<?php
/* vim: set expandtab tabstop=4 shiftwidth=4 softtabstop=4: */

/**
 * Pure-PHP arbitrary precision integer arithmetic library.
 *
 * Supports base-2, base-10, base-16, and base-256 numbers.  Uses the GMP or BCMath extensions, if available,
 * and an internal implementation, otherwise.
 *
 * PHP versions 4 and 5
 *
 * {@internal (all DocBlock comments regarding implementation - such as the one that follows - refer to the
 * {@link MATH_BIGINTEGER_MODE_INTERNAL MATH_BIGINTEGER_MODE_INTERNAL} mode)
 *
 * Math_BigInteger uses base-2**26 to perform operations such as multiplication and division and
 * base-2**52 (ie. two base 2**26 digits) to perform addition and subtraction.  Because the largest possible
 * value when multiplying two base-2**26 numbers together is a base-2**52 number, double precision floating
 * point numbers - numbers that should be supported on most hardware and whose significand is 53 bits - are
 * used.  As a consequence, bitwise operators such as >> and << cannot be used, nor can the modulo operator %,
 * which only supports integers.  Although this fact will slow this library down, the fact that such a high
 * base is being used should more than compensate.
 *
 * When PHP version 6 is officially released, we'll be able to use 64-bit integers.  This should, once again,
 * allow bitwise operators, and will increase the maximum possible base to 2**31 (or 2**62 for addition /
 * subtraction).
 *
 * Numbers are stored in {@link http://en.wikipedia.org/wiki/Endianness little endian} format.  ie.
 * (new Math_BigInteger(pow(2, 26)))->value = array(0, 1)
 *
 * Useful resources are as follows:
 *
 *  - {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf Handbook of Applied Cryptography (HAC)}
 *  - {@link http://math.libtomcrypt.com/files/tommath.pdf Multi-Precision Math (MPM)}
 *  - Java's BigInteger classes.  See /j2se/src/share/classes/java/math in jdk-1_5_0-src-jrl.zip
 *
 * Here's an example of how to use this library:
 * <code>
 * <?php
 *    include('Math/BigInteger.php');
 *
 *    $a = new Math_BigInteger(2);
 *    $b = new Math_BigInteger(3);
 *
 *    $c = $a->add($b);
 *
 *    echo $c->toString(); // outputs 5
 * ?>
 * </code>
 *
 * LICENSE: This library is free software; you can redistribute it and/or
 * modify it under the terms of the GNU Lesser General Public
 * License as published by the Free Software Foundation; either
 * version 2.1 of the License, or (at your option) any later version.
 *
 * This library is distributed in the hope that it will be useful,
 * but WITHOUT ANY WARRANTY; without even the implied warranty of
 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
 * Lesser General Public License for more details.
 *
 * You should have received a copy of the GNU Lesser General Public
 * License along with this library; if not, write to the Free Software
 * Foundation, Inc., 59 Temple Place, Suite 330, Boston,
 * MA  02111-1307  USA
 *
 * @category   Math
 * @package    Math_BigInteger
 * @author     Jim Wigginton <terrafrost@php.net>
 * @copyright  MMVI Jim Wigginton
 * @license    http://www.gnu.org/licenses/lgpl.txt
 * @version    $Id: BigInteger.php,v 1.1 2011/03/03 08:46:14 eldy Exp $
 * @link       http://pear.php.net/package/Math_BigInteger
 */

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

		$temp = $this->copy();

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

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

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

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

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

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

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

		$temp = $this->copy();
		$temp->is_negative = false;

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

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

		return $result;
	}

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

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

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

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

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

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

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

				return $this->_normalize($temp);
		}

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

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

		return $this->_normalize($result);
	}

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

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

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

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

			return $temp;
		}

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

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

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

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

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

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

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

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

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

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

				return $this->_normalize($temp);
		}

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

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

		return $this->_normalize($result);
	}

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

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

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

			return $temp;
		}

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

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

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

			$x_negative = !$x_negative;

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

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

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

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

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

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

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

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

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

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

				return $this->_normalize($temp);
		}

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

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

		return $this->_normalize($product);
	}

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

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

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

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

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

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

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

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

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

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

		$carry = 0;

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

		$product_value[$j] = $carry;

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

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

			$product_value[$k] = $carry;
		}

		return $product_value;
	}

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

		if ($m < MATH_BIGINTEGER_KARATSUBA_CUTOFF) {
			return $this->_regularMultiply($x_value, $y_value);
		}

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

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

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

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

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

		return $xy[MATH_BIGINTEGER_VALUE];
	}

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

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

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

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

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

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

		return $square_value;
	}

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

		if ($m < MATH_BIGINTEGER_KARATSUBA_CUTOFF) {
			return $this->_baseSquare($value);
		}

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

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

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

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

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

		return $xx[MATH_BIGINTEGER_VALUE];
	}

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

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

				if (gmp_sign($remainder->value) < 0) {
					$remainder->value = gmp_add($remainder->value, gmp_abs($y->value));
				}

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

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

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

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

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

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

		$x = $this->copy();
		$y = $y->copy();

		$x_sign = $x->is_negative;
		$y_sign = $y->is_negative;

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

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

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

		if ( $diff < 0 ) {
			// if $x is negative, "add" $y.
			if ( $x_sign ) {
				$x = $y->subtract($x);
			}
			return array($this->_normalize(new Math_BigInteger()), $this->_normalize($x));
		}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

			if ($x->compare($zero) < 0) {
				$temp_value = array_merge($adjust, $y_value);
				$x = $x->add($temp);

				--$quotient_value[$q_index];
			}

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

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

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

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

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

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

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

		return array($result, $carry);
	}

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

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

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

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

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

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

				return $this->_normalize($temp);
		}

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

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

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

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

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

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

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

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

		$y1 = $mod2->modInverse($mod1);
		$y2 = $mod1->modInverse($mod2);

		$result = $part1->multiply($mod2);
		$result = $result->multiply($y1);

		$temp = $part2->multiply($mod1);
		$temp = $temp->multiply($y2);

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

		return $this->_normalize($result);
	}

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

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

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

		$e_length = strlen($e_bits);

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

		$n_value = $n->value;

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

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

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

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

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

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

				$i+=$j + 1;
			}
		}

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

		return $temp;
	}

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

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

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

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

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

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

		$m_length = count($m);

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

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

		// n = 2 * m.length

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

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

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

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

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

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

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

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

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

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

		return $result[MATH_BIGINTEGER_VALUE];
	}

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

		$n_length = count($n);

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

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

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

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

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

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

		return $result[MATH_BIGINTEGER_VALUE];
	}

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

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

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

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

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

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

		$carry = 0;

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

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

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

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

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

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

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

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

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

		$k = count($n);

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

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

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

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

		return $result[MATH_BIGINTEGER_VALUE];
	}

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

		static $cache = array(
			MATH_BIGINTEGER_VARIABLE => array(),
			MATH_BIGINTEGER_DATA => array()
		);

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

				$u = $this->value;
				$v = $n->value;

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

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

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

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

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

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

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

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

		$u = $x->copy();
		$v = $y->copy();

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

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

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

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

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

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

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

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

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

		return $temp;
	}

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

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

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

		$result = $x_negative ? -1 : 1;

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

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

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

		return 0;
	}

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

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

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

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

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

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

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

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

		$result = $this->copy();

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

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

		for ($i = 0; $i < $length; ++$i) {
			$result->value[$i] = $result->value[$i] & $x->value[$i];
		}

		return $this->_normalize($result);
	}

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

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

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

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

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

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

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

		return $this->_normalize($result);
	}

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

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

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

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

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

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

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

		return $this->_normalize($result);
	}

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

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

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

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

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

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

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

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

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

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

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

		return $this->_normalize($temp);
	}

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

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

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

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

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

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

		return $this->_normalize($temp);
	}

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

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

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

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

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

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

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

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

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

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

		if (!$compare) {
			return $this->_normalize($min);
		} elseif ($compare < 0) {
			// if $min is bigger then $max, swap $min and $max
			$temp = $max;
			$max = $min;
			$min = $temp;
		}

		$generator = $this->generator;

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

			return false;
		}

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

		$start = time();

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

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

		$initial_x = $x->copy();

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

			if ($x->isPrime()) {
				return $x;
			}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

		$carry = 0;

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

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

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

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

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

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

		$carry = 0;

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

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

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

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

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

				return $result;
		}

		$value = &$result->value;

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

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

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

			for ($i = 0; $i < $length; ++$i) {
				$value[$i] = $value[$i] & $this->bitmask->value[$i];
			}
		}

		return $result;
	}

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

		return $value;
	}

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

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

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

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

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

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

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

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

		$remainder = chr($carry >> $carry_shift) . $remainder;

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

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

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

	/**
	 * Converts bytes to 32-bit integers
	 *
	 * @param String $x
	 * @return Integer
	 * @access private
	 */
	function _bytes2int($x)
	{
		$temp = unpack('Nint', str_pad($x, 4, chr(0), STR_PAD_LEFT));
		return $temp['int'];
	}
}