1 files changed, 2162 insertions, 0 deletions
diff --git a/phpBB/includes/sftp/biginteger.php b/phpBB/includes/sftp/biginteger.php
new file mode 100644
index 0000000000..d9b90dacb8
--- /dev/null
+++ b/phpBB/includes/sftp/biginteger.php
@@ -0,0 +1,2162 @@
+<?php
+/**
+*
+* @package sftp
+* @version $Id$
+* @copyright (c) 2006 phpBB Group
+* @license http://opensource.org/licenses/gpl-license.php GNU Public License
+*
+*/
+
+/**
+* @ignore
+*/
+if (!defined('IN_PHPBB'))
+{
+	exit;
+}
+
+/**
+* Code from http://phpseclib.sourceforge.net/
+*
+* Modified by phpBB Group to meet our coding standards
+* and being able to integrate into phpBB
+*
+* Pure-PHP arbitrary precission integer arithmetic library
+*
+* Copyright 2007-2009 TerraFrost <terrafrost@php.net>
+* Copyright 2009+ phpBB
+*
+* @package sftp
+* @author  TerraFrost <terrafrost@php.net>
+*/
+
+/**#@+
+ * @access private
+ * @see biginteger::_sliding_window()
+ */
+/**
+ * @see biginteger::_montgomery()
+ * @see biginteger::_undo_montgomery()
+ */
+define('MATH_BIGINTEGER_MONTGOMERY', 0);
+/**
+ * @see biginteger::_barrett()
+ */
+define('MATH_BIGINTEGER_BARRETT', 1);
+/**
+ * @see biginteger::_mod2()
+ */
+define('MATH_BIGINTEGER_POWEROF2', 2);
+/**
+ * @see biginteger::_remainder()
+ */
+define('MATH_BIGINTEGER_CLASSIC', 3);
+/**
+ * @see biginteger::_copy()
+ */
+define('MATH_BIGINTEGER_NONE', 4);
+/**#@-*/
+
+/**#@+
+ * @access private
+ * @see biginteger::_montgomery()
+ * @see biginteger::_barrett()
+ */
+/**
+ * $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);
+/**#@-*/
+
+/**#@+
+ * @access private
+ * @see biginteger::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);
+/**#@-*/
+
+/**
+ * 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 1.0.0RC3
+ * @access  public
+ * @package biginteger
+ */
+class 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;
+
+	/**
+	 * 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.
+	 *
+	 * @param optional $x base-10 number or base-$base number if $base set.
+	 * @param optional integer $base
+	 * @return biginteger
+	 * @access public
+	 */
+	function __construct($x = 0, $base = 10)
+	{
+		if ( !defined('MATH_BIGINTEGER_MODE') )
+		{
+			switch (true)
+			{
+				case extension_loaded('gmp'):
+					define('MATH_BIGINTEGER_MODE', MATH_BIGINTEGER_MODE_GMP);
+					break;
+				case extension_loaded('bcmath'):
+					define('MATH_BIGINTEGER_MODE', MATH_BIGINTEGER_MODE_BCMATH);
+					break;
+				default:
+					define('MATH_BIGINTEGER_MODE', MATH_BIGINTEGER_MODE_INTERNAL);
+			}
+		}
+
+		switch ( MATH_BIGINTEGER_MODE )
+		{
+			case MATH_BIGINTEGER_MODE_GMP:
+				$this->value = gmp_init(0);
+				break;
+			case MATH_BIGINTEGER_MODE_BCMATH:
+				$this->value = '0';
+				break;
+			default:
+				$this->value = array();
+		}
+
+		if ($x === 0)
+		{
+			return;
+		}
+
+		switch ($base)
+		{
+			case -256:
+				if (ord($x[0]) & 0x80)
+				{
+					$x = ~$x;
+					$this->is_negative = true;
+				}
+			case  256:
+				switch ( MATH_BIGINTEGER_MODE )
+				{
+					case MATH_BIGINTEGER_MODE_GMP:
+						$temp = unpack('H*hex', $x);
+						$sign = $this->is_negative ? '-' : '';
+						$this->value = gmp_init($sign . '0x' . $temp['hex']);
+						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'); // 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])));
+						}
+
+						if ($this->is_negative) {
+							$this->value = '-' . $this->value;
+						}
+
+						break;
+					// converts a base-2**8 (big endian / msb) number to base-2**26 (little endian / lsb)
+					case MATH_BIGINTEGER_MODE_INTERNAL:
+						while (strlen($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 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 biginteger(pack('H*', $x), 256);
+						$this->value = $this->is_negative ? '-' . $temp->value : $temp->value;
+						$this->is_negative = false;
+						break;
+					case MATH_BIGINTEGER_MODE_INTERNAL:
+						$x = ( strlen($x) & 1 ) ? '0' . $x : $x;
+						$temp = new biginteger(pack('H*', $x), 256);
+						$this->value = $temp->value;
+				}
+
+				if ($is_negative)
+				{
+					$temp = $this->add(new 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;
+					case MATH_BIGINTEGER_MODE_INTERNAL:
+						$temp = new 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 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 = $temp->add(new biginteger($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;
+				}
+
+				$temp = new biginteger($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.
+	 *
+	 * @param Boolean $twos_compliment
+	 * @return String
+	 * @access public
+	 * @internal Converts a base-2**26 number to base-2**8
+	 */
+	function to_bytes($twos_compliment = false)
+	{
+		if ($twos_compliment)
+		{
+			$comparison = $this->compare(new biginteger());
+			if ($comparison == 0)
+			{
+				return '';
+			}
+
+			$temp = $comparison < 0 ? $this->add(new biginteger(1)) : $this->_copy();
+			$bytes = $temp->to_bytes();
+
+			if (empty($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 '';
+				}
+
+				$temp = gmp_strval(gmp_abs($this->value), 16);
+				$temp = ( strlen($temp) & 1 ) ? '0' . $temp : $temp;
+
+				return ltrim(pack('H*', $temp), chr(0));
+			case MATH_BIGINTEGER_MODE_BCMATH:
+				if ($this->value === '0')
+				{
+					return '';
+				}
+
+				$value = '';
+				$current = $this->value;
+
+				if ($current[0] == '-')
+				{
+					$current = substr($current, 1);
+				}
+
+				// we don't do four bytes at a time because then numbers larger than 1<<31 would be negative
+				// two's complimented numbers, which would break chr.
+				while (bccomp($current, '0') > 0)
+				{
+					$temp = bcmod($current, 0x1000000);
+					$value = chr($temp >> 16) . chr($temp >> 8) . chr($temp) . $value;
+					$current = bcdiv($current, 0x1000000);
+				}
+
+				return ltrim($value, chr(0));
+		}
+
+		if (!count($this->value))
+		{
+			return '';
+		}
+
+		$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 $result;
+	}
+
+	/**
+	 * Converts a BigInteger to a base-10 number.
+	 *
+	 * @return String
+	 * @access public
+	 * @internal Converts a base-2**26 number to base-10**7 (which is pretty much base-10)
+	 */
+	function to_string()
+	{
+		switch ( MATH_BIGINTEGER_MODE )
+		{
+			case MATH_BIGINTEGER_MODE_GMP:
+				return gmp_strval($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 biginteger();
+		$divisor->value = array(10000000); // eg. 10**7
+		while (count($temp->value))
+		{
+			list($temp, $mod) = $temp->divide($divisor);
+			$result = str_pad($this->_bytes2int($mod->to_bytes()), 7, '0', STR_PAD_LEFT) . $result;
+		}
+		$result = ltrim($result, '0');
+
+		if ($this->is_negative)
+		{
+			$result = '-' . $result;
+		}
+
+		return $result;
+	}
+
+	/**
+	 *  __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->to_string();
+	}
+
+	/**
+	 * Adds two BigIntegers.
+	 *
+	 * @param biginteger $y
+	 * @return biginteger
+	 * @access public
+	 * @internal Performs base-2**52 addition
+	 */
+	function add($y)
+	{
+		switch ( MATH_BIGINTEGER_MODE )
+		{
+			case MATH_BIGINTEGER_MODE_GMP:
+				$temp = new biginteger();
+				$temp->value = gmp_add($this->value, $y->value);
+
+				return $temp;
+			case MATH_BIGINTEGER_MODE_BCMATH:
+				$temp = new biginteger();
+				$temp->value = bcadd($this->value, $y->value);
+
+				return $temp;
+		}
+
+		// subtract, if appropriate
+		if ( $this->is_negative != $y->is_negative )
+		{
+			// is $y the negative number?
+			$y_negative = $this->compare($y) > 0;
+
+			$temp = $this->_copy();
+			$y = $y->_copy();
+			$temp->is_negative = $y->is_negative = false;
+
+			$diff = $temp->compare($y);
+			if ( !$diff )
+			{
+				return new biginteger();
+			}
+
+			$temp = $temp->subtract($y);
+
+			$temp->is_negative = ($diff > 0) ? !$y_negative : $y_negative;
+
+			return $temp;
+		}
+
+		$result = new biginteger();
+		$carry = 0;
+
+		$size = max(count($this->value), count($y->value));
+		$size+= $size & 1; // rounds $size to the nearest 2.
+
+		$x = array_pad($this->value, $size,0);
+		$y = array_pad($y->value, $size, 0);
+
+		for ($i = 0; $i < $size - 1; $i+=2)
+		{
+			$sum = $x[$i + 1] * 0x4000000 + $x[$i] + $y[$i + 1] * 0x4000000 + $y[$i] + $carry;
+			$carry = $sum >= 4503599627370496; // eg. floor($sum / 2**52); only possible values (in any base) are 0 and 1
+			$sum = $carry ? $sum - 4503599627370496 : $sum;
+
+			$temp = floor($sum / 0x4000000);
+
+			$result->value[] = $sum - 0x4000000 * $temp; // eg. a faster alternative to fmod($sum, 0x4000000)
+			$result->value[] = $temp;
+		}
+
+		if ($carry)
+		{
+			$result->value[] = $carry;
+		}
+
+		$result->is_negative = $this->is_negative;
+
+		return $result->_normalize();
+	}
+
+	/**
+	 * Subtracts two BigIntegers.
+	 *
+	 * @param biginteger $y
+	 * @return biginteger
+	 * @access public
+	 * @internal Performs base-2**52 subtraction
+	 */
+	function subtract($y)
+	{
+		switch ( MATH_BIGINTEGER_MODE )
+		{
+			case MATH_BIGINTEGER_MODE_GMP:
+				$temp = new biginteger();
+				$temp->value = gmp_sub($this->value, $y->value);
+
+				return $temp;
+			case MATH_BIGINTEGER_MODE_BCMATH:
+				$temp = new biginteger();
+				$temp->value = bcsub($this->value, $y->value);
+
+				return $temp;
+		}
+
+		// add, if appropriate
+		if ( $this->is_negative != $y->is_negative )
+		{
+			$is_negative = $y->compare($this) > 0;
+
+			$temp = $this->_copy();
+			$y = $y->_copy();
+			$temp->is_negative = $y->is_negative = false;
+
+			$temp = $temp->add($y);
+
+			$temp->is_negative = $is_negative;
+
+			return $temp;
+		}
+
+		$diff = $this->compare($y);
+
+		if ( !$diff )
+		{
+			return new biginteger();
+		}
+
+		// switch $this and $y around, if appropriate.
+		if ( (!$this->is_negative && $diff < 0) || ($this->is_negative && $diff > 0) )
+		{
+			$is_negative = $y->is_negative;
+
+			$temp = $this->_copy();
+			$y = $y->_copy();
+			$temp->is_negative = $y->is_negative = false;
+
+			$temp = $y->subtract($temp);
+			$temp->is_negative = !$is_negative;
+
+			return $temp;
+		}
+
+		$result = new biginteger();
+		$carry = 0;
+
+		$size = max(count($this->value), count($y->value));
+		$size+= $size % 2;
+
+		$x = array_pad($this->value, $size, 0);
+		$y = array_pad($y->value, $size, 0);
+
+		for ($i = 0; $i < $size - 1; $i+=2)
+		{
+			$sum = $x[$i + 1] * 0x4000000 + $x[$i] - $y[$i + 1] * 0x4000000 - $y[$i] + $carry;
+			$carry = $sum < 0 ? -1 : 0; // eg. floor($sum / 2**52); only possible values (in any base) are 0 and 1
+			$sum = $carry ? $sum + 4503599627370496 : $sum;
+
+			$temp = floor($sum / 0x4000000);
+
+			$result->value[] = $sum - 0x4000000 * $temp;
+			$result->value[] = $temp;
+		}
+
+		// $carry shouldn't be anything other than zero, at this point, since we already made sure that $this
+		// was bigger than $y.
+
+		$result->is_negative = $this->is_negative;
+
+		return $result->_normalize();
+	}
+
+	/**
+	 * Multiplies two BigIntegers
+	 *
+	 * @param biginteger $x
+	 * @return biginteger
+	 * @access public
+	 * @internal Modeled after 'multiply' in MutableBigInteger.java.
+	 */
+	function multiply($x)
+	{
+		switch ( MATH_BIGINTEGER_MODE )
+		{
+			case MATH_BIGINTEGER_MODE_GMP:
+				$temp = new biginteger();
+				$temp->value = gmp_mul($this->value, $x->value);
+
+				return $temp;
+			case MATH_BIGINTEGER_MODE_BCMATH:
+				$temp = new biginteger();
+				$temp->value = bcmul($this->value, $x->value);
+
+				return $temp;
+		}
+
+		if ( !$this->compare($x) )
+		{
+			return $this->_square();
+		}
+
+		$this_length = count($this->value);
+		$x_length = count($x->value);
+
+		if ( !$this_length || !$x_length ) // a 0 is being multiplied
+		{
+			return new biginteger();
+		}
+
+		$product = new biginteger();
+		$product->value = $this->_array_repeat(0, $this_length + $x_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;
+		$i = 0;
+
+		for ($j = 0, $k = $i; $j < $this_length; $j++, $k++)
+		{
+			$temp = $product->value[$k] + $this->value[$j] * $x->value[$i] + $carry;
+			$carry = floor($temp / 0x4000000);
+			$product->value[$k] = $temp - 0x4000000 * $carry;
+		}
+
+		$product->value[$k] = $carry;
+
+		// the above for loop is what the previous comment was talking about.  the
+		// following for loop is the "one with nested for loops"
+
+		for ($i = 1; $i < $x_length; $i++)
+		{
+			$carry = 0;
+
+			for ($j = 0, $k = $i; $j < $this_length; $j++, $k++)
+			{
+				$temp = $product->value[$k] + $this->value[$j] * $x->value[$i] + $carry;
+				$carry = floor($temp / 0x4000000);
+				$product->value[$k] = $temp - 0x4000000 * $carry;
+			}
+
+			$product->value[$k] = $carry;
+		}
+
+		$product->is_negative = $this->is_negative != $x->is_negative;
+
+		return $product->_normalize();
+	}
+
+	/**
+	 * Squares a BigInteger
+	 *
+	 * Squaring can be done faster than multiplying a number by itself can be.  See
+	 * {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=7 HAC 14.2.4} /
+	 * {@link http://math.libtomcrypt.com/files/tommath.pdf#page=141 MPM 5.3} for more information.
+	 *
+	 * @return biginteger
+	 * @access private
+	 */
+	function _square()
+	{
+		if ( empty($this->value) )
+		{
+			return new biginteger();
+		}
+
+		$max_index = count($this->value) - 1;
+
+		$square = new biginteger();
+		$square->value = $this->_array_repeat(0, 2 * $max_index);
+
+		for ($i = 0; $i <= $max_index; $i++)
+		{
+			$temp = $square->value[2 * $i] + $this->value[$i] * $this->value[$i];
+			$carry = floor($temp / 0x4000000);
+			$square->value[2 * $i] = $temp - 0x4000000 * $carry;
+
+			// note how we start from $i+1 instead of 0 as we do in multiplication.
+			for ($j = $i + 1; $j <= $max_index; $j++)
+			{
+				$temp = $square->value[$i + $j] + 2 * $this->value[$j] * $this->value[$i] + $carry;
+				$carry = floor($temp / 0x4000000);
+				$square->value[$i + $j] = $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->_normalize();
+	}
+
+	/**
+	 * 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).
+	 *
+	 * @param biginteger $y
+	 * @return Array
+	 * @access public
+	 * @internal This function is based off of {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=9 HAC 14.20}
+	 *	with a slight variation due to the fact that this script, initially, did not support negative numbers.  Now,
+	 *	it does, but I don't want to change that which already works.
+	 */
+	function divide($y)
+	{
+		switch ( MATH_BIGINTEGER_MODE )
+		{
+			case MATH_BIGINTEGER_MODE_GMP:
+				$quotient = new biginteger();
+				$remainder = new biginteger();
+
+				list($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($y->value));
+				}
+
+				return array($quotient, $remainder);
+			case MATH_BIGINTEGER_MODE_BCMATH:
+				$quotient = new biginteger();
+				$remainder = new biginteger();
+
+				$quotient->value = bcdiv($this->value, $y->value);
+				$remainder->value = bcmod($this->value, $y->value);
+
+				if ($remainder->value[0] == '-')
+				{
+					$remainder->value = bcadd($remainder->value, $y->value[0] == '-' ? substr($y->value, 1) : $y->value);
+				}
+
+				return array($quotient, $remainder);
+		}
+
+		$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 biginteger();
+			$temp->value = array(1);
+			$temp->is_negative = $x_sign != $y_sign;
+			return array($temp, new biginteger());
+		}
+
+		if ( $diff < 0 )
+		{
+			// if $x is negative, "add" $y.
+			if ( $x_sign )
+			{
+				$x = $y->subtract($x);
+			}
+			return array(new biginteger(), $x);
+		}
+
+		// normalize $x and $y as described in HAC 14.23 / 14.24
+		// (incidently, i haven't been able to find a definitive example showing that this
+		// results in worth-while speedup, but whatever)
+		$msb = $y->value[count($y->value) - 1];
+		for ($shift = 0; !($msb & 0x2000000); $shift++)
+		{
+			$msb <<= 1;
+		}
+		$x->_lshift($shift);
+		$y->_lshift($shift);
+
+		$x_max = count($x->value) - 1;
+		$y_max = count($y->value) - 1;
+
+		$quotient = new biginteger();
+		$quotient->value = $this->_array_repeat(0, $x_max - $y_max + 1);
+
+		// $temp = $y << ($x_max - $y_max-1) in base 2**26
+		$temp = new biginteger();
+		$temp->value = array_merge($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 = array(
+				$x->value[$i],
+				( $i > 0 ) ? $x->value[$i - 1] : 0,
+				( $i - 1 > 0 ) ? $x->value[$i - 2] : 0
+			);
+			$y_value = array(
+				$y->value[$y_max],
+				( $y_max > 0 ) ? $y_max - 1 : 0
+			);
+
+
+			$q_index = $i - $y_max - 1;
+			if ($x_value[0] == $y_value[0])
+			{
+				$quotient->value[$q_index] = 0x3FFFFFF;
+			}
+			else
+			{
+				$quotient->value[$q_index] = floor(
+					($x_value[0] * 0x4000000 + $x_value[1])
+					/
+					$y_value[0]
+				);
+			}
+
+			$temp = new biginteger();
+			$temp->value = array($y_value[1], $y_value[0]);
+
+			$lhs = new biginteger();
+			$lhs->value = array($quotient->value[$q_index]);
+			$lhs = $lhs->multiply($temp);
+
+			$rhs = new biginteger();
+			$rhs->value = array($x_value[2], $x_value[1], $x_value[0]);
+			
+			while ( $lhs->compare($rhs) > 0 )
+			{
+				$quotient->value[$q_index]--;
+
+				$lhs = new biginteger();
+				$lhs->value = array($quotient->value[$q_index]);
+				$lhs = $lhs->multiply($temp);
+			}
+
+			$corrector = new biginteger();
+			$temp = new biginteger();
+			$corrector->value = $temp->value = $this->_array_repeat(0, $q_index);
+			$temp->value[] = $quotient->value[$q_index];
+
+			$temp = $temp->multiply($y);
+
+			if ( $x->compare($temp) < 0 )
+			{
+				$corrector->value[] = 1;
+				$x = $x->add($corrector->multiply($y));
+				$quotient->value[$q_index]--;
+			}
+
+			$x = $x->subtract($temp);
+			$x_max = count($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($quotient->_normalize(), $x);
+	}
+
+	/**
+	 * Performs modular exponentiation.
+	 *
+	 * @param biginteger $e
+	 * @param biginteger $n
+	 * @return biginteger
+	 * @access public
+	 * @internal The most naive approach to modular exponentiation has very unreasonable requirements, and
+	 *	and although the approach involving repeated squaring does vastly better, it, too, is impractical
+	 *	for our purposes.  The reason being that division - by far the most complicated and time-consuming
+	 *	of the basic operations (eg. +,-,*,/) - occurs multiple times within it.
+	 *
+	 *	Modular reductions resolve this issue.  Although an individual modular reduction takes more time
+	 *	then an individual division, when performed in succession (with the same modulo), they're a lot faster.
+	 *
+	 *	The two most commonly used modular reductions are Barrett and Montgomery reduction.  Montgomery reduction,
+	 *	although faster, only works when the gcd of the modulo and of the base being used is 1.  In RSA, when the
+	 *	base is a power of two, the modulo - a product of two primes - is always going to have a gcd of 1 (because
+	 *	the product of two odd numbers is odd), but what about when RSA isn't used?
+	 *
+	 *	In contrast, Barrett reduction has no such constraint.  As such, some bigint implementations perform a
+	 *	Barrett reduction after every operation in the modpow function.  Others perform Barrett reductions when the
+	 *	modulo is even and Montgomery reductions when the modulo is odd.  BigInteger.java's modPow method, however,
+	 *	uses a trick involving the Chinese Remainder Theorem to factor the even modulo into two numbers - one odd and
+	 *	the other, a power of two - and recombine them, later.  This is the method that this modPow function uses.
+	 *	{@link http://islab.oregonstate.edu/papers/j34monex.pdf Montgomery Reduction with Even Modulus} elaborates.
+	 */
+	function mod_pow($e, $n)
+	{
+		$n = $n->abs();
+		if ($e->compare(new biginteger()) < 0)
+		{
+			$e = $e->abs();
+
+			$temp = $this->modInverse($n);
+			if ($temp === false)
+			{
+				return false;
+			}
+
+			return $temp->modPow($e, $n);
+		}
+
+		switch ( MATH_BIGINTEGER_MODE )
+		{
+			case MATH_BIGINTEGER_MODE_GMP:
+				$temp = new biginteger();
+				$temp->value = gmp_powm($this->value, $e->value, $n->value);
+
+				return $temp;
+			case MATH_BIGINTEGER_MODE_BCMATH:
+				$temp = new biginteger();
+				$temp->value = bcpowmod($this->value, $e->value, $n->value);
+
+				return $temp;
+		}
+
+		if ( empty($e->value) )
+		{
+			$temp = new biginteger();
+			$temp->value = array(1);
+			return $temp;
+		}
+
+		if ( $e->value == array(1) )
+		{
+			list(, $temp) = $this->divide($n);
+			return $temp;
+		}
+
+		if ( $e->value == array(2) )
+		{
+			$temp = $this->_square();
+			list(, $temp) = $temp->divide($n);
+			return $temp;
+		}
+
+		// is the modulo odd?
+		if ( $n->value[0] & 1 )
+		{
+			return $this->_slidingWindow($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 biginteger();
+		$mod2->value = array(1);
+		$mod2->_lshift($j);
+
+		$part1 = ( $mod1->value != array(1) ) ? $this->_slidingWindow($e, $mod1, MATH_BIGINTEGER_MONTGOMERY) : new biginteger();
+		$part2 = $this->_sliding_window($e, $mod2, MATH_BIGINTEGER_POWEROF2);
+
+		$y1 = $mod2->mod_inverse($mod1);
+		$y2 = $mod1->mod_inverse($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 $result;
+	}
+
+	/**
+	 * 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 biginteger $e
+	 * @param biginteger $n
+	 * @param Integer $mode
+	 * @return biginteger
+	 * @access private
+	 */
+	function _sliding_window($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
+
+		$e_length = count($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++, $i++);
+
+		switch ($mode)
+		{
+			case MATH_BIGINTEGER_MONTGOMERY:
+				$reduce = '_montgomery';
+				$undo = '_undo_montgomery';
+				break;
+			case MATH_BIGINTEGER_BARRETT:
+				$reduce = '_barrett';
+				$undo = '_barrett';
+				break;
+			case MATH_BIGINTEGER_POWEROF2:
+				$reduce = '_mod2';
+				$undo = '_mod2';
+				break;
+			case MATH_BIGINTEGER_CLASSIC:
+				$reduce = '_remainder';
+				$undo = '_remainder';
+				break;
+			case MATH_BIGINTEGER_NONE:
+				// ie. do no modular reduction.  useful if you want to just do pow as opposed to modPow.
+				$reduce = '_copy';
+				$undo = '_copy';
+				break;
+			default:
+				// an invalid $mode was provided
+		}
+
+		// precompute $this^0 through $this^$window_size
+		$powers = array();
+		$powers[1] = $this->$undo($n);
+		$powers[2] = $powers[1]->_square();
+		$powers[2] = $powers[2]->$reduce($n);
+
+		// we do every other number since substr($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++)
+		{
+			$powers[2 * $i + 1] = $powers[2 * $i - 1]->multiply($powers[2]);
+			$powers[2 * $i + 1] = $powers[2 * $i + 1]->$reduce($n);
+		}
+
+		$result = new biginteger();
+		$result->value = array(1);
+		$result = $result->$undo($n);
+
+		for ($i = 0; $i < $e_length; )
+		{
+			if ( !$e_bits[$i] )
+			{
+				$result = $result->_square();
+				$result = $result->$reduce($n);
+				$i++;
+			}
+			else
+			{
+				for ($j = $window_size - 1; $j >= 0; $j--)
+				{
+					if ( $e_bits[$i + $j] )
+					{
+						break;
+					}
+				}
+
+				for ($k = 0; $k <= $j; $k++) // eg. the length of substr($e_bits, $i, $j+1)
+				{
+					$result = $result->_square();
+					$result = $result->$reduce($n);
+				}
+
+				$result = $result->multiply($powers[bindec(substr($e_bits, $i, $j + 1))]);
+				$result = $result->$reduce($n);
+
+				$i+=$j + 1;
+			}
+		}
+
+		$result = $result->$reduce($n);
+		return $result->_normalize();
+	}
+
+	/**
+	 * Remainder
+	 *
+	 * A wrapper for the divide function.
+	 *
+	 * @see divide()
+	 * @see _slidingWindow()
+	 * @access private
+	 * @param biginteger
+	 * @return biginteger
+	 */
+	function _remainder($n)
+	{
+		list(, $temp) = $this->divide($n);
+		return $temp;
+	}
+
+	/**
+	 * 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 biginteger
+	 * @return biginteger
+	 */
+	function _mod2($n)
+	{
+		$temp = new biginteger();
+		$temp->value = array(1);
+		return $this->bitwise_and($n->subtract($temp));
+	}
+
+	/**
+	 * Barrett Modular Reduction
+	 *
+	 * See {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=14 HAC 14.3.3} /
+	 * {@link http://math.libtomcrypt.com/files/tommath.pdf#page=165 MPM 6.2.5} for more information.  Modified slightly,
+	 * so as not to require negative numbers (initially, this script didn't support negative numbers).
+	 *
+	 * @see _slidingWindow()
+	 * @access private
+	 * @param biginteger
+	 * @return biginteger
+	 */
+	function _barrett($n)
+	{
+		static $cache;
+
+		$n_length = count($n->value);
+
+		if ( !isset($cache[MATH_BIGINTEGER_VARIABLE]) || $n->compare($cache[MATH_BIGINTEGER_VARIABLE]) )
+		{
+			$cache[MATH_BIGINTEGER_VARIABLE] = $n;
+			$temp = new biginteger();
+			$temp->value = $this->_array_repeat(0, 2 * $n_length);
+			$temp->value[] = 1;
+			list($cache[MATH_BIGINTEGER_DATA], ) = $temp->divide($n);
+		}
+
+		$temp = new biginteger();
+		$temp->value = array_slice($this->value, $n_length - 1);
+		$temp = $temp->multiply($cache[MATH_BIGINTEGER_DATA]);
+		$temp->value = array_slice($temp->value, $n_length + 1);
+
+		$result = new biginteger();
+		$result->value = array_slice($this->value, 0, $n_length + 1);
+		$temp = $temp->multiply($n);
+		$temp->value = array_slice($temp->value, 0, $n_length + 1);
+
+		if ($result->compare($temp) < 0)
+		{
+			$corrector = new biginteger();
+			$corrector->value = $this->_array_repeat(0, $n_length + 1);
+			$corrector->value[] = 1;
+			$result = $result->add($corrector);
+		}
+
+		$result = $result->subtract($temp);
+		while ($result->compare($n) > 0)
+		{
+			$result = $result->subtract($n);
+		}
+
+		return $result;
+	}
+
+	/**
+	 * Montgomery Modular Reduction
+	 *
+	 * ($this->_montgomery($n))->_undoMontgomery($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($n, 2) must be equal to one for this function
+	 * to work correctly.
+	 *
+	 * @see _undoMontgomery()
+	 * @see _slidingWindow()
+	 * @access private
+	 * @param biginteger
+	 * @return biginteger
+	 */
+	function _montgomery($n)
+	{
+		static $cache;
+
+		if ( !isset($cache[MATH_BIGINTEGER_VARIABLE]) || $n->compare($cache[MATH_BIGINTEGER_VARIABLE]) )
+		{
+			$cache[MATH_BIGINTEGER_VARIABLE] = $n;
+			$cache[MATH_BIGINTEGER_DATA] = $n->_mod_inverse67108864();
+		}
+
+		$result = $this->_copy();
+
+		$n_length = count($n->value);
+
+		for ($i = 0; $i < $n_length; $i++)
+		{
+			$temp = new biginteger();
+			$temp->value = array(
+				($result->value[$i] * $cache[MATH_BIGINTEGER_DATA]) & 0x3FFFFFF
+			);
+			$temp = $temp->multiply($n);
+			$temp->value = array_merge($this->_array_repeat(0, $i), $temp->value);
+			$result = $result->add($temp);
+		}
+
+		$result->value = array_slice($result->value, $n_length);
+
+		if ($result->compare($n) >= 0)
+		{
+			$result = $result->subtract($n);
+		}
+
+		return $result->_normalize();
+	}
+
+	/**
+	 * Undo Montgomery Modular Reduction
+	 *
+	 * @see _montgomery()
+	 * @see _slidingWindow()
+	 * @access private
+	 * @param biginteger
+	 * @return biginteger
+	 */
+	function _undo_montgomery($n)
+	{
+		$temp = new biginteger();
+		$temp->value = array_merge($this->_array_repeat(0, count($n->value)), $this->value);
+		list(, $temp) = $temp->divide($n);
+		return $temp->_normalize();
+	}
+
+	/**
+	 * 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
+	 * @return Integer
+	 */
+	function _mod_inverse67108864() // 2**26 == 67108864
+	{
+		$x = -$this->value[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.
+	 *
+	 * @param biginteger $n
+	 * @return mixed false, if no modular inverse exists, biginteger, otherwise.
+	 * @access public
+	 * @internal Calculates the modular inverse of $this mod $n using the binary xGCD algorithim described in
+	 *	{@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=19 HAC 14.61}.  As the text above 14.61 notes,
+	 *	the more traditional algorithim requires "relatively costly multiple-precision divisions".  See
+	 *	{@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=21 HAC 14.64} for more information.
+	 */
+	function mod_inverse($n)
+	{
+		switch ( MATH_BIGINTEGER_MODE )
+		{
+			case MATH_BIGINTEGER_MODE_GMP:
+				$temp = new biginteger();
+				$temp->value = gmp_invert($this->value, $n->value);
+
+				return ( $temp->value === false ) ? false : $temp;
+			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.
+
+				// if $x is less than 0, the first character of $x is a '-', so we'll remove it.  we can do this because
+				// $x mod $n == $x mod -$n.
+				$n = (bccomp($n->value, '0') < 0) ? substr($n->value, 1) : $n->value;
+
+				if (bccomp($this->value,'0') < 0)
+				{
+					$negated_this = new biginteger();
+					$negated_this->value = substr($this->value, 1);
+
+					$temp = $negated_this->mod_inverse(new biginteger($n));
+
+					if ($temp === false)
+					{
+						return false;
+					}
+
+					$temp->value = bcsub($n, $temp->value);
+
+					return $temp;
+				}
+
+				$u = $this->value;
+				$v = $n;
+
+				$a = '1';
+				$c = '0';
+
+				while (true)
+				{
+					$q = bcdiv($u, $v);
+					$temp = $u;
+					$u = $v;
+					$v = bcsub($temp, bcmul($v, $q));
+
+					if (bccomp($v, '0') == 0) {
+						break;
+					}
+
+					$temp = $a;
+					$a = $c;
+					$c = bcsub($temp, bcmul($c, $q));
+				}
+
+				$temp = new biginteger();
+				$temp->value = (bccomp($c, '0') < 0) ? bcadd($c, $n) : $c;
+
+				// $u contains the gcd of $this and $n
+				return (bccomp($u,'1') == 0) ? $temp : false;
+		}
+
+		// if $this and $n are even, return false.
+		if ( !($this->value[0]&1) && !($n->value[0]&1) )
+		{
+			return false;
+		}
+
+		$n = $n->_copy();
+		$n->is_negative = false;
+
+		if ($this->compare(new biginteger()) < 0)
+		{
+			// is_negative is currently true.  since we need it to be false, we'll just set it to false, temporarily,
+			// and reset it as true, later.
+			$this->is_negative = false;
+
+			$temp = $this->mod_inverse($n);
+
+			if ($temp === false)
+			{
+				return false;
+			}
+
+			$temp = $n->subtract($temp);
+
+			$this->is_negative = true;
+
+			return $temp;
+		}
+
+		$u = $n->_copy();
+		$x = $this;
+		//list(, $x) = $this->divide($n);
+		$v = $x->_copy();
+
+		$a = new biginteger();
+		$b = new biginteger();
+		$c = new biginteger();
+		$d = new biginteger();
+
+		$a->value = $d->value = array(1);
+
+		while ( !empty($u->value) )
+		{
+			while ( !($u->value[0] & 1) )
+			{
+				$u->_rshift(1);
+				if ( ($a->value[0] & 1) || ($b->value[0] & 1) )
+				{
+					$a = $a->add($x);
+					$b = $b->subtract($n);
+				}
+				$a->_rshift(1);
+				$b->_rshift(1);
+			}
+
+			while ( !($v->value[0] & 1) )
+			{
+				$v->_rshift(1);
+				if ( ($c->value[0] & 1) || ($d->value[0] & 1) )
+				{
+					$c = $c->add($x);
+					$d = $d->subtract($n);
+				}
+				$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);
+			}
+
+			$u->_normalize();
+		}
+
+		// at this point, $v == gcd($this, $n).  if it's not equal to 1, no modular inverse exists.
+		if ( $v->value != array(1) )
+		{
+			return false;
+		}
+
+		$d = ($d->compare(new biginteger()) < 0) ? $d->add($n) : $d;
+
+		return ($this->is_negative) ? $n->subtract($d) : $d;
+	}
+
+	/**
+	 * Absolute value.
+	 *
+	 * @return biginteger
+	 * @access public
+	 */
+	function abs()
+	{
+		$temp = new 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) ? substr($this->value, 1) : $this->value;
+				break;
+			default:
+				$temp->value = $this->value;
+		}
+
+		return $temp;
+	}
+
+	/**
+	 * Compares two numbers.
+	 *
+	 * @param 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
+	 * @internal Could return $this->sub($x), but that's not as fast as what we do do.
+	 */
+	function compare($x)
+	{
+		switch ( MATH_BIGINTEGER_MODE )
+		{
+			case MATH_BIGINTEGER_MODE_GMP:
+				return gmp_cmp($this->value, $x->value);
+			case MATH_BIGINTEGER_MODE_BCMATH:
+				return bccomp($this->value, $x->value);
+		}
+
+		$this->_normalize();
+		$x->_normalize();
+
+		if ( $this->is_negative != $x->is_negative )
+		{
+			return ( !$this->is_negative && $x->is_negative ) ? 1 : -1;
+		}
+
+		$result = $this->is_negative ? -1 : 1;
+
+		if ( count($this->value) != count($x->value) )
+		{
+			return ( count($this->value) > count($x->value) ) ? $result : -$result;
+		}
+
+		for ($i = count($this->value) - 1; $i >= 0; $i--)
+		{
+			if ($this->value[$i] != $x->value[$i])
+			{
+				return ( $this->value[$i] > $x->value[$i] ) ? $result : -$result;
+			}
+		}
+
+		return 0;
+	}
+
+	/**
+	 * Returns a copy of $this
+	 *
+	 * 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://www.php.net/manual/en/language.oop5.basic.php#51624}
+	 *
+	 * @access private
+	 * @return biginteger
+	 */
+	function _copy()
+	{
+		$temp = new biginteger();
+		$temp->value = $this->value;
+		$temp->is_negative = $this->is_negative;
+		return $temp;
+	}
+
+	/**
+	 * Logical And
+	 *
+	 * @param biginteger $x
+	 * @access public
+	 * @internal Implemented per a request by Lluis Pamies i Juarez <lluis _a_ pamies.cat>
+	 * @return biginteger
+	 */
+	function bitwise_and($x)
+	{
+		switch ( MATH_BIGINTEGER_MODE )
+		{
+			case MATH_BIGINTEGER_MODE_GMP:
+				$temp = new biginteger();
+				$temp->value = gmp_and($this->value, $x->value);
+
+				return $temp;
+			case MATH_BIGINTEGER_MODE_BCMATH:
+				return new biginteger($this->to_bytes() & $x->to_bytes(), 256);
+		}
+
+		$result = new biginteger();
+
+		$x_length = count($x->value);
+		for ($i = 0; $i < $x_length; $i++)
+		{
+			$result->value[] = $this->value[$i] & $x->value[$i];
+		}
+
+		return $result->_normalize();
+	}
+
+	/**
+	 * Logical Or
+	 *
+	 * @param biginteger $x
+	 * @access public
+	 * @internal Implemented per a request by Lluis Pamies i Juarez <lluis _a_ pamies.cat>
+	 * @return biginteger
+	 */
+	function bitwise_or($x)
+	{
+		switch ( MATH_BIGINTEGER_MODE )
+		{
+			case MATH_BIGINTEGER_MODE_GMP:
+				$temp = new biginteger();
+				$temp->value = gmp_or($this->value, $x->value);
+
+				return $temp;
+			case MATH_BIGINTEGER_MODE_BCMATH:
+				return new biginteger($this->to_bytes() | $x->to_bytes(), 256);
+		}
+
+		$result = $this->_copy();
+
+		$x_length = count($x->value);
+		for ($i = 0; $i < $x_length; $i++)
+		{
+			$result->value[$i] = $this->value[$i] | $x->value[$i];
+		}
+
+		return $result->_normalize();
+	}
+
+	/**
+	 * Logical Exclusive-Or
+	 *
+	 * @param biginteger $x
+	 * @access public
+	 * @internal Implemented per a request by Lluis Pamies i Juarez <lluis _a_ pamies.cat>
+	 * @return biginteger
+	 */
+	function bitwise_xor($x)
+	{
+		switch ( MATH_BIGINTEGER_MODE )
+		{
+			case MATH_BIGINTEGER_MODE_GMP:
+				$temp = new biginteger();
+				$temp->value = gmp_xor($this->value, $x->value);
+
+				return $temp;
+			case MATH_BIGINTEGER_MODE_BCMATH:
+				return new biginteger($this->to_bytes() ^ $x->to_bytes(), 256);
+		}
+
+		$result = $this->_copy();
+
+		$x_length = count($x->value);
+		for ($i = 0; $i < $x_length; $i++)
+		{
+			$result->value[$i] = $this->value[$i] ^ $x->value[$i];
+		}
+
+		return $result->_normalize();
+	}
+
+	/**
+	 * Logical Not
+	 *
+	 * Although integers can be converted to and from various bases with relative ease, there is one piece
+	 * of information that is lost during such conversions.  The number of leading zeros that number had
+	 * or should have in any given base.  Per that, if you convert 1 from decimal to binary, there's no
+	 * way to know just how many leading zero's there should be.  In truth, there could be any number.
+	 *
+	 * Normally, the number of leading zero's is unimportant.  When doing "not", however, it is.  The "not"
+	 * of 1 on an 8-bit representation of 1 is 1111 1110.  The "not" of 1 on a 16-bit representation of 1 is
+	 * 1111 1111 1111 1110.  When doing it on a number that's preceeded by an infinite number of zero's, it's
+	 * infinite.
+	 *
+	 * This function assumes that there are no leading zero's - that the bit-representation being used is
+	 * equal to the minimum number of required bits, unless otherwise specified in the optional parameter,
+	 * where the optional parameter represents the bit-representation being used.  If the specified
+	 * bit-representation is smaller than the minimum number of bits required to represent the number, the
+	 * latter will be used as the bit-representation.
+	 *
+	 * @param $bits Integer
+	 * @access public
+	 * @internal Implemented per a request by Lluis Pamies i Juarez <lluis _a_ pamies.cat>
+	 * @return biginteger
+	 */
+	function bitwise_not($bits = -1)
+	{
+		// calculuate "not" without regard to $bits
+		$temp = ~$this->to_bytes();
+		$msb = decbin(ord($temp[0]));
+		$msb = substr($msb, strpos($msb, '0'));
+		$temp[0] = chr(bindec($msb));
+
+		// see if we need to add extra leading 1's
+		$current_bits = strlen($msb) + 8 * strlen($temp) - 8;
+		$new_bits = $bits - $current_bits;
+		if ($new_bits <= 0)
+		{
+			return new 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($bits / 8), chr(0), STR_PAD_LEFT);
+
+		return new biginteger($leading_ones | $temp, 256);
+	}
+
+	/**
+	 * Logical Right Shift
+	 *
+	 * Shifts BigInteger's by $shift bits, effectively dividing by 2**$shift.
+	 *
+	 * @param Integer $shift
+	 * @return biginteger
+	 * @access public
+	 * @internal The only version that yields any speed increases is the internal version.
+	 */
+	function bitwise_right_shift($shift)
+	{
+		$temp = new biginteger();
+
+		switch ( MATH_BIGINTEGER_MODE )
+		{
+			case MATH_BIGINTEGER_MODE_GMP:
+				static $two;
+
+				if (empty($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));
+
+				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 $temp;
+	}
+
+	/**
+	 * Logical Left Shift
+	 *
+	 * Shifts BigInteger's by $shift bits, effectively multiplying by 2**$shift.
+	 *
+	 * @param Integer $shift
+	 * @return biginteger
+	 * @access public
+	 * @internal The only version that yields any speed increases is the internal version.
+	 */
+	function bitwise_left_shift($shift)
+	{
+		$temp = new biginteger();
+
+		switch ( MATH_BIGINTEGER_MODE )
+		{
+			case MATH_BIGINTEGER_MODE_GMP:
+				static $two;
+
+				if (empty($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));
+
+				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 $temp;
+	}
+
+	/**
+	 * Generate a random number
+	 *
+	 * $generator should be the name of a random number 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
+	 * done before this function is called.
+	 *
+	 * @param optional Integer $min
+	 * @param optional Integer $max
+	 * @param optional String $generator
+	 * @return biginteger
+	 * @access public
+	 */
+	function random($min = false, $max = false, $generator = 'mt_rand')
+	{
+		if ($min === false)
+		{
+			$min = new biginteger(0);
+		}
+
+		if ($max === false)
+		{
+			$max = new biginteger(0x7FFFFFFF);
+		}
+
+		$compare = $max->compare($min);
+
+		if (!$compare)
+		{
+			return $min;
+		}
+		else if ($compare < 0)
+		{
+			// if $min is bigger then $max, swap $min and $max
+			$temp = $max;
+			$max = $min;
+			$min = $temp;
+		}
+
+		$max = $max->subtract($min);
+		$max = ltrim($max->to_bytes(), chr(0));
+		$size = strlen($max) - 1;
+		$random = '';
+
+		$bytes = $size & 3;
+		for ($i = 0; $i < $bytes; $i++)
+		{
+			$random.= chr($generator(0, 255));
+		}
+
+		$blocks = $size >> 2;
+		for ($i = 0; $i < $blocks; $i++)
+		{
+			$random.= pack('N', $generator(-2147483648, 0x7FFFFFFF));
+		}
+
+		$temp = new biginteger($random, 256);
+		if ($temp->compare(new biginteger(substr($max, 1), 256)) > 0)
+		{
+			$random = chr($generator(0, ord($max[0]) - 1)) . $random;
+		}
+		else
+		{
+			$random = chr($generator(0, ord($max[0])	)) . $random;
+		}
+
+		$random = new biginteger($random, 256);
+
+		return $random->add($min);
+	}
+
+	/**
+	 * Logical Left Shift
+	 *
+	 * Shifts BigInteger's by $shift bits.
+	 *
+	 * @param Integer $shift
+	 * @access private
+	 */
+	function _lshift($shift)
+	{
+		if ( $shift == 0 )
+		{
+			return;
+		}
+
+		$num_digits = floor($shift / 26);
+		$shift %= 26;
+		$shift = 1 << $shift;
+
+		$carry = 0;
+
+		for ($i = 0; $i < count($this->value); $i++)
+		{
+			$temp = $this->value[$i] * $shift + $carry;
+			$carry = floor($temp / 0x4000000);
+			$this->value[$i] = $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)
+		{
+			$this->_normalize();
+		}
+
+		$num_digits = floor($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;
+		}
+
+		$this->_normalize();
+	}
+
+	/**
+	 * Normalize
+	 *
+	 * Deletes leading zeros.
+	 *
+	 * @see divide()
+	 * @return Math_BigInteger
+	 * @access private
+	 */
+	function _normalize()
+	{
+		if ( !count($this->value) )
+		{
+			return $this;
+		}
+
+		for ($i=count($this->value) - 1; $i >= 0; $i--)
+		{
+			if ( $this->value[$i] )
+			{
+				break;
+			}
+			unset($this->value[$i]);
+		}
+
+		return $this;
+	}
+
+	/**
+	 * 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'];
+	}
+}
+
+?>
+\ No newline at end of file