Source file src/math/big/nat.go

     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  	"internal/byteorder"
    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  type nat []Word
    35  
    36  var (
    37  	natOne  = nat{1}
    38  	natTwo  = nat{2}
    39  	natFive = nat{5}
    40  	natTen  = nat{10}
    41  )
    42  
    43  func (z nat) String() string {
    44  	return "0x" + string(z.itoa(false, 16))
    45  }
    46  
    47  func (z nat) norm() nat {
    48  	i := len(z)
    49  	for i > 0 && z[i-1] == 0 {
    50  		i--
    51  	}
    52  	return z[0:i]
    53  }
    54  
    55  func (z nat) make(n int) nat {
    56  	if n <= cap(z) {
    57  		return z[:n] // reuse z
    58  	}
    59  	if n == 1 {
    60  		// Most nats start small and stay that way; don't over-allocate.
    61  		return make(nat, 1)
    62  	}
    63  	// Choosing a good value for e has significant performance impact
    64  	// because it increases the chance that a value can be reused.
    65  	const e = 4 // extra capacity
    66  	return make(nat, n, n+e)
    67  }
    68  
    69  func (z nat) setWord(x Word) nat {
    70  	if x == 0 {
    71  		return z[:0]
    72  	}
    73  	z = z.make(1)
    74  	z[0] = x
    75  	return z
    76  }
    77  
    78  func (z nat) setUint64(x uint64) nat {
    79  	// single-word value
    80  	if w := Word(x); uint64(w) == x {
    81  		return z.setWord(w)
    82  	}
    83  	// 2-word value
    84  	z = z.make(2)
    85  	z[1] = Word(x >> 32)
    86  	z[0] = Word(x)
    87  	return z
    88  }
    89  
    90  func (z nat) set(x nat) nat {
    91  	z = z.make(len(x))
    92  	copy(z, x)
    93  	return z
    94  }
    95  
    96  func (z nat) add(x, y nat) nat {
    97  	m := len(x)
    98  	n := len(y)
    99  
   100  	switch {
   101  	case m < n:
   102  		return z.add(y, x)
   103  	case m == 0:
   104  		// n == 0 because m >= n; result is 0
   105  		return z[:0]
   106  	case n == 0:
   107  		// result is x
   108  		return z.set(x)
   109  	}
   110  	// m > 0
   111  
   112  	z = z.make(m + 1)
   113  	c := addVV(z[0:n], x, y)
   114  	if m > n {
   115  		c = addVW(z[n:m], x[n:], c)
   116  	}
   117  	z[m] = c
   118  
   119  	return z.norm()
   120  }
   121  
   122  func (z nat) sub(x, y nat) nat {
   123  	m := len(x)
   124  	n := len(y)
   125  
   126  	switch {
   127  	case m < n:
   128  		panic("underflow")
   129  	case m == 0:
   130  		// n == 0 because m >= n; result is 0
   131  		return z[:0]
   132  	case n == 0:
   133  		// result is x
   134  		return z.set(x)
   135  	}
   136  	// m > 0
   137  
   138  	z = z.make(m)
   139  	c := subVV(z[0:n], x, y)
   140  	if m > n {
   141  		c = subVW(z[n:], x[n:], c)
   142  	}
   143  	if c != 0 {
   144  		panic("underflow")
   145  	}
   146  
   147  	return z.norm()
   148  }
   149  
   150  func (x nat) cmp(y nat) (r int) {
   151  	m := len(x)
   152  	n := len(y)
   153  	if m != n || m == 0 {
   154  		switch {
   155  		case m < n:
   156  			r = -1
   157  		case m > n:
   158  			r = 1
   159  		}
   160  		return
   161  	}
   162  
   163  	i := m - 1
   164  	for i > 0 && x[i] == y[i] {
   165  		i--
   166  	}
   167  
   168  	switch {
   169  	case x[i] < y[i]:
   170  		r = -1
   171  	case x[i] > y[i]:
   172  		r = 1
   173  	}
   174  	return
   175  }
   176  
   177  func (z nat) mulAddWW(x nat, y, r Word) nat {
   178  	m := len(x)
   179  	if m == 0 || y == 0 {
   180  		return z.setWord(r) // result is r
   181  	}
   182  	// m > 0
   183  
   184  	z = z.make(m + 1)
   185  	z[m] = mulAddVWW(z[0:m], x, y, r)
   186  
   187  	return z.norm()
   188  }
   189  
   190  // basicMul multiplies x and y and leaves the result in z.
   191  // The (non-normalized) result is placed in z[0 : len(x) + len(y)].
   192  func basicMul(z, x, y nat) {
   193  	clear(z[0 : len(x)+len(y)]) // initialize z
   194  	for i, d := range y {
   195  		if d != 0 {
   196  			z[len(x)+i] = addMulVVW(z[i:i+len(x)], x, d)
   197  		}
   198  	}
   199  }
   200  
   201  // montgomery computes z mod m = x*y*2**(-n*_W) mod m,
   202  // assuming k = -1/m mod 2**_W.
   203  // z is used for storing the result which is returned;
   204  // z must not alias x, y or m.
   205  // See Gueron, "Efficient Software Implementations of Modular Exponentiation".
   206  // https://eprint.iacr.org/2011/239.pdf
   207  // In the terminology of that paper, this is an "Almost Montgomery Multiplication":
   208  // x and y are required to satisfy 0 <= z < 2**(n*_W) and then the result
   209  // z is guaranteed to satisfy 0 <= z < 2**(n*_W), but it may not be < m.
   210  func (z nat) montgomery(x, y, m nat, k Word, n int) nat {
   211  	// This code assumes x, y, m are all the same length, n.
   212  	// (required by addMulVVW and the for loop).
   213  	// It also assumes that x, y are already reduced mod m,
   214  	// or else the result will not be properly reduced.
   215  	if len(x) != n || len(y) != n || len(m) != n {
   216  		panic("math/big: mismatched montgomery number lengths")
   217  	}
   218  	z = z.make(n * 2)
   219  	clear(z)
   220  	var c Word
   221  	for i := 0; i < n; i++ {
   222  		d := y[i]
   223  		c2 := addMulVVW(z[i:n+i], x, d)
   224  		t := z[i] * k
   225  		c3 := addMulVVW(z[i:n+i], m, t)
   226  		cx := c + c2
   227  		cy := cx + c3
   228  		z[n+i] = cy
   229  		if cx < c2 || cy < c3 {
   230  			c = 1
   231  		} else {
   232  			c = 0
   233  		}
   234  	}
   235  	if c != 0 {
   236  		subVV(z[:n], z[n:], m)
   237  	} else {
   238  		copy(z[:n], z[n:])
   239  	}
   240  	return z[:n]
   241  }
   242  
   243  // Fast version of z[0:n+n>>1].add(z[0:n+n>>1], x[0:n]) w/o bounds checks.
   244  // Factored out for readability - do not use outside karatsuba.
   245  func karatsubaAdd(z, x nat, n int) {
   246  	if c := addVV(z[0:n], z, x); c != 0 {
   247  		addVW(z[n:n+n>>1], z[n:], c)
   248  	}
   249  }
   250  
   251  // Like karatsubaAdd, but does subtract.
   252  func karatsubaSub(z, x nat, n int) {
   253  	if c := subVV(z[0:n], z, x); c != 0 {
   254  		subVW(z[n:n+n>>1], z[n:], c)
   255  	}
   256  }
   257  
   258  // Operands that are shorter than karatsubaThreshold are multiplied using
   259  // "grade school" multiplication; for longer operands the Karatsuba algorithm
   260  // is used.
   261  var karatsubaThreshold = 40 // computed by calibrate_test.go
   262  
   263  // karatsuba multiplies x and y and leaves the result in z.
   264  // Both x and y must have the same length n and n must be a
   265  // power of 2. The result vector z must have len(z) >= 6*n.
   266  // The (non-normalized) result is placed in z[0 : 2*n].
   267  func karatsuba(z, x, y nat) {
   268  	n := len(y)
   269  
   270  	// Switch to basic multiplication if numbers are odd or small.
   271  	// (n is always even if karatsubaThreshold is even, but be
   272  	// conservative)
   273  	if n&1 != 0 || n < karatsubaThreshold || n < 2 {
   274  		basicMul(z, x, y)
   275  		return
   276  	}
   277  	// n&1 == 0 && n >= karatsubaThreshold && n >= 2
   278  
   279  	// Karatsuba multiplication is based on the observation that
   280  	// for two numbers x and y with:
   281  	//
   282  	//   x = x1*b + x0
   283  	//   y = y1*b + y0
   284  	//
   285  	// the product x*y can be obtained with 3 products z2, z1, z0
   286  	// instead of 4:
   287  	//
   288  	//   x*y = x1*y1*b*b + (x1*y0 + x0*y1)*b + x0*y0
   289  	//       =    z2*b*b +              z1*b +    z0
   290  	//
   291  	// with:
   292  	//
   293  	//   xd = x1 - x0
   294  	//   yd = y0 - y1
   295  	//
   296  	//   z1 =      xd*yd                    + z2 + z0
   297  	//      = (x1-x0)*(y0 - y1)             + z2 + z0
   298  	//      = x1*y0 - x1*y1 - x0*y0 + x0*y1 + z2 + z0
   299  	//      = x1*y0 -    z2 -    z0 + x0*y1 + z2 + z0
   300  	//      = x1*y0                 + x0*y1
   301  
   302  	// split x, y into "digits"
   303  	n2 := n >> 1              // n2 >= 1
   304  	x1, x0 := x[n2:], x[0:n2] // x = x1*b + y0
   305  	y1, y0 := y[n2:], y[0:n2] // y = y1*b + y0
   306  
   307  	// z is used for the result and temporary storage:
   308  	//
   309  	//   6*n     5*n     4*n     3*n     2*n     1*n     0*n
   310  	// z = [z2 copy|z0 copy| xd*yd | yd:xd | x1*y1 | x0*y0 ]
   311  	//
   312  	// For each recursive call of karatsuba, an unused slice of
   313  	// z is passed in that has (at least) half the length of the
   314  	// caller's z.
   315  
   316  	// compute z0 and z2 with the result "in place" in z
   317  	karatsuba(z, x0, y0)     // z0 = x0*y0
   318  	karatsuba(z[n:], x1, y1) // z2 = x1*y1
   319  
   320  	// compute xd (or the negative value if underflow occurs)
   321  	s := 1 // sign of product xd*yd
   322  	xd := z[2*n : 2*n+n2]
   323  	if subVV(xd, x1, x0) != 0 { // x1-x0
   324  		s = -s
   325  		subVV(xd, x0, x1) // x0-x1
   326  	}
   327  
   328  	// compute yd (or the negative value if underflow occurs)
   329  	yd := z[2*n+n2 : 3*n]
   330  	if subVV(yd, y0, y1) != 0 { // y0-y1
   331  		s = -s
   332  		subVV(yd, y1, y0) // y1-y0
   333  	}
   334  
   335  	// p = (x1-x0)*(y0-y1) == x1*y0 - x1*y1 - x0*y0 + x0*y1 for s > 0
   336  	// p = (x0-x1)*(y0-y1) == x0*y0 - x0*y1 - x1*y0 + x1*y1 for s < 0
   337  	p := z[n*3:]
   338  	karatsuba(p, xd, yd)
   339  
   340  	// save original z2:z0
   341  	// (ok to use upper half of z since we're done recurring)
   342  	r := z[n*4:]
   343  	copy(r, z[:n*2])
   344  
   345  	// add up all partial products
   346  	//
   347  	//   2*n     n     0
   348  	// z = [ z2  | z0  ]
   349  	//   +    [ z0  ]
   350  	//   +    [ z2  ]
   351  	//   +    [  p  ]
   352  	//
   353  	karatsubaAdd(z[n2:], r, n)
   354  	karatsubaAdd(z[n2:], r[n:], n)
   355  	if s > 0 {
   356  		karatsubaAdd(z[n2:], p, n)
   357  	} else {
   358  		karatsubaSub(z[n2:], p, n)
   359  	}
   360  }
   361  
   362  // alias reports whether x and y share the same base array.
   363  //
   364  // Note: alias assumes that the capacity of underlying arrays
   365  // is never changed for nat values; i.e. that there are
   366  // no 3-operand slice expressions in this code (or worse,
   367  // reflect-based operations to the same effect).
   368  func alias(x, y nat) bool {
   369  	return cap(x) > 0 && cap(y) > 0 && &x[0:cap(x)][cap(x)-1] == &y[0:cap(y)][cap(y)-1]
   370  }
   371  
   372  // addAt implements z += x<<(_W*i); z must be long enough.
   373  // (we don't use nat.add because we need z to stay the same
   374  // slice, and we don't need to normalize z after each addition)
   375  func addAt(z, x nat, i int) {
   376  	if n := len(x); n > 0 {
   377  		if c := addVV(z[i:i+n], z[i:], x); c != 0 {
   378  			j := i + n
   379  			if j < len(z) {
   380  				addVW(z[j:], z[j:], c)
   381  			}
   382  		}
   383  	}
   384  }
   385  
   386  // karatsubaLen computes an approximation to the maximum k <= n such that
   387  // k = p<<i for a number p <= threshold and an i >= 0. Thus, the
   388  // result is the largest number that can be divided repeatedly by 2 before
   389  // becoming about the value of threshold.
   390  func karatsubaLen(n, threshold int) int {
   391  	i := uint(0)
   392  	for n > threshold {
   393  		n >>= 1
   394  		i++
   395  	}
   396  	return n << i
   397  }
   398  
   399  func (z nat) mul(x, y nat) nat {
   400  	m := len(x)
   401  	n := len(y)
   402  
   403  	switch {
   404  	case m < n:
   405  		return z.mul(y, x)
   406  	case m == 0 || n == 0:
   407  		return z[:0]
   408  	case n == 1:
   409  		return z.mulAddWW(x, y[0], 0)
   410  	}
   411  	// m >= n > 1
   412  
   413  	// determine if z can be reused
   414  	if alias(z, x) || alias(z, y) {
   415  		z = nil // z is an alias for x or y - cannot reuse
   416  	}
   417  
   418  	// use basic multiplication if the numbers are small
   419  	if n < karatsubaThreshold {
   420  		z = z.make(m + n)
   421  		basicMul(z, x, y)
   422  		return z.norm()
   423  	}
   424  	// m >= n && n >= karatsubaThreshold && n >= 2
   425  
   426  	// determine Karatsuba length k such that
   427  	//
   428  	//   x = xh*b + x0  (0 <= x0 < b)
   429  	//   y = yh*b + y0  (0 <= y0 < b)
   430  	//   b = 1<<(_W*k)  ("base" of digits xi, yi)
   431  	//
   432  	k := karatsubaLen(n, karatsubaThreshold)
   433  	// k <= n
   434  
   435  	// multiply x0 and y0 via Karatsuba
   436  	x0 := x[0:k]              // x0 is not normalized
   437  	y0 := y[0:k]              // y0 is not normalized
   438  	z = z.make(max(6*k, m+n)) // enough space for karatsuba of x0*y0 and full result of x*y
   439  	karatsuba(z, x0, y0)
   440  	z = z[0 : m+n] // z has final length but may be incomplete
   441  	clear(z[2*k:]) // upper portion of z is garbage (and 2*k <= m+n since k <= n <= m)
   442  
   443  	// If xh != 0 or yh != 0, add the missing terms to z. For
   444  	//
   445  	//   xh = xi*b^i + ... + x2*b^2 + x1*b (0 <= xi < b)
   446  	//   yh =                         y1*b (0 <= y1 < b)
   447  	//
   448  	// the missing terms are
   449  	//
   450  	//   x0*y1*b and xi*y0*b^i, xi*y1*b^(i+1) for i > 0
   451  	//
   452  	// since all the yi for i > 1 are 0 by choice of k: If any of them
   453  	// were > 0, then yh >= b^2 and thus y >= b^2. Then k' = k*2 would
   454  	// be a larger valid threshold contradicting the assumption about k.
   455  	//
   456  	if k < n || m != n {
   457  		tp := getNat(3 * k)
   458  		t := *tp
   459  
   460  		// add x0*y1*b
   461  		x0 := x0.norm()
   462  		y1 := y[k:]       // y1 is normalized because y is
   463  		t = t.mul(x0, y1) // update t so we don't lose t's underlying array
   464  		addAt(z, t, k)
   465  
   466  		// add xi*y0<<i, xi*y1*b<<(i+k)
   467  		y0 := y0.norm()
   468  		for i := k; i < len(x); i += k {
   469  			xi := x[i:]
   470  			if len(xi) > k {
   471  				xi = xi[:k]
   472  			}
   473  			xi = xi.norm()
   474  			t = t.mul(xi, y0)
   475  			addAt(z, t, i)
   476  			t = t.mul(xi, y1)
   477  			addAt(z, t, i+k)
   478  		}
   479  
   480  		putNat(tp)
   481  	}
   482  
   483  	return z.norm()
   484  }
   485  
   486  // basicSqr sets z = x*x and is asymptotically faster than basicMul
   487  // by about a factor of 2, but slower for small arguments due to overhead.
   488  // Requirements: len(x) > 0, len(z) == 2*len(x)
   489  // The (non-normalized) result is placed in z.
   490  func basicSqr(z, x nat) {
   491  	n := len(x)
   492  	tp := getNat(2 * n)
   493  	t := *tp // temporary variable to hold the products
   494  	clear(t)
   495  	z[1], z[0] = mulWW(x[0], x[0]) // the initial square
   496  	for i := 1; i < n; i++ {
   497  		d := x[i]
   498  		// z collects the squares x[i] * x[i]
   499  		z[2*i+1], z[2*i] = mulWW(d, d)
   500  		// t collects the products x[i] * x[j] where j < i
   501  		t[2*i] = addMulVVW(t[i:2*i], x[0:i], d)
   502  	}
   503  	t[2*n-1] = shlVU(t[1:2*n-1], t[1:2*n-1], 1) // double the j < i products
   504  	addVV(z, z, t)                              // combine the result
   505  	putNat(tp)
   506  }
   507  
   508  // karatsubaSqr squares x and leaves the result in z.
   509  // len(x) must be a power of 2 and len(z) >= 6*len(x).
   510  // The (non-normalized) result is placed in z[0 : 2*len(x)].
   511  //
   512  // The algorithm and the layout of z are the same as for karatsuba.
   513  func karatsubaSqr(z, x nat) {
   514  	n := len(x)
   515  
   516  	if n&1 != 0 || n < karatsubaSqrThreshold || n < 2 {
   517  		basicSqr(z[:2*n], x)
   518  		return
   519  	}
   520  
   521  	n2 := n >> 1
   522  	x1, x0 := x[n2:], x[0:n2]
   523  
   524  	karatsubaSqr(z, x0)
   525  	karatsubaSqr(z[n:], x1)
   526  
   527  	// s = sign(xd*yd) == -1 for xd != 0; s == 1 for xd == 0
   528  	xd := z[2*n : 2*n+n2]
   529  	if subVV(xd, x1, x0) != 0 {
   530  		subVV(xd, x0, x1)
   531  	}
   532  
   533  	p := z[n*3:]
   534  	karatsubaSqr(p, xd)
   535  
   536  	r := z[n*4:]
   537  	copy(r, z[:n*2])
   538  
   539  	karatsubaAdd(z[n2:], r, n)
   540  	karatsubaAdd(z[n2:], r[n:], n)
   541  	karatsubaSub(z[n2:], p, n) // s == -1 for p != 0; s == 1 for p == 0
   542  }
   543  
   544  // Operands that are shorter than basicSqrThreshold are squared using
   545  // "grade school" multiplication; for operands longer than karatsubaSqrThreshold
   546  // we use the Karatsuba algorithm optimized for x == y.
   547  var basicSqrThreshold = 20      // computed by calibrate_test.go
   548  var karatsubaSqrThreshold = 260 // computed by calibrate_test.go
   549  
   550  // z = x*x
   551  func (z nat) sqr(x nat) nat {
   552  	n := len(x)
   553  	switch {
   554  	case n == 0:
   555  		return z[:0]
   556  	case n == 1:
   557  		d := x[0]
   558  		z = z.make(2)
   559  		z[1], z[0] = mulWW(d, d)
   560  		return z.norm()
   561  	}
   562  
   563  	if alias(z, x) {
   564  		z = nil // z is an alias for x - cannot reuse
   565  	}
   566  
   567  	if n < basicSqrThreshold {
   568  		z = z.make(2 * n)
   569  		basicMul(z, x, x)
   570  		return z.norm()
   571  	}
   572  	if n < karatsubaSqrThreshold {
   573  		z = z.make(2 * n)
   574  		basicSqr(z, x)
   575  		return z.norm()
   576  	}
   577  
   578  	// Use Karatsuba multiplication optimized for x == y.
   579  	// The algorithm and layout of z are the same as for mul.
   580  
   581  	// z = (x1*b + x0)^2 = x1^2*b^2 + 2*x1*x0*b + x0^2
   582  
   583  	k := karatsubaLen(n, karatsubaSqrThreshold)
   584  
   585  	x0 := x[0:k]
   586  	z = z.make(max(6*k, 2*n))
   587  	karatsubaSqr(z, x0) // z = x0^2
   588  	z = z[0 : 2*n]
   589  	clear(z[2*k:])
   590  
   591  	if k < n {
   592  		tp := getNat(2 * k)
   593  		t := *tp
   594  		x0 := x0.norm()
   595  		x1 := x[k:]
   596  		t = t.mul(x0, x1)
   597  		addAt(z, t, k)
   598  		addAt(z, t, k) // z = 2*x1*x0*b + x0^2
   599  		t = t.sqr(x1)
   600  		addAt(z, t, 2*k) // z = x1^2*b^2 + 2*x1*x0*b + x0^2
   601  		putNat(tp)
   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-a)/2 // avoid overflow
   622  	return z.mul(nat(nil).mulRange(a, m), nat(nil).mulRange(m+1, b))
   623  }
   624  
   625  // getNat returns a *nat of len n. The contents may not be zero.
   626  // The pool holds *nat to avoid allocation when converting to interface{}.
   627  func getNat(n int) *nat {
   628  	var z *nat
   629  	if v := natPool.Get(); v != nil {
   630  		z = v.(*nat)
   631  	}
   632  	if z == nil {
   633  		z = new(nat)
   634  	}
   635  	*z = z.make(n)
   636  	if n > 0 {
   637  		(*z)[0] = 0xfedcb // break code expecting zero
   638  	}
   639  	return z
   640  }
   641  
   642  func putNat(x *nat) {
   643  	natPool.Put(x)
   644  }
   645  
   646  var natPool sync.Pool
   647  
   648  // bitLen returns the length of x in bits.
   649  // Unlike most methods, it works even if x is not normalized.
   650  func (x nat) bitLen() int {
   651  	// This function is used in cryptographic operations. It must not leak
   652  	// anything but the Int's sign and bit size through side-channels. Any
   653  	// changes must be reviewed by a security expert.
   654  	if i := len(x) - 1; i >= 0 {
   655  		// bits.Len uses a lookup table for the low-order bits on some
   656  		// architectures. Neutralize any input-dependent behavior by setting all
   657  		// bits after the first one bit.
   658  		top := uint(x[i])
   659  		top |= top >> 1
   660  		top |= top >> 2
   661  		top |= top >> 4
   662  		top |= top >> 8
   663  		top |= top >> 16
   664  		top |= top >> 16 >> 16 // ">> 32" doesn't compile on 32-bit architectures
   665  		return i*_W + bits.Len(top)
   666  	}
   667  	return 0
   668  }
   669  
   670  // trailingZeroBits returns the number of consecutive least significant zero
   671  // bits of x.
   672  func (x nat) trailingZeroBits() uint {
   673  	if len(x) == 0 {
   674  		return 0
   675  	}
   676  	var i uint
   677  	for x[i] == 0 {
   678  		i++
   679  	}
   680  	// x[i] != 0
   681  	return i*_W + uint(bits.TrailingZeros(uint(x[i])))
   682  }
   683  
   684  // isPow2 returns i, true when x == 2**i and 0, false otherwise.
   685  func (x nat) isPow2() (uint, bool) {
   686  	var i uint
   687  	for x[i] == 0 {
   688  		i++
   689  	}
   690  	if i == uint(len(x))-1 && x[i]&(x[i]-1) == 0 {
   691  		return i*_W + uint(bits.TrailingZeros(uint(x[i]))), true
   692  	}
   693  	return 0, false
   694  }
   695  
   696  func same(x, y nat) bool {
   697  	return len(x) == len(y) && len(x) > 0 && &x[0] == &y[0]
   698  }
   699  
   700  // z = x << s
   701  func (z nat) shl(x nat, s uint) nat {
   702  	if s == 0 {
   703  		if same(z, x) {
   704  			return z
   705  		}
   706  		if !alias(z, x) {
   707  			return z.set(x)
   708  		}
   709  	}
   710  
   711  	m := len(x)
   712  	if m == 0 {
   713  		return z[:0]
   714  	}
   715  	// m > 0
   716  
   717  	n := m + int(s/_W)
   718  	z = z.make(n + 1)
   719  	z[n] = shlVU(z[n-m:n], x, s%_W)
   720  	clear(z[0 : n-m])
   721  
   722  	return z.norm()
   723  }
   724  
   725  // z = x >> s
   726  func (z nat) shr(x nat, s uint) nat {
   727  	if s == 0 {
   728  		if same(z, x) {
   729  			return z
   730  		}
   731  		if !alias(z, x) {
   732  			return z.set(x)
   733  		}
   734  	}
   735  
   736  	m := len(x)
   737  	n := m - int(s/_W)
   738  	if n <= 0 {
   739  		return z[:0]
   740  	}
   741  	// n > 0
   742  
   743  	z = z.make(n)
   744  	shrVU(z, x[m-n:], s%_W)
   745  
   746  	return z.norm()
   747  }
   748  
   749  func (z nat) setBit(x nat, i uint, b uint) nat {
   750  	j := int(i / _W)
   751  	m := Word(1) << (i % _W)
   752  	n := len(x)
   753  	switch b {
   754  	case 0:
   755  		z = z.make(n)
   756  		copy(z, x)
   757  		if j >= n {
   758  			// no need to grow
   759  			return z
   760  		}
   761  		z[j] &^= m
   762  		return z.norm()
   763  	case 1:
   764  		if j >= n {
   765  			z = z.make(j + 1)
   766  			clear(z[n:])
   767  		} else {
   768  			z = z.make(n)
   769  		}
   770  		copy(z, x)
   771  		z[j] |= m
   772  		// no need to normalize
   773  		return z
   774  	}
   775  	panic("set bit is not 0 or 1")
   776  }
   777  
   778  // bit returns the value of the i'th bit, with lsb == bit 0.
   779  func (x nat) bit(i uint) uint {
   780  	j := i / _W
   781  	if j >= uint(len(x)) {
   782  		return 0
   783  	}
   784  	// 0 <= j < len(x)
   785  	return uint(x[j] >> (i % _W) & 1)
   786  }
   787  
   788  // sticky returns 1 if there's a 1 bit within the
   789  // i least significant bits, otherwise it returns 0.
   790  func (x nat) sticky(i uint) uint {
   791  	j := i / _W
   792  	if j >= uint(len(x)) {
   793  		if len(x) == 0 {
   794  			return 0
   795  		}
   796  		return 1
   797  	}
   798  	// 0 <= j < len(x)
   799  	for _, x := range x[:j] {
   800  		if x != 0 {
   801  			return 1
   802  		}
   803  	}
   804  	if x[j]<<(_W-i%_W) != 0 {
   805  		return 1
   806  	}
   807  	return 0
   808  }
   809  
   810  func (z nat) and(x, y nat) nat {
   811  	m := len(x)
   812  	n := len(y)
   813  	if m > n {
   814  		m = n
   815  	}
   816  	// m <= n
   817  
   818  	z = z.make(m)
   819  	for i := 0; i < m; i++ {
   820  		z[i] = x[i] & y[i]
   821  	}
   822  
   823  	return z.norm()
   824  }
   825  
   826  // trunc returns z = x mod 2ⁿ.
   827  func (z nat) trunc(x nat, n uint) nat {
   828  	w := (n + _W - 1) / _W
   829  	if uint(len(x)) < w {
   830  		return z.set(x)
   831  	}
   832  	z = z.make(int(w))
   833  	copy(z, x)
   834  	if n%_W != 0 {
   835  		z[len(z)-1] &= 1<<(n%_W) - 1
   836  	}
   837  	return z.norm()
   838  }
   839  
   840  func (z nat) andNot(x, y nat) nat {
   841  	m := len(x)
   842  	n := len(y)
   843  	if n > m {
   844  		n = m
   845  	}
   846  	// m >= n
   847  
   848  	z = z.make(m)
   849  	for i := 0; i < n; i++ {
   850  		z[i] = x[i] &^ y[i]
   851  	}
   852  	copy(z[n:m], x[n:m])
   853  
   854  	return z.norm()
   855  }
   856  
   857  func (z nat) or(x, y nat) nat {
   858  	m := len(x)
   859  	n := len(y)
   860  	s := x
   861  	if m < n {
   862  		n, m = m, n
   863  		s = y
   864  	}
   865  	// m >= n
   866  
   867  	z = z.make(m)
   868  	for i := 0; i < n; i++ {
   869  		z[i] = x[i] | y[i]
   870  	}
   871  	copy(z[n:m], s[n:m])
   872  
   873  	return z.norm()
   874  }
   875  
   876  func (z nat) xor(x, y nat) nat {
   877  	m := len(x)
   878  	n := len(y)
   879  	s := x
   880  	if m < n {
   881  		n, m = m, n
   882  		s = y
   883  	}
   884  	// m >= n
   885  
   886  	z = z.make(m)
   887  	for i := 0; i < n; i++ {
   888  		z[i] = x[i] ^ y[i]
   889  	}
   890  	copy(z[n:m], s[n:m])
   891  
   892  	return z.norm()
   893  }
   894  
   895  // random creates a random integer in [0..limit), using the space in z if
   896  // possible. n is the bit length of limit.
   897  func (z nat) random(rand *rand.Rand, limit nat, n int) nat {
   898  	if alias(z, limit) {
   899  		z = nil // z is an alias for limit - cannot reuse
   900  	}
   901  	z = z.make(len(limit))
   902  
   903  	bitLengthOfMSW := uint(n % _W)
   904  	if bitLengthOfMSW == 0 {
   905  		bitLengthOfMSW = _W
   906  	}
   907  	mask := Word((1 << bitLengthOfMSW) - 1)
   908  
   909  	for {
   910  		switch _W {
   911  		case 32:
   912  			for i := range z {
   913  				z[i] = Word(rand.Uint32())
   914  			}
   915  		case 64:
   916  			for i := range z {
   917  				z[i] = Word(rand.Uint32()) | Word(rand.Uint32())<<32
   918  			}
   919  		default:
   920  			panic("unknown word size")
   921  		}
   922  		z[len(limit)-1] &= mask
   923  		if z.cmp(limit) < 0 {
   924  			break
   925  		}
   926  	}
   927  
   928  	return z.norm()
   929  }
   930  
   931  // If m != 0 (i.e., len(m) != 0), expNN sets z to x**y mod m;
   932  // otherwise it sets z to x**y. The result is the value of z.
   933  func (z nat) expNN(x, y, m nat, slow bool) nat {
   934  	if alias(z, x) || alias(z, y) {
   935  		// We cannot allow in-place modification of x or y.
   936  		z = nil
   937  	}
   938  
   939  	// x**y mod 1 == 0
   940  	if len(m) == 1 && m[0] == 1 {
   941  		return z.setWord(0)
   942  	}
   943  	// m == 0 || m > 1
   944  
   945  	// x**0 == 1
   946  	if len(y) == 0 {
   947  		return z.setWord(1)
   948  	}
   949  	// y > 0
   950  
   951  	// 0**y = 0
   952  	if len(x) == 0 {
   953  		return z.setWord(0)
   954  	}
   955  	// x > 0
   956  
   957  	// 1**y = 1
   958  	if len(x) == 1 && x[0] == 1 {
   959  		return z.setWord(1)
   960  	}
   961  	// x > 1
   962  
   963  	// x**1 == x
   964  	if len(y) == 1 && y[0] == 1 {
   965  		if len(m) != 0 {
   966  			return z.rem(x, m)
   967  		}
   968  		return z.set(x)
   969  	}
   970  	// y > 1
   971  
   972  	if len(m) != 0 {
   973  		// We likely end up being as long as the modulus.
   974  		z = z.make(len(m))
   975  
   976  		// If the exponent is large, we use the Montgomery method for odd values,
   977  		// and a 4-bit, windowed exponentiation for powers of two,
   978  		// and a CRT-decomposed Montgomery method for the remaining values
   979  		// (even values times non-trivial odd values, which decompose into one
   980  		// instance of each of the first two cases).
   981  		if len(y) > 1 && !slow {
   982  			if m[0]&1 == 1 {
   983  				return z.expNNMontgomery(x, y, m)
   984  			}
   985  			if logM, ok := m.isPow2(); ok {
   986  				return z.expNNWindowed(x, y, logM)
   987  			}
   988  			return z.expNNMontgomeryEven(x, y, m)
   989  		}
   990  	}
   991  
   992  	z = z.set(x)
   993  	v := y[len(y)-1] // v > 0 because y is normalized and y > 0
   994  	shift := nlz(v) + 1
   995  	v <<= shift
   996  	var q nat
   997  
   998  	const mask = 1 << (_W - 1)
   999  
  1000  	// We walk through the bits of the exponent one by one. Each time we
  1001  	// see a bit, we square, thus doubling the power. If the bit is a one,
  1002  	// we also multiply by x, thus adding one to the power.
  1003  
  1004  	w := _W - int(shift)
  1005  	// zz and r are used to avoid allocating in mul and div as
  1006  	// otherwise the arguments would alias.
  1007  	var zz, r nat
  1008  	for j := 0; j < w; j++ {
  1009  		zz = zz.sqr(z)
  1010  		zz, z = z, zz
  1011  
  1012  		if v&mask != 0 {
  1013  			zz = zz.mul(z, x)
  1014  			zz, z = z, zz
  1015  		}
  1016  
  1017  		if len(m) != 0 {
  1018  			zz, r = zz.div(r, z, m)
  1019  			zz, r, q, z = q, z, zz, r
  1020  		}
  1021  
  1022  		v <<= 1
  1023  	}
  1024  
  1025  	for i := len(y) - 2; i >= 0; i-- {
  1026  		v = y[i]
  1027  
  1028  		for j := 0; j < _W; j++ {
  1029  			zz = zz.sqr(z)
  1030  			zz, z = z, zz
  1031  
  1032  			if v&mask != 0 {
  1033  				zz = zz.mul(z, x)
  1034  				zz, z = z, zz
  1035  			}
  1036  
  1037  			if len(m) != 0 {
  1038  				zz, r = zz.div(r, z, m)
  1039  				zz, r, q, z = q, z, zz, r
  1040  			}
  1041  
  1042  			v <<= 1
  1043  		}
  1044  	}
  1045  
  1046  	return z.norm()
  1047  }
  1048  
  1049  // expNNMontgomeryEven calculates x**y mod m where m = m1 × m2 for m1 = 2ⁿ and m2 odd.
  1050  // It uses two recursive calls to expNN for x**y mod m1 and x**y mod m2
  1051  // and then uses the Chinese Remainder Theorem to combine the results.
  1052  // The recursive call using m1 will use expNNWindowed,
  1053  // while the recursive call using m2 will use expNNMontgomery.
  1054  // For more details, see Ç. K. Koç, “Montgomery Reduction with Even Modulus”,
  1055  // IEE Proceedings: Computers and Digital Techniques, 141(5) 314-316, September 1994.
  1056  // http://www.people.vcu.edu/~jwang3/CMSC691/j34monex.pdf
  1057  func (z nat) expNNMontgomeryEven(x, y, m nat) nat {
  1058  	// Split m = m₁ × m₂ where m₁ = 2ⁿ
  1059  	n := m.trailingZeroBits()
  1060  	m1 := nat(nil).shl(natOne, n)
  1061  	m2 := nat(nil).shr(m, n)
  1062  
  1063  	// We want z = x**y mod m.
  1064  	// z₁ = x**y mod m1 = (x**y mod m) mod m1 = z mod m1
  1065  	// z₂ = x**y mod m2 = (x**y mod m) mod m2 = z mod m2
  1066  	// (We are using the math/big convention for names here,
  1067  	// where the computation is z = x**y mod m, so its parts are z1 and z2.
  1068  	// The paper is computing x = a**e mod n; it refers to these as x2 and z1.)
  1069  	z1 := nat(nil).expNN(x, y, m1, false)
  1070  	z2 := nat(nil).expNN(x, y, m2, false)
  1071  
  1072  	// Reconstruct z from z₁, z₂ using CRT, using algorithm from paper,
  1073  	// which uses only a single modInverse (and an easy one at that).
  1074  	//	p = (z₁ - z₂) × m₂⁻¹ (mod m₁)
  1075  	//	z = z₂ + p × m₂
  1076  	// The final addition is in range because:
  1077  	//	z = z₂ + p × m₂
  1078  	//	  ≤ z₂ + (m₁-1) × m₂
  1079  	//	  < m₂ + (m₁-1) × m₂
  1080  	//	  = m₁ × m₂
  1081  	//	  = m.
  1082  	z = z.set(z2)
  1083  
  1084  	// Compute (z₁ - z₂) mod m1 [m1 == 2**n] into z1.
  1085  	z1 = z1.subMod2N(z1, z2, n)
  1086  
  1087  	// Reuse z2 for p = (z₁ - z₂) [in z1] * m2⁻¹ (mod m₁ [= 2ⁿ]).
  1088  	m2inv := nat(nil).modInverse(m2, m1)
  1089  	z2 = z2.mul(z1, m2inv)
  1090  	z2 = z2.trunc(z2, n)
  1091  
  1092  	// Reuse z1 for p * m2.
  1093  	z = z.add(z, z1.mul(z2, m2))
  1094  
  1095  	return z
  1096  }
  1097  
  1098  // expNNWindowed calculates x**y mod m using a fixed, 4-bit window,
  1099  // where m = 2**logM.
  1100  func (z nat) expNNWindowed(x, y nat, logM uint) nat {
  1101  	if len(y) <= 1 {
  1102  		panic("big: misuse of expNNWindowed")
  1103  	}
  1104  	if x[0]&1 == 0 {
  1105  		// len(y) > 1, so y  > logM.
  1106  		// x is even, so x**y is a multiple of 2**y which is a multiple of 2**logM.
  1107  		return z.setWord(0)
  1108  	}
  1109  	if logM == 1 {
  1110  		return z.setWord(1)
  1111  	}
  1112  
  1113  	// zz is used to avoid allocating in mul as otherwise
  1114  	// the arguments would alias.
  1115  	w := int((logM + _W - 1) / _W)
  1116  	zzp := getNat(w)
  1117  	zz := *zzp
  1118  
  1119  	const n = 4
  1120  	// powers[i] contains x^i.
  1121  	var powers [1 << n]*nat
  1122  	for i := range powers {
  1123  		powers[i] = getNat(w)
  1124  	}
  1125  	*powers[0] = powers[0].set(natOne)
  1126  	*powers[1] = powers[1].trunc(x, logM)
  1127  	for i := 2; i < 1<<n; i += 2 {
  1128  		p2, p, p1 := powers[i/2], powers[i], powers[i+1]
  1129  		*p = p.sqr(*p2)
  1130  		*p = p.trunc(*p, logM)
  1131  		*p1 = p1.mul(*p, x)
  1132  		*p1 = p1.trunc(*p1, logM)
  1133  	}
  1134  
  1135  	// Because phi(2**logM) = 2**(logM-1), x**(2**(logM-1)) = 1,
  1136  	// so we can compute x**(y mod 2**(logM-1)) instead of x**y.
  1137  	// That is, we can throw away all but the bottom logM-1 bits of y.
  1138  	// Instead of allocating a new y, we start reading y at the right word
  1139  	// and truncate it appropriately at the start of the loop.
  1140  	i := len(y) - 1
  1141  	mtop := int((logM - 2) / _W) // -2 because the top word of N bits is the (N-1)/W'th word.
  1142  	mmask := ^Word(0)
  1143  	if mbits := (logM - 1) & (_W - 1); mbits != 0 {
  1144  		mmask = (1 << mbits) - 1
  1145  	}
  1146  	if i > mtop {
  1147  		i = mtop
  1148  	}
  1149  	advance := false
  1150  	z = z.setWord(1)
  1151  	for ; i >= 0; i-- {
  1152  		yi := y[i]
  1153  		if i == mtop {
  1154  			yi &= mmask
  1155  		}
  1156  		for j := 0; j < _W; j += n {
  1157  			if advance {
  1158  				// Account for use of 4 bits in previous iteration.
  1159  				// Unrolled loop for significant performance
  1160  				// gain. Use go test -bench=".*" in crypto/rsa
  1161  				// to check performance before making changes.
  1162  				zz = zz.sqr(z)
  1163  				zz, z = z, zz
  1164  				z = z.trunc(z, logM)
  1165  
  1166  				zz = zz.sqr(z)
  1167  				zz, z = z, zz
  1168  				z = z.trunc(z, logM)
  1169  
  1170  				zz = zz.sqr(z)
  1171  				zz, z = z, zz
  1172  				z = z.trunc(z, logM)
  1173  
  1174  				zz = zz.sqr(z)
  1175  				zz, z = z, zz
  1176  				z = z.trunc(z, logM)
  1177  			}
  1178  
  1179  			zz = zz.mul(z, *powers[yi>>(_W-n)])
  1180  			zz, z = z, zz
  1181  			z = z.trunc(z, logM)
  1182  
  1183  			yi <<= n
  1184  			advance = true
  1185  		}
  1186  	}
  1187  
  1188  	*zzp = zz
  1189  	putNat(zzp)
  1190  	for i := range powers {
  1191  		putNat(powers[i])
  1192  	}
  1193  
  1194  	return z.norm()
  1195  }
  1196  
  1197  // expNNMontgomery calculates x**y mod m using a fixed, 4-bit window.
  1198  // Uses Montgomery representation.
  1199  func (z nat) expNNMontgomery(x, y, m nat) nat {
  1200  	numWords := len(m)
  1201  
  1202  	// We want the lengths of x and m to be equal.
  1203  	// It is OK if x >= m as long as len(x) == len(m).
  1204  	if len(x) > numWords {
  1205  		_, x = nat(nil).div(nil, x, m)
  1206  		// Note: now len(x) <= numWords, not guaranteed ==.
  1207  	}
  1208  	if len(x) < numWords {
  1209  		rr := make(nat, numWords)
  1210  		copy(rr, x)
  1211  		x = rr
  1212  	}
  1213  
  1214  	// Ideally the precomputations would be performed outside, and reused
  1215  	// k0 = -m**-1 mod 2**_W. Algorithm from: Dumas, J.G. "On Newton–Raphson
  1216  	// Iteration for Multiplicative Inverses Modulo Prime Powers".
  1217  	k0 := 2 - m[0]
  1218  	t := m[0] - 1
  1219  	for i := 1; i < _W; i <<= 1 {
  1220  		t *= t
  1221  		k0 *= (t + 1)
  1222  	}
  1223  	k0 = -k0
  1224  
  1225  	// RR = 2**(2*_W*len(m)) mod m
  1226  	RR := nat(nil).setWord(1)
  1227  	zz := nat(nil).shl(RR, uint(2*numWords*_W))
  1228  	_, RR = nat(nil).div(RR, zz, m)
  1229  	if len(RR) < numWords {
  1230  		zz = zz.make(numWords)
  1231  		copy(zz, RR)
  1232  		RR = zz
  1233  	}
  1234  	// one = 1, with equal length to that of m
  1235  	one := make(nat, numWords)
  1236  	one[0] = 1
  1237  
  1238  	const n = 4
  1239  	// powers[i] contains x^i
  1240  	var powers [1 << n]nat
  1241  	powers[0] = powers[0].montgomery(one, RR, m, k0, numWords)
  1242  	powers[1] = powers[1].montgomery(x, RR, m, k0, numWords)
  1243  	for i := 2; i < 1<<n; i++ {
  1244  		powers[i] = powers[i].montgomery(powers[i-1], powers[1], m, k0, numWords)
  1245  	}
  1246  
  1247  	// initialize z = 1 (Montgomery 1)
  1248  	z = z.make(numWords)
  1249  	copy(z, powers[0])
  1250  
  1251  	zz = zz.make(numWords)
  1252  
  1253  	// same windowed exponent, but with Montgomery multiplications
  1254  	for i := len(y) - 1; i >= 0; i-- {
  1255  		yi := y[i]
  1256  		for j := 0; j < _W; j += n {
  1257  			if i != len(y)-1 || j != 0 {
  1258  				zz = zz.montgomery(z, z, m, k0, numWords)
  1259  				z = z.montgomery(zz, zz, m, k0, numWords)
  1260  				zz = zz.montgomery(z, z, m, k0, numWords)
  1261  				z = z.montgomery(zz, zz, m, k0, numWords)
  1262  			}
  1263  			zz = zz.montgomery(z, powers[yi>>(_W-n)], m, k0, numWords)
  1264  			z, zz = zz, z
  1265  			yi <<= n
  1266  		}
  1267  	}
  1268  	// convert to regular number
  1269  	zz = zz.montgomery(z, one, m, k0, numWords)
  1270  
  1271  	// One last reduction, just in case.
  1272  	// See golang.org/issue/13907.
  1273  	if zz.cmp(m) >= 0 {
  1274  		// Common case is m has high bit set; in that case,
  1275  		// since zz is the same length as m, there can be just
  1276  		// one multiple of m to remove. Just subtract.
  1277  		// We think that the subtract should be sufficient in general,
  1278  		// so do that unconditionally, but double-check,
  1279  		// in case our beliefs are wrong.
  1280  		// The div is not expected to be reached.
  1281  		zz = zz.sub(zz, m)
  1282  		if zz.cmp(m) >= 0 {
  1283  			_, zz = nat(nil).div(nil, zz, m)
  1284  		}
  1285  	}
  1286  
  1287  	return zz.norm()
  1288  }
  1289  
  1290  // bytes writes the value of z into buf using big-endian encoding.
  1291  // The value of z is encoded in the slice buf[i:]. If the value of z
  1292  // cannot be represented in buf, bytes panics. The number i of unused
  1293  // bytes at the beginning of buf is returned as result.
  1294  func (z nat) bytes(buf []byte) (i int) {
  1295  	// This function is used in cryptographic operations. It must not leak
  1296  	// anything but the Int's sign and bit size through side-channels. Any
  1297  	// changes must be reviewed by a security expert.
  1298  	i = len(buf)
  1299  	for _, d := range z {
  1300  		for j := 0; j < _S; j++ {
  1301  			i--
  1302  			if i >= 0 {
  1303  				buf[i] = byte(d)
  1304  			} else if byte(d) != 0 {
  1305  				panic("math/big: buffer too small to fit value")
  1306  			}
  1307  			d >>= 8
  1308  		}
  1309  	}
  1310  
  1311  	if i < 0 {
  1312  		i = 0
  1313  	}
  1314  	for i < len(buf) && buf[i] == 0 {
  1315  		i++
  1316  	}
  1317  
  1318  	return
  1319  }
  1320  
  1321  // bigEndianWord returns the contents of buf interpreted as a big-endian encoded Word value.
  1322  func bigEndianWord(buf []byte) Word {
  1323  	if _W == 64 {
  1324  		return Word(byteorder.BeUint64(buf))
  1325  	}
  1326  	return Word(byteorder.BeUint32(buf))
  1327  }
  1328  
  1329  // setBytes interprets buf as the bytes of a big-endian unsigned
  1330  // integer, sets z to that value, and returns z.
  1331  func (z nat) setBytes(buf []byte) nat {
  1332  	z = z.make((len(buf) + _S - 1) / _S)
  1333  
  1334  	i := len(buf)
  1335  	for k := 0; i >= _S; k++ {
  1336  		z[k] = bigEndianWord(buf[i-_S : i])
  1337  		i -= _S
  1338  	}
  1339  	if i > 0 {
  1340  		var d Word
  1341  		for s := uint(0); i > 0; s += 8 {
  1342  			d |= Word(buf[i-1]) << s
  1343  			i--
  1344  		}
  1345  		z[len(z)-1] = d
  1346  	}
  1347  
  1348  	return z.norm()
  1349  }
  1350  
  1351  // sqrt sets z = ⌊√x⌋
  1352  func (z nat) sqrt(x nat) nat {
  1353  	if x.cmp(natOne) <= 0 {
  1354  		return z.set(x)
  1355  	}
  1356  	if alias(z, x) {
  1357  		z = nil
  1358  	}
  1359  
  1360  	// Start with value known to be too large and repeat "z = ⌊(z + ⌊x/z⌋)/2⌋" until it stops getting smaller.
  1361  	// See Brent and Zimmermann, Modern Computer Arithmetic, Algorithm 1.13 (SqrtInt).
  1362  	// https://members.loria.fr/PZimmermann/mca/pub226.html
  1363  	// If x is one less than a perfect square, the sequence oscillates between the correct z and z+1;
  1364  	// otherwise it converges to the correct z and stays there.
  1365  	var z1, z2 nat
  1366  	z1 = z
  1367  	z1 = z1.setUint64(1)
  1368  	z1 = z1.shl(z1, uint(x.bitLen()+1)/2) // must be ≥ √x
  1369  	for n := 0; ; n++ {
  1370  		z2, _ = z2.div(nil, x, z1)
  1371  		z2 = z2.add(z2, z1)
  1372  		z2 = z2.shr(z2, 1)
  1373  		if z2.cmp(z1) >= 0 {
  1374  			// z1 is answer.
  1375  			// Figure out whether z1 or z2 is currently aliased to z by looking at loop count.
  1376  			if n&1 == 0 {
  1377  				return z1
  1378  			}
  1379  			return z.set(z1)
  1380  		}
  1381  		z1, z2 = z2, z1
  1382  	}
  1383  }
  1384  
  1385  // subMod2N returns z = (x - y) mod 2ⁿ.
  1386  func (z nat) subMod2N(x, y nat, n uint) nat {
  1387  	if uint(x.bitLen()) > n {
  1388  		if alias(z, x) {
  1389  			// ok to overwrite x in place
  1390  			x = x.trunc(x, n)
  1391  		} else {
  1392  			x = nat(nil).trunc(x, n)
  1393  		}
  1394  	}
  1395  	if uint(y.bitLen()) > n {
  1396  		if alias(z, y) {
  1397  			// ok to overwrite y in place
  1398  			y = y.trunc(y, n)
  1399  		} else {
  1400  			y = nat(nil).trunc(y, n)
  1401  		}
  1402  	}
  1403  	if x.cmp(y) >= 0 {
  1404  		return z.sub(x, y)
  1405  	}
  1406  	// x - y < 0; x - y mod 2ⁿ = x - y + 2ⁿ = 2ⁿ - (y - x) = 1 + 2ⁿ-1 - (y - x) = 1 + ^(y - x).
  1407  	z = z.sub(y, x)
  1408  	for uint(len(z))*_W < n {
  1409  		z = append(z, 0)
  1410  	}
  1411  	for i := range z {
  1412  		z[i] = ^z[i]
  1413  	}
  1414  	z = z.trunc(z, n)
  1415  	return z.add(z, natOne)
  1416  }
  1417  

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