/fuel/core/vendor/phpseclib/Math/BigInteger.php

<?php

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



namespace PHPSecLib;



/**

 * 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 >> and << 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)))->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:

 * <code>

 * <?php

 *    include('Math/BigInteger.php');

 *

 *    $a = new Math_BigInteger(2);

 *    $b = new Math_BigInteger(3);

 *

 *    $c = $a->add($b);

 *

 *    echo $c->toString(); // outputs 5

 * ?>

 * </code>

 *

 * 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 <terrafrost@php.net>

 * @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 <terrafrost@php.net>

 * @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->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->value,

     * however, $this->hex is only calculated when $this->__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:

     * <code>

     * <?php

     *    include('Math/BigInteger.php');

     *

     *    $a = new Math_BigInteger('0x32', 16); // 50 in base-16

     *

     *    echo $a->toString(); // outputs 50

     * ?>

     * </code>

     *

     * @param optional $x base-10 number or base-$base number if $base set.

     * @param optional integer $base

     * @return Math_BigInteger

     * @access public

     */

    function __construct($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($x) && get_resource_type($x) == 'GMP integer') {

                    $this->value = $x;

                    return;

                }

                $this->value = gmp_init(0);

                break;

            case MATH_BIGINTEGER_MODE_BCMATH:

                $this->value = '0';

                break;

            default:

                $this->value = array();

        }



        if (empty($x)) {

            return;

        }



        switch ($base) {

            case -256:

                if (ord($x[0]) & 0x80) {

                    $x = ~$x;

                    $this->is_negative = true;

                }

            case  256:

                switch ( MATH_BIGINTEGER_MODE ) {

                    case MATH_BIGINTEGER_MODE_GMP:

                        $sign = $this->is_negative ? '-' : '';

                        $this->value = gmp_init($sign . '0x' . bin2hex($x));

                        break;

                    case MATH_BIGINTEGER_MODE_BCMATH:

                        // round $len to the nearest 4 (thanks, DavidMJ!)

                        $len = (strlen($x) + 3) & 0xFFFFFFFC;



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



                        for ($i = 0; $i < $len; $i+= 4) {

                            $this->value = bcmul($this->value, '4294967296', 0); // 4294967296 == 2**32

                            $this->value = bcadd($this->value, 0x1000000 * ord($x[$i]) + ((ord($x[$i + 1]) << 16) | (ord($x[$i + 2]) << 8) | ord($x[$i + 3])), 0);

                        }



                        if ($this->is_negative) {

                            $this->value = '-' . $this->value;

                        }



                        break;

                    // converts a base-2**8 (big endian / msb) number to base-2**26 (little endian / lsb)

                    default:

                        while (strlen($x)) {

                            $this->value[] = $this->_bytes2int($this->_base256_rshift($x, 26));

                        }

                }



                if ($this->is_negative) {

                    if (MATH_BIGINTEGER_MODE != MATH_BIGINTEGER_MODE_INTERNAL) {

                        $this->is_negative = false;

                    }

                    $temp = $this->add(new Math_BigInteger('-1'));

                    $this->value = $temp->value;

                }

                break;

            case  16:

            case -16:

                if ($base > 0 && $x[0] == '-') {

                    $this->is_negative = true;

                    $x = substr($x, 1);

                }



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



                $is_negative = false;

                if ($base < 0 && hexdec($x[0]) >= 8) {

                    $this->is_negative = $is_negative = true;

                    $x = bin2hex(~pack('H*', $x));

                }



                switch ( MATH_BIGINTEGER_MODE ) {

                    case MATH_BIGINTEGER_MODE_GMP:

                        $temp = $this->is_negative ? '-0x' . $x : '0x' . $x;

                        $this->value = gmp_init($temp);

                        $this->is_negative = false;

                        break;

                    case MATH_BIGINTEGER_MODE_BCMATH:

                        $x = ( strlen($x) & 1 ) ? '0' . $x : $x;

                        $temp = new Math_BigInteger(pack('H*', $x), 256);

                        $this->value = $this->is_negative ? '-' . $temp->value : $temp->value;

                        $this->is_negative = false;

                        break;

                    default:

                        $x = ( strlen($x) & 1 ) ? '0' . $x : $x;

                        $temp = new Math_BigInteger(pack('H*', $x), 256);

                        $this->value = $temp->value;

                }



                if ($is_negative) {

                    $temp = $this->add(new Math_BigInteger('-1'));

                    $this->value = $temp->value;

                }

                break;

            case  10:

            case -10:

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



                switch ( MATH_BIGINTEGER_MODE ) {

                    case MATH_BIGINTEGER_MODE_GMP:

                        $this->value = gmp_init($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->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->value = array(10000000);



                        if ($x[0] == '-') {

                            $this->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 = $temp->multiply($multiplier);

                            $temp = $temp->add(new Math_BigInteger($this->_int2bytes(substr($x, 0, 7)), 256));

                            $x = substr($x, 7);

                        }



                        $this->value = $temp->value;

                }

                break;

            case  2: // base-2 support originally implemented by Lluis Pamies - thanks!

            case -2:

                if ($base > 0 && $x[0] == '-') {

                    $this->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->is_negative) {

                    $str = '-' . $str;

                }



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

                $this->value = $temp->value;

                $this->is_negative = $temp->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:

     * <code>

     * <?php

     *    include('Math/BigInteger.php');

     *

     *    $a = new Math_BigInteger('65');

     *

     *    echo $a->toBytes(); // outputs chr(65)

     * ?>

     * </code>

     *

     * @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->compare(new Math_BigInteger());

            if ($comparison == 0) {

                return $this->precision > 0 ? str_repeat(chr(0), ($this->precision + 1) >> 3) : '';

            }



            $temp = $comparison < 0 ? $this->add(new Math_BigInteger(1)) : $this->copy();

            $bytes = $temp->toBytes();



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

                $bytes = chr(0);

            }



            if (ord($bytes[0]) & 0x80) {

                $bytes = chr(0) . $bytes;

            }



            return $comparison < 0 ? ~$bytes : $bytes;

        }



        switch ( MATH_BIGINTEGER_MODE ) {

            case MATH_BIGINTEGER_MODE_GMP:

                if (gmp_cmp($this->value, gmp_init(0)) == 0) {

                    return $this->precision > 0 ? str_repeat(chr(0), ($this->precision + 1) >> 3) : '';

                }



                $temp = gmp_strval(gmp_abs($this->value), 16);

                $temp = ( strlen($temp) & 1 ) ? '0' . $temp : $temp;

                $temp = pack('H*', $temp);



                return $this->precision > 0 ?

                    substr(str_pad($temp, $this->precision >> 3, chr(0), STR_PAD_LEFT), -($this->precision >> 3)) :

                    ltrim($temp, chr(0));

            case MATH_BIGINTEGER_MODE_BCMATH:

                if ($this->value === '0') {

                    return $this->precision > 0 ? str_repeat(chr(0), ($this->precision + 1) >> 3) : '';

                }



                $value = '';

                $current = $this->value;



                if ($current[0] == '-') {

                    $current = substr($current, 1);

                }



                while (bccomp($current, '0', 0) > 0) {

                    $temp = bcmod($current, '16777216');

                    $value = chr($temp >> 16) . chr($temp >> 8) . chr($temp) . $value;

                    $current = bcdiv($current, '16777216', 0);

                }



                return $this->precision > 0 ?

                    substr(str_pad($value, $this->precision >> 3, chr(0), STR_PAD_LEFT), -($this->precision >> 3)) :

                    ltrim($value, chr(0));

        }



        if (!count($this->value)) {

            return $this->precision > 0 ? str_repeat(chr(0), ($this->precision + 1) >> 3) : '';

        }

        $result = $this->_int2bytes($this->value[count($this->value) - 1]);



        $temp = $this->copy();



        for ($i = count($temp->value) - 2; $i >= 0; --$i) {

            $temp->_base256_lshift($result, 26);

            $result = $result | str_pad($temp->_int2bytes($temp->value[$i]), strlen($result), chr(0), STR_PAD_LEFT);

        }



        return $this->precision > 0 ?

            str_pad(substr($result, -(($this->precision + 7) >> 3)), ($this->precision + 7) >> 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:

     * <code>

     * <?php

     *    include('Math/BigInteger.php');

     *

     *    $a = new Math_BigInteger('65');

     *

     *    echo $a->toHex(); // outputs '41'

     * ?>

     * </code>

     *

     * @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($this->toBytes($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:

     * <code>

     * <?php

     *    include('Math/BigInteger.php');

     *

     *    $a = new Math_BigInteger('65');

     *

     *    echo $a->toBits(); // outputs '1000001'

     * ?>

     * </code>

     *

     * @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 = $this->toHex($twos_compliment);

        $bits = '';

        for ($i = 0; $i < strlen($hex); $i+=8) {

            $bits.= str_pad(decbin(hexdec(substr($hex, $i, 8))), 32, '0', STR_PAD_LEFT);

        }

        return $this->precision > 0 ? substr($bits, -$this->precision) : ltrim($bits, '0');

    }



    /**

     * Converts a BigInteger to a base-10 number.

     *

     * Here's an example:

     * <code>

     * <?php

     *    include('Math/BigInteger.php');

     *

     *    $a = new Math_BigInteger('50');

     *

     *    echo $a->toString(); // outputs 50

     * ?>

     * </code>

     *

     * @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->value);

            case MATH_BIGINTEGER_MODE_BCMATH:

                if ($this->value === '0') {

                    return '0';

                }



                return ltrim($this->value, '0');

        }



        if (!count($this->value)) {

            return '0';

        }



        $temp = $this->copy();

        $temp->is_negative = false;



        $divisor = new Math_BigInteger();

        $divisor->value = array(10000000); // eg. 10**7

        $result = '';

        while (count($temp->value)) {

            list($temp, $mod) = $temp->divide($divisor);

            $result = str_pad(isset($mod->value[0]) ? $mod->value[0] : '', 7, '0', STR_PAD_LEFT) . $result;

        }

        $result = ltrim($result, '0');

        if (empty($result)) {

            $result = '0';

        }



        if ($this->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->value = $this->value;

        $temp->is_negative = $this->is_negative;

        $temp->generator = $this->generator;

        $temp->precision = $this->precision;

        $temp->bitmask = $this->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->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->copy();

    }



    /**

     *  __sleep() magic method

     *

     * Will be called, automatically, when serialize() is called on a Math_BigInteger object.

     *

     * @see __wakeup()

     * @access public

     */

    function __sleep()

    {

        $this->hex = $this->toHex(true);

        $vars = array('hex');

        if ($this->generator != 'mt_rand') {

            $vars[] = 'generator';

        }

        if ($this->precision > 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->hex, -16);

        $this->value = $temp->value;

        $this->is_negative = $temp->is_negative;

        $this->setRandomGenerator($this->generator);

        if ($this->precision > 0) {

            // recalculate $this->bitmask

            $this->setPrecision($this->precision);

        }

    }



    /**

     * Adds two BigIntegers.

     *

     * Here's an example:

     * <code>

     * <?php

     *    include('Math/BigInteger.php');

     *

     *    $a = new Math_BigInteger('10');

     *    $b = new Math_BigInteger('20');

     *

     *    $c = $a->add($b);

     *

     *    echo $c->toString(); // outputs 30

     * ?>

     * </code>

     *

     * @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();

                $temp->value = gmp_add($this->value, $y->value);



                return $this->_normalize($temp);

            case MATH_BIGINTEGER_MODE_BCMATH:

                $temp = new Math_BigInteger();

                $temp->value = bcadd($this->value, $y->value, 0);



                return $this->_normalize($temp);

        }



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



        $result = new Math_BigInteger();

        $result->value = $temp[MATH_BIGINTEGER_VALUE];

        $result->is_negative = $temp[MATH_BIGINTEGER_SIGN];



        return $this->_normalize($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 => $y_value,

                MATH_BIGINTEGER_SIGN => $y_negative

            );

        } else if ($y_size == 0) {

            return array(

                MATH_BIGINTEGER_VALUE => $x_value,

                MATH_BIGINTEGER_SIGN => $x_negative

            );

        }



        // subtract, if appropriate

        if ( $x_negative != $y_negative ) {

            if ( $x_value == $y_value ) {

                return array(

                    MATH_BIGINTEGER_VALUE => array(),

                    MATH_BIGINTEGER_SIGN => false

                );

            }



            $temp = $this->_subtract($x_value, false, $y_value, false);

            $temp[MATH_BIGINTEGER_SIGN] = $this->_compare($x_value, false, $y_value, false) > 0 ?

                                          $x_negative : $y_negative;



            return $temp;

        }



        if ($x_size < $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 < $size; $i+=2, $j+=2) {

            $sum = $x_value[$j] * 0x4000000 + $x_value[$i] + $y_value[$j] * 0x4000000 + $y_value[$i] + $carry;

            $carry = $sum >= MATH_BIGINTEGER_MAX_DIGIT52; // eg. floor($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 >= 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 => $this->_trim($value),

            MATH_BIGINTEGER_SIGN => $x_negative

        );

    }



    /**

     * Subtracts two BigIntegers.

     *

     * Here's an example:

     * <code>

     * <?php

     *    include('Math/BigInteger.php');

     *

     *    $a = new Math_BigInteger('10');

     *    $b = new Math_BigInteger('20');

     *

     *    $c = $a->subtract($b);

     *

     *    echo $c->toString(); // outputs -10

     * ?>

     * </code>

     *

     * @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();

                $temp->value = gmp_sub($this->value, $y->value);



                return $this->_normalize($temp);

            case MATH_BIGINTEGER_MODE_BCMATH:

                $temp = new Math_BigInteger();

                $temp->value = bcsub($this->value, $y->value, 0);



                return $this->_normalize($temp);

        }



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



        $result = new Math_BigInteger();

        $result->value = $temp[MATH_BIGINTEGER_VALUE];

        $result->is_negative = $temp[MATH_BIGINTEGER_SIGN];



        return $this->_normalize($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 => $y_value,

                MATH_BIGINTEGER_SIGN => !$y_negative

            );

        } else if ($y_size == 0) {

            return array(

                MATH_BIGINTEGER_VALUE => $x_value,

                MATH_BIGINTEGER_SIGN => $x_negative

            );

        }



        // add, if appropriate (ie. -$x - +$y or +$x - -$y)

        if ( $x_negative != $y_negative ) {

            $temp = $this->_add($x_value, false, $y_value, false);

            $temp[MATH_BIGINTEGER_SIGN] = $x_negative;



            return $temp;

        }



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



        if ( !$diff ) {

            return array(

                MATH_BIGINTEGER_VALUE => array(),

                MATH_BIGINTEGER_SIGN => false

            );

        }



        // switch $x and $y around, if appropriate.

        if ( (!$x_negative && $diff < 0) || ($x_negative && $diff > 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 < $y_size; $i+=2, $j+=2) {

            $sum = $x_value[$j] * 0x4000000 + $x_value[$i] - $y_value[$j] * 0x4000000 - $y_value[$i] - $carry;

            $carry = $sum < 0; // eg. floor($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 < 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 => $this->_trim($x_value),

            MATH_BIGINTEGER_SIGN => $x_negative

        );

    }



    /**

     * Multiplies two BigIntegers

     *

     * Here's an example:

     * <code>

     * <?php

     *    include('Math/BigInteger.php');

     *

     *    $a = new Math_BigInteger('10');

     *    $b = new Math_BigInteger('20');

     *

     *    $c = $a->multiply($b);

     *

     *    echo $c->toString(); // outputs 200

     * ?>

     * </code>

     *

     * @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();

                $temp->value = gmp_mul($this->value, $x->value);



                return $this->_normalize($temp);

            case MATH_BIGINTEGER_MODE_BCMATH:

                $temp = new Math_BigInteger();

                $temp->value = bcmul($this->value, $x->value, 0);



                return $this->_normalize($temp);

        }



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



        $product = new Math_BigInteger();

        $product->value = $temp[MATH_BIGINTEGER_VALUE];

        $product->is_negative = $temp[MATH_BIGINTEGER_SIGN];



        return $this->_normalize($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 => $this->_square($x_value),

        //        MATH_BIGINTEGER_SIGN => $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 => array(),

                MATH_BIGINTEGER_SIGN => false

            );

        }



        return array(

            MATH_BIGINTEGER_VALUE => min($x_length, $y_length) < 2 * MATH_BIGINTEGER_KARATSUBA_CUTOFF ?

                $this->_trim($this->_regularMultiply($x_value, $y_value)) :

                $this->_trim($this->_karatsuba($x_value, $y_value)),

            MATH_BIGINTEGER_SIGN => $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 < $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->_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, $product->value[$k] is going

        // to always be 0



        $carry = 0;



        for ($j = 0; $j < $x_length; ++$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 "one with nested for loops"

        for ($i = 1; $i < $y_length; ++$i) {

            $carry = 0;



            for ($j = 0, $k = $i; $j < $x_length; ++$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) >> 1, count($y_value) >> 1);



        if ($m < MATH_BIGINTEGER_KARATSUBA_CUTOFF) {

            return $this->_regularMultiply($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 = $this->_karatsuba($x1, $y1);

        $z0 = $this->_karatsuba($x0, $y0);



        $z1 = $this->_add($x1, false, $x0, false);

        $temp = $this->_add($y1, false, $y0, false);

        $z1 = $this->_karatsuba($z1[MATH_BIGINTEGER_VALUE], $temp[MATH_BIGINTEGER_VALUE]);

        $temp = $this->_add($z2, false, $z0, false);

        $z1 = $this->_subtract($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 = $this->_add($z2, false, $z1[MATH_BIGINTEGER_VALUE], $z1[MATH_BIGINTEGER_SIGN]);

        $xy = $this->_add($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) < 2 * MATH_BIGINTEGER_KARATSUBA_CUTOFF ?

            $this->_trim($this->_baseSquare($x)) :

            $this->_trim($this->_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 = $this->_array_repeat(0, 2 * count($value));



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

            $i2 = $i << 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 <= $max_index; ++$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 "squaring" 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) >> 1;



        if ($m < MATH_BIGINTEGER_KARATSUBA_CUTOFF) {

            return $this->_baseSquare($value);

        }



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

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



        $z2 = $this->_karatsubaSquare($x1);

        $z0 = $this->_karatsubaSquare($x0);



        $z1 = $this->_add($x1, false, $x0, false);

        $z1 = $this->_karatsubaSquare($z1[MATH_BIGINTEGER_VALUE]);

        $temp = $this->_add($z2, false, $z0, false);

        $z1 = $this->_subtract($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 = $this->_add($z2, false, $z1[MATH_BIGINTEGER_VALUE], $z1[MATH_BIGINTEGER_SIGN]);

        $xx = $this->_add($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

     * "common residue".  If the remainder would be positive, the "common residue" and the remainder are the

     * same.  If the remainder would be negative, the "common residue" is equal to the sum of the remainder

     * and the divisor (basically, the "common residue" is the first positive modulo).

     *

     * Here's an example:

     * <code>

     * <?php

     *    include('Math/BigInteger.php');

     *

     *    $a = new Math_BigInteger('10');

     *    $b = new Math_BigInteger('20');

     *

     *    list($quotient, $remainder) = $a->divide($b);

     *

     *    echo $quotient->toString(); // outputs 0

     *    echo "\r\n";

     *    echo $remainder->toString(); // outputs 10

     * ?>

     * </code>

     *

     * @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->value, $remainder->value) = gmp_div_qr($this->value, $y->value);



                if (gmp_sign($remainder->value) < 0) {

                    $remainder->value = gmp_add($remainder->value, gmp_abs($y->value));

                }



                return array($this->_normalize($quotient), $this->_normalize($remainder));

            case MATH_BIGINTEGER_MODE_BCMATH:

                $quotient = new Math_BigInteger();

                $remainder = new Math_BigInteger();



                $quotient->value = bcdiv($this->value, $y->value, 0);

                $remainder->value = bcmod($this->value, $y->value);



                if ($remainder->value[0] == '-') {

                    $remainder->value = bcadd($remainder->value, $y->value[0] == '-' ? substr($y->value, 1) : $y->value, 0);

                }



                return array($this->_normalize($quotient), $this->_normalize($remainder));

        }



        if (count($y->value) == 1) {

            list($q, $r) = $this->_divide_digit($this->value, $y->value[0]);

            $quotient = new Math_BigInteger();

            $remainder = new Math_BigInteger();

            $quotient->value = $q;

            $remainder->value = array($r);

            $quotient->is_negative = $this->is_negative != $y->is_negative;

            return array($this->_normalize($quotient), $this->_normalize($remainder));

        }



        static $zero;

        if ( !isset($zero) ) {

            $zero = new Math_BigInteger();

        }



        $x = $this->copy();

        $y = $y->copy();



        $x_sign = $x->is_negative;

        $y_sign = $y->is_negative;



        $x->is_negative = $y->is_negative = false;



        $diff = $x->compare($y);



        if ( !$diff ) {

            $temp = new Math_BigInteger();

            $temp->value = array(1);

            $temp->is_negative = $x_sign != $y_sign;

            return array($this->_normalize($temp), $this->_normalize(new Math_BigInteger()));

        }



        if ( $diff < 0 ) {

            // if $x is negative, "add" $y.

            if ( $x_sign ) {

                $x = $y->subtract($x);

            }

            return array($this->_normalize(new Math_BigInteger()), $this->_normalize($x));

        }



        // normalize $x and $y as described in HAC 14.23 / 14.24

        $msb = $y->value[count($y->value) - 1];

        for ($shift = 0; !($msb & 0x2000000); ++$shift) {

            $msb <<= 1;

        }

        $x->_lshift($shift);

        $y->_lshift($shift);

        $y_value = &$y->value;



        $x_max = count($x->value) - 1;

        $y_max = count($y->value) - 1;



        $quotient = new Math_BigInteger();

        $quotient_value = &$quotient->value;

        $quotient_value = $this->_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();

        }

        $temp_value = &$temp->value;

        $rhs_value =  &$rhs->value;



        // $temp = $y << ($x_max - $y_max-1) in base 2**26

        $temp_value = array_merge($this->_array_repeat(0, $x_max - $y_max), $y_value);



        while ( $x->compare($temp) >= 0 ) {

            // calculate the "common residue"

            ++$quotient_value[$x_max - $y_max];

            $x = $x->subtract($temp);

            $x_max = count($x->value) - 1;

        }



        for ($i = $x_max; $i >= $y_max + 1; --$i) {

            $x_value = &$x->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 > 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]);



            $lhs->value = array($quotient_value[$q_index]);

            $lhs = $lhs->multiply($temp);



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



            while ( $lhs->compare($rhs) > 0 ) {

                --$quotient_value[$q_index];



                $lhs->value = array($quotient_value[$q_index]);

                $lhs = $lhs->multiply($temp);

            }



            $adjust = $this->_array_repeat(0, $q_index);

            $temp_value = array($quotient_value[$q_index]);

            $temp = $temp->multiply($y);

            $temp_value = &$temp->value;

            $temp_value = array_merge($adjust, $temp_value);



            $x = $x->subtract($temp);



            if ($x->compare($zero) < 0) {

                $temp_value = array_merge($adjust, $y_value);

                $x = $x->add($temp);



                --$quotient_value[$q_index];

            }



            $x_max = count($x_value) - 1;

        }



        // unnormalize the remainder

        $x->_rshift($shift);



        $quotient->is_negative = $x_sign != $y_sign;



        // calculate the "common residue", if appropriate

        if ( $x_sign ) {

            $y->_rshift($shift);

            $x = $y->subtract($x);

        }



        return array($this->_normalize($quotient), $this->_normalize($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; $i >= 0; --$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:

     * <code>

     * <?php

     *    include('Math/BigInteger.php');

     *

     *    $a = new Math_BigInteger('10');

     *    $b = new Math_BigInteger('20');

     *    $c = new Math_BigInteger('30');

     *

     *    $c = $a->modPow($b, $c);

     *

     *    echo $c->toString(); // outputs 10

     * ?>

     * </code>

     *

     * @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->bitmask !== false && $this->bitmask->compare($n) < 0 ? $this->bitmask : $n->abs();



        if ($e->compare(new Math_BigInteger()) < 0) {

            $e = $e->abs();



            $temp = $this->modInverse($n);

            if ($temp === false) {

                return false;

            }



            return $this->_normalize($temp->modPow($e, $n));

        }



        switch ( MATH_BIGINTEGER_MODE ) {

            case MATH_BIGINTEGER_MODE_GMP:

                $temp = new Math_BigInteger();

                $temp->value = gmp_powm($this->value, $e->value, $n->value);



                return $this->_normalize($temp);

            case MATH_BIGINTEGER_MODE_BCMATH:

                $temp = new Math_BigInteger();

                $temp->value = bcpowmod($this->value, $e->value, $n->value, 0);



                return $this->_normalize($temp);

        }



        if ( empty($e->value) ) {

            $temp = new Math_BigInteger();

            $temp->value = array(1);

            return $this->_normalize($temp);

        }



        if ( $e->value == array(1) ) {

            list(, $temp) = $this->divide($n);

            return $this->_normalize($temp);

        }



        if ( $e->value == array(2) ) {

            $temp = new Math_BigInteger();

            $temp->value = $this->_square($this->value);

            list(, $temp) = $temp->divide($n);

            return $this->_normalize($temp);

        }



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



        // is the modulo odd?

        if ( $n->value[0] & 1 ) {

            return $this->_normalize($this->_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; $i < count($n->value); ++$i) {

            if ( $n->value[$i] ) {

                $temp = decbin($n->value[$i]);

                $j = strlen($temp) - strrpos($temp, '1') - 1;

                $j+= 26 * $i;

                break;

            }

        }

        // at this point, 2^$j * $n/(2^$j) == $n



        $mod1 = $n->copy();

        $mod1->_rshift($j);

        $mod2 = new Math_BigInteger();

        $mod2->value = array(1);

        $mod2->_lshift($j);



        $part1 = ( $mod1->value != array(1) ) ? $this->_slidingWindow($e, $mod1, MATH_BIGINTEGER_MONTGOMERY) : new Math_BigInteger();

        $part2 = $this->_slidingWindow($e, $mod2, MATH_BIGINTEGER_POWEROF2);



        $y1 = $mod2->modInverse($mod1);

        $y2 = $mod1->modInverse($mod2);



        $result = $part1->multiply($mod2);

        $result = $result->multiply($y1);



        $temp = $part2->multiply($mod1);

        $temp = $temp->multiply($y2);



        $result = $result->add($temp);

        list(, $result) = $result->divide($n);



        return $this->_normalize($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 $this->modPow($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->value;

        $e_length = count($e_value) - 1;

        $e_bits = decbin($e_value[$e_length]);

        for ($i = $e_length - 1; $i >= 0; --$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 > $window_ranges[$i] && $i < count($window_ranges); ++$window_size, ++$i);



        $n_value = $n->value;



        // precompute $this^0 through $this^$window_size

        $powers = array();

        $powers[1] = $this->_prepareReduce($this->value, $n_value, $mode);

        $powers[2] = $this->_squareReduce($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.

        $temp = 1 << ($window_size - 1);

        for ($i = 1; $i < $temp; ++$i) {

            $i2 = $i << 1;

            $powers[$i2 + 1] = $this->_multiplyReduce($powers[$i2 - 1], $powers[2], $n_value, $mode);

        }



        $result = array(1);

        $result = $this->_prepareReduce($result, $n_value, $mode);



        for ($i = 0; $i < $e_length; ) {

            if ( !$e_bits[$i] ) {

                $result = $this->_squareReduce($result, $n_value, $mode);

                ++$i;

            } else {

                for ($j = $window_size - 1; $j > 0; --$j) {

                    if ( !empty($e_bits[$i + $j]) ) {

                        break;

                    }

                }



                for ($k = 0; $k <= $j; ++$k) {// eg. the length of substr($e_bits, $i, $j+1)

                    $result = $this->_squareReduce($result, $n_value, $mode);

                }



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



                $i+=$j + 1;

            }

        }



        $temp = new Math_BigInteger();

        $temp->value = $this->_reduce($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 $this->_montgomery($x, $n);

            case MATH_BIGINTEGER_BARRETT:

                return $this->_barrett($x, $n);

            case MATH_BIGINTEGER_POWEROF2:

                $lhs = new Math_BigInteger();

                $lhs->value = $x;

                $rhs = new Math_BigInteger();

                $rhs->value = $n;

                return $x->_mod2($n);

            case MATH_BIGINTEGER_CLASSIC:

                $lhs = new Math_BigInteger();

                $lhs->value = $x;

                $rhs = new Math_BigInteger();

                $rhs->value = $n;

                list(, $temp) = $lhs->divide($rhs);

                return $temp->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 $this->_prepMontgomery($x, $n);

        }

        return $this->_reduce($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 $this->_montgomeryMultiply($x, $y, $n);

        }

        $temp = $this->_multiply($x, false, $y, false);

        return $this->_reduce($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 $this->_montgomeryMultiply($x, $x, $n);

        }

        return $this->_reduce($this->_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 $x & ($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->value = array(1);

        return $this->bitwise_and($n->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 "folding", 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, "the idea [behind folding] is to find a value x' such that x (mod m) = x' (mod m), with x' being smaller than x."

     *

     * Unfortunately, the "Barrett Reduction with Folding" 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 >> 1) + (x >> 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 => array(),

            MATH_BIGINTEGER_DATA => array()

        );



        $m_length = count($m);



        // if ($this->_compare($n, $this->_square($m)) >= 0) {

        if (count($n) > 2 * $m_length) {

            $lhs = new Math_BigInteger();

            $rhs = new Math_BigInteger();

            $lhs->value = $n;

            $rhs->value = $m;

            list(, $temp) = $lhs->divide($rhs);

            return $temp->value;

        }



        // if (m.length >> 1) + 2 <= m.length then m is too small and n can't be reduced

        if ($m_length < 5) {

            return $this->_regularBarrett($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();

            $lhs_value = &$lhs->value;

            $lhs_value = $this->_array_repeat(0, $m_length + ($m_length >> 1));

            $lhs_value[] = 1;

            $rhs = new Math_BigInteger();

            $rhs->value = $m;



            list($u, $m1) = $lhs->divide($rhs);

            $u = $u->value;

            $m1 = $m1->value;



            $cache[MATH_BIGINTEGER_DATA][] = array(

                'u' => $u, // m.length >> 1 (technically (m.length >> 1) + 1)

                'm1'=> $m1 // m.length

            );

        } else {

            extract($cache[MATH_BIGINTEGER_DATA][$key]);

        }



        $cutoff = $m_length + ($m_length >> 1);

        $lsd = array_slice($n, 0, $cutoff); // m.length + (m.length >> 1)

        $msd = array_slice($n, $cutoff);    // m.length >> 1

        $lsd = $this->_trim($lsd);

        $temp = $this->_multiply($msd, false, $m1, false);

        $n = $this->_add($lsd, false, $temp[MATH_BIGINTEGER_VALUE], false); // m.length + (m.length >> 1) + 1



        if ($m_length & 1) {

            return $this->_regularBarrett($n[MATH_BIGINTEGER_VALUE], $m);

        }



        // (m.length + (m.length >> 1) + 1) - (m.length - 1) == (m.length >> 1) + 2

        $temp = array_slice($n[MATH_BIGINTEGER_VALUE], $m_length - 1);

        // if even: ((m.length >> 1) + 2) + (m.length >> 1) == m.length + 2

        // if odd:  ((m.length >> 1) + 2) + (m.length >> 1) == (m.length - 1) + 2 == m.length + 1

        $temp = $this->_multiply($temp, false, $u, false);

        // if even: (m.length + 2) - ((m.length >> 1) + 1) = m.length - (m.length >> 1) + 1

        // if odd:  (m.length + 1) - ((m.length >> 1) + 1) = m.length - (m.length >> 1)

        $temp = array_slice($temp[MATH_BIGINTEGER_VALUE], ($m_length >> 1) + 1);

        // if even: (m.length - (m.length >> 1) + 1) + m.length = 2 * m.length - (m.length >> 1) + 1

        // if odd:  (m.length - (m.length >> 1)) + m.length     = 2 * m.length - (m.length >> 1)

        $temp = $this->_multiply($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 >> 1) digit

        // number from a m.length + (m.length >> 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 = $this->_subtract($n[MATH_BIGINTEGER_VALUE], false, $temp[MATH_BIGINTEGER_VALUE], false);



        while ($this->_compare($result[MATH_BIGINTEGER_VALUE], $result[MATH_BIGINTEGER_SIGN], $m, false) >= 0) {

            $result = $this->_subtract($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 => array(),

            MATH_BIGINTEGER_DATA => array()

        );



        $n_length = count($n);



        if (count($x) > 2 * $n_length) {

            $lhs = new Math_BigInteger();

            $rhs = new Math_BigInteger();

            $lhs->value = $x;

            $rhs->value = $n;

            list(, $temp) = $lhs->divide($rhs);

            return $temp->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();

            $lhs_value = &$lhs->value;

            $lhs_value = $this->_array_repeat(0, 2 * $n_length);

            $lhs_value[] = 1;

            $rhs = new Math_BigInteger();

            $rhs->value = $n;

            list($temp, ) = $lhs->divide($rhs); // m.length

            $cache[MATH_BIGINTEGER_DATA][] = $temp->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 = $this->_multiply($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 = $this->_multiplyLower($temp, false, $n, false, $n_length + 1);

        // $temp == array_slice($temp->_multiply($temp, false, $n, false)->value, 0, $n_length + 1)



        if ($this->_compare($result, false, $temp[MATH_BIGINTEGER_VALUE], $temp[MATH_BIGINTEGER_SIGN]) < 0) {

            $corrector_value = $this->_array_repeat(0, $n_length + 1);

            $corrector_value[] = 1;

            $result = $this->_add($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 = $this->_subtract($result, false, $temp[MATH_BIGINTEGER_VALUE], $temp[MATH_BIGINTEGER_SIGN]);

        while ($this->_compare($result[MATH_BIGINTEGER_VALUE], $result[MATH_BIGINTEGER_SIGN], $n, false) > 0) {

            $result = $this->_subtract($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->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 => array(),

                MATH_BIGINTEGER_SIGN => false

            );

        }



        if ( $x_length < $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->_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, $product->value[$k] is going

        // to always be 0



        $carry = 0;



        for ($j = 0; $j < $x_length; ++$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 < $stop) {

            $product_value[$j] = $carry;

        }



        // the above for loop is what the previous comment was talking about.  the

        // following for loop is the "one with nested for loops"



        for ($i = 1; $i < $y_length; ++$i) {

            $carry = 0;



            for ($j = 0, $k = $i; $j < $x_length && $k < $stop; ++$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 < $stop) {

                $product_value[$k] = $carry;

            }

        }



        return array(

            MATH_BIGINTEGER_VALUE => $this->_trim($product_value),

            MATH_BIGINTEGER_SIGN => $x_negative != $y_negative

        );

    }



    /**

     * Montgomery Modular Reduction

     *

     * ($x->_prepMontgomery($n))->_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 => array(),

            MATH_BIGINTEGER_DATA => 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][] = $this->_modInverse67108864($n);

        }



        $k = count($n);



        $result = array(MATH_BIGINTEGER_VALUE => $x);



        for ($i = 0; $i < $k; ++$i) {

            $temp = $result[MATH_BIGINTEGER_VALUE][$i] * $cache[MATH_BIGINTEGER_DATA][$key];

            $temp = (int) ($temp - 0x4000000 * ((int) ($temp / 0x4000000)));

            $temp = $this->_regularMultiply(array($temp), $n);

            $temp = array_merge($this->_array_repeat(0, $i), $temp);

            $result = $this->_add($result[MATH_BIGINTEGER_VALUE], false, $temp, false);

        }



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



        if ($this->_compare($result, false, $n, false) >= 0) {

            $result = $this->_subtract($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 = $this->_multiply($x, false, $y, false);

        return $this->_montgomery($temp[MATH_BIGINTEGER_VALUE], $m);



        static $cache = array(

            MATH_BIGINTEGER_VARIABLE => array(),

            MATH_BIGINTEGER_DATA => 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][] = $this->_modInverse67108864($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 => $this->_array_repeat(0, $n + 1));

        for ($i = 0; $i < $n; ++$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 = $this->_add($this->_regularMultiply(array($x[$i]), $y), false, $this->_regularMultiply(array($temp), $m), false);

            $a = $this->_add($a[MATH_BIGINTEGER_VALUE], false, $temp[MATH_BIGINTEGER_VALUE], false);

            $a[MATH_BIGINTEGER_VALUE] = array_slice($a[MATH_BIGINTEGER_VALUE], 1);

        }

        if ($this->_compare($a[MATH_BIGINTEGER_VALUE], false, $m, false) >= 0) {

            $a = $this->_subtract($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();

        $lhs->value = array_merge($this->_array_repeat(0, count($n)), $x);

        $rhs = new Math_BigInteger();

        $rhs->value = $n;



        list(, $temp) = $lhs->divide($rhs);

        return $temp->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 "residue" returned would be the so-called "common residue".  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 & 0x3; // x**-1 mod 2**2

        $result = ($result * (2 - $x * $result)) & 0xF; // x**-1 mod 2**4

        $result = ($result * (2 - ($x & 0xFF) * $result))  & 0xFF; // x**-1 mod 2**8

        $result = ($result * ((2 - ($x & 0xFFFF) * $result) & 0xFFFF)) & 0xFFFF; // x**-1 mod 2**16

        $result = fmod($result * (2 - fmod($x * $result, 0x4000000)), 0x4000000); // x**-1 mod 2**26

        return $result & 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:

     * <code>

     * <?php

     *    include('Math/BigInteger.php');

     *

     *    $a = new Math_BigInteger(30);

     *    $b = new Math_BigInteger(17);

     *

     *    $c = $a->modInverse($b);

     *    echo $c->toString(); // outputs 4

     *

     *    echo "\r\n";

     *

     *    $d = $a->multiply($c);

     *    list(, $d) = $d->divide($b);

     *    echo $d; // outputs 1 (as per the definition of modular inverse)

     * ?>

     * </code>

     *

     * @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();

                $temp->value = gmp_invert($this->value, $n->value);



                return ( $temp->value === false ) ? false : $this->_normalize($temp);

        }



        static $zero, $one;

        if (!isset($zero)) {

            $zero = new Math_BigInteger();

            $one = new Math_BigInteger(1);

        }



        // $x mod $n == $x mod -$n.

        $n = $n->abs();



        if ($this->compare($zero) < 0) {

            $temp = $this->abs();

            $temp = $temp->modInverse($n);

            return $negated === false ? false : $this->_normalize($n->subtract($temp));

        }



        extract($this->extendedGCD($n));



        if (!$gcd->equals($one)) {

            return false;

        }



        $x = $x->compare($zero) < 0 ? $x->add($n) : $x;



        return $this->compare($zero) < 0 ? $this->_normalize($n->subtract($x)) : $this->_normalize($x);

    }



    /**

     * Calculates the greatest common divisor and Bézout's identity.

     *

     * Say you have 693 and 609.  The GCD is 21.  Bézout'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 Bézout's identity - Wikipedia} for more information.

     *

     * Here's an example:

     * <code>

     * <?php

     *    include('Math/BigInteger.php');

     *

     *    $a = new Math_BigInteger(693);

     *    $b = new Math_BigInteger(609);

     *

     *    extract($a->extendedGCD($b));

     *

     *    echo $gcd->toString() . "\r\n"; // outputs 21

     *    echo $a->toString() * $x->toString() + $b->toString() * $y->toString(); // outputs 21

     * ?>

     * </code>

     *

     * @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 "relatively costly multiple-precision divisions".

     */

    function extendedGCD($n)

    {

        switch ( MATH_BIGINTEGER_MODE ) {

            case MATH_BIGINTEGER_MODE_GMP:

                extract(gmp_gcdext($this->value, $n->value));



                return array(

                    'gcd' => $this->_normalize(new Math_BigInteger($g)),

                    'x'   => $this->_normalize(new Math_BigInteger($s)),

                    'y'   => $this->_normalize(new Math_BigInteger($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->value;

                $v = $n->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' => $this->_normalize(new Math_BigInteger($u)),

                    'x'   => $this->_normalize(new Math_BigInteger($a)),

                    'y'   => $this->_normalize(new Math_BigInteger($b))

                );

        }



        $y = $n->copy();

        $x = $this->copy();

        $g = new Math_BigInteger();

        $g->value = array(1);



        while ( !(($x->value[0] & 1)|| ($y->value[0] & 1)) ) {

            $x->_rshift(1);

            $y->_rshift(1);

            $g->_lshift(1);

        }



        $u = $x->copy();

        $v = $y->copy();



        $a = new Math_BigInteger();

        $b = new Math_BigInteger();

        $c = new Math_BigInteger();

        $d = new Math_BigInteger();



        $a->value = $d->value = $g->value = array(1);

        $b->value = $c->value = array();



        while ( !empty($u->value) ) {

            while ( !($u->value[0] & 1) ) {

                $u->_rshift(1);

                if ( (!empty($a->value) && ($a->value[0] & 1)) || (!empty($b->value) && ($b->value[0] & 1)) ) {

                    $a = $a->add($y);

                    $b = $b->subtract($x);

                }

                $a->_rshift(1);

                $b->_rshift(1);

            }



            while ( !($v->value[0] & 1) ) {

                $v->_rshift(1);

                if ( (!empty($d->value) && ($d->value[0] & 1)) || (!empty($c->value) && ($c->value[0] & 1)) ) {

                    $c = $c->add($y);

                    $d = $d->subtract($x);

                }

                $c->_rshift(1);

                $d->_rshift(1);

            }



            if ($u->compare($v) >= 0) {

                $u = $u->subtract($v);

                $a = $a->subtract($c);

                $b = $b->subtract($d);

            } else {

                $v = $v->subtract($u);

                $c = $c->subtract($a);

                $d = $d->subtract($b);

            }

        }



        return array(

            'gcd' => $this->_normalize($g->multiply($v)),

            'x'   => $this->_normalize($c),

            'y'   => $this->_normalize($d)

        );

    }



    /**

     * Calculates the greatest common divisor

     *

     * Say you have 693 and 609.  The GCD is 21.

     *

     * Here's an example:

     * <code>

     * <?php

     *    include('Math/BigInteger.php');

     *

     *    $a = new Math_BigInteger(693);

     *    $b = new Math_BigInteger(609);

     *

     *    $gcd = a->extendedGCD($b);

     *

     *    echo $gcd->toString() . "\r\n"; // outputs 21

     * ?>

     * </code>

     *

     * @param Math_BigInteger $n

     * @return Math_BigInteger

     * @access public

     */

    function gcd($n)

    {

        extract($this->extendedGCD($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:

                $temp->value = gmp_abs($this->value);

                break;

            case MATH_BIGINTEGER_MODE_BCMATH:

                $temp->value = (bccomp($this->value, '0', 0) < 0) ? substr($this->value, 1) : $this->value;

                break;

            default:

                $temp->value = $this->value;

        }



        return $temp;

    }



    /**

     * Compares two numbers.

     *

     * Although one might think !$x->compare($y) means $x != $y, it, in fact, means the opposite.  The reason for this is

     * demonstrated thusly:

     *

     * $x  > $y: $x->compare($y)  > 0

     * $x  < $y: $x->compare($y)  < 0

     * $x == $y: $x->compare($y) == 0

     *

     * Note how the same comparison operator is used.  If you want to test for equality, use $x->equals($y).

     *

     * @param Math_BigInteger $x

     * @return Integer < 0 if $this is less than $x; > 0 if $this is greater than $x, and 0 if they are equal.

     * @access public

     * @see equals()

     * @internal Could return $this->subtract($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->value, $y->value);

            case MATH_BIGINTEGER_MODE_BCMATH:

                return bccomp($this->value, $y->value, 0);

        }



        return $this->_compare($this->value, $this->is_negative, $y->value, $y->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 && $y_negative ) ? 1 : -1;

        }



        $result = $x_negative ? -1 : 1;



        if ( count($x_value) != count($y_value) ) {

            return ( count($x_value) > count($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; $i >= 0; --$i) {

            if ($x_value[$i] != $y_value[$i]) {

                return ( $x_value[$i] > $y_value[$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->value, $x->value) == 0;

            default:

                return $this->value === $x->value && $this->is_negative == $x->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->precision = $bits;

        if ( MATH_BIGINTEGER_MODE != MATH_BIGINTEGER_MODE_BCMATH ) {

            $this->bitmask = new Math_BigInteger(chr((1 << ($bits & 0x7)) - 1) . str_repeat(chr(0xFF), $bits >> 3), 256);

        } else {

            $this->bitmask = new Math_BigInteger(bcpow('2', $bits, 0));

        }



        $temp = $this->_normalize($this);

        $this->value = $temp->value;

    }



    /**

     * Logical And

     *

     * @param Math_BigInteger $x

     * @access public

     * @internal Implemented per a request by Lluis Pamies i Juarez <lluis _a_ pamies.cat>

     * @return Math_BigInteger

     */

    function bitwise_and($x)

    {

        switch ( MATH_BIGINTEGER_MODE ) {

            case MATH_BIGINTEGER_MODE_GMP:

                $temp = new Math_BigInteger();

                $temp->value = gmp_and($this->value, $x->value);



                return $this->_normalize($temp);

            case MATH_BIGINTEGER_MODE_BCMATH:

                $left = $this->toBytes();

                $right = $x->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 $this->_normalize(new Math_BigInteger($left & $right, 256));

        }



        $result = $this->copy();



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



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



        for ($i = 0; $i < $length; ++$i) {

            $result->value[$i] = $result->value[$i] & $x->value[$i];

        }



        return $this->_normalize($result);

    }



    /**

     * Logical Or

     *

     * @param Math_BigInteger $x

     * @access public

     * @internal Implemented per a request by Lluis Pamies i Juarez <lluis _a_ pamies.cat>

     * @return Math_BigInteger

     */

    function bitwise_or($x)

    {

        switch ( MATH_BIGINTEGER_MODE ) {

            case MATH_BIGINTEGER_MODE_GMP:

                $temp = new Math_BigInteger();

                $temp->value = gmp_or($this->value, $x->value);



                return $this->_normalize($temp);

            case MATH_BIGINTEGER_MODE_BCMATH:

                $left = $this->toBytes();

                $right = $x->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 $this->_normalize(new Math_BigInteger($left | $right, 256));

        }



        $length = max(count($this->value), count($x->value));

        $result = $this->copy();

        $result->value = array_pad($result->value, 0, $length);

        $x->value = array_pad($x->value, 0, $length);



        for ($i = 0; $i < $length; ++$i) {

            $result->value[$i] = $this->value[$i] | $x->value[$i];

        }



        return $this->_normalize($result);

    }



    /**

     * Logical Exclusive-Or

     *

     * @param Math_BigInteger $x

     * @access public

     * @internal Implemented per a request by Lluis Pamies i Juarez <lluis _a_ pamies.cat>

     * @return Math_BigInteger

     */

    function bitwise_xor($x)

    {

        switch ( MATH_BIGINTEGER_MODE ) {

            case MATH_BIGINTEGER_MODE_GMP:

                $temp = new Math_BigInteger();

                $temp->value = gmp_xor($this->value, $x->value);



                return $this->_normalize($temp);

            case MATH_BIGINTEGER_MODE_BCMATH:

                $left = $this->toBytes();

                $right = $x->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 $this->_normalize(new Math_BigInteger($left ^ $right, 256));

        }



        $length = max(count($this->value), count($x->value));

        $result = $this->copy();

        $result->value = array_pad($result->value, 0, $length);

        $x->value = array_pad($x->value, 0, $length);



        for ($i = 0; $i < $length; ++$i) {

            $result->value[$i] = $this->value[$i] ^ $x->value[$i];

        }



        return $this->_normalize($result);

    }



    /**

     * Logical Not

     *

     * @access public

     * @internal Implemented per a request by Lluis Pamies i Juarez <lluis _a_ pamies.cat>

     * @return Math_BigInteger

     */

    function bitwise_not()

    {

        // calculuate "not" without regard to $this->precision

        // (will always result in a smaller number.  ie. ~1 isn't 1111 1110 - it's 0)

        $temp = $this->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->precision - $current_bits;

        if ($new_bits <= 0) {

            return $this->_normalize(new Math_BigInteger($temp, 256));

        }



        // generate as many leading 1's as we need to.

        $leading_ones = chr((1 << ($new_bits & 0x7)) - 1) . str_repeat(chr(0xFF), $new_bits >> 3);

        $this->_base256_lshift($leading_ones, $current_bits);



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



        return $this->_normalize(new Math_BigInteger($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');

                }



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



                break;

            case MATH_BIGINTEGER_MODE_BCMATH:

                $temp->value = bcdiv($this->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->value = $this->value;

                $temp->_rshift($shift);

        }



        return $this->_normalize($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');

                }



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



                break;

            case MATH_BIGINTEGER_MODE_BCMATH:

                $temp->value = bcmul($this->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->value = $this->value;

                $temp->_lshift($shift);

        }



        return $this->_normalize($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->toBytes();



        if ($this->precision > 0) {

            $precision = $this->precision;

            if ( MATH_BIGINTEGER_MODE == MATH_BIGINTEGER_MODE_BCMATH ) {

                $mask = $this->bitmask->subtract(new Math_BigInteger(1));

                $mask = $mask->toBytes();

            } else {

                $mask = $this->bitmask->toBytes();

            }

        } else {

            $temp = ord($bits[0]);

            for ($i = 0; $temp >> $i; ++$i);

            $precision = 8 * strlen($bits) - 8 + $i;

            $mask = chr((1 << ($precision & 0x7)) - 1) . str_repeat(chr(0xFF), $precision >> 3);

        }



        if ($shift < 0) {

            $shift+= $precision;

        }

        $shift%= $precision;



        if (!$shift) {

            return $this->copy();

        }



        $left = $this->bitwise_leftShift($shift);

        $left = $left->bitwise_and(new Math_BigInteger($mask, 256));

        $right = $this->bitwise_rightShift($precision - $shift);

        $result = MATH_BIGINTEGER_MODE != MATH_BIGINTEGER_MODE_BCMATH ? $left->bitwise_or($right) : $left->add($right);

        return $this->_normalize($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 $this->bitwise_leftRotate(-$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->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 = $max->compare($min);



        if (!$compare) {

            return $this->_normalize($min);

        } else if ($compare < 0) {

            // if $min is bigger then $max, swap $min and $max

            $temp = $max;

            $max = $min;

            $min = $temp;

        }



        $generator = $this->generator;



        $max = $max->subtract($min);

        $max = ltrim($max->toBytes(), chr(0));

        $size = strlen($max) - 1;

        $random = '';



        $bytes = $size & 1;

        for ($i = 0; $i < $bytes; ++$i) {

            $random.= chr($generator(0, 255));

        }



        $blocks = $size >> 1;

        for ($i = 0; $i < $blocks; ++$i) {

            // mt_rand(-2147483648, 0x7FFFFFFF) always produces -2147483648 on some systems

            $random.= pack('n', $generator(0, 0xFFFF));

        }



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

        if ($temp->compare(new Math_BigInteger(substr($max, 1), 256)) > 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 $this->_normalize($random->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 = $max->compare($min);



        if (!$compare) {

            return $min;

        } else if ($compare < 0) {

            // if $min is bigger then $max, swap $min and $max

            $temp = $max;

            $max = $min;

            $min = $temp;

        }



        // gmp_nextprime() requires PHP 5 >= 5.2.0 per <http://php.net/gmp-nextprime>.

        if ( MATH_BIGINTEGER_MODE == MATH_BIGINTEGER_MODE_GMP && 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 = $this->random($min, $max);



            $x->value = gmp_nextprime($x->value);



            if ($x->compare($max) <= 0) {

                return $x;

            }



            $x->value = gmp_nextprime($min->value);



            if ($x->compare($max) <= 0) {

                return $x;

            }



            return false;

        }



        static $one, $two;

        if (!isset($one)) {

            $one = new Math_BigInteger(1);

            $two = new Math_BigInteger(2);

        }



        $start = time();



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

        if ($x->equals($two)) {

            return $x;

        }



        $x->_make_odd();

        if ($x->compare($max) > 0) {

            // if $x > $max then $max is even and if $min == $max then no prime number exists between the specified range

            if ($min->equals($max)) {

                return false;

            }

            $x = $min->copy();

            $x->_make_odd();

        }



        $initial_x = $x->copy();



        while (true) {

            if ($timeout !== false && time() - $start > $timeout) {

                return false;

            }



            if ($x->isPrime()) {

                return $x;

            }



            $x = $x->add($two);



            if ($x->compare($max) > 0) {

                $x = $min->copy();

                if ($x->equals($two)) {

                    return $x;

                }

                $x->_make_odd();

            }



            if ($x->equals($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->value, 0);

                break;

            case MATH_BIGINTEGER_MODE_BCMATH:

                if ($this->value[strlen($this->value) - 1] % 2 == 0) {

                    $this->value = bcadd($this->value, '1');

                }

                break;

            default:

                $this->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->toBytes());



        if (!$t) {

            // see HAC 4.49 "Note (controlling the error probability)"

                 if ($length >= 163) { $t =  2; } // floor(1300 / 8)

            else if ($length >= 106) { $t =  3; } // floor( 850 / 8)

            else if ($length >= 81 ) { $t =  4; } // floor( 650 / 8)

            else if ($length >= 68 ) { $t =  5; } // floor( 550 / 8)

            else if ($length >= 56 ) { $t =  6; } // floor( 450 / 8)

            else if ($length >= 50 ) { $t =  7; } // floor( 400 / 8)

            else if ($length >= 43 ) { $t =  8; } // floor( 350 / 8)

            else if ($length >= 37 ) { $t =  9; } // floor( 300 / 8)

            else if ($length >= 31 ) { $t = 12; } // floor( 250 / 8)

            else if ($length >= 25 ) { $t = 15; } // floor( 200 / 8)

            else if ($length >= 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->value, $t) != 0;

            case MATH_BIGINTEGER_MODE_BCMATH:

                if ($this->value === '2') {

                    return true;

                }

                if ($this->value[strlen($this->value) - 1] % 2 == 0) {

                    return false;

                }

                break;

            default:

                if ($this->value == array(2)) {

                    return true;

                }

                if (~$this->value[0] & 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; $i < count($primes); ++$i) {

                    $primes[$i] = new Math_BigInteger($primes[$i]);

                }

            }



            $zero = new Math_BigInteger();

            $one = new Math_BigInteger(1);

            $two = new Math_BigInteger(2);

        }



        if ($this->equals($one)) {

            return false;

        }



        // see HAC 4.4.1 "Random search for probable primes"

        if ( MATH_BIGINTEGER_MODE != MATH_BIGINTEGER_MODE_INTERNAL ) {

            foreach ($primes as $prime) {

                list(, $r) = $this->divide($prime);

                if ($r->equals($zero)) {

                    return $this->equals($prime);

                }

            }

        } else {

            $value = $this->value;

            foreach ($primes as $prime) {

                list(, $r) = $this->_divide_digit($value, $prime);

                if (!$r) {

                    return count($value) == 1 && $value[0] == $prime;

                }

            }

        }



        $n   = $this->copy();

        $n_1 = $n->subtract($one);

        $n_2 = $n->subtract($two);



        $r = $n_1->copy();

        $r_value = $r->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 $this->equals($one) check earlier

            while ($r->value[strlen($r->value) - 1] % 2 == 0) {

                $r->value = bcdiv($r->value, '2', 0);

                ++$s;

            }

        } else {

            for ($i = 0, $r_length = count($r_value); $i < $r_length; ++$i) {

                $temp = ~$r_value[$i] & 0xFFFFFF;

                for ($j = 1; ($temp >> $j) & 1; ++$j);

                if ($j != 25) {

                    break;

                }

            }

            $s = 26 * $i + $j - 1;

            $r->_rshift($s);

        }



        for ($i = 0; $i < $t; ++$i) {

            $a = $this->random($two, $n_2);

            $y = $a->modPow($r, $n);



            if (!$y->equals($one) && !$y->equals($n_1)) {

                for ($j = 1; $j < $s && !$y->equals($n_1); ++$j) {

                    $y = $y->modPow($two, $n);

                    if ($y->equals($one)) {

                        return false;

                    }

                }



                if (!$y->equals($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 << $shift;



        $carry = 0;



        for ($i = 0; $i < count($this->value); ++$i) {

            $temp = $this->value[$i] * $shift + $carry;

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

            $this->value[$i] = (int) ($temp - $carry * 0x4000000);

        }



        if ( $carry ) {

            $this->value[] = $carry;

        }



        while ($num_digits--) {

            array_unshift($this->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 << $shift) - 1;



        if ( $num_digits ) {

            $this->value = array_slice($this->value, $num_digits);

        }



        $carry = 0;



        for ($i = count($this->value) - 1; $i >= 0; --$i) {

            $temp = $this->value[$i] >> $shift | $carry;

            $carry = ($this->value[$i] & $carry_mask) << $carry_shift;

            $this->value[$i] = $temp;

        }



        $this->value = $this->_trim($this->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->precision = $this->precision;

        $result->bitmask = $this->bitmask;



        switch ( MATH_BIGINTEGER_MODE ) {

            case MATH_BIGINTEGER_MODE_GMP:

                if (!empty($result->bitmask->value)) {

                    $result->value = gmp_and($result->value, $result->bitmask->value);

                }



                return $result;

            case MATH_BIGINTEGER_MODE_BCMATH:

                if (!empty($result->bitmask->value)) {

                    $result->value = bcmod($result->value, $result->bitmask->value);

                }



                return $result;

        }



        $value = &$result->value;



        if ( !count($value) ) {

            return $result;

        }



        $value = $this->_trim($value);



        if (!empty($result->bitmask->value)) {

            $length = min(count($value), count($this->bitmask->value));

            $value = array_slice($value, 0, $length);



            for ($i = 0; $i < $length; ++$i) {

                $value[$i] = $value[$i] & $this->bitmask->value[$i];

            }

        }



        return $result;

    }



    /**

     * Trim

     *

     * Removes leading zeros

     *

     * @return Math_BigInteger

     * @access private

     */

    function _trim($value)

    {

        for ($i = count($value) - 1; $i >= 0; --$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(&$x, $shift)

    {

        if ($shift == 0) {

            return;

        }



        $num_bytes = $shift >> 3; // eg. floor($shift/8)

        $shift &= 7; // eg. $shift % 8



        $carry = 0;

        for ($i = strlen($x) - 1; $i >= 0; --$i) {

            $temp = ord($x[$i]) << $shift | $carry;

            $x[$i] = chr($temp);

            $carry = $temp >> 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(&$x, $shift)

    {

        if ($shift == 0) {

            $x = ltrim($x, chr(0));

            return '';

        }



        $num_bytes = $shift >> 3; // eg. floor($shift/8)

        $shift &= 7; // eg. $shift % 8



        $remainder = '';

        if ($num_bytes) {

            $start = $num_bytes > strlen($x) ? -strlen($x) : -$num_bytes;

            $remainder = substr($x, $start);

            $x = substr($x, 0, -$num_bytes);

        }



        $carry = 0;

        $carry_shift = 8 - $shift;

        for ($i = 0; $i < strlen($x); ++$i) {

            $temp = (ord($x[$i]) >> $shift) | $carry;

            $carry = (ord($x[$i]) << $carry_shift) & 0xFF;

            $x[$i] = chr($temp);

        }

        $x = ltrim($x, chr(0));



        $remainder = chr($carry >> $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'];

    }

}