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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  	// Choosing a good value for e has significant performance impact
    62  	// because it increases the chance that a value can be reused.
    63  	const e = 4 // extra capacity
    64  	return make(nat, n, n+e)
    65  }
    66  
    67  func (z nat) setWord(x Word) nat {
    68  	if x == 0 {
    69  		return z[:0]
    70  	}
    71  	z = z.make(1)
    72  	z[0] = x
    73  	return z
    74  }
    75  
    76  func (z nat) setUint64(x uint64) nat {
    77  	// single-word value
    78  	if w := Word(x); uint64(w) == x {
    79  		return z.setWord(w)
    80  	}
    81  	// 2-word value
    82  	z = z.make(2)
    83  	z[1] = Word(x >> 32)
    84  	z[0] = Word(x)
    85  	return z
    86  }
    87  
    88  func (z nat) set(x nat) nat {
    89  	z = z.make(len(x))
    90  	copy(z, x)
    91  	return z
    92  }
    93  
    94  func (z nat) add(x, y nat) nat {
    95  	m := len(x)
    96  	n := len(y)
    97  
    98  	switch {
    99  	case m < n:
   100  		return z.add(y, x)
   101  	case m == 0:
   102  		// n == 0 because m >= n; result is 0
   103  		return z[:0]
   104  	case n == 0:
   105  		// result is x
   106  		return z.set(x)
   107  	}
   108  	// m > 0
   109  
   110  	z = z.make(m + 1)
   111  	c := addVV(z[0:n], x, y)
   112  	if m > n {
   113  		c = addVW(z[n:m], x[n:], c)
   114  	}
   115  	z[m] = c
   116  
   117  	return z.norm()
   118  }
   119  
   120  func (z nat) sub(x, y nat) nat {
   121  	m := len(x)
   122  	n := len(y)
   123  
   124  	switch {
   125  	case m < n:
   126  		panic("underflow")
   127  	case m == 0:
   128  		// n == 0 because m >= n; result is 0
   129  		return z[:0]
   130  	case n == 0:
   131  		// result is x
   132  		return z.set(x)
   133  	}
   134  	// m > 0
   135  
   136  	z = z.make(m)
   137  	c := subVV(z[0:n], x, y)
   138  	if m > n {
   139  		c = subVW(z[n:], x[n:], c)
   140  	}
   141  	if c != 0 {
   142  		panic("underflow")
   143  	}
   144  
   145  	return z.norm()
   146  }
   147  
   148  func (x nat) cmp(y nat) (r int) {
   149  	m := len(x)
   150  	n := len(y)
   151  	if m != n || m == 0 {
   152  		switch {
   153  		case m < n:
   154  			r = -1
   155  		case m > n:
   156  			r = 1
   157  		}
   158  		return
   159  	}
   160  
   161  	i := m - 1
   162  	for i > 0 && x[i] == y[i] {
   163  		i--
   164  	}
   165  
   166  	switch {
   167  	case x[i] < y[i]:
   168  		r = -1
   169  	case x[i] > y[i]:
   170  		r = 1
   171  	}
   172  	return
   173  }
   174  
   175  func (z nat) mulAddWW(x nat, y, r Word) nat {
   176  	m := len(x)
   177  	if m == 0 || y == 0 {
   178  		return z.setWord(r) // result is r
   179  	}
   180  	// m > 0
   181  
   182  	z = z.make(m + 1)
   183  	z[m] = mulAddVWW(z[0:m], x, y, r)
   184  
   185  	return z.norm()
   186  }
   187  
   188  // basicMul multiplies x and y and leaves the result in z.
   189  // The (non-normalized) result is placed in z[0 : len(x) + len(y)].
   190  func basicMul(z, x, y nat) {
   191  	z[0 : len(x)+len(y)].clear() // initialize z
   192  	for i, d := range y {
   193  		if d != 0 {
   194  			z[len(x)+i] = addMulVVW(z[i:i+len(x)], x, d)
   195  		}
   196  	}
   197  }
   198  
   199  // montgomery computes z mod m = x*y*2**(-n*_W) mod m,
   200  // assuming k = -1/m mod 2**_W.
   201  // z is used for storing the result which is returned;
   202  // z must not alias x, y or m.
   203  // See Gueron, "Efficient Software Implementations of Modular Exponentiation".
   204  // https://eprint.iacr.org/2011/239.pdf
   205  // In the terminology of that paper, this is an "Almost Montgomery Multiplication":
   206  // x and y are required to satisfy 0 <= z < 2**(n*_W) and then the result
   207  // z is guaranteed to satisfy 0 <= z < 2**(n*_W), but it may not be < m.
   208  func (z nat) montgomery(x, y, m nat, k Word, n int) nat {
   209  	// This code assumes x, y, m are all the same length, n.
   210  	// (required by addMulVVW and the for loop).
   211  	// It also assumes that x, y are already reduced mod m,
   212  	// or else the result will not be properly reduced.
   213  	if len(x) != n || len(y) != n || len(m) != n {
   214  		panic("math/big: mismatched montgomery number lengths")
   215  	}
   216  	z = z.make(n * 2)
   217  	z.clear()
   218  	var c Word
   219  	for i := 0; i < n; i++ {
   220  		d := y[i]
   221  		c2 := addMulVVW(z[i:n+i], x, d)
   222  		t := z[i] * k
   223  		c3 := addMulVVW(z[i:n+i], m, t)
   224  		cx := c + c2
   225  		cy := cx + c3
   226  		z[n+i] = cy
   227  		if cx < c2 || cy < c3 {
   228  			c = 1
   229  		} else {
   230  			c = 0
   231  		}
   232  	}
   233  	if c != 0 {
   234  		subVV(z[:n], z[n:], m)
   235  	} else {
   236  		copy(z[:n], z[n:])
   237  	}
   238  	return z[:n]
   239  }
   240  
   241  // Fast version of z[0:n+n>>1].add(z[0:n+n>>1], x[0:n]) w/o bounds checks.
   242  // Factored out for readability - do not use outside karatsuba.
   243  func karatsubaAdd(z, x nat, n int) {
   244  	if c := addVV(z[0:n], z, x); c != 0 {
   245  		addVW(z[n:n+n>>1], z[n:], c)
   246  	}
   247  }
   248  
   249  // Like karatsubaAdd, but does subtract.
   250  func karatsubaSub(z, x nat, n int) {
   251  	if c := subVV(z[0:n], z, x); c != 0 {
   252  		subVW(z[n:n+n>>1], z[n:], c)
   253  	}
   254  }
   255  
   256  // Operands that are shorter than karatsubaThreshold are multiplied using
   257  // "grade school" multiplication; for longer operands the Karatsuba algorithm
   258  // is used.
   259  var karatsubaThreshold = 40 // computed by calibrate_test.go
   260  
   261  // karatsuba multiplies x and y and leaves the result in z.
   262  // Both x and y must have the same length n and n must be a
   263  // power of 2. The result vector z must have len(z) >= 6*n.
   264  // The (non-normalized) result is placed in z[0 : 2*n].
   265  func karatsuba(z, x, y nat) {
   266  	n := len(y)
   267  
   268  	// Switch to basic multiplication if numbers are odd or small.
   269  	// (n is always even if karatsubaThreshold is even, but be
   270  	// conservative)
   271  	if n&1 != 0 || n < karatsubaThreshold || n < 2 {
   272  		basicMul(z, x, y)
   273  		return
   274  	}
   275  	// n&1 == 0 && n >= karatsubaThreshold && n >= 2
   276  
   277  	// Karatsuba multiplication is based on the observation that
   278  	// for two numbers x and y with:
   279  	//
   280  	//   x = x1*b + x0
   281  	//   y = y1*b + y0
   282  	//
   283  	// the product x*y can be obtained with 3 products z2, z1, z0
   284  	// instead of 4:
   285  	//
   286  	//   x*y = x1*y1*b*b + (x1*y0 + x0*y1)*b + x0*y0
   287  	//       =    z2*b*b +              z1*b +    z0
   288  	//
   289  	// with:
   290  	//
   291  	//   xd = x1 - x0
   292  	//   yd = y0 - y1
   293  	//
   294  	//   z1 =      xd*yd                    + z2 + z0
   295  	//      = (x1-x0)*(y0 - y1)             + z2 + z0
   296  	//      = x1*y0 - x1*y1 - x0*y0 + x0*y1 + z2 + z0
   297  	//      = x1*y0 -    z2 -    z0 + x0*y1 + z2 + z0
   298  	//      = x1*y0                 + x0*y1
   299  
   300  	// split x, y into "digits"
   301  	n2 := n >> 1              // n2 >= 1
   302  	x1, x0 := x[n2:], x[0:n2] // x = x1*b + y0
   303  	y1, y0 := y[n2:], y[0:n2] // y = y1*b + y0
   304  
   305  	// z is used for the result and temporary storage:
   306  	//
   307  	//   6*n     5*n     4*n     3*n     2*n     1*n     0*n
   308  	// z = [z2 copy|z0 copy| xd*yd | yd:xd | x1*y1 | x0*y0 ]
   309  	//
   310  	// For each recursive call of karatsuba, an unused slice of
   311  	// z is passed in that has (at least) half the length of the
   312  	// caller's z.
   313  
   314  	// compute z0 and z2 with the result "in place" in z
   315  	karatsuba(z, x0, y0)     // z0 = x0*y0
   316  	karatsuba(z[n:], x1, y1) // z2 = x1*y1
   317  
   318  	// compute xd (or the negative value if underflow occurs)
   319  	s := 1 // sign of product xd*yd
   320  	xd := z[2*n : 2*n+n2]
   321  	if subVV(xd, x1, x0) != 0 { // x1-x0
   322  		s = -s
   323  		subVV(xd, x0, x1) // x0-x1
   324  	}
   325  
   326  	// compute yd (or the negative value if underflow occurs)
   327  	yd := z[2*n+n2 : 3*n]
   328  	if subVV(yd, y0, y1) != 0 { // y0-y1
   329  		s = -s
   330  		subVV(yd, y1, y0) // y1-y0
   331  	}
   332  
   333  	// p = (x1-x0)*(y0-y1) == x1*y0 - x1*y1 - x0*y0 + x0*y1 for s > 0
   334  	// p = (x0-x1)*(y0-y1) == x0*y0 - x0*y1 - x1*y0 + x1*y1 for s < 0
   335  	p := z[n*3:]
   336  	karatsuba(p, xd, yd)
   337  
   338  	// save original z2:z0
   339  	// (ok to use upper half of z since we're done recursing)
   340  	r := z[n*4:]
   341  	copy(r, z[:n*2])
   342  
   343  	// add up all partial products
   344  	//
   345  	//   2*n     n     0
   346  	// z = [ z2  | z0  ]
   347  	//   +    [ z0  ]
   348  	//   +    [ z2  ]
   349  	//   +    [  p  ]
   350  	//
   351  	karatsubaAdd(z[n2:], r, n)
   352  	karatsubaAdd(z[n2:], r[n:], n)
   353  	if s > 0 {
   354  		karatsubaAdd(z[n2:], p, n)
   355  	} else {
   356  		karatsubaSub(z[n2:], p, n)
   357  	}
   358  }
   359  
   360  // alias reports whether x and y share the same base array.
   361  // Note: alias assumes that the capacity of underlying arrays
   362  //       is never changed for nat values; i.e. that there are
   363  //       no 3-operand slice expressions in this code (or worse,
   364  //       reflect-based operations to the same effect).
   365  func alias(x, y nat) bool {
   366  	return cap(x) > 0 && cap(y) > 0 && &x[0:cap(x)][cap(x)-1] == &y[0:cap(y)][cap(y)-1]
   367  }
   368  
   369  // addAt implements z += x<<(_W*i); z must be long enough.
   370  // (we don't use nat.add because we need z to stay the same
   371  // slice, and we don't need to normalize z after each addition)
   372  func addAt(z, x nat, i int) {
   373  	if n := len(x); n > 0 {
   374  		if c := addVV(z[i:i+n], z[i:], x); c != 0 {
   375  			j := i + n
   376  			if j < len(z) {
   377  				addVW(z[j:], z[j:], c)
   378  			}
   379  		}
   380  	}
   381  }
   382  
   383  func max(x, y int) int {
   384  	if x > y {
   385  		return x
   386  	}
   387  	return y
   388  }
   389  
   390  // karatsubaLen computes an approximation to the maximum k <= n such that
   391  // k = p<<i for a number p <= threshold and an i >= 0. Thus, the
   392  // result is the largest number that can be divided repeatedly by 2 before
   393  // becoming about the value of threshold.
   394  func karatsubaLen(n, threshold int) int {
   395  	i := uint(0)
   396  	for n > threshold {
   397  		n >>= 1
   398  		i++
   399  	}
   400  	return n << i
   401  }
   402  
   403  func (z nat) mul(x, y nat) nat {
   404  	m := len(x)
   405  	n := len(y)
   406  
   407  	switch {
   408  	case m < n:
   409  		return z.mul(y, x)
   410  	case m == 0 || n == 0:
   411  		return z[:0]
   412  	case n == 1:
   413  		return z.mulAddWW(x, y[0], 0)
   414  	}
   415  	// m >= n > 1
   416  
   417  	// determine if z can be reused
   418  	if alias(z, x) || alias(z, y) {
   419  		z = nil // z is an alias for x or y - cannot reuse
   420  	}
   421  
   422  	// use basic multiplication if the numbers are small
   423  	if n < karatsubaThreshold {
   424  		z = z.make(m + n)
   425  		basicMul(z, x, y)
   426  		return z.norm()
   427  	}
   428  	// m >= n && n >= karatsubaThreshold && n >= 2
   429  
   430  	// determine Karatsuba length k such that
   431  	//
   432  	//   x = xh*b + x0  (0 <= x0 < b)
   433  	//   y = yh*b + y0  (0 <= y0 < b)
   434  	//   b = 1<<(_W*k)  ("base" of digits xi, yi)
   435  	//
   436  	k := karatsubaLen(n, karatsubaThreshold)
   437  	// k <= n
   438  
   439  	// multiply x0 and y0 via Karatsuba
   440  	x0 := x[0:k]              // x0 is not normalized
   441  	y0 := y[0:k]              // y0 is not normalized
   442  	z = z.make(max(6*k, m+n)) // enough space for karatsuba of x0*y0 and full result of x*y
   443  	karatsuba(z, x0, y0)
   444  	z = z[0 : m+n]  // z has final length but may be incomplete
   445  	z[2*k:].clear() // upper portion of z is garbage (and 2*k <= m+n since k <= n <= m)
   446  
   447  	// If xh != 0 or yh != 0, add the missing terms to z. For
   448  	//
   449  	//   xh = xi*b^i + ... + x2*b^2 + x1*b (0 <= xi < b)
   450  	//   yh =                         y1*b (0 <= y1 < b)
   451  	//
   452  	// the missing terms are
   453  	//
   454  	//   x0*y1*b and xi*y0*b^i, xi*y1*b^(i+1) for i > 0
   455  	//
   456  	// since all the yi for i > 1 are 0 by choice of k: If any of them
   457  	// were > 0, then yh >= b^2 and thus y >= b^2. Then k' = k*2 would
   458  	// be a larger valid threshold contradicting the assumption about k.
   459  	//
   460  	if k < n || m != n {
   461  		var t nat
   462  
   463  		// add x0*y1*b
   464  		x0 := x0.norm()
   465  		y1 := y[k:]       // y1 is normalized because y is
   466  		t = t.mul(x0, y1) // update t so we don't lose t's underlying array
   467  		addAt(z, t, k)
   468  
   469  		// add xi*y0<<i, xi*y1*b<<(i+k)
   470  		y0 := y0.norm()
   471  		for i := k; i < len(x); i += k {
   472  			xi := x[i:]
   473  			if len(xi) > k {
   474  				xi = xi[:k]
   475  			}
   476  			xi = xi.norm()
   477  			t = t.mul(xi, y0)
   478  			addAt(z, t, i)
   479  			t = t.mul(xi, y1)
   480  			addAt(z, t, i+k)
   481  		}
   482  	}
   483  
   484  	return z.norm()
   485  }
   486  
   487  // basicSqr sets z = x*x and is asymptotically faster than basicMul
   488  // by about a factor of 2, but slower for small arguments due to overhead.
   489  // Requirements: len(x) > 0, len(z) == 2*len(x)
   490  // The (non-normalized) result is placed in z.
   491  func basicSqr(z, x nat) {
   492  	n := len(x)
   493  	t := make(nat, 2*n)            // temporary variable to hold the products
   494  	z[1], z[0] = mulWW(x[0], x[0]) // the initial square
   495  	for i := 1; i < n; i++ {
   496  		d := x[i]
   497  		// z collects the squares x[i] * x[i]
   498  		z[2*i+1], z[2*i] = mulWW(d, d)
   499  		// t collects the products x[i] * x[j] where j < i
   500  		t[2*i] = addMulVVW(t[i:2*i], x[0:i], d)
   501  	}
   502  	t[2*n-1] = shlVU(t[1:2*n-1], t[1:2*n-1], 1) // double the j < i products
   503  	addVV(z, z, t)                              // combine the result
   504  }
   505  
   506  // karatsubaSqr squares x and leaves the result in z.
   507  // len(x) must be a power of 2 and len(z) >= 6*len(x).
   508  // The (non-normalized) result is placed in z[0 : 2*len(x)].
   509  //
   510  // The algorithm and the layout of z are the same as for karatsuba.
   511  func karatsubaSqr(z, x nat) {
   512  	n := len(x)
   513  
   514  	if n&1 != 0 || n < karatsubaSqrThreshold || n < 2 {
   515  		basicSqr(z[:2*n], x)
   516  		return
   517  	}
   518  
   519  	n2 := n >> 1
   520  	x1, x0 := x[n2:], x[0:n2]
   521  
   522  	karatsubaSqr(z, x0)
   523  	karatsubaSqr(z[n:], x1)
   524  
   525  	// s = sign(xd*yd) == -1 for xd != 0; s == 1 for xd == 0
   526  	xd := z[2*n : 2*n+n2]
   527  	if subVV(xd, x1, x0) != 0 {
   528  		subVV(xd, x0, x1)
   529  	}
   530  
   531  	p := z[n*3:]
   532  	karatsubaSqr(p, xd)
   533  
   534  	r := z[n*4:]
   535  	copy(r, z[:n*2])
   536  
   537  	karatsubaAdd(z[n2:], r, n)
   538  	karatsubaAdd(z[n2:], r[n:], n)
   539  	karatsubaSub(z[n2:], p, n) // s == -1 for p != 0; s == 1 for p == 0
   540  }
   541  
   542  // Operands that are shorter than basicSqrThreshold are squared using
   543  // "grade school" multiplication; for operands longer than karatsubaSqrThreshold
   544  // we use the Karatsuba algorithm optimized for x == y.
   545  var basicSqrThreshold = 20      // computed by calibrate_test.go
   546  var karatsubaSqrThreshold = 260 // computed by calibrate_test.go
   547  
   548  // z = x*x
   549  func (z nat) sqr(x nat) nat {
   550  	n := len(x)
   551  	switch {
   552  	case n == 0:
   553  		return z[:0]
   554  	case n == 1:
   555  		d := x[0]
   556  		z = z.make(2)
   557  		z[1], z[0] = mulWW(d, d)
   558  		return z.norm()
   559  	}
   560  
   561  	if alias(z, x) {
   562  		z = nil // z is an alias for x - cannot reuse
   563  	}
   564  
   565  	if n < basicSqrThreshold {
   566  		z = z.make(2 * n)
   567  		basicMul(z, x, x)
   568  		return z.norm()
   569  	}
   570  	if n < karatsubaSqrThreshold {
   571  		z = z.make(2 * n)
   572  		basicSqr(z, x)
   573  		return z.norm()
   574  	}
   575  
   576  	// Use Karatsuba multiplication optimized for x == y.
   577  	// The algorithm and layout of z are the same as for mul.
   578  
   579  	// z = (x1*b + x0)^2 = x1^2*b^2 + 2*x1*x0*b + x0^2
   580  
   581  	k := karatsubaLen(n, karatsubaSqrThreshold)
   582  
   583  	x0 := x[0:k]
   584  	z = z.make(max(6*k, 2*n))
   585  	karatsubaSqr(z, x0) // z = x0^2
   586  	z = z[0 : 2*n]
   587  	z[2*k:].clear()
   588  
   589  	if k < n {
   590  		var t nat
   591  		x0 := x0.norm()
   592  		x1 := x[k:]
   593  		t = t.mul(x0, x1)
   594  		addAt(z, t, k)
   595  		addAt(z, t, k) // z = 2*x1*x0*b + x0^2
   596  		t = t.sqr(x1)
   597  		addAt(z, t, 2*k) // z = x1^2*b^2 + 2*x1*x0*b + x0^2
   598  	}
   599  
   600  	return z.norm()
   601  }
   602  
   603  // mulRange computes the product of all the unsigned integers in the
   604  // range [a, b] inclusively. If a > b (empty range), the result is 1.
   605  func (z nat) mulRange(a, b uint64) nat {
   606  	switch {
   607  	case a == 0:
   608  		// cut long ranges short (optimization)
   609  		return z.setUint64(0)
   610  	case a > b:
   611  		return z.setUint64(1)
   612  	case a == b:
   613  		return z.setUint64(a)
   614  	case a+1 == b:
   615  		return z.mul(nat(nil).setUint64(a), nat(nil).setUint64(b))
   616  	}
   617  	m := (a + b) / 2
   618  	return z.mul(nat(nil).mulRange(a, m), nat(nil).mulRange(m+1, b))
   619  }
   620  
   621  // q = (x-r)/y, with 0 <= r < y
   622  func (z nat) divW(x nat, y Word) (q nat, r Word) {
   623  	m := len(x)
   624  	switch {
   625  	case y == 0:
   626  		panic("division by zero")
   627  	case y == 1:
   628  		q = z.set(x) // result is x
   629  		return
   630  	case m == 0:
   631  		q = z[:0] // result is 0
   632  		return
   633  	}
   634  	// m > 0
   635  	z = z.make(m)
   636  	r = divWVW(z, 0, x, y)
   637  	q = z.norm()
   638  	return
   639  }
   640  
   641  func (z nat) div(z2, u, v nat) (q, r nat) {
   642  	if len(v) == 0 {
   643  		panic("division by zero")
   644  	}
   645  
   646  	if u.cmp(v) < 0 {
   647  		q = z[:0]
   648  		r = z2.set(u)
   649  		return
   650  	}
   651  
   652  	if len(v) == 1 {
   653  		var r2 Word
   654  		q, r2 = z.divW(u, v[0])
   655  		r = z2.setWord(r2)
   656  		return
   657  	}
   658  
   659  	q, r = z.divLarge(z2, u, v)
   660  	return
   661  }
   662  
   663  // getNat returns a *nat of len n. The contents may not be zero.
   664  // The pool holds *nat to avoid allocation when converting to interface{}.
   665  func getNat(n int) *nat {
   666  	var z *nat
   667  	if v := natPool.Get(); v != nil {
   668  		z = v.(*nat)
   669  	}
   670  	if z == nil {
   671  		z = new(nat)
   672  	}
   673  	*z = z.make(n)
   674  	return z
   675  }
   676  
   677  func putNat(x *nat) {
   678  	natPool.Put(x)
   679  }
   680  
   681  var natPool sync.Pool
   682  
   683  // q = (uIn-r)/v, with 0 <= r < y
   684  // Uses z as storage for q, and u as storage for r if possible.
   685  // See Knuth, Volume 2, section 4.3.1, Algorithm D.
   686  // Preconditions:
   687  //    len(v) >= 2
   688  //    len(uIn) >= len(v)
   689  func (z nat) divLarge(u, uIn, v nat) (q, r nat) {
   690  	n := len(v)
   691  	m := len(uIn) - n
   692  
   693  	// determine if z can be reused
   694  	// TODO(gri) should find a better solution - this if statement
   695  	//           is very costly (see e.g. time pidigits -s -n 10000)
   696  	if alias(z, u) || alias(z, uIn) || alias(z, v) {
   697  		z = nil // z is an alias for u or uIn or v - cannot reuse
   698  	}
   699  	q = z.make(m + 1)
   700  
   701  	qhatvp := getNat(n + 1)
   702  	qhatv := *qhatvp
   703  	if alias(u, uIn) || alias(u, v) {
   704  		u = nil // u is an alias for uIn or v - cannot reuse
   705  	}
   706  	u = u.make(len(uIn) + 1)
   707  	u.clear() // TODO(gri) no need to clear if we allocated a new u
   708  
   709  	// D1.
   710  	var v1p *nat
   711  	shift := nlz(v[n-1])
   712  	if shift > 0 {
   713  		// do not modify v, it may be used by another goroutine simultaneously
   714  		v1p = getNat(n)
   715  		v1 := *v1p
   716  		shlVU(v1, v, shift)
   717  		v = v1
   718  	}
   719  	u[len(uIn)] = shlVU(u[0:len(uIn)], uIn, shift)
   720  
   721  	// D2.
   722  	vn1 := v[n-1]
   723  	for j := m; j >= 0; j-- {
   724  		// D3.
   725  		qhat := Word(_M)
   726  		if ujn := u[j+n]; ujn != vn1 {
   727  			var rhat Word
   728  			qhat, rhat = divWW(ujn, u[j+n-1], vn1)
   729  
   730  			// x1 | x2 = q̂v_{n-2}
   731  			vn2 := v[n-2]
   732  			x1, x2 := mulWW(qhat, vn2)
   733  			// test if q̂v_{n-2} > br̂ + u_{j+n-2}
   734  			ujn2 := u[j+n-2]
   735  			for greaterThan(x1, x2, rhat, ujn2) {
   736  				qhat--
   737  				prevRhat := rhat
   738  				rhat += vn1
   739  				// v[n-1] >= 0, so this tests for overflow.
   740  				if rhat < prevRhat {
   741  					break
   742  				}
   743  				x1, x2 = mulWW(qhat, vn2)
   744  			}
   745  		}
   746  
   747  		// D4.
   748  		qhatv[n] = mulAddVWW(qhatv[0:n], v, qhat, 0)
   749  
   750  		c := subVV(u[j:j+len(qhatv)], u[j:], qhatv)
   751  		if c != 0 {
   752  			c := addVV(u[j:j+n], u[j:], v)
   753  			u[j+n] += c
   754  			qhat--
   755  		}
   756  
   757  		q[j] = qhat
   758  	}
   759  	if v1p != nil {
   760  		putNat(v1p)
   761  	}
   762  	putNat(qhatvp)
   763  
   764  	q = q.norm()
   765  	shrVU(u, u, shift)
   766  	r = u.norm()
   767  
   768  	return q, r
   769  }
   770  
   771  // Length of x in bits. x must be normalized.
   772  func (x nat) bitLen() int {
   773  	if i := len(x) - 1; i >= 0 {
   774  		return i*_W + bits.Len(uint(x[i]))
   775  	}
   776  	return 0
   777  }
   778  
   779  // trailingZeroBits returns the number of consecutive least significant zero
   780  // bits of x.
   781  func (x nat) trailingZeroBits() uint {
   782  	if len(x) == 0 {
   783  		return 0
   784  	}
   785  	var i uint
   786  	for x[i] == 0 {
   787  		i++
   788  	}
   789  	// x[i] != 0
   790  	return i*_W + uint(bits.TrailingZeros(uint(x[i])))
   791  }
   792  
   793  func same(x, y nat) bool {
   794  	return len(x) == len(y) && len(x) > 0 && &x[0] == &y[0]
   795  }
   796  
   797  // z = x << s
   798  func (z nat) shl(x nat, s uint) nat {
   799  	if s == 0 {
   800  		if same(z, x) {
   801  			return z
   802  		}
   803  		if !alias(z, x) {
   804  			return z.set(x)
   805  		}
   806  	}
   807  
   808  	m := len(x)
   809  	if m == 0 {
   810  		return z[:0]
   811  	}
   812  	// m > 0
   813  
   814  	n := m + int(s/_W)
   815  	z = z.make(n + 1)
   816  	z[n] = shlVU(z[n-m:n], x, s%_W)
   817  	z[0 : n-m].clear()
   818  
   819  	return z.norm()
   820  }
   821  
   822  // z = x >> s
   823  func (z nat) shr(x nat, s uint) nat {
   824  	if s == 0 {
   825  		if same(z, x) {
   826  			return z
   827  		}
   828  		if !alias(z, x) {
   829  			return z.set(x)
   830  		}
   831  	}
   832  
   833  	m := len(x)
   834  	n := m - int(s/_W)
   835  	if n <= 0 {
   836  		return z[:0]
   837  	}
   838  	// n > 0
   839  
   840  	z = z.make(n)
   841  	shrVU(z, x[m-n:], s%_W)
   842  
   843  	return z.norm()
   844  }
   845  
   846  func (z nat) setBit(x nat, i uint, b uint) nat {
   847  	j := int(i / _W)
   848  	m := Word(1) << (i % _W)
   849  	n := len(x)
   850  	switch b {
   851  	case 0:
   852  		z = z.make(n)
   853  		copy(z, x)
   854  		if j >= n {
   855  			// no need to grow
   856  			return z
   857  		}
   858  		z[j] &^= m
   859  		return z.norm()
   860  	case 1:
   861  		if j >= n {
   862  			z = z.make(j + 1)
   863  			z[n:].clear()
   864  		} else {
   865  			z = z.make(n)
   866  		}
   867  		copy(z, x)
   868  		z[j] |= m
   869  		// no need to normalize
   870  		return z
   871  	}
   872  	panic("set bit is not 0 or 1")
   873  }
   874  
   875  // bit returns the value of the i'th bit, with lsb == bit 0.
   876  func (x nat) bit(i uint) uint {
   877  	j := i / _W
   878  	if j >= uint(len(x)) {
   879  		return 0
   880  	}
   881  	// 0 <= j < len(x)
   882  	return uint(x[j] >> (i % _W) & 1)
   883  }
   884  
   885  // sticky returns 1 if there's a 1 bit within the
   886  // i least significant bits, otherwise it returns 0.
   887  func (x nat) sticky(i uint) uint {
   888  	j := i / _W
   889  	if j >= uint(len(x)) {
   890  		if len(x) == 0 {
   891  			return 0
   892  		}
   893  		return 1
   894  	}
   895  	// 0 <= j < len(x)
   896  	for _, x := range x[:j] {
   897  		if x != 0 {
   898  			return 1
   899  		}
   900  	}
   901  	if x[j]<<(_W-i%_W) != 0 {
   902  		return 1
   903  	}
   904  	return 0
   905  }
   906  
   907  func (z nat) and(x, y nat) nat {
   908  	m := len(x)
   909  	n := len(y)
   910  	if m > n {
   911  		m = n
   912  	}
   913  	// m <= n
   914  
   915  	z = z.make(m)
   916  	for i := 0; i < m; i++ {
   917  		z[i] = x[i] & y[i]
   918  	}
   919  
   920  	return z.norm()
   921  }
   922  
   923  func (z nat) andNot(x, y nat) nat {
   924  	m := len(x)
   925  	n := len(y)
   926  	if n > m {
   927  		n = m
   928  	}
   929  	// m >= n
   930  
   931  	z = z.make(m)
   932  	for i := 0; i < n; i++ {
   933  		z[i] = x[i] &^ y[i]
   934  	}
   935  	copy(z[n:m], x[n:m])
   936  
   937  	return z.norm()
   938  }
   939  
   940  func (z nat) or(x, y nat) nat {
   941  	m := len(x)
   942  	n := len(y)
   943  	s := x
   944  	if m < n {
   945  		n, m = m, n
   946  		s = y
   947  	}
   948  	// m >= n
   949  
   950  	z = z.make(m)
   951  	for i := 0; i < n; i++ {
   952  		z[i] = x[i] | y[i]
   953  	}
   954  	copy(z[n:m], s[n:m])
   955  
   956  	return z.norm()
   957  }
   958  
   959  func (z nat) xor(x, y nat) nat {
   960  	m := len(x)
   961  	n := len(y)
   962  	s := x
   963  	if m < n {
   964  		n, m = m, n
   965  		s = y
   966  	}
   967  	// m >= n
   968  
   969  	z = z.make(m)
   970  	for i := 0; i < n; i++ {
   971  		z[i] = x[i] ^ y[i]
   972  	}
   973  	copy(z[n:m], s[n:m])
   974  
   975  	return z.norm()
   976  }
   977  
   978  // greaterThan reports whether (x1<<_W + x2) > (y1<<_W + y2)
   979  func greaterThan(x1, x2, y1, y2 Word) bool {
   980  	return x1 > y1 || x1 == y1 && x2 > y2
   981  }
   982  
   983  // modW returns x % d.
   984  func (x nat) modW(d Word) (r Word) {
   985  	// TODO(agl): we don't actually need to store the q value.
   986  	var q nat
   987  	q = q.make(len(x))
   988  	return divWVW(q, 0, x, d)
   989  }
   990  
   991  // random creates a random integer in [0..limit), using the space in z if
   992  // possible. n is the bit length of limit.
   993  func (z nat) random(rand *rand.Rand, limit nat, n int) nat {
   994  	if alias(z, limit) {
   995  		z = nil // z is an alias for limit - cannot reuse
   996  	}
   997  	z = z.make(len(limit))
   998  
   999  	bitLengthOfMSW := uint(n % _W)
  1000  	if bitLengthOfMSW == 0 {
  1001  		bitLengthOfMSW = _W
  1002  	}
  1003  	mask := Word((1 << bitLengthOfMSW) - 1)
  1004  
  1005  	for {
  1006  		switch _W {
  1007  		case 32:
  1008  			for i := range z {
  1009  				z[i] = Word(rand.Uint32())
  1010  			}
  1011  		case 64:
  1012  			for i := range z {
  1013  				z[i] = Word(rand.Uint32()) | Word(rand.Uint32())<<32
  1014  			}
  1015  		default:
  1016  			panic("unknown word size")
  1017  		}
  1018  		z[len(limit)-1] &= mask
  1019  		if z.cmp(limit) < 0 {
  1020  			break
  1021  		}
  1022  	}
  1023  
  1024  	return z.norm()
  1025  }
  1026  
  1027  // If m != 0 (i.e., len(m) != 0), expNN sets z to x**y mod m;
  1028  // otherwise it sets z to x**y. The result is the value of z.
  1029  func (z nat) expNN(x, y, m nat) nat {
  1030  	if alias(z, x) || alias(z, y) {
  1031  		// We cannot allow in-place modification of x or y.
  1032  		z = nil
  1033  	}
  1034  
  1035  	// x**y mod 1 == 0
  1036  	if len(m) == 1 && m[0] == 1 {
  1037  		return z.setWord(0)
  1038  	}
  1039  	// m == 0 || m > 1
  1040  
  1041  	// x**0 == 1
  1042  	if len(y) == 0 {
  1043  		return z.setWord(1)
  1044  	}
  1045  	// y > 0
  1046  
  1047  	// x**1 mod m == x mod m
  1048  	if len(y) == 1 && y[0] == 1 && len(m) != 0 {
  1049  		_, z = nat(nil).div(z, x, m)
  1050  		return z
  1051  	}
  1052  	// y > 1
  1053  
  1054  	if len(m) != 0 {
  1055  		// We likely end up being as long as the modulus.
  1056  		z = z.make(len(m))
  1057  	}
  1058  	z = z.set(x)
  1059  
  1060  	// If the base is non-trivial and the exponent is large, we use
  1061  	// 4-bit, windowed exponentiation. This involves precomputing 14 values
  1062  	// (x^2...x^15) but then reduces the number of multiply-reduces by a
  1063  	// third. Even for a 32-bit exponent, this reduces the number of
  1064  	// operations. Uses Montgomery method for odd moduli.
  1065  	if x.cmp(natOne) > 0 && len(y) > 1 && len(m) > 0 {
  1066  		if m[0]&1 == 1 {
  1067  			return z.expNNMontgomery(x, y, m)
  1068  		}
  1069  		return z.expNNWindowed(x, y, m)
  1070  	}
  1071  
  1072  	v := y[len(y)-1] // v > 0 because y is normalized and y > 0
  1073  	shift := nlz(v) + 1
  1074  	v <<= shift
  1075  	var q nat
  1076  
  1077  	const mask = 1 << (_W - 1)
  1078  
  1079  	// We walk through the bits of the exponent one by one. Each time we
  1080  	// see a bit, we square, thus doubling the power. If the bit is a one,
  1081  	// we also multiply by x, thus adding one to the power.
  1082  
  1083  	w := _W - int(shift)
  1084  	// zz and r are used to avoid allocating in mul and div as
  1085  	// otherwise the arguments would alias.
  1086  	var zz, r nat
  1087  	for j := 0; j < w; j++ {
  1088  		zz = zz.sqr(z)
  1089  		zz, z = z, zz
  1090  
  1091  		if v&mask != 0 {
  1092  			zz = zz.mul(z, x)
  1093  			zz, z = z, zz
  1094  		}
  1095  
  1096  		if len(m) != 0 {
  1097  			zz, r = zz.div(r, z, m)
  1098  			zz, r, q, z = q, z, zz, r
  1099  		}
  1100  
  1101  		v <<= 1
  1102  	}
  1103  
  1104  	for i := len(y) - 2; i >= 0; i-- {
  1105  		v = y[i]
  1106  
  1107  		for j := 0; j < _W; j++ {
  1108  			zz = zz.sqr(z)
  1109  			zz, z = z, zz
  1110  
  1111  			if v&mask != 0 {
  1112  				zz = zz.mul(z, x)
  1113  				zz, z = z, zz
  1114  			}
  1115  
  1116  			if len(m) != 0 {
  1117  				zz, r = zz.div(r, z, m)
  1118  				zz, r, q, z = q, z, zz, r
  1119  			}
  1120  
  1121  			v <<= 1
  1122  		}
  1123  	}
  1124  
  1125  	return z.norm()
  1126  }
  1127  
  1128  // expNNWindowed calculates x**y mod m using a fixed, 4-bit window.
  1129  func (z nat) expNNWindowed(x, y, m nat) nat {
  1130  	// zz and r are used to avoid allocating in mul and div as otherwise
  1131  	// the arguments would alias.
  1132  	var zz, r nat
  1133  
  1134  	const n = 4
  1135  	// powers[i] contains x^i.
  1136  	var powers [1 << n]nat
  1137  	powers[0] = natOne
  1138  	powers[1] = x
  1139  	for i := 2; i < 1<<n; i += 2 {
  1140  		p2, p, p1 := &powers[i/2], &powers[i], &powers[i+1]
  1141  		*p = p.sqr(*p2)
  1142  		zz, r = zz.div(r, *p, m)
  1143  		*p, r = r, *p
  1144  		*p1 = p1.mul(*p, x)
  1145  		zz, r = zz.div(r, *p1, m)
  1146  		*p1, r = r, *p1
  1147  	}
  1148  
  1149  	z = z.setWord(1)
  1150  
  1151  	for i := len(y) - 1; i >= 0; i-- {
  1152  		yi := y[i]
  1153  		for j := 0; j < _W; j += n {
  1154  			if i != len(y)-1 || j != 0 {
  1155  				// Unrolled loop for significant performance
  1156  				// gain. Use go test -bench=".*" in crypto/rsa
  1157  				// to check performance before making changes.
  1158  				zz = zz.sqr(z)
  1159  				zz, z = z, zz
  1160  				zz, r = zz.div(r, z, m)
  1161  				z, r = r, z
  1162  
  1163  				zz = zz.sqr(z)
  1164  				zz, z = z, zz
  1165  				zz, r = zz.div(r, z, m)
  1166  				z, r = r, z
  1167  
  1168  				zz = zz.sqr(z)
  1169  				zz, z = z, zz
  1170  				zz, r = zz.div(r, z, m)
  1171  				z, r = r, z
  1172  
  1173  				zz = zz.sqr(z)
  1174  				zz, z = z, zz
  1175  				zz, r = zz.div(r, z, m)
  1176  				z, r = r, z
  1177  			}
  1178  
  1179  			zz = zz.mul(z, powers[yi>>(_W-n)])
  1180  			zz, z = z, zz
  1181  			zz, r = zz.div(r, z, m)
  1182  			z, r = r, z
  1183  
  1184  			yi <<= n
  1185  		}
  1186  	}
  1187  
  1188  	return z.norm()
  1189  }
  1190  
  1191  // expNNMontgomery calculates x**y mod m using a fixed, 4-bit window.
  1192  // Uses Montgomery representation.
  1193  func (z nat) expNNMontgomery(x, y, m nat) nat {
  1194  	numWords := len(m)
  1195  
  1196  	// We want the lengths of x and m to be equal.
  1197  	// It is OK if x >= m as long as len(x) == len(m).
  1198  	if len(x) > numWords {
  1199  		_, x = nat(nil).div(nil, x, m)
  1200  		// Note: now len(x) <= numWords, not guaranteed ==.
  1201  	}
  1202  	if len(x) < numWords {
  1203  		rr := make(nat, numWords)
  1204  		copy(rr, x)
  1205  		x = rr
  1206  	}
  1207  
  1208  	// Ideally the precomputations would be performed outside, and reused
  1209  	// k0 = -m**-1 mod 2**_W. Algorithm from: Dumas, J.G. "On Newton–Raphson
  1210  	// Iteration for Multiplicative Inverses Modulo Prime Powers".
  1211  	k0 := 2 - m[0]
  1212  	t := m[0] - 1
  1213  	for i := 1; i < _W; i <<= 1 {
  1214  		t *= t
  1215  		k0 *= (t + 1)
  1216  	}
  1217  	k0 = -k0
  1218  
  1219  	// RR = 2**(2*_W*len(m)) mod m
  1220  	RR := nat(nil).setWord(1)
  1221  	zz := nat(nil).shl(RR, uint(2*numWords*_W))
  1222  	_, RR = nat(nil).div(RR, zz, m)
  1223  	if len(RR) < numWords {
  1224  		zz = zz.make(numWords)
  1225  		copy(zz, RR)
  1226  		RR = zz
  1227  	}
  1228  	// one = 1, with equal length to that of m
  1229  	one := make(nat, numWords)
  1230  	one[0] = 1
  1231  
  1232  	const n = 4
  1233  	// powers[i] contains x^i
  1234  	var powers [1 << n]nat
  1235  	powers[0] = powers[0].montgomery(one, RR, m, k0, numWords)
  1236  	powers[1] = powers[1].montgomery(x, RR, m, k0, numWords)
  1237  	for i := 2; i < 1<<n; i++ {
  1238  		powers[i] = powers[i].montgomery(powers[i-1], powers[1], m, k0, numWords)
  1239  	}
  1240  
  1241  	// initialize z = 1 (Montgomery 1)
  1242  	z = z.make(numWords)
  1243  	copy(z, powers[0])
  1244  
  1245  	zz = zz.make(numWords)
  1246  
  1247  	// same windowed exponent, but with Montgomery multiplications
  1248  	for i := len(y) - 1; i >= 0; i-- {
  1249  		yi := y[i]
  1250  		for j := 0; j < _W; j += n {
  1251  			if i != len(y)-1 || j != 0 {
  1252  				zz = zz.montgomery(z, z, m, k0, numWords)
  1253  				z = z.montgomery(zz, zz, m, k0, numWords)
  1254  				zz = zz.montgomery(z, z, m, k0, numWords)
  1255  				z = z.montgomery(zz, zz, m, k0, numWords)
  1256  			}
  1257  			zz = zz.montgomery(z, powers[yi>>(_W-n)], m, k0, numWords)
  1258  			z, zz = zz, z
  1259  			yi <<= n
  1260  		}
  1261  	}
  1262  	// convert to regular number
  1263  	zz = zz.montgomery(z, one, m, k0, numWords)
  1264  
  1265  	// One last reduction, just in case.
  1266  	// See golang.org/issue/13907.
  1267  	if zz.cmp(m) >= 0 {
  1268  		// Common case is m has high bit set; in that case,
  1269  		// since zz is the same length as m, there can be just
  1270  		// one multiple of m to remove. Just subtract.
  1271  		// We think that the subtract should be sufficient in general,
  1272  		// so do that unconditionally, but double-check,
  1273  		// in case our beliefs are wrong.
  1274  		// The div is not expected to be reached.
  1275  		zz = zz.sub(zz, m)
  1276  		if zz.cmp(m) >= 0 {
  1277  			_, zz = nat(nil).div(nil, zz, m)
  1278  		}
  1279  	}
  1280  
  1281  	return zz.norm()
  1282  }
  1283  
  1284  // bytes writes the value of z into buf using big-endian encoding.
  1285  // len(buf) must be >= len(z)*_S. The value of z is encoded in the
  1286  // slice buf[i:]. The number i of unused bytes at the beginning of
  1287  // buf is returned as result.
  1288  func (z nat) bytes(buf []byte) (i int) {
  1289  	i = len(buf)
  1290  	for _, d := range z {
  1291  		for j := 0; j < _S; j++ {
  1292  			i--
  1293  			buf[i] = byte(d)
  1294  			d >>= 8
  1295  		}
  1296  	}
  1297  
  1298  	for i < len(buf) && buf[i] == 0 {
  1299  		i++
  1300  	}
  1301  
  1302  	return
  1303  }
  1304  
  1305  // bigEndianWord returns the contents of buf interpreted as a big-endian encoded Word value.
  1306  func bigEndianWord(buf []byte) Word {
  1307  	if _W == 64 {
  1308  		return Word(binary.BigEndian.Uint64(buf))
  1309  	}
  1310  	return Word(binary.BigEndian.Uint32(buf))
  1311  }
  1312  
  1313  // setBytes interprets buf as the bytes of a big-endian unsigned
  1314  // integer, sets z to that value, and returns z.
  1315  func (z nat) setBytes(buf []byte) nat {
  1316  	z = z.make((len(buf) + _S - 1) / _S)
  1317  
  1318  	i := len(buf)
  1319  	for k := 0; i >= _S; k++ {
  1320  		z[k] = bigEndianWord(buf[i-_S : i])
  1321  		i -= _S
  1322  	}
  1323  	if i > 0 {
  1324  		var d Word
  1325  		for s := uint(0); i > 0; s += 8 {
  1326  			d |= Word(buf[i-1]) << s
  1327  			i--
  1328  		}
  1329  		z[len(z)-1] = d
  1330  	}
  1331  
  1332  	return z.norm()
  1333  }
  1334  
  1335  // sqrt sets z = ⌊√x⌋
  1336  func (z nat) sqrt(x nat) nat {
  1337  	if x.cmp(natOne) <= 0 {
  1338  		return z.set(x)
  1339  	}
  1340  	if alias(z, x) {
  1341  		z = nil
  1342  	}
  1343  
  1344  	// Start with value known to be too large and repeat "z = ⌊(z + ⌊x/z⌋)/2⌋" until it stops getting smaller.
  1345  	// See Brent and Zimmermann, Modern Computer Arithmetic, Algorithm 1.13 (SqrtInt).
  1346  	// https://members.loria.fr/PZimmermann/mca/pub226.html
  1347  	// If x is one less than a perfect square, the sequence oscillates between the correct z and z+1;
  1348  	// otherwise it converges to the correct z and stays there.
  1349  	var z1, z2 nat
  1350  	z1 = z
  1351  	z1 = z1.setUint64(1)
  1352  	z1 = z1.shl(z1, uint(x.bitLen()/2+1)) // must be ≥ √x
  1353  	for n := 0; ; n++ {
  1354  		z2, _ = z2.div(nil, x, z1)
  1355  		z2 = z2.add(z2, z1)
  1356  		z2 = z2.shr(z2, 1)
  1357  		if z2.cmp(z1) >= 0 {
  1358  			// z1 is answer.
  1359  			// Figure out whether z1 or z2 is currently aliased to z by looking at loop count.
  1360  			if n&1 == 0 {
  1361  				return z1
  1362  			}
  1363  			return z.set(z1)
  1364  		}
  1365  		z1, z2 = z2, z1
  1366  	}
  1367  }
  1368  

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