Last commit: Sat Oct 31 00:43:29 2020 +0100	Jan Dankert	New: Support for OpenId Connect; Removed: Support for LDAP.


      <?php
      
      /**
       * 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 version 5
      *
      * {@internal (all DocBlock comments regarding implementation - such as the one that follows - refer to the
      * {@link self::MODE_INTERNAL self::MODE_INTERNAL} mode)
      *
      * 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.
      *
      * Numbers are stored in {@link http://en.wikipedia.org/wiki/Endianness little endian} format.  ie.
      * (new \phpseclib\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
      *    $a = new \phpseclib\Math\BigInteger(2);
      *    $b = new \phpseclib\Math\BigInteger(3);
      *
      *    $c = $a->add($b);
      *
      *    echo $c->toString(); // outputs 5
      * ?>
      * </code>
      *
      * @category  Math
      * @package   BigInteger
      * @author    Jim Wigginton <terrafrost@php.net>
      * @copyright 2006 Jim Wigginton
      * @license   http://www.opensource.org/licenses/mit-license.html  MIT License
      */
     
     namespace phpseclib\Math;
     
     use phpseclib\Crypt\Random;
     
     /**
      * Pure-PHP arbitrary precision integer arithmetic library. Supports base-2, base-10, base-16, and base-256
      * numbers.
      *
      * @package BigInteger
      * @author  Jim Wigginton <terrafrost@php.net>
      * @access  public
      */
     class BigInteger
     {
         /**#@+
          * Reduction constants
          *
          * @access private
          * @see BigInteger::_reduce()
          */
         /**
          * @see BigInteger::_montgomery()
          * @see BigInteger::_prepMontgomery()
          */
         const MONTGOMERY = 0;
         /**
          * @see BigInteger::_barrett()
          */
         const BARRETT = 1;
         /**
          * @see BigInteger::_mod2()
          */
         const POWEROF2 = 2;
         /**
          * @see BigInteger::_remainder()
          */
         const CLASSIC = 3;
         /**
          * @see BigInteger::__clone()
          */
         const NONE = 4;
         /**#@-*/
     
         /**#@+
          * Array constants
          *
          * Rather than create a thousands and thousands of new 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[self::VALUE] contains the value.
         */
        const VALUE = 0;
        /**
         * $result[self::SIGN] contains the sign.
         */
        const SIGN = 1;
        /**#@-*/
    
        /**#@+
         * @access private
         * @see BigInteger::_montgomery()
         * @see BigInteger::_barrett()
        */
        /**
         * Cache constants
         *
         * $cache[self::VARIABLE] tells us whether or not the cached data is still valid.
         */
        const VARIABLE = 0;
        /**
         * $cache[self::DATA] contains the cached data.
         */
        const DATA = 1;
        /**#@-*/
    
        /**#@+
         * Mode constants.
         *
         * @access private
         * @see BigInteger::__construct()
        */
        /**
         * To use the pure-PHP implementation
         */
        const MODE_INTERNAL = 1;
        /**
         * To use the BCMath library
         *
         * (if enabled; otherwise, the internal implementation will be used)
         */
        const MODE_BCMATH = 2;
        /**
         * To use the GMP library
         *
         * (if present; otherwise, either the BCMath or the internal implementation will be used)
         */
        const MODE_GMP = 3;
        /**#@-*/
    
        /**
         * Karatsuba Cutoff
         *
         * At what point do we switch between Karatsuba multiplication and schoolbook long multiplication?
         *
         * @access private
         */
        const KARATSUBA_CUTOFF = 25;
    
        /**#@+
         * Static properties used by the pure-PHP implementation.
         *
         * @see __construct()
         */
        protected static $base;
        protected static $baseFull;
        protected static $maxDigit;
        protected static $msb;
    
        /**
         * $max10 in greatest $max10Len satisfying
         * $max10 = 10**$max10Len <= 2**$base.
         */
        protected static $max10;
    
        /**
         * $max10Len in greatest $max10Len satisfying
         * $max10 = 10**$max10Len <= 2**$base.
         */
        protected static $max10Len;
        protected static $maxDigit2;
        /**#@-*/
    
        /**
         * Holds the BigInteger's value.
         *
         * @var array
         * @access private
         */
        var $value;
    
        /**
         * Holds the BigInteger's magnitude.
         *
         * @var bool
         * @access private
         */
        var $is_negative = false;
    
        /**
         * Precision
         *
         * @see self::setPrecision()
         * @access private
         */
        var $precision = -1;
    
        /**
         * Precision Bitmask
         *
         * @see self::setPrecision()
         * @access private
         */
        var $bitmask = false;
    
        /**
         * Mode independent 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 self::__sleep()
         * @see self::__wakeup()
         * @var string
         * @access private
         */
        var $hex;
    
        /**
         * Converts base-2, base-10, base-16, and binary strings (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
         *    $a = new \phpseclib\Math\BigInteger('0x32', 16); // 50 in base-16
         *
         *    echo $a->toString(); // outputs 50
         * ?>
         * </code>
         *
         * @param $x base-10 number or base-$base number if $base set.
         * @param int $base
         * @return \phpseclib\Math\BigInteger
         * @access public
         */
        function __construct($x = 0, $base = 10)
        {
            if (!defined('MATH_BIGINTEGER_MODE')) {
                switch (true) {
                    case extension_loaded('gmp'):
                        define('MATH_BIGINTEGER_MODE', self::MODE_GMP);
                        break;
                    case extension_loaded('bcmath'):
                        define('MATH_BIGINTEGER_MODE', self::MODE_BCMATH);
                        break;
                    default:
                        define('MATH_BIGINTEGER_MODE', self::MODE_INTERNAL);
                }
            }
    
            if (extension_loaded('openssl') && !defined('MATH_BIGINTEGER_OPENSSL_DISABLE') && !defined('MATH_BIGINTEGER_OPENSSL_ENABLED')) {
                // some versions of XAMPP have mismatched versions of OpenSSL which causes it not to work
                $versions = array();
    
                // avoid generating errors (even with suppression) when phpinfo() is disabled (common in production systems)
                if (strpos(ini_get('disable_functions'), 'phpinfo') === false) {
                    ob_start();
                    @phpinfo();
                    $content = ob_get_contents();
                    ob_end_clean();
    
                    preg_match_all('#OpenSSL (Header|Library) Version(.*)#im', $content, $matches);
    
                    if (!empty($matches[1])) {
                        for ($i = 0; $i < count($matches[1]); $i++) {
                            $fullVersion = trim(str_replace('=>', '', strip_tags($matches[2][$i])));
    
                            // Remove letter part in OpenSSL version
                            if (!preg_match('/(\d+\.\d+\.\d+)/i', $fullVersion, $m)) {
                                $versions[$matches[1][$i]] = $fullVersion;
                            } else {
                                $versions[$matches[1][$i]] = $m[0];
                            }
                        }
                    }
                }
    
                // it doesn't appear that OpenSSL versions were reported upon until PHP 5.3+
                switch (true) {
                    case !isset($versions['Header']):
                    case !isset($versions['Library']):
                    case $versions['Header'] == $versions['Library']:
                    case version_compare($versions['Header'], '1.0.0') >= 0 && version_compare($versions['Library'], '1.0.0') >= 0:
                        define('MATH_BIGINTEGER_OPENSSL_ENABLED', true);
                        break;
                    default:
                        define('MATH_BIGINTEGER_OPENSSL_DISABLE', true);
                }
            }
    
            if (!defined('PHP_INT_SIZE')) {
                define('PHP_INT_SIZE', 4);
            }
    
            if (empty(self::$base) && MATH_BIGINTEGER_MODE == self::MODE_INTERNAL) {
                switch (PHP_INT_SIZE) {
                    case 8: // use 64-bit integers if int size is 8 bytes
                        self::$base      = 31;
                        self::$baseFull  = 0x80000000;
                        self::$maxDigit  = 0x7FFFFFFF;
                        self::$msb       = 0x40000000;
                        self::$max10     = 1000000000;
                        self::$max10Len  = 9;
                        self::$maxDigit2 = pow(2, 62);
                        break;
                    //case 4: // use 64-bit floats if int size is 4 bytes
                    default:
                        self::$base      = 26;
                        self::$baseFull  = 0x4000000;
                        self::$maxDigit  = 0x3FFFFFF;
                        self::$msb       = 0x2000000;
                        self::$max10     = 10000000;
                        self::$max10Len  = 7;
                        self::$maxDigit2 = pow(2, 52); // pow() prevents truncation
                }
            }
    
            switch (MATH_BIGINTEGER_MODE) {
                case self::MODE_GMP:
                    switch (true) {
                        case is_resource($x) && get_resource_type($x) == 'GMP integer':
                        // PHP 5.6 switched GMP from using resources to objects
                        case $x instanceof \GMP:
                            $this->value = $x;
                            return;
                    }
                    $this->value = gmp_init(0);
                    break;
                case self::MODE_BCMATH:
                    $this->value = '0';
                    break;
                default:
                    $this->value = array();
            }
    
            // '0' counts as empty() but when the base is 256 '0' is equal to ord('0') or 48
            // '0' is the only value like this per http://php.net/empty
            if (empty($x) && (abs($base) != 256 || $x !== '0')) {
                return;
            }
    
            switch ($base) {
                case -256:
                    if (ord($x[0]) & 0x80) {
                        $x = ~$x;
                        $this->is_negative = true;
                    }
                case 256:
                    switch (MATH_BIGINTEGER_MODE) {
                        case self::MODE_GMP:
                            $this->value = function_exists('gmp_import') ?
                                gmp_import($x) :
                                gmp_init('0x' . bin2hex($x));
                            if ($this->is_negative) {
                                $this->value = gmp_neg($this->value);
                            }
                            break;
                        case self::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($x, self::$base));
                            }
                    }
    
                    if ($this->is_negative) {
                        if (MATH_BIGINTEGER_MODE != self::MODE_INTERNAL) {
                            $this->is_negative = false;
                        }
                        $temp = $this->add(new static('-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 self::MODE_GMP:
                            $temp = $this->is_negative ? '-0x' . $x : '0x' . $x;
                            $this->value = gmp_init($temp);
                            $this->is_negative = false;
                            break;
                        case self::MODE_BCMATH:
                            $x = (strlen($x) & 1) ? '0' . $x : $x;
                            $temp = new static(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 static(pack('H*', $x), 256);
                            $this->value = $temp->value;
                    }
    
                    if ($is_negative) {
                        $temp = $this->add(new static('-1'));
                        $this->value = $temp->value;
                    }
                    break;
                case 10:
                case -10:
                    // (?<!^)(?:-).*: find any -'s that aren't at the beginning and then any characters that follow that
                    // (?<=^|-)0*: find any 0's that are preceded by the start of the string or by a - (ie. octals)
                    // [^-0-9].*: find any non-numeric characters and then any characters that follow that
                    $x = preg_replace('#(?<!^)(?:-).*|(?<=^|-)0*|[^-0-9].*#', '', $x);
                    if (!strlen($x) || $x == '-') {
                        $x = '0';
                    }
    
                    switch (MATH_BIGINTEGER_MODE) {
                        case self::MODE_GMP:
                            $this->value = gmp_init($x);
                            break;
                        case self::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 = $x === '-' ? '0' : (string) $x;
                            break;
                        default:
                            $temp = new static();
    
                            $multiplier = new static();
                            $multiplier->value = array(self::$max10);
    
                            if ($x[0] == '-') {
                                $this->is_negative = true;
                                $x = substr($x, 1);
                            }
    
                            $x = str_pad($x, strlen($x) + ((self::$max10Len - 1) * strlen($x)) % self::$max10Len, 0, STR_PAD_LEFT);
                            while (strlen($x)) {
                                $temp = $temp->multiply($multiplier);
                                $temp = $temp->add(new static($this->_int2bytes(substr($x, 0, self::$max10Len)), 256));
                                $x = substr($x, self::$max10Len);
                            }
    
                            $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;
                    }
    
                    $temp = new static($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
         *    $a = new \phpseclib\Math\BigInteger('65');
         *
         *    echo $a->toBytes(); // outputs chr(65)
         * ?>
         * </code>
         *
         * @param bool $twos_compliment
         * @return string
         * @access public
         * @internal Converts a base-2**26 number to base-2**8
         */
        function toBytes($twos_compliment = false)
        {
            if ($twos_compliment) {
                $comparison = $this->compare(new static());
                if ($comparison == 0) {
                    return $this->precision > 0 ? str_repeat(chr(0), ($this->precision + 1) >> 3) : '';
                }
    
                $temp = $comparison < 0 ? $this->add(new static(1)) : $this->copy();
                $bytes = $temp->toBytes();
    
                if (!strlen($bytes)) { // eg. if the number we're trying to convert is -1
                    $bytes = chr(0);
                }
    
                if ($this->precision <= 0 && (ord($bytes[0]) & 0x80)) {
                    $bytes = chr(0) . $bytes;
                }
    
                return $comparison < 0 ? ~$bytes : $bytes;
            }
    
            switch (MATH_BIGINTEGER_MODE) {
                case self::MODE_GMP:
                    if (gmp_cmp($this->value, gmp_init(0)) == 0) {
                        return $this->precision > 0 ? str_repeat(chr(0), ($this->precision + 1) >> 3) : '';
                    }
    
                    if (function_exists('gmp_export')) {
                        $temp = gmp_export($this->value);
                    } else {
                        $temp = gmp_strval(gmp_abs($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($temp, chr(0));
                case self::MODE_BCMATH:
                    if ($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, self::$base);
                $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
         *    $a = new \phpseclib\Math\BigInteger('65');
         *
         *    echo $a->toHex(); // outputs '41'
         * ?>
         * </code>
         *
         * @param bool $twos_compliment
         * @return string
         * @access public
         * @internal Converts a base-2**26 number to base-2**8
         */
        function toHex($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
         *    $a = new \phpseclib\Math\BigInteger('65');
         *
         *    echo $a->toBits(); // outputs '1000001'
         * ?>
         * </code>
         *
         * @param bool $twos_compliment
         * @return string
         * @access public
         * @internal Converts a base-2**26 number to base-2**2
         */
        function toBits($twos_compliment = false)
        {
            $hex = $this->toHex($twos_compliment);
            $bits = '';
            for ($i = strlen($hex) - 8, $start = strlen($hex) & 7; $i >= $start; $i-=8) {
                $bits = str_pad(decbin(hexdec(substr($hex, $i, 8))), 32, '0', STR_PAD_LEFT) . $bits;
            }
            if ($start) { // hexdec('') == 0
                $bits = str_pad(decbin(hexdec(substr($hex, 0, $start))), 8, '0', STR_PAD_LEFT) . $bits;
            }
            $result = $this->precision > 0 ? substr($bits, -$this->precision) : ltrim($bits, '0');
    
            if ($twos_compliment && $this->compare(new static()) > 0 && $this->precision <= 0) {
                return '0' . $result;
            }
    
            return $result;
        }
    
        /**
         * Converts a BigInteger to a base-10 number.
         *
         * Here's an example:
         * <code>
         * <?php
         *    $a = new \phpseclib\Math\BigInteger('50');
         *
         *    echo $a->toString(); // outputs 50
         * ?>
         * </code>
         *
         * @return string
         * @access public
         * @internal Converts a base-2**26 number to base-10**7 (which is pretty much base-10)
         */
        function toString()
        {
            switch (MATH_BIGINTEGER_MODE) {
                case self::MODE_GMP:
                    return gmp_strval($this->value);
                case self::MODE_BCMATH:
                    if ($this->value === '0') {
                        return '0';
                    }
    
                    return ltrim($this->value, '0');
            }
    
            if (!count($this->value)) {
                return '0';
            }
    
            $temp = $this->copy();
            $temp->bitmask = false;
            $temp->is_negative = false;
    
            $divisor = new static();
            $divisor->value = array(self::$max10);
            $result = '';
            while (count($temp->value)) {
                list($temp, $mod) = $temp->divide($divisor);
                $result = str_pad(isset($mod->value[0]) ? $mod->value[0] : '', self::$max10Len, '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 self::__clone()
         * @return \phpseclib\Math\BigInteger
         */
        function copy()
        {
            $temp = new static();
            $temp->value = $this->value;
            $temp->is_negative = $this->is_negative;
            $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 BigInteger::__toString() directly in PHP5, you cannot call 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 BigInteger::copy(), instead.
         *
         * @access public
         * @see self::copy()
         * @return \phpseclib\Math\BigInteger
         */
        function __clone()
        {
            return $this->copy();
        }
    
        /**
         *  __sleep() magic method
         *
         * Will be called, automatically, when serialize() is called on a BigInteger object.
         *
         * @see self::__wakeup()
         * @access public
         */
        function __sleep()
        {
            $this->hex = $this->toHex(true);
            $vars = array('hex');
            if ($this->precision > 0) {
                $vars[] = 'precision';
            }
            return $vars;
        }
    
        /**
         *  __wakeup() magic method
         *
         * Will be called, automatically, when unserialize() is called on a BigInteger object.
         *
         * @see self::__sleep()
         * @access public
         */
        function __wakeup()
        {
            $temp = new static($this->hex, -16);
            $this->value = $temp->value;
            $this->is_negative = $temp->is_negative;
            if ($this->precision > 0) {
                // recalculate $this->bitmask
                $this->setPrecision($this->precision);
            }
        }
    
        /**
         *  __debugInfo() magic method
         *
         * Will be called, automatically, when print_r() or var_dump() are called
         *
         * @access public
         */
        function __debugInfo()
        {
            $opts = array();
            switch (MATH_BIGINTEGER_MODE) {
                case self::MODE_GMP:
                    $engine = 'gmp';
                    break;
                case self::MODE_BCMATH:
                    $engine = 'bcmath';
                    break;
                case self::MODE_INTERNAL:
                    $engine = 'internal';
                    $opts[] = PHP_INT_SIZE == 8 ? '64-bit' : '32-bit';
            }
            if (MATH_BIGINTEGER_MODE != self::MODE_GMP && defined('MATH_BIGINTEGER_OPENSSL_ENABLED')) {
                $opts[] = 'OpenSSL';
            }
            if (!empty($opts)) {
                $engine.= ' (' . implode('.', $opts) . ')';
            }
            return array(
                'value' => '0x' . $this->toHex(true),
                'engine' => $engine
            );
        }
    
        /**
         * Adds two BigIntegers.
         *
         * Here's an example:
         * <code>
         * <?php
         *    $a = new \phpseclib\Math\BigInteger('10');
         *    $b = new \phpseclib\Math\BigInteger('20');
         *
         *    $c = $a->add($b);
         *
         *    echo $c->toString(); // outputs 30
         * ?>
         * </code>
         *
         * @param \phpseclib\Math\BigInteger $y
         * @return \phpseclib\Math\BigInteger
         * @access public
         * @internal Performs base-2**52 addition
         */
        function add($y)
        {
            switch (MATH_BIGINTEGER_MODE) {
                case self::MODE_GMP:
                    $temp = new static();
                    $temp->value = gmp_add($this->value, $y->value);
    
                    return $this->_normalize($temp);
                case self::MODE_BCMATH:
                    $temp = new static();
                    $temp->value = bcadd($this->value, $y->value, 0);
    
                    return $this->_normalize($temp);
            }
    
            $temp = $this->_add($this->value, $this->is_negative, $y->value, $y->is_negative);
    
            $result = new static();
            $result->value = $temp[self::VALUE];
            $result->is_negative = $temp[self::SIGN];
    
            return $this->_normalize($result);
        }
    
        /**
         * Performs addition.
         *
         * @param array $x_value
         * @param bool $x_negative
         * @param array $y_value
         * @param bool $y_negative
         * @return array
         * @access private
         */
        function _add($x_value, $x_negative, $y_value, $y_negative)
        {
            $x_size = count($x_value);
            $y_size = count($y_value);
    
            if ($x_size == 0) {
                return array(
                    self::VALUE => $y_value,
                    self::SIGN => $y_negative
                );
            } elseif ($y_size == 0) {
                return array(
                    self::VALUE => $x_value,
                    self::SIGN => $x_negative
                );
            }
    
            // subtract, if appropriate
            if ($x_negative != $y_negative) {
                if ($x_value == $y_value) {
                    return array(
                        self::VALUE => array(),
                        self::SIGN => false
                    );
                }
    
                $temp = $this->_subtract($x_value, false, $y_value, false);
                $temp[self::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;
            }
    
            $value[count($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] * self::$baseFull + $x_value[$i] + $y_value[$j] * self::$baseFull + $y_value[$i] + $carry;
                $carry = $sum >= self::$maxDigit2; // eg. floor($sum / 2**52); only possible values (in any base) are 0 and 1
                $sum = $carry ? $sum - self::$maxDigit2 : $sum;
    
                $temp = self::$base === 26 ? intval($sum / 0x4000000) : ($sum >> 31);
    
                $value[$i] = (int) ($sum - self::$baseFull * $temp); // eg. a faster alternative to fmod($sum, 0x4000000)
                $value[$j] = $temp;
            }
    
            if ($j == $size) { // ie. if $y_size is odd
                $sum = $x_value[$i] + $y_value[$i] + $carry;
                $carry = $sum >= self::$baseFull;
                $value[$i] = $carry ? $sum - self::$baseFull : $sum;
                ++$i; // ie. let $i = $j since we've just done $value[$i]
            }
    
            if ($carry) {
                for (; $value[$i] == self::$maxDigit; ++$i) {
                    $value[$i] = 0;
                }
                ++$value[$i];
            }
    
            return array(
                self::VALUE => $this->_trim($value),
                self::SIGN => $x_negative
            );
        }
    
        /**
         * Subtracts two BigIntegers.
         *
         * Here's an example:
         * <code>
         * <?php
         *    $a = new \phpseclib\Math\BigInteger('10');
         *    $b = new \phpseclib\Math\BigInteger('20');
         *
         *    $c = $a->subtract($b);
         *
         *    echo $c->toString(); // outputs -10
         * ?>
         * </code>
         *
         * @param \phpseclib\Math\BigInteger $y
        * @return \phpseclib\Math\BigInteger
        * @access public
        * @internal Performs base-2**52 subtraction
        */
       function subtract($y)
       {
           switch (MATH_BIGINTEGER_MODE) {
               case self::MODE_GMP:
                   $temp = new static();
                   $temp->value = gmp_sub($this->value, $y->value);
   
                   return $this->_normalize($temp);
               case self::MODE_BCMATH:
                   $temp = new static();
                   $temp->value = bcsub($this->value, $y->value, 0);
   
                   return $this->_normalize($temp);
           }
   
           $temp = $this->_subtract($this->value, $this->is_negative, $y->value, $y->is_negative);
   
           $result = new static();
           $result->value = $temp[self::VALUE];
           $result->is_negative = $temp[self::SIGN];
   
           return $this->_normalize($result);
       }
   
       /**
        * Performs subtraction.
        *
        * @param array $x_value
        * @param bool $x_negative
        * @param array $y_value
        * @param bool $y_negative
        * @return array
        * @access private
        */
       function _subtract($x_value, $x_negative, $y_value, $y_negative)
       {
           $x_size = count($x_value);
           $y_size = count($y_value);
   
           if ($x_size == 0) {
               return array(
                   self::VALUE => $y_value,
                   self::SIGN => !$y_negative
               );
           } elseif ($y_size == 0) {
               return array(
                   self::VALUE => $x_value,
                   self::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[self::SIGN] = $x_negative;
   
               return $temp;
           }
   
           $diff = $this->_compare($x_value, $x_negative, $y_value, $y_negative);
   
           if (!$diff) {
               return array(
                   self::VALUE => array(),
                   self::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] * self::$baseFull + $x_value[$i] - $y_value[$j] * self::$baseFull - $y_value[$i] - $carry;
               $carry = $sum < 0; // eg. floor($sum / 2**52); only possible values (in any base) are 0 and 1
               $sum = $carry ? $sum + self::$maxDigit2 : $sum;
   
               $temp = self::$base === 26 ? intval($sum / 0x4000000) : ($sum >> 31);
   
               $x_value[$i] = (int) ($sum - self::$baseFull * $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 ? $sum + self::$baseFull : $sum;
               ++$i;
           }
   
           if ($carry) {
               for (; !$x_value[$i]; ++$i) {
                   $x_value[$i] = self::$maxDigit;
               }
               --$x_value[$i];
           }
   
           return array(
               self::VALUE => $this->_trim($x_value),
               self::SIGN => $x_negative
           );
       }
   
       /**
        * Multiplies two BigIntegers
        *
        * Here's an example:
        * <code>
        * <?php
        *    $a = new \phpseclib\Math\BigInteger('10');
        *    $b = new \phpseclib\Math\BigInteger('20');
        *
        *    $c = $a->multiply($b);
        *
        *    echo $c->toString(); // outputs 200
        * ?>
        * </code>
        *
        * @param \phpseclib\Math\BigInteger $x
        * @return \phpseclib\Math\BigInteger
        * @access public
        */
       function multiply($x)
       {
           switch (MATH_BIGINTEGER_MODE) {
               case self::MODE_GMP:
                   $temp = new static();
                   $temp->value = gmp_mul($this->value, $x->value);
   
                   return $this->_normalize($temp);
               case self::MODE_BCMATH:
                   $temp = new static();
                   $temp->value = bcmul($this->value, $x->value, 0);
   
                   return $this->_normalize($temp);
           }
   
           $temp = $this->_multiply($this->value, $this->is_negative, $x->value, $x->is_negative);
   
           $product = new static();
           $product->value = $temp[self::VALUE];
           $product->is_negative = $temp[self::SIGN];
   
           return $this->_normalize($product);
       }
   
       /**
        * Performs multiplication.
        *
        * @param array $x_value
        * @param bool $x_negative
        * @param array $y_value
        * @param bool $y_negative
        * @return array
        * @access private
        */
       function _multiply($x_value, $x_negative, $y_value, $y_negative)
       {
           //if ( $x_value == $y_value ) {
           //    return array(
           //        self::VALUE => $this->_square($x_value),
           //        self::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(
                   self::VALUE => array(),
                   self::SIGN => false
               );
           }
   
           return array(
               self::VALUE => min($x_length, $y_length) < 2 * self::KARATSUBA_CUTOFF ?
                   $this->_trim($this->_regularMultiply($x_value, $y_value)) :
                   $this->_trim($this->_karatsuba($x_value, $y_value)),
               self::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
               $carry = self::$base === 26 ? intval($temp / 0x4000000) : ($temp >> 31);
               $product_value[$j] = (int) ($temp - self::$baseFull * $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;
                   $carry = self::$base === 26 ? intval($temp / 0x4000000) : ($temp >> 31);
                   $product_value[$k] = (int) ($temp - self::$baseFull * $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 < self::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[self::VALUE], $temp[self::VALUE]);
           $temp = $this->_add($z2, false, $z0, false);
           $z1 = $this->_subtract($z1, false, $temp[self::VALUE], false);
   
           $z2 = array_merge(array_fill(0, 2 * $m, 0), $z2);
           $z1[self::VALUE] = array_merge(array_fill(0, $m, 0), $z1[self::VALUE]);
   
           $xy = $this->_add($z2, false, $z1[self::VALUE], $z1[self::SIGN]);
           $xy = $this->_add($xy[self::VALUE], $xy[self::SIGN], $z0, false);
   
           return $xy[self::VALUE];
       }
   
       /**
        * Performs squaring
        *
        * @param array $x
        * @return array
        * @access private
        */
       function _square($x = false)
       {
           return count($x) < 2 * self::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];
               $carry = self::$base === 26 ? intval($temp / 0x4000000) : ($temp >> 31);
               $square_value[$i2] = (int) ($temp - self::$baseFull * $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;
                   $carry = self::$base === 26 ? intval($temp / 0x4000000) : ($temp >> 31);
                   $square_value[$k] = (int) ($temp - self::$baseFull * $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 < self::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[self::VALUE]);
           $temp = $this->_add($z2, false, $z0, false);
           $z1 = $this->_subtract($z1, false, $temp[self::VALUE], false);
   
           $z2 = array_merge(array_fill(0, 2 * $m, 0), $z2);
           $z1[self::VALUE] = array_merge(array_fill(0, $m, 0), $z1[self::VALUE]);
   
           $xx = $this->_add($z2, false, $z1[self::VALUE], $z1[self::SIGN]);
           $xx = $this->_add($xx[self::VALUE], $xx[self::SIGN], $z0, false);
   
           return $xx[self::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
        *    $a = new \phpseclib\Math\BigInteger('10');
        *    $b = new \phpseclib\Math\BigInteger('20');
        *
        *    list($quotient, $remainder) = $a->divide($b);
        *
        *    echo $quotient->toString(); // outputs 0
        *    echo "\r\n";
        *    echo $remainder->toString(); // outputs 10
        * ?>
        * </code>
        *
        * @param \phpseclib\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 self::MODE_GMP:
                   $quotient = new static();
                   $remainder = new static();
   
                   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 self::MODE_BCMATH:
                   $quotient = new static();
                   $remainder = new static();
   
                   $quotient->value = bcdiv($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 static();
               $remainder = new static();
               $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 static();
           }
   
           $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 static();
               $temp->value = array(1);
               $temp->is_negative = $x_sign != $y_sign;
               return array($this->_normalize($temp), $this->_normalize(new static()));
           }
   
           if ($diff < 0) {
               // if $x is negative, "add" $y.
               if ($x_sign) {
                   $x = $y->subtract($x);
               }
               return array($this->_normalize(new static()), $this->_normalize($x));
           }
   
           // normalize $x and $y as described in HAC 14.23 / 14.24
           $msb = $y->value[count($y->value) - 1];
           for ($shift = 0; !($msb & self::$msb); ++$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 static();
           $quotient_value = &$quotient->value;
           $quotient_value = $this->_array_repeat(0, $x_max - $y_max + 1);
   
           static $temp, $lhs, $rhs;
           if (!isset($temp)) {
               $temp = new static();
               $lhs =  new static();
               $rhs =  new static();
           }
           $temp_value = &$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] = self::$maxDigit;
               } else {
                   $quotient_value[$q_index] = $this->_safe_divide(
                       $x_window[0] * self::$baseFull + $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;
               if (count($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) {
               $temp = self::$baseFull * $carry + $dividend[$i];
               $result[$i] = $this->_safe_divide($temp, $divisor);
               $carry = (int) ($temp - $divisor * $result[$i]);
           }
   
           return array($result, $carry);
       }
   
       /**
        * Performs modular exponentiation.
        *
        * Here's an example:
        * <code>
        * <?php
        *    $a = new \phpseclib\Math\BigInteger('10');
        *    $b = new \phpseclib\Math\BigInteger('20');
        *    $c = new \phpseclib\Math\BigInteger('30');
        *
        *    $c = $a->modPow($b, $c);
        *
        *    echo $c->toString(); // outputs 10
        * ?>
        * </code>
        *
        * @param \phpseclib\Math\BigInteger $e
        * @param \phpseclib\Math\BigInteger $n
        * @return \phpseclib\Math\BigInteger
        * @access public
        * @internal The most naive approach to modular exponentiation has very unreasonable requirements, and
        *    and although the approach involving repeated squaring does vastly better, it, too, is impractical
        *    for our purposes.  The reason being that division - by far the most complicated and time-consuming
        *    of the basic operations (eg. +,-,*,/) - occurs multiple times within it.
        *
        *    Modular reductions resolve this issue.  Although an individual modular reduction takes more time
        *    then an individual division, when performed in succession (with the same modulo), they're a lot faster.
        *
        *    The two most commonly used modular reductions are Barrett and Montgomery reduction.  Montgomery reduction,
        *    although faster, only works when the gcd of the modulo and of the base being used is 1.  In RSA, when the
        *    base is a power of two, the modulo - a product of two primes - is always going to have a gcd of 1 (because
        *    the product of two odd numbers is odd), but what about when RSA isn't used?
        *
        *    In contrast, Barrett reduction has no such constraint.  As such, some bigint implementations perform a
        *    Barrett reduction after every operation in the modpow function.  Others perform Barrett reductions when the
        *    modulo is even and Montgomery reductions when the modulo is odd.  BigInteger.java's modPow method, however,
        *    uses a trick involving the Chinese Remainder Theorem to factor the even modulo into two numbers - one odd and
        *    the other, a power of two - and recombine them, later.  This is the method that this modPow function uses.
        *    {@link http://islab.oregonstate.edu/papers/j34monex.pdf Montgomery Reduction with Even Modulus} elaborates.
        */
       function modPow($e, $n)
       {
           $n = $this->bitmask !== false && $this->bitmask->compare($n) < 0 ? $this->bitmask : $n->abs();
   
           if ($e->compare(new static()) < 0) {
               $e = $e->abs();
   
               $temp = $this->modInverse($n);
               if ($temp === false) {
                   return false;
               }
   
               return $this->_normalize($temp->modPow($e, $n));
           }
   
           if (MATH_BIGINTEGER_MODE == self::MODE_GMP) {
               $temp = new static();
               $temp->value = gmp_powm($this->value, $e->value, $n->value);
   
               return $this->_normalize($temp);
           }
   
           if ($this->compare(new static()) < 0 || $this->compare($n) > 0) {
               list(, $temp) = $this->divide($n);
               return $temp->modPow($e, $n);
           }
   
           if (defined('MATH_BIGINTEGER_OPENSSL_ENABLED')) {
               $components = array(
                   'modulus' => $n->toBytes(true),
                   'publicExponent' => $e->toBytes(true)
               );
   
               $components = array(
                   'modulus' => pack('Ca*a*', 2, $this->_encodeASN1Length(strlen($components['modulus'])), $components['modulus']),
                   'publicExponent' => pack('Ca*a*', 2, $this->_encodeASN1Length(strlen($components['publicExponent'])), $components['publicExponent'])
               );
   
               $RSAPublicKey = pack(
                   'Ca*a*a*',
                   48,
                   $this->_encodeASN1Length(strlen($components['modulus']) + strlen($components['publicExponent'])),
                   $components['modulus'],
                   $components['publicExponent']
               );
   
               $rsaOID = pack('H*', '300d06092a864886f70d0101010500'); // hex version of MA0GCSqGSIb3DQEBAQUA
               $RSAPublicKey = chr(0) . $RSAPublicKey;
               $RSAPublicKey = chr(3) . $this->_encodeASN1Length(strlen($RSAPublicKey)) . $RSAPublicKey;
   
               $encapsulated = pack(
                   'Ca*a*',
                   48,
                   $this->_encodeASN1Length(strlen($rsaOID . $RSAPublicKey)),
                   $rsaOID . $RSAPublicKey
               );
   
               $RSAPublicKey = "-----BEGIN PUBLIC KEY-----\r\n" .
                                chunk_split(base64_encode($encapsulated)) .
                                '-----END PUBLIC KEY-----';
   
               $plaintext = str_pad($this->toBytes(), strlen($n->toBytes(true)) - 1, "\0", STR_PAD_LEFT);
   
               if (openssl_public_encrypt($plaintext, $result, $RSAPublicKey, OPENSSL_NO_PADDING)) {
                   return new static($result, 256);
               }
           }
   
           if (MATH_BIGINTEGER_MODE == self::MODE_BCMATH) {
               $temp = new static();
               $temp->value = bcpowmod($this->value, $e->value, $n->value, 0);
   
               return $this->_normalize($temp);
           }
   
           if (empty($e->value)) {
               $temp = new static();
               $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 static();
               $temp->value = $this->_square($this->value);
               list(, $temp) = $temp->divide($n);
               return $this->_normalize($temp);
           }
   
           return $this->_normalize($this->_slidingWindow($e, $n, self::BARRETT));
   
           // the following code, although not callable, can be run independently of the above code
           // although the above code performed better in my benchmarks the following could might
           // perform better under different circumstances. in lieu of deleting it it's just been
           // made uncallable
   
           // is the modulo odd?
           if ($n->value[0] & 1) {
               return $this->_normalize($this->_slidingWindow($e, $n, self::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 static();
           $mod2->value = array(1);
           $mod2->_lshift($j);
   
           $part1 = ($mod1->value != array(1)) ? $this->_slidingWindow($e, $mod1, self::MONTGOMERY) : new static();
           $part2 = $this->_slidingWindow($e, $mod2, self::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 modPow().
        *
        * @param \phpseclib\Math\BigInteger $e
        * @param \phpseclib\Math\BigInteger $n
        * @return \phpseclib\Math\BigInteger
        * @access public
        */
       function powMod($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 \phpseclib\Math\BigInteger $e
        * @param \phpseclib\Math\BigInteger $n
        * @param int $mode
        * @return \phpseclib\Math\BigInteger
        * @access private
        */
       function _slidingWindow($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[$i]), self::$base, '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; $i < count($window_ranges) && $e_length > $window_ranges[$i]; ++$window_size, ++$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;
                       }
                   }
   
                   // eg. the length of substr($e_bits, $i, $j + 1)
                   for ($k = 0; $k <= $j; ++$k) {
                       $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 static();
           $temp->value = $this->_reduce($result, $n_value, $mode);
   
           return $temp;
       }
   
       /**
        * Modular reduction
        *
        * For most $modes this will return the remainder.
        *
        * @see self::_slidingWindow()
        * @access private
        * @param array $x
        * @param array $n
        * @param int $mode
        * @return array
        */
       function _reduce($x, $n, $mode)
       {
           switch ($mode) {
               case self::MONTGOMERY:
                   return $this->_montgomery($x, $n);
               case self::BARRETT:
                   return $this->_barrett($x, $n);
               case self::POWEROF2:
                   $lhs = new static();
                   $lhs->value = $x;
                   $rhs = new static();
                   $rhs->value = $n;
                   return $x->_mod2($n);
               case self::CLASSIC:
                   $lhs = new static();
                   $lhs->value = $x;
                   $rhs = new static();
                   $rhs->value = $n;
                   list(, $temp) = $lhs->divide($rhs);
                   return $temp->value;
               case self::NONE:
                   return $x;
               default:
                   // an invalid $mode was provided
           }
       }
   
       /**
        * Modular reduction preperation
        *
        * @see self::_slidingWindow()
        * @access private
        * @param array $x
        * @param array $n
        * @param int $mode
        * @return array
        */
       function _prepareReduce($x, $n, $mode)
       {
           if ($mode == self::MONTGOMERY) {
               return $this->_prepMontgomery($x, $n);
           }
           return $this->_reduce($x, $n, $mode);
       }
   
       /**
        * Modular multiply
        *
        * @see self::_slidingWindow()
        * @access private
        * @param array $x
        * @param array $y
        * @param array $n
        * @param int $mode
        * @return array
        */
       function _multiplyReduce($x, $y, $n, $mode)
       {
           if ($mode == self::MONTGOMERY) {
               return $this->_montgomeryMultiply($x, $y, $n);
           }
           $temp = $this->_multiply($x, false, $y, false);
           return $this->_reduce($temp[self::VALUE], $n, $mode);
       }
   
       /**
        * Modular square
        *
        * @see self::_slidingWindow()
        * @access private
        * @param array $x
        * @param array $n
        * @param int $mode
        * @return array
        */
       function _squareReduce($x, $n, $mode)
       {
           if ($mode == self::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 self::_slidingWindow()
        * @access private
        * @param \phpseclib\Math\BigInteger
        * @return \phpseclib\Math\BigInteger
        */
       function _mod2($n)
       {
           $temp = new static();
           $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 self::_slidingWindow()
        * @access private
        * @param array $n
        * @param array $m
        * @return array
        */
       function _barrett($n, $m)
       {
           static $cache = array(
               self::VARIABLE => array(),
               self::DATA => array()
           );
   
           $m_length = count($m);
   
           // if ($this->_compare($n, $this->_square($m)) >= 0) {
           if (count($n) > 2 * $m_length) {
               $lhs = new static();
               $rhs = new static();
               $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[self::VARIABLE])) === false) {
               $key = count($cache[self::VARIABLE]);
               $cache[self::VARIABLE][] = $m;
   
               $lhs = new static();
               $lhs_value = &$lhs->value;
               $lhs_value = $this->_array_repeat(0, $m_length + ($m_length >> 1));
               $lhs_value[] = 1;
               $rhs = new static();
               $rhs->value = $m;
   
               list($u, $m1) = $lhs->divide($rhs);
               $u = $u->value;
               $m1 = $m1->value;
   
               $cache[self::DATA][] = array(
                   'u' => $u, // m.length >> 1 (technically (m.length >> 1) + 1)
                   'm1'=> $m1 // m.length
               );
           } else {
               extract($cache[self::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[self::VALUE], false); // m.length + (m.length >> 1) + 1
   
           if ($m_length & 1) {
               return $this->_regularBarrett($n[self::VALUE], $m);
           }
   
           // (m.length + (m.length >> 1) + 1) - (m.length - 1) == (m.length >> 1) + 2
           $temp = array_slice($n[self::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[self::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[self::VALUE], false, $temp[self::VALUE], false);
   
           while ($this->_compare($result[self::VALUE], $result[self::SIGN], $m, false) >= 0) {
               $result = $this->_subtract($result[self::VALUE], $result[self::SIGN], $m, false);
           }
   
           return $result[self::VALUE];
       }
   
       /**
        * (Regular) Barrett Modular Reduction
        *
        * For numbers with more than four digits BigInteger::_barrett() is faster.  The difference between that and this
        * is that this function does not fold the denominator into a smaller form.
        *
        * @see self::_slidingWindow()
        * @access private
        * @param array $x
        * @param array $n
        * @return array
        */
       function _regularBarrett($x, $n)
       {
           static $cache = array(
               self::VARIABLE => array(),
               self::DATA => array()
           );
   
           $n_length = count($n);
   
           if (count($x) > 2 * $n_length) {
               $lhs = new static();
               $rhs = new static();
               $lhs->value = $x;
               $rhs->value = $n;
               list(, $temp) = $lhs->divide($rhs);
               return $temp->value;
           }
   
           if (($key = array_search($n, $cache[self::VARIABLE])) === false) {
               $key = count($cache[self::VARIABLE]);
               $cache[self::VARIABLE][] = $n;
               $lhs = new static();
               $lhs_value = &$lhs->value;
               $lhs_value = $this->_array_repeat(0, 2 * $n_length);
               $lhs_value[] = 1;
               $rhs = new static();
               $rhs->value = $n;
               list($temp, ) = $lhs->divide($rhs); // m.length
               $cache[self::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, $cache[self::DATA][$key], false);
           // (2 * m.length + 1) - (m.length - 1) = m.length + 2
           $temp = array_slice($temp[self::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[self::VALUE], $temp[self::SIGN]) < 0) {
               $corrector_value = $this->_array_repeat(0, $n_length + 1);
               $corrector_value[count($corrector_value)] = 1;
               $result = $this->_add($result, false, $corrector_value, false);
               $result = $result[self::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[self::VALUE], $temp[self::SIGN]);
           while ($this->_compare($result[self::VALUE], $result[self::SIGN], $n, false) > 0) {
               $result = $this->_subtract($result[self::VALUE], $result[self::SIGN], $n, false);
           }
   
           return $result[self::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 self::_regularBarrett()
        * @param array $x_value
        * @param bool $x_negative
        * @param array $y_value
        * @param bool $y_negative
        * @param int $stop
        * @return array
        * @access private
        */
       function _multiplyLower($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(
                   self::VALUE => array(),
                   self::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
               $carry = self::$base === 26 ? intval($temp / 0x4000000) : ($temp >> 31);
               $product_value[$j] = (int) ($temp - self::$baseFull * $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;
                   $carry = self::$base === 26 ? intval($temp / 0x4000000) : ($temp >> 31);
                   $product_value[$k] = (int) ($temp - self::$baseFull * $carry);
               }
   
               if ($k < $stop) {
                   $product_value[$k] = $carry;
               }
           }
   
           return array(
               self::VALUE => $this->_trim($product_value),
               self::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 self::_prepMontgomery()
        * @see self::_slidingWindow()
        * @access private
        * @param array $x
        * @param array $n
        * @return array
        */
       function _montgomery($x, $n)
       {
           static $cache = array(
               self::VARIABLE => array(),
               self::DATA => array()
           );
   
           if (($key = array_search($n, $cache[self::VARIABLE])) === false) {
               $key = count($cache[self::VARIABLE]);
               $cache[self::VARIABLE][] = $x;
               $cache[self::DATA][] = $this->_modInverse67108864($n);
           }
   
           $k = count($n);
   
           $result = array(self::VALUE => $x);
   
           for ($i = 0; $i < $k; ++$i) {
               $temp = $result[self::VALUE][$i] * $cache[self::DATA][$key];
               $temp = $temp - self::$baseFull * (self::$base === 26 ? intval($temp / 0x4000000) : ($temp >> 31));
               $temp = $this->_regularMultiply(array($temp), $n);
               $temp = array_merge($this->_array_repeat(0, $i), $temp);
               $result = $this->_add($result[self::VALUE], false, $temp, false);
           }
   
           $result[self::VALUE] = array_slice($result[self::VALUE], $k);
   
           if ($this->_compare($result, false, $n, false) >= 0) {
               $result = $this->_subtract($result[self::VALUE], false, $n, false);
           }
   
           return $result[self::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 self::_prepMontgomery()
        * @see self::_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[self::VALUE], $m);
   
           // the following code, although not callable, can be run independently of the above code
           // although the above code performed better in my benchmarks the following could might
           // perform better under different circumstances. in lieu of deleting it it's just been
           // made uncallable
   
           static $cache = array(
               self::VARIABLE => array(),
               self::DATA => array()
           );
   
           if (($key = array_search($m, $cache[self::VARIABLE])) === false) {
               $key = count($cache[self::VARIABLE]);
               $cache[self::VARIABLE][] = $m;
               $cache[self::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(self::VALUE => $this->_array_repeat(0, $n + 1));
           for ($i = 0; $i < $n; ++$i) {
               $temp = $a[self::VALUE][0] + $x[$i] * $y[0];
               $temp = $temp - self::$baseFull * (self::$base === 26 ? intval($temp / 0x4000000) : ($temp >> 31));
               $temp = $temp * $cache[self::DATA][$key];
               $temp = $temp - self::$baseFull * (self::$base === 26 ? intval($temp / 0x4000000) : ($temp >> 31));
               $temp = $this->_add($this->_regularMultiply(array($x[$i]), $y), false, $this->_regularMultiply(array($temp), $m), false);
               $a = $this->_add($a[self::VALUE], false, $temp[self::VALUE], false);
               $a[self::VALUE] = array_slice($a[self::VALUE], 1);
           }
           if ($this->_compare($a[self::VALUE], false, $m, false) >= 0) {
               $a = $this->_subtract($a[self::VALUE], false, $m, false);
           }
           return $a[self::VALUE];
       }
   
       /**
        * Prepare a number for use in Montgomery Modular Reductions
        *
        * @see self::_montgomery()
        * @see self::_slidingWindow()
        * @access private
        * @param array $x
        * @param array $n
        * @return array
        */
       function _prepMontgomery($x, $n)
       {
           $lhs = new static();
           $lhs->value = array_merge($this->_array_repeat(0, count($n)), $x);
           $rhs = new static();
           $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 self::_montgomery()
        * @access private
        * @param array $x
        * @return int
        */
       function _modInverse67108864($x) // 2**26 == 67,108,864
       {
           $x = -$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 * $result, self::$baseFull)), self::$baseFull); // x**-1 mod 2**26
           return $result & self::$maxDigit;
       }
   
       /**
        * 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
        *    $a = new \phpseclib\Math\BigInteger(30);
        *    $b = new \phpseclib\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 \phpseclib\Math\BigInteger $n
        * @return \phpseclib\Math\BigInteger|false
        * @access public
        * @internal See {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=21 HAC 14.64} for more information.
        */
       function modInverse($n)
       {
           switch (MATH_BIGINTEGER_MODE) {
               case self::MODE_GMP:
                   $temp = new static();
                   $temp->value = gmp_invert($this->value, $n->value);
   
                   return ($temp->value === false) ? false : $this->_normalize($temp);
           }
   
           static $zero, $one;
           if (!isset($zero)) {
               $zero = new static();
               $one = new static(1);
           }
   
           // $x mod -$n == $x mod $n.
           $n = $n->abs();
   
           if ($this->compare($zero) < 0) {
               $temp = $this->abs();
               $temp = $temp->modInverse($n);
               return $this->_normalize($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 Bezout's identity.
        *
        * Say you have 693 and 609.  The GCD is 21.  Bezout'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 dependent upon which mode is in use.  See
        * {@link http://en.wikipedia.org/wiki/B%C3%A9zout%27s_identity Bezout's identity - Wikipedia} for more information.
        *
        * Here's an example:
        * <code>
        * <?php
        *    $a = new \phpseclib\Math\BigInteger(693);
        *    $b = new \phpseclib\Math\BigInteger(609);
        *
        *    extract($a->extendedGCD($b));
        *
        *    echo $gcd->toString() . "\r\n"; // outputs 21
        *    echo $a->toString() * $x->toString() + $b->toString() * $y->toString(); // outputs 21
        * ?>
        * </code>
        *
        * @param \phpseclib\Math\BigInteger $n
        * @return \phpseclib\Math\BigInteger
        * @access public
        * @internal Calculates the GCD using the binary xGCD algorithim described in
        *    {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=19 HAC 14.61}.  As the text above 14.61 notes,
        *    the more traditional algorithim requires "relatively costly multiple-precision divisions".
        */
       function extendedGCD($n)
       {
           switch (MATH_BIGINTEGER_MODE) {
               case self::MODE_GMP:
                   extract(gmp_gcdext($this->value, $n->value));
   
                   return array(
                       'gcd' => $this->_normalize(new static($g)),
                       'x'   => $this->_normalize(new static($s)),
                       'y'   => $this->_normalize(new static($t))
                   );
               case self::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' => $this->_normalize(new static($u)),
                       'x'   => $this->_normalize(new static($a)),
                       'y'   => $this->_normalize(new static($b))
                   );
           }
   
           $y = $n->copy();
           $x = $this->copy();
           $g = new static();
           $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 static();
           $b = new static();
           $c = new static();
           $d = new static();
   
           $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
        *    $a = new \phpseclib\Math\BigInteger(693);
        *    $b = new \phpseclib\Math\BigInteger(609);
        *
        *    $gcd = a->extendedGCD($b);
        *
        *    echo $gcd->toString() . "\r\n"; // outputs 21
        * ?>
        * </code>
        *
        * @param \phpseclib\Math\BigInteger $n
        * @return \phpseclib\Math\BigInteger
        * @access public
        */
       function gcd($n)
       {
           extract($this->extendedGCD($n));
           return $gcd;
       }
   
       /**
        * Absolute value.
        *
        * @return \phpseclib\Math\BigInteger
        * @access public
        */
       function abs()
       {
           $temp = new static();
   
           switch (MATH_BIGINTEGER_MODE) {
               case self::MODE_GMP:
                   $temp->value = gmp_abs($this->value);
                   break;
               case self::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 \phpseclib\Math\BigInteger $y
        * @return int < 0 if $this is less than $y; > 0 if $this is greater than $y, and 0 if they are equal.
        * @access public
        * @see self::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 self::MODE_GMP:
                   $r = gmp_cmp($this->value, $y->value);
                   if ($r < -1) {
                       $r = -1;
                   }
                   if ($r > 1) {
                       $r = 1;
                   }
                   return $r;
               case self::MODE_BCMATH:
                   return bccomp($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 bool $x_negative
        * @param array $y_value
        * @param bool $y_negative
        * @return int
        * @see self::compare()
        * @access private
        */
       function _compare($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 BigInteger::compare()
        *
        * @param \phpseclib\Math\BigInteger $x
        * @return bool
        * @access public
        * @see self::compare()
        */
       function equals($x)
       {
           switch (MATH_BIGINTEGER_MODE) {
               case self::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 int $bits
        * @access public
        */
       function setPrecision($bits)
       {
           $this->precision = $bits;
           if (MATH_BIGINTEGER_MODE != self::MODE_BCMATH) {
               $this->bitmask = new static(chr((1 << ($bits & 0x7)) - 1) . str_repeat(chr(0xFF), $bits >> 3), 256);
           } else {
               $this->bitmask = new static(bcpow('2', $bits, 0));
           }
   
           $temp = $this->_normalize($this);
           $this->value = $temp->value;
       }
   
       /**
        * Logical And
        *
        * @param \phpseclib\Math\BigInteger $x
        * @access public
        * @internal Implemented per a request by Lluis Pamies i Juarez <lluis _a_ pamies.cat>
        * @return \phpseclib\Math\BigInteger
        */
       function bitwise_and($x)
       {
           switch (MATH_BIGINTEGER_MODE) {
               case self::MODE_GMP:
                   $temp = new static();
                   $temp->value = gmp_and($this->value, $x->value);
   
                   return $this->_normalize($temp);
               case self::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 $this->_normalize(new static($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]&= $x->value[$i];
           }
   
           return $this->_normalize($result);
       }
   
       /**
        * Logical Or
        *
        * @param \phpseclib\Math\BigInteger $x
        * @access public
        * @internal Implemented per a request by Lluis Pamies i Juarez <lluis _a_ pamies.cat>
        * @return \phpseclib\Math\BigInteger
        */
       function bitwise_or($x)
       {
           switch (MATH_BIGINTEGER_MODE) {
               case self::MODE_GMP:
                   $temp = new static();
                   $temp->value = gmp_or($this->value, $x->value);
   
                   return $this->_normalize($temp);
               case self::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 $this->_normalize(new static($left | $right, 256));
           }
   
           $length = max(count($this->value), count($x->value));
           $result = $this->copy();
           $result->value = array_pad($result->value, $length, 0);
           $x->value = array_pad($x->value, $length, 0);
   
           for ($i = 0; $i < $length; ++$i) {
               $result->value[$i]|= $x->value[$i];
           }
   
           return $this->_normalize($result);
       }
   
       /**
        * Logical Exclusive-Or
        *
        * @param \phpseclib\Math\BigInteger $x
        * @access public
        * @internal Implemented per a request by Lluis Pamies i Juarez <lluis _a_ pamies.cat>
        * @return \phpseclib\Math\BigInteger
        */
       function bitwise_xor($x)
       {
           switch (MATH_BIGINTEGER_MODE) {
               case self::MODE_GMP:
                   $temp = new static();
                   $temp->value = gmp_xor(gmp_abs($this->value), gmp_abs($x->value));
                   return $this->_normalize($temp);
               case self::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 $this->_normalize(new static($left ^ $right, 256));
           }
   
           $length = max(count($this->value), count($x->value));
           $result = $this->copy();
           $result->is_negative = false;
           $result->value = array_pad($result->value, $length, 0);
           $x->value = array_pad($x->value, $length, 0);
   
           for ($i = 0; $i < $length; ++$i) {
               $result->value[$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 \phpseclib\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 = $this->toBytes();
           if ($temp == '') {
               return $this->_normalize(new static());
           }
           $pre_msb = decbin(ord($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 $this->_normalize(new static($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, strlen($leading_ones), chr(0), STR_PAD_LEFT);
   
           return $this->_normalize(new static($leading_ones | $temp, 256));
       }
   
       /**
        * Logical Right Shift
        *
        * Shifts BigInteger's by $shift bits, effectively dividing by 2**$shift.
        *
        * @param int $shift
        * @return \phpseclib\Math\BigInteger
        * @access public
        * @internal The only version that yields any speed increases is the internal version.
        */
       function bitwise_rightShift($shift)
       {
           $temp = new static();
   
           switch (MATH_BIGINTEGER_MODE) {
               case self::MODE_GMP:
                   static $two;
   
                   if (!isset($two)) {
                       $two = gmp_init('2');
                   }
   
                   $temp->value = gmp_div_q($this->value, gmp_pow($two, $shift));
   
                   break;
               case self::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 int $shift
        * @return \phpseclib\Math\BigInteger
        * @access public
        * @internal The only version that yields any speed increases is the internal version.
        */
       function bitwise_leftShift($shift)
       {
           $temp = new static();
   
           switch (MATH_BIGINTEGER_MODE) {
               case self::MODE_GMP:
                   static $two;
   
                   if (!isset($two)) {
                       $two = gmp_init('2');
                   }
   
                   $temp->value = gmp_mul($this->value, gmp_pow($two, $shift));
   
                   break;
               case self::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 int $shift
        * @return \phpseclib\Math\BigInteger
        * @access public
        */
       function bitwise_leftRotate($shift)
       {
           $bits = $this->toBytes();
   
           if ($this->precision > 0) {
               $precision = $this->precision;
               if (MATH_BIGINTEGER_MODE == self::MODE_BCMATH) {
                   $mask = $this->bitmask->subtract(new static(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 = $left->bitwise_and(new static($mask, 256));
           $right = $this->bitwise_rightShift($precision - $shift);
           $result = MATH_BIGINTEGER_MODE != self::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 int $shift
        * @return \phpseclib\Math\BigInteger
        * @access public
        */
       function bitwise_rightRotate($shift)
       {
           return $this->bitwise_leftRotate(-$shift);
       }
   
       /**
        * Generates a random BigInteger
        *
        * Byte length is equal to $length. Uses \phpseclib\Crypt\Random if it's loaded and mt_rand if it's not.
        *
        * @param int $length
        * @return \phpseclib\Math\BigInteger
        * @access private
        */
       function _random_number_helper($size)
       {
           if (class_exists('\phpseclib\Crypt\Random')) {
               $random = Random::string($size);
           } else {
               $random = '';
   
               if ($size & 1) {
                   $random.= chr(mt_rand(0, 255));
               }
   
               $blocks = $size >> 1;
               for ($i = 0; $i < $blocks; ++$i) {
                   // mt_rand(-2147483648, 0x7FFFFFFF) always produces -2147483648 on some systems
                   $random.= pack('n', mt_rand(0, 0xFFFF));
               }
           }
   
           return new static($random, 256);
       }
   
       /**
        * Generate a random number
        *
        * Returns a random number between $min and $max where $min and $max
        * can be defined using one of the two methods:
        *
        * $min->random($max)
        * $max->random($min)
        *
        * @param \phpseclib\Math\BigInteger $arg1
        * @param \phpseclib\Math\BigInteger $arg2
        * @return \phpseclib\Math\BigInteger
        * @access public
        * @internal The API for creating random numbers used to be $a->random($min, $max), where $a was a BigInteger object.
        *           That method is still supported for BC purposes.
        */
       function random($arg1, $arg2 = false)
       {
           if ($arg1 === false) {
               return false;
           }
   
           if ($arg2 === false) {
               $max = $arg1;
               $min = $this;
           } else {
               $min = $arg1;
               $max = $arg2;
           }
   
           $compare = $max->compare($min);
   
           if (!$compare) {
               return $this->_normalize($min);
           } elseif ($compare < 0) {
               // if $min is bigger then $max, swap $min and $max
               $temp = $max;
               $max = $min;
               $min = $temp;
           }
   
           static $one;
           if (!isset($one)) {
               $one = new static(1);
           }
   
           $max = $max->subtract($min->subtract($one));
           $size = strlen(ltrim($max->toBytes(), chr(0)));
   
           /*
               doing $random % $max doesn't work because some numbers will be more likely to occur than others.
               eg. if $max is 140 and $random's max is 255 then that'd mean both $random = 5 and $random = 145
               would produce 5 whereas the only value of random that could produce 139 would be 139. ie.
               not all numbers would be equally likely. some would be more likely than others.
   
               creating a whole new random number until you find one that is within the range doesn't work
               because, for sufficiently small ranges, the likelihood that you'd get a number within that range
               would be pretty small. eg. with $random's max being 255 and if your $max being 1 the probability
               would be pretty high that $random would be greater than $max.
   
               phpseclib works around this using the technique described here:
   
               http://crypto.stackexchange.com/questions/5708/creating-a-small-number-from-a-cryptographically-secure-random-string
           */
           $random_max = new static(chr(1) . str_repeat("\0", $size), 256);
           $random = $this->_random_number_helper($size);
   
           list($max_multiple) = $random_max->divide($max);
           $max_multiple = $max_multiple->multiply($max);
   
           while ($random->compare($max_multiple) >= 0) {
               $random = $random->subtract($max_multiple);
               $random_max = $random_max->subtract($max_multiple);
               $random = $random->bitwise_leftShift(8);
               $random = $random->add($this->_random_number_helper(1));
               $random_max = $random_max->bitwise_leftShift(8);
               list($max_multiple) = $random_max->divide($max);
               $max_multiple = $max_multiple->multiply($max);
           }
           list(, $random) = $random->divide($max);
   
           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 \phpseclib\Math\BigInteger $arg1
        * @param \phpseclib\Math\BigInteger $arg2
        * @param int $timeout
        * @return Math_BigInteger|false
        * @access public
        * @internal See {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap4.pdf#page=15 HAC 4.44}.
        */
       function randomPrime($arg1, $arg2 = false, $timeout = false)
       {
           if ($arg1 === false) {
               return false;
           }
   
           if ($arg2 === false) {
               $max = $arg1;
               $min = $this;
           } else {
               $min = $arg1;
               $max = $arg2;
           }
   
           $compare = $max->compare($min);
   
           if (!$compare) {
               return $min->isPrime() ? $min : false;
           } elseif ($compare < 0) {
               // if $min is bigger then $max, swap $min and $max
               $temp = $max;
               $max = $min;
               $min = $temp;
           }
   
           static $one, $two;
           if (!isset($one)) {
               $one = new static(1);
               $two = new static(2);
           }
   
           $start = time();
   
           $x = $this->random($min, $max);
   
           // gmp_nextprime() requires PHP 5 >= 5.2.0 per <http://php.net/gmp-nextprime>.
           if (MATH_BIGINTEGER_MODE == self::MODE_GMP && extension_loaded('gmp')) {
               $p = new static();
               $p->value = gmp_nextprime($x->value);
   
               if ($p->compare($max) <= 0) {
                   return $p;
               }
   
               if (!$min->equals($x)) {
                   $x = $x->subtract($one);
               }
   
               return $x->randomPrime($min, $x);
           }
   
           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 self::randomPrime()
        * @access private
        */
       function _make_odd()
       {
           switch (MATH_BIGINTEGER_MODE) {
               case self::MODE_GMP:
                   gmp_setbit($this->value, 0);
                   break;
               case self::MODE_BCMATH:
                   if ($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.  BigInteger::randomPrime() can be distributed across multiple pageloads
        * on a website instead of just one.
        *
        * @param \phpseclib\Math\BigInteger $t
        * @return bool
        * @access public
        * @internal Uses the
        *     {@link http://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test Miller-Rabin primality test}.  See
        *     {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap4.pdf#page=8 HAC 4.24}.
        */
       function isPrime($t = false)
       {
           $length = strlen($this->toBytes());
   
           if (!$t) {
               // see HAC 4.49 "Note (controlling the error probability)"
               // @codingStandardsIgnoreStart
                    if ($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; }
               // @codingStandardsIgnoreEnd
           }
   
           // ie. gmp_testbit($this, 0)
           // ie. isEven() or !isOdd()
           switch (MATH_BIGINTEGER_MODE) {
               case self::MODE_GMP:
                   return gmp_prob_prime($this->value, $t) != 0;
               case self::MODE_BCMATH:
                   if ($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 != self::MODE_INTERNAL) {
                   for ($i = 0; $i < count($primes); ++$i) {
                       $primes[$i] = new static($primes[$i]);
                   }
               }
   
               $zero = new static();
               $one = new static(1);
               $two = new static(2);
           }
   
           if ($this->equals($one)) {
               return false;
           }
   
           // see HAC 4.4.1 "Random search for probable primes"
           if (MATH_BIGINTEGER_MODE != self::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 == self::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;
               $r->_rshift($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 int $shift
        * @access private
        */
       function _lshift($shift)
       {
           if ($shift == 0) {
               return;
           }
   
           $num_digits = (int) ($shift / self::$base);
           $shift %= self::$base;
           $shift = 1 << $shift;
   
           $carry = 0;
   
           for ($i = 0; $i < count($this->value); ++$i) {
               $temp = $this->value[$i] * $shift + $carry;
               $carry = self::$base === 26 ? intval($temp / 0x4000000) : ($temp >> 31);
               $this->value[$i] = (int) ($temp - $carry * self::$baseFull);
           }
   
           if ($carry) {
               $this->value[count($this->value)] = $carry;
           }
   
           while ($num_digits--) {
               array_unshift($this->value, 0);
           }
       }
   
       /**
        * Logical Right Shift
        *
        * Shifts BigInteger's by $shift bits.
        *
        * @param int $shift
        * @access private
        */
       function _rshift($shift)
       {
           if ($shift == 0) {
               return;
           }
   
           $num_digits = (int) ($shift / self::$base);
           $shift %= self::$base;
           $carry_shift = self::$base - $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 \phpseclib\Math\BigInteger
        * @return \phpseclib\Math\BigInteger
        * @see self::_trim()
        * @access private
        */
       function _normalize($result)
       {
           $result->precision = $this->precision;
           $result->bitmask = $this->bitmask;
   
           switch (MATH_BIGINTEGER_MODE) {
               case self::MODE_GMP:
                   if ($this->bitmask !== false) {
                       $flip = gmp_cmp($result->value, gmp_init(0)) < 0;
                       if ($flip) {
                           $result->value = gmp_neg($result->value);
                       }
                       $result->value = gmp_and($result->value, $result->bitmask->value);
                       if ($flip) {
                           $result->value = gmp_neg($result->value);
                       }
                   }
   
                   return $result;
               case self::MODE_BCMATH:
                   if (!empty($result->bitmask->value)) {
                       $result->value = bcmod($result->value, $result->bitmask->value);
                   }
   
                   return $result;
           }
   
           $value = &$result->value;
   
           if (!count($value)) {
               $result->is_negative = false;
               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
        *
        * @param array $value
        * @return \phpseclib\Math\BigInteger
        * @access private
        */
       function _trim($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 int $x
        * @return string
        * @access private
        */
       function _int2bytes($x)
       {
           return ltrim(pack('N', $x), chr(0));
       }
   
       /**
        * Converts bytes to 32-bit integers
        *
        * @param string $x
        * @return int
        * @access private
        */
       function _bytes2int($x)
       {
           $temp = unpack('Nint', str_pad($x, 4, chr(0), STR_PAD_LEFT));
           return $temp['int'];
       }
   
       /**
        * DER-encode an integer
        *
        * The ability to DER-encode integers is needed to create RSA public keys for use with OpenSSL
        *
        * @see self::modPow()
        * @access private
        * @param int $length
        * @return string
        */
       function _encodeASN1Length($length)
       {
           if ($length <= 0x7F) {
               return chr($length);
           }
   
           $temp = ltrim(pack('N', $length), chr(0));
           return pack('Ca*', 0x80 | strlen($temp), $temp);
       }
   
       /**
        * Single digit division
        *
        * Even if int64 is being used the division operator will return a float64 value
        * if the dividend is not evenly divisible by the divisor. Since a float64 doesn't
        * have the precision of int64 this is a problem so, when int64 is being used,
        * we'll guarantee that the dividend is divisible by first subtracting the remainder.
        *
        * @access private
        * @param int $x
        * @param int $y
        * @return int
        */
       function _safe_divide($x, $y)
       {
           if (self::$base === 26) {
               return (int) ($x / $y);
           }
   
           // self::$base === 31
           return ($x - ($x % $y)) / $y;
       }
   }


Download modules/phpseclib/Math/BigInteger.class.php

History
Sat, 31 Oct 2020 00:43:29 +0100	Jan Dankert	New: Support for OpenId Connect; Removed: Support for LDAP.