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  	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:
   105  		return z.add(y, x)
   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 {
   250  		addVW(z[n:n+n>>1], z[n:], c)
   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
   289  	// instead of 4:
   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  	//
   356  	karatsubaAdd(z[n2:], r, n)
   357  	karatsubaAdd(z[n2:], r[n:], n)
   358  	if s > 0 {
   359  		karatsubaAdd(z[n2:], p, n)
   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) {
   382  				addVW(z[j:], z[j:], c)
   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:
   418  		return z.mulAddWW(x, y[0], 0)
   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  		var t nat
   467  
   468  		// add x0*y1*b
   469  		x0 := x0.norm()
   470  		y1 := y[k:]       // y1 is normalized because y is
   471  		t = t.mul(x0, y1) // update t so we don't lose t's underlying array
   472  		addAt(z, t, k)
   473  
   474  		// add xi*y0<<i, xi*y1*b<<(i+k)
   475  		y0 := y0.norm()
   476  		for i := k; i < len(x); i += k {
   477  			xi := x[i:]
   478  			if len(xi) > k {
   479  				xi = xi[:k]
   480  			}
   481  			xi = xi.norm()
   482  			t = t.mul(xi, y0)
   483  			addAt(z, t, i)
   484  			t = t.mul(xi, y1)
   485  			addAt(z, t, i+k)
   486  		}
   487  	}
   488  
   489  	return z.norm()
   490  }
   491  
   492  // basicSqr sets z = x*x and is asymptotically faster than basicMul
   493  // by about a factor of 2, but slower for small arguments due to overhead.
   494  // Requirements: len(x) > 0, len(z) == 2*len(x)
   495  // The (non-normalized) result is placed in z.
   496  func basicSqr(z, x nat) {
   497  	n := len(x)
   498  	t := make(nat, 2*n)            // temporary variable to hold the products
   499  	z[1], z[0] = mulWW(x[0], x[0]) // the initial square
   500  	for i := 1; i < n; i++ {
   501  		d := x[i]
   502  		// z collects the squares x[i] * x[i]
   503  		z[2*i+1], z[2*i] = mulWW(d, d)
   504  		// t collects the products x[i] * x[j] where j < i
   505  		t[2*i] = addMulVVW(t[i:2*i], x[0:i], d)
   506  	}
   507  	t[2*n-1] = shlVU(t[1:2*n-1], t[1:2*n-1], 1) // double the j < i products
   508  	addVV(z, z, t)                              // combine the result
   509  }
   510  
   511  // karatsubaSqr squares x and leaves the result in z.
   512  // len(x) must be a power of 2 and len(z) >= 6*len(x).
   513  // The (non-normalized) result is placed in z[0 : 2*len(x)].
   514  //
   515  // The algorithm and the layout of z are the same as for karatsuba.
   516  func karatsubaSqr(z, x nat) {
   517  	n := len(x)
   518  
   519  	if n&1 != 0 || n < karatsubaSqrThreshold || n < 2 {
   520  		basicSqr(z[:2*n], x)
   521  		return
   522  	}
   523  
   524  	n2 := n >> 1
   525  	x1, x0 := x[n2:], x[0:n2]
   526  
   527  	karatsubaSqr(z, x0)
   528  	karatsubaSqr(z[n:], x1)
   529  
   530  	// s = sign(xd*yd) == -1 for xd != 0; s == 1 for xd == 0
   531  	xd := z[2*n : 2*n+n2]
   532  	if subVV(xd, x1, x0) != 0 {
   533  		subVV(xd, x0, x1)
   534  	}
   535  
   536  	p := z[n*3:]
   537  	karatsubaSqr(p, xd)
   538  
   539  	r := z[n*4:]
   540  	copy(r, z[:n*2])
   541  
   542  	karatsubaAdd(z[n2:], r, n)
   543  	karatsubaAdd(z[n2:], r[n:], n)
   544  	karatsubaSub(z[n2:], p, n) // s == -1 for p != 0; s == 1 for p == 0
   545  }
   546  
   547  // Operands that are shorter than basicSqrThreshold are squared using
   548  // "grade school" multiplication; for operands longer than karatsubaSqrThreshold
   549  // we use the Karatsuba algorithm optimized for x == y.
   550  var basicSqrThreshold = 20      // computed by calibrate_test.go
   551  var karatsubaSqrThreshold = 260 // computed by calibrate_test.go
   552  
   553  // z = x*x
   554  func (z nat) sqr(x nat) nat {
   555  	n := len(x)
   556  	switch {
   557  	case n == 0:
   558  		return z[:0]
   559  	case n == 1:
   560  		d := x[0]
   561  		z = z.make(2)
   562  		z[1], z[0] = mulWW(d, d)
   563  		return z.norm()
   564  	}
   565  
   566  	if alias(z, x) {
   567  		z = nil // z is an alias for x - cannot reuse
   568  	}
   569  
   570  	if n < basicSqrThreshold {
   571  		z = z.make(2 * n)
   572  		basicMul(z, x, x)
   573  		return z.norm()
   574  	}
   575  	if n < karatsubaSqrThreshold {
   576  		z = z.make(2 * n)
   577  		basicSqr(z, x)
   578  		return z.norm()
   579  	}
   580  
   581  	// Use Karatsuba multiplication optimized for x == y.
   582  	// The algorithm and layout of z are the same as for mul.
   583  
   584  	// z = (x1*b + x0)^2 = x1^2*b^2 + 2*x1*x0*b + x0^2
   585  
   586  	k := karatsubaLen(n, karatsubaSqrThreshold)
   587  
   588  	x0 := x[0:k]
   589  	z = z.make(max(6*k, 2*n))
   590  	karatsubaSqr(z, x0) // z = x0^2
   591  	z = z[0 : 2*n]
   592  	z[2*k:].clear()
   593  
   594  	if k < n {
   595  		var t nat
   596  		x0 := x0.norm()
   597  		x1 := x[k:]
   598  		t = t.mul(x0, x1)
   599  		addAt(z, t, k)
   600  		addAt(z, t, k) // z = 2*x1*x0*b + x0^2
   601  		t = t.sqr(x1)
   602  		addAt(z, t, 2*k) // z = x1^2*b^2 + 2*x1*x0*b + x0^2
   603  	}
   604  
   605  	return z.norm()
   606  }
   607  
   608  // mulRange computes the product of all the unsigned integers in the
   609  // range [a, b] inclusively. If a > b (empty range), the result is 1.
   610  func (z nat) mulRange(a, b uint64) nat {
   611  	switch {
   612  	case a == 0:
   613  		// cut long ranges short (optimization)
   614  		return z.setUint64(0)
   615  	case a > b:
   616  		return z.setUint64(1)
   617  	case a == b:
   618  		return z.setUint64(a)
   619  	case a+1 == b:
   620  		return z.mul(nat(nil).setUint64(a), nat(nil).setUint64(b))
   621  	}
   622  	m := (a + b) / 2
   623  	return z.mul(nat(nil).mulRange(a, m), nat(nil).mulRange(m+1, b))
   624  }
   625  
   626  // q = (x-r)/y, with 0 <= r < y
   627  func (z nat) divW(x nat, y Word) (q nat, r Word) {
   628  	m := len(x)
   629  	switch {
   630  	case y == 0:
   631  		panic("division by zero")
   632  	case y == 1:
   633  		q = z.set(x) // result is x
   634  		return
   635  	case m == 0:
   636  		q = z[:0] // result is 0
   637  		return
   638  	}
   639  	// m > 0
   640  	z = z.make(m)
   641  	r = divWVW(z, 0, x, y)
   642  	q = z.norm()
   643  	return
   644  }
   645  
   646  func (z nat) div(z2, u, v nat) (q, r nat) {
   647  	if len(v) == 0 {
   648  		panic("division by zero")
   649  	}
   650  
   651  	if u.cmp(v) < 0 {
   652  		q = z[:0]
   653  		r = z2.set(u)
   654  		return
   655  	}
   656  
   657  	if len(v) == 1 {
   658  		var r2 Word
   659  		q, r2 = z.divW(u, v[0])
   660  		r = z2.setWord(r2)
   661  		return
   662  	}
   663  
   664  	q, r = z.divLarge(z2, u, v)
   665  	return
   666  }
   667  
   668  // getNat returns a *nat of len n. The contents may not be zero.
   669  // The pool holds *nat to avoid allocation when converting to interface{}.
   670  func getNat(n int) *nat {
   671  	var z *nat
   672  	if v := natPool.Get(); v != nil {
   673  		z = v.(*nat)
   674  	}
   675  	if z == nil {
   676  		z = new(nat)
   677  	}
   678  	*z = z.make(n)
   679  	return z
   680  }
   681  
   682  func putNat(x *nat) {
   683  	natPool.Put(x)
   684  }
   685  
   686  var natPool sync.Pool
   687  
   688  // q = (uIn-r)/vIn, with 0 <= r < y
   689  // Uses z as storage for q, and u as storage for r if possible.
   690  // See Knuth, Volume 2, section 4.3.1, Algorithm D.
   691  // Preconditions:
   692  //    len(vIn) >= 2
   693  //    len(uIn) >= len(vIn)
   694  //    u must not alias z
   695  func (z nat) divLarge(u, uIn, vIn nat) (q, r nat) {
   696  	n := len(vIn)
   697  	m := len(uIn) - n
   698  
   699  	// D1.
   700  	shift := nlz(vIn[n-1])
   701  	// do not modify vIn, it may be used by another goroutine simultaneously
   702  	vp := getNat(n)
   703  	v := *vp
   704  	shlVU(v, vIn, shift)
   705  
   706  	// u may safely alias uIn or vIn, the value of uIn is used to set u and vIn was already used
   707  	u = u.make(len(uIn) + 1)
   708  	u[len(uIn)] = shlVU(u[0:len(uIn)], uIn, shift)
   709  
   710  	// z may safely alias uIn or vIn, both values were used already
   711  	if alias(z, u) {
   712  		z = nil // z is an alias for u - cannot reuse
   713  	}
   714  	q = z.make(m + 1)
   715  
   716  	qhatvp := getNat(n + 1)
   717  	qhatv := *qhatvp
   718  
   719  	// D2.
   720  	vn1 := v[n-1]
   721  	for j := m; j >= 0; j-- {
   722  		// D3.
   723  		qhat := Word(_M)
   724  		if ujn := u[j+n]; ujn != vn1 {
   725  			var rhat Word
   726  			qhat, rhat = divWW(ujn, u[j+n-1], vn1)
   727  
   728  			// x1 | x2 = q̂v_{n-2}
   729  			vn2 := v[n-2]
   730  			x1, x2 := mulWW(qhat, vn2)
   731  			// test if q̂v_{n-2} > br̂ + u_{j+n-2}
   732  			ujn2 := u[j+n-2]
   733  			for greaterThan(x1, x2, rhat, ujn2) {
   734  				qhat--
   735  				prevRhat := rhat
   736  				rhat += vn1
   737  				// v[n-1] >= 0, so this tests for overflow.
   738  				if rhat < prevRhat {
   739  					break
   740  				}
   741  				x1, x2 = mulWW(qhat, vn2)
   742  			}
   743  		}
   744  
   745  		// D4.
   746  		qhatv[n] = mulAddVWW(qhatv[0:n], v, qhat, 0)
   747  
   748  		c := subVV(u[j:j+len(qhatv)], u[j:], qhatv)
   749  		if c != 0 {
   750  			c := addVV(u[j:j+n], u[j:], v)
   751  			u[j+n] += c
   752  			qhat--
   753  		}
   754  
   755  		q[j] = qhat
   756  	}
   757  
   758  	putNat(vp)
   759  	putNat(qhatvp)
   760  
   761  	q = q.norm()
   762  	shrVU(u, u, shift)
   763  	r = u.norm()
   764  
   765  	return q, r
   766  }
   767  
   768  // Length of x in bits. x must be normalized.
   769  func (x nat) bitLen() int {
   770  	if i := len(x) - 1; i >= 0 {
   771  		return i*_W + bits.Len(uint(x[i]))
   772  	}
   773  	return 0
   774  }
   775  
   776  // trailingZeroBits returns the number of consecutive least significant zero
   777  // bits of x.
   778  func (x nat) trailingZeroBits() uint {
   779  	if len(x) == 0 {
   780  		return 0
   781  	}
   782  	var i uint
   783  	for x[i] == 0 {
   784  		i++
   785  	}
   786  	// x[i] != 0
   787  	return i*_W + uint(bits.TrailingZeros(uint(x[i])))
   788  }
   789  
   790  func same(x, y nat) bool {
   791  	return len(x) == len(y) && len(x) > 0 && &x[0] == &y[0]
   792  }
   793  
   794  // z = x << s
   795  func (z nat) shl(x nat, s uint) nat {
   796  	if s == 0 {
   797  		if same(z, x) {
   798  			return z
   799  		}
   800  		if !alias(z, x) {
   801  			return z.set(x)
   802  		}
   803  	}
   804  
   805  	m := len(x)
   806  	if m == 0 {
   807  		return z[:0]
   808  	}
   809  	// m > 0
   810  
   811  	n := m + int(s/_W)
   812  	z = z.make(n + 1)
   813  	z[n] = shlVU(z[n-m:n], x, s%_W)
   814  	z[0 : n-m].clear()
   815  
   816  	return z.norm()
   817  }
   818  
   819  // z = x >> s
   820  func (z nat) shr(x nat, s uint) nat {
   821  	if s == 0 {
   822  		if same(z, x) {
   823  			return z
   824  		}
   825  		if !alias(z, x) {
   826  			return z.set(x)
   827  		}
   828  	}
   829  
   830  	m := len(x)
   831  	n := m - int(s/_W)
   832  	if n <= 0 {
   833  		return z[:0]
   834  	}
   835  	// n > 0
   836  
   837  	z = z.make(n)
   838  	shrVU(z, x[m-n:], s%_W)
   839  
   840  	return z.norm()
   841  }
   842  
   843  func (z nat) setBit(x nat, i uint, b uint) nat {
   844  	j := int(i / _W)
   845  	m := Word(1) << (i % _W)
   846  	n := len(x)
   847  	switch b {
   848  	case 0:
   849  		z = z.make(n)
   850  		copy(z, x)
   851  		if j >= n {
   852  			// no need to grow
   853  			return z
   854  		}
   855  		z[j] &^= m
   856  		return z.norm()
   857  	case 1:
   858  		if j >= n {
   859  			z = z.make(j + 1)
   860  			z[n:].clear()
   861  		} else {
   862  			z = z.make(n)
   863  		}
   864  		copy(z, x)
   865  		z[j] |= m
   866  		// no need to normalize
   867  		return z
   868  	}
   869  	panic("set bit is not 0 or 1")
   870  }
   871  
   872  // bit returns the value of the i'th bit, with lsb == bit 0.
   873  func (x nat) bit(i uint) uint {
   874  	j := i / _W
   875  	if j >= uint(len(x)) {
   876  		return 0
   877  	}
   878  	// 0 <= j < len(x)
   879  	return uint(x[j] >> (i % _W) & 1)
   880  }
   881  
   882  // sticky returns 1 if there's a 1 bit within the
   883  // i least significant bits, otherwise it returns 0.
   884  func (x nat) sticky(i uint) uint {
   885  	j := i / _W
   886  	if j >= uint(len(x)) {
   887  		if len(x) == 0 {
   888  			return 0
   889  		}
   890  		return 1
   891  	}
   892  	// 0 <= j < len(x)
   893  	for _, x := range x[:j] {
   894  		if x != 0 {
   895  			return 1
   896  		}
   897  	}
   898  	if x[j]<<(_W-i%_W) != 0 {
   899  		return 1
   900  	}
   901  	return 0
   902  }
   903  
   904  func (z nat) and(x, y nat) nat {
   905  	m := len(x)
   906  	n := len(y)
   907  	if m > n {
   908  		m = n
   909  	}
   910  	// m <= n
   911  
   912  	z = z.make(m)
   913  	for i := 0; i < m; i++ {
   914  		z[i] = x[i] & y[i]
   915  	}
   916  
   917  	return z.norm()
   918  }
   919  
   920  func (z nat) andNot(x, y nat) nat {
   921  	m := len(x)
   922  	n := len(y)
   923  	if n > m {
   924  		n = m
   925  	}
   926  	// m >= n
   927  
   928  	z = z.make(m)
   929  	for i := 0; i < n; i++ {
   930  		z[i] = x[i] &^ y[i]
   931  	}
   932  	copy(z[n:m], x[n:m])
   933  
   934  	return z.norm()
   935  }
   936  
   937  func (z nat) or(x, y nat) nat {
   938  	m := len(x)
   939  	n := len(y)
   940  	s := x
   941  	if m < n {
   942  		n, m = m, n
   943  		s = y
   944  	}
   945  	// m >= n
   946  
   947  	z = z.make(m)
   948  	for i := 0; i < n; i++ {
   949  		z[i] = x[i] | y[i]
   950  	}
   951  	copy(z[n:m], s[n:m])
   952  
   953  	return z.norm()
   954  }
   955  
   956  func (z nat) xor(x, y nat) nat {
   957  	m := len(x)
   958  	n := len(y)
   959  	s := x
   960  	if m < n {
   961  		n, m = m, n
   962  		s = y
   963  	}
   964  	// m >= n
   965  
   966  	z = z.make(m)
   967  	for i := 0; i < n; i++ {
   968  		z[i] = x[i] ^ y[i]
   969  	}
   970  	copy(z[n:m], s[n:m])
   971  
   972  	return z.norm()
   973  }
   974  
   975  // greaterThan reports whether (x1<<_W + x2) > (y1<<_W + y2)
   976  func greaterThan(x1, x2, y1, y2 Word) bool {
   977  	return x1 > y1 || x1 == y1 && x2 > y2
   978  }
   979  
   980  // modW returns x % d.
   981  func (x nat) modW(d Word) (r Word) {
   982  	// TODO(agl): we don't actually need to store the q value.
   983  	var q nat
   984  	q = q.make(len(x))
   985  	return divWVW(q, 0, x, d)
   986  }
   987  
   988  // random creates a random integer in [0..limit), using the space in z if
   989  // possible. n is the bit length of limit.
   990  func (z nat) random(rand *rand.Rand, limit nat, n int) nat {
   991  	if alias(z, limit) {
   992  		z = nil // z is an alias for limit - cannot reuse
   993  	}
   994  	z = z.make(len(limit))
   995  
   996  	bitLengthOfMSW := uint(n % _W)
   997  	if bitLengthOfMSW == 0 {
   998  		bitLengthOfMSW = _W
   999  	}
  1000  	mask := Word((1 << bitLengthOfMSW) - 1)
  1001  
  1002  	for {
  1003  		switch _W {
  1004  		case 32:
  1005  			for i := range z {
  1006  				z[i] = Word(rand.Uint32())
  1007  			}
  1008  		case 64:
  1009  			for i := range z {
  1010  				z[i] = Word(rand.Uint32()) | Word(rand.Uint32())<<32
  1011  			}
  1012  		default:
  1013  			panic("unknown word size")
  1014  		}
  1015  		z[len(limit)-1] &= mask
  1016  		if z.cmp(limit) < 0 {
  1017  			break
  1018  		}
  1019  	}
  1020  
  1021  	return z.norm()
  1022  }
  1023  
  1024  // If m != 0 (i.e., len(m) != 0), expNN sets z to x**y mod m;
  1025  // otherwise it sets z to x**y. The result is the value of z.
  1026  func (z nat) expNN(x, y, m nat) nat {
  1027  	if alias(z, x) || alias(z, y) {
  1028  		// We cannot allow in-place modification of x or y.
  1029  		z = nil
  1030  	}
  1031  
  1032  	// x**y mod 1 == 0
  1033  	if len(m) == 1 && m[0] == 1 {
  1034  		return z.setWord(0)
  1035  	}
  1036  	// m == 0 || m > 1
  1037  
  1038  	// x**0 == 1
  1039  	if len(y) == 0 {
  1040  		return z.setWord(1)
  1041  	}
  1042  	// y > 0
  1043  
  1044  	// x**1 mod m == x mod m
  1045  	if len(y) == 1 && y[0] == 1 && len(m) != 0 {
  1046  		_, z = nat(nil).div(z, x, m)
  1047  		return z
  1048  	}
  1049  	// y > 1
  1050  
  1051  	if len(m) != 0 {
  1052  		// We likely end up being as long as the modulus.
  1053  		z = z.make(len(m))
  1054  	}
  1055  	z = z.set(x)
  1056  
  1057  	// If the base is non-trivial and the exponent is large, we use
  1058  	// 4-bit, windowed exponentiation. This involves precomputing 14 values
  1059  	// (x^2...x^15) but then reduces the number of multiply-reduces by a
  1060  	// third. Even for a 32-bit exponent, this reduces the number of
  1061  	// operations. Uses Montgomery method for odd moduli.
  1062  	if x.cmp(natOne) > 0 && len(y) > 1 && len(m) > 0 {
  1063  		if m[0]&1 == 1 {
  1064  			return z.expNNMontgomery(x, y, m)
  1065  		}
  1066  		return z.expNNWindowed(x, y, m)
  1067  	}
  1068  
  1069  	v := y[len(y)-1] // v > 0 because y is normalized and y > 0
  1070  	shift := nlz(v) + 1
  1071  	v <<= shift
  1072  	var q nat
  1073  
  1074  	const mask = 1 << (_W - 1)
  1075  
  1076  	// We walk through the bits of the exponent one by one. Each time we
  1077  	// see a bit, we square, thus doubling the power. If the bit is a one,
  1078  	// we also multiply by x, thus adding one to the power.
  1079  
  1080  	w := _W - int(shift)
  1081  	// zz and r are used to avoid allocating in mul and div as
  1082  	// otherwise the arguments would alias.
  1083  	var zz, r nat
  1084  	for j := 0; j < w; j++ {
  1085  		zz = zz.sqr(z)
  1086  		zz, z = z, zz
  1087  
  1088  		if v&mask != 0 {
  1089  			zz = zz.mul(z, x)
  1090  			zz, z = z, zz
  1091  		}
  1092  
  1093  		if len(m) != 0 {
  1094  			zz, r = zz.div(r, z, m)
  1095  			zz, r, q, z = q, z, zz, r
  1096  		}
  1097  
  1098  		v <<= 1
  1099  	}
  1100  
  1101  	for i := len(y) - 2; i >= 0; i-- {
  1102  		v = y[i]
  1103  
  1104  		for j := 0; j < _W; j++ {
  1105  			zz = zz.sqr(z)
  1106  			zz, z = z, zz
  1107  
  1108  			if v&mask != 0 {
  1109  				zz = zz.mul(z, x)
  1110  				zz, z = z, zz
  1111  			}
  1112  
  1113  			if len(m) != 0 {
  1114  				zz, r = zz.div(r, z, m)
  1115  				zz, r, q, z = q, z, zz, r
  1116  			}
  1117  
  1118  			v <<= 1
  1119  		}
  1120  	}
  1121  
  1122  	return z.norm()
  1123  }
  1124  
  1125  // expNNWindowed calculates x**y mod m using a fixed, 4-bit window.
  1126  func (z nat) expNNWindowed(x, y, m nat) nat {
  1127  	// zz and r are used to avoid allocating in mul and div as otherwise
  1128  	// the arguments would alias.
  1129  	var zz, r nat
  1130  
  1131  	const n = 4
  1132  	// powers[i] contains x^i.
  1133  	var powers [1 << n]nat
  1134  	powers[0] = natOne
  1135  	powers[1] = x
  1136  	for i := 2; i < 1<<n; i += 2 {
  1137  		p2, p, p1 := &powers[i/2], &powers[i], &powers[i+1]
  1138  		*p = p.sqr(*p2)
  1139  		zz, r = zz.div(r, *p, m)
  1140  		*p, r = r, *p
  1141  		*p1 = p1.mul(*p, x)
  1142  		zz, r = zz.div(r, *p1, m)
  1143  		*p1, r = r, *p1
  1144  	}
  1145  
  1146  	z = z.setWord(1)
  1147  
  1148  	for i := len(y) - 1; i >= 0; i-- {
  1149  		yi := y[i]
  1150  		for j := 0; j < _W; j += n {
  1151  			if i != len(y)-1 || j != 0 {
  1152  				// Unrolled loop for significant performance
  1153  				// gain. Use go test -bench=".*" in crypto/rsa
  1154  				// to check performance before making changes.
  1155  				zz = zz.sqr(z)
  1156  				zz, z = z, zz
  1157  				zz, r = zz.div(r, z, m)
  1158  				z, r = r, z
  1159  
  1160  				zz = zz.sqr(z)
  1161  				zz, z = z, zz
  1162  				zz, r = zz.div(r, z, m)
  1163  				z, r = r, z
  1164  
  1165  				zz = zz.sqr(z)
  1166  				zz, z = z, zz
  1167  				zz, r = zz.div(r, z, m)
  1168  				z, r = r, z
  1169  
  1170  				zz = zz.sqr(z)
  1171  				zz, z = z, zz
  1172  				zz, r = zz.div(r, z, m)
  1173  				z, r = r, z
  1174  			}
  1175  
  1176  			zz = zz.mul(z, powers[yi>>(_W-n)])
  1177  			zz, z = z, zz
  1178  			zz, r = zz.div(r, z, m)
  1179  			z, r = r, z
  1180  
  1181  			yi <<= n
  1182  		}
  1183  	}
  1184  
  1185  	return z.norm()
  1186  }
  1187  
  1188  // expNNMontgomery calculates x**y mod m using a fixed, 4-bit window.
  1189  // Uses Montgomery representation.
  1190  func (z nat) expNNMontgomery(x, y, m nat) nat {
  1191  	numWords := len(m)
  1192  
  1193  	// We want the lengths of x and m to be equal.
  1194  	// It is OK if x >= m as long as len(x) == len(m).
  1195  	if len(x) > numWords {
  1196  		_, x = nat(nil).div(nil, x, m)
  1197  		// Note: now len(x) <= numWords, not guaranteed ==.
  1198  	}
  1199  	if len(x) < numWords {
  1200  		rr := make(nat, numWords)
  1201  		copy(rr, x)
  1202  		x = rr
  1203  	}
  1204  
  1205  	// Ideally the precomputations would be performed outside, and reused
  1206  	// k0 = -m**-1 mod 2**_W. Algorithm from: Dumas, J.G. "On Newton–Raphson
  1207  	// Iteration for Multiplicative Inverses Modulo Prime Powers".
  1208  	k0 := 2 - m[0]
  1209  	t := m[0] - 1
  1210  	for i := 1; i < _W; i <<= 1 {
  1211  		t *= t
  1212  		k0 *= (t + 1)
  1213  	}
  1214  	k0 = -k0
  1215  
  1216  	// RR = 2**(2*_W*len(m)) mod m
  1217  	RR := nat(nil).setWord(1)
  1218  	zz := nat(nil).shl(RR, uint(2*numWords*_W))
  1219  	_, RR = nat(nil).div(RR, zz, m)
  1220  	if len(RR) < numWords {
  1221  		zz = zz.make(numWords)
  1222  		copy(zz, RR)
  1223  		RR = zz
  1224  	}
  1225  	// one = 1, with equal length to that of m
  1226  	one := make(nat, numWords)
  1227  	one[0] = 1
  1228  
  1229  	const n = 4
  1230  	// powers[i] contains x^i
  1231  	var powers [1 << n]nat
  1232  	powers[0] = powers[0].montgomery(one, RR, m, k0, numWords)
  1233  	powers[1] = powers[1].montgomery(x, RR, m, k0, numWords)
  1234  	for i := 2; i < 1<<n; i++ {
  1235  		powers[i] = powers[i].montgomery(powers[i-1], powers[1], m, k0, numWords)
  1236  	}
  1237  
  1238  	// initialize z = 1 (Montgomery 1)
  1239  	z = z.make(numWords)
  1240  	copy(z, powers[0])
  1241  
  1242  	zz = zz.make(numWords)
  1243  
  1244  	// same windowed exponent, but with Montgomery multiplications
  1245  	for i := len(y) - 1; i >= 0; i-- {
  1246  		yi := y[i]
  1247  		for j := 0; j < _W; j += n {
  1248  			if i != len(y)-1 || j != 0 {
  1249  				zz = zz.montgomery(z, z, m, k0, numWords)
  1250  				z = z.montgomery(zz, zz, m, k0, numWords)
  1251  				zz = zz.montgomery(z, z, m, k0, numWords)
  1252  				z = z.montgomery(zz, zz, m, k0, numWords)
  1253  			}
  1254  			zz = zz.montgomery(z, powers[yi>>(_W-n)], m, k0, numWords)
  1255  			z, zz = zz, z
  1256  			yi <<= n
  1257  		}
  1258  	}
  1259  	// convert to regular number
  1260  	zz = zz.montgomery(z, one, m, k0, numWords)
  1261  
  1262  	// One last reduction, just in case.
  1263  	// See golang.org/issue/13907.
  1264  	if zz.cmp(m) >= 0 {
  1265  		// Common case is m has high bit set; in that case,
  1266  		// since zz is the same length as m, there can be just
  1267  		// one multiple of m to remove. Just subtract.
  1268  		// We think that the subtract should be sufficient in general,
  1269  		// so do that unconditionally, but double-check,
  1270  		// in case our beliefs are wrong.
  1271  		// The div is not expected to be reached.
  1272  		zz = zz.sub(zz, m)
  1273  		if zz.cmp(m) >= 0 {
  1274  			_, zz = nat(nil).div(nil, zz, m)
  1275  		}
  1276  	}
  1277  
  1278  	return zz.norm()
  1279  }
  1280  
  1281  // bytes writes the value of z into buf using big-endian encoding.
  1282  // len(buf) must be >= len(z)*_S. The value of z is encoded in the
  1283  // slice buf[i:]. The number i of unused bytes at the beginning of
  1284  // buf is returned as result.
  1285  func (z nat) bytes(buf []byte) (i int) {
  1286  	i = len(buf)
  1287  	for _, d := range z {
  1288  		for j := 0; j < _S; j++ {
  1289  			i--
  1290  			buf[i] = byte(d)
  1291  			d >>= 8
  1292  		}
  1293  	}
  1294  
  1295  	for i < len(buf) && buf[i] == 0 {
  1296  		i++
  1297  	}
  1298  
  1299  	return
  1300  }
  1301  
  1302  // bigEndianWord returns the contents of buf interpreted as a big-endian encoded Word value.
  1303  func bigEndianWord(buf []byte) Word {
  1304  	if _W == 64 {
  1305  		return Word(binary.BigEndian.Uint64(buf))
  1306  	}
  1307  	return Word(binary.BigEndian.Uint32(buf))
  1308  }
  1309  
  1310  // setBytes interprets buf as the bytes of a big-endian unsigned
  1311  // integer, sets z to that value, and returns z.
  1312  func (z nat) setBytes(buf []byte) nat {
  1313  	z = z.make((len(buf) + _S - 1) / _S)
  1314  
  1315  	i := len(buf)
  1316  	for k := 0; i >= _S; k++ {
  1317  		z[k] = bigEndianWord(buf[i-_S : i])
  1318  		i -= _S
  1319  	}
  1320  	if i > 0 {
  1321  		var d Word
  1322  		for s := uint(0); i > 0; s += 8 {
  1323  			d |= Word(buf[i-1]) << s
  1324  			i--
  1325  		}
  1326  		z[len(z)-1] = d
  1327  	}
  1328  
  1329  	return z.norm()
  1330  }
  1331  
  1332  // sqrt sets z = ⌊√x⌋
  1333  func (z nat) sqrt(x nat) nat {
  1334  	if x.cmp(natOne) <= 0 {
  1335  		return z.set(x)
  1336  	}
  1337  	if alias(z, x) {
  1338  		z = nil
  1339  	}
  1340  
  1341  	// Start with value known to be too large and repeat "z = ⌊(z + ⌊x/z⌋)/2⌋" until it stops getting smaller.
  1342  	// See Brent and Zimmermann, Modern Computer Arithmetic, Algorithm 1.13 (SqrtInt).
  1343  	// https://members.loria.fr/PZimmermann/mca/pub226.html
  1344  	// If x is one less than a perfect square, the sequence oscillates between the correct z and z+1;
  1345  	// otherwise it converges to the correct z and stays there.
  1346  	var z1, z2 nat
  1347  	z1 = z
  1348  	z1 = z1.setUint64(1)
  1349  	z1 = z1.shl(z1, uint(x.bitLen()+1)/2) // must be ≥ √x
  1350  	for n := 0; ; n++ {
  1351  		z2, _ = z2.div(nil, x, z1)
  1352  		z2 = z2.add(z2, z1)
  1353  		z2 = z2.shr(z2, 1)
  1354  		if z2.cmp(z1) >= 0 {
  1355  			// z1 is answer.
  1356  			// Figure out whether z1 or z2 is currently aliased to z by looking at loop count.
  1357  			if n&1 == 0 {
  1358  				return z1
  1359  			}
  1360  			return z.set(z1)
  1361  		}
  1362  		z1, z2 = z2, z1
  1363  	}
  1364  }
  1365  

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