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  //
    26  // Note that methods may leak the Int's value through timing side-channels.
    27  // Because of this and because of the scope and complexity of the
    28  // implementation, Int is not well-suited to implement cryptographic operations.
    29  // The standard library avoids exposing non-trivial Int methods to
    30  // attacker-controlled inputs and the determination of whether a bug in math/big
    31  // is considered a security vulnerability might depend on the impact on the
    32  // standard library.
    33  type Int struct {
    34  	neg bool // sign
    35  	abs nat  // absolute value of the integer
    36  }
    37  
    38  var intOne = &Int{false, natOne}
    39  
    40  // Sign returns:
    41  //
    42  //	-1 if x <  0
    43  //	 0 if x == 0
    44  //	+1 if x >  0
    45  func (x *Int) Sign() int {
    46  	// This function is used in cryptographic operations. It must not leak
    47  	// anything but the Int's sign and bit size through side-channels. Any
    48  	// changes must be reviewed by a security expert.
    49  	if len(x.abs) == 0 {
    50  		return 0
    51  	}
    52  	if x.neg {
    53  		return -1
    54  	}
    55  	return 1
    56  }
    57  
    58  // SetInt64 sets z to x and returns z.
    59  func (z *Int) SetInt64(x int64) *Int {
    60  	neg := false
    61  	if x < 0 {
    62  		neg = true
    63  		x = -x
    64  	}
    65  	z.abs = z.abs.setUint64(uint64(x))
    66  	z.neg = neg
    67  	return z
    68  }
    69  
    70  // SetUint64 sets z to x and returns z.
    71  func (z *Int) SetUint64(x uint64) *Int {
    72  	z.abs = z.abs.setUint64(x)
    73  	z.neg = false
    74  	return z
    75  }
    76  
    77  // NewInt allocates and returns a new [Int] set to x.
    78  func NewInt(x int64) *Int {
    79  	// This code is arranged to be inlineable and produce
    80  	// zero allocations when inlined. See issue 29951.
    81  	u := uint64(x)
    82  	if x < 0 {
    83  		u = -u
    84  	}
    85  	var abs []Word
    86  	if x == 0 {
    87  	} else if _W == 32 && u>>32 != 0 {
    88  		abs = []Word{Word(u), Word(u >> 32)}
    89  	} else {
    90  		abs = []Word{Word(u)}
    91  	}
    92  	return &Int{neg: x < 0, abs: abs}
    93  }
    94  
    95  // Set sets z to x and returns z.
    96  func (z *Int) Set(x *Int) *Int {
    97  	if z != x {
    98  		z.abs = z.abs.set(x.abs)
    99  		z.neg = x.neg
   100  	}
   101  	return z
   102  }
   103  
   104  // Bits provides raw (unchecked but fast) access to x by returning its
   105  // absolute value as a little-endian [Word] slice. The result and x share
   106  // the same underlying array.
   107  // Bits is intended to support implementation of missing low-level [Int]
   108  // functionality outside this package; it should be avoided otherwise.
   109  func (x *Int) Bits() []Word {
   110  	// This function is used in cryptographic operations. It must not leak
   111  	// anything but the Int's sign and bit size through side-channels. Any
   112  	// changes must be reviewed by a security expert.
   113  	return x.abs
   114  }
   115  
   116  // SetBits provides raw (unchecked but fast) access to z by setting its
   117  // value to abs, interpreted as a little-endian [Word] slice, and returning
   118  // z. The result and abs share the same underlying array.
   119  // SetBits is intended to support implementation of missing low-level [Int]
   120  // functionality outside this package; it should be avoided otherwise.
   121  func (z *Int) SetBits(abs []Word) *Int {
   122  	z.abs = nat(abs).norm()
   123  	z.neg = false
   124  	return z
   125  }
   126  
   127  // Abs sets z to |x| (the absolute value of x) and returns z.
   128  func (z *Int) Abs(x *Int) *Int {
   129  	z.Set(x)
   130  	z.neg = false
   131  	return z
   132  }
   133  
   134  // Neg sets z to -x and returns z.
   135  func (z *Int) Neg(x *Int) *Int {
   136  	z.Set(x)
   137  	z.neg = len(z.abs) > 0 && !z.neg // 0 has no sign
   138  	return z
   139  }
   140  
   141  // Add sets z to the sum x+y and returns z.
   142  func (z *Int) Add(x, y *Int) *Int {
   143  	neg := x.neg
   144  	if x.neg == y.neg {
   145  		// x + y == x + y
   146  		// (-x) + (-y) == -(x + y)
   147  		z.abs = z.abs.add(x.abs, y.abs)
   148  	} else {
   149  		// x + (-y) == x - y == -(y - x)
   150  		// (-x) + y == y - x == -(x - y)
   151  		if x.abs.cmp(y.abs) >= 0 {
   152  			z.abs = z.abs.sub(x.abs, y.abs)
   153  		} else {
   154  			neg = !neg
   155  			z.abs = z.abs.sub(y.abs, x.abs)
   156  		}
   157  	}
   158  	z.neg = len(z.abs) > 0 && neg // 0 has no sign
   159  	return z
   160  }
   161  
   162  // Sub sets z to the difference x-y and returns z.
   163  func (z *Int) Sub(x, y *Int) *Int {
   164  	neg := x.neg
   165  	if x.neg != y.neg {
   166  		// x - (-y) == x + y
   167  		// (-x) - y == -(x + y)
   168  		z.abs = z.abs.add(x.abs, y.abs)
   169  	} else {
   170  		// x - y == x - y == -(y - x)
   171  		// (-x) - (-y) == y - x == -(x - y)
   172  		if x.abs.cmp(y.abs) >= 0 {
   173  			z.abs = z.abs.sub(x.abs, y.abs)
   174  		} else {
   175  			neg = !neg
   176  			z.abs = z.abs.sub(y.abs, x.abs)
   177  		}
   178  	}
   179  	z.neg = len(z.abs) > 0 && neg // 0 has no sign
   180  	return z
   181  }
   182  
   183  // Mul sets z to the product x*y and returns z.
   184  func (z *Int) Mul(x, y *Int) *Int {
   185  	// x * y == x * y
   186  	// x * (-y) == -(x * y)
   187  	// (-x) * y == -(x * y)
   188  	// (-x) * (-y) == x * y
   189  	if x == y {
   190  		z.abs = z.abs.sqr(x.abs)
   191  		z.neg = false
   192  		return z
   193  	}
   194  	z.abs = z.abs.mul(x.abs, y.abs)
   195  	z.neg = len(z.abs) > 0 && x.neg != y.neg // 0 has no sign
   196  	return z
   197  }
   198  
   199  // MulRange sets z to the product of all integers
   200  // in the range [a, b] inclusively and returns z.
   201  // If a > b (empty range), the result is 1.
   202  func (z *Int) MulRange(a, b int64) *Int {
   203  	switch {
   204  	case a > b:
   205  		return z.SetInt64(1) // empty range
   206  	case a <= 0 && b >= 0:
   207  		return z.SetInt64(0) // range includes 0
   208  	}
   209  	// a <= b && (b < 0 || a > 0)
   210  
   211  	neg := false
   212  	if a < 0 {
   213  		neg = (b-a)&1 == 0
   214  		a, b = -b, -a
   215  	}
   216  
   217  	z.abs = z.abs.mulRange(uint64(a), uint64(b))
   218  	z.neg = neg
   219  	return z
   220  }
   221  
   222  // Binomial sets z to the binomial coefficient C(n, k) and returns z.
   223  func (z *Int) Binomial(n, k int64) *Int {
   224  	if k > n {
   225  		return z.SetInt64(0)
   226  	}
   227  	// reduce the number of multiplications by reducing k
   228  	if k > n-k {
   229  		k = n - k // C(n, k) == C(n, n-k)
   230  	}
   231  	// C(n, k) == n * (n-1) * ... * (n-k+1) / k * (k-1) * ... * 1
   232  	//         == n * (n-1) * ... * (n-k+1) / 1 * (1+1) * ... * k
   233  	//
   234  	// Using the multiplicative formula produces smaller values
   235  	// at each step, requiring fewer allocations and computations:
   236  	//
   237  	// z = 1
   238  	// for i := 0; i < k; i = i+1 {
   239  	//     z *= n-i
   240  	//     z /= i+1
   241  	// }
   242  	//
   243  	// finally to avoid computing i+1 twice per loop:
   244  	//
   245  	// z = 1
   246  	// i := 0
   247  	// for i < k {
   248  	//     z *= n-i
   249  	//     i++
   250  	//     z /= i
   251  	// }
   252  	var N, K, i, t Int
   253  	N.SetInt64(n)
   254  	K.SetInt64(k)
   255  	z.Set(intOne)
   256  	for i.Cmp(&K) < 0 {
   257  		z.Mul(z, t.Sub(&N, &i))
   258  		i.Add(&i, intOne)
   259  		z.Quo(z, &i)
   260  	}
   261  	return z
   262  }
   263  
   264  // Quo sets z to the quotient x/y for y != 0 and returns z.
   265  // If y == 0, a division-by-zero run-time panic occurs.
   266  // Quo implements truncated division (like Go); see [Int.QuoRem] for more details.
   267  func (z *Int) Quo(x, y *Int) *Int {
   268  	z.abs, _ = z.abs.div(nil, x.abs, y.abs)
   269  	z.neg = len(z.abs) > 0 && x.neg != y.neg // 0 has no sign
   270  	return z
   271  }
   272  
   273  // Rem sets z to the remainder x%y for y != 0 and returns z.
   274  // If y == 0, a division-by-zero run-time panic occurs.
   275  // Rem implements truncated modulus (like Go); see [Int.QuoRem] for more details.
   276  func (z *Int) Rem(x, y *Int) *Int {
   277  	_, z.abs = nat(nil).div(z.abs, x.abs, y.abs)
   278  	z.neg = len(z.abs) > 0 && x.neg // 0 has no sign
   279  	return z
   280  }
   281  
   282  // QuoRem sets z to the quotient x/y and r to the remainder x%y
   283  // and returns the pair (z, r) for y != 0.
   284  // If y == 0, a division-by-zero run-time panic occurs.
   285  //
   286  // QuoRem implements T-division and modulus (like Go):
   287  //
   288  //	q = x/y      with the result truncated to zero
   289  //	r = x - y*q
   290  //
   291  // (See Daan Leijen, “Division and Modulus for Computer Scientists”.)
   292  // See DivMod for Euclidean division and modulus (unlike Go).
   293  func (z *Int) QuoRem(x, y, r *Int) (*Int, *Int) {
   294  	z.abs, r.abs = z.abs.div(r.abs, x.abs, y.abs)
   295  	z.neg, r.neg = len(z.abs) > 0 && x.neg != y.neg, len(r.abs) > 0 && x.neg // 0 has no sign
   296  	return z, r
   297  }
   298  
   299  // Div sets z to the quotient x/y for y != 0 and returns z.
   300  // If y == 0, a division-by-zero run-time panic occurs.
   301  // Div implements Euclidean division (unlike Go); see [Int.DivMod] for more details.
   302  func (z *Int) Div(x, y *Int) *Int {
   303  	y_neg := y.neg // z may be an alias for y
   304  	var r Int
   305  	z.QuoRem(x, y, &r)
   306  	if r.neg {
   307  		if y_neg {
   308  			z.Add(z, intOne)
   309  		} else {
   310  			z.Sub(z, intOne)
   311  		}
   312  	}
   313  	return z
   314  }
   315  
   316  // Mod sets z to the modulus x%y for y != 0 and returns z.
   317  // If y == 0, a division-by-zero run-time panic occurs.
   318  // Mod implements Euclidean modulus (unlike Go); see [Int.DivMod] for more details.
   319  func (z *Int) Mod(x, y *Int) *Int {
   320  	y0 := y // save y
   321  	if z == y || alias(z.abs, y.abs) {
   322  		y0 = new(Int).Set(y)
   323  	}
   324  	var q Int
   325  	q.QuoRem(x, y, z)
   326  	if z.neg {
   327  		if y0.neg {
   328  			z.Sub(z, y0)
   329  		} else {
   330  			z.Add(z, y0)
   331  		}
   332  	}
   333  	return z
   334  }
   335  
   336  // DivMod sets z to the quotient x div y and m to the modulus x mod y
   337  // and returns the pair (z, m) for y != 0.
   338  // If y == 0, a division-by-zero run-time panic occurs.
   339  //
   340  // DivMod implements Euclidean division and modulus (unlike Go):
   341  //
   342  //	q = x div y  such that
   343  //	m = x - y*q  with 0 <= m < |y|
   344  //
   345  // (See Raymond T. Boute, “The Euclidean definition of the functions
   346  // div and mod”. ACM Transactions on Programming Languages and
   347  // Systems (TOPLAS), 14(2):127-144, New York, NY, USA, 4/1992.
   348  // ACM press.)
   349  // See [Int.QuoRem] for T-division and modulus (like Go).
   350  func (z *Int) DivMod(x, y, m *Int) (*Int, *Int) {
   351  	y0 := y // save y
   352  	if z == y || alias(z.abs, y.abs) {
   353  		y0 = new(Int).Set(y)
   354  	}
   355  	z.QuoRem(x, y, m)
   356  	if m.neg {
   357  		if y0.neg {
   358  			z.Add(z, intOne)
   359  			m.Sub(m, y0)
   360  		} else {
   361  			z.Sub(z, intOne)
   362  			m.Add(m, y0)
   363  		}
   364  	}
   365  	return z, m
   366  }
   367  
   368  // Cmp compares x and y and returns:
   369  //
   370  //	-1 if x <  y
   371  //	 0 if x == y
   372  //	+1 if x >  y
   373  func (x *Int) Cmp(y *Int) (r int) {
   374  	// x cmp y == x cmp y
   375  	// x cmp (-y) == x
   376  	// (-x) cmp y == y
   377  	// (-x) cmp (-y) == -(x cmp y)
   378  	switch {
   379  	case x == y:
   380  		// nothing to do
   381  	case x.neg == y.neg:
   382  		r = x.abs.cmp(y.abs)
   383  		if x.neg {
   384  			r = -r
   385  		}
   386  	case x.neg:
   387  		r = -1
   388  	default:
   389  		r = 1
   390  	}
   391  	return
   392  }
   393  
   394  // CmpAbs compares the absolute values of x and y and returns:
   395  //
   396  //	-1 if |x| <  |y|
   397  //	 0 if |x| == |y|
   398  //	+1 if |x| >  |y|
   399  func (x *Int) CmpAbs(y *Int) int {
   400  	return x.abs.cmp(y.abs)
   401  }
   402  
   403  // low32 returns the least significant 32 bits of x.
   404  func low32(x nat) uint32 {
   405  	if len(x) == 0 {
   406  		return 0
   407  	}
   408  	return uint32(x[0])
   409  }
   410  
   411  // low64 returns the least significant 64 bits of x.
   412  func low64(x nat) uint64 {
   413  	if len(x) == 0 {
   414  		return 0
   415  	}
   416  	v := uint64(x[0])
   417  	if _W == 32 && len(x) > 1 {
   418  		return uint64(x[1])<<32 | v
   419  	}
   420  	return v
   421  }
   422  
   423  // Int64 returns the int64 representation of x.
   424  // If x cannot be represented in an int64, the result is undefined.
   425  func (x *Int) Int64() int64 {
   426  	v := int64(low64(x.abs))
   427  	if x.neg {
   428  		v = -v
   429  	}
   430  	return v
   431  }
   432  
   433  // Uint64 returns the uint64 representation of x.
   434  // If x cannot be represented in a uint64, the result is undefined.
   435  func (x *Int) Uint64() uint64 {
   436  	return low64(x.abs)
   437  }
   438  
   439  // IsInt64 reports whether x can be represented as an int64.
   440  func (x *Int) IsInt64() bool {
   441  	if len(x.abs) <= 64/_W {
   442  		w := int64(low64(x.abs))
   443  		return w >= 0 || x.neg && w == -w
   444  	}
   445  	return false
   446  }
   447  
   448  // IsUint64 reports whether x can be represented as a uint64.
   449  func (x *Int) IsUint64() bool {
   450  	return !x.neg && len(x.abs) <= 64/_W
   451  }
   452  
   453  // Float64 returns the float64 value nearest x,
   454  // and an indication of any rounding that occurred.
   455  func (x *Int) Float64() (float64, Accuracy) {
   456  	n := x.abs.bitLen() // NB: still uses slow crypto impl!
   457  	if n == 0 {
   458  		return 0.0, Exact
   459  	}
   460  
   461  	// Fast path: no more than 53 significant bits.
   462  	if n <= 53 || n < 64 && n-int(x.abs.trailingZeroBits()) <= 53 {
   463  		f := float64(low64(x.abs))
   464  		if x.neg {
   465  			f = -f
   466  		}
   467  		return f, Exact
   468  	}
   469  
   470  	return new(Float).SetInt(x).Float64()
   471  }
   472  
   473  // SetString sets z to the value of s, interpreted in the given base,
   474  // and returns z and a boolean indicating success. The entire string
   475  // (not just a prefix) must be valid for success. If SetString fails,
   476  // the value of z is undefined but the returned value is nil.
   477  //
   478  // The base argument must be 0 or a value between 2 and [MaxBase].
   479  // For base 0, the number prefix determines the actual base: A prefix of
   480  // “0b” or “0B” selects base 2, “0”, “0o” or “0O” selects base 8,
   481  // and “0x” or “0X” selects base 16. Otherwise, the selected base is 10
   482  // and no prefix is accepted.
   483  //
   484  // For bases <= 36, lower and upper case letters are considered the same:
   485  // The letters 'a' to 'z' and 'A' to 'Z' represent digit values 10 to 35.
   486  // For bases > 36, the upper case letters 'A' to 'Z' represent the digit
   487  // values 36 to 61.
   488  //
   489  // For base 0, an underscore character “_” may appear between a base
   490  // prefix and an adjacent digit, and between successive digits; such
   491  // underscores do not change the value of the number.
   492  // Incorrect placement of underscores is reported as an error if there
   493  // are no other errors. If base != 0, underscores are not recognized
   494  // and act like any other character that is not a valid digit.
   495  func (z *Int) SetString(s string, base int) (*Int, bool) {
   496  	return z.setFromScanner(strings.NewReader(s), base)
   497  }
   498  
   499  // setFromScanner implements SetString given an io.ByteScanner.
   500  // For documentation see comments of SetString.
   501  func (z *Int) setFromScanner(r io.ByteScanner, base int) (*Int, bool) {
   502  	if _, _, err := z.scan(r, base); err != nil {
   503  		return nil, false
   504  	}
   505  	// entire content must have been consumed
   506  	if _, err := r.ReadByte(); err != io.EOF {
   507  		return nil, false
   508  	}
   509  	return z, true // err == io.EOF => scan consumed all content of r
   510  }
   511  
   512  // SetBytes interprets buf as the bytes of a big-endian unsigned
   513  // integer, sets z to that value, and returns z.
   514  func (z *Int) SetBytes(buf []byte) *Int {
   515  	z.abs = z.abs.setBytes(buf)
   516  	z.neg = false
   517  	return z
   518  }
   519  
   520  // Bytes returns the absolute value of x as a big-endian byte slice.
   521  //
   522  // To use a fixed length slice, or a preallocated one, use [Int.FillBytes].
   523  func (x *Int) Bytes() []byte {
   524  	// This function is used in cryptographic operations. It must not leak
   525  	// anything but the Int's sign and bit size through side-channels. Any
   526  	// changes must be reviewed by a security expert.
   527  	buf := make([]byte, len(x.abs)*_S)
   528  	return buf[x.abs.bytes(buf):]
   529  }
   530  
   531  // FillBytes sets buf to the absolute value of x, storing it as a zero-extended
   532  // big-endian byte slice, and returns buf.
   533  //
   534  // If the absolute value of x doesn't fit in buf, FillBytes will panic.
   535  func (x *Int) FillBytes(buf []byte) []byte {
   536  	// Clear whole buffer. (This gets optimized into a memclr.)
   537  	for i := range buf {
   538  		buf[i] = 0
   539  	}
   540  	x.abs.bytes(buf)
   541  	return buf
   542  }
   543  
   544  // BitLen returns the length of the absolute value of x in bits.
   545  // The bit length of 0 is 0.
   546  func (x *Int) BitLen() int {
   547  	// This function is used in cryptographic operations. It must not leak
   548  	// anything but the Int's sign and bit size through side-channels. Any
   549  	// changes must be reviewed by a security expert.
   550  	return x.abs.bitLen()
   551  }
   552  
   553  // TrailingZeroBits returns the number of consecutive least significant zero
   554  // bits of |x|.
   555  func (x *Int) TrailingZeroBits() uint {
   556  	return x.abs.trailingZeroBits()
   557  }
   558  
   559  // Exp sets z = x**y mod |m| (i.e. the sign of m is ignored), and returns z.
   560  // If m == nil or m == 0, z = x**y unless y <= 0 then z = 1. If m != 0, y < 0,
   561  // and x and m are not relatively prime, z is unchanged and nil is returned.
   562  //
   563  // Modular exponentiation of inputs of a particular size is not a
   564  // cryptographically constant-time operation.
   565  func (z *Int) Exp(x, y, m *Int) *Int {
   566  	return z.exp(x, y, m, false)
   567  }
   568  
   569  func (z *Int) expSlow(x, y, m *Int) *Int {
   570  	return z.exp(x, y, m, true)
   571  }
   572  
   573  func (z *Int) exp(x, y, m *Int, slow bool) *Int {
   574  	// See Knuth, volume 2, section 4.6.3.
   575  	xWords := x.abs
   576  	if y.neg {
   577  		if m == nil || len(m.abs) == 0 {
   578  			return z.SetInt64(1)
   579  		}
   580  		// for y < 0: x**y mod m == (x**(-1))**|y| mod m
   581  		inverse := new(Int).ModInverse(x, m)
   582  		if inverse == nil {
   583  			return nil
   584  		}
   585  		xWords = inverse.abs
   586  	}
   587  	yWords := y.abs
   588  
   589  	var mWords nat
   590  	if m != nil {
   591  		if z == m || alias(z.abs, m.abs) {
   592  			m = new(Int).Set(m)
   593  		}
   594  		mWords = m.abs // m.abs may be nil for m == 0
   595  	}
   596  
   597  	z.abs = z.abs.expNN(xWords, yWords, mWords, slow)
   598  	z.neg = len(z.abs) > 0 && x.neg && len(yWords) > 0 && yWords[0]&1 == 1 // 0 has no sign
   599  	if z.neg && len(mWords) > 0 {
   600  		// make modulus result positive
   601  		z.abs = z.abs.sub(mWords, z.abs) // z == x**y mod |m| && 0 <= z < |m|
   602  		z.neg = false
   603  	}
   604  
   605  	return z
   606  }
   607  
   608  // GCD sets z to the greatest common divisor of a and b and returns z.
   609  // If x or y are not nil, GCD sets their value such that z = a*x + b*y.
   610  //
   611  // a and b may be positive, zero or negative. (Before Go 1.14 both had
   612  // to be > 0.) Regardless of the signs of a and b, z is always >= 0.
   613  //
   614  // If a == b == 0, GCD sets z = x = y = 0.
   615  //
   616  // If a == 0 and b != 0, GCD sets z = |b|, x = 0, y = sign(b) * 1.
   617  //
   618  // If a != 0 and b == 0, GCD sets z = |a|, x = sign(a) * 1, y = 0.
   619  func (z *Int) GCD(x, y, a, b *Int) *Int {
   620  	if len(a.abs) == 0 || len(b.abs) == 0 {
   621  		lenA, lenB, negA, negB := len(a.abs), len(b.abs), a.neg, b.neg
   622  		if lenA == 0 {
   623  			z.Set(b)
   624  		} else {
   625  			z.Set(a)
   626  		}
   627  		z.neg = false
   628  		if x != nil {
   629  			if lenA == 0 {
   630  				x.SetUint64(0)
   631  			} else {
   632  				x.SetUint64(1)
   633  				x.neg = negA
   634  			}
   635  		}
   636  		if y != nil {
   637  			if lenB == 0 {
   638  				y.SetUint64(0)
   639  			} else {
   640  				y.SetUint64(1)
   641  				y.neg = negB
   642  			}
   643  		}
   644  		return z
   645  	}
   646  
   647  	return z.lehmerGCD(x, y, a, b)
   648  }
   649  
   650  // lehmerSimulate attempts to simulate several Euclidean update steps
   651  // using the leading digits of A and B.  It returns u0, u1, v0, v1
   652  // such that A and B can be updated as:
   653  //
   654  //	A = u0*A + v0*B
   655  //	B = u1*A + v1*B
   656  //
   657  // Requirements: A >= B and len(B.abs) >= 2
   658  // Since we are calculating with full words to avoid overflow,
   659  // we use 'even' to track the sign of the cosequences.
   660  // For even iterations: u0, v1 >= 0 && u1, v0 <= 0
   661  // For odd  iterations: u0, v1 <= 0 && u1, v0 >= 0
   662  func lehmerSimulate(A, B *Int) (u0, u1, v0, v1 Word, even bool) {
   663  	// initialize the digits
   664  	var a1, a2, u2, v2 Word
   665  
   666  	m := len(B.abs) // m >= 2
   667  	n := len(A.abs) // n >= m >= 2
   668  
   669  	// extract the top Word of bits from A and B
   670  	h := nlz(A.abs[n-1])
   671  	a1 = A.abs[n-1]<<h | A.abs[n-2]>>(_W-h)
   672  	// B may have implicit zero words in the high bits if the lengths differ
   673  	switch {
   674  	case n == m:
   675  		a2 = B.abs[n-1]<<h | B.abs[n-2]>>(_W-h)
   676  	case n == m+1:
   677  		a2 = B.abs[n-2] >> (_W - h)
   678  	default:
   679  		a2 = 0
   680  	}
   681  
   682  	// Since we are calculating with full words to avoid overflow,
   683  	// we use 'even' to track the sign of the cosequences.
   684  	// For even iterations: u0, v1 >= 0 && u1, v0 <= 0
   685  	// For odd  iterations: u0, v1 <= 0 && u1, v0 >= 0
   686  	// The first iteration starts with k=1 (odd).
   687  	even = false
   688  	// variables to track the cosequences
   689  	u0, u1, u2 = 0, 1, 0
   690  	v0, v1, v2 = 0, 0, 1
   691  
   692  	// Calculate the quotient and cosequences using Collins' stopping condition.
   693  	// Note that overflow of a Word is not possible when computing the remainder
   694  	// sequence and cosequences since the cosequence size is bounded by the input size.
   695  	// See section 4.2 of Jebelean for details.
   696  	for a2 >= v2 && a1-a2 >= v1+v2 {
   697  		q, r := a1/a2, a1%a2
   698  		a1, a2 = a2, r
   699  		u0, u1, u2 = u1, u2, u1+q*u2
   700  		v0, v1, v2 = v1, v2, v1+q*v2
   701  		even = !even
   702  	}
   703  	return
   704  }
   705  
   706  // lehmerUpdate updates the inputs A and B such that:
   707  //
   708  //	A = u0*A + v0*B
   709  //	B = u1*A + v1*B
   710  //
   711  // where the signs of u0, u1, v0, v1 are given by even
   712  // For even == true: u0, v1 >= 0 && u1, v0 <= 0
   713  // For even == false: u0, v1 <= 0 && u1, v0 >= 0
   714  // q, r, s, t are temporary variables to avoid allocations in the multiplication.
   715  func lehmerUpdate(A, B, q, r, s, t *Int, u0, u1, v0, v1 Word, even bool) {
   716  
   717  	t.abs = t.abs.setWord(u0)
   718  	s.abs = s.abs.setWord(v0)
   719  	t.neg = !even
   720  	s.neg = even
   721  
   722  	t.Mul(A, t)
   723  	s.Mul(B, s)
   724  
   725  	r.abs = r.abs.setWord(u1)
   726  	q.abs = q.abs.setWord(v1)
   727  	r.neg = even
   728  	q.neg = !even
   729  
   730  	r.Mul(A, r)
   731  	q.Mul(B, q)
   732  
   733  	A.Add(t, s)
   734  	B.Add(r, q)
   735  }
   736  
   737  // euclidUpdate performs a single step of the Euclidean GCD algorithm
   738  // if extended is true, it also updates the cosequence Ua, Ub.
   739  func euclidUpdate(A, B, Ua, Ub, q, r, s, t *Int, extended bool) {
   740  	q, r = q.QuoRem(A, B, r)
   741  
   742  	*A, *B, *r = *B, *r, *A
   743  
   744  	if extended {
   745  		// Ua, Ub = Ub, Ua - q*Ub
   746  		t.Set(Ub)
   747  		s.Mul(Ub, q)
   748  		Ub.Sub(Ua, s)
   749  		Ua.Set(t)
   750  	}
   751  }
   752  
   753  // lehmerGCD sets z to the greatest common divisor of a and b,
   754  // which both must be != 0, and returns z.
   755  // If x or y are not nil, their values are set such that z = a*x + b*y.
   756  // See Knuth, The Art of Computer Programming, Vol. 2, Section 4.5.2, Algorithm L.
   757  // This implementation uses the improved condition by Collins requiring only one
   758  // quotient and avoiding the possibility of single Word overflow.
   759  // See Jebelean, "Improving the multiprecision Euclidean algorithm",
   760  // Design and Implementation of Symbolic Computation Systems, pp 45-58.
   761  // The cosequences are updated according to Algorithm 10.45 from
   762  // Cohen et al. "Handbook of Elliptic and Hyperelliptic Curve Cryptography" pp 192.
   763  func (z *Int) lehmerGCD(x, y, a, b *Int) *Int {
   764  	var A, B, Ua, Ub *Int
   765  
   766  	A = new(Int).Abs(a)
   767  	B = new(Int).Abs(b)
   768  
   769  	extended := x != nil || y != nil
   770  
   771  	if extended {
   772  		// Ua (Ub) tracks how many times input a has been accumulated into A (B).
   773  		Ua = new(Int).SetInt64(1)
   774  		Ub = new(Int)
   775  	}
   776  
   777  	// temp variables for multiprecision update
   778  	q := new(Int)
   779  	r := new(Int)
   780  	s := new(Int)
   781  	t := new(Int)
   782  
   783  	// ensure A >= B
   784  	if A.abs.cmp(B.abs) < 0 {
   785  		A, B = B, A
   786  		Ub, Ua = Ua, Ub
   787  	}
   788  
   789  	// loop invariant A >= B
   790  	for len(B.abs) > 1 {
   791  		// Attempt to calculate in single-precision using leading words of A and B.
   792  		u0, u1, v0, v1, even := lehmerSimulate(A, B)
   793  
   794  		// multiprecision Step
   795  		if v0 != 0 {
   796  			// Simulate the effect of the single-precision steps using the cosequences.
   797  			// A = u0*A + v0*B
   798  			// B = u1*A + v1*B
   799  			lehmerUpdate(A, B, q, r, s, t, u0, u1, v0, v1, even)
   800  
   801  			if extended {
   802  				// Ua = u0*Ua + v0*Ub
   803  				// Ub = u1*Ua + v1*Ub
   804  				lehmerUpdate(Ua, Ub, q, r, s, t, u0, u1, v0, v1, even)
   805  			}
   806  
   807  		} else {
   808  			// Single-digit calculations failed to simulate any quotients.
   809  			// Do a standard Euclidean step.
   810  			euclidUpdate(A, B, Ua, Ub, q, r, s, t, extended)
   811  		}
   812  	}
   813  
   814  	if len(B.abs) > 0 {
   815  		// extended Euclidean algorithm base case if B is a single Word
   816  		if len(A.abs) > 1 {
   817  			// A is longer than a single Word, so one update is needed.
   818  			euclidUpdate(A, B, Ua, Ub, q, r, s, t, extended)
   819  		}
   820  		if len(B.abs) > 0 {
   821  			// A and B are both a single Word.
   822  			aWord, bWord := A.abs[0], B.abs[0]
   823  			if extended {
   824  				var ua, ub, va, vb Word
   825  				ua, ub = 1, 0
   826  				va, vb = 0, 1
   827  				even := true
   828  				for bWord != 0 {
   829  					q, r := aWord/bWord, aWord%bWord
   830  					aWord, bWord = bWord, r
   831  					ua, ub = ub, ua+q*ub
   832  					va, vb = vb, va+q*vb
   833  					even = !even
   834  				}
   835  
   836  				t.abs = t.abs.setWord(ua)
   837  				s.abs = s.abs.setWord(va)
   838  				t.neg = !even
   839  				s.neg = even
   840  
   841  				t.Mul(Ua, t)
   842  				s.Mul(Ub, s)
   843  
   844  				Ua.Add(t, s)
   845  			} else {
   846  				for bWord != 0 {
   847  					aWord, bWord = bWord, aWord%bWord
   848  				}
   849  			}
   850  			A.abs[0] = aWord
   851  		}
   852  	}
   853  	negA := a.neg
   854  	if y != nil {
   855  		// avoid aliasing b needed in the division below
   856  		if y == b {
   857  			B.Set(b)
   858  		} else {
   859  			B = b
   860  		}
   861  		// y = (z - a*x)/b
   862  		y.Mul(a, Ua) // y can safely alias a
   863  		if negA {
   864  			y.neg = !y.neg
   865  		}
   866  		y.Sub(A, y)
   867  		y.Div(y, B)
   868  	}
   869  
   870  	if x != nil {
   871  		*x = *Ua
   872  		if negA {
   873  			x.neg = !x.neg
   874  		}
   875  	}
   876  
   877  	*z = *A
   878  
   879  	return z
   880  }
   881  
   882  // Rand sets z to a pseudo-random number in [0, n) and returns z.
   883  //
   884  // As this uses the [math/rand] package, it must not be used for
   885  // security-sensitive work. Use [crypto/rand.Int] instead.
   886  func (z *Int) Rand(rnd *rand.Rand, n *Int) *Int {
   887  	// z.neg is not modified before the if check, because z and n might alias.
   888  	if n.neg || len(n.abs) == 0 {
   889  		z.neg = false
   890  		z.abs = nil
   891  		return z
   892  	}
   893  	z.neg = false
   894  	z.abs = z.abs.random(rnd, n.abs, n.abs.bitLen())
   895  	return z
   896  }
   897  
   898  // ModInverse sets z to the multiplicative inverse of g in the ring ℤ/nℤ
   899  // and returns z. If g and n are not relatively prime, g has no multiplicative
   900  // inverse in the ring ℤ/nℤ.  In this case, z is unchanged and the return value
   901  // is nil. If n == 0, a division-by-zero run-time panic occurs.
   902  func (z *Int) ModInverse(g, n *Int) *Int {
   903  	// GCD expects parameters a and b to be > 0.
   904  	if n.neg {
   905  		var n2 Int
   906  		n = n2.Neg(n)
   907  	}
   908  	if g.neg {
   909  		var g2 Int
   910  		g = g2.Mod(g, n)
   911  	}
   912  	var d, x Int
   913  	d.GCD(&x, nil, g, n)
   914  
   915  	// if and only if d==1, g and n are relatively prime
   916  	if d.Cmp(intOne) != 0 {
   917  		return nil
   918  	}
   919  
   920  	// x and y are such that g*x + n*y = 1, therefore x is the inverse element,
   921  	// but it may be negative, so convert to the range 0 <= z < |n|
   922  	if x.neg {
   923  		z.Add(&x, n)
   924  	} else {
   925  		z.Set(&x)
   926  	}
   927  	return z
   928  }
   929  
   930  func (z nat) modInverse(g, n nat) nat {
   931  	// TODO(rsc): ModInverse should be implemented in terms of this function.
   932  	return (&Int{abs: z}).ModInverse(&Int{abs: g}, &Int{abs: n}).abs
   933  }
   934  
   935  // Jacobi returns the Jacobi symbol (x/y), either +1, -1, or 0.
   936  // The y argument must be an odd integer.
   937  func Jacobi(x, y *Int) int {
   938  	if len(y.abs) == 0 || y.abs[0]&1 == 0 {
   939  		panic(fmt.Sprintf("big: invalid 2nd argument to Int.Jacobi: need odd integer but got %s", y.String()))
   940  	}
   941  
   942  	// We use the formulation described in chapter 2, section 2.4,
   943  	// "The Yacas Book of Algorithms":
   944  	// http://yacas.sourceforge.net/Algo.book.pdf
   945  
   946  	var a, b, c Int
   947  	a.Set(x)
   948  	b.Set(y)
   949  	j := 1
   950  
   951  	if b.neg {
   952  		if a.neg {
   953  			j = -1
   954  		}
   955  		b.neg = false
   956  	}
   957  
   958  	for {
   959  		if b.Cmp(intOne) == 0 {
   960  			return j
   961  		}
   962  		if len(a.abs) == 0 {
   963  			return 0
   964  		}
   965  		a.Mod(&a, &b)
   966  		if len(a.abs) == 0 {
   967  			return 0
   968  		}
   969  		// a > 0
   970  
   971  		// handle factors of 2 in 'a'
   972  		s := a.abs.trailingZeroBits()
   973  		if s&1 != 0 {
   974  			bmod8 := b.abs[0] & 7
   975  			if bmod8 == 3 || bmod8 == 5 {
   976  				j = -j
   977  			}
   978  		}
   979  		c.Rsh(&a, s) // a = 2^s*c
   980  
   981  		// swap numerator and denominator
   982  		if b.abs[0]&3 == 3 && c.abs[0]&3 == 3 {
   983  			j = -j
   984  		}
   985  		a.Set(&b)
   986  		b.Set(&c)
   987  	}
   988  }
   989  
   990  // modSqrt3Mod4 uses the identity
   991  //
   992  //	   (a^((p+1)/4))^2  mod p
   993  //	== u^(p+1)          mod p
   994  //	== u^2              mod p
   995  //
   996  // to calculate the square root of any quadratic residue mod p quickly for 3
   997  // mod 4 primes.
   998  func (z *Int) modSqrt3Mod4Prime(x, p *Int) *Int {
   999  	e := new(Int).Add(p, intOne) // e = p + 1
  1000  	e.Rsh(e, 2)                  // e = (p + 1) / 4
  1001  	z.Exp(x, e, p)               // z = x^e mod p
  1002  	return z
  1003  }
  1004  
  1005  // modSqrt5Mod8Prime uses Atkin's observation that 2 is not a square mod p
  1006  //
  1007  //	alpha ==  (2*a)^((p-5)/8)    mod p
  1008  //	beta  ==  2*a*alpha^2        mod p  is a square root of -1
  1009  //	b     ==  a*alpha*(beta-1)   mod p  is a square root of a
  1010  //
  1011  // to calculate the square root of any quadratic residue mod p quickly for 5
  1012  // mod 8 primes.
  1013  func (z *Int) modSqrt5Mod8Prime(x, p *Int) *Int {
  1014  	// p == 5 mod 8 implies p = e*8 + 5
  1015  	// e is the quotient and 5 the remainder on division by 8
  1016  	e := new(Int).Rsh(p, 3)  // e = (p - 5) / 8
  1017  	tx := new(Int).Lsh(x, 1) // tx = 2*x
  1018  	alpha := new(Int).Exp(tx, e, p)
  1019  	beta := new(Int).Mul(alpha, alpha)
  1020  	beta.Mod(beta, p)
  1021  	beta.Mul(beta, tx)
  1022  	beta.Mod(beta, p)
  1023  	beta.Sub(beta, intOne)
  1024  	beta.Mul(beta, x)
  1025  	beta.Mod(beta, p)
  1026  	beta.Mul(beta, alpha)
  1027  	z.Mod(beta, p)
  1028  	return z
  1029  }
  1030  
  1031  // modSqrtTonelliShanks uses the Tonelli-Shanks algorithm to find the square
  1032  // root of a quadratic residue modulo any prime.
  1033  func (z *Int) modSqrtTonelliShanks(x, p *Int) *Int {
  1034  	// Break p-1 into s*2^e such that s is odd.
  1035  	var s Int
  1036  	s.Sub(p, intOne)
  1037  	e := s.abs.trailingZeroBits()
  1038  	s.Rsh(&s, e)
  1039  
  1040  	// find some non-square n
  1041  	var n Int
  1042  	n.SetInt64(2)
  1043  	for Jacobi(&n, p) != -1 {
  1044  		n.Add(&n, intOne)
  1045  	}
  1046  
  1047  	// Core of the Tonelli-Shanks algorithm. Follows the description in
  1048  	// section 6 of "Square roots from 1; 24, 51, 10 to Dan Shanks" by Ezra
  1049  	// Brown:
  1050  	// https://www.maa.org/sites/default/files/pdf/upload_library/22/Polya/07468342.di020786.02p0470a.pdf
  1051  	var y, b, g, t Int
  1052  	y.Add(&s, intOne)
  1053  	y.Rsh(&y, 1)
  1054  	y.Exp(x, &y, p)  // y = x^((s+1)/2)
  1055  	b.Exp(x, &s, p)  // b = x^s
  1056  	g.Exp(&n, &s, p) // g = n^s
  1057  	r := e
  1058  	for {
  1059  		// find the least m such that ord_p(b) = 2^m
  1060  		var m uint
  1061  		t.Set(&b)
  1062  		for t.Cmp(intOne) != 0 {
  1063  			t.Mul(&t, &t).Mod(&t, p)
  1064  			m++
  1065  		}
  1066  
  1067  		if m == 0 {
  1068  			return z.Set(&y)
  1069  		}
  1070  
  1071  		t.SetInt64(0).SetBit(&t, int(r-m-1), 1).Exp(&g, &t, p)
  1072  		// t = g^(2^(r-m-1)) mod p
  1073  		g.Mul(&t, &t).Mod(&g, p) // g = g^(2^(r-m)) mod p
  1074  		y.Mul(&y, &t).Mod(&y, p)
  1075  		b.Mul(&b, &g).Mod(&b, p)
  1076  		r = m
  1077  	}
  1078  }
  1079  
  1080  // ModSqrt sets z to a square root of x mod p if such a square root exists, and
  1081  // returns z. The modulus p must be an odd prime. If x is not a square mod p,
  1082  // ModSqrt leaves z unchanged and returns nil. This function panics if p is
  1083  // not an odd integer, its behavior is undefined if p is odd but not prime.
  1084  func (z *Int) ModSqrt(x, p *Int) *Int {
  1085  	switch Jacobi(x, p) {
  1086  	case -1:
  1087  		return nil // x is not a square mod p
  1088  	case 0:
  1089  		return z.SetInt64(0) // sqrt(0) mod p = 0
  1090  	case 1:
  1091  		break
  1092  	}
  1093  	if x.neg || x.Cmp(p) >= 0 { // ensure 0 <= x < p
  1094  		x = new(Int).Mod(x, p)
  1095  	}
  1096  
  1097  	switch {
  1098  	case p.abs[0]%4 == 3:
  1099  		// Check whether p is 3 mod 4, and if so, use the faster algorithm.
  1100  		return z.modSqrt3Mod4Prime(x, p)
  1101  	case p.abs[0]%8 == 5:
  1102  		// Check whether p is 5 mod 8, use Atkin's algorithm.
  1103  		return z.modSqrt5Mod8Prime(x, p)
  1104  	default:
  1105  		// Otherwise, use Tonelli-Shanks.
  1106  		return z.modSqrtTonelliShanks(x, p)
  1107  	}
  1108  }
  1109  
  1110  // Lsh sets z = x << n and returns z.
  1111  func (z *Int) Lsh(x *Int, n uint) *Int {
  1112  	z.abs = z.abs.shl(x.abs, n)
  1113  	z.neg = x.neg
  1114  	return z
  1115  }
  1116  
  1117  // Rsh sets z = x >> n and returns z.
  1118  func (z *Int) Rsh(x *Int, n uint) *Int {
  1119  	if x.neg {
  1120  		// (-x) >> s == ^(x-1) >> s == ^((x-1) >> s) == -(((x-1) >> s) + 1)
  1121  		t := z.abs.sub(x.abs, natOne) // no underflow because |x| > 0
  1122  		t = t.shr(t, n)
  1123  		z.abs = t.add(t, natOne)
  1124  		z.neg = true // z cannot be zero if x is negative
  1125  		return z
  1126  	}
  1127  
  1128  	z.abs = z.abs.shr(x.abs, n)
  1129  	z.neg = false
  1130  	return z
  1131  }
  1132  
  1133  // Bit returns the value of the i'th bit of x. That is, it
  1134  // returns (x>>i)&1. The bit index i must be >= 0.
  1135  func (x *Int) Bit(i int) uint {
  1136  	if i == 0 {
  1137  		// optimization for common case: odd/even test of x
  1138  		if len(x.abs) > 0 {
  1139  			return uint(x.abs[0] & 1) // bit 0 is same for -x
  1140  		}
  1141  		return 0
  1142  	}
  1143  	if i < 0 {
  1144  		panic("negative bit index")
  1145  	}
  1146  	if x.neg {
  1147  		t := nat(nil).sub(x.abs, natOne)
  1148  		return t.bit(uint(i)) ^ 1
  1149  	}
  1150  
  1151  	return x.abs.bit(uint(i))
  1152  }
  1153  
  1154  // SetBit sets z to x, with x's i'th bit set to b (0 or 1).
  1155  // That is, if b is 1 SetBit sets z = x | (1 << i);
  1156  // if b is 0 SetBit sets z = x &^ (1 << i). If b is not 0 or 1,
  1157  // SetBit will panic.
  1158  func (z *Int) SetBit(x *Int, i int, b uint) *Int {
  1159  	if i < 0 {
  1160  		panic("negative bit index")
  1161  	}
  1162  	if x.neg {
  1163  		t := z.abs.sub(x.abs, natOne)
  1164  		t = t.setBit(t, uint(i), b^1)
  1165  		z.abs = t.add(t, natOne)
  1166  		z.neg = len(z.abs) > 0
  1167  		return z
  1168  	}
  1169  	z.abs = z.abs.setBit(x.abs, uint(i), b)
  1170  	z.neg = false
  1171  	return z
  1172  }
  1173  
  1174  // And sets z = x & y and returns z.
  1175  func (z *Int) And(x, y *Int) *Int {
  1176  	if x.neg == y.neg {
  1177  		if x.neg {
  1178  			// (-x) & (-y) == ^(x-1) & ^(y-1) == ^((x-1) | (y-1)) == -(((x-1) | (y-1)) + 1)
  1179  			x1 := nat(nil).sub(x.abs, natOne)
  1180  			y1 := nat(nil).sub(y.abs, natOne)
  1181  			z.abs = z.abs.add(z.abs.or(x1, y1), natOne)
  1182  			z.neg = true // z cannot be zero if x and y are negative
  1183  			return z
  1184  		}
  1185  
  1186  		// x & y == x & y
  1187  		z.abs = z.abs.and(x.abs, y.abs)
  1188  		z.neg = false
  1189  		return z
  1190  	}
  1191  
  1192  	// x.neg != y.neg
  1193  	if x.neg {
  1194  		x, y = y, x // & is symmetric
  1195  	}
  1196  
  1197  	// x & (-y) == x & ^(y-1) == x &^ (y-1)
  1198  	y1 := nat(nil).sub(y.abs, natOne)
  1199  	z.abs = z.abs.andNot(x.abs, y1)
  1200  	z.neg = false
  1201  	return z
  1202  }
  1203  
  1204  // AndNot sets z = x &^ y and returns z.
  1205  func (z *Int) AndNot(x, y *Int) *Int {
  1206  	if x.neg == y.neg {
  1207  		if x.neg {
  1208  			// (-x) &^ (-y) == ^(x-1) &^ ^(y-1) == ^(x-1) & (y-1) == (y-1) &^ (x-1)
  1209  			x1 := nat(nil).sub(x.abs, natOne)
  1210  			y1 := nat(nil).sub(y.abs, natOne)
  1211  			z.abs = z.abs.andNot(y1, x1)
  1212  			z.neg = false
  1213  			return z
  1214  		}
  1215  
  1216  		// x &^ y == x &^ y
  1217  		z.abs = z.abs.andNot(x.abs, y.abs)
  1218  		z.neg = false
  1219  		return z
  1220  	}
  1221  
  1222  	if x.neg {
  1223  		// (-x) &^ y == ^(x-1) &^ y == ^(x-1) & ^y == ^((x-1) | y) == -(((x-1) | y) + 1)
  1224  		x1 := nat(nil).sub(x.abs, natOne)
  1225  		z.abs = z.abs.add(z.abs.or(x1, y.abs), natOne)
  1226  		z.neg = true // z cannot be zero if x is negative and y is positive
  1227  		return z
  1228  	}
  1229  
  1230  	// x &^ (-y) == x &^ ^(y-1) == x & (y-1)
  1231  	y1 := nat(nil).sub(y.abs, natOne)
  1232  	z.abs = z.abs.and(x.abs, y1)
  1233  	z.neg = false
  1234  	return z
  1235  }
  1236  
  1237  // Or sets z = x | y and returns z.
  1238  func (z *Int) Or(x, y *Int) *Int {
  1239  	if x.neg == y.neg {
  1240  		if x.neg {
  1241  			// (-x) | (-y) == ^(x-1) | ^(y-1) == ^((x-1) & (y-1)) == -(((x-1) & (y-1)) + 1)
  1242  			x1 := nat(nil).sub(x.abs, natOne)
  1243  			y1 := nat(nil).sub(y.abs, natOne)
  1244  			z.abs = z.abs.add(z.abs.and(x1, y1), natOne)
  1245  			z.neg = true // z cannot be zero if x and y are negative
  1246  			return z
  1247  		}
  1248  
  1249  		// x | y == x | y
  1250  		z.abs = z.abs.or(x.abs, y.abs)
  1251  		z.neg = false
  1252  		return z
  1253  	}
  1254  
  1255  	// x.neg != y.neg
  1256  	if x.neg {
  1257  		x, y = y, x // | is symmetric
  1258  	}
  1259  
  1260  	// x | (-y) == x | ^(y-1) == ^((y-1) &^ x) == -(^((y-1) &^ x) + 1)
  1261  	y1 := nat(nil).sub(y.abs, natOne)
  1262  	z.abs = z.abs.add(z.abs.andNot(y1, x.abs), natOne)
  1263  	z.neg = true // z cannot be zero if one of x or y is negative
  1264  	return z
  1265  }
  1266  
  1267  // Xor sets z = x ^ y and returns z.
  1268  func (z *Int) Xor(x, y *Int) *Int {
  1269  	if x.neg == y.neg {
  1270  		if x.neg {
  1271  			// (-x) ^ (-y) == ^(x-1) ^ ^(y-1) == (x-1) ^ (y-1)
  1272  			x1 := nat(nil).sub(x.abs, natOne)
  1273  			y1 := nat(nil).sub(y.abs, natOne)
  1274  			z.abs = z.abs.xor(x1, y1)
  1275  			z.neg = false
  1276  			return z
  1277  		}
  1278  
  1279  		// x ^ y == x ^ y
  1280  		z.abs = z.abs.xor(x.abs, y.abs)
  1281  		z.neg = false
  1282  		return z
  1283  	}
  1284  
  1285  	// x.neg != y.neg
  1286  	if x.neg {
  1287  		x, y = y, x // ^ is symmetric
  1288  	}
  1289  
  1290  	// x ^ (-y) == x ^ ^(y-1) == ^(x ^ (y-1)) == -((x ^ (y-1)) + 1)
  1291  	y1 := nat(nil).sub(y.abs, natOne)
  1292  	z.abs = z.abs.add(z.abs.xor(x.abs, y1), natOne)
  1293  	z.neg = true // z cannot be zero if only one of x or y is negative
  1294  	return z
  1295  }
  1296  
  1297  // Not sets z = ^x and returns z.
  1298  func (z *Int) Not(x *Int) *Int {
  1299  	if x.neg {
  1300  		// ^(-x) == ^(^(x-1)) == x-1
  1301  		z.abs = z.abs.sub(x.abs, natOne)
  1302  		z.neg = false
  1303  		return z
  1304  	}
  1305  
  1306  	// ^x == -x-1 == -(x+1)
  1307  	z.abs = z.abs.add(x.abs, natOne)
  1308  	z.neg = true // z cannot be zero if x is positive
  1309  	return z
  1310  }
  1311  
  1312  // Sqrt sets z to ⌊√x⌋, the largest integer such that z² ≤ x, and returns z.
  1313  // It panics if x is negative.
  1314  func (z *Int) Sqrt(x *Int) *Int {
  1315  	if x.neg {
  1316  		panic("square root of negative number")
  1317  	}
  1318  	z.neg = false
  1319  	z.abs = z.abs.sqrt(x.abs)
  1320  	return z
  1321  }
  1322  

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