
1124  				zz = zz.montgomery(z, z, m, k0, numWords)
1125  				z = z.montgomery(zz, zz, m, k0, numWords)
1126  			}
1127  			zz = zz.montgomery(z, powers[yi>>(_W-n)], m, k0, numWords)
1128  			z, zz = zz, z
1129  			yi <<= n
1130  		}
1131  	}
1132  	// convert to regular number
1133  	zz = zz.montgomery(z, one, m, k0, numWords)
1134
1135  	// One last reduction, just in case.
1136  	// See golang.org/issue/13907.
1137  	if zz.cmp(m) >= 0 {
1138  		// Common case is m has high bit set; in that case,
1139  		// since zz is the same length as m, there can be just
1140  		// one multiple of m to remove. Just subtract.
1141  		// We think that the subtract should be sufficient in general,
1142  		// so do that unconditionally, but double-check,
1143  		// in case our beliefs are wrong.
1144  		// The div is not expected to be reached.
1145  		zz = zz.sub(zz, m)
1146  		if zz.cmp(m) >= 0 {
1147  			_, zz = nat(nil).div(nil, zz, m)
1148  		}
1149  	}
1150
1151  	return zz.norm()
1152  }
1153
1154  // bytes writes the value of z into buf using big-endian encoding.
1155  // The value of z is encoded in the slice buf[i:]. If the value of z
1156  // cannot be represented in buf, bytes panics. The number i of unused
1157  // bytes at the beginning of buf is returned as result.
1158  func (z nat) bytes(buf []byte) (i int) {
1159  	i = len(buf)
1160  	for _, d := range z {
1161  		for j := 0; j < _S; j++ {
1162  			i--
1163  			if i >= 0 {
1164  				buf[i] = byte(d)
1165  			} else if byte(d) != 0 {
1166  				panic("math/big: buffer too small to fit value")
1167  			}
1168  			d >>= 8
1169  		}
1170  	}
1171
1172  	if i < 0 {
1173  		i = 0
1174  	}
1175  	for i < len(buf) && buf[i] == 0 {
1176  		i++
1177  	}
1178
1179  	return
1180  }
1181
1182  // bigEndianWord returns the contents of buf interpreted as a big-endian encoded Word value.
1183  func bigEndianWord(buf []byte) Word {
1184  	if _W == 64 {
1185  		return Word(binary.BigEndian.Uint64(buf))
1186  	}
1187  	return Word(binary.BigEndian.Uint32(buf))
1188  }
1189
1190  // setBytes interprets buf as the bytes of a big-endian unsigned
1191  // integer, sets z to that value, and returns z.
1192  func (z nat) setBytes(buf []byte) nat {
1193  	z = z.make((len(buf) + _S - 1) / _S)
1194
1195  	i := len(buf)
1196  	for k := 0; i >= _S; k++ {
1197  		z[k] = bigEndianWord(buf[i-_S : i])
1198  		i -= _S
1199  	}
1200  	if i > 0 {
1201  		var d Word
1202  		for s := uint(0); i > 0; s += 8 {
1203  			d |= Word(buf[i-1]) << s
1204  			i--
1205  		}
1206  		z[len(z)-1] = d
1207  	}
1208
1209  	return z.norm()
1210  }
1211
1212  // sqrt sets z = ⌊√x⌋
1213  func (z nat) sqrt(x nat) nat {
1214  	if x.cmp(natOne) <= 0 {
1215  		return z.set(x)
1216  	}
1217  	if alias(z, x) {
1218  		z = nil
1219  	}
1220
1221  	// Start with value known to be too large and repeat "z = ⌊(z + ⌊x/z⌋)/2⌋" until it stops getting smaller.
1222  	// See Brent and Zimmermann, Modern Computer Arithmetic, Algorithm 1.13 (SqrtInt).
1223  	// https://members.loria.fr/PZimmermann/mca/pub226.html
1224  	// If x is one less than a perfect square, the sequence oscillates between the correct z and z+1;
1225  	// otherwise it converges to the correct z and stays there.
1226  	var z1, z2 nat
1227  	z1 = z
1228  	z1 = z1.setUint64(1)
1229  	z1 = z1.shl(z1, uint(x.bitLen()+1)/2) // must be ≥ √x
1230  	for n := 0; ; n++ {
1231  		z2, _ = z2.div(nil, x, z1)
1233  		z2 = z2.shr(z2, 1)
1234  		if z2.cmp(z1) >= 0 {