openrat-cms

BigInteger.class.php (126017B)

---

 <?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;
     }
 }