Subversion Repositories openrisc

// Copyright 2009 The Go Authors. All rights reserved.

// Use of this source code is governed by a BSD-style

// license that can be found in the LICENSE file.

// This file implements signed multi-precision integers.

package big

import (

        "errors"

        "fmt"

        "io"

        "math/rand"

        "strings"

)

// An Int represents a signed multi-precision integer.

// The zero value for an Int represents the value 0.

type Int struct {

        neg bool // sign

        abs nat  // absolute value of the integer

}

var intOne = &Int{false, natOne}

// Sign returns:

//

//      -1 if x <  0

//       0 if x == 0

//      +1 if x >  0

//

func (x *Int) Sign() int {

        if len(x.abs) == 0 {

                return 0

        }

        if x.neg {

                return -1

        }

        return 1

}

// SetInt64 sets z to x and returns z.

func (z *Int) SetInt64(x int64) *Int {

        neg := false

        if x < 0 {

                neg = true

                x = -x

        }

        z.abs = z.abs.setUint64(uint64(x))

        z.neg = neg

        return z

}

// NewInt allocates and returns a new Int set to x.

func NewInt(x int64) *Int {

        return new(Int).SetInt64(x)

}

// Set sets z to x and returns z.

func (z *Int) Set(x *Int) *Int {

        if z != x {

                z.abs = z.abs.set(x.abs)

                z.neg = x.neg

        }

        return z

}

// Bits provides raw (unchecked but fast) access to x by returning its

// absolute value as a little-endian Word slice. The result and x share

// the same underlying array.

// Bits is intended to support implementation of missing low-level Int

// functionality outside this package; it should be avoided otherwise.

func (x *Int) Bits() []Word {

        return x.abs

}

// SetBits provides raw (unchecked but fast) access to z by setting its

// value to abs, interpreted as a little-endian Word slice, and returning

// z. The result and abs share the same underlying array.

// SetBits is intended to support implementation of missing low-level Int

// functionality outside this package; it should be avoided otherwise.

func (z *Int) SetBits(abs []Word) *Int {

        z.abs = nat(abs).norm()

        z.neg = false

        return z

}

// Abs sets z to |x| (the absolute value of x) and returns z.

func (z *Int) Abs(x *Int) *Int {

        z.Set(x)

        z.neg = false

        return z

}

// Neg sets z to -x and returns z.

func (z *Int) Neg(x *Int) *Int {

        z.Set(x)

        z.neg = len(z.abs) > 0 && !z.neg // 0 has no sign

        return z

}

// Add sets z to the sum x+y and returns z.

func (z *Int) Add(x, y *Int) *Int {

        neg := x.neg

        if x.neg == y.neg {

                // x + y == x + y

                // (-x) + (-y) == -(x + y)

                z.abs = z.abs.add(x.abs, y.abs)

        } else {

                // x + (-y) == x - y == -(y - x)

                // (-x) + y == y - x == -(x - y)

                if x.abs.cmp(y.abs) >= 0 {

                        z.abs = z.abs.sub(x.abs, y.abs)

                } else {

                        neg = !neg

                        z.abs = z.abs.sub(y.abs, x.abs)

                }

        }

        z.neg = len(z.abs) > 0 && neg // 0 has no sign

        return z

}

// Sub sets z to the difference x-y and returns z.

func (z *Int) Sub(x, y *Int) *Int {

        neg := x.neg

        if x.neg != y.neg {

                // x - (-y) == x + y

                // (-x) - y == -(x + y)

                z.abs = z.abs.add(x.abs, y.abs)

        } else {

                // x - y == x - y == -(y - x)

                // (-x) - (-y) == y - x == -(x - y)

                if x.abs.cmp(y.abs) >= 0 {

                        z.abs = z.abs.sub(x.abs, y.abs)

                } else {

                        neg = !neg

                        z.abs = z.abs.sub(y.abs, x.abs)

                }

        }

        z.neg = len(z.abs) > 0 && neg // 0 has no sign

        return z

}

// Mul sets z to the product x*y and returns z.

func (z *Int) Mul(x, y *Int) *Int {

        // x * y == x * y

        // x * (-y) == -(x * y)

        // (-x) * y == -(x * y)

        // (-x) * (-y) == x * y

        z.abs = z.abs.mul(x.abs, y.abs)

        z.neg = len(z.abs) > 0 && x.neg != y.neg // 0 has no sign

        return z

}

// MulRange sets z to the product of all integers

// in the range [a, b] inclusively and returns z.

// If a > b (empty range), the result is 1.

func (z *Int) MulRange(a, b int64) *Int {

        switch {

        case a > b:

                return z.SetInt64(1) // empty range

        case a <= 0 && b >= 0:

                return z.SetInt64(0) // range includes 0

        }

        // a <= b && (b < 0 || a > 0)

        neg := false

        if a < 0 {

                neg = (b-a)&1 == 0

                a, b = -b, -a

        }

        z.abs = z.abs.mulRange(uint64(a), uint64(b))

        z.neg = neg

        return z

}

// Binomial sets z to the binomial coefficient of (n, k) and returns z.

func (z *Int) Binomial(n, k int64) *Int {

        var a, b Int

        a.MulRange(n-k+1, n)

        b.MulRange(1, k)

        return z.Quo(&a, &b)

}

// Quo sets z to the quotient x/y for y != 0 and returns z.

// If y == 0, a division-by-zero run-time panic occurs.

// Quo implements truncated division (like Go); see QuoRem for more details.

func (z *Int) Quo(x, y *Int) *Int {

        z.abs, _ = z.abs.div(nil, x.abs, y.abs)

        z.neg = len(z.abs) > 0 && x.neg != y.neg // 0 has no sign

        return z

}

// Rem sets z to the remainder x%y for y != 0 and returns z.

// If y == 0, a division-by-zero run-time panic occurs.

// Rem implements truncated modulus (like Go); see QuoRem for more details.

func (z *Int) Rem(x, y *Int) *Int {

        _, z.abs = nat(nil).div(z.abs, x.abs, y.abs)

        z.neg = len(z.abs) > 0 && x.neg // 0 has no sign

        return z

}

// QuoRem sets z to the quotient x/y and r to the remainder x%y

// and returns the pair (z, r) for y != 0.

// If y == 0, a division-by-zero run-time panic occurs.

//

// QuoRem implements T-division and modulus (like Go):

//

//      q = x/y      with the result truncated to zero

//      r = x - y*q

//

// (See Daan Leijen, ``Division and Modulus for Computer Scientists''.)

// See DivMod for Euclidean division and modulus (unlike Go).

//

func (z *Int) QuoRem(x, y, r *Int) (*Int, *Int) {

        z.abs, r.abs = z.abs.div(r.abs, x.abs, y.abs)

        z.neg, r.neg = len(z.abs) > 0 && x.neg != y.neg, len(r.abs) > 0 && x.neg // 0 has no sign

        return z, r

}

// Div sets z to the quotient x/y for y != 0 and returns z.

// If y == 0, a division-by-zero run-time panic occurs.

// Div implements Euclidean division (unlike Go); see DivMod for more details.

func (z *Int) Div(x, y *Int) *Int {

        y_neg := y.neg // z may be an alias for y

        var r Int

        z.QuoRem(x, y, &r)

        if r.neg {

                if y_neg {

                        z.Add(z, intOne)

                } else {

                        z.Sub(z, intOne)

                }

        }

        return z

}

// Mod sets z to the modulus x%y for y != 0 and returns z.

// If y == 0, a division-by-zero run-time panic occurs.

// Mod implements Euclidean modulus (unlike Go); see DivMod for more details.

func (z *Int) Mod(x, y *Int) *Int {

        y0 := y // save y

        if z == y || alias(z.abs, y.abs) {

                y0 = new(Int).Set(y)

        }

        var q Int

        q.QuoRem(x, y, z)

        if z.neg {

                if y0.neg {

                        z.Sub(z, y0)

                } else {

                        z.Add(z, y0)

                }

        }

        return z

}

// DivMod sets z to the quotient x div y and m to the modulus x mod y

// and returns the pair (z, m) for y != 0.

// If y == 0, a division-by-zero run-time panic occurs.

//

// DivMod implements Euclidean division and modulus (unlike Go):

//

//      q = x div y  such that

//      m = x - y*q  with 0 <= m < |q|

//

// (See Raymond T. Boute, ``The Euclidean definition of the functions

// div and mod''. ACM Transactions on Programming Languages and

// Systems (TOPLAS), 14(2):127-144, New York, NY, USA, 4/1992.

// ACM press.)

// See QuoRem for T-division and modulus (like Go).

//

func (z *Int) DivMod(x, y, m *Int) (*Int, *Int) {

        y0 := y // save y

        if z == y || alias(z.abs, y.abs) {

                y0 = new(Int).Set(y)

        }

        z.QuoRem(x, y, m)

        if m.neg {

                if y0.neg {

                        z.Add(z, intOne)

                        m.Sub(m, y0)

                } else {

                        z.Sub(z, intOne)

                        m.Add(m, y0)

                }

        }

        return z, m

}

// Cmp compares x and y and returns:

//

//   -1 if x <  y

//    0 if x == y

//   +1 if x >  y

//

func (x *Int) Cmp(y *Int) (r int) {

        // x cmp y == x cmp y

        // x cmp (-y) == x

        // (-x) cmp y == y

        // (-x) cmp (-y) == -(x cmp y)

        switch {

        case x.neg == y.neg:

                r = x.abs.cmp(y.abs)

                if x.neg {

                        r = -r

                }

        case x.neg:

                r = -1

        default:

                r = 1

        }

        return

}

func (x *Int) String() string {

        switch {

        case x == nil:

                return ""

        case x.neg:

                return "-" + x.abs.decimalString()

        }

        return x.abs.decimalString()

}

func charset(ch rune) string {

        switch ch {

        case 'b':

                return lowercaseDigits[0:2]

        case 'o':

                return lowercaseDigits[0:8]

        case 'd', 's', 'v':

                return lowercaseDigits[0:10]

        case 'x':

                return lowercaseDigits[0:16]

        case 'X':

                return uppercaseDigits[0:16]

        }

        return "" // unknown format

}

// write count copies of text to s

func writeMultiple(s fmt.State, text string, count int) {

        if len(text) > 0 {

                b := []byte(text)

                for ; count > 0; count-- {

                        s.Write(b)

                }

        }

}

// Format is a support routine for fmt.Formatter. It accepts

// the formats 'b' (binary), 'o' (octal), 'd' (decimal), 'x'

// (lowercase hexadecimal), and 'X' (uppercase hexadecimal).

// Also supported are the full suite of package fmt's format

// verbs for integral types, including '+', '-', and ' '

// for sign control, '#' for leading zero in octal and for

// hexadecimal, a leading "0x" or "0X" for "%#x" and "%#X"

// respectively, specification of minimum digits precision,

// output field width, space or zero padding, and left or

// right justification.

//

func (x *Int) Format(s fmt.State, ch rune) {

        cs := charset(ch)

        // special cases

        switch {

        case cs == "":

                // unknown format

                fmt.Fprintf(s, "%%!%c(big.Int=%s)", ch, x.String())

                return

        case x == nil:

                fmt.Fprint(s, "")

                return

        }

        // determine sign character

        sign := ""

        switch {

        case x.neg:

                sign = "-"

        case s.Flag('+'): // supersedes ' ' when both specified

                sign = "+"

        case s.Flag(' '):

                sign = " "

        }

        // determine prefix characters for indicating output base

        prefix := ""

        if s.Flag('#') {

                switch ch {

                case 'o': // octal

                        prefix = "0"

                case 'x': // hexadecimal

                        prefix = "0x"

                case 'X':

                        prefix = "0X"

                }

        }

        // determine digits with base set by len(cs) and digit characters from cs

        digits := x.abs.string(cs)

        // number of characters for the three classes of number padding

        var left int   // space characters to left of digits for right justification ("%8d")

        var zeroes int // zero characters (actually cs[0]) as left-most digits ("%.8d")

        var right int  // space characters to right of digits for left justification ("%-8d")

        // determine number padding from precision: the least number of digits to output

        precision, precisionSet := s.Precision()

        if precisionSet {

                switch {

                case len(digits) < precision:

                        zeroes = precision - len(digits) // count of zero padding

                case digits == "0" && precision == 0:

                        return // print nothing if zero value (x == 0) and zero precision ("." or ".0")

                }

        }

        // determine field pad from width: the least number of characters to output

        length := len(sign) + len(prefix) + zeroes + len(digits)

        if width, widthSet := s.Width(); widthSet && length < width { // pad as specified

                switch d := width - length; {

                case s.Flag('-'):

                        // pad on the right with spaces; supersedes '0' when both specified

                        right = d

                case s.Flag('0') && !precisionSet:

                        // pad with zeroes unless precision also specified

                        zeroes = d

                default:

                        // pad on the left with spaces

                        left = d

                }

        }

        // print number as [left pad][sign][prefix][zero pad][digits][right pad]

        writeMultiple(s, " ", left)

        writeMultiple(s, sign, 1)

        writeMultiple(s, prefix, 1)

        writeMultiple(s, "0", zeroes)

        writeMultiple(s, digits, 1)

        writeMultiple(s, " ", right)

}

// scan sets z to the integer value corresponding to the longest possible prefix

// read from r representing a signed integer number in a given conversion base.

// It returns z, the actual conversion base used, and an error, if any. In the

// error case, the value of z is undefined but the returned value is nil. The

// syntax follows the syntax of integer literals in Go.

//

// The base argument must be 0 or a value from 2 through MaxBase. If the base

// is 0, the string prefix determines the actual conversion base. A prefix of

// ``0x'' or ``0X'' selects base 16; the ``0'' prefix selects base 8, and a

// ``0b'' or ``0B'' prefix selects base 2. Otherwise the selected base is 10.

//

func (z *Int) scan(r io.RuneScanner, base int) (*Int, int, error) {

        // determine sign

        ch, _, err := r.ReadRune()

        if err != nil {

                return nil, 0, err

        }

        neg := false

        switch ch {

        case '-':

                neg = true

        case '+': // nothing to do

        default:

                r.UnreadRune()

        }

        // determine mantissa

        z.abs, base, err = z.abs.scan(r, base)

        if err != nil {

                return nil, base, err

        }

        z.neg = len(z.abs) > 0 && neg // 0 has no sign

        return z, base, nil

}

// Scan is a support routine for fmt.Scanner; it sets z to the value of

// the scanned number. It accepts the formats 'b' (binary), 'o' (octal),

// 'd' (decimal), 'x' (lowercase hexadecimal), and 'X' (uppercase hexadecimal).

func (z *Int) Scan(s fmt.ScanState, ch rune) error {

        s.SkipSpace() // skip leading space characters

        base := 0

        switch ch {

        case 'b':

                base = 2

        case 'o':

                base = 8

        case 'd':

                base = 10

        case 'x', 'X':

                base = 16

        case 's', 'v':

                // let scan determine the base

        default:

                return errors.New("Int.Scan: invalid verb")

        }

        _, _, err := z.scan(s, base)

        return err

}

// Int64 returns the int64 representation of x.

// If x cannot be represented in an int64, the result is undefined.

func (x *Int) Int64() int64 {

        if len(x.abs) == 0 {

                return 0

        }

        v := int64(x.abs[0])

        if _W == 32 && len(x.abs) > 1 {

                v |= int64(x.abs[1]) << 32

        }

        if x.neg {

                v = -v

        }

        return v

}

// SetString sets z to the value of s, interpreted in the given base,

// and returns z and a boolean indicating success. If SetString fails,

// the value of z is undefined but the returned value is nil.

//

// The base argument must be 0 or a value from 2 through MaxBase. If the base

// is 0, the string prefix determines the actual conversion base. A prefix of

// ``0x'' or ``0X'' selects base 16; the ``0'' prefix selects base 8, and a

// ``0b'' or ``0B'' prefix selects base 2. Otherwise the selected base is 10.

//

func (z *Int) SetString(s string, base int) (*Int, bool) {

        r := strings.NewReader(s)

        _, _, err := z.scan(r, base)

        if err != nil {

                return nil, false

        }

        _, _, err = r.ReadRune()

        if err != io.EOF {

                return nil, false

        }

        return z, true // err == io.EOF => scan consumed all of s

}

// SetBytes interprets buf as the bytes of a big-endian unsigned

// integer, sets z to that value, and returns z.

func (z *Int) SetBytes(buf []byte) *Int {

        z.abs = z.abs.setBytes(buf)

        z.neg = false

        return z

}

// Bytes returns the absolute value of z as a big-endian byte slice.

func (x *Int) Bytes() []byte {

        buf := make([]byte, len(x.abs)*_S)

        return buf[x.abs.bytes(buf):]

}

// BitLen returns the length of the absolute value of z in bits.

// The bit length of 0 is 0.

func (x *Int) BitLen() int {

        return x.abs.bitLen()

}

// Exp sets z = x**y mod m and returns z. If m is nil, z = x**y.

// See Knuth, volume 2, section 4.6.3.

func (z *Int) Exp(x, y, m *Int) *Int {

        if y.neg || len(y.abs) == 0 {

                neg := x.neg

                z.SetInt64(1)

                z.neg = neg

                return z

        }

        var mWords nat

        if m != nil {

                mWords = m.abs

        }

        z.abs = z.abs.expNN(x.abs, y.abs, mWords)

        z.neg = len(z.abs) > 0 && x.neg && y.abs[0]&1 == 1 // 0 has no sign

        return z

}

// GCD sets z to the greatest common divisor of a and b, which must be

// positive numbers, and returns z.

// If x and y are not nil, GCD sets x and y such that z = a*x + b*y.

// If either a or b is not positive, GCD sets z = x = y = 0.

func (z *Int) GCD(x, y, a, b *Int) *Int {

        if a.neg || b.neg {

                z.SetInt64(0)

                if x != nil {

                        x.SetInt64(0)

                }

                if y != nil {

                        y.SetInt64(0)

                }

                return z

        }

        A := new(Int).Set(a)

        B := new(Int).Set(b)

        X := new(Int)

        Y := new(Int).SetInt64(1)

        lastX := new(Int).SetInt64(1)

        lastY := new(Int)

        q := new(Int)

        temp := new(Int)

        for len(B.abs) > 0 {

                r := new(Int)

                q, r = q.QuoRem(A, B, r)

                A, B = B, r

                temp.Set(X)

                X.Mul(X, q)

                X.neg = !X.neg

                X.Add(X, lastX)

                lastX.Set(temp)

                temp.Set(Y)

                Y.Mul(Y, q)

                Y.neg = !Y.neg

                Y.Add(Y, lastY)

                lastY.Set(temp)

        }

        if x != nil {

                *x = *lastX

        }

        if y != nil {

                *y = *lastY

        }

        *z = *A

        return z

}

// ProbablyPrime performs n Miller-Rabin tests to check whether x is prime.

// If it returns true, x is prime with probability 1 - 1/4^n.

// If it returns false, x is not prime.

func (x *Int) ProbablyPrime(n int) bool {

        return !x.neg && x.abs.probablyPrime(n)

}

// Rand sets z to a pseudo-random number in [0, n) and returns z.

func (z *Int) Rand(rnd *rand.Rand, n *Int) *Int {

        z.neg = false

        if n.neg == true || len(n.abs) == 0 {

                z.abs = nil

                return z

        }

        z.abs = z.abs.random(rnd, n.abs, n.abs.bitLen())

        return z

}

// ModInverse sets z to the multiplicative inverse of g in the group ℤ/pℤ (where

// p is a prime) and returns z.

func (z *Int) ModInverse(g, p *Int) *Int {

        var d Int

        d.GCD(z, nil, g, p)

        // x and y are such that g*x + p*y = d. Since p is prime, d = 1. Taking

        // that modulo p results in g*x = 1, therefore x is the inverse element.

        if z.neg {

                z.Add(z, p)

        }

        return z

}

// Lsh sets z = x << n and returns z.

func (z *Int) Lsh(x *Int, n uint) *Int {

        z.abs = z.abs.shl(x.abs, n)

        z.neg = x.neg

        return z

}

// Rsh sets z = x >> n and returns z.

func (z *Int) Rsh(x *Int, n uint) *Int {

        if x.neg {

                // (-x) >> s == ^(x-1) >> s == ^((x-1) >> s) == -(((x-1) >> s) + 1)

                t := z.abs.sub(x.abs, natOne) // no underflow because |x| > 0

                t = t.shr(t, n)

                z.abs = t.add(t, natOne)

                z.neg = true // z cannot be zero if x is negative

                return z

        }

        z.abs = z.abs.shr(x.abs, n)

        z.neg = false

        return z

}

// Bit returns the value of the i'th bit of x. That is, it

// returns (x>>i)&1. The bit index i must be >= 0.

func (x *Int) Bit(i int) uint {

        if i < 0 {

                panic("negative bit index")

        }

        if x.neg {

                t := nat(nil).sub(x.abs, natOne)

                return t.bit(uint(i)) ^ 1

        }

        return x.abs.bit(uint(i))

}

// SetBit sets z to x, with x's i'th bit set to b (0 or 1).

// That is, if bit is 1 SetBit sets z = x | (1 << i);

// if bit is 0 it sets z = x &^ (1 << i). If bit is not 0 or 1,

// SetBit will panic.

func (z *Int) SetBit(x *Int, i int, b uint) *Int {

        if i < 0 {

                panic("negative bit index")

        }

        if x.neg {

                t := z.abs.sub(x.abs, natOne)

                t = t.setBit(t, uint(i), b^1)

                z.abs = t.add(t, natOne)

                z.neg = len(z.abs) > 0

                return z

        }

        z.abs = z.abs.setBit(x.abs, uint(i), b)

        z.neg = false

        return z

}

// And sets z = x & y and returns z.

func (z *Int) And(x, y *Int) *Int {

        if x.neg == y.neg {

                if x.neg {

                        // (-x) & (-y) == ^(x-1) & ^(y-1) == ^((x-1) | (y-1)) == -(((x-1) | (y-1)) + 1)

                        x1 := nat(nil).sub(x.abs, natOne)

                        y1 := nat(nil).sub(y.abs, natOne)

                        z.abs = z.abs.add(z.abs.or(x1, y1), natOne)

                        z.neg = true // z cannot be zero if x and y are negative

                        return z

                }

                // x & y == x & y

                z.abs = z.abs.and(x.abs, y.abs)

                z.neg = false

                return z

        }

        // x.neg != y.neg

        if x.neg {

                x, y = y, x // & is symmetric

        }

        // x & (-y) == x & ^(y-1) == x &^ (y-1)

        y1 := nat(nil).sub(y.abs, natOne)

        z.abs = z.abs.andNot(x.abs, y1)

        z.neg = false

        return z

}

// AndNot sets z = x &^ y and returns z.

func (z *Int) AndNot(x, y *Int) *Int {

        if x.neg == y.neg {

                if x.neg {

                        // (-x) &^ (-y) == ^(x-1) &^ ^(y-1) == ^(x-1) & (y-1) == (y-1) &^ (x-1)

                        x1 := nat(nil).sub(x.abs, natOne)

                        y1 := nat(nil).sub(y.abs, natOne)

                        z.abs = z.abs.andNot(y1, x1)

                        z.neg = false

                        return z

                }

                // x &^ y == x &^ y

                z.abs = z.abs.andNot(x.abs, y.abs)

                z.neg = false

                return z

        }

        if x.neg {

                // (-x) &^ y == ^(x-1) &^ y == ^(x-1) & ^y == ^((x-1) | y) == -(((x-1) | y) + 1)

                x1 := nat(nil).sub(x.abs, natOne)

                z.abs = z.abs.add(z.abs.or(x1, y.abs), natOne)

                z.neg = true // z cannot be zero if x is negative and y is positive

                return z

        }

        // x &^ (-y) == x &^ ^(y-1) == x & (y-1)

        y1 := nat(nil).add(y.abs, natOne)

        z.abs = z.abs.and(x.abs, y1)

        z.neg = false

        return z

}

// Or sets z = x | y and returns z.

func (z *Int) Or(x, y *Int) *Int {

        if x.neg == y.neg {

                if x.neg {

                        // (-x) | (-y) == ^(x-1) | ^(y-1) == ^((x-1) & (y-1)) == -(((x-1) & (y-1)) + 1)

                        x1 := nat(nil).sub(x.abs, natOne)

                        y1 := nat(nil).sub(y.abs, natOne)

                        z.abs = z.abs.add(z.abs.and(x1, y1), natOne)

                        z.neg = true // z cannot be zero if x and y are negative

                        return z

                }

                // x | y == x | y

                z.abs = z.abs.or(x.abs, y.abs)

                z.neg = false

                return z

        }

        // x.neg != y.neg

        if x.neg {

                x, y = y, x // | is symmetric

        }

        // x | (-y) == x | ^(y-1) == ^((y-1) &^ x) == -(^((y-1) &^ x) + 1)

        y1 := nat(nil).sub(y.abs, natOne)

        z.abs = z.abs.add(z.abs.andNot(y1, x.abs), natOne)

        z.neg = true // z cannot be zero if one of x or y is negative

        return z

}

// Xor sets z = x ^ y and returns z.

func (z *Int) Xor(x, y *Int) *Int {

        if x.neg == y.neg {

                if x.neg {

                        // (-x) ^ (-y) == ^(x-1) ^ ^(y-1) == (x-1) ^ (y-1)

                        x1 := nat(nil).sub(x.abs, natOne)

                        y1 := nat(nil).sub(y.abs, natOne)

                        z.abs = z.abs.xor(x1, y1)

                        z.neg = false

                        return z

                }

                // x ^ y == x ^ y

                z.abs = z.abs.xor(x.abs, y.abs)

                z.neg = false

                return z

        }

        // x.neg != y.neg

        if x.neg {

                x, y = y, x // ^ is symmetric

        }

        // x ^ (-y) == x ^ ^(y-1) == ^(x ^ (y-1)) == -((x ^ (y-1)) + 1)

        y1 := nat(nil).sub(y.abs, natOne)

        z.abs = z.abs.add(z.abs.xor(x.abs, y1), natOne)

        z.neg = true // z cannot be zero if only one of x or y is negative

        return z

}

// Not sets z = ^x and returns z.

func (z *Int) Not(x *Int) *Int {

        if x.neg {

                // ^(-x) == ^(^(x-1)) == x-1

                z.abs = z.abs.sub(x.abs, natOne)

                z.neg = false

                return z

        }

        // ^x == -x-1 == -(x+1)

        z.abs = z.abs.add(x.abs, natOne)

        z.neg = true // z cannot be zero if x is positive

        return z

}

// Gob codec version. Permits backward-compatible changes to the encoding.

const intGobVersion byte = 1

// GobEncode implements the gob.GobEncoder interface.

func (x *Int) GobEncode() ([]byte, error) {

        buf := make([]byte, 1+len(x.abs)*_S) // extra byte for version and sign bit

        i := x.abs.bytes(buf) - 1            // i >= 0

        b := intGobVersion << 1              // make space for sign bit

        if x.neg {

                b |= 1

        }

        buf[i] = b

        return buf[i:], nil

}

// GobDecode implements the gob.GobDecoder interface.

func (z *Int) GobDecode(buf []byte) error {

        if len(buf) == 0 {

                return errors.New("Int.GobDecode: no data")

        }

        b := buf[0]

        if b>>1 != intGobVersion {

                return errors.New(fmt.Sprintf("Int.GobDecode: encoding version %d not supported", b>>1))

        }

        z.neg = b&1 != 0

        z.abs = z.abs.setBytes(buf[1:])

        return nil

}