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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
     3  // license that can be found in the LICENSE file.
     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  	natTen = nat{10}
    41  )
    42  
    43  func (z nat) clear() {
    44  	for i := range z {
    45  		z[i] = 0
    46  	}
    47  }
    48  
    49  func (z nat) norm() nat {
    50  	i := len(z)
    51  	for i > 0 && z[i-1] == 0 {
    52  		i--
    53  	}
    54  	return z[0:i]
    55  }
    56  
    57  func (z nat) make(n int) nat {
    58  	if n <= cap(z) {
    59  		return z[:n] // reuse z
    60  	}
    61  	if n == 1 {
    62  		// Most nats start small and stay that way; don't over-allocate.
    63  		return make(nat, 1)
    64  	}
    65  	// Choosing a good value for e has significant performance impact
    66  	// because it increases the chance that a value can be reused.
    67  	const e = 4 // extra capacity
    68  	return make(nat, n, n+e)
    69  }
    70  
    71  func (z nat) setWord(x Word) nat {
    72  	if x == 0 {
    73  		return z[:0]
    74  	}
    75  	z = z.make(1)
    76  	z[0] = x
    77  	return z
    78  }
    79  
    80  func (z nat) setUint64(x uint64) nat {
    81  	// single-word value
    82  	if w := Word(x); uint64(w) == x {
    83  		return z.setWord(w)
    84  	}
    85  	// 2-word value
    86  	z = z.make(2)
    87  	z[1] = Word(x >> 32)
    88  	z[0] = Word(x)
    89  	return z
    90  }
    91  
    92  func (z nat) set(x nat) nat {
    93  	z = z.make(len(x))
    94  	copy(z, x)
    95  	return z
    96  }
    97  
    98  func (z nat) add(x, y nat) nat {
    99  	m := len(x)
   100  	n := len(y)
   101  
   102  	switch {
   103  	case m < n:
   104  		return z.add(y, x)
   105  	case m == 0:
   106  		// n == 0 because m >= n; result is 0
   107  		return z[:0]
   108  	case n == 0:
   109  		// result is x
   110  		return z.set(x)
   111  	}
   112  	// m > 0
   113  
   114  	z = z.make(m + 1)
   115  	c := addVV(z[0:n], x, y)
   116  	if m > n {
   117  		c = addVW(z[n:m], x[n:], c)
   118  	}
   119  	z[m] = c
   120  
   121  	return z.norm()
   122  }
   123  
   124  func (z nat) sub(x, y nat) nat {
   125  	m := len(x)
   126  	n := len(y)
   127  
   128  	switch {
   129  	case m < n:
   130  		panic("underflow")
   131  	case m == 0:
   132  		// n == 0 because m >= n; result is 0
   133  		return z[:0]
   134  	case n == 0:
   135  		// result is x
   136  		return z.set(x)
   137  	}
   138  	// m > 0
   139  
   140  	z = z.make(m)
   141  	c := subVV(z[0:n], x, y)
   142  	if m > n {
   143  		c = subVW(z[n:], x[n:], c)
   144  	}
   145  	if c != 0 {
   146  		panic("underflow")
   147  	}
   148  
   149  	return z.norm()
   150  }
   151  
   152  func (x nat) cmp(y nat) (r int) {
   153  	m := len(x)
   154  	n := len(y)
   155  	if m != n || m == 0 {
   156  		switch {
   157  		case m < n:
   158  			r = -1
   159  		case m > n:
   160  			r = 1
   161  		}
   162  		return
   163  	}
   164  
   165  	i := m - 1
   166  	for i > 0 && x[i] == y[i] {
   167  		i--
   168  	}
   169  
   170  	switch {
   171  	case x[i] < y[i]:
   172  		r = -1
   173  	case x[i] > y[i]:
   174  		r = 1
   175  	}
   176  	return
   177  }
   178  
   179  func (z nat) mulAddWW(x nat, y, r Word) nat {
   180  	m := len(x)
   181  	if m == 0 || y == 0 {
   182  		return z.setWord(r) // result is r
   183  	}
   184  	// m > 0
   185  
   186  	z = z.make(m + 1)
   187  	z[m] = mulAddVWW(z[0:m], x, y, r)
   188  
   189  	return z.norm()
   190  }
   191  
   192  // basicMul multiplies x and y and leaves the result in z.
   193  // The (non-normalized) result is placed in z[0 : len(x) + len(y)].
   194  func basicMul(z, x, y nat) {
   195  	z[0 : len(x)+len(y)].clear() // initialize z
   196  	for i, d := range y {
   197  		if d != 0 {
   198  			z[len(x)+i] = addMulVVW(z[i:i+len(x)], x, d)
   199  		}
   200  	}
   201  }
   202  
   203  // montgomery computes z mod m = x*y*2**(-n*_W) mod m,
   204  // assuming k = -1/m mod 2**_W.
   205  // z is used for storing the result which is returned;
   206  // z must not alias x, y or m.
   207  // See Gueron, "Efficient Software Implementations of Modular Exponentiation".
   208  // https://eprint.iacr.org/2011/239.pdf
   209  // In the terminology of that paper, this is an "Almost Montgomery Multiplication":
   210  // x and y are required to satisfy 0 <= z < 2**(n*_W) and then the result
   211  // z is guaranteed to satisfy 0 <= z < 2**(n*_W), but it may not be < m.
   212  func (z nat) montgomery(x, y, m nat, k Word, n int) nat {
   213  	// This code assumes x, y, m are all the same length, n.
   214  	// (required by addMulVVW and the for loop).
   215  	// It also assumes that x, y are already reduced mod m,
   216  	// or else the result will not be properly reduced.
   217  	if len(x) != n || len(y) != n || len(m) != n {
   218  		panic("math/big: mismatched montgomery number lengths")
   219  	}
   220  	z = z.make(n * 2)
   221  	z.clear()
   222  	var c Word
   223  	for i := 0; i < n; i++ {
   224  		d := y[i]
   225  		c2 := addMulVVW(z[i:n+i], x, d)
   226  		t := z[i] * k
   227  		c3 := addMulVVW(z[i:n+i], m, t)
   228  		cx := c + c2
   229  		cy := cx + c3
   230  		z[n+i] = cy
   231  		if cx < c2 || cy < c3 {
   232  			c = 1
   233  		} else {
   234  			c = 0
   235  		}
   236  	}
   237  	if c != 0 {
   238  		subVV(z[:n], z[n:], m)
   239  	} else {
   240  		copy(z[:n], z[n:])
   241  	}
   242  	return z[:n]
   243  }
   244  
   245  // Fast version of z[0:n+n>>1].add(z[0:n+n>>1], x[0:n]) w/o bounds checks.
   246  // Factored out for readability - do not use outside karatsuba.
   247  func karatsubaAdd(z, x nat, n int) {
   248  	if c := addVV(z[0:n], z, x); c != 0 {
   249  		addVW(z[n:n+n>>1], z[n:], c)
   250  	}
   251  }
   252  
   253  // Like karatsubaAdd, but does subtract.
   254  func karatsubaSub(z, x nat, n int) {
   255  	if c := subVV(z[0:n], z, x); c != 0 {
   256  		subVW(z[n:n+n>>1], z[n:], c)
   257  	}
   258  }
   259  
   260  // Operands that are shorter than karatsubaThreshold are multiplied using
   261  // "grade school" multiplication; for longer operands the Karatsuba algorithm
   262  // is used.
   263  var karatsubaThreshold = 40 // computed by calibrate_test.go
   264  
   265  // karatsuba multiplies x and y and leaves the result in z.
   266  // Both x and y must have the same length n and n must be a
   267  // power of 2. The result vector z must have len(z) >= 6*n.
   268  // The (non-normalized) result is placed in z[0 : 2*n].
   269  func karatsuba(z, x, y nat) {
   270  	n := len(y)
   271  
   272  	// Switch to basic multiplication if numbers are odd or small.
   273  	// (n is always even if karatsubaThreshold is even, but be
   274  	// conservative)
   275  	if n&1 != 0 || n < karatsubaThreshold || n < 2 {
   276  		basicMul(z, x, y)
   277  		return
   278  	}
   279  	// n&1 == 0 && n >= karatsubaThreshold && n >= 2
   280  
   281  	// Karatsuba multiplication is based on the observation that
   282  	// for two numbers x and y with:
   283  	//
   284  	//   x = x1*b + x0
   285  	//   y = y1*b + y0
   286  	//
   287  	// the product x*y can be obtained with 3 products z2, z1, z0
   288  	// instead of 4:
   289  	//
   290  	//   x*y = x1*y1*b*b + (x1*y0 + x0*y1)*b + x0*y0
   291  	//       =    z2*b*b +              z1*b +    z0
   292  	//
   293  	// with:
   294  	//
   295  	//   xd = x1 - x0
   296  	//   yd = y0 - y1
   297  	//
   298  	//   z1 =      xd*yd                    + z2 + z0
   299  	//      = (x1-x0)*(y0 - y1)             + z2 + z0
   300  	//      = x1*y0 - x1*y1 - x0*y0 + x0*y1 + z2 + z0
   301  	//      = x1*y0 -    z2 -    z0 + x0*y1 + z2 + z0
   302  	//      = x1*y0                 + x0*y1
   303  
   304  	// split x, y into "digits"
   305  	n2 := n >> 1              // n2 >= 1
   306  	x1, x0 := x[n2:], x[0:n2] // x = x1*b + y0
   307  	y1, y0 := y[n2:], y[0:n2] // y = y1*b + y0
   308  
   309  	// z is used for the result and temporary storage:
   310  	//
   311  	//   6*n     5*n     4*n     3*n     2*n     1*n     0*n
   312  	// z = [z2 copy|z0 copy| xd*yd | yd:xd | x1*y1 | x0*y0 ]
   313  	//
   314  	// For each recursive call of karatsuba, an unused slice of
   315  	// z is passed in that has (at least) half the length of the
   316  	// caller's z.
   317  
   318  	// compute z0 and z2 with the result "in place" in z
   319  	karatsuba(z, x0, y0)     // z0 = x0*y0
   320  	karatsuba(z[n:], x1, y1) // z2 = x1*y1
   321  
   322  	// compute xd (or the negative value if underflow occurs)
   323  	s := 1 // sign of product xd*yd
   324  	xd := z[2*n : 2*n+n2]
   325  	if subVV(xd, x1, x0) != 0 { // x1-x0
   326  		s = -s
   327  		subVV(xd, x0, x1) // x0-x1
   328  	}
   329  
   330  	// compute yd (or the negative value if underflow occurs)
   331  	yd := z[2*n+n2 : 3*n]
   332  	if subVV(yd, y0, y1) != 0 { // y0-y1
   333  		s = -s
   334  		subVV(yd, y1, y0) // y1-y0
   335  	}
   336  
   337  	// p = (x1-x0)*(y0-y1) == x1*y0 - x1*y1 - x0*y0 + x0*y1 for s > 0
   338  	// p = (x0-x1)*(y0-y1) == x0*y0 - x0*y1 - x1*y0 + x1*y1 for s < 0
   339  	p := z[n*3:]
   340  	karatsuba(p, xd, yd)
   341  
   342  	// save original z2:z0
   343  	// (ok to use upper half of z since we're done recursing)
   344  	r := z[n*4:]
   345  	copy(r, z[:n*2])
   346  
   347  	// add up all partial products
   348  	//
   349  	//   2*n     n     0
   350  	// z = [ z2  | z0  ]
   351  	//   +    [ z0  ]
   352  	//   +    [ z2  ]
   353  	//   +    [  p  ]
   354  	//
   355  	karatsubaAdd(z[n2:], r, n)
   356  	karatsubaAdd(z[n2:], r[n:], n)
   357  	if s > 0 {
   358  		karatsubaAdd(z[n2:], p, n)
   359  	} else {
   360  		karatsubaSub(z[n2:], p, n)
   361  	}
   362  }
   363  
   364  // alias reports whether x and y share the same base array.
   365  // Note: alias assumes that the capacity of underlying arrays
   366  //       is never changed for nat values; i.e. that there are
   367  //       no 3-operand slice expressions in this code (or worse,
   368  //       reflect-based operations to the same effect).
   369  func alias(x, y nat) bool {
   370  	return cap(x) > 0 && cap(y) > 0 && &x[0:cap(x)][cap(x)-1] == &y[0:cap(y)][cap(y)-1]
   371  }
   372  
   373  // addAt implements z += x<<(_W*i); z must be long enough.
   374  // (we don't use nat.add because we need z to stay the same
   375  // slice, and we don't need to normalize z after each addition)
   376  func addAt(z, x nat, i int) {
   377  	if n := len(x); n > 0 {
   378  		if c := addVV(z[i:i+n], z[i:], x); c != 0 {
   379  			j := i + n
   380  			if j < len(z) {
   381  				addVW(z[j:], z[j:], c)
   382  			}
   383  		}
   384  	}
   385  }
   386  
   387  func max(x, y int) int {
   388  	if x > y {
   389  		return x
   390  	}
   391  	return y
   392  }
   393  
   394  // karatsubaLen computes an approximation to the maximum k <= n such that
   395  // k = p<<i for a number p <= threshold and an i >= 0. Thus, the
   396  // result is the largest number that can be divided repeatedly by 2 before
   397  // becoming about the value of threshold.
   398  func karatsubaLen(n, threshold int) int {
   399  	i := uint(0)
   400  	for n > threshold {
   401  		n >>= 1
   402  		i++
   403  	}
   404  	return n << i
   405  }
   406  
   407  func (z nat) mul(x, y nat) nat {
   408  	m := len(x)
   409  	n := len(y)
   410  
   411  	switch {
   412  	case m < n:
   413  		return z.mul(y, x)
   414  	case m == 0 || n == 0:
   415  		return z[:0]
   416  	case n == 1:
   417  		return z.mulAddWW(x, y[0], 0)
   418  	}
   419  	// m >= n > 1
   420  
   421  	// determine if z can be reused
   422  	if alias(z, x) || alias(z, y) {
   423  		z = nil // z is an alias for x or y - cannot reuse
   424  	}
   425  
   426  	// use basic multiplication if the numbers are small
   427  	if n < karatsubaThreshold {
   428  		z = z.make(m + n)
   429  		basicMul(z, x, y)
   430  		return z.norm()
   431  	}
   432  	// m >= n && n >= karatsubaThreshold && n >= 2
   433  
   434  	// determine Karatsuba length k such that
   435  	//
   436  	//   x = xh*b + x0  (0 <= x0 < b)
   437  	//   y = yh*b + y0  (0 <= y0 < b)
   438  	//   b = 1<<(_W*k)  ("base" of digits xi, yi)
   439  	//
   440  	k := karatsubaLen(n, karatsubaThreshold)
   441  	// k <= n
   442  
   443  	// multiply x0 and y0 via Karatsuba
   444  	x0 := x[0:k]              // x0 is not normalized
   445  	y0 := y[0:k]              // y0 is not normalized
   446  	z = z.make(max(6*k, m+n)) // enough space for karatsuba of x0*y0 and full result of x*y
   447  	karatsuba(z, x0, y0)
   448  	z = z[0 : m+n]  // z has final length but may be incomplete
   449  	z[2*k:].clear() // upper portion of z is garbage (and 2*k <= m+n since k <= n <= m)
   450  
   451  	// If xh != 0 or yh != 0, add the missing terms to z. For
   452  	//
   453  	//   xh = xi*b^i + ... + x2*b^2 + x1*b (0 <= xi < b)
   454  	//   yh =                         y1*b (0 <= y1 < b)
   455  	//
   456  	// the missing terms are
   457  	//
   458  	//   x0*y1*b and xi*y0*b^i, xi*y1*b^(i+1) for i > 0
   459  	//
   460  	// since all the yi for i > 1 are 0 by choice of k: If any of them
   461  	// were > 0, then yh >= b^2 and thus y >= b^2. Then k' = k*2 would
   462  	// be a larger valid threshold contradicting the assumption about k.
   463  	//
   464  	if k < n || m != n {
   465  		var t nat
   466  
   467  		// add x0*y1*b
   468  		x0 := x0.norm()
   469  		y1 := y[k:]       // y1 is normalized because y is
   470  		t = t.mul(x0, y1) // update t so we don't lose t's underlying array
   471  		addAt(z, t, k)
   472  
   473  		// add xi*y0<<i, xi*y1*b<<(i+k)
   474  		y0 := y0.norm()
   475  		for i := k; i < len(x); i += k {
   476  			xi := x[i:]
   477  			if len(xi) > k {
   478  				xi = xi[:k]
   479  			}
   480  			xi = xi.norm()
   481  			t = t.mul(xi, y0)
   482  			addAt(z, t, i)
   483  			t = t.mul(xi, y1)
   484  			addAt(z, t, i+k)
   485  		}
   486  	}
   487  
   488  	return z.norm()
   489  }
   490  
   491  // basicSqr sets z = x*x and is asymptotically faster than basicMul
   492  // by about a factor of 2, but slower for small arguments due to overhead.
   493  // Requirements: len(x) > 0, len(z) == 2*len(x)
   494  // The (non-normalized) result is placed in z.
   495  func basicSqr(z, x nat) {
   496  	n := len(x)
   497  	t := make(nat, 2*n)            // temporary variable to hold the products
   498  	z[1], z[0] = mulWW(x[0], x[0]) // the initial square
   499  	for i := 1; i < n; i++ {
   500  		d := x[i]
   501  		// z collects the squares x[i] * x[i]
   502  		z[2*i+1], z[2*i] = mulWW(d, d)
   503  		// t collects the products x[i] * x[j] where j < i
   504  		t[2*i] = addMulVVW(t[i:2*i], x[0:i], d)
   505  	}
   506  	t[2*n-1] = shlVU(t[1:2*n-1], t[1:2*n-1], 1) // double the j < i products
   507  	addVV(z, z, t)                              // combine the result
   508  }
   509  
   510  // karatsubaSqr squares x and leaves the result in z.
   511  // len(x) must be a power of 2 and len(z) >= 6*len(x).
   512  // The (non-normalized) result is placed in z[0 : 2*len(x)].
   513  //
   514  // The algorithm and the layout of z are the same as for karatsuba.
   515  func karatsubaSqr(z, x nat) {
   516  	n := len(x)
   517  
   518  	if n&1 != 0 || n < karatsubaSqrThreshold || n < 2 {
   519  		basicSqr(z[:2*n], x)
   520  		return
   521  	}
   522  
   523  	n2 := n >> 1
   524  	x1, x0 := x[n2:], x[0:n2]
   525  
   526  	karatsubaSqr(z, x0)
   527  	karatsubaSqr(z[n:], x1)
   528  
   529  	// s = sign(xd*yd) == -1 for xd != 0; s == 1 for xd == 0
   530  	xd := z[2*n : 2*n+n2]
   531  	if subVV(xd, x1, x0) != 0 {
   532  		subVV(xd, x0, x1)
   533  	}
   534  
   535  	p := z[n*3:]
   536  	karatsubaSqr(p, xd)
   537  
   538  	r := z[n*4:]
   539  	copy(r, z[:n*2])
   540  
   541  	karatsubaAdd(z[n2:], r, n)
   542  	karatsubaAdd(z[n2:], r[n:], n)
   543  	karatsubaSub(z[n2:], p, n) // s == -1 for p != 0; s == 1 for p == 0
   544  }
   545  
   546  // Operands that are shorter than basicSqrThreshold are squared using
   547  // "grade school" multiplication; for operands longer than karatsubaSqrThreshold
   548  // we use the Karatsuba algorithm optimized for x == y.
   549  var basicSqrThreshold = 20      // computed by calibrate_test.go
   550  var karatsubaSqrThreshold = 260 // computed by calibrate_test.go
   551  
   552  // z = x*x
   553  func (z nat) sqr(x nat) nat {
   554  	n := len(x)
   555  	switch {
   556  	case n == 0:
   557  		return z[:0]
   558  	case n == 1:
   559  		d := x[0]
   560  		z = z.make(2)
   561  		z[1], z[0] = mulWW(d, d)
   562  		return z.norm()
   563  	}
   564  
   565  	if alias(z, x) {
   566  		z = nil // z is an alias for x - cannot reuse
   567  	}
   568  
   569  	if n < basicSqrThreshold {
   570  		z = z.make(2 * n)
   571  		basicMul(z, x, x)
   572  		return z.norm()
   573  	}
   574  	if n < karatsubaSqrThreshold {
   575  		z = z.make(2 * n)
   576  		basicSqr(z, x)
   577  		return z.norm()
   578  	}
   579  
   580  	// Use Karatsuba multiplication optimized for x == y.
   581  	// The algorithm and layout of z are the same as for mul.
   582  
   583  	// z = (x1*b + x0)^2 = x1^2*b^2 + 2*x1*x0*b + x0^2
   584  
   585  	k := karatsubaLen(n, karatsubaSqrThreshold)
   586  
   587  	x0 := x[0:k]
   588  	z = z.make(max(6*k, 2*n))
   589  	karatsubaSqr(z, x0) // z = x0^2
   590  	z = z[0 : 2*n]
   591  	z[2*k:].clear()
   592  
   593  	if k < n {
   594  		var t nat
   595  		x0 := x0.norm()
   596  		x1 := x[k:]
   597  		t = t.mul(x0, x1)
   598  		addAt(z, t, k)
   599  		addAt(z, t, k) // z = 2*x1*x0*b + x0^2
   600  		t = t.sqr(x1)
   601  		addAt(z, t, 2*k) // z = x1^2*b^2 + 2*x1*x0*b + x0^2
   602  	}
   603  
   604  	return z.norm()
   605  }
   606  
   607  // mulRange computes the product of all the unsigned integers in the
   608  // range [a, b] inclusively. If a > b (empty range), the result is 1.
   609  func (z nat) mulRange(a, b uint64) nat {
   610  	switch {
   611  	case a == 0:
   612  		// cut long ranges short (optimization)
   613  		return z.setUint64(0)
   614  	case a > b:
   615  		return z.setUint64(1)
   616  	case a == b:
   617  		return z.setUint64(a)
   618  	case a+1 == b:
   619  		return z.mul(nat(nil).setUint64(a), nat(nil).setUint64(b))
   620  	}
   621  	m := (a + b) / 2
   622  	return z.mul(nat(nil).mulRange(a, m), nat(nil).mulRange(m+1, b))
   623  }
   624  
   625  // q = (x-r)/y, with 0 <= r < y
   626  func (z nat) divW(x nat, y Word) (q nat, r Word) {
   627  	m := len(x)
   628  	switch {
   629  	case y == 0:
   630  		panic("division by zero")
   631  	case y == 1:
   632  		q = z.set(x) // result is x
   633  		return
   634  	case m == 0:
   635  		q = z[:0] // result is 0
   636  		return
   637  	}
   638  	// m > 0
   639  	z = z.make(m)
   640  	r = divWVW(z, 0, x, y)
   641  	q = z.norm()
   642  	return
   643  }
   644  
   645  func (z nat) div(z2, u, v nat) (q, r nat) {
   646  	if len(v) == 0 {
   647  		panic("division by zero")
   648  	}
   649  
   650  	if u.cmp(v) < 0 {
   651  		q = z[:0]
   652  		r = z2.set(u)
   653  		return
   654  	}
   655  
   656  	if len(v) == 1 {
   657  		var r2 Word
   658  		q, r2 = z.divW(u, v[0])
   659  		r = z2.setWord(r2)
   660  		return
   661  	}
   662  
   663  	q, r = z.divLarge(z2, u, v)
   664  	return
   665  }
   666  
   667  // getNat returns a *nat of len n. The contents may not be zero.
   668  // The pool holds *nat to avoid allocation when converting to interface{}.
   669  func getNat(n int) *nat {
   670  	var z *nat
   671  	if v := natPool.Get(); v != nil {
   672  		z = v.(*nat)
   673  	}
   674  	if z == nil {
   675  		z = new(nat)
   676  	}
   677  	*z = z.make(n)
   678  	return z
   679  }
   680  
   681  func putNat(x *nat) {
   682  	natPool.Put(x)
   683  }
   684  
   685  var natPool sync.Pool
   686  
   687  // q = (uIn-r)/vIn, with 0 <= r < y
   688  // Uses z as storage for q, and u as storage for r if possible.
   689  // See Knuth, Volume 2, section 4.3.1, Algorithm D.
   690  // Preconditions:
   691  //    len(vIn) >= 2
   692  //    len(uIn) >= len(vIn)
   693  //    u must not alias z
   694  func (z nat) divLarge(u, uIn, vIn nat) (q, r nat) {
   695  	n := len(vIn)
   696  	m := len(uIn) - n
   697  
   698  	// D1.
   699  	shift := nlz(vIn[n-1])
   700  	// do not modify vIn, it may be used by another goroutine simultaneously
   701  	vp := getNat(n)
   702  	v := *vp
   703  	shlVU(v, vIn, shift)
   704  
   705  	// u may safely alias uIn or vIn, the value of uIn is used to set u and vIn was already used
   706  	u = u.make(len(uIn) + 1)
   707  	u[len(uIn)] = shlVU(u[0:len(uIn)], uIn, shift)
   708  
   709  	// z may safely alias uIn or vIn, both values were used already
   710  	if alias(z, u) {
   711  		z = nil // z is an alias for u - cannot reuse
   712  	}
   713  	q = z.make(m + 1)
   714  
   715  	qhatvp := getNat(n + 1)
   716  	qhatv := *qhatvp
   717  
   718  	// D2.
   719  	vn1 := v[n-1]
   720  	for j := m; j >= 0; j-- {
   721  		// D3.
   722  		qhat := Word(_M)
   723  		if ujn := u[j+n]; ujn != vn1 {
   724  			var rhat Word
   725  			qhat, rhat = divWW(ujn, u[j+n-1], vn1)
   726  
   727  			// x1 | x2 = q̂v_{n-2}
   728  			vn2 := v[n-2]
   729  			x1, x2 := mulWW(qhat, vn2)
   730  			// test if q̂v_{n-2} > br̂ + u_{j+n-2}
   731  			ujn2 := u[j+n-2]
   732  			for greaterThan(x1, x2, rhat, ujn2) {
   733  				qhat--
   734  				prevRhat := rhat
   735  				rhat += vn1
   736  				// v[n-1] >= 0, so this tests for overflow.
   737  				if rhat < prevRhat {
   738  					break
   739  				}
   740  				x1, x2 = mulWW(qhat, vn2)
   741  			}
   742  		}
   743  
   744  		// D4.
   745  		qhatv[n] = mulAddVWW(qhatv[0:n], v, qhat, 0)
   746  
   747  		c := subVV(u[j:j+len(qhatv)], u[j:], qhatv)
   748  		if c != 0 {
   749  			c := addVV(u[j:j+n], u[j:], v)
   750  			u[j+n] += c
   751  			qhat--
   752  		}
   753  
   754  		q[j] = qhat
   755  	}
   756  
   757  	putNat(vp)
   758  	putNat(qhatvp)
   759  
   760  	q = q.norm()
   761  	shrVU(u, u, shift)
   762  	r = u.norm()
   763  
   764  	return q, r
   765  }
   766  
   767  // Length of x in bits. x must be normalized.
   768  func (x nat) bitLen() int {
   769  	if i := len(x) - 1; i >= 0 {
   770  		return i*_W + bits.Len(uint(x[i]))
   771  	}
   772  	return 0
   773  }
   774  
   775  // trailingZeroBits returns the number of consecutive least significant zero
   776  // bits of x.
   777  func (x nat) trailingZeroBits() uint {
   778  	if len(x) == 0 {
   779  		return 0
   780  	}
   781  	var i uint
   782  	for x[i] == 0 {
   783  		i++
   784  	}
   785  	// x[i] != 0
   786  	return i*_W + uint(bits.TrailingZeros(uint(x[i])))
   787  }
   788  
   789  func same(x, y nat) bool {
   790  	return len(x) == len(y) && len(x) > 0 && &x[0] == &y[0]
   791  }
   792  
   793  // z = x << s
   794  func (z nat) shl(x nat, s uint) nat {
   795  	if s == 0 {
   796  		if same(z, x) {
   797  			return z
   798  		}
   799  		if !alias(z, x) {
   800  			return z.set(x)
   801  		}
   802  	}
   803  
   804  	m := len(x)
   805  	if m == 0 {
   806  		return z[:0]
   807  	}
   808  	// m > 0
   809  
   810  	n := m + int(s/_W)
   811  	z = z.make(n + 1)
   812  	z[n] = shlVU(z[n-m:n], x, s%_W)
   813  	z[0 : n-m].clear()
   814  
   815  	return z.norm()
   816  }
   817  
   818  // z = x >> s
   819  func (z nat) shr(x nat, s uint) nat {
   820  	if s == 0 {
   821  		if same(z, x) {
   822  			return z
   823  		}
   824  		if !alias(z, x) {
   825  			return z.set(x)
   826  		}
   827  	}
   828  
   829  	m := len(x)
   830  	n := m - int(s/_W)
   831  	if n <= 0 {
   832  		return z[:0]
   833  	}
   834  	// n > 0
   835  
   836  	z = z.make(n)
   837  	shrVU(z, x[m-n:], s%_W)
   838  
   839  	return z.norm()
   840  }
   841  
   842  func (z nat) setBit(x nat, i uint, b uint) nat {
   843  	j := int(i / _W)
   844  	m := Word(1) << (i % _W)
   845  	n := len(x)
   846  	switch b {
   847  	case 0:
   848  		z = z.make(n)
   849  		copy(z, x)
   850  		if j >= n {
   851  			// no need to grow
   852  			return z
   853  		}
   854  		z[j] &^= m
   855  		return z.norm()
   856  	case 1:
   857  		if j >= n {
   858  			z = z.make(j + 1)
   859  			z[n:].clear()
   860  		} else {
   861  			z = z.make(n)
   862  		}
   863  		copy(z, x)
   864  		z[j] |= m
   865  		// no need to normalize
   866  		return z
   867  	}
   868  	panic("set bit is not 0 or 1")
   869  }
   870  
   871  // bit returns the value of the i'th bit, with lsb == bit 0.
   872  func (x nat) bit(i uint) uint {
   873  	j := i / _W
   874  	if j >= uint(len(x)) {
   875  		return 0
   876  	}
   877  	// 0 <= j < len(x)
   878  	return uint(x[j] >> (i % _W) & 1)
   879  }
   880  
   881  // sticky returns 1 if there's a 1 bit within the
   882  // i least significant bits, otherwise it returns 0.
   883  func (x nat) sticky(i uint) uint {
   884  	j := i / _W
   885  	if j >= uint(len(x)) {
   886  		if len(x) == 0 {
   887  			return 0
   888  		}
   889  		return 1
   890  	}
   891  	// 0 <= j < len(x)
   892  	for _, x := range x[:j] {
   893  		if x != 0 {
   894  			return 1
   895  		}
   896  	}
   897  	if x[j]<<(_W-i%_W) != 0 {
   898  		return 1
   899  	}
   900  	return 0
   901  }
   902  
   903  func (z nat) and(x, y nat) nat {
   904  	m := len(x)
   905  	n := len(y)
   906  	if m > n {
   907  		m = n
   908  	}
   909  	// m <= n
   910  
   911  	z = z.make(m)
   912  	for i := 0; i < m; i++ {
   913  		z[i] = x[i] & y[i]
   914  	}
   915  
   916  	return z.norm()
   917  }
   918  
   919  func (z nat) andNot(x, y nat) nat {
   920  	m := len(x)
   921  	n := len(y)
   922  	if n > m {
   923  		n = m
   924  	}
   925  	// m >= n
   926  
   927  	z = z.make(m)
   928  	for i := 0; i < n; i++ {
   929  		z[i] = x[i] &^ y[i]
   930  	}
   931  	copy(z[n:m], x[n:m])
   932  
   933  	return z.norm()
   934  }
   935  
   936  func (z nat) or(x, y nat) nat {
   937  	m := len(x)
   938  	n := len(y)
   939  	s := x
   940  	if m < n {
   941  		n, m = m, n
   942  		s = y
   943  	}
   944  	// m >= n
   945  
   946  	z = z.make(m)
   947  	for i := 0; i < n; i++ {
   948  		z[i] = x[i] | y[i]
   949  	}
   950  	copy(z[n:m], s[n:m])
   951  
   952  	return z.norm()
   953  }
   954  
   955  func (z nat) xor(x, y nat) nat {
   956  	m := len(x)
   957  	n := len(y)
   958  	s := x
   959  	if m < n {
   960  		n, m = m, n
   961  		s = y
   962  	}
   963  	// m >= n
   964  
   965  	z = z.make(m)
   966  	for i := 0; i < n; i++ {
   967  		z[i] = x[i] ^ y[i]
   968  	}
   969  	copy(z[n:m], s[n:m])
   970  
   971  	return z.norm()
   972  }
   973  
   974  // greaterThan reports whether (x1<<_W + x2) > (y1<<_W + y2)
   975  func greaterThan(x1, x2, y1, y2 Word) bool {
   976  	return x1 > y1 || x1 == y1 && x2 > y2
   977  }
   978  
   979  // modW returns x % d.
   980  func (x nat) modW(d Word) (r Word) {
   981  	// TODO(agl): we don't actually need to store the q value.
   982  	var q nat
   983  	q = q.make(len(x))
   984  	return divWVW(q, 0, x, d)
   985  }
   986  
   987  // random creates a random integer in [0..limit), using the space in z if
   988  // possible. n is the bit length of limit.
   989  func (z nat) random(rand *rand.Rand, limit nat, n int) nat {
   990  	if alias(z, limit) {
   991  		z = nil // z is an alias for limit - cannot reuse
   992  	}
   993  	z = z.make(len(limit))
   994  
   995  	bitLengthOfMSW := uint(n % _W)
   996  	if bitLengthOfMSW == 0 {
   997  		bitLengthOfMSW = _W
   998  	}
   999  	mask := Word((1 << bitLengthOfMSW) - 1)
  1000  
  1001  	for {
  1002  		switch _W {
  1003  		case 32:
  1004  			for i := range z {
  1005  				z[i] = Word(rand.Uint32())
  1006  			}
  1007  		case 64:
  1008  			for i := range z {
  1009  				z[i] = Word(rand.Uint32()) | Word(rand.Uint32())<<32
  1010  			}
  1011  		default:
  1012  			panic("unknown word size")
  1013  		}
  1014  		z[len(limit)-1] &= mask
  1015  		if z.cmp(limit) < 0 {
  1016  			break
  1017  		}
  1018  	}
  1019  
  1020  	return z.norm()
  1021  }
  1022  
  1023  // If m != 0 (i.e., len(m) != 0), expNN sets z to x**y mod m;
  1024  // otherwise it sets z to x**y. The result is the value of z.
  1025  func (z nat) expNN(x, y, m nat) nat {
  1026  	if alias(z, x) || alias(z, y) {
  1027  		// We cannot allow in-place modification of x or y.
  1028  		z = nil
  1029  	}
  1030  
  1031  	// x**y mod 1 == 0
  1032  	if len(m) == 1 && m[0] == 1 {
  1033  		return z.setWord(0)
  1034  	}
  1035  	// m == 0 || m > 1
  1036  
  1037  	// x**0 == 1
  1038  	if len(y) == 0 {
  1039  		return z.setWord(1)
  1040  	}
  1041  	// y > 0
  1042  
  1043  	// x**1 mod m == x mod m
  1044  	if len(y) == 1 && y[0] == 1 && len(m) != 0 {
  1045  		_, z = nat(nil).div(z, x, m)
  1046  		return z
  1047  	}
  1048  	// y > 1
  1049  
  1050  	if len(m) != 0 {
  1051  		// We likely end up being as long as the modulus.
  1052  		z = z.make(len(m))
  1053  	}
  1054  	z = z.set(x)
  1055  
  1056  	// If the base is non-trivial and the exponent is large, we use
  1057  	// 4-bit, windowed exponentiation. This involves precomputing 14 values
  1058  	// (x^2...x^15) but then reduces the number of multiply-reduces by a
  1059  	// third. Even for a 32-bit exponent, this reduces the number of
  1060  	// operations. Uses Montgomery method for odd moduli.
  1061  	if x.cmp(natOne) > 0 && len(y) > 1 && len(m) > 0 {
  1062  		if m[0]&1 == 1 {
  1063  			return z.expNNMontgomery(x, y, m)
  1064  		}
  1065  		return z.expNNWindowed(x, y, m)
  1066  	}
  1067  
  1068  	v := y[len(y)-1] // v > 0 because y is normalized and y > 0
  1069  	shift := nlz(v) + 1
  1070  	v <<= shift
  1071  	var q nat
  1072  
  1073  	const mask = 1 << (_W - 1)
  1074  
  1075  	// We walk through the bits of the exponent one by one. Each time we
  1076  	// see a bit, we square, thus doubling the power. If the bit is a one,
  1077  	// we also multiply by x, thus adding one to the power.
  1078  
  1079  	w := _W - int(shift)
  1080  	// zz and r are used to avoid allocating in mul and div as
  1081  	// otherwise the arguments would alias.
  1082  	var zz, r nat
  1083  	for j := 0; j < w; j++ {
  1084  		zz = zz.sqr(z)
  1085  		zz, z = z, zz
  1086  
  1087  		if v&mask != 0 {
  1088  			zz = zz.mul(z, x)
  1089  			zz, z = z, zz
  1090  		}
  1091  
  1092  		if len(m) != 0 {
  1093  			zz, r = zz.div(r, z, m)
  1094  			zz, r, q, z = q, z, zz, r
  1095  		}
  1096  
  1097  		v <<= 1
  1098  	}
  1099  
  1100  	for i := len(y) - 2; i >= 0; i-- {
  1101  		v = y[i]
  1102  
  1103  		for j := 0; j < _W; j++ {
  1104  			zz = zz.sqr(z)
  1105  			zz, z = z, zz
  1106  
  1107  			if v&mask != 0 {
  1108  				zz = zz.mul(z, x)
  1109  				zz, z = z, zz
  1110  			}
  1111  
  1112  			if len(m) != 0 {
  1113  				zz, r = zz.div(r, z, m)
  1114  				zz, r, q, z = q, z, zz, r
  1115  			}
  1116  
  1117  			v <<= 1
  1118  		}
  1119  	}
  1120  
  1121  	return z.norm()
  1122  }
  1123  
  1124  // expNNWindowed calculates x**y mod m using a fixed, 4-bit window.
  1125  func (z nat) expNNWindowed(x, y, m nat) nat {
  1126  	// zz and r are used to avoid allocating in mul and div as otherwise
  1127  	// the arguments would alias.
  1128  	var zz, r nat
  1129  
  1130  	const n = 4
  1131  	// powers[i] contains x^i.
  1132  	var powers [1 << n]nat
  1133  	powers[0] = natOne
  1134  	powers[1] = x
  1135  	for i := 2; i < 1<<n; i += 2 {
  1136  		p2, p, p1 := &powers[i/2], &powers[i], &powers[i+1]
  1137  		*p = p.sqr(*p2)
  1138  		zz, r = zz.div(r, *p, m)
  1139  		*p, r = r, *p
  1140  		*p1 = p1.mul(*p, x)
  1141  		zz, r = zz.div(r, *p1, m)
  1142  		*p1, r = r, *p1
  1143  	}
  1144  
  1145  	z = z.setWord(1)
  1146  
  1147  	for i := len(y) - 1; i >= 0; i-- {
  1148  		yi := y[i]
  1149  		for j := 0; j < _W; j += n {
  1150  			if i != len(y)-1 || j != 0 {
  1151  				// Unrolled loop for significant performance
  1152  				// gain. Use go test -bench=".*" in crypto/rsa
  1153  				// to check performance before making changes.
  1154  				zz = zz.sqr(z)
  1155  				zz, z = z, zz
  1156  				zz, r = zz.div(r, z, m)
  1157  				z, r = r, z
  1158  
  1159  				zz = zz.sqr(z)
  1160  				zz, z = z, zz
  1161  				zz, r = zz.div(r, z, m)
  1162  				z, r = r, z
  1163  
  1164  				zz = zz.sqr(z)
  1165  				zz, z = z, zz
  1166  				zz, r = zz.div(r, z, m)
  1167  				z, r = r, z
  1168  
  1169  				zz = zz.sqr(z)
  1170  				zz, z = z, zz
  1171  				zz, r = zz.div(r, z, m)
  1172  				z, r = r, z
  1173  			}
  1174  
  1175  			zz = zz.mul(z, powers[yi>>(_W-n)])
  1176  			zz, z = z, zz
  1177  			zz, r = zz.div(r, z, m)
  1178  			z, r = r, z
  1179  
  1180  			yi <<= n
  1181  		}
  1182  	}
  1183  
  1184  	return z.norm()
  1185  }
  1186  
  1187  // expNNMontgomery calculates x**y mod m using a fixed, 4-bit window.
  1188  // Uses Montgomery representation.
  1189  func (z nat) expNNMontgomery(x, y, m nat) nat {
  1190  	numWords := len(m)
  1191  
  1192  	// We want the lengths of x and m to be equal.
  1193  	// It is OK if x >= m as long as len(x) == len(m).
  1194  	if len(x) > numWords {
  1195  		_, x = nat(nil).div(nil, x, m)
  1196  		// Note: now len(x) <= numWords, not guaranteed ==.
  1197  	}
  1198  	if len(x) < numWords {
  1199  		rr := make(nat, numWords)
  1200  		copy(rr, x)
  1201  		x = rr
  1202  	}
  1203  
  1204  	// Ideally the precomputations would be performed outside, and reused
  1205  	// k0 = -m**-1 mod 2**_W. Algorithm from: Dumas, J.G. "On Newton–Raphson
  1206  	// Iteration for Multiplicative Inverses Modulo Prime Powers".
  1207  	k0 := 2 - m[0]
  1208  	t := m[0] - 1
  1209  	for i := 1; i < _W; i <<= 1 {
  1210  		t *= t
  1211  		k0 *= (t + 1)
  1212  	}
  1213  	k0 = -k0
  1214  
  1215  	// RR = 2**(2*_W*len(m)) mod m
  1216  	RR := nat(nil).setWord(1)
  1217  	zz := nat(nil).shl(RR, uint(2*numWords*_W))
  1218  	_, RR = nat(nil).div(RR, zz, m)
  1219  	if len(RR) < numWords {
  1220  		zz = zz.make(numWords)
  1221  		copy(zz, RR)
  1222  		RR = zz
  1223  	}
  1224  	// one = 1, with equal length to that of m
  1225  	one := make(nat, numWords)
  1226  	one[0] = 1
  1227  
  1228  	const n = 4
  1229  	// powers[i] contains x^i
  1230  	var powers [1 << n]nat
  1231  	powers[0] = powers[0].montgomery(one, RR, m, k0, numWords)
  1232  	powers[1] = powers[1].montgomery(x, RR, m, k0, numWords)
  1233  	for i := 2; i < 1<<n; i++ {
  1234  		powers[i] = powers[i].montgomery(powers[i-1], powers[1], m, k0, numWords)
  1235  	}
  1236  
  1237  	// initialize z = 1 (Montgomery 1)
  1238  	z = z.make(numWords)
  1239  	copy(z, powers[0])
  1240  
  1241  	zz = zz.make(numWords)
  1242  
  1243  	// same windowed exponent, but with Montgomery multiplications
  1244  	for i := len(y) - 1; i >= 0; i-- {
  1245  		yi := y[i]
  1246  		for j := 0; j < _W; j += n {
  1247  			if i != len(y)-1 || j != 0 {
  1248  				zz = zz.montgomery(z, z, m, k0, numWords)
  1249  				z = z.montgomery(zz, zz, m, k0, numWords)
  1250  				zz = zz.montgomery(z, z, m, k0, numWords)
  1251  				z = z.montgomery(zz, zz, m, k0, numWords)
  1252  			}
  1253  			zz = zz.montgomery(z, powers[yi>>(_W-n)], m, k0, numWords)
  1254  			z, zz = zz, z
  1255  			yi <<= n
  1256  		}
  1257  	}
  1258  	// convert to regular number
  1259  	zz = zz.montgomery(z, one, m, k0, numWords)
  1260  
  1261  	// One last reduction, just in case.
  1262  	// See golang.org/issue/13907.
  1263  	if zz.cmp(m) >= 0 {
  1264  		// Common case is m has high bit set; in that case,
  1265  		// since zz is the same length as m, there can be just
  1266  		// one multiple of m to remove. Just subtract.
  1267  		// We think that the subtract should be sufficient in general,
  1268  		// so do that unconditionally, but double-check,
  1269  		// in case our beliefs are wrong.
  1270  		// The div is not expected to be reached.
  1271  		zz = zz.sub(zz, m)
  1272  		if zz.cmp(m) >= 0 {
  1273  			_, zz = nat(nil).div(nil, zz, m)
  1274  		}
  1275  	}
  1276  
  1277  	return zz.norm()
  1278  }
  1279  
  1280  // bytes writes the value of z into buf using big-endian encoding.
  1281  // len(buf) must be >= len(z)*_S. The value of z is encoded in the
  1282  // slice buf[i:]. The number i of unused bytes at the beginning of
  1283  // buf is returned as result.
  1284  func (z nat) bytes(buf []byte) (i int) {
  1285  	i = len(buf)
  1286  	for _, d := range z {
  1287  		for j := 0; j < _S; j++ {
  1288  			i--
  1289  			buf[i] = byte(d)
  1290  			d >>= 8
  1291  		}
  1292  	}
  1293  
  1294  	for i < len(buf) && buf[i] == 0 {
  1295  		i++
  1296  	}
  1297  
  1298  	return
  1299  }
  1300  
  1301  // bigEndianWord returns the contents of buf interpreted as a big-endian encoded Word value.
  1302  func bigEndianWord(buf []byte) Word {
  1303  	if _W == 64 {
  1304  		return Word(binary.BigEndian.Uint64(buf))
  1305  	}
  1306  	return Word(binary.BigEndian.Uint32(buf))
  1307  }
  1308  
  1309  // setBytes interprets buf as the bytes of a big-endian unsigned
  1310  // integer, sets z to that value, and returns z.
  1311  func (z nat) setBytes(buf []byte) nat {
  1312  	z = z.make((len(buf) + _S - 1) / _S)
  1313  
  1314  	i := len(buf)
  1315  	for k := 0; i >= _S; k++ {
  1316  		z[k] = bigEndianWord(buf[i-_S : i])
  1317  		i -= _S
  1318  	}
  1319  	if i > 0 {
  1320  		var d Word
  1321  		for s := uint(0); i > 0; s += 8 {
  1322  			d |= Word(buf[i-1]) << s
  1323  			i--
  1324  		}
  1325  		z[len(z)-1] = d
  1326  	}
  1327  
  1328  	return z.norm()
  1329  }
  1330  
  1331  // sqrt sets z = ⌊√x⌋
  1332  func (z nat) sqrt(x nat) nat {
  1333  	if x.cmp(natOne) <= 0 {
  1334  		return z.set(x)
  1335  	}
  1336  	if alias(z, x) {
  1337  		z = nil
  1338  	}
  1339  
  1340  	// Start with value known to be too large and repeat "z = ⌊(z + ⌊x/z⌋)/2⌋" until it stops getting smaller.
  1341  	// See Brent and Zimmermann, Modern Computer Arithmetic, Algorithm 1.13 (SqrtInt).
  1342  	// https://members.loria.fr/PZimmermann/mca/pub226.html
  1343  	// If x is one less than a perfect square, the sequence oscillates between the correct z and z+1;
  1344  	// otherwise it converges to the correct z and stays there.
  1345  	var z1, z2 nat
  1346  	z1 = z
  1347  	z1 = z1.setUint64(1)
  1348  	z1 = z1.shl(z1, uint(x.bitLen()/2+1)) // must be ≥ √x
  1349  	for n := 0; ; n++ {
  1350  		z2, _ = z2.div(nil, x, z1)
  1351  		z2 = z2.add(z2, z1)
  1352  		z2 = z2.shr(z2, 1)
  1353  		if z2.cmp(z1) >= 0 {
  1354  			// z1 is answer.
  1355  			// Figure out whether z1 or z2 is currently aliased to z by looking at loop count.
  1356  			if n&1 == 0 {
  1357  				return z1
  1358  			}
  1359  			return z.set(z1)
  1360  		}
  1361  		z1, z2 = z2, z1
  1362  	}
  1363  }
  1364  

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