Source file src/math/big/int.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 signed multi-precision integers.
     6  
     7  package big
     8  
     9  import (
    10  	"fmt"
    11  	"io"
    12  	"math/rand"
    13  	"strings"
    14  )
    15  
    16  // An Int represents a signed multi-precision integer.
    17  // The zero value for an Int represents the value 0.
    18  type Int struct {
    19  	neg bool // sign
    20  	abs nat  // absolute value of the integer
    21  }
    22  
    23  var intOne = &Int{false, natOne}
    24  
    25  // Sign returns:
    26  //
    27  //	-1 if x <  0
    28  //	 0 if x == 0
    29  //	+1 if x >  0
    30  //
    31  func (x *Int) Sign() int {
    32  	if len(x.abs) == 0 {
    33  		return 0
    34  	}
    35  	if x.neg {
    36  		return -1
    37  	}
    38  	return 1
    39  }
    40  
    41  // SetInt64 sets z to x and returns z.
    42  func (z *Int) SetInt64(x int64) *Int {
    43  	neg := false
    44  	if x < 0 {
    45  		neg = true
    46  		x = -x
    47  	}
    48  	z.abs = z.abs.setUint64(uint64(x))
    49  	z.neg = neg
    50  	return z
    51  }
    52  
    53  // SetUint64 sets z to x and returns z.
    54  func (z *Int) SetUint64(x uint64) *Int {
    55  	z.abs = z.abs.setUint64(x)
    56  	z.neg = false
    57  	return z
    58  }
    59  
    60  // NewInt allocates and returns a new Int set to x.
    61  func NewInt(x int64) *Int {
    62  	return new(Int).SetInt64(x)
    63  }
    64  
    65  // Set sets z to x and returns z.
    66  func (z *Int) Set(x *Int) *Int {
    67  	if z != x {
    68  		z.abs = z.abs.set(x.abs)
    69  		z.neg = x.neg
    70  	}
    71  	return z
    72  }
    73  
    74  // Bits provides raw (unchecked but fast) access to x by returning its
    75  // absolute value as a little-endian Word slice. The result and x share
    76  // the same underlying array.
    77  // Bits is intended to support implementation of missing low-level Int
    78  // functionality outside this package; it should be avoided otherwise.
    79  func (x *Int) Bits() []Word {
    80  	return x.abs
    81  }
    82  
    83  // SetBits provides raw (unchecked but fast) access to z by setting its
    84  // value to abs, interpreted as a little-endian Word slice, and returning
    85  // z. The result and abs share the same underlying array.
    86  // SetBits is intended to support implementation of missing low-level Int
    87  // functionality outside this package; it should be avoided otherwise.
    88  func (z *Int) SetBits(abs []Word) *Int {
    89  	z.abs = nat(abs).norm()
    90  	z.neg = false
    91  	return z
    92  }
    93  
    94  // Abs sets z to |x| (the absolute value of x) and returns z.
    95  func (z *Int) Abs(x *Int) *Int {
    96  	z.Set(x)
    97  	z.neg = false
    98  	return z
    99  }
   100  
   101  // Neg sets z to -x and returns z.
   102  func (z *Int) Neg(x *Int) *Int {
   103  	z.Set(x)
   104  	z.neg = len(z.abs) > 0 && !z.neg // 0 has no sign
   105  	return z
   106  }
   107  
   108  // Add sets z to the sum x+y and returns z.
   109  func (z *Int) Add(x, y *Int) *Int {
   110  	neg := x.neg
   111  	if x.neg == y.neg {
   112  		// x + y == x + y
   113  		// (-x) + (-y) == -(x + y)
   114  		z.abs = z.abs.add(x.abs, y.abs)
   115  	} else {
   116  		// x + (-y) == x - y == -(y - x)
   117  		// (-x) + y == y - x == -(x - y)
   118  		if x.abs.cmp(y.abs) >= 0 {
   119  			z.abs = z.abs.sub(x.abs, y.abs)
   120  		} else {
   121  			neg = !neg
   122  			z.abs = z.abs.sub(y.abs, x.abs)
   123  		}
   124  	}
   125  	z.neg = len(z.abs) > 0 && neg // 0 has no sign
   126  	return z
   127  }
   128  
   129  // Sub sets z to the difference x-y and returns z.
   130  func (z *Int) Sub(x, y *Int) *Int {
   131  	neg := x.neg
   132  	if x.neg != y.neg {
   133  		// x - (-y) == x + y
   134  		// (-x) - y == -(x + y)
   135  		z.abs = z.abs.add(x.abs, y.abs)
   136  	} else {
   137  		// x - y == x - y == -(y - x)
   138  		// (-x) - (-y) == y - x == -(x - y)
   139  		if x.abs.cmp(y.abs) >= 0 {
   140  			z.abs = z.abs.sub(x.abs, y.abs)
   141  		} else {
   142  			neg = !neg
   143  			z.abs = z.abs.sub(y.abs, x.abs)
   144  		}
   145  	}
   146  	z.neg = len(z.abs) > 0 && neg // 0 has no sign
   147  	return z
   148  }
   149  
   150  // Mul sets z to the product x*y and returns z.
   151  func (z *Int) Mul(x, y *Int) *Int {
   152  	// x * y == x * y
   153  	// x * (-y) == -(x * y)
   154  	// (-x) * y == -(x * y)
   155  	// (-x) * (-y) == x * y
   156  	if x == y {
   157  		z.abs = z.abs.sqr(x.abs)
   158  		z.neg = false
   159  		return z
   160  	}
   161  	z.abs = z.abs.mul(x.abs, y.abs)
   162  	z.neg = len(z.abs) > 0 && x.neg != y.neg // 0 has no sign
   163  	return z
   164  }
   165  
   166  // MulRange sets z to the product of all integers
   167  // in the range [a, b] inclusively and returns z.
   168  // If a > b (empty range), the result is 1.
   169  func (z *Int) MulRange(a, b int64) *Int {
   170  	switch {
   171  	case a > b:
   172  		return z.SetInt64(1) // empty range
   173  	case a <= 0 && b >= 0:
   174  		return z.SetInt64(0) // range includes 0
   175  	}
   176  	// a <= b && (b < 0 || a > 0)
   177  
   178  	neg := false
   179  	if a < 0 {
   180  		neg = (b-a)&1 == 0
   181  		a, b = -b, -a
   182  	}
   183  
   184  	z.abs = z.abs.mulRange(uint64(a), uint64(b))
   185  	z.neg = neg
   186  	return z
   187  }
   188  
   189  // Binomial sets z to the binomial coefficient of (n, k) and returns z.
   190  func (z *Int) Binomial(n, k int64) *Int {
   191  	// reduce the number of multiplications by reducing k
   192  	if n/2 < k && k <= n {
   193  		k = n - k // Binomial(n, k) == Binomial(n, n-k)
   194  	}
   195  	var a, b Int
   196  	a.MulRange(n-k+1, n)
   197  	b.MulRange(1, k)
   198  	return z.Quo(&a, &b)
   199  }
   200  
   201  // Quo sets z to the quotient x/y for y != 0 and returns z.
   202  // If y == 0, a division-by-zero run-time panic occurs.
   203  // Quo implements truncated division (like Go); see QuoRem for more details.
   204  func (z *Int) Quo(x, y *Int) *Int {
   205  	z.abs, _ = z.abs.div(nil, x.abs, y.abs)
   206  	z.neg = len(z.abs) > 0 && x.neg != y.neg // 0 has no sign
   207  	return z
   208  }
   209  
   210  // Rem sets z to the remainder x%y for y != 0 and returns z.
   211  // If y == 0, a division-by-zero run-time panic occurs.
   212  // Rem implements truncated modulus (like Go); see QuoRem for more details.
   213  func (z *Int) Rem(x, y *Int) *Int {
   214  	_, z.abs = nat(nil).div(z.abs, x.abs, y.abs)
   215  	z.neg = len(z.abs) > 0 && x.neg // 0 has no sign
   216  	return z
   217  }
   218  
   219  // QuoRem sets z to the quotient x/y and r to the remainder x%y
   220  // and returns the pair (z, r) for y != 0.
   221  // If y == 0, a division-by-zero run-time panic occurs.
   222  //
   223  // QuoRem implements T-division and modulus (like Go):
   224  //
   225  //	q = x/y      with the result truncated to zero
   226  //	r = x - y*q
   227  //
   228  // (See Daan Leijen, ``Division and Modulus for Computer Scientists''.)
   229  // See DivMod for Euclidean division and modulus (unlike Go).
   230  //
   231  func (z *Int) QuoRem(x, y, r *Int) (*Int, *Int) {
   232  	z.abs, r.abs = z.abs.div(r.abs, x.abs, y.abs)
   233  	z.neg, r.neg = len(z.abs) > 0 && x.neg != y.neg, len(r.abs) > 0 && x.neg // 0 has no sign
   234  	return z, r
   235  }
   236  
   237  // Div sets z to the quotient x/y for y != 0 and returns z.
   238  // If y == 0, a division-by-zero run-time panic occurs.
   239  // Div implements Euclidean division (unlike Go); see DivMod for more details.
   240  func (z *Int) Div(x, y *Int) *Int {
   241  	y_neg := y.neg // z may be an alias for y
   242  	var r Int
   243  	z.QuoRem(x, y, &r)
   244  	if r.neg {
   245  		if y_neg {
   246  			z.Add(z, intOne)
   247  		} else {
   248  			z.Sub(z, intOne)
   249  		}
   250  	}
   251  	return z
   252  }
   253  
   254  // Mod sets z to the modulus x%y for y != 0 and returns z.
   255  // If y == 0, a division-by-zero run-time panic occurs.
   256  // Mod implements Euclidean modulus (unlike Go); see DivMod for more details.
   257  func (z *Int) Mod(x, y *Int) *Int {
   258  	y0 := y // save y
   259  	if z == y || alias(z.abs, y.abs) {
   260  		y0 = new(Int).Set(y)
   261  	}
   262  	var q Int
   263  	q.QuoRem(x, y, z)
   264  	if z.neg {
   265  		if y0.neg {
   266  			z.Sub(z, y0)
   267  		} else {
   268  			z.Add(z, y0)
   269  		}
   270  	}
   271  	return z
   272  }
   273  
   274  // DivMod sets z to the quotient x div y and m to the modulus x mod y
   275  // and returns the pair (z, m) for y != 0.
   276  // If y == 0, a division-by-zero run-time panic occurs.
   277  //
   278  // DivMod implements Euclidean division and modulus (unlike Go):
   279  //
   280  //	q = x div y  such that
   281  //	m = x - y*q  with 0 <= m < |y|
   282  //
   283  // (See Raymond T. Boute, ``The Euclidean definition of the functions
   284  // div and mod''. ACM Transactions on Programming Languages and
   285  // Systems (TOPLAS), 14(2):127-144, New York, NY, USA, 4/1992.
   286  // ACM press.)
   287  // See QuoRem for T-division and modulus (like Go).
   288  //
   289  func (z *Int) DivMod(x, y, m *Int) (*Int, *Int) {
   290  	y0 := y // save y
   291  	if z == y || alias(z.abs, y.abs) {
   292  		y0 = new(Int).Set(y)
   293  	}
   294  	z.QuoRem(x, y, m)
   295  	if m.neg {
   296  		if y0.neg {
   297  			z.Add(z, intOne)
   298  			m.Sub(m, y0)
   299  		} else {
   300  			z.Sub(z, intOne)
   301  			m.Add(m, y0)
   302  		}
   303  	}
   304  	return z, m
   305  }
   306  
   307  // Cmp compares x and y and returns:
   308  //
   309  //   -1 if x <  y
   310  //    0 if x == y
   311  //   +1 if x >  y
   312  //
   313  func (x *Int) Cmp(y *Int) (r int) {
   314  	// x cmp y == x cmp y
   315  	// x cmp (-y) == x
   316  	// (-x) cmp y == y
   317  	// (-x) cmp (-y) == -(x cmp y)
   318  	switch {
   319  	case x.neg == y.neg:
   320  		r = x.abs.cmp(y.abs)
   321  		if x.neg {
   322  			r = -r
   323  		}
   324  	case x.neg:
   325  		r = -1
   326  	default:
   327  		r = 1
   328  	}
   329  	return
   330  }
   331  
   332  // CmpAbs compares the absolute values of x and y and returns:
   333  //
   334  //   -1 if |x| <  |y|
   335  //    0 if |x| == |y|
   336  //   +1 if |x| >  |y|
   337  //
   338  func (x *Int) CmpAbs(y *Int) int {
   339  	return x.abs.cmp(y.abs)
   340  }
   341  
   342  // low32 returns the least significant 32 bits of x.
   343  func low32(x nat) uint32 {
   344  	if len(x) == 0 {
   345  		return 0
   346  	}
   347  	return uint32(x[0])
   348  }
   349  
   350  // low64 returns the least significant 64 bits of x.
   351  func low64(x nat) uint64 {
   352  	if len(x) == 0 {
   353  		return 0
   354  	}
   355  	v := uint64(x[0])
   356  	if _W == 32 && len(x) > 1 {
   357  		return uint64(x[1])<<32 | v
   358  	}
   359  	return v
   360  }
   361  
   362  // Int64 returns the int64 representation of x.
   363  // If x cannot be represented in an int64, the result is undefined.
   364  func (x *Int) Int64() int64 {
   365  	v := int64(low64(x.abs))
   366  	if x.neg {
   367  		v = -v
   368  	}
   369  	return v
   370  }
   371  
   372  // Uint64 returns the uint64 representation of x.
   373  // If x cannot be represented in a uint64, the result is undefined.
   374  func (x *Int) Uint64() uint64 {
   375  	return low64(x.abs)
   376  }
   377  
   378  // IsInt64 reports whether x can be represented as an int64.
   379  func (x *Int) IsInt64() bool {
   380  	if len(x.abs) <= 64/_W {
   381  		w := int64(low64(x.abs))
   382  		return w >= 0 || x.neg && w == -w
   383  	}
   384  	return false
   385  }
   386  
   387  // IsUint64 reports whether x can be represented as a uint64.
   388  func (x *Int) IsUint64() bool {
   389  	return !x.neg && len(x.abs) <= 64/_W
   390  }
   391  
   392  // SetString sets z to the value of s, interpreted in the given base,
   393  // and returns z and a boolean indicating success. The entire string
   394  // (not just a prefix) must be valid for success. If SetString fails,
   395  // the value of z is undefined but the returned value is nil.
   396  //
   397  // The base argument must be 0 or a value between 2 and MaxBase. If the base
   398  // is 0, the string prefix determines the actual conversion base. A prefix of
   399  // ``0x'' or ``0X'' selects base 16; the ``0'' prefix selects base 8, and a
   400  // ``0b'' or ``0B'' prefix selects base 2. Otherwise the selected base is 10.
   401  //
   402  // For bases <= 36, lower and upper case letters are considered the same:
   403  // The letters 'a' to 'z' and 'A' to 'Z' represent digit values 10 to 35.
   404  // For bases > 36, the upper case letters 'A' to 'Z' represent the digit
   405  // values 36 to 61.
   406  //
   407  func (z *Int) SetString(s string, base int) (*Int, bool) {
   408  	return z.setFromScanner(strings.NewReader(s), base)
   409  }
   410  
   411  // setFromScanner implements SetString given an io.BytesScanner.
   412  // For documentation see comments of SetString.
   413  func (z *Int) setFromScanner(r io.ByteScanner, base int) (*Int, bool) {
   414  	if _, _, err := z.scan(r, base); err != nil {
   415  		return nil, false
   416  	}
   417  	// entire content must have been consumed
   418  	if _, err := r.ReadByte(); err != io.EOF {
   419  		return nil, false
   420  	}
   421  	return z, true // err == io.EOF => scan consumed all content of r
   422  }
   423  
   424  // SetBytes interprets buf as the bytes of a big-endian unsigned
   425  // integer, sets z to that value, and returns z.
   426  func (z *Int) SetBytes(buf []byte) *Int {
   427  	z.abs = z.abs.setBytes(buf)
   428  	z.neg = false
   429  	return z
   430  }
   431  
   432  // Bytes returns the absolute value of x as a big-endian byte slice.
   433  func (x *Int) Bytes() []byte {
   434  	buf := make([]byte, len(x.abs)*_S)
   435  	return buf[x.abs.bytes(buf):]
   436  }
   437  
   438  // BitLen returns the length of the absolute value of x in bits.
   439  // The bit length of 0 is 0.
   440  func (x *Int) BitLen() int {
   441  	return x.abs.bitLen()
   442  }
   443  
   444  // Exp sets z = x**y mod |m| (i.e. the sign of m is ignored), and returns z.
   445  // If m == nil or m == 0, z = x**y unless y <= 0 then z = 1.
   446  //
   447  // Modular exponentation of inputs of a particular size is not a
   448  // cryptographically constant-time operation.
   449  func (z *Int) Exp(x, y, m *Int) *Int {
   450  	// See Knuth, volume 2, section 4.6.3.
   451  	xWords := x.abs
   452  	if y.neg {
   453  		if m == nil || len(m.abs) == 0 {
   454  			return z.SetInt64(1)
   455  		}
   456  		// for y < 0: x**y mod m == (x**(-1))**|y| mod m
   457  		xWords = new(Int).ModInverse(x, m).abs
   458  	}
   459  	yWords := y.abs
   460  
   461  	var mWords nat
   462  	if m != nil {
   463  		mWords = m.abs // m.abs may be nil for m == 0
   464  	}
   465  
   466  	z.abs = z.abs.expNN(xWords, yWords, mWords)
   467  	z.neg = len(z.abs) > 0 && x.neg && len(yWords) > 0 && yWords[0]&1 == 1 // 0 has no sign
   468  	if z.neg && len(mWords) > 0 {
   469  		// make modulus result positive
   470  		z.abs = z.abs.sub(mWords, z.abs) // z == x**y mod |m| && 0 <= z < |m|
   471  		z.neg = false
   472  	}
   473  
   474  	return z
   475  }
   476  
   477  // GCD sets z to the greatest common divisor of a and b, which both must
   478  // be > 0, and returns z.
   479  // If x or y are not nil, GCD sets their value such that z = a*x + b*y.
   480  // If either a or b is <= 0, GCD sets z = x = y = 0.
   481  func (z *Int) GCD(x, y, a, b *Int) *Int {
   482  	if a.Sign() <= 0 || b.Sign() <= 0 {
   483  		z.SetInt64(0)
   484  		if x != nil {
   485  			x.SetInt64(0)
   486  		}
   487  		if y != nil {
   488  			y.SetInt64(0)
   489  		}
   490  		return z
   491  	}
   492  
   493  	return z.lehmerGCD(x, y, a, b)
   494  }
   495  
   496  // lehmerSimulate attempts to simulate several Euclidean update steps
   497  // using the leading digits of A and B.  It returns u0, u1, v0, v1
   498  // such that A and B can be updated as:
   499  //		A = u0*A + v0*B
   500  //		B = u1*A + v1*B
   501  // Requirements: A >= B and len(B.abs) >= 2
   502  // Since we are calculating with full words to avoid overflow,
   503  // we use 'even' to track the sign of the cosequences.
   504  // For even iterations: u0, v1 >= 0 && u1, v0 <= 0
   505  // For odd  iterations: u0, v1 <= 0 && u1, v0 >= 0
   506  func lehmerSimulate(A, B *Int) (u0, u1, v0, v1 Word, even bool) {
   507  	// initialize the digits
   508  	var a1, a2, u2, v2 Word
   509  
   510  	m := len(B.abs) // m >= 2
   511  	n := len(A.abs) // n >= m >= 2
   512  
   513  	// extract the top Word of bits from A and B
   514  	h := nlz(A.abs[n-1])
   515  	a1 = A.abs[n-1]<<h | A.abs[n-2]>>(_W-h)
   516  	// B may have implicit zero words in the high bits if the lengths differ
   517  	switch {
   518  	case n == m:
   519  		a2 = B.abs[n-1]<<h | B.abs[n-2]>>(_W-h)
   520  	case n == m+1:
   521  		a2 = B.abs[n-2] >> (_W - h)
   522  	default:
   523  		a2 = 0
   524  	}
   525  
   526  	// Since we are calculating with full words to avoid overflow,
   527  	// we use 'even' to track the sign of the cosequences.
   528  	// For even iterations: u0, v1 >= 0 && u1, v0 <= 0
   529  	// For odd  iterations: u0, v1 <= 0 && u1, v0 >= 0
   530  	// The first iteration starts with k=1 (odd).
   531  	even = false
   532  	// variables to track the cosequences
   533  	u0, u1, u2 = 0, 1, 0
   534  	v0, v1, v2 = 0, 0, 1
   535  
   536  	// Calculate the quotient and cosequences using Collins' stopping condition.
   537  	// Note that overflow of a Word is not possible when computing the remainder
   538  	// sequence and cosequences since the cosequence size is bounded by the input size.
   539  	// See section 4.2 of Jebelean for details.
   540  	for a2 >= v2 && a1-a2 >= v1+v2 {
   541  		q, r := a1/a2, a1%a2
   542  		a1, a2 = a2, r
   543  		u0, u1, u2 = u1, u2, u1+q*u2
   544  		v0, v1, v2 = v1, v2, v1+q*v2
   545  		even = !even
   546  	}
   547  	return
   548  }
   549  
   550  // lehmerUpdate updates the inputs A and B such that:
   551  //		A = u0*A + v0*B
   552  //		B = u1*A + v1*B
   553  // where the signs of u0, u1, v0, v1 are given by even
   554  // For even == true: u0, v1 >= 0 && u1, v0 <= 0
   555  // For even == false: u0, v1 <= 0 && u1, v0 >= 0
   556  // q, r, s, t are temporary variables to avoid allocations in the multiplication
   557  func lehmerUpdate(A, B, q, r, s, t *Int, u0, u1, v0, v1 Word, even bool) {
   558  
   559  	t.abs = t.abs.setWord(u0)
   560  	s.abs = s.abs.setWord(v0)
   561  	t.neg = !even
   562  	s.neg = even
   563  
   564  	t.Mul(A, t)
   565  	s.Mul(B, s)
   566  
   567  	r.abs = r.abs.setWord(u1)
   568  	q.abs = q.abs.setWord(v1)
   569  	r.neg = even
   570  	q.neg = !even
   571  
   572  	r.Mul(A, r)
   573  	q.Mul(B, q)
   574  
   575  	A.Add(t, s)
   576  	B.Add(r, q)
   577  }
   578  
   579  // euclidUpdate performs a single step of the Euclidean GCD algorithm
   580  // if extended is true, it also updates the cosequence Ua, Ub
   581  func euclidUpdate(A, B, Ua, Ub, q, r, s, t *Int, extended bool) {
   582  	q, r = q.QuoRem(A, B, r)
   583  
   584  	*A, *B, *r = *B, *r, *A
   585  
   586  	if extended {
   587  		// Ua, Ub = Ub, Ua - q*Ub
   588  		t.Set(Ub)
   589  		s.Mul(Ub, q)
   590  		Ub.Sub(Ua, s)
   591  		Ua.Set(t)
   592  	}
   593  }
   594  
   595  // lehmerGCD sets z to the greatest common divisor of a and b,
   596  // which both must be > 0, and returns z.
   597  // If x or y are not nil, their values are set such that z = a*x + b*y.
   598  // See Knuth, The Art of Computer Programming, Vol. 2, Section 4.5.2, Algorithm L.
   599  // This implementation uses the improved condition by Collins requiring only one
   600  // quotient and avoiding the possibility of single Word overflow.
   601  // See Jebelean, "Improving the multiprecision Euclidean algorithm",
   602  // Design and Implementation of Symbolic Computation Systems, pp 45-58.
   603  // The cosequences are updated according to Algorithm 10.45 from
   604  // Cohen et al. "Handbook of Elliptic and Hyperelliptic Curve Cryptography" pp 192.
   605  func (z *Int) lehmerGCD(x, y, a, b *Int) *Int {
   606  	var A, B, Ua, Ub *Int
   607  
   608  	A = new(Int).Set(a)
   609  	B = new(Int).Set(b)
   610  
   611  	extended := x != nil || y != nil
   612  
   613  	if extended {
   614  		// Ua (Ub) tracks how many times input a has been accumulated into A (B).
   615  		Ua = new(Int).SetInt64(1)
   616  		Ub = new(Int)
   617  	}
   618  
   619  	// temp variables for multiprecision update
   620  	q := new(Int)
   621  	r := new(Int)
   622  	s := new(Int)
   623  	t := new(Int)
   624  
   625  	// ensure A >= B
   626  	if A.abs.cmp(B.abs) < 0 {
   627  		A, B = B, A
   628  		Ub, Ua = Ua, Ub
   629  	}
   630  
   631  	// loop invariant A >= B
   632  	for len(B.abs) > 1 {
   633  		// Attempt to calculate in single-precision using leading words of A and B.
   634  		u0, u1, v0, v1, even := lehmerSimulate(A, B)
   635  
   636  		// multiprecision Step
   637  		if v0 != 0 {
   638  			// Simulate the effect of the single-precision steps using the cosequences.
   639  			// A = u0*A + v0*B
   640  			// B = u1*A + v1*B
   641  			lehmerUpdate(A, B, q, r, s, t, u0, u1, v0, v1, even)
   642  
   643  			if extended {
   644  				// Ua = u0*Ua + v0*Ub
   645  				// Ub = u1*Ua + v1*Ub
   646  				lehmerUpdate(Ua, Ub, q, r, s, t, u0, u1, v0, v1, even)
   647  			}
   648  
   649  		} else {
   650  			// Single-digit calculations failed to simulate any quotients.
   651  			// Do a standard Euclidean step.
   652  			euclidUpdate(A, B, Ua, Ub, q, r, s, t, extended)
   653  		}
   654  	}
   655  
   656  	if len(B.abs) > 0 {
   657  		// extended Euclidean algorithm base case if B is a single Word
   658  		if len(A.abs) > 1 {
   659  			// A is longer than a single Word, so one update is needed.
   660  			euclidUpdate(A, B, Ua, Ub, q, r, s, t, extended)
   661  		}
   662  		if len(B.abs) > 0 {
   663  			// A and B are both a single Word.
   664  			aWord, bWord := A.abs[0], B.abs[0]
   665  			if extended {
   666  				var ua, ub, va, vb Word
   667  				ua, ub = 1, 0
   668  				va, vb = 0, 1
   669  				even := true
   670  				for bWord != 0 {
   671  					q, r := aWord/bWord, aWord%bWord
   672  					aWord, bWord = bWord, r
   673  					ua, ub = ub, ua+q*ub
   674  					va, vb = vb, va+q*vb
   675  					even = !even
   676  				}
   677  
   678  				t.abs = t.abs.setWord(ua)
   679  				s.abs = s.abs.setWord(va)
   680  				t.neg = !even
   681  				s.neg = even
   682  
   683  				t.Mul(Ua, t)
   684  				s.Mul(Ub, s)
   685  
   686  				Ua.Add(t, s)
   687  			} else {
   688  				for bWord != 0 {
   689  					aWord, bWord = bWord, aWord%bWord
   690  				}
   691  			}
   692  			A.abs[0] = aWord
   693  		}
   694  	}
   695  
   696  	if x != nil {
   697  		*x = *Ua
   698  	}
   699  
   700  	if y != nil {
   701  		// y = (z - a*x)/b
   702  		y.Mul(a, Ua)
   703  		y.Sub(A, y)
   704  		y.Div(y, b)
   705  	}
   706  
   707  	*z = *A
   708  
   709  	return z
   710  }
   711  
   712  // Rand sets z to a pseudo-random number in [0, n) and returns z.
   713  //
   714  // As this uses the math/rand package, it must not be used for
   715  // security-sensitive work. Use crypto/rand.Int instead.
   716  func (z *Int) Rand(rnd *rand.Rand, n *Int) *Int {
   717  	z.neg = false
   718  	if n.neg || len(n.abs) == 0 {
   719  		z.abs = nil
   720  		return z
   721  	}
   722  	z.abs = z.abs.random(rnd, n.abs, n.abs.bitLen())
   723  	return z
   724  }
   725  
   726  // ModInverse sets z to the multiplicative inverse of g in the ring ℤ/nℤ
   727  // and returns z. If g and n are not relatively prime, g has no multiplicative
   728  // inverse in the ring ℤ/nℤ.  In this case, z is unchanged and the return value
   729  // is nil.
   730  func (z *Int) ModInverse(g, n *Int) *Int {
   731  	// GCD expects parameters a and b to be > 0.
   732  	if n.neg {
   733  		var n2 Int
   734  		n = n2.Neg(n)
   735  	}
   736  	if g.neg {
   737  		var g2 Int
   738  		g = g2.Mod(g, n)
   739  	}
   740  	var d, x Int
   741  	d.GCD(&x, nil, g, n)
   742  
   743  	// if and only if d==1, g and n are relatively prime
   744  	if d.Cmp(intOne) != 0 {
   745  		return nil
   746  	}
   747  
   748  	// x and y are such that g*x + n*y = 1, therefore x is the inverse element,
   749  	// but it may be negative, so convert to the range 0 <= z < |n|
   750  	if x.neg {
   751  		z.Add(&x, n)
   752  	} else {
   753  		z.Set(&x)
   754  	}
   755  	return z
   756  }
   757  
   758  // Jacobi returns the Jacobi symbol (x/y), either +1, -1, or 0.
   759  // The y argument must be an odd integer.
   760  func Jacobi(x, y *Int) int {
   761  	if len(y.abs) == 0 || y.abs[0]&1 == 0 {
   762  		panic(fmt.Sprintf("big: invalid 2nd argument to Int.Jacobi: need odd integer but got %s", y))
   763  	}
   764  
   765  	// We use the formulation described in chapter 2, section 2.4,
   766  	// "The Yacas Book of Algorithms":
   767  	// http://yacas.sourceforge.net/Algo.book.pdf
   768  
   769  	var a, b, c Int
   770  	a.Set(x)
   771  	b.Set(y)
   772  	j := 1
   773  
   774  	if b.neg {
   775  		if a.neg {
   776  			j = -1
   777  		}
   778  		b.neg = false
   779  	}
   780  
   781  	for {
   782  		if b.Cmp(intOne) == 0 {
   783  			return j
   784  		}
   785  		if len(a.abs) == 0 {
   786  			return 0
   787  		}
   788  		a.Mod(&a, &b)
   789  		if len(a.abs) == 0 {
   790  			return 0
   791  		}
   792  		// a > 0
   793  
   794  		// handle factors of 2 in 'a'
   795  		s := a.abs.trailingZeroBits()
   796  		if s&1 != 0 {
   797  			bmod8 := b.abs[0] & 7
   798  			if bmod8 == 3 || bmod8 == 5 {
   799  				j = -j
   800  			}
   801  		}
   802  		c.Rsh(&a, s) // a = 2^s*c
   803  
   804  		// swap numerator and denominator
   805  		if b.abs[0]&3 == 3 && c.abs[0]&3 == 3 {
   806  			j = -j
   807  		}
   808  		a.Set(&b)
   809  		b.Set(&c)
   810  	}
   811  }
   812  
   813  // modSqrt3Mod4 uses the identity
   814  //      (a^((p+1)/4))^2  mod p
   815  //   == u^(p+1)          mod p
   816  //   == u^2              mod p
   817  // to calculate the square root of any quadratic residue mod p quickly for 3
   818  // mod 4 primes.
   819  func (z *Int) modSqrt3Mod4Prime(x, p *Int) *Int {
   820  	e := new(Int).Add(p, intOne) // e = p + 1
   821  	e.Rsh(e, 2)                  // e = (p + 1) / 4
   822  	z.Exp(x, e, p)               // z = x^e mod p
   823  	return z
   824  }
   825  
   826  // modSqrt5Mod8 uses Atkin's observation that 2 is not a square mod p
   827  //   alpha ==  (2*a)^((p-5)/8)    mod p
   828  //   beta  ==  2*a*alpha^2        mod p  is a square root of -1
   829  //   b     ==  a*alpha*(beta-1)   mod p  is a square root of a
   830  // to calculate the square root of any quadratic residue mod p quickly for 5
   831  // mod 8 primes.
   832  func (z *Int) modSqrt5Mod8Prime(x, p *Int) *Int {
   833  	// p == 5 mod 8 implies p = e*8 + 5
   834  	// e is the quotient and 5 the remainder on division by 8
   835  	e := new(Int).Rsh(p, 3)  // e = (p - 5) / 8
   836  	tx := new(Int).Lsh(x, 1) // tx = 2*x
   837  	alpha := new(Int).Exp(tx, e, p)
   838  	beta := new(Int).Mul(alpha, alpha)
   839  	beta.Mod(beta, p)
   840  	beta.Mul(beta, tx)
   841  	beta.Mod(beta, p)
   842  	beta.Sub(beta, intOne)
   843  	beta.Mul(beta, x)
   844  	beta.Mod(beta, p)
   845  	beta.Mul(beta, alpha)
   846  	z.Mod(beta, p)
   847  	return z
   848  }
   849  
   850  // modSqrtTonelliShanks uses the Tonelli-Shanks algorithm to find the square
   851  // root of a quadratic residue modulo any prime.
   852  func (z *Int) modSqrtTonelliShanks(x, p *Int) *Int {
   853  	// Break p-1 into s*2^e such that s is odd.
   854  	var s Int
   855  	s.Sub(p, intOne)
   856  	e := s.abs.trailingZeroBits()
   857  	s.Rsh(&s, e)
   858  
   859  	// find some non-square n
   860  	var n Int
   861  	n.SetInt64(2)
   862  	for Jacobi(&n, p) != -1 {
   863  		n.Add(&n, intOne)
   864  	}
   865  
   866  	// Core of the Tonelli-Shanks algorithm. Follows the description in
   867  	// section 6 of "Square roots from 1; 24, 51, 10 to Dan Shanks" by Ezra
   868  	// Brown:
   869  	// https://www.maa.org/sites/default/files/pdf/upload_library/22/Polya/07468342.di020786.02p0470a.pdf
   870  	var y, b, g, t Int
   871  	y.Add(&s, intOne)
   872  	y.Rsh(&y, 1)
   873  	y.Exp(x, &y, p)  // y = x^((s+1)/2)
   874  	b.Exp(x, &s, p)  // b = x^s
   875  	g.Exp(&n, &s, p) // g = n^s
   876  	r := e
   877  	for {
   878  		// find the least m such that ord_p(b) = 2^m
   879  		var m uint
   880  		t.Set(&b)
   881  		for t.Cmp(intOne) != 0 {
   882  			t.Mul(&t, &t).Mod(&t, p)
   883  			m++
   884  		}
   885  
   886  		if m == 0 {
   887  			return z.Set(&y)
   888  		}
   889  
   890  		t.SetInt64(0).SetBit(&t, int(r-m-1), 1).Exp(&g, &t, p)
   891  		// t = g^(2^(r-m-1)) mod p
   892  		g.Mul(&t, &t).Mod(&g, p) // g = g^(2^(r-m)) mod p
   893  		y.Mul(&y, &t).Mod(&y, p)
   894  		b.Mul(&b, &g).Mod(&b, p)
   895  		r = m
   896  	}
   897  }
   898  
   899  // ModSqrt sets z to a square root of x mod p if such a square root exists, and
   900  // returns z. The modulus p must be an odd prime. If x is not a square mod p,
   901  // ModSqrt leaves z unchanged and returns nil. This function panics if p is
   902  // not an odd integer.
   903  func (z *Int) ModSqrt(x, p *Int) *Int {
   904  	switch Jacobi(x, p) {
   905  	case -1:
   906  		return nil // x is not a square mod p
   907  	case 0:
   908  		return z.SetInt64(0) // sqrt(0) mod p = 0
   909  	case 1:
   910  		break
   911  	}
   912  	if x.neg || x.Cmp(p) >= 0 { // ensure 0 <= x < p
   913  		x = new(Int).Mod(x, p)
   914  	}
   915  
   916  	switch {
   917  	case p.abs[0]%4 == 3:
   918  		// Check whether p is 3 mod 4, and if so, use the faster algorithm.
   919  		return z.modSqrt3Mod4Prime(x, p)
   920  	case p.abs[0]%8 == 5:
   921  		// Check whether p is 5 mod 8, use Atkin's algorithm.
   922  		return z.modSqrt5Mod8Prime(x, p)
   923  	default:
   924  		// Otherwise, use Tonelli-Shanks.
   925  		return z.modSqrtTonelliShanks(x, p)
   926  	}
   927  }
   928  
   929  // Lsh sets z = x << n and returns z.
   930  func (z *Int) Lsh(x *Int, n uint) *Int {
   931  	z.abs = z.abs.shl(x.abs, n)
   932  	z.neg = x.neg
   933  	return z
   934  }
   935  
   936  // Rsh sets z = x >> n and returns z.
   937  func (z *Int) Rsh(x *Int, n uint) *Int {
   938  	if x.neg {
   939  		// (-x) >> s == ^(x-1) >> s == ^((x-1) >> s) == -(((x-1) >> s) + 1)
   940  		t := z.abs.sub(x.abs, natOne) // no underflow because |x| > 0
   941  		t = t.shr(t, n)
   942  		z.abs = t.add(t, natOne)
   943  		z.neg = true // z cannot be zero if x is negative
   944  		return z
   945  	}
   946  
   947  	z.abs = z.abs.shr(x.abs, n)
   948  	z.neg = false
   949  	return z
   950  }
   951  
   952  // Bit returns the value of the i'th bit of x. That is, it
   953  // returns (x>>i)&1. The bit index i must be >= 0.
   954  func (x *Int) Bit(i int) uint {
   955  	if i == 0 {
   956  		// optimization for common case: odd/even test of x
   957  		if len(x.abs) > 0 {
   958  			return uint(x.abs[0] & 1) // bit 0 is same for -x
   959  		}
   960  		return 0
   961  	}
   962  	if i < 0 {
   963  		panic("negative bit index")
   964  	}
   965  	if x.neg {
   966  		t := nat(nil).sub(x.abs, natOne)
   967  		return t.bit(uint(i)) ^ 1
   968  	}
   969  
   970  	return x.abs.bit(uint(i))
   971  }
   972  
   973  // SetBit sets z to x, with x's i'th bit set to b (0 or 1).
   974  // That is, if b is 1 SetBit sets z = x | (1 << i);
   975  // if b is 0 SetBit sets z = x &^ (1 << i). If b is not 0 or 1,
   976  // SetBit will panic.
   977  func (z *Int) SetBit(x *Int, i int, b uint) *Int {
   978  	if i < 0 {
   979  		panic("negative bit index")
   980  	}
   981  	if x.neg {
   982  		t := z.abs.sub(x.abs, natOne)
   983  		t = t.setBit(t, uint(i), b^1)
   984  		z.abs = t.add(t, natOne)
   985  		z.neg = len(z.abs) > 0
   986  		return z
   987  	}
   988  	z.abs = z.abs.setBit(x.abs, uint(i), b)
   989  	z.neg = false
   990  	return z
   991  }
   992  
   993  // And sets z = x & y and returns z.
   994  func (z *Int) And(x, y *Int) *Int {
   995  	if x.neg == y.neg {
   996  		if x.neg {
   997  			// (-x) & (-y) == ^(x-1) & ^(y-1) == ^((x-1) | (y-1)) == -(((x-1) | (y-1)) + 1)
   998  			x1 := nat(nil).sub(x.abs, natOne)
   999  			y1 := nat(nil).sub(y.abs, natOne)
  1000  			z.abs = z.abs.add(z.abs.or(x1, y1), natOne)
  1001  			z.neg = true // z cannot be zero if x and y are negative
  1002  			return z
  1003  		}
  1004  
  1005  		// x & y == x & y
  1006  		z.abs = z.abs.and(x.abs, y.abs)
  1007  		z.neg = false
  1008  		return z
  1009  	}
  1010  
  1011  	// x.neg != y.neg
  1012  	if x.neg {
  1013  		x, y = y, x // & is symmetric
  1014  	}
  1015  
  1016  	// x & (-y) == x & ^(y-1) == x &^ (y-1)
  1017  	y1 := nat(nil).sub(y.abs, natOne)
  1018  	z.abs = z.abs.andNot(x.abs, y1)
  1019  	z.neg = false
  1020  	return z
  1021  }
  1022  
  1023  // AndNot sets z = x &^ y and returns z.
  1024  func (z *Int) AndNot(x, y *Int) *Int {
  1025  	if x.neg == y.neg {
  1026  		if x.neg {
  1027  			// (-x) &^ (-y) == ^(x-1) &^ ^(y-1) == ^(x-1) & (y-1) == (y-1) &^ (x-1)
  1028  			x1 := nat(nil).sub(x.abs, natOne)
  1029  			y1 := nat(nil).sub(y.abs, natOne)
  1030  			z.abs = z.abs.andNot(y1, x1)
  1031  			z.neg = false
  1032  			return z
  1033  		}
  1034  
  1035  		// x &^ y == x &^ y
  1036  		z.abs = z.abs.andNot(x.abs, y.abs)
  1037  		z.neg = false
  1038  		return z
  1039  	}
  1040  
  1041  	if x.neg {
  1042  		// (-x) &^ y == ^(x-1) &^ y == ^(x-1) & ^y == ^((x-1) | y) == -(((x-1) | y) + 1)
  1043  		x1 := nat(nil).sub(x.abs, natOne)
  1044  		z.abs = z.abs.add(z.abs.or(x1, y.abs), natOne)
  1045  		z.neg = true // z cannot be zero if x is negative and y is positive
  1046  		return z
  1047  	}
  1048  
  1049  	// x &^ (-y) == x &^ ^(y-1) == x & (y-1)
  1050  	y1 := nat(nil).sub(y.abs, natOne)
  1051  	z.abs = z.abs.and(x.abs, y1)
  1052  	z.neg = false
  1053  	return z
  1054  }
  1055  
  1056  // Or sets z = x | y and returns z.
  1057  func (z *Int) Or(x, y *Int) *Int {
  1058  	if x.neg == y.neg {
  1059  		if x.neg {
  1060  			// (-x) | (-y) == ^(x-1) | ^(y-1) == ^((x-1) & (y-1)) == -(((x-1) & (y-1)) + 1)
  1061  			x1 := nat(nil).sub(x.abs, natOne)
  1062  			y1 := nat(nil).sub(y.abs, natOne)
  1063  			z.abs = z.abs.add(z.abs.and(x1, y1), natOne)
  1064  			z.neg = true // z cannot be zero if x and y are negative
  1065  			return z
  1066  		}
  1067  
  1068  		// x | y == x | y
  1069  		z.abs = z.abs.or(x.abs, y.abs)
  1070  		z.neg = false
  1071  		return z
  1072  	}
  1073  
  1074  	// x.neg != y.neg
  1075  	if x.neg {
  1076  		x, y = y, x // | is symmetric
  1077  	}
  1078  
  1079  	// x | (-y) == x | ^(y-1) == ^((y-1) &^ x) == -(^((y-1) &^ x) + 1)
  1080  	y1 := nat(nil).sub(y.abs, natOne)
  1081  	z.abs = z.abs.add(z.abs.andNot(y1, x.abs), natOne)
  1082  	z.neg = true // z cannot be zero if one of x or y is negative
  1083  	return z
  1084  }
  1085  
  1086  // Xor sets z = x ^ y and returns z.
  1087  func (z *Int) Xor(x, y *Int) *Int {
  1088  	if x.neg == y.neg {
  1089  		if x.neg {
  1090  			// (-x) ^ (-y) == ^(x-1) ^ ^(y-1) == (x-1) ^ (y-1)
  1091  			x1 := nat(nil).sub(x.abs, natOne)
  1092  			y1 := nat(nil).sub(y.abs, natOne)
  1093  			z.abs = z.abs.xor(x1, y1)
  1094  			z.neg = false
  1095  			return z
  1096  		}
  1097  
  1098  		// x ^ y == x ^ y
  1099  		z.abs = z.abs.xor(x.abs, y.abs)
  1100  		z.neg = false
  1101  		return z
  1102  	}
  1103  
  1104  	// x.neg != y.neg
  1105  	if x.neg {
  1106  		x, y = y, x // ^ is symmetric
  1107  	}
  1108  
  1109  	// x ^ (-y) == x ^ ^(y-1) == ^(x ^ (y-1)) == -((x ^ (y-1)) + 1)
  1110  	y1 := nat(nil).sub(y.abs, natOne)
  1111  	z.abs = z.abs.add(z.abs.xor(x.abs, y1), natOne)
  1112  	z.neg = true // z cannot be zero if only one of x or y is negative
  1113  	return z
  1114  }
  1115  
  1116  // Not sets z = ^x and returns z.
  1117  func (z *Int) Not(x *Int) *Int {
  1118  	if x.neg {
  1119  		// ^(-x) == ^(^(x-1)) == x-1
  1120  		z.abs = z.abs.sub(x.abs, natOne)
  1121  		z.neg = false
  1122  		return z
  1123  	}
  1124  
  1125  	// ^x == -x-1 == -(x+1)
  1126  	z.abs = z.abs.add(x.abs, natOne)
  1127  	z.neg = true // z cannot be zero if x is positive
  1128  	return z
  1129  }
  1130  
  1131  // Sqrt sets z to ⌊√x⌋, the largest integer such that z² ≤ x, and returns z.
  1132  // It panics if x is negative.
  1133  func (z *Int) Sqrt(x *Int) *Int {
  1134  	if x.neg {
  1135  		panic("square root of negative number")
  1136  	}
  1137  	z.neg = false
  1138  	z.abs = z.abs.sqrt(x.abs)
  1139  	return z
  1140  }
  1141  

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