/lib/Phpseclib/Math/BigInteger.php

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<?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)
 *
 * 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).
 *
 * 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
 *
 * One idea for optimization is to use the comba method to reduce the number of operations performed.
 * MPM uses this quite extensively.  The following URL elaborates:
 *
 * {@link http://www.everything2.com/index.pl?node_id=1736418}}}
 *
 * Here's an example of how to use this library:
 * <code>
 * <?php
 *    include('Math/BigInteger.php');
 *
 *    $a = new BigInteger(2);
 *    $b = new 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    BigInteger
 * @author     Jim Wigginton <terrafrost@php.net>
 * @copyright  MMVI Jim Wigginton
 * @license    http://www.gnu.org/licenses/lgpl.txt
 * @version    $Id: BigInteger.php 81 2010-05-19 21:56:16Z admin $
 * @link       http://pear.php.net/package/BigInteger
 */

namespace Phpseclib\Math;


/**#@+
 * @access private
 * @see BigInteger::_slidingWindow()
 */
/**
 * @see BigInteger::_montgomery()
 * @see BigInteger::_prepMontgomery()
 */
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::__clone()
 */
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);
/**#@-*/

/**
 * 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', 15);

/**
 * 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;

    /**
     * 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;

    /**
     * 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 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 $base
     * @return BigInteger
     * @access public
     */
    function __construct($x = 0, $base = 10)
    {
        if ( !defined('MATH_BIGINTEGER_MODE') ) {
            switch (true) {
                case extension_loaded('gmp'):
                    define('MATH_BIGINTEGER_MODE', MATH_BIGINTEGER_MODE_GMP);
                    break;
                case extension_loaded('bcmath'):
                    define('MATH_BIGINTEGER_MODE', MATH_BIGINTEGER_MODE_BCMATH);
                    break;
                default:
                    define('MATH_BIGINTEGER_MODE', MATH_BIGINTEGER_MODE_INTERNAL);
            }
        }

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

        if ($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:
                        $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'); // 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)
                    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 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;
                    default:
                        $x = ( strlen($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;
                    default:
                        $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.
     *
     * Here's an example:
     * <code>
     * <?php
     *    include('Math/BigInteger.php');
     *
     *    $a = new 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 = $this->compare(new BigInteger());
            if ($comparison == 0) {
                return $this->precision > 0 ? str_repeat(chr(0), ($this->precision + 1) >> 3) : '';
            }

            $temp = $comparison < 0 ? $this->add(new 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);
                }

                // 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 $this->precision > 0 ?
                    substr(str_pad($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 $this->precision > 0 ?
            substr(str_pad($result, $this->precision >> 3, chr(0), STR_PAD_LEFT), -($this->precision >> 3)) :
            $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 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 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 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 BigInteger();
        $divisor->value = array(10000000); // eg. 10**7
        $result = '';
        while (count($temp->value)) {
            list($temp, $mod) = $temp->divide($divisor);
            $result = str_pad($mod->value[0], 7, '0', STR_PAD_LEFT) . $result;
        }
        $result = ltrim($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 BigInteger
     */
    function copy()
    {
        $temp = new 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 BigInteger::__toString() directly in PHP5, you cannot call BigInteger::__clone()
     * directly in PHP5.  You can in PHP4 since it's not a magic method, but in PHP5, you have to call it by using the PHP5
     * only syntax of $y = clone $x.  As such, if you're trying to write an application that works on both PHP4 and PHP5,
     * call BigInteger::copy(), instead.
     *
     * @access public
     * @see copy()
     * @return BigInteger
     */
    function __clone()
    {
        return $this->copy();
    }

    /**
     * Adds two BigIntegers.
     *
     * Here's an example:
     * <code>
     * <?php
     *    include('Math/BigInteger.php');
     *
     *    $a = new BigInteger('10');
     *    $b = new BigInteger('20');
     *
     *    $c = $a->add($b);
     *
     *    echo $c->toString(); // outputs 30
     * ?>
     * </code>
     *
     * @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 $this->_normalize($temp);
            case MATH_BIGINTEGER_MODE_BCMATH:
                $temp = new BigInteger();
                $temp->value = bcadd($this->value, $y->value);

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

        $this_size = count($this->value);
        $y_size = count($y->value);

        if ($this_size == 0) {
            return $y->copy();
        } else if ($y_size == 0) {
            return $this->copy();
        }

        // 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 ) {
                $temp = new BigInteger();
                return $this->_normalize($temp);
            }

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

            $temp->is_negative = ($diff > 0) ? !$y_negative : $y_negative;

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

        $result = new BigInteger();
        $carry = 0;

        $size = max($this_size, $y_size);
        $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 = $sum >= MATH_BIGINTEGER_MAX_DIGIT52; // eg. floor($sum / 2**52); only possible values (in any base) are 0 and 1
            $sum = $carry ? $sum - MATH_BIGINTEGER_MAX_DIGIT52 : $sum;

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

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

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

        $result->is_negative = $this->is_negative;

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

    /**
     * Subtracts two BigIntegers.
     *
     * Here's an example:
     * <code>
     * <?php
     *    include('Math/BigInteger.php');
     *
     *    $a = new BigInteger('10');
     *    $b = new BigInteger('20');
     *
     *    $c = $a->subtract($b);
     *
     *    echo $c->toString(); // outputs -10
     * ?>
     * </code>
     *
     * @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 $this->_normalize($temp);
            case MATH_BIGINTEGER_MODE_BCMATH:
                $temp = new BigInteger();
                $temp->value = bcsub($this->value, $y->value);

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

        $this_size = count($this->value);
        $y_size = count($y->value);

        if ($this_size == 0) {
            $temp = $y->copy();
            $temp->is_negative = !$temp->is_negative;
            return $temp;
        } else if ($y_size == 0) {
            return $this->copy();
        }

        // add, if appropriate (ie. -$x - +$y or +$x - -$y)
        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 $this->_normalize($temp);
        }

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

        if ( !$diff ) {
            $temp = new BigInteger();
            return $this->_normalize($temp);
        }

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

        $result = new BigInteger();
        $carry = 0;

        $size = max($this_size, $y_size);
        $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 + MATH_BIGINTEGER_MAX_DIGIT52 : $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 $this->_normalize($result);
    }

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

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

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

        static $cutoff = false;
        if ($cutoff === false) {
            $cutoff = 2 * MATH_BIGINTEGER_KARATSUBA_CUTOFF;
        }

        if ( $this->equals($x) ) {
            return $this->_square();
        }

        $this_length = count($this->value);
        $x_length = count($x->value);

        if ( !$this_length || !$x_length ) { // a 0 is being multiplied
            $temp = new BigInteger();
            return $this->_normalize($temp);
        }

        $product = min($this_length, $x_length) < $cutoff ? $this->_multiply($x) : $this->_karatsuba($x);

        $product->is_negative = $this->is_negative != $x->is_negative;

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

    /**
     * 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 _barrett()
     * @param BigInteger $x
     * @return BigInteger
     * @access private
     */
    function _multiplyLower($x, $stop)
    {
        $this_length = count($this->value);
        $x_length = count($x->value);

        if ( !$this_length || !$x_length ) { // a 0 is being multiplied
            return new BigInteger();
        }

        if ( $this_length < $x_length ) {
            return $x->_multiplyLower($this, $stop);
        }

        $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;

        for ($j = 0; $j < $this_length; $j++) { // ie. $i = 0, $k = $i
            $temp = $this->value[$j] * $x->value[0] + $carry; // $product->value[$k] == 0
            $carry = floor($temp / 0x4000000);
            $product->value[$j] = $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 < $x_length; $i++) {
            $carry = 0;

            for ($j = 0, $k = $i; $j < $this_length && $k < $stop; $j++, $k++) {
                $temp = $product->value[$k] + $this->value[$j] * $x->value[$i] + $carry;
                $carry = floor($temp / 0x4000000);
                $product->value[$k] = $temp - 0x4000000 * $carry;
            }

            if ($k < $stop) {
                $product->value[$k] = $carry;
            }
        }

        $product->is_negative = $this->is_negative != $x->is_negative;

        return $product;
    }

    /**
     * Performs long multiplication on two BigIntegers
     *
     * Modeled after 'multiply' in MutableBigInteger.java.
     *
     * @param BigInteger $x
     * @return BigInteger
     * @access private
     */
    function _multiply($x)
    {
        $this_length = count($this->value);
        $x_length = count($x->value);

        if ( !$this_length || !$x_length ) { // a 0 is being multiplied
            return new BigInteger();
        }

        if ( $this_length < $x_length ) {
            return $x->_multiply($this);
        }

        $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;

        for ($j = 0; $j < $this_length; $j++) { // ie. $i = 0
            $temp = $this->value[$j] * $x->value[0] + $carry; // $product->value[$k] == 0
            $carry = floor($temp / 0x4000000);
            $product->value[$j] = $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 < $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 $this->_normalize($product);
    }

    /**
     * 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 BigInteger $y
     * @return BigInteger
     * @access private
     */
    function _karatsuba($y)
    {
        $x = $this->copy();

        $m = min(count($x->value) >> 1, count($y->value) >> 1);

        if ($m < MATH_BIGINTEGER_KARATSUBA_CUTOFF) {
            return $x->_multiply($y);
        }

        $x1 = new BigInteger();
        $x0 = new BigInteger();
        $y1 = new BigInteger();
        $y0 = new BigInteger();

        $x1->value = array_slice($x->value, $m);
        $x0->value = array_slice($x->value, 0, $m);
        $y1->value = array_slice($y->value, $m);
        $y0->value = array_slice($y->value, 0, $m);

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

        $z1 = $x1->add($x0);
        $z1 = $z1->_karatsuba($y1->add($y0));
        $z1 = $z1->subtract($z2->add($z0));

        $z2->value = array_merge(array_fill(0, 2 * $m, 0), $z2->value);
        $z1->value = array_merge(array_fill(0,     $m, 0), $z1->value);

        $xy = $z2->add($z1);
        $xy = $xy->add($z0);

        return $xy;
    }

    /**
     * Squares a BigInteger
     *
     * @return BigInteger
     * @access private
     */
    function _square()
    {
        static $cutoff = false;
        if ($cutoff === false) {
            $cutoff = 2 * MATH_BIGINTEGER_KARATSUBA_CUTOFF;
        }

        return count($this->value) < $cutoff ? $this->_baseSquare() : $this->_karatsubaSquare();
    }

    /**
     * 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.
     *
     * @return BigInteger
     * @access private
     */
    function _baseSquare()
    {
        if ( empty($this->value) ) {
            return new BigInteger();
        }

        $square = new BigInteger();
        $square->value = $this->_array_repeat(0, 2 * count($this->value));

        for ($i = 0, $max_index = count($this->value) - 1; $i <= $max_index; $i++) {
            $i2 = 2 * $i;

            $temp = $square->value[$i2] + $this->value[$i] * $this->value[$i];
            $carry = floor($temp / 0x4000000);
            $square->value[$i2] = $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 * $this->value[$j] * $this->value[$i] + $carry;
                $carry = floor($temp / 0x4000000);
                $square->value[$k] = $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;
    }

    /**
     * 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 BigInteger $y
     * @return BigInteger
     * @access private
     */
    function _karatsubaSquare()
    {
        $m = count($this->value) >> 1;

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

        $x1 = new BigInteger();
        $x0 = new BigInteger();

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

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

        $z1 = $x1->add($x0);
        $z1 = $z1->_karatsubaSquare();
        $z1 = $z1->subtract($z2->add($z0));

        $z2->value = array_merge(array_fill(0, 2 * $m, 0), $z2->value);
        $z1->value = array_merge(array_fill(0,     $m, 0), $z1->value);

        $xx = $z2->add($z1);
        $xx = $xx->add($z0);

        return $xx;
    }

    /**
     * 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 BigInteger('10');
     *    $b = new BigInteger('20');
     *
     *    list($quotient, $remainder) = $a->divide($b);
     *
     *    echo $quotient->toString(); // outputs 0
     *    echo "\r\n";
     *    echo $remainder->toString(); // outputs 10
     * ?>
     * </code>
     *
     * @param BigInteger $y
     * @return Array
     * @access public
     * @internal This function is based off of {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=9 HAC 14.20}.
     */
    function divide($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($this->_normalize($quotient), $this->_normalize($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($this->_normalize($quotient), $this->_normalize($remainder));
        }

        if (count($y->value) == 1) {
            $temp = $this->_divide_digit($y->value[0]);
            $temp[0]->is_negative = $this->is_negative != $y->is_negative;
            return array($this->_normalize($temp[0]), $this->_normalize($temp[1]));
        }

        static $zero;
        if (!isset($zero)) {
            $zero = new 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 BigInteger();
            $temp->value = array(1);
            $temp->is_negative = $x_sign != $y_sign;
            return array($this->_normalize($temp), $this->_normalize(new BigInteger()));
        }

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

        $x_max = count($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 ) ? $x->value[$i - 2] : 0
            );
            $y_value = array(
                $y->value[$y_max],
                ( $y_max > 0 ) ? $y->value[$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);
            }

            $adjust = $this->_array_repeat(0, $q_index);
            $temp = new BigInteger();
            $temp->value = array($quotient->value[$q_index]);
            $temp = $temp->multiply($y);
            $temp->value = array_merge($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 BigInteger $divisor
     * @return Array
     * @access public
     */
    function _divide_digit($divisor)
    {
        $carry = 0;
        $result = new BigInteger();

        for ($i = count($this->value) - 1; $i >= 0; $i--) {
            $temp = 0x4000000 * $carry + $this->value[$i];
            $result->value[$i] = floor($temp / $divisor);
            $carry = fmod($temp, $divisor);
        }

        $remainder = new BigInteger();
        $remainder->value = array($carry);

        return array($result, $remainder);
    }

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

        if ($e->compare(new 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 BigInteger();
                $temp->value = gmp_powm($this->value, $e->value, $n->value);

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

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

        if ( empty($e->value) ) {
            $temp = new 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 = $this->_square();
            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)
        …