/vendor/phpseclib/Math/BigInteger.php

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

namespace PHPSecLib\Math;

/**
 * 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.
 *
 * Numbers are stored in {@link http://en.wikipedia.org/wiki/Endianness little endian} format.  ie.
 * (new \PHPSecLib\Math\BigInteger(pow(2, 26)))->value = array(0, 1)
 *
 * Useful resources are as follows:
 *
 *  - {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf Handbook of Applied Cryptography (HAC)}
 *  - {@link http://math.libtomcrypt.com/files/tommath.pdf Multi-Precision Math (MPM)}
 *  - Java's BigInteger classes.  See /j2se/src/share/classes/java/math in jdk-1_5_0-src-jrl.zip
 *
 * Here's an example of how to use this library:
 * <code>
 * <?php
 *    $a = new \PHPSecLib\Math\BigInteger(2);
 *    $b = new \PHPSecLib\Math\BigInteger(3);
 *
 *    $c = $a->add($b);
 *
 *    echo $c->toString(); // outputs 5
 * ?>
 * </code>
 *
 * LICENSE: Permission is hereby granted, free of charge, to any person obtaining a copy
 * of this software and associated documentation files (the "Software"), to deal
 * in the Software without restriction, including without limitation the rights
 * to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
 * copies of the Software, and to permit persons to whom the Software is
 * furnished to do so, subject to the following conditions:
 *
 * The above copyright notice and this permission notice shall be included in
 * all copies or substantial portions of the Software.
 *
 * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
 * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
 * FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
 * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
 * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
 * OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
 * THE SOFTWARE.
 *
 * @category  Math
 * @package   Math_BigInteger
 * @author    Jim Wigginton <terrafrost@php.net>
 * @copyright MMVI Jim Wigginton
 * @license   http://www.opensource.org/licenses/mit-license.html  MIT License
 * @link      http://pear.php.net/package/Math_BigInteger
 */

use \phpseclib\Crypt\Random;

/**#@+
 * Reduction constants
 *
 * @access private
 * @see BigInteger::_reduce()
 */
/**
 * @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);
/**#@-*/

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

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

/**#@+
 * Mode constants.
 *
 * @access private
 * @see BigInteger::__construct()
 */
/**
 * 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);
/**#@-*/

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

/**
 * Pure-PHP arbitrary precision integer arithmetic library. Supports base-2, base-10, base-16, and base-256
 * numbers.
 *
 * @package Math_BigInteger
 * @author  Jim Wigginton <terrafrost@php.net>
 * @access  public
 */
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
     *
     * @access private
     */
    var $generator = 'mt_rand';

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

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

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

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

        if (function_exists('openssl_public_encrypt') && !defined('MATH_BIGINTEGER_OPENSSL_DISABLE') && !defined('MATH_BIGINTEGER_OPENSSL_ENABLED')) {
            // some versions of XAMPP have mismatched versions of OpenSSL which causes it not to work
            ob_start();
            @phpinfo();
            $content = ob_get_contents();
            ob_end_clean();

            preg_match_all('#OpenSSL (Header|Library) Version(.*)#im', $content, $matches);

            $versions = array();
            if (!empty($matches[1])) {
                for ($i = 0; $i < count($matches[1]); $i++) {
                    $versions[$matches[1][$i]] = trim(str_replace('=>', '', strip_tags($matches[2][$i])));
                }
            }

            // it doesn't appear that OpenSSL versions were reported upon until PHP 5.3+
            switch (true) {
                case !isset($versions['Header']):
                case !isset($versions['Library']):
                case $versions['Header'] == $versions['Library']:
                    define('MATH_BIGINTEGER_OPENSSL_ENABLED', true);
                    break;
                default:
                    define('MATH_BIGINTEGER_OPENSSL_DISABLE', true);
            }
        }

        if (!defined('PHP_INT_SIZE')) {
            define('PHP_INT_SIZE', 4);
        }

        if (!defined('MATH_BIGINTEGER_BASE') && MATH_BIGINTEGER_MODE == MATH_BIGINTEGER_MODE_INTERNAL) {
            switch (PHP_INT_SIZE) {
                case 8: // use 64-bit integers if int size is 8 bytes
                    define('MATH_BIGINTEGER_BASE',       31);
                    define('MATH_BIGINTEGER_BASE_FULL',  0x80000000);
                    define('MATH_BIGINTEGER_MAX_DIGIT',  0x7FFFFFFF);
                    define('MATH_BIGINTEGER_MSB',        0x40000000);
                    // 10**9 is the closest we can get to 2**31 without passing it
                    define('MATH_BIGINTEGER_MAX10',      1000000000);
                    define('MATH_BIGINTEGER_MAX10_LEN',  9);
                    // the largest digit that may be used in addition / subtraction
                    define('MATH_BIGINTEGER_MAX_DIGIT2', pow(2, 62));
                    break;
                //case 4: // use 64-bit floats if int size is 4 bytes
                default:
                    define('MATH_BIGINTEGER_BASE',       26);
                    define('MATH_BIGINTEGER_BASE_FULL',  0x4000000);
                    define('MATH_BIGINTEGER_MAX_DIGIT',  0x3FFFFFF);
                    define('MATH_BIGINTEGER_MSB',        0x2000000);
                    // 10**7 is the closest to 2**26 without passing it
                    define('MATH_BIGINTEGER_MAX10',      10000000);
                    define('MATH_BIGINTEGER_MAX10_LEN',  7);
                    // 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.
                    define('MATH_BIGINTEGER_MAX_DIGIT2', pow(2, 52));
            }
        }

        switch ( MATH_BIGINTEGER_MODE ) {
            case MATH_BIGINTEGER_MODE_GMP:
                switch (true) {
                    case is_resource($x) && get_resource_type($x) == 'GMP integer':
                    // PHP 5.6 switched GMP from using resources to objects
                    case is_object($x) && get_class($x) == 'GMP':
                        $this->value = $x;
                        return;
                }
                $this->value = gmp_init(0);
                break;
            case MATH_BIGINTEGER_MODE_BCMATH:
                $this->value = '0';
                break;
            default:
                $this->value = array();
        }

        // '0' counts as empty() but when the base is 256 '0' is equal to ord('0') or 48
        // '0' is the only value like this per http://php.net/empty
        if (empty($x) && (abs($base) != 256 || $x !== '0')) {
            return;
        }

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

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

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

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

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

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

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

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

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

                if ($is_negative) {
                    $temp = $this->add(new static('-1'));
                    $this->value = $temp->value;
                }
                break;
            case  10:
            case -10:
                // (?<!^)(?:-).*: find any -'s that aren't at the beginning and then any characters that follow that
                // (?<=^|-)0*: find any 0's that are preceded by the start of the string or by a - (ie. octals)
                // [^-0-9].*: find any non-numeric characters and then any characters that follow that
                $x = preg_replace('#(?<!^)(?:-).*|(?<=^|-)0*|[^-0-9].*#', '', $x);

                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 = $x === '-' ? '0' : (string) $x;
                        break;
                    default:
                        $temp = new static();

                        $multiplier = new static();
                        $multiplier->value = array(MATH_BIGINTEGER_MAX10);

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

                        $x = str_pad($x, strlen($x) + ((MATH_BIGINTEGER_MAX10_LEN - 1) * strlen($x)) % MATH_BIGINTEGER_MAX10_LEN, 0, STR_PAD_LEFT);
                        while (strlen($x)) {
                            $temp = $temp->multiply($multiplier);
                            $temp = $temp->add(new static($this->_int2bytes(substr($x, 0, MATH_BIGINTEGER_MAX10_LEN)), 256));
                            $x = substr($x, MATH_BIGINTEGER_MAX10_LEN);
                        }

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

        $temp = $this->copy();

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

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

    /**
     * Converts a BigInteger to a hex string (eg. base-16)).
     *
     * Negative numbers are saved as positive numbers, unless $twos_compliment is set to true, at which point, they're
     * saved as two's compliment.
     *
     * Here's an example:
     * <code>
     * <?php
     *    $a = new \PHPSecLib\Math\BigInteger('65');
     *
     *    echo $a->toHex(); // outputs '41'
     * ?>
     * </code>
     *
     * @param 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
     *    $a = new \PHPSecLib\Math\BigInteger('65');
     *
     *    echo $a->toBits(); // outputs '1000001'
     * ?>
     * </code>
     *
     * @param Boolean $twos_compliment
     * @return String
     * @access public
     * @internal Converts a base-2**26 number to base-2**2
     */
    function toBits($twos_compliment = false)
    {
        $hex = $this->toHex($twos_compliment);
        $bits = '';
        for ($i = strlen($hex) - 8, $start = strlen($hex) & 7; $i >= $start; $i-=8) {
            $bits = str_pad(decbin(hexdec(substr($hex, $i, 8))), 32, '0', STR_PAD_LEFT) . $bits;
        }
        if ($start) { // hexdec('') == 0
            $bits = str_pad(decbin(hexdec(substr($hex, 0, $start))), 8, '0', STR_PAD_LEFT) . $bits;
        }
        $result = $this->precision > 0 ? substr($bits, -$this->precision) : ltrim($bits, '0');

        if ($twos_compliment && $this->compare(new static()) > 0 && $this->precision <= 0) {
            return '0' . $result;
        }

        return $result;
    }

    /**
     * Converts a BigInteger to a base-10 number.
     *
     * Here's an example:
     * <code>
     * <?php
     *    $a = new \PHPSecLib\Math\BigInteger('50');
     *
     *    echo $a->toString(); // outputs 50
     * ?>
     * </code>
     *
     * @return String
     * @access public
     * @internal Converts a base-2**26 number to base-10**7 (which is pretty much base-10)
     */
    function toString()
    {
        switch ( MATH_BIGINTEGER_MODE ) {
            case MATH_BIGINTEGER_MODE_GMP:
                return gmp_strval($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 static();
        $divisor->value = array(MATH_BIGINTEGER_MAX10);
        $result = '';
        while (count($temp->value)) {
            list($temp, $mod) = $temp->divide($divisor);
            $result = str_pad(isset($mod->value[0]) ? $mod->value[0] : '', MATH_BIGINTEGER_MAX10_LEN, '0', STR_PAD_LEFT) . $result;
        }
        $result = ltrim($result, '0');
        if (empty($result)) {
            $result = '0';
        }

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

        return $result;
    }

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

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

    }

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

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

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

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

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

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

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

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

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

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

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

            return $temp;
        }

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

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

        $carry = 0;
        for ($i = 0, $j = 1; $j < $size; $i+=2, $j+=2) {
            $sum = $x_value[$j] * MATH_BIGINTEGER_BASE_FULL + $x_value[$i] + $y_value[$j] * MATH_BIGINTEGER_BASE_FULL + $y_value[$i] + $carry;
            $carry = $sum >= MATH_BIGINTEGER_MAX_DIGIT2; // eg. floor($sum / 2**52); only possible values (in any base) are 0 and 1
            $sum = $carry ? $sum - MATH_BIGINTEGER_MAX_DIGIT2 : $sum;

            $temp = MATH_BIGINTEGER_BASE === 26 ? intval($sum / 0x4000000) : ($sum >> 31);

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

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

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

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

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

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

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

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

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

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

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

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

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

            return $temp;
        }

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

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

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

            $x_negative = !$x_negative;

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

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

        $carry = 0;
        for ($i = 0, $j = 1; $j < $y_size; $i+=2, $j+=2) {
            $sum = $x_value[$j] * MATH_BIGINTEGER_BASE_FULL + $x_value[$i] - $y_value[$j] * MATH_BIGINTEGER_BASE_FULL - $y_value[$i] - $carry;
            $carry = $sum < 0; // eg. floor($sum / 2**52); only possible values (in any base) are 0 and 1
            $sum = $carry ? $sum + MATH_BIGINTEGER_MAX_DIGIT2 : $sum;

            $temp = MATH_BIGINTEGER_BASE === 26 ? intval($sum / 0x4000000) : ($sum >> 31);

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

        $carry = 0;

        for ($j = 0; $j < $x_length; ++$j) { // ie. $i = 0
            $temp = $x_value[$j] * $y_value[0] + $carry; // $product_value[$k] == 0
            $carry = MATH_BIGINTEGER_BASE === 26 ? intval($temp / 0x4000000) : ($temp >> 31);
            $product_value[$j] = (int) ($temp - MATH_BIGINTEGER_BASE_FULL * $carry);
        }

        $product_value[$j] = $carry;

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

            for ($j = 0, $k = $i; $j < $x_length; ++$j, ++$k) {
                $temp = $product_value[$k] + $x_value[$j] * $y_value[$i] + $carry;
                $carry = MATH_BIGINTEGER_BASE === 26 ? intval($temp / 0x4000000) : ($temp >> 31);
                $product_value[$k] = (int) ($temp - MATH_BIGINTEGER_BASE_FULL * $carry);
            }

            $product_value[$k] = $carry;
        }

        return $product_value;
    }

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

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

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

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

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

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

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

        return $xy[MATH_BIGINTEGER_VALUE];
    }

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

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

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

            $temp = $square_value[$i2] + $value[$i] * $value[$i];
            $carry = MATH_BIGINTEGER_BASE === 26 ? intval($temp / 0x4000000) : ($temp >> 31);
            $square_value[$i2] = (int) ($temp - MATH_BIGINTEGER_BASE_FULL * $carry);

            // note how we start from $i+1 instead of 0 as we do in multiplication.
            for ($j = $i + 1, $k = $i2 + 1; $j <= $max_index; ++$j, ++$k) {
                $temp = $square_value[$k] + 2 * $value[$j] * $value[$i] + $carry;
                $carry = MATH_BIGINTEGER_BASE === 26 ? intval($temp / 0x4000000) : ($temp >> 31);
                $square_value[$k] = (int) ($temp - MATH_BIGINTEGER_BASE_FULL * $carry);
            }

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

        return $square_value;
    }

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

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

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

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

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

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

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

        return $xx[MATH_BIGINTEGER_VALUE];
    }

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

        $quotient = new static();
  …