Source file src/pkg/math/big/int.go

     1	// Copyright 2009 The Go Authors. All rights reserved.
     2	// Use of this source code is governed by a BSD-style
     3	// license that can be found in the LICENSE file.
     4	
     5	// This file implements signed multi-precision integers.
     6	
     7	package big
     8	
     9	import (
    10		"errors"
    11		"fmt"
    12		"io"
    13		"math/rand"
    14		"strings"
    15	)
    16	
    17	// An Int represents a signed multi-precision integer.
    18	// The zero value for an Int represents the value 0.
    19	type Int struct {
    20		neg bool // sign
    21		abs nat  // absolute value of the integer
    22	}
    23	
    24	var intOne = &Int{false, natOne}
    25	
    26	// Sign returns:
    27	//
    28	//	-1 if x <  0
    29	//	 0 if x == 0
    30	//	+1 if x >  0
    31	//
    32	func (x *Int) Sign() int {
    33		if len(x.abs) == 0 {
    34			return 0
    35		}
    36		if x.neg {
    37			return -1
    38		}
    39		return 1
    40	}
    41	
    42	// SetInt64 sets z to x and returns z.
    43	func (z *Int) SetInt64(x int64) *Int {
    44		neg := false
    45		if x < 0 {
    46			neg = true
    47			x = -x
    48		}
    49		z.abs = z.abs.setUint64(uint64(x))
    50		z.neg = neg
    51		return z
    52	}
    53	
    54	// NewInt allocates and returns a new Int set to x.
    55	func NewInt(x int64) *Int {
    56		return new(Int).SetInt64(x)
    57	}
    58	
    59	// Set sets z to x and returns z.
    60	func (z *Int) Set(x *Int) *Int {
    61		if z != x {
    62			z.abs = z.abs.set(x.abs)
    63			z.neg = x.neg
    64		}
    65		return z
    66	}
    67	
    68	// Bits provides raw (unchecked but fast) access to x by returning its
    69	// absolute value as a little-endian Word slice. The result and x share
    70	// the same underlying array.
    71	// Bits is intended to support implementation of missing low-level Int
    72	// functionality outside this package; it should be avoided otherwise.
    73	func (x *Int) Bits() []Word {
    74		return x.abs
    75	}
    76	
    77	// SetBits provides raw (unchecked but fast) access to z by setting its
    78	// value to abs, interpreted as a little-endian Word slice, and returning
    79	// z. The result and abs share the same underlying array.
    80	// SetBits is intended to support implementation of missing low-level Int
    81	// functionality outside this package; it should be avoided otherwise.
    82	func (z *Int) SetBits(abs []Word) *Int {
    83		z.abs = nat(abs).norm()
    84		z.neg = false
    85		return z
    86	}
    87	
    88	// Abs sets z to |x| (the absolute value of x) and returns z.
    89	func (z *Int) Abs(x *Int) *Int {
    90		z.Set(x)
    91		z.neg = false
    92		return z
    93	}
    94	
    95	// Neg sets z to -x and returns z.
    96	func (z *Int) Neg(x *Int) *Int {
    97		z.Set(x)
    98		z.neg = len(z.abs) > 0 && !z.neg // 0 has no sign
    99		return z
   100	}
   101	
   102	// Add sets z to the sum x+y and returns z.
   103	func (z *Int) Add(x, y *Int) *Int {
   104		neg := x.neg
   105		if x.neg == y.neg {
   106			// x + y == x + y
   107			// (-x) + (-y) == -(x + y)
   108			z.abs = z.abs.add(x.abs, y.abs)
   109		} else {
   110			// x + (-y) == x - y == -(y - x)
   111			// (-x) + y == y - x == -(x - y)
   112			if x.abs.cmp(y.abs) >= 0 {
   113				z.abs = z.abs.sub(x.abs, y.abs)
   114			} else {
   115				neg = !neg
   116				z.abs = z.abs.sub(y.abs, x.abs)
   117			}
   118		}
   119		z.neg = len(z.abs) > 0 && neg // 0 has no sign
   120		return z
   121	}
   122	
   123	// Sub sets z to the difference x-y and returns z.
   124	func (z *Int) Sub(x, y *Int) *Int {
   125		neg := x.neg
   126		if x.neg != y.neg {
   127			// x - (-y) == x + y
   128			// (-x) - y == -(x + y)
   129			z.abs = z.abs.add(x.abs, y.abs)
   130		} else {
   131			// x - y == x - y == -(y - x)
   132			// (-x) - (-y) == y - x == -(x - y)
   133			if x.abs.cmp(y.abs) >= 0 {
   134				z.abs = z.abs.sub(x.abs, y.abs)
   135			} else {
   136				neg = !neg
   137				z.abs = z.abs.sub(y.abs, x.abs)
   138			}
   139		}
   140		z.neg = len(z.abs) > 0 && neg // 0 has no sign
   141		return z
   142	}
   143	
   144	// Mul sets z to the product x*y and returns z.
   145	func (z *Int) Mul(x, y *Int) *Int {
   146		// x * y == x * y
   147		// x * (-y) == -(x * y)
   148		// (-x) * y == -(x * y)
   149		// (-x) * (-y) == x * y
   150		z.abs = z.abs.mul(x.abs, y.abs)
   151		z.neg = len(z.abs) > 0 && x.neg != y.neg // 0 has no sign
   152		return z
   153	}
   154	
   155	// MulRange sets z to the product of all integers
   156	// in the range [a, b] inclusively and returns z.
   157	// If a > b (empty range), the result is 1.
   158	func (z *Int) MulRange(a, b int64) *Int {
   159		switch {
   160		case a > b:
   161			return z.SetInt64(1) // empty range
   162		case a <= 0 && b >= 0:
   163			return z.SetInt64(0) // range includes 0
   164		}
   165		// a <= b && (b < 0 || a > 0)
   166	
   167		neg := false
   168		if a < 0 {
   169			neg = (b-a)&1 == 0
   170			a, b = -b, -a
   171		}
   172	
   173		z.abs = z.abs.mulRange(uint64(a), uint64(b))
   174		z.neg = neg
   175		return z
   176	}
   177	
   178	// Binomial sets z to the binomial coefficient of (n, k) and returns z.
   179	func (z *Int) Binomial(n, k int64) *Int {
   180		var a, b Int
   181		a.MulRange(n-k+1, n)
   182		b.MulRange(1, k)
   183		return z.Quo(&a, &b)
   184	}
   185	
   186	// Quo sets z to the quotient x/y for y != 0 and returns z.
   187	// If y == 0, a division-by-zero run-time panic occurs.
   188	// Quo implements truncated division (like Go); see QuoRem for more details.
   189	func (z *Int) Quo(x, y *Int) *Int {
   190		z.abs, _ = z.abs.div(nil, x.abs, y.abs)
   191		z.neg = len(z.abs) > 0 && x.neg != y.neg // 0 has no sign
   192		return z
   193	}
   194	
   195	// Rem sets z to the remainder x%y for y != 0 and returns z.
   196	// If y == 0, a division-by-zero run-time panic occurs.
   197	// Rem implements truncated modulus (like Go); see QuoRem for more details.
   198	func (z *Int) Rem(x, y *Int) *Int {
   199		_, z.abs = nat(nil).div(z.abs, x.abs, y.abs)
   200		z.neg = len(z.abs) > 0 && x.neg // 0 has no sign
   201		return z
   202	}
   203	
   204	// QuoRem sets z to the quotient x/y and r to the remainder x%y
   205	// and returns the pair (z, r) for y != 0.
   206	// If y == 0, a division-by-zero run-time panic occurs.
   207	//
   208	// QuoRem implements T-division and modulus (like Go):
   209	//
   210	//	q = x/y      with the result truncated to zero
   211	//	r = x - y*q
   212	//
   213	// (See Daan Leijen, ``Division and Modulus for Computer Scientists''.)
   214	// See DivMod for Euclidean division and modulus (unlike Go).
   215	//
   216	func (z *Int) QuoRem(x, y, r *Int) (*Int, *Int) {
   217		z.abs, r.abs = z.abs.div(r.abs, x.abs, y.abs)
   218		z.neg, r.neg = len(z.abs) > 0 && x.neg != y.neg, len(r.abs) > 0 && x.neg // 0 has no sign
   219		return z, r
   220	}
   221	
   222	// Div sets z to the quotient x/y for y != 0 and returns z.
   223	// If y == 0, a division-by-zero run-time panic occurs.
   224	// Div implements Euclidean division (unlike Go); see DivMod for more details.
   225	func (z *Int) Div(x, y *Int) *Int {
   226		y_neg := y.neg // z may be an alias for y
   227		var r Int
   228		z.QuoRem(x, y, &r)
   229		if r.neg {
   230			if y_neg {
   231				z.Add(z, intOne)
   232			} else {
   233				z.Sub(z, intOne)
   234			}
   235		}
   236		return z
   237	}
   238	
   239	// Mod sets z to the modulus x%y for y != 0 and returns z.
   240	// If y == 0, a division-by-zero run-time panic occurs.
   241	// Mod implements Euclidean modulus (unlike Go); see DivMod for more details.
   242	func (z *Int) Mod(x, y *Int) *Int {
   243		y0 := y // save y
   244		if z == y || alias(z.abs, y.abs) {
   245			y0 = new(Int).Set(y)
   246		}
   247		var q Int
   248		q.QuoRem(x, y, z)
   249		if z.neg {
   250			if y0.neg {
   251				z.Sub(z, y0)
   252			} else {
   253				z.Add(z, y0)
   254			}
   255		}
   256		return z
   257	}
   258	
   259	// DivMod sets z to the quotient x div y and m to the modulus x mod y
   260	// and returns the pair (z, m) for y != 0.
   261	// If y == 0, a division-by-zero run-time panic occurs.
   262	//
   263	// DivMod implements Euclidean division and modulus (unlike Go):
   264	//
   265	//	q = x div y  such that
   266	//	m = x - y*q  with 0 <= m < |q|
   267	//
   268	// (See Raymond T. Boute, ``The Euclidean definition of the functions
   269	// div and mod''. ACM Transactions on Programming Languages and
   270	// Systems (TOPLAS), 14(2):127-144, New York, NY, USA, 4/1992.
   271	// ACM press.)
   272	// See QuoRem for T-division and modulus (like Go).
   273	//
   274	func (z *Int) DivMod(x, y, m *Int) (*Int, *Int) {
   275		y0 := y // save y
   276		if z == y || alias(z.abs, y.abs) {
   277			y0 = new(Int).Set(y)
   278		}
   279		z.QuoRem(x, y, m)
   280		if m.neg {
   281			if y0.neg {
   282				z.Add(z, intOne)
   283				m.Sub(m, y0)
   284			} else {
   285				z.Sub(z, intOne)
   286				m.Add(m, y0)
   287			}
   288		}
   289		return z, m
   290	}
   291	
   292	// Cmp compares x and y and returns:
   293	//
   294	//   -1 if x <  y
   295	//    0 if x == y
   296	//   +1 if x >  y
   297	//
   298	func (x *Int) Cmp(y *Int) (r int) {
   299		// x cmp y == x cmp y
   300		// x cmp (-y) == x
   301		// (-x) cmp y == y
   302		// (-x) cmp (-y) == -(x cmp y)
   303		switch {
   304		case x.neg == y.neg:
   305			r = x.abs.cmp(y.abs)
   306			if x.neg {
   307				r = -r
   308			}
   309		case x.neg:
   310			r = -1
   311		default:
   312			r = 1
   313		}
   314		return
   315	}
   316	
   317	func (x *Int) String() string {
   318		switch {
   319		case x == nil:
   320			return "<nil>"
   321		case x.neg:
   322			return "-" + x.abs.decimalString()
   323		}
   324		return x.abs.decimalString()
   325	}
   326	
   327	func charset(ch rune) string {
   328		switch ch {
   329		case 'b':
   330			return lowercaseDigits[0:2]
   331		case 'o':
   332			return lowercaseDigits[0:8]
   333		case 'd', 's', 'v':
   334			return lowercaseDigits[0:10]
   335		case 'x':
   336			return lowercaseDigits[0:16]
   337		case 'X':
   338			return uppercaseDigits[0:16]
   339		}
   340		return "" // unknown format
   341	}
   342	
   343	// write count copies of text to s
   344	func writeMultiple(s fmt.State, text string, count int) {
   345		if len(text) > 0 {
   346			b := []byte(text)
   347			for ; count > 0; count-- {
   348				s.Write(b)
   349			}
   350		}
   351	}
   352	
   353	// Format is a support routine for fmt.Formatter. It accepts
   354	// the formats 'b' (binary), 'o' (octal), 'd' (decimal), 'x'
   355	// (lowercase hexadecimal), and 'X' (uppercase hexadecimal).
   356	// Also supported are the full suite of package fmt's format
   357	// verbs for integral types, including '+', '-', and ' '
   358	// for sign control, '#' for leading zero in octal and for
   359	// hexadecimal, a leading "0x" or "0X" for "%#x" and "%#X"
   360	// respectively, specification of minimum digits precision,
   361	// output field width, space or zero padding, and left or
   362	// right justification.
   363	//
   364	func (x *Int) Format(s fmt.State, ch rune) {
   365		cs := charset(ch)
   366	
   367		// special cases
   368		switch {
   369		case cs == "":
   370			// unknown format
   371			fmt.Fprintf(s, "%%!%c(big.Int=%s)", ch, x.String())
   372			return
   373		case x == nil:
   374			fmt.Fprint(s, "<nil>")
   375			return
   376		}
   377	
   378		// determine sign character
   379		sign := ""
   380		switch {
   381		case x.neg:
   382			sign = "-"
   383		case s.Flag('+'): // supersedes ' ' when both specified
   384			sign = "+"
   385		case s.Flag(' '):
   386			sign = " "
   387		}
   388	
   389		// determine prefix characters for indicating output base
   390		prefix := ""
   391		if s.Flag('#') {
   392			switch ch {
   393			case 'o': // octal
   394				prefix = "0"
   395			case 'x': // hexadecimal
   396				prefix = "0x"
   397			case 'X':
   398				prefix = "0X"
   399			}
   400		}
   401	
   402		// determine digits with base set by len(cs) and digit characters from cs
   403		digits := x.abs.string(cs)
   404	
   405		// number of characters for the three classes of number padding
   406		var left int   // space characters to left of digits for right justification ("%8d")
   407		var zeroes int // zero characters (actually cs[0]) as left-most digits ("%.8d")
   408		var right int  // space characters to right of digits for left justification ("%-8d")
   409	
   410		// determine number padding from precision: the least number of digits to output
   411		precision, precisionSet := s.Precision()
   412		if precisionSet {
   413			switch {
   414			case len(digits) < precision:
   415				zeroes = precision - len(digits) // count of zero padding 
   416			case digits == "0" && precision == 0:
   417				return // print nothing if zero value (x == 0) and zero precision ("." or ".0")
   418			}
   419		}
   420	
   421		// determine field pad from width: the least number of characters to output
   422		length := len(sign) + len(prefix) + zeroes + len(digits)
   423		if width, widthSet := s.Width(); widthSet && length < width { // pad as specified
   424			switch d := width - length; {
   425			case s.Flag('-'):
   426				// pad on the right with spaces; supersedes '0' when both specified
   427				right = d
   428			case s.Flag('0') && !precisionSet:
   429				// pad with zeroes unless precision also specified
   430				zeroes = d
   431			default:
   432				// pad on the left with spaces
   433				left = d
   434			}
   435		}
   436	
   437		// print number as [left pad][sign][prefix][zero pad][digits][right pad]
   438		writeMultiple(s, " ", left)
   439		writeMultiple(s, sign, 1)
   440		writeMultiple(s, prefix, 1)
   441		writeMultiple(s, "0", zeroes)
   442		writeMultiple(s, digits, 1)
   443		writeMultiple(s, " ", right)
   444	}
   445	
   446	// scan sets z to the integer value corresponding to the longest possible prefix
   447	// read from r representing a signed integer number in a given conversion base.
   448	// It returns z, the actual conversion base used, and an error, if any. In the
   449	// error case, the value of z is undefined but the returned value is nil. The
   450	// syntax follows the syntax of integer literals in Go.
   451	//
   452	// The base argument must be 0 or a value from 2 through MaxBase. If the base
   453	// is 0, the string prefix determines the actual conversion base. A prefix of
   454	// ``0x'' or ``0X'' selects base 16; the ``0'' prefix selects base 8, and a
   455	// ``0b'' or ``0B'' prefix selects base 2. Otherwise the selected base is 10.
   456	//
   457	func (z *Int) scan(r io.RuneScanner, base int) (*Int, int, error) {
   458		// determine sign
   459		ch, _, err := r.ReadRune()
   460		if err != nil {
   461			return nil, 0, err
   462		}
   463		neg := false
   464		switch ch {
   465		case '-':
   466			neg = true
   467		case '+': // nothing to do
   468		default:
   469			r.UnreadRune()
   470		}
   471	
   472		// determine mantissa
   473		z.abs, base, err = z.abs.scan(r, base)
   474		if err != nil {
   475			return nil, base, err
   476		}
   477		z.neg = len(z.abs) > 0 && neg // 0 has no sign
   478	
   479		return z, base, nil
   480	}
   481	
   482	// Scan is a support routine for fmt.Scanner; it sets z to the value of
   483	// the scanned number. It accepts the formats 'b' (binary), 'o' (octal),
   484	// 'd' (decimal), 'x' (lowercase hexadecimal), and 'X' (uppercase hexadecimal).
   485	func (z *Int) Scan(s fmt.ScanState, ch rune) error {
   486		s.SkipSpace() // skip leading space characters
   487		base := 0
   488		switch ch {
   489		case 'b':
   490			base = 2
   491		case 'o':
   492			base = 8
   493		case 'd':
   494			base = 10
   495		case 'x', 'X':
   496			base = 16
   497		case 's', 'v':
   498			// let scan determine the base
   499		default:
   500			return errors.New("Int.Scan: invalid verb")
   501		}
   502		_, _, err := z.scan(s, base)
   503		return err
   504	}
   505	
   506	// Int64 returns the int64 representation of x.
   507	// If x cannot be represented in an int64, the result is undefined.
   508	func (x *Int) Int64() int64 {
   509		if len(x.abs) == 0 {
   510			return 0
   511		}
   512		v := int64(x.abs[0])
   513		if _W == 32 && len(x.abs) > 1 {
   514			v |= int64(x.abs[1]) << 32
   515		}
   516		if x.neg {
   517			v = -v
   518		}
   519		return v
   520	}
   521	
   522	// SetString sets z to the value of s, interpreted in the given base,
   523	// and returns z and a boolean indicating success. If SetString fails,
   524	// the value of z is undefined but the returned value is nil.
   525	//
   526	// The base argument must be 0 or a value from 2 through MaxBase. If the base
   527	// is 0, the string prefix determines the actual conversion base. A prefix of
   528	// ``0x'' or ``0X'' selects base 16; the ``0'' prefix selects base 8, and a
   529	// ``0b'' or ``0B'' prefix selects base 2. Otherwise the selected base is 10.
   530	//
   531	func (z *Int) SetString(s string, base int) (*Int, bool) {
   532		r := strings.NewReader(s)
   533		_, _, err := z.scan(r, base)
   534		if err != nil {
   535			return nil, false
   536		}
   537		_, _, err = r.ReadRune()
   538		if err != io.EOF {
   539			return nil, false
   540		}
   541		return z, true // err == io.EOF => scan consumed all of s
   542	}
   543	
   544	// SetBytes interprets buf as the bytes of a big-endian unsigned
   545	// integer, sets z to that value, and returns z.
   546	func (z *Int) SetBytes(buf []byte) *Int {
   547		z.abs = z.abs.setBytes(buf)
   548		z.neg = false
   549		return z
   550	}
   551	
   552	// Bytes returns the absolute value of z as a big-endian byte slice.
   553	func (x *Int) Bytes() []byte {
   554		buf := make([]byte, len(x.abs)*_S)
   555		return buf[x.abs.bytes(buf):]
   556	}
   557	
   558	// BitLen returns the length of the absolute value of z in bits.
   559	// The bit length of 0 is 0.
   560	func (x *Int) BitLen() int {
   561		return x.abs.bitLen()
   562	}
   563	
   564	// Exp sets z = x**y mod m and returns z. If m is nil, z = x**y.
   565	// See Knuth, volume 2, section 4.6.3.
   566	func (z *Int) Exp(x, y, m *Int) *Int {
   567		if y.neg || len(y.abs) == 0 {
   568			neg := x.neg
   569			z.SetInt64(1)
   570			z.neg = neg
   571			return z
   572		}
   573	
   574		var mWords nat
   575		if m != nil {
   576			mWords = m.abs
   577		}
   578	
   579		z.abs = z.abs.expNN(x.abs, y.abs, mWords)
   580		z.neg = len(z.abs) > 0 && x.neg && y.abs[0]&1 == 1 // 0 has no sign
   581		return z
   582	}
   583	
   584	// GCD sets z to the greatest common divisor of a and b, which must be
   585	// positive numbers, and returns z.
   586	// If x and y are not nil, GCD sets x and y such that z = a*x + b*y.
   587	// If either a or b is not positive, GCD sets z = x = y = 0.
   588	func (z *Int) GCD(x, y, a, b *Int) *Int {
   589		if a.neg || b.neg {
   590			z.SetInt64(0)
   591			if x != nil {
   592				x.SetInt64(0)
   593			}
   594			if y != nil {
   595				y.SetInt64(0)
   596			}
   597			return z
   598		}
   599	
   600		A := new(Int).Set(a)
   601		B := new(Int).Set(b)
   602	
   603		X := new(Int)
   604		Y := new(Int).SetInt64(1)
   605	
   606		lastX := new(Int).SetInt64(1)
   607		lastY := new(Int)
   608	
   609		q := new(Int)
   610		temp := new(Int)
   611	
   612		for len(B.abs) > 0 {
   613			r := new(Int)
   614			q, r = q.QuoRem(A, B, r)
   615	
   616			A, B = B, r
   617	
   618			temp.Set(X)
   619			X.Mul(X, q)
   620			X.neg = !X.neg
   621			X.Add(X, lastX)
   622			lastX.Set(temp)
   623	
   624			temp.Set(Y)
   625			Y.Mul(Y, q)
   626			Y.neg = !Y.neg
   627			Y.Add(Y, lastY)
   628			lastY.Set(temp)
   629		}
   630	
   631		if x != nil {
   632			*x = *lastX
   633		}
   634	
   635		if y != nil {
   636			*y = *lastY
   637		}
   638	
   639		*z = *A
   640		return z
   641	}
   642	
   643	// ProbablyPrime performs n Miller-Rabin tests to check whether x is prime.
   644	// If it returns true, x is prime with probability 1 - 1/4^n.
   645	// If it returns false, x is not prime.
   646	func (x *Int) ProbablyPrime(n int) bool {
   647		return !x.neg && x.abs.probablyPrime(n)
   648	}
   649	
   650	// Rand sets z to a pseudo-random number in [0, n) and returns z.
   651	func (z *Int) Rand(rnd *rand.Rand, n *Int) *Int {
   652		z.neg = false
   653		if n.neg == true || len(n.abs) == 0 {
   654			z.abs = nil
   655			return z
   656		}
   657		z.abs = z.abs.random(rnd, n.abs, n.abs.bitLen())
   658		return z
   659	}
   660	
   661	// ModInverse sets z to the multiplicative inverse of g in the group ℤ/pℤ (where
   662	// p is a prime) and returns z.
   663	func (z *Int) ModInverse(g, p *Int) *Int {
   664		var d Int
   665		d.GCD(z, nil, g, p)
   666		// x and y are such that g*x + p*y = d. Since p is prime, d = 1. Taking
   667		// that modulo p results in g*x = 1, therefore x is the inverse element.
   668		if z.neg {
   669			z.Add(z, p)
   670		}
   671		return z
   672	}
   673	
   674	// Lsh sets z = x << n and returns z.
   675	func (z *Int) Lsh(x *Int, n uint) *Int {
   676		z.abs = z.abs.shl(x.abs, n)
   677		z.neg = x.neg
   678		return z
   679	}
   680	
   681	// Rsh sets z = x >> n and returns z.
   682	func (z *Int) Rsh(x *Int, n uint) *Int {
   683		if x.neg {
   684			// (-x) >> s == ^(x-1) >> s == ^((x-1) >> s) == -(((x-1) >> s) + 1)
   685			t := z.abs.sub(x.abs, natOne) // no underflow because |x| > 0
   686			t = t.shr(t, n)
   687			z.abs = t.add(t, natOne)
   688			z.neg = true // z cannot be zero if x is negative
   689			return z
   690		}
   691	
   692		z.abs = z.abs.shr(x.abs, n)
   693		z.neg = false
   694		return z
   695	}
   696	
   697	// Bit returns the value of the i'th bit of x. That is, it
   698	// returns (x>>i)&1. The bit index i must be >= 0.
   699	func (x *Int) Bit(i int) uint {
   700		if i < 0 {
   701			panic("negative bit index")
   702		}
   703		if x.neg {
   704			t := nat(nil).sub(x.abs, natOne)
   705			return t.bit(uint(i)) ^ 1
   706		}
   707	
   708		return x.abs.bit(uint(i))
   709	}
   710	
   711	// SetBit sets z to x, with x's i'th bit set to b (0 or 1).
   712	// That is, if bit is 1 SetBit sets z = x | (1 << i);
   713	// if bit is 0 it sets z = x &^ (1 << i). If bit is not 0 or 1,
   714	// SetBit will panic.
   715	func (z *Int) SetBit(x *Int, i int, b uint) *Int {
   716		if i < 0 {
   717			panic("negative bit index")
   718		}
   719		if x.neg {
   720			t := z.abs.sub(x.abs, natOne)
   721			t = t.setBit(t, uint(i), b^1)
   722			z.abs = t.add(t, natOne)
   723			z.neg = len(z.abs) > 0
   724			return z
   725		}
   726		z.abs = z.abs.setBit(x.abs, uint(i), b)
   727		z.neg = false
   728		return z
   729	}
   730	
   731	// And sets z = x & y and returns z.
   732	func (z *Int) And(x, y *Int) *Int {
   733		if x.neg == y.neg {
   734			if x.neg {
   735				// (-x) & (-y) == ^(x-1) & ^(y-1) == ^((x-1) | (y-1)) == -(((x-1) | (y-1)) + 1)
   736				x1 := nat(nil).sub(x.abs, natOne)
   737				y1 := nat(nil).sub(y.abs, natOne)
   738				z.abs = z.abs.add(z.abs.or(x1, y1), natOne)
   739				z.neg = true // z cannot be zero if x and y are negative
   740				return z
   741			}
   742	
   743			// x & y == x & y
   744			z.abs = z.abs.and(x.abs, y.abs)
   745			z.neg = false
   746			return z
   747		}
   748	
   749		// x.neg != y.neg
   750		if x.neg {
   751			x, y = y, x // & is symmetric
   752		}
   753	
   754		// x & (-y) == x & ^(y-1) == x &^ (y-1)
   755		y1 := nat(nil).sub(y.abs, natOne)
   756		z.abs = z.abs.andNot(x.abs, y1)
   757		z.neg = false
   758		return z
   759	}
   760	
   761	// AndNot sets z = x &^ y and returns z.
   762	func (z *Int) AndNot(x, y *Int) *Int {
   763		if x.neg == y.neg {
   764			if x.neg {
   765				// (-x) &^ (-y) == ^(x-1) &^ ^(y-1) == ^(x-1) & (y-1) == (y-1) &^ (x-1)
   766				x1 := nat(nil).sub(x.abs, natOne)
   767				y1 := nat(nil).sub(y.abs, natOne)
   768				z.abs = z.abs.andNot(y1, x1)
   769				z.neg = false
   770				return z
   771			}
   772	
   773			// x &^ y == x &^ y
   774			z.abs = z.abs.andNot(x.abs, y.abs)
   775			z.neg = false
   776			return z
   777		}
   778	
   779		if x.neg {
   780			// (-x) &^ y == ^(x-1) &^ y == ^(x-1) & ^y == ^((x-1) | y) == -(((x-1) | y) + 1)
   781			x1 := nat(nil).sub(x.abs, natOne)
   782			z.abs = z.abs.add(z.abs.or(x1, y.abs), natOne)
   783			z.neg = true // z cannot be zero if x is negative and y is positive
   784			return z
   785		}
   786	
   787		// x &^ (-y) == x &^ ^(y-1) == x & (y-1)
   788		y1 := nat(nil).add(y.abs, natOne)
   789		z.abs = z.abs.and(x.abs, y1)
   790		z.neg = false
   791		return z
   792	}
   793	
   794	// Or sets z = x | y and returns z.
   795	func (z *Int) Or(x, y *Int) *Int {
   796		if x.neg == y.neg {
   797			if x.neg {
   798				// (-x) | (-y) == ^(x-1) | ^(y-1) == ^((x-1) & (y-1)) == -(((x-1) & (y-1)) + 1)
   799				x1 := nat(nil).sub(x.abs, natOne)
   800				y1 := nat(nil).sub(y.abs, natOne)
   801				z.abs = z.abs.add(z.abs.and(x1, y1), natOne)
   802				z.neg = true // z cannot be zero if x and y are negative
   803				return z
   804			}
   805	
   806			// x | y == x | y
   807			z.abs = z.abs.or(x.abs, y.abs)
   808			z.neg = false
   809			return z
   810		}
   811	
   812		// x.neg != y.neg
   813		if x.neg {
   814			x, y = y, x // | is symmetric
   815		}
   816	
   817		// x | (-y) == x | ^(y-1) == ^((y-1) &^ x) == -(^((y-1) &^ x) + 1)
   818		y1 := nat(nil).sub(y.abs, natOne)
   819		z.abs = z.abs.add(z.abs.andNot(y1, x.abs), natOne)
   820		z.neg = true // z cannot be zero if one of x or y is negative
   821		return z
   822	}
   823	
   824	// Xor sets z = x ^ y and returns z.
   825	func (z *Int) Xor(x, y *Int) *Int {
   826		if x.neg == y.neg {
   827			if x.neg {
   828				// (-x) ^ (-y) == ^(x-1) ^ ^(y-1) == (x-1) ^ (y-1)
   829				x1 := nat(nil).sub(x.abs, natOne)
   830				y1 := nat(nil).sub(y.abs, natOne)
   831				z.abs = z.abs.xor(x1, y1)
   832				z.neg = false
   833				return z
   834			}
   835	
   836			// x ^ y == x ^ y
   837			z.abs = z.abs.xor(x.abs, y.abs)
   838			z.neg = false
   839			return z
   840		}
   841	
   842		// x.neg != y.neg
   843		if x.neg {
   844			x, y = y, x // ^ is symmetric
   845		}
   846	
   847		// x ^ (-y) == x ^ ^(y-1) == ^(x ^ (y-1)) == -((x ^ (y-1)) + 1)
   848		y1 := nat(nil).sub(y.abs, natOne)
   849		z.abs = z.abs.add(z.abs.xor(x.abs, y1), natOne)
   850		z.neg = true // z cannot be zero if only one of x or y is negative
   851		return z
   852	}
   853	
   854	// Not sets z = ^x and returns z.
   855	func (z *Int) Not(x *Int) *Int {
   856		if x.neg {
   857			// ^(-x) == ^(^(x-1)) == x-1
   858			z.abs = z.abs.sub(x.abs, natOne)
   859			z.neg = false
   860			return z
   861		}
   862	
   863		// ^x == -x-1 == -(x+1)
   864		z.abs = z.abs.add(x.abs, natOne)
   865		z.neg = true // z cannot be zero if x is positive
   866		return z
   867	}
   868	
   869	// Gob codec version. Permits backward-compatible changes to the encoding.
   870	const intGobVersion byte = 1
   871	
   872	// GobEncode implements the gob.GobEncoder interface.
   873	func (x *Int) GobEncode() ([]byte, error) {
   874		buf := make([]byte, 1+len(x.abs)*_S) // extra byte for version and sign bit
   875		i := x.abs.bytes(buf) - 1            // i >= 0
   876		b := intGobVersion << 1              // make space for sign bit
   877		if x.neg {
   878			b |= 1
   879		}
   880		buf[i] = b
   881		return buf[i:], nil
   882	}
   883	
   884	// GobDecode implements the gob.GobDecoder interface.
   885	func (z *Int) GobDecode(buf []byte) error {
   886		if len(buf) == 0 {
   887			return errors.New("Int.GobDecode: no data")
   888		}
   889		b := buf[0]
   890		if b>>1 != intGobVersion {
   891			return errors.New(fmt.Sprintf("Int.GobDecode: encoding version %d not supported", b>>1))
   892		}
   893		z.neg = b&1 != 0
   894		z.abs = z.abs.setBytes(buf[1:])
   895		return nil
   896	}