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  //
    19  // Operations always take pointer arguments (*Int) rather
    20  // than Int values, and each unique Int value requires
    21  // its own unique *Int pointer. To "copy" an Int value,
    22  // an existing (or newly allocated) Int must be set to
    23  // a new value using the Int.Set method; shallow copies
    24  // of Ints are not supported and may lead to errors.
    25  type Int struct {
    26  	neg bool // sign
    27  	abs nat  // absolute value of the integer
    28  }
    29  
    30  var intOne = &Int{false, natOne}
    31  
    32  // Sign returns:
    33  //
    34  //	-1 if x <  0
    35  //	 0 if x == 0
    36  //	+1 if x >  0
    37  //
    38  func (x *Int) Sign() int {
    39  	if len(x.abs) == 0 {
    40  		return 0
    41  	}
    42  	if x.neg {
    43  		return -1
    44  	}
    45  	return 1
    46  }
    47  
    48  // SetInt64 sets z to x and returns z.
    49  func (z *Int) SetInt64(x int64) *Int {
    50  	neg := false
    51  	if x < 0 {
    52  		neg = true
    53  		x = -x
    54  	}
    55  	z.abs = z.abs.setUint64(uint64(x))
    56  	z.neg = neg
    57  	return z
    58  }
    59  
    60  // SetUint64 sets z to x and returns z.
    61  func (z *Int) SetUint64(x uint64) *Int {
    62  	z.abs = z.abs.setUint64(x)
    63  	z.neg = false
    64  	return z
    65  }
    66  
    67  // NewInt allocates and returns a new Int set to x.
    68  func NewInt(x int64) *Int {
    69  	return new(Int).SetInt64(x)
    70  }
    71  
    72  // Set sets z to x and returns z.
    73  func (z *Int) Set(x *Int) *Int {
    74  	if z != x {
    75  		z.abs = z.abs.set(x.abs)
    76  		z.neg = x.neg
    77  	}
    78  	return z
    79  }
    80  
    81  // Bits provides raw (unchecked but fast) access to x by returning its
    82  // absolute value as a little-endian Word slice. The result and x share
    83  // the same underlying array.
    84  // Bits is intended to support implementation of missing low-level Int
    85  // functionality outside this package; it should be avoided otherwise.
    86  func (x *Int) Bits() []Word {
    87  	return x.abs
    88  }
    89  
    90  // SetBits provides raw (unchecked but fast) access to z by setting its
    91  // value to abs, interpreted as a little-endian Word slice, and returning
    92  // z. The result and abs share the same underlying array.
    93  // SetBits is intended to support implementation of missing low-level Int
    94  // functionality outside this package; it should be avoided otherwise.
    95  func (z *Int) SetBits(abs []Word) *Int {
    96  	z.abs = nat(abs).norm()
    97  	z.neg = false
    98  	return z
    99  }
   100  
   101  // Abs sets z to |x| (the absolute value of x) and returns z.
   102  func (z *Int) Abs(x *Int) *Int {
   103  	z.Set(x)
   104  	z.neg = false
   105  	return z
   106  }
   107  
   108  // Neg sets z to -x and returns z.
   109  func (z *Int) Neg(x *Int) *Int {
   110  	z.Set(x)
   111  	z.neg = len(z.abs) > 0 && !z.neg // 0 has no sign
   112  	return z
   113  }
   114  
   115  // Add sets z to the sum x+y and returns z.
   116  func (z *Int) Add(x, y *Int) *Int {
   117  	neg := x.neg
   118  	if x.neg == y.neg {
   119  		// x + y == x + y
   120  		// (-x) + (-y) == -(x + y)
   121  		z.abs = z.abs.add(x.abs, y.abs)
   122  	} else {
   123  		// x + (-y) == x - y == -(y - x)
   124  		// (-x) + y == y - x == -(x - y)
   125  		if x.abs.cmp(y.abs) >= 0 {
   126  			z.abs = z.abs.sub(x.abs, y.abs)
   127  		} else {
   128  			neg = !neg
   129  			z.abs = z.abs.sub(y.abs, x.abs)
   130  		}
   131  	}
   132  	z.neg = len(z.abs) > 0 && neg // 0 has no sign
   133  	return z
   134  }
   135  
   136  // Sub sets z to the difference x-y and returns z.
   137  func (z *Int) Sub(x, y *Int) *Int {
   138  	neg := x.neg
   139  	if x.neg != y.neg {
   140  		// x - (-y) == x + y
   141  		// (-x) - y == -(x + y)
   142  		z.abs = z.abs.add(x.abs, y.abs)
   143  	} else {
   144  		// x - y == x - y == -(y - x)
   145  		// (-x) - (-y) == y - x == -(x - y)
   146  		if x.abs.cmp(y.abs) >= 0 {
   147  			z.abs = z.abs.sub(x.abs, y.abs)
   148  		} else {
   149  			neg = !neg
   150  			z.abs = z.abs.sub(y.abs, x.abs)
   151  		}
   152  	}
   153  	z.neg = len(z.abs) > 0 && neg // 0 has no sign
   154  	return z
   155  }
   156  
   157  // Mul sets z to the product x*y and returns z.
   158  func (z *Int) Mul(x, y *Int) *Int {
   159  	// x * y == x * y
   160  	// x * (-y) == -(x * y)
   161  	// (-x) * y == -(x * y)
   162  	// (-x) * (-y) == x * y
   163  	if x == y {
   164  		z.abs = z.abs.sqr(x.abs)
   165  		z.neg = false
   166  		return z
   167  	}
   168  	z.abs = z.abs.mul(x.abs, y.abs)
   169  	z.neg = len(z.abs) > 0 && x.neg != y.neg // 0 has no sign
   170  	return z
   171  }
   172  
   173  // MulRange sets z to the product of all integers
   174  // in the range [a, b] inclusively and returns z.
   175  // If a > b (empty range), the result is 1.
   176  func (z *Int) MulRange(a, b int64) *Int {
   177  	switch {
   178  	case a > b:
   179  		return z.SetInt64(1) // empty range
   180  	case a <= 0 && b >= 0:
   181  		return z.SetInt64(0) // range includes 0
   182  	}
   183  	// a <= b && (b < 0 || a > 0)
   184  
   185  	neg := false
   186  	if a < 0 {
   187  		neg = (b-a)&1 == 0
   188  		a, b = -b, -a
   189  	}
   190  
   191  	z.abs = z.abs.mulRange(uint64(a), uint64(b))
   192  	z.neg = neg
   193  	return z
   194  }
   195  
   196  // Binomial sets z to the binomial coefficient of (n, k) and returns z.
   197  func (z *Int) Binomial(n, k int64) *Int {
   198  	// reduce the number of multiplications by reducing k
   199  	if n/2 < k && k <= n {
   200  		k = n - k // Binomial(n, k) == Binomial(n, n-k)
   201  	}
   202  	var a, b Int
   203  	a.MulRange(n-k+1, n)
   204  	b.MulRange(1, k)
   205  	return z.Quo(&a, &b)
   206  }
   207  
   208  // Quo sets z to the quotient x/y for y != 0 and returns z.
   209  // If y == 0, a division-by-zero run-time panic occurs.
   210  // Quo implements truncated division (like Go); see QuoRem for more details.
   211  func (z *Int) Quo(x, y *Int) *Int {
   212  	z.abs, _ = z.abs.div(nil, x.abs, y.abs)
   213  	z.neg = len(z.abs) > 0 && x.neg != y.neg // 0 has no sign
   214  	return z
   215  }
   216  
   217  // Rem sets z to the remainder x%y for y != 0 and returns z.
   218  // If y == 0, a division-by-zero run-time panic occurs.
   219  // Rem implements truncated modulus (like Go); see QuoRem for more details.
   220  func (z *Int) Rem(x, y *Int) *Int {
   221  	_, z.abs = nat(nil).div(z.abs, x.abs, y.abs)
   222  	z.neg = len(z.abs) > 0 && x.neg // 0 has no sign
   223  	return z
   224  }
   225  
   226  // QuoRem sets z to the quotient x/y and r to the remainder x%y
   227  // and returns the pair (z, r) for y != 0.
   228  // If y == 0, a division-by-zero run-time panic occurs.
   229  //
   230  // QuoRem implements T-division and modulus (like Go):
   231  //
   232  //	q = x/y      with the result truncated to zero
   233  //	r = x - y*q
   234  //
   235  // (See Daan Leijen, ``Division and Modulus for Computer Scientists''.)
   236  // See DivMod for Euclidean division and modulus (unlike Go).
   237  //
   238  func (z *Int) QuoRem(x, y, r *Int) (*Int, *Int) {
   239  	z.abs, r.abs = z.abs.div(r.abs, x.abs, y.abs)
   240  	z.neg, r.neg = len(z.abs) > 0 && x.neg != y.neg, len(r.abs) > 0 && x.neg // 0 has no sign
   241  	return z, r
   242  }
   243  
   244  // Div sets z to the quotient x/y for y != 0 and returns z.
   245  // If y == 0, a division-by-zero run-time panic occurs.
   246  // Div implements Euclidean division (unlike Go); see DivMod for more details.
   247  func (z *Int) Div(x, y *Int) *Int {
   248  	y_neg := y.neg // z may be an alias for y
   249  	var r Int
   250  	z.QuoRem(x, y, &r)
   251  	if r.neg {
   252  		if y_neg {
   253  			z.Add(z, intOne)
   254  		} else {
   255  			z.Sub(z, intOne)
   256  		}
   257  	}
   258  	return z
   259  }
   260  
   261  // Mod sets z to the modulus x%y for y != 0 and returns z.
   262  // If y == 0, a division-by-zero run-time panic occurs.
   263  // Mod implements Euclidean modulus (unlike Go); see DivMod for more details.
   264  func (z *Int) Mod(x, y *Int) *Int {
   265  	y0 := y // save y
   266  	if z == y || alias(z.abs, y.abs) {
   267  		y0 = new(Int).Set(y)
   268  	}
   269  	var q Int
   270  	q.QuoRem(x, y, z)
   271  	if z.neg {
   272  		if y0.neg {
   273  			z.Sub(z, y0)
   274  		} else {
   275  			z.Add(z, y0)
   276  		}
   277  	}
   278  	return z
   279  }
   280  
   281  // DivMod sets z to the quotient x div y and m to the modulus x mod y
   282  // and returns the pair (z, m) for y != 0.
   283  // If y == 0, a division-by-zero run-time panic occurs.
   284  //
   285  // DivMod implements Euclidean division and modulus (unlike Go):
   286  //
   287  //	q = x div y  such that
   288  //	m = x - y*q  with 0 <= m < |y|
   289  //
   290  // (See Raymond T. Boute, ``The Euclidean definition of the functions
   291  // div and mod''. ACM Transactions on Programming Languages and
   292  // Systems (TOPLAS), 14(2):127-144, New York, NY, USA, 4/1992.
   293  // ACM press.)
   294  // See QuoRem for T-division and modulus (like Go).
   295  //
   296  func (z *Int) DivMod(x, y, m *Int) (*Int, *Int) {
   297  	y0 := y // save y
   298  	if z == y || alias(z.abs, y.abs) {
   299  		y0 = new(Int).Set(y)
   300  	}
   301  	z.QuoRem(x, y, m)
   302  	if m.neg {
   303  		if y0.neg {
   304  			z.Add(z, intOne)
   305  			m.Sub(m, y0)
   306  		} else {
   307  			z.Sub(z, intOne)
   308  			m.Add(m, y0)
   309  		}
   310  	}
   311  	return z, m
   312  }
   313  
   314  // Cmp compares x and y and returns:
   315  //
   316  //   -1 if x <  y
   317  //    0 if x == y
   318  //   +1 if x >  y
   319  //
   320  func (x *Int) Cmp(y *Int) (r int) {
   321  	// x cmp y == x cmp y
   322  	// x cmp (-y) == x
   323  	// (-x) cmp y == y
   324  	// (-x) cmp (-y) == -(x cmp y)
   325  	switch {
   326  	case x == y:
   327  		// nothing to do
   328  	case x.neg == y.neg:
   329  		r = x.abs.cmp(y.abs)
   330  		if x.neg {
   331  			r = -r
   332  		}
   333  	case x.neg:
   334  		r = -1
   335  	default:
   336  		r = 1
   337  	}
   338  	return
   339  }
   340  
   341  // CmpAbs compares the absolute values of x and y and returns:
   342  //
   343  //   -1 if |x| <  |y|
   344  //    0 if |x| == |y|
   345  //   +1 if |x| >  |y|
   346  //
   347  func (x *Int) CmpAbs(y *Int) int {
   348  	return x.abs.cmp(y.abs)
   349  }
   350  
   351  // low32 returns the least significant 32 bits of x.
   352  func low32(x nat) uint32 {
   353  	if len(x) == 0 {
   354  		return 0
   355  	}
   356  	return uint32(x[0])
   357  }
   358  
   359  // low64 returns the least significant 64 bits of x.
   360  func low64(x nat) uint64 {
   361  	if len(x) == 0 {
   362  		return 0
   363  	}
   364  	v := uint64(x[0])
   365  	if _W == 32 && len(x) > 1 {
   366  		return uint64(x[1])<<32 | v
   367  	}
   368  	return v
   369  }
   370  
   371  // Int64 returns the int64 representation of x.
   372  // If x cannot be represented in an int64, the result is undefined.
   373  func (x *Int) Int64() int64 {
   374  	v := int64(low64(x.abs))
   375  	if x.neg {
   376  		v = -v
   377  	}
   378  	return v
   379  }
   380  
   381  // Uint64 returns the uint64 representation of x.
   382  // If x cannot be represented in a uint64, the result is undefined.
   383  func (x *Int) Uint64() uint64 {
   384  	return low64(x.abs)
   385  }
   386  
   387  // IsInt64 reports whether x can be represented as an int64.
   388  func (x *Int) IsInt64() bool {
   389  	if len(x.abs) <= 64/_W {
   390  		w := int64(low64(x.abs))
   391  		return w >= 0 || x.neg && w == -w
   392  	}
   393  	return false
   394  }
   395  
   396  // IsUint64 reports whether x can be represented as a uint64.
   397  func (x *Int) IsUint64() bool {
   398  	return !x.neg && len(x.abs) <= 64/_W
   399  }
   400  
   401  // SetString sets z to the value of s, interpreted in the given base,
   402  // and returns z and a boolean indicating success. The entire string
   403  // (not just a prefix) must be valid for success. If SetString fails,
   404  // the value of z is undefined but the returned value is nil.
   405  //
   406  // The base argument must be 0 or a value between 2 and MaxBase.
   407  // For base 0, the number prefix determines the actual base: A prefix of
   408  // ``0b'' or ``0B'' selects base 2, ``0'', ``0o'' or ``0O'' selects base 8,
   409  // and ``0x'' or ``0X'' selects base 16. Otherwise, the selected base is 10
   410  // and no prefix is accepted.
   411  //
   412  // For bases <= 36, lower and upper case letters are considered the same:
   413  // The letters 'a' to 'z' and 'A' to 'Z' represent digit values 10 to 35.
   414  // For bases > 36, the upper case letters 'A' to 'Z' represent the digit
   415  // values 36 to 61.
   416  //
   417  // For base 0, an underscore character ``_'' may appear between a base
   418  // prefix and an adjacent digit, and between successive digits; such
   419  // underscores do not change the value of the number.
   420  // Incorrect placement of underscores is reported as an error if there
   421  // are no other errors. If base != 0, underscores are not recognized
   422  // and act like any other character that is not a valid digit.
   423  //
   424  func (z *Int) SetString(s string, base int) (*Int, bool) {
   425  	return z.setFromScanner(strings.NewReader(s), base)
   426  }
   427  
   428  // setFromScanner implements SetString given an io.BytesScanner.
   429  // For documentation see comments of SetString.
   430  func (z *Int) setFromScanner(r io.ByteScanner, base int) (*Int, bool) {
   431  	if _, _, err := z.scan(r, base); err != nil {
   432  		return nil, false
   433  	}
   434  	// entire content must have been consumed
   435  	if _, err := r.ReadByte(); err != io.EOF {
   436  		return nil, false
   437  	}
   438  	return z, true // err == io.EOF => scan consumed all content of r
   439  }
   440  
   441  // SetBytes interprets buf as the bytes of a big-endian unsigned
   442  // integer, sets z to that value, and returns z.
   443  func (z *Int) SetBytes(buf []byte) *Int {
   444  	z.abs = z.abs.setBytes(buf)
   445  	z.neg = false
   446  	return z
   447  }
   448  
   449  // Bytes returns the absolute value of x as a big-endian byte slice.
   450  func (x *Int) Bytes() []byte {
   451  	buf := make([]byte, len(x.abs)*_S)
   452  	return buf[x.abs.bytes(buf):]
   453  }
   454  
   455  // BitLen returns the length of the absolute value of x in bits.
   456  // The bit length of 0 is 0.
   457  func (x *Int) BitLen() int {
   458  	return x.abs.bitLen()
   459  }
   460  
   461  // TrailingZeroBits returns the number of consecutive least significant zero
   462  // bits of |x|.
   463  func (x *Int) TrailingZeroBits() uint {
   464  	return x.abs.trailingZeroBits()
   465  }
   466  
   467  // Exp sets z = x**y mod |m| (i.e. the sign of m is ignored), and returns z.
   468  // If m == nil or m == 0, z = x**y unless y <= 0 then z = 1. If m > 0, y < 0,
   469  // and x and n are not relatively prime, z is unchanged and nil is returned.
   470  //
   471  // Modular exponentiation of inputs of a particular size is not a
   472  // cryptographically constant-time operation.
   473  func (z *Int) Exp(x, y, m *Int) *Int {
   474  	// See Knuth, volume 2, section 4.6.3.
   475  	xWords := x.abs
   476  	if y.neg {
   477  		if m == nil || len(m.abs) == 0 {
   478  			return z.SetInt64(1)
   479  		}
   480  		// for y < 0: x**y mod m == (x**(-1))**|y| mod m
   481  		inverse := new(Int).ModInverse(x, m)
   482  		if inverse == nil {
   483  			return nil
   484  		}
   485  		xWords = inverse.abs
   486  	}
   487  	yWords := y.abs
   488  
   489  	var mWords nat
   490  	if m != nil {
   491  		mWords = m.abs // m.abs may be nil for m == 0
   492  	}
   493  
   494  	z.abs = z.abs.expNN(xWords, yWords, mWords)
   495  	z.neg = len(z.abs) > 0 && x.neg && len(yWords) > 0 && yWords[0]&1 == 1 // 0 has no sign
   496  	if z.neg && len(mWords) > 0 {
   497  		// make modulus result positive
   498  		z.abs = z.abs.sub(mWords, z.abs) // z == x**y mod |m| && 0 <= z < |m|
   499  		z.neg = false
   500  	}
   501  
   502  	return z
   503  }
   504  
   505  // GCD sets z to the greatest common divisor of a and b and returns z.
   506  // If x or y are not nil, GCD sets their value such that z = a*x + b*y.
   507  //
   508  // a and b may be positive, zero or negative. (Before Go 1.14 both had
   509  // to be > 0.) Regardless of the signs of a and b, z is always >= 0.
   510  //
   511  // If a == b == 0, GCD sets z = x = y = 0.
   512  //
   513  // If a == 0 and b != 0, GCD sets z = |b|, x = 0, y = sign(b) * 1.
   514  //
   515  // If a != 0 and b == 0, GCD sets z = |a|, x = sign(a) * 1, y = 0.
   516  func (z *Int) GCD(x, y, a, b *Int) *Int {
   517  	if len(a.abs) == 0 || len(b.abs) == 0 {
   518  		lenA, lenB, negA, negB := len(a.abs), len(b.abs), a.neg, b.neg
   519  		if lenA == 0 {
   520  			z.Set(b)
   521  		} else {
   522  			z.Set(a)
   523  		}
   524  		z.neg = false
   525  		if x != nil {
   526  			if lenA == 0 {
   527  				x.SetUint64(0)
   528  			} else {
   529  				x.SetUint64(1)
   530  				x.neg = negA
   531  			}
   532  		}
   533  		if y != nil {
   534  			if lenB == 0 {
   535  				y.SetUint64(0)
   536  			} else {
   537  				y.SetUint64(1)
   538  				y.neg = negB
   539  			}
   540  		}
   541  		return z
   542  	}
   543  
   544  	return z.lehmerGCD(x, y, a, b)
   545  }
   546  
   547  // lehmerSimulate attempts to simulate several Euclidean update steps
   548  // using the leading digits of A and B.  It returns u0, u1, v0, v1
   549  // such that A and B can be updated as:
   550  //		A = u0*A + v0*B
   551  //		B = u1*A + v1*B
   552  // Requirements: A >= B and len(B.abs) >= 2
   553  // Since we are calculating with full words to avoid overflow,
   554  // we use 'even' to track the sign of the cosequences.
   555  // For even iterations: u0, v1 >= 0 && u1, v0 <= 0
   556  // For odd  iterations: u0, v1 <= 0 && u1, v0 >= 0
   557  func lehmerSimulate(A, B *Int) (u0, u1, v0, v1 Word, even bool) {
   558  	// initialize the digits
   559  	var a1, a2, u2, v2 Word
   560  
   561  	m := len(B.abs) // m >= 2
   562  	n := len(A.abs) // n >= m >= 2
   563  
   564  	// extract the top Word of bits from A and B
   565  	h := nlz(A.abs[n-1])
   566  	a1 = A.abs[n-1]<<h | A.abs[n-2]>>(_W-h)
   567  	// B may have implicit zero words in the high bits if the lengths differ
   568  	switch {
   569  	case n == m:
   570  		a2 = B.abs[n-1]<<h | B.abs[n-2]>>(_W-h)
   571  	case n == m+1:
   572  		a2 = B.abs[n-2] >> (_W - h)
   573  	default:
   574  		a2 = 0
   575  	}
   576  
   577  	// Since we are calculating with full words to avoid overflow,
   578  	// we use 'even' to track the sign of the cosequences.
   579  	// For even iterations: u0, v1 >= 0 && u1, v0 <= 0
   580  	// For odd  iterations: u0, v1 <= 0 && u1, v0 >= 0
   581  	// The first iteration starts with k=1 (odd).
   582  	even = false
   583  	// variables to track the cosequences
   584  	u0, u1, u2 = 0, 1, 0
   585  	v0, v1, v2 = 0, 0, 1
   586  
   587  	// Calculate the quotient and cosequences using Collins' stopping condition.
   588  	// Note that overflow of a Word is not possible when computing the remainder
   589  	// sequence and cosequences since the cosequence size is bounded by the input size.
   590  	// See section 4.2 of Jebelean for details.
   591  	for a2 >= v2 && a1-a2 >= v1+v2 {
   592  		q, r := a1/a2, a1%a2
   593  		a1, a2 = a2, r
   594  		u0, u1, u2 = u1, u2, u1+q*u2
   595  		v0, v1, v2 = v1, v2, v1+q*v2
   596  		even = !even
   597  	}
   598  	return
   599  }
   600  
   601  // lehmerUpdate updates the inputs A and B such that:
   602  //		A = u0*A + v0*B
   603  //		B = u1*A + v1*B
   604  // where the signs of u0, u1, v0, v1 are given by even
   605  // For even == true: u0, v1 >= 0 && u1, v0 <= 0
   606  // For even == false: u0, v1 <= 0 && u1, v0 >= 0
   607  // q, r, s, t are temporary variables to avoid allocations in the multiplication
   608  func lehmerUpdate(A, B, q, r, s, t *Int, u0, u1, v0, v1 Word, even bool) {
   609  
   610  	t.abs = t.abs.setWord(u0)
   611  	s.abs = s.abs.setWord(v0)
   612  	t.neg = !even
   613  	s.neg = even
   614  
   615  	t.Mul(A, t)
   616  	s.Mul(B, s)
   617  
   618  	r.abs = r.abs.setWord(u1)
   619  	q.abs = q.abs.setWord(v1)
   620  	r.neg = even
   621  	q.neg = !even
   622  
   623  	r.Mul(A, r)
   624  	q.Mul(B, q)
   625  
   626  	A.Add(t, s)
   627  	B.Add(r, q)
   628  }
   629  
   630  // euclidUpdate performs a single step of the Euclidean GCD algorithm
   631  // if extended is true, it also updates the cosequence Ua, Ub
   632  func euclidUpdate(A, B, Ua, Ub, q, r, s, t *Int, extended bool) {
   633  	q, r = q.QuoRem(A, B, r)
   634  
   635  	*A, *B, *r = *B, *r, *A
   636  
   637  	if extended {
   638  		// Ua, Ub = Ub, Ua - q*Ub
   639  		t.Set(Ub)
   640  		s.Mul(Ub, q)
   641  		Ub.Sub(Ua, s)
   642  		Ua.Set(t)
   643  	}
   644  }
   645  
   646  // lehmerGCD sets z to the greatest common divisor of a and b,
   647  // which both must be != 0, and returns z.
   648  // If x or y are not nil, their values are set such that z = a*x + b*y.
   649  // See Knuth, The Art of Computer Programming, Vol. 2, Section 4.5.2, Algorithm L.
   650  // This implementation uses the improved condition by Collins requiring only one
   651  // quotient and avoiding the possibility of single Word overflow.
   652  // See Jebelean, "Improving the multiprecision Euclidean algorithm",
   653  // Design and Implementation of Symbolic Computation Systems, pp 45-58.
   654  // The cosequences are updated according to Algorithm 10.45 from
   655  // Cohen et al. "Handbook of Elliptic and Hyperelliptic Curve Cryptography" pp 192.
   656  func (z *Int) lehmerGCD(x, y, a, b *Int) *Int {
   657  	var A, B, Ua, Ub *Int
   658  
   659  	A = new(Int).Abs(a)
   660  	B = new(Int).Abs(b)
   661  
   662  	extended := x != nil || y != nil
   663  
   664  	if extended {
   665  		// Ua (Ub) tracks how many times input a has been accumulated into A (B).
   666  		Ua = new(Int).SetInt64(1)
   667  		Ub = new(Int)
   668  	}
   669  
   670  	// temp variables for multiprecision update
   671  	q := new(Int)
   672  	r := new(Int)
   673  	s := new(Int)
   674  	t := new(Int)
   675  
   676  	// ensure A >= B
   677  	if A.abs.cmp(B.abs) < 0 {
   678  		A, B = B, A
   679  		Ub, Ua = Ua, Ub
   680  	}
   681  
   682  	// loop invariant A >= B
   683  	for len(B.abs) > 1 {
   684  		// Attempt to calculate in single-precision using leading words of A and B.
   685  		u0, u1, v0, v1, even := lehmerSimulate(A, B)
   686  
   687  		// multiprecision Step
   688  		if v0 != 0 {
   689  			// Simulate the effect of the single-precision steps using the cosequences.
   690  			// A = u0*A + v0*B
   691  			// B = u1*A + v1*B
   692  			lehmerUpdate(A, B, q, r, s, t, u0, u1, v0, v1, even)
   693  
   694  			if extended {
   695  				// Ua = u0*Ua + v0*Ub
   696  				// Ub = u1*Ua + v1*Ub
   697  				lehmerUpdate(Ua, Ub, q, r, s, t, u0, u1, v0, v1, even)
   698  			}
   699  
   700  		} else {
   701  			// Single-digit calculations failed to simulate any quotients.
   702  			// Do a standard Euclidean step.
   703  			euclidUpdate(A, B, Ua, Ub, q, r, s, t, extended)
   704  		}
   705  	}
   706  
   707  	if len(B.abs) > 0 {
   708  		// extended Euclidean algorithm base case if B is a single Word
   709  		if len(A.abs) > 1 {
   710  			// A is longer than a single Word, so one update is needed.
   711  			euclidUpdate(A, B, Ua, Ub, q, r, s, t, extended)
   712  		}
   713  		if len(B.abs) > 0 {
   714  			// A and B are both a single Word.
   715  			aWord, bWord := A.abs[0], B.abs[0]
   716  			if extended {
   717  				var ua, ub, va, vb Word
   718  				ua, ub = 1, 0
   719  				va, vb = 0, 1
   720  				even := true
   721  				for bWord != 0 {
   722  					q, r := aWord/bWord, aWord%bWord
   723  					aWord, bWord = bWord, r
   724  					ua, ub = ub, ua+q*ub
   725  					va, vb = vb, va+q*vb
   726  					even = !even
   727  				}
   728  
   729  				t.abs = t.abs.setWord(ua)
   730  				s.abs = s.abs.setWord(va)
   731  				t.neg = !even
   732  				s.neg = even
   733  
   734  				t.Mul(Ua, t)
   735  				s.Mul(Ub, s)
   736  
   737  				Ua.Add(t, s)
   738  			} else {
   739  				for bWord != 0 {
   740  					aWord, bWord = bWord, aWord%bWord
   741  				}
   742  			}
   743  			A.abs[0] = aWord
   744  		}
   745  	}
   746  	negA := a.neg
   747  	if y != nil {
   748  		// avoid aliasing b needed in the division below
   749  		if y == b {
   750  			B.Set(b)
   751  		} else {
   752  			B = b
   753  		}
   754  		// y = (z - a*x)/b
   755  		y.Mul(a, Ua) // y can safely alias a
   756  		if negA {
   757  			y.neg = !y.neg
   758  		}
   759  		y.Sub(A, y)
   760  		y.Div(y, B)
   761  	}
   762  
   763  	if x != nil {
   764  		*x = *Ua
   765  		if negA {
   766  			x.neg = !x.neg
   767  		}
   768  	}
   769  
   770  	*z = *A
   771  
   772  	return z
   773  }
   774  
   775  // Rand sets z to a pseudo-random number in [0, n) and returns z.
   776  //
   777  // As this uses the math/rand package, it must not be used for
   778  // security-sensitive work. Use crypto/rand.Int instead.
   779  func (z *Int) Rand(rnd *rand.Rand, n *Int) *Int {
   780  	z.neg = false
   781  	if n.neg || len(n.abs) == 0 {
   782  		z.abs = nil
   783  		return z
   784  	}
   785  	z.abs = z.abs.random(rnd, n.abs, n.abs.bitLen())
   786  	return z
   787  }
   788  
   789  // ModInverse sets z to the multiplicative inverse of g in the ring ℤ/nℤ
   790  // and returns z. If g and n are not relatively prime, g has no multiplicative
   791  // inverse in the ring ℤ/nℤ.  In this case, z is unchanged and the return value
   792  // is nil.
   793  func (z *Int) ModInverse(g, n *Int) *Int {
   794  	// GCD expects parameters a and b to be > 0.
   795  	if n.neg {
   796  		var n2 Int
   797  		n = n2.Neg(n)
   798  	}
   799  	if g.neg {
   800  		var g2 Int
   801  		g = g2.Mod(g, n)
   802  	}
   803  	var d, x Int
   804  	d.GCD(&x, nil, g, n)
   805  
   806  	// if and only if d==1, g and n are relatively prime
   807  	if d.Cmp(intOne) != 0 {
   808  		return nil
   809  	}
   810  
   811  	// x and y are such that g*x + n*y = 1, therefore x is the inverse element,
   812  	// but it may be negative, so convert to the range 0 <= z < |n|
   813  	if x.neg {
   814  		z.Add(&x, n)
   815  	} else {
   816  		z.Set(&x)
   817  	}
   818  	return z
   819  }
   820  
   821  // Jacobi returns the Jacobi symbol (x/y), either +1, -1, or 0.
   822  // The y argument must be an odd integer.
   823  func Jacobi(x, y *Int) int {
   824  	if len(y.abs) == 0 || y.abs[0]&1 == 0 {
   825  		panic(fmt.Sprintf("big: invalid 2nd argument to Int.Jacobi: need odd integer but got %s", y))
   826  	}
   827  
   828  	// We use the formulation described in chapter 2, section 2.4,
   829  	// "The Yacas Book of Algorithms":
   830  	// http://yacas.sourceforge.net/Algo.book.pdf
   831  
   832  	var a, b, c Int
   833  	a.Set(x)
   834  	b.Set(y)
   835  	j := 1
   836  
   837  	if b.neg {
   838  		if a.neg {
   839  			j = -1
   840  		}
   841  		b.neg = false
   842  	}
   843  
   844  	for {
   845  		if b.Cmp(intOne) == 0 {
   846  			return j
   847  		}
   848  		if len(a.abs) == 0 {
   849  			return 0
   850  		}
   851  		a.Mod(&a, &b)
   852  		if len(a.abs) == 0 {
   853  			return 0
   854  		}
   855  		// a > 0
   856  
   857  		// handle factors of 2 in 'a'
   858  		s := a.abs.trailingZeroBits()
   859  		if s&1 != 0 {
   860  			bmod8 := b.abs[0] & 7
   861  			if bmod8 == 3 || bmod8 == 5 {
   862  				j = -j
   863  			}
   864  		}
   865  		c.Rsh(&a, s) // a = 2^s*c
   866  
   867  		// swap numerator and denominator
   868  		if b.abs[0]&3 == 3 && c.abs[0]&3 == 3 {
   869  			j = -j
   870  		}
   871  		a.Set(&b)
   872  		b.Set(&c)
   873  	}
   874  }
   875  
   876  // modSqrt3Mod4 uses the identity
   877  //      (a^((p+1)/4))^2  mod p
   878  //   == u^(p+1)          mod p
   879  //   == u^2              mod p
   880  // to calculate the square root of any quadratic residue mod p quickly for 3
   881  // mod 4 primes.
   882  func (z *Int) modSqrt3Mod4Prime(x, p *Int) *Int {
   883  	e := new(Int).Add(p, intOne) // e = p + 1
   884  	e.Rsh(e, 2)                  // e = (p + 1) / 4
   885  	z.Exp(x, e, p)               // z = x^e mod p
   886  	return z
   887  }
   888  
   889  // modSqrt5Mod8 uses Atkin's observation that 2 is not a square mod p
   890  //   alpha ==  (2*a)^((p-5)/8)    mod p
   891  //   beta  ==  2*a*alpha^2        mod p  is a square root of -1
   892  //   b     ==  a*alpha*(beta-1)   mod p  is a square root of a
   893  // to calculate the square root of any quadratic residue mod p quickly for 5
   894  // mod 8 primes.
   895  func (z *Int) modSqrt5Mod8Prime(x, p *Int) *Int {
   896  	// p == 5 mod 8 implies p = e*8 + 5
   897  	// e is the quotient and 5 the remainder on division by 8
   898  	e := new(Int).Rsh(p, 3)  // e = (p - 5) / 8
   899  	tx := new(Int).Lsh(x, 1) // tx = 2*x
   900  	alpha := new(Int).Exp(tx, e, p)
   901  	beta := new(Int).Mul(alpha, alpha)
   902  	beta.Mod(beta, p)
   903  	beta.Mul(beta, tx)
   904  	beta.Mod(beta, p)
   905  	beta.Sub(beta, intOne)
   906  	beta.Mul(beta, x)
   907  	beta.Mod(beta, p)
   908  	beta.Mul(beta, alpha)
   909  	z.Mod(beta, p)
   910  	return z
   911  }
   912  
   913  // modSqrtTonelliShanks uses the Tonelli-Shanks algorithm to find the square
   914  // root of a quadratic residue modulo any prime.
   915  func (z *Int) modSqrtTonelliShanks(x, p *Int) *Int {
   916  	// Break p-1 into s*2^e such that s is odd.
   917  	var s Int
   918  	s.Sub(p, intOne)
   919  	e := s.abs.trailingZeroBits()
   920  	s.Rsh(&s, e)
   921  
   922  	// find some non-square n
   923  	var n Int
   924  	n.SetInt64(2)
   925  	for Jacobi(&n, p) != -1 {
   926  		n.Add(&n, intOne)
   927  	}
   928  
   929  	// Core of the Tonelli-Shanks algorithm. Follows the description in
   930  	// section 6 of "Square roots from 1; 24, 51, 10 to Dan Shanks" by Ezra
   931  	// Brown:
   932  	// https://www.maa.org/sites/default/files/pdf/upload_library/22/Polya/07468342.di020786.02p0470a.pdf
   933  	var y, b, g, t Int
   934  	y.Add(&s, intOne)
   935  	y.Rsh(&y, 1)
   936  	y.Exp(x, &y, p)  // y = x^((s+1)/2)
   937  	b.Exp(x, &s, p)  // b = x^s
   938  	g.Exp(&n, &s, p) // g = n^s
   939  	r := e
   940  	for {
   941  		// find the least m such that ord_p(b) = 2^m
   942  		var m uint
   943  		t.Set(&b)
   944  		for t.Cmp(intOne) != 0 {
   945  			t.Mul(&t, &t).Mod(&t, p)
   946  			m++
   947  		}
   948  
   949  		if m == 0 {
   950  			return z.Set(&y)
   951  		}
   952  
   953  		t.SetInt64(0).SetBit(&t, int(r-m-1), 1).Exp(&g, &t, p)
   954  		// t = g^(2^(r-m-1)) mod p
   955  		g.Mul(&t, &t).Mod(&g, p) // g = g^(2^(r-m)) mod p
   956  		y.Mul(&y, &t).Mod(&y, p)
   957  		b.Mul(&b, &g).Mod(&b, p)
   958  		r = m
   959  	}
   960  }
   961  
   962  // ModSqrt sets z to a square root of x mod p if such a square root exists, and
   963  // returns z. The modulus p must be an odd prime. If x is not a square mod p,
   964  // ModSqrt leaves z unchanged and returns nil. This function panics if p is
   965  // not an odd integer.
   966  func (z *Int) ModSqrt(x, p *Int) *Int {
   967  	switch Jacobi(x, p) {
   968  	case -1:
   969  		return nil // x is not a square mod p
   970  	case 0:
   971  		return z.SetInt64(0) // sqrt(0) mod p = 0
   972  	case 1:
   973  		break
   974  	}
   975  	if x.neg || x.Cmp(p) >= 0 { // ensure 0 <= x < p
   976  		x = new(Int).Mod(x, p)
   977  	}
   978  
   979  	switch {
   980  	case p.abs[0]%4 == 3:
   981  		// Check whether p is 3 mod 4, and if so, use the faster algorithm.
   982  		return z.modSqrt3Mod4Prime(x, p)
   983  	case p.abs[0]%8 == 5:
   984  		// Check whether p is 5 mod 8, use Atkin's algorithm.
   985  		return z.modSqrt5Mod8Prime(x, p)
   986  	default:
   987  		// Otherwise, use Tonelli-Shanks.
   988  		return z.modSqrtTonelliShanks(x, p)
   989  	}
   990  }
   991  
   992  // Lsh sets z = x << n and returns z.
   993  func (z *Int) Lsh(x *Int, n uint) *Int {
   994  	z.abs = z.abs.shl(x.abs, n)
   995  	z.neg = x.neg
   996  	return z
   997  }
   998  
   999  // Rsh sets z = x >> n and returns z.
  1000  func (z *Int) Rsh(x *Int, n uint) *Int {
  1001  	if x.neg {
  1002  		// (-x) >> s == ^(x-1) >> s == ^((x-1) >> s) == -(((x-1) >> s) + 1)
  1003  		t := z.abs.sub(x.abs, natOne) // no underflow because |x| > 0
  1004  		t = t.shr(t, n)
  1005  		z.abs = t.add(t, natOne)
  1006  		z.neg = true // z cannot be zero if x is negative
  1007  		return z
  1008  	}
  1009  
  1010  	z.abs = z.abs.shr(x.abs, n)
  1011  	z.neg = false
  1012  	return z
  1013  }
  1014  
  1015  // Bit returns the value of the i'th bit of x. That is, it
  1016  // returns (x>>i)&1. The bit index i must be >= 0.
  1017  func (x *Int) Bit(i int) uint {
  1018  	if i == 0 {
  1019  		// optimization for common case: odd/even test of x
  1020  		if len(x.abs) > 0 {
  1021  			return uint(x.abs[0] & 1) // bit 0 is same for -x
  1022  		}
  1023  		return 0
  1024  	}
  1025  	if i < 0 {
  1026  		panic("negative bit index")
  1027  	}
  1028  	if x.neg {
  1029  		t := nat(nil).sub(x.abs, natOne)
  1030  		return t.bit(uint(i)) ^ 1
  1031  	}
  1032  
  1033  	return x.abs.bit(uint(i))
  1034  }
  1035  
  1036  // SetBit sets z to x, with x's i'th bit set to b (0 or 1).
  1037  // That is, if b is 1 SetBit sets z = x | (1 << i);
  1038  // if b is 0 SetBit sets z = x &^ (1 << i). If b is not 0 or 1,
  1039  // SetBit will panic.
  1040  func (z *Int) SetBit(x *Int, i int, b uint) *Int {
  1041  	if i < 0 {
  1042  		panic("negative bit index")
  1043  	}
  1044  	if x.neg {
  1045  		t := z.abs.sub(x.abs, natOne)
  1046  		t = t.setBit(t, uint(i), b^1)
  1047  		z.abs = t.add(t, natOne)
  1048  		z.neg = len(z.abs) > 0
  1049  		return z
  1050  	}
  1051  	z.abs = z.abs.setBit(x.abs, uint(i), b)
  1052  	z.neg = false
  1053  	return z
  1054  }
  1055  
  1056  // And sets z = x & y and returns z.
  1057  func (z *Int) And(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.or(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.and(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) == x &^ (y-1)
  1080  	y1 := nat(nil).sub(y.abs, natOne)
  1081  	z.abs = z.abs.andNot(x.abs, y1)
  1082  	z.neg = false
  1083  	return z
  1084  }
  1085  
  1086  // AndNot sets z = x &^ y and returns z.
  1087  func (z *Int) AndNot(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) == (y-1) &^ (x-1)
  1091  			x1 := nat(nil).sub(x.abs, natOne)
  1092  			y1 := nat(nil).sub(y.abs, natOne)
  1093  			z.abs = z.abs.andNot(y1, x1)
  1094  			z.neg = false
  1095  			return z
  1096  		}
  1097  
  1098  		// x &^ y == x &^ y
  1099  		z.abs = z.abs.andNot(x.abs, y.abs)
  1100  		z.neg = false
  1101  		return z
  1102  	}
  1103  
  1104  	if x.neg {
  1105  		// (-x) &^ y == ^(x-1) &^ y == ^(x-1) & ^y == ^((x-1) | y) == -(((x-1) | y) + 1)
  1106  		x1 := nat(nil).sub(x.abs, natOne)
  1107  		z.abs = z.abs.add(z.abs.or(x1, y.abs), natOne)
  1108  		z.neg = true // z cannot be zero if x is negative and y is positive
  1109  		return z
  1110  	}
  1111  
  1112  	// x &^ (-y) == x &^ ^(y-1) == x & (y-1)
  1113  	y1 := nat(nil).sub(y.abs, natOne)
  1114  	z.abs = z.abs.and(x.abs, y1)
  1115  	z.neg = false
  1116  	return z
  1117  }
  1118  
  1119  // Or sets z = x | y and returns z.
  1120  func (z *Int) Or(x, y *Int) *Int {
  1121  	if x.neg == y.neg {
  1122  		if x.neg {
  1123  			// (-x) | (-y) == ^(x-1) | ^(y-1) == ^((x-1) & (y-1)) == -(((x-1) & (y-1)) + 1)
  1124  			x1 := nat(nil).sub(x.abs, natOne)
  1125  			y1 := nat(nil).sub(y.abs, natOne)
  1126  			z.abs = z.abs.add(z.abs.and(x1, y1), natOne)
  1127  			z.neg = true // z cannot be zero if x and y are negative
  1128  			return z
  1129  		}
  1130  
  1131  		// x | y == x | y
  1132  		z.abs = z.abs.or(x.abs, y.abs)
  1133  		z.neg = false
  1134  		return z
  1135  	}
  1136  
  1137  	// x.neg != y.neg
  1138  	if x.neg {
  1139  		x, y = y, x // | is symmetric
  1140  	}
  1141  
  1142  	// x | (-y) == x | ^(y-1) == ^((y-1) &^ x) == -(^((y-1) &^ x) + 1)
  1143  	y1 := nat(nil).sub(y.abs, natOne)
  1144  	z.abs = z.abs.add(z.abs.andNot(y1, x.abs), natOne)
  1145  	z.neg = true // z cannot be zero if one of x or y is negative
  1146  	return z
  1147  }
  1148  
  1149  // Xor sets z = x ^ y and returns z.
  1150  func (z *Int) Xor(x, y *Int) *Int {
  1151  	if x.neg == y.neg {
  1152  		if x.neg {
  1153  			// (-x) ^ (-y) == ^(x-1) ^ ^(y-1) == (x-1) ^ (y-1)
  1154  			x1 := nat(nil).sub(x.abs, natOne)
  1155  			y1 := nat(nil).sub(y.abs, natOne)
  1156  			z.abs = z.abs.xor(x1, y1)
  1157  			z.neg = false
  1158  			return z
  1159  		}
  1160  
  1161  		// x ^ y == x ^ y
  1162  		z.abs = z.abs.xor(x.abs, y.abs)
  1163  		z.neg = false
  1164  		return z
  1165  	}
  1166  
  1167  	// x.neg != y.neg
  1168  	if x.neg {
  1169  		x, y = y, x // ^ is symmetric
  1170  	}
  1171  
  1172  	// x ^ (-y) == x ^ ^(y-1) == ^(x ^ (y-1)) == -((x ^ (y-1)) + 1)
  1173  	y1 := nat(nil).sub(y.abs, natOne)
  1174  	z.abs = z.abs.add(z.abs.xor(x.abs, y1), natOne)
  1175  	z.neg = true // z cannot be zero if only one of x or y is negative
  1176  	return z
  1177  }
  1178  
  1179  // Not sets z = ^x and returns z.
  1180  func (z *Int) Not(x *Int) *Int {
  1181  	if x.neg {
  1182  		// ^(-x) == ^(^(x-1)) == x-1
  1183  		z.abs = z.abs.sub(x.abs, natOne)
  1184  		z.neg = false
  1185  		return z
  1186  	}
  1187  
  1188  	// ^x == -x-1 == -(x+1)
  1189  	z.abs = z.abs.add(x.abs, natOne)
  1190  	z.neg = true // z cannot be zero if x is positive
  1191  	return z
  1192  }
  1193  
  1194  // Sqrt sets z to ⌊√x⌋, the largest integer such that z² ≤ x, and returns z.
  1195  // It panics if x is negative.
  1196  func (z *Int) Sqrt(x *Int) *Int {
  1197  	if x.neg {
  1198  		panic("square root of negative number")
  1199  	}
  1200  	z.neg = false
  1201  	z.abs = z.abs.sqrt(x.abs)
  1202  	return z
  1203  }
  1204  

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