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// 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 "<nil>"
        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, "<nil>")
                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
}