/package/app/dwh/etlsource/scripts/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)

 *

 * Math_BigInteger uses base-2**26 to perform operations such as multiplication and division and

 * base-2**52 (ie. two base 2**26 digits) to perform addition and subtraction.  Because the largest possible

 * value when multiplying two base-2**26 numbers together is a base-2**52 number, double precision floating

 * point numbers - numbers that should be supported on most hardware and whose significand is 53 bits - are

 * used.  As a consequence, bitwise operators such as >> and << cannot be used, nor can the modulo operator %,

 * which only supports integers.  Although this fact will slow this library down, the fact that such a high

 * base is being used should more than compensate.

 *

 * When PHP version 6 is officially released, we'll be able to use 64-bit integers.  This should, once again,

 * allow bitwise operators, and will increase the maximum possible base to 2**31 (or 2**62 for addition /

 * subtraction).

 *

 * 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 a quick 'n dirty example of how to use this library:

 * <code>

 * <?php

 *    include('Math/BigInteger.php');

 *

 *    $a = new Math_BigInteger(2);

 *    $b = new Math_BigInteger(3);

 *

 *    $c = $a->add($b);

 *

 *    echo $c->toString(); // outputs 5

 * ?>

 * </code>

 *

 * LICENSE: This library is free software; you can redistribute it and/or

 * modify it under the terms of the GNU Lesser General Public

 * License as published by the Free Software Foundation; either

 * version 2.1 of the License, or (at your option) any later version.

 *

 * This library is distributed in the hope that it will be useful,

 * but WITHOUT ANY WARRANTY; without even the implied warranty of

 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU

 * Lesser General Public License for more details.

 *

 * You should have received a copy of the GNU Lesser General Public

 * License along with this library; if not, write to the Free Software

 * Foundation, Inc., 59 Temple Place, Suite 330, Boston,

 * MA  02111-1307  USA

 *

 * @category   Math

 * @package    Math_BigInteger

 * @author     Jim Wigginton <terrafrost@php.net>

 * @copyright  MMVI Jim Wigginton

 * @license    http://www.gnu.org/licenses/lgpl.txt

 * @version    CVS: $Id: BigInteger.php,v 1.4 2007/01/29 17:19:11 terrafrost Exp $

 * @link       http://pear.php.net/package/Math_BigInteger

 */



// array_fill requires PHP 4.2.0+, per http://php.net/function.array_fill

// see http://www.frostjedi.com/terra/scripts/pear/array_fill.phps

// require_once 'PHP/Compat/Function/array_fill.php';



/**#@+

 * @access private

 * @see Math_BigInteger::_slidingWindow()

 */

/**

 * @see Math_BigInteger::_montgomery()

 * @see Math_BigInteger::_undoMontgomery()

 */

define('MATH_BIGINTEGER_MONTGOMERY', 0);

/**

 * @see Math_BigInteger::_barrett()

 */

define('MATH_BIGINTEGER_BARRETT', 1);

/**

 * @see Math_BigInteger::_mod2()

 */

define('MATH_BIGINTEGER_POWEROF2', 2);

/**

 * @see Math_BigInteger::_remainder()

 */

define('MATH_BIGINTEGER_CLASSIC', 3);

/**

 * @see Math_BigInteger::_copy()

 */

define('MATH_BIGINTEGER_NONE', 4);

/**#@-*/



/**#@+

 * @access private

 * @see Math_BigInteger::_montgomery()

 * @see Math_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 Math_BigInteger::Math_BigInteger()

 */

/**

 * To use the pure-PHP implementation

 */

define('MATH_BIGINTEGER_MODE_INTERNAL', 1);

/**

 * To use the BCMath library

 *

 * (if enabled; otherwise, the internal implementation will be used)

 */

define('MATH_BIGINTEGER_MODE_BCMATH', 2);

/**

 * To use the GMP library

 *

 * (if present; otherwise, either the BCMath or the internal implementation will be used)

 */

define('MATH_BIGINTEGER_MODE_GMP', 3);

/**#@-*/



/**

 * Pure-PHP arbitrary precission integer arithmetic library. Supports base-2, base-10, base-16, and base-256

 * numbers.  Negative numbers are supported in all publically accessable functions save for modPow

 * and modInverse.

 *

 * @author  Jim Wigginton <terrafrost@php.net>

 * @version 1.0.0RC3

 * @access  public

 * @package Math_BigInteger

 */

class Math_BigInteger {

    /**

     * Holds the BigInteger's value.

     *

     * @var Array

     * @access private

     */

    var $value;



    /**

     * Holds the BigInteger's magnitude.

     *

     * @var Boolean

     * @access private

     */

    var $is_negative = false;



    /**

     * Converts base-2, base-10, base-16, and binary strings (eg. base-256) to BigIntegers.

     *

     * Here's a quick 'n dirty example:

     * <code>

     * <?php

     *    include('Math/BigInteger.php');

     *

     *    $a = new Math_BigInteger('0x32', 16); // 50 in base-16

     *

     *    echo $a->toString(); // outputs 50

     * ?>

     * </code>

     *

     * @param optional $x base-10 number or base-$base number if $base set.

     * @param optional integer $base

     * @return Math_BigInteger

     * @access public

     */

    function Math_BigInteger($x = 0, $base = 10)

    {

        if ( !defined('MATH_BIGINTEGER_MODE') ) {

            switch (true) {

                case extension_loaded('gmp'):

                    define('MATH_BIGINTEGER_MODE', MATH_BIGINTEGER_MODE_GMP);

                    break;

                case extension_loaded('bcmath'):

                    define('MATH_BIGINTEGER_MODE', MATH_BIGINTEGER_MODE_BCMATH);

                    break;

                default:

                    define('MATH_BIGINTEGER_MODE', MATH_BIGINTEGER_MODE_INTERNAL);

            }

        }



        switch ( MATH_BIGINTEGER_MODE ) {

            case MATH_BIGINTEGER_MODE_GMP:

                $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:

                switch ( MATH_BIGINTEGER_MODE ) {

                    case MATH_BIGINTEGER_MODE_GMP:

                        $temp = unpack('H*hex', $x);

                        $this->value = gmp_init('0x' . $temp['hex']);

                        break;

                    case MATH_BIGINTEGER_MODE_BCMATH:

                        $len = strlen($x);

                        $len+= (3 * $len) % 4; // rounds $len to the nearest 4.



                        $x = str_pad($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})));

                        }



                        break;

                    // converts a base-2**8 (big endian / msb) number to base-2**26 (little endian / lsb)

                    case MATH_BIGINTEGER_MODE_INTERNAL:

                        while (strlen($x)) {

                            $this->value[] = $this->_bytes2int($this->_base256_rshift($x, 26));

                        }

                }

                break;

            case 16:

                if ($x{0} == '-') {

                    $this->is_negative = true;

                    $x = substr($x, 1);

                }



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



                switch ( MATH_BIGINTEGER_MODE ) {

                    case MATH_BIGINTEGER_MODE_GMP:

                        $temp = $this->is_negative ? '-0x' . $x : '0x' . $x;

                        $this->value = gmp_init($temp);

                        $this->is_negative = false;

                        break;

                    case MATH_BIGINTEGER_MODE_BCMATH:

                        $x = ( strlen($x) & 1 ) ? '0' . $x : $x;

                        $temp = new Math_BigInteger(pack('H*', $x), 256);

                        $this->value = $this->is_negative ? '-' . $temp->value : $temp->value;

                        $this->is_negative = false;

                        break;

                    case MATH_BIGINTEGER_MODE_INTERNAL:

                        $x = ( strlen($x) & 1 ) ? '0' . $x : $x;

                        $temp = new Math_BigInteger(pack('H*', $x), 256);

                        $this->value = $temp->value;

                }

                break;

            case 10:

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



                switch ( MATH_BIGINTEGER_MODE ) {

                    case MATH_BIGINTEGER_MODE_GMP:

                        $this->value = gmp_init($x);

                        break;

                    case MATH_BIGINTEGER_MODE_BCMATH:

                        // explicitly casting $x to a string is necessary, here, since doing $x{0} on -1 yields different

                        // results then doing it on '-1' does (modInverse does $x{0})

                        $this->value = (string) $x;

                        break;

                    case MATH_BIGINTEGER_MODE_INTERNAL:

                        $temp = new Math_BigInteger();



                        // array(10000000) is 10**7 in base-2**26.  10**7 is the closest to 2**26 we can get without passing it.

                        $multiplier = new Math_BigInteger();

                        $multiplier->value = array(10000000);



                        if ($x{0} == '-') {

                            $this->is_negative = true;

                            $x = substr($x, 1);

                        }



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



                        while (strlen($x)) {

                            $temp = $temp->multiply($multiplier);

                            $temp = $temp->add(new Math_BigInteger($this->_int2bytes(substr($x, 0, 7)), 256));

                            $x = substr($x, 7);

                        }



                        $this->value = $temp->value;

                }

                break;

            case 2: // base-2 support originally implemented by Lluis Pamies - thanks!

                if (MATH_BIGINTEGER_MODE == MATH_BIGINTEGER_MODE_GMP) {

                    $this->value = gmp_init($x, 2);

                    break;

                }



                if ($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);

                }



                $temp = new Math_BigInteger($str, 16);

                $this->value = $temp->value;



                if (MATH_BINGINTEGER_MODE == MATH_BIGINTEGER_MODE_BCMATH && $this->is_negative) {

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

                    $this->is_negative = false;

                }



                break;

            default:

                // base not supported, so we'll let $this == 0

        }

    }



    /**

     * Converts a BigInteger to a byte string (eg. base-256).

     *

     * Here's a quick 'n dirty example:

     * <code>

     * <?php

     *    include('Math/BigInteger.php');

     *

     *    $a = new Math_BigInteger('65');

     *

     *    echo $a->toBytes(); // outputs chr(65)

     * ?>

     * </code>

     *

     * @return String

     * @access public

     * @internal Converts a base-2**26 number to base-2**8

     */

    function toBytes()

    {

        switch ( MATH_BIGINTEGER_MODE ) {

            case MATH_BIGINTEGER_MODE_GMP:

                if (gmp_cmp($this->value, gmp_init(0)) == 0) {

                    return '';

                }



                $temp = gmp_strval(gmp_abs($this->value), 16);

                $temp = ( strlen($temp) & 1 ) ? '0' . $temp : $temp;



                return ltrim(pack('H*', $temp), chr(0));

            case MATH_BIGINTEGER_MODE_BCMATH:

                if ($this->value === '0') {

                    return '';

                }



                $value = '';

                $current = $this->value;                 



                // we don't do four bytes at a time because then numbers larger than 1<<31 would be negative

                // two's complimented numbers, which would break chr.

                while (bccomp($current, '0') > 0) {

                    $temp = bcmod($current, 0x1000000);

                    $value = chr($temp >> 16) . chr($temp >> 8) . chr($temp) . $value;

                    $current = bcdiv($current, 0x1000000);

                }



                return ltrim($value, chr(0));

        }



        if (!count($this->value)) {

            return '';

        }



        $result = $this->_int2bytes($this->value[count($this->value) - 1]);



        $temp = $this->_copy();



        for ($i = count($temp->value) - 2; $i >= 0; $i--) {

            $temp->_base256_lshift($result, 26);

            $result = $result | str_pad($temp->_int2bytes($temp->value[$i]), strlen($result), chr(0), STR_PAD_LEFT);

        }



        return $result;

    }



    /**

     * Converts a BigInteger to a base-10 number.

     *

     * Here's a quick 'n dirty example:

     * <code>

     * <?php

     *    include('Math/BigInteger.php');

     *

     *    $a = new Math_BigInteger('50');

     *

     *    echo $a->toString(); // outputs 50

     * ?>

     * </code>

     *

     * @return String

     * @access public

     * @internal Converts a base-2**26 number to base-10**7 (which is pretty much base-10)

     */

    function toString()

    {

        switch ( MATH_BIGINTEGER_MODE ) {

            case MATH_BIGINTEGER_MODE_GMP:

                return gmp_strval($this->value);

            case MATH_BIGINTEGER_MODE_BCMATH:

                if ($this->value === '0') {

                    return '0';

                }



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

        }



        if (!count($this->value)) {

            return '0';

        }



        $result = ($this->is_negative) ? '-' : '';



        $temp = $this->_copy();



        $divisor = new Math_BigInteger();

        $divisor->value = array(10000000); // eg. 10**7



        while (count($temp->value)) {

            list($temp, $mod) = $temp->divide($divisor);

            $result = str_pad($this->_bytes2int($mod->toBytes()), 7, '0', STR_PAD_LEFT) . $result;

        }



        return ltrim($result, '0');

    }



    /**

     * Adds two BigIntegers.

     *

     * Here's a quick 'n dirty example:

     * <code>

     * <?php

     *    include('Math/BigInteger.php');

     *

     *    $a = new Math_BigInteger('10');

     *    $b = new Math_BigInteger('20');

     *

     *    $c = $a->add($b);

     *

     *    echo $c->toString(); // outputs 30

     * ?>

     * </code>

     *

     * @param Math_BigInteger $y

     * @return Math_BigInteger

     * @access public

     * @internal Performs base-2**52 addition

     */

    function add($y)

    {

        switch ( MATH_BIGINTEGER_MODE ) {

            case MATH_BIGINTEGER_MODE_GMP:

                $temp = new Math_BigInteger();

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



                return $temp;

            case MATH_BIGINTEGER_MODE_BCMATH:

                $temp = new Math_BigInteger();

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



                return $temp;

        }



        // subtract, if appropriate

        if ( $this->is_negative != $y->is_negative ) {

            // is $y the negative number?

            $y_negative = $this->compare($y) > 0;



            $temp = $this->_copy();

            $y = $y->_copy();

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



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

            if ( !$diff ) {

                return new Math_BigInteger();

            }



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



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



            return $temp;

        }



        $result = new Math_BigInteger();

        $carry = 0;



        $size = max(count($this->value), count($y->value));

        $size+= $size % 2; // rounds $size to the nearest 2.



        $x = array_pad($this->value, $size,0);

        $y = array_pad($y->value, $size, 0);



        for ($i = 0; $i < $size - 1; $i+=2) {

            $sum = $x[$i + 1] * 0x4000000 + $x[$i] + $y[$i + 1] * 0x4000000 + $y[$i] + $carry;

            $carry = $sum >= 4503599627370496; // eg. floor($sum / 2**52); only possible values (in any base) are 0 and 1

            $sum = $carry ? $sum - 4503599627370496 : $sum;



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



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

            $result->value[] = $temp;

        }



        if ($carry) {

            $result->value[] = $carry;

        }



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



        return $result->_normalize();

    }



    /**

     * Subtracts two BigIntegers.

     *

     * Here's a quick 'n dirty example:

     * <code>

     * <?php

     *    include('Math/BigInteger.php');

     *

     *    $a = new Math_BigInteger('10');

     *    $b = new Math_BigInteger('20');

     *

     *    $c = $a->subtract($b);

     *

     *    echo $c->toString(); // outputs -10

     * ?>

     * </code>

     *

     * @param Math_BigInteger $y

     * @return Math_BigInteger

     * @access public

     * @internal Performs base-2**52 subtraction

     */

    function subtract($y)

    {

        switch ( MATH_BIGINTEGER_MODE ) {

            case MATH_BIGINTEGER_MODE_GMP:

                $temp = new Math_BigInteger();

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



                return $temp;

            case MATH_BIGINTEGER_MODE_BCMATH:

                $temp = new Math_BigInteger();

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



                return $temp;

        }



        // add, if appropriate

        if ( $this->is_negative != $y->is_negative ) {

            $is_negative = $y->compare($this) > 0;



            $temp = $this->_copy();

            $y = $y->_copy();

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



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



            $temp->is_negative = $is_negative;



            return $temp;

        }



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



        if ( !$diff ) {

            return new Math_BigInteger();

        }



        // switch $this and $y around, if appropriate.

        if ( (!$this->is_negative && $diff < 0) || ($this->is_negative && $diff > 0) ) {

            $is_negative = $y->is_negative;



            $temp = $this->_copy();

            $y = $y->_copy();

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



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

            $temp->is_negative = !$is_negative;



            return $temp;

        }



        $result = new Math_BigInteger();

        $carry = 0;



        $size = max(count($this->value), count($y->value));

        $size+= $size % 2;



        $x = array_pad($this->value, $size, 0);

        $y = array_pad($y->value, $size, 0);



        for ($i = 0; $i < $size - 1;$i+=2) {

            $sum = $x[$i + 1] * 0x4000000 + $x[$i] - $y[$i + 1] * 0x4000000 - $y[$i] + $carry;

            $carry = $sum < 0 ? -1 : 0; // eg. floor($sum / 2**52); only possible values (in any base) are 0 and 1

            $sum = $carry ? $sum + 4503599627370496 : $sum;



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



            $result->value[] = $sum - 0x4000000 * $temp;

            $result->value[] = $temp;

        }



        // $carry shouldn't be anything other than zero, at this point, since we already made sure that $this

        // was bigger than $y.



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



        return $result->_normalize();

    }



    /**

     * Multiplies two BigIntegers

     *

     * Here's a quick 'n dirty example:

     * <code>

     * <?php

     *    include('Math/BigInteger.php');

     *

     *    $a = new Math_BigInteger('10');

     *    $b = new Math_BigInteger('20');

     *

     *    $c = $a->multiply($b);

     *

     *    echo $c->toString(); // outputs 200

     * ?>

     * </code>

     *

     * @param Math_BigInteger $x

     * @return Math_BigInteger

     * @access public

     * @internal Modeled after 'multiply' in MutableBigInteger.java.

     */

    function multiply($x)

    {

        switch ( MATH_BIGINTEGER_MODE ) {

            case MATH_BIGINTEGER_MODE_GMP:

                $temp = new Math_BigInteger();

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



                return $temp;

            case MATH_BIGINTEGER_MODE_BCMATH:

                $temp = new Math_BigInteger();

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



                return $temp;

        }



        if ( !$this->compare($x) ) {

            return $this->_square();

        }



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

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



        if ( !$this_length || !$x_length ) { // a 0 is being multiplied

            return new Math_BigInteger();

        }



        $product = new Math_BigInteger();

        $product->value = $this->_array_repeat(0, $this_length + $x_length);



        // the following for loop could be removed if the for loop following it

        // (the one with nested for loops) initially set $i to 0, but

        // doing so would also make the result in one set of unnecessary adds,

        // since on the outermost loops first pass, $product->value[$k] is going

        // to always be 0



        $carry = 0;

        $i = 0;



        for ($j = 0, $k = $i; $j < $this_length; $j++, $k++) {

            $temp = $product->value[$k] + $this->value[$j] * $x->value[$i] + $carry;

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

            $product->value[$k] = $temp - 0x4000000 * $carry;

        }



        $product->value[$k] = $carry;





        // the above for loop is what the previous comment was talking about.  the

        // following for loop is the "one with nested for loops"



        for ($i = 1; $i < $x_length; $i++) {

            $carry = 0;



            for ($j = 0, $k = $i; $j < $this_length; $j++, $k++) {

                $temp = $product->value[$k] + $this->value[$j] * $x->value[$i] + $carry;

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

                $product->value[$k] = $temp - 0x4000000 * $carry;

            }



            $product->value[$k] = $carry;

        }



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



        return $product->_normalize();

    }



    /**

     * Squares a BigInteger

     *

     * Squaring can be done faster than multiplying a number by itself can be.  See

     * {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=7 HAC 14.2.4} /

     * {@link http://math.libtomcrypt.com/files/tommath.pdf#page=141 MPM 5.3} for more information.

     *

     * @return Math_BigInteger

     * @access private

     */

    function _square()

    {

        if ( empty($this->value) ) {

            return new Math_BigInteger();

        }



        $max_index = count($this->value) - 1;



        $square = new Math_BigInteger();

        $square->value = $this->_array_repeat(0, 2 * $max_index);



        for ($i = 0; $i <= $max_index; $i++) {

            $temp = $square->value[2 * $i] + $this->value[$i] * $this->value[$i];

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

            $square->value[2 * $i] = $temp - 0x4000000 * $carry;



            // note how we start from $i+1 instead of 0 as we do in multiplication.

            for ($j = $i + 1; $j <= $max_index; $j++) {

                $temp = $square->value[$i + $j] + 2 * $this->value[$j] * $this->value[$i] + $carry;

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

                $square->value[$i + $j] = $temp - 0x4000000 * $carry;

            }



            // the following line can yield values larger 2**15.  at this point, PHP should switch

            // over to floats.

            $square->value[$i + $max_index + 1] = $carry;

        }



        return $square->_normalize();

    }



    /**

     * Divides two BigIntegers.

     *

     * Returns an array whose first element contains the quotient and whose second element contains the

     * "common residue".  If the remainder would be positive, the "common residue" and the remainder are the

     * same.  If the remainder would be negative, the "common residue" is equal to the sum of the remainder

     * and the divisor.

     *

     * Here's a quick 'n dirty example:

     * <code>

     * <?php

     *    include('Math/BigInteger.php');

     *

     *    $a = new Math_BigInteger('10');

     *    $b = new Math_BigInteger('20');

     *

     *    list($quotient,$remainder) = $a->divide($b);

     *

     *    echo $quotient->toString(); // outputs 0

     *    echo "\r\n";

     *    echo $remainder->toString(); // outputs 10

     * ?>

     * </code>

     *

     * @param Math_BigInteger $y

     * @return Array

     * @access public

     * @internal This function is based off of {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=9 HAC 14.20}

     *    with a slight variation due to the fact that this script, initially, did not support negative numbers.  Now,

     *    it does, but I don't want to change that which already works.

     */

    function divide($y)

    {

        switch ( MATH_BIGINTEGER_MODE ) {

            case MATH_BIGINTEGER_MODE_GMP:

                $quotient = new Math_BigInteger();

                $remainder = new Math_BigInteger();



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



                if (gmp_sign($remainder->value) < 0) {

                    $remainder->value = gmp_add($remainder->value, gmp_abs($y->value));

                }



                return array($quotient, $remainder);

            case MATH_BIGINTEGER_MODE_BCMATH:

                $quotient = new Math_BigInteger();

                $remainder = new Math_BigInteger();



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

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



                if ($remainder->value{0} == '-') {

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

                }



                return array($quotient, $remainder);

        }



        $x = $this->_copy();

        $y = $y->_copy();



        $x_sign = $x->is_negative;

        $y_sign = $y->is_negative;



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



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



        if ( !$diff ) {

            $temp = new Math_BigInteger();

            $temp->value = array(1);

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

            return array($temp, new Math_BigInteger());

        }



        if ( $diff < 0 ) {

            // if $x is negative, "add" $y.

            if ( $x_sign ) {

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

            }

            return array(new Math_BigInteger(), $x);

        }



        // normalize $x and $y as described in HAC 14.23 / 14.24

        // (incidently, i haven't been able to find a definitive example showing that this

        // results in worth-while speedup, but whatever)

        $msb = $y->value[count($y->value) - 1];

        for ($shift = 0; !($msb & 0x2000000); $shift++) {

            $msb <<= 1;

        }

        $x->_lshift($shift);

        $y->_lshift($shift);



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

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



        $quotient = new Math_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 Math_BigInteger();

        $temp->value = array_merge($this->_array_repeat(0, $x_max - $y_max), $y->value);



        while ( $x->compare($temp) >= 0 ) {

            // calculate the "common residue"

            $quotient->value[$x_max - $y_max]++;

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

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

        }



        for ($i = $x_max; $i >= $y_max + 1; $i--) {

            $x_value = array(

                $x->value[$i],

                ( $i > 0 ) ? $x->value[$i - 1] : 0,

                ( $i - 1 > 0 ) ? $x->value[$i - 2] : 0

            );

            $y_value = array(

                $y->value[$y_max],

                ( $y_max > 0 ) ? $y_max - 1 : 0

            );





            $q_index = $i - $y_max - 1;

            if ($x_value[0] == $y_value[0]) {

                $quotient->value[$q_index] = 0x3FFFFFF;

            } else {

                $quotient->value[$q_index] = floor(

                    ($x_value[0] * 0x4000000 + $x_value[1])

                    /

                    $y_value[0]

                );

            }



            $temp = new Math_BigInteger();

            $temp->value = array($y_value[1], $y_value[0]);



            $lhs = new Math_BigInteger();

            $lhs->value = array($quotient->value[$q_index]);

            $lhs = $lhs->multiply($temp);



            $rhs = new Math_BigInteger();

            $rhs->value = array($x_value[2], $x_value[1], $x_value[0]);

            

            while ( $lhs->compare($rhs) > 0 ) {

                $quotient->value[$q_index]--;



                $lhs = new Math_BigInteger();

                $lhs->value = array($quotient->value[$q_index]);

                $lhs = $lhs->multiply($temp);

            }



            $corrector = new Math_BigInteger();

            $temp = new Math_BigInteger();

            $corrector->value = $temp->value = $this->_array_repeat(0, $q_index);

            $temp->value[] = $quotient->value[$q_index];



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



            if ( $x->compare($temp) < 0 ) {

                $corrector->value[] = 1;

                $x = $x->add($corrector->multiply($y));

                $quotient->value[$q_index]--;

            }



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

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

        }



        // unnormalize the remainder

        $x->_rshift($shift);



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



        // calculate the "common residue", if appropriate

        if ( $x_sign ) {

            $y->_rshift($shift);

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

        }



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

    }



    /**

     * Performs modular exponentiation.

     *

     * Here's a quick 'n dirty example:

     * <code>

     * <?php

     *    include('Math/BigInteger.php');

     *

     *    $a = new Math_BigInteger('10');

     *    $b = new Math_BigInteger('20');

     *    $c = new Math_BigInteger('30');

     *

     *    $c = $a->modPow($b, $c);

     *

     *    echo $c->toString(); // outputs 10

     * ?>

     * </code>

     *

     * @param Math_BigInteger $e

     * @param Math_BigInteger $n

     * @return Math_BigInteger

     * @access public

     * @internal The most naive approach to modular exponentiation has very unreasonable requirements, and

     *    and although the approach involving repeated squaring does vastly better, it, too, is impractical

     *    for our purposes.  The reason being that division - by far the most complicated and time-consuming

     *    of the basic operations (eg. +,-,*,/) - occurs multiple times within it.

     *

     *    Modular reductions resolve this issue.  Although an individual modular reduction takes more time

     *    then an individual division, when performed in succession (with the same modulo), they're a lot faster.

     *

     *    The two most commonly used modular reductions are Barrett and Montgomery reduction.  Montgomery reduction,

     *    although faster, only works when the gcd of the modulo and of the base being used is 1.  In RSA, when the

     *    base is a power of two, the modulo - a product of two primes - is always going to have a gcd of 1 (because

     *    the product of two odd numbers is odd), but what about when RSA isn't used?

     *

     *    In contrast, Barrett reduction has no such constraint.  As such, some bigint implementations perform a

     *    Barrett reduction after every operation in the modpow function.  Others perform Barrett reductions when the

     *    modulo is even and Montgomery reductions when the modulo is odd.  BigInteger.java's modPow method, however,

     *    uses a trick involving the Chinese Remainder Theorem to factor the even modulo into two numbers - one odd and

     *    the other, a power of two - and recombine them, later.  This is the method that this modPow function uses.

     *    {@link http://islab.oregonstate.edu/papers/j34monex.pdf Montgomery Reduction with Even Modulus} elaborates.

     */

    function modPow($e, $n)

    {

        $n = $n->abs();

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

            $e = $e->abs();



            $temp = $this->modInverse($n);

            if ($temp === false) {

                return false;

            }



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

        }



        switch ( MATH_BIGINTEGER_MODE ) {

            case MATH_BIGINTEGER_MODE_GMP:

                $temp = new Math_BigInteger();

                $temp->value = gmp_powm($this->value, $e->value, $n->value);



                return $temp;

            case MATH_BIGINTEGER_MODE_BCMATH:

                // even though the last parameter is optional, according to php.net, it's not optional in

                // PHP_Compat 1.5.0 when running PHP 4.

                $temp = new Math_BigInteger();

                $temp->value = bcpowmod($this->value, $e->value, $n->value, 0);



                return $temp;

        }



        if ( empty($e->value) ) {

            $temp = new Math_BigInteger();

            $temp->value = array(1);

            return $temp;

        }



        if ( $e->value == array(1) ) {

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

            return $temp;

        }



        if ( $e->value == array(2) ) {

            $temp = $this->_square();

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

            return $temp;

        }



        // is the modulo odd?

        if ( $n->value[0] & 1 ) {

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

        }

        // if it's not, it's even



        // find the lowest set bit (eg. the max pow of 2 that divides $n)

        for ($i = 0; $i < count($n->value); $i++) {

            if ( $n->value[$i] ) {

                $temp = decbin($n->value[$i]);

                $j = strlen($temp) - strrpos($temp, '1') - 1;

                $j+= 26 * $i;

                break;

            }

        }

        // at this point, 2^$j * $n/(2^$j) == $n



        $mod1 = $n->_copy();

        $mod1->_rshift($j);

        $mod2 = new Math_BigInteger();

        $mod2->value = array(1);

        $mod2->_lshift($j);



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

        $part2 = $this->_slidingWindow($e, $mod2, MATH_BIGINTEGER_POWEROF2);



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

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



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

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



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

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



        $result = $result->add($temp);

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



        return $result;

    }



    /**

     * Sliding Window k-ary Modular Exponentiation

     *

     * Based on {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=27 HAC 14.85} /

     * {@link http://math.libtomcrypt.com/files/tommath.pdf#page=210 MPM 7.7}.  In a departure from those algorithims,

     * however, this function performs a modular reduction after every multiplication and squaring operation.

     * As such, this function has the same preconditions that the reductions being used do.

     *

     * The window size is calculated in the same fashion that the window size in BigInteger.java's oddModPow

     * function is.

     *

     * @param Math_BigInteger $e

     * @param Math_BigInteger $n

     * @param Integer $mode

     * @return Math_BigInteger

     * @access private

     */

    function _slidingWindow($e, $n, $mode)

    {

        static $window_ranges = array(7, 25, 81, 241, 673, 1793);



        $e_length = count($e->value) - 1;

        $e_bits = decbin($e->value[$e_length]);

        for ($i = $e_length - 1; $i >= 0; $i--) {

            $e_bits.= str_pad(decbin($e->value[$i]), 26, '0', STR_PAD_LEFT);

        }

        $e_length = strlen($e_bits);



        // calculate the appropriate window size.

        // $window_size == 3 if $window_ranges is between 25 and 81, for example.

        for ($i = 0, $window_size = 1; $e_length > $window_ranges[$i] && $i < count($window_ranges); $window_size++, $i++);



        switch ($mode) {

            case MATH_BIGINTEGER_MONTGOMERY:

                $reduce = '_montgomery';

                $undo = '_undoMontgomery';

                break;

            case MATH_BIGINTEGER_BARRETT:

                $reduce = '_barrett';

                $undo = '_barrett';

                break;

            case MATH_BIGINTEGER_POWEROF2:

                $reduce = '_mod2';

                $undo = '_mod2';

                break;

            case MATH_BIGINTEGER_CLASSIC:

                $reduce = '_remainder';

                $undo = '_remainder';

                break;

            case MATH_BIGINTEGER_NONE:

                // ie. do no modular reduction.  useful if you want to just do pow as opposed to modPow.

                $reduce = '_copy';

                $undo = '_copy';

                break;

            default:

                // an invalid $mode was provided

        }



        // precompute $this^0 through $this^$window_size

        $powers = array();

        $powers[1] = $this->$undo($n);

        $powers[2] = $powers[1]->_square();

        $powers[2] = $powers[2]->$reduce($n);



        // we do every other number since substr($e_bits, $i, $j+1) (see below) is supposed to end

        // in a 1.  ie. it's supposed to be odd.

        $temp = 1 << ($window_size - 1);

        for ($i = 1; $i < $temp; $i++) {

            $powers[2 * $i + 1] = $powers[2 * $i - 1]->multiply($powers[2]);

            $powers[2 * $i + 1] = $powers[2 * $i + 1]->$reduce($n);

        }



        $result = new Math_BigInteger();

        $result->value = array(1);

        $result = $result->$undo($n);



        for ($i = 0; $i < $e_length; ) {

            if ( !$e_bits{$i} ) {

                $result = $result->_square();

                $result = $result->$reduce($n);

                $i++;

            } else {

                for ($j = $window_size - 1; $j >= 0; $j--) {

                    if ( $e_bits{$i + $j} ) {

                        break;

                    }

                }



                for ($k = 0; $k <= $j; $k++) {// eg. the length of substr($e_bits, $i, $j+1)

                    $result = $result->_square();

                    $result = $result->$reduce($n);

                }



                $result = $result->multiply($powers[bindec(substr($e_bits, $i, $j + 1))]);

                $result = $result->$reduce($n);



                $i+=$j + 1;

            }

        }



        $result = $result->$reduce($n);

        return $result->_normalize();

    }



    /**

     * Remainder

     *

     * A wrapper for the divide function.

     *

     * @see divide()

     * @see _slidingWindow()

     * @access private

     * @param Math_BigInteger

     * @return Math_BigInteger

     */

    function _remainder($n)

    {

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

        return $temp;

    }



    /**

     * Modulos for Powers of Two

     *

     * Calculates $x%$n, where $n = 2**$e, for some $e.  Since this is basically the same as doing $x & ($n-1),

     * we'll just use this function as a wrapper for doing that.

     *

     * @see _slidingWindow()

     * @access private

     * @param Math_BigInteger

     * @return Math_BigInteger

     */

    function _mod2($n)

    {

        $temp = new Math_BigInteger();

        $temp->value = array(1);

        return $this->bitwise_and($n->subtract($temp));

    }



    /**

     * Barrett Modular Reduction

     *

     * See {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=14 HAC 14.3.3} /

     * {@link http://math.libtomcrypt.com/files/tommath.pdf#page=165 MPM 6.2.5} for more information.  Modified slightly,

     * so as not to require negative numbers (initially, this script didn't support negative numbers).

     *

     * @see _slidingWindow()

     * @access private

     * @param Math_BigInteger

     * @return Math_BigInteger

     */

    function _barrett($n)

    {

        static $cache;



        $n_length = count($n->value);



        if ( !isset($cache[MATH_BIGINTEGER_VARIABLE]) || $n->compare($cache[MATH_BIGINTEGER_VARIABLE]) ) {

            $cache[MATH_BIGINTEGER_VARIABLE] = $n;

            $temp = new Math_BigInteger();

            $temp->value = $this->_array_repeat(0, 2 * $n_length);

            $temp->value[] = 1;

            list($cache[MATH_BIGINTEGER_DATA], ) = $temp->divide($n);

        }



        $temp = new Math_BigInteger();

        $temp->value = array_slice($this->value, $n_length - 1);

        $temp = $temp->multiply($cache[MATH_BIGINTEGER_DATA]);

        $temp->value = array_slice($temp->value, $n_length + 1);



        $result = new Math_BigInteger();

        $result->value = array_slice($this->value, 0, $n_length + 1);

        $temp = $temp->multiply($n);

        $temp->value = array_slice($temp->value, 0, $n_length + 1);



        if ($result->compare($temp) < 0) {

            $corrector = new Math_BigInteger();

            $corrector->value = $this->_array_repeat(0, $n_length + 1);

            $corrector->value[] = 1;

            $result = $result->add($corrector);

        }



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

        while ($result->compare($n) > 0) {

            $result = $result->subtract($n);

        }



        return $result;

    }



    /**

     * Montgomery Modular Reduction

     *

     * ($this->_montgomery($n))->_undoMontgomery($n) yields $x%$n.

     * {@link http://math.libtomcrypt.com/files/tommath.pdf#page=170 MPM 6.3} provides insights on how this can be

     * improved upon (basically, by using the comba method).  gcd($n, 2) must be equal to one for this function

     * to work correctly.

     *

     * @see _undoMontgomery()

     * @see _slidingWindow()

     * @access private

     * @param Math_BigInteger

     * @return Math_BigInteger

     */

    function _montgomery($n)

    {

        static $cache;



        if ( !isset($cache[MATH_BIGINTEGER_VARIABLE]) || $n->compare($cache[MATH_BIGINTEGER_VARIABLE]) ) {

            $cache[MATH_BIGINTEGER_VARIABLE] = $n;

            $cache[MATH_BIGINTEGER_DATA] = $n->_modInverse67108864();

        }



        $result = $this->_copy();



        $n_length = count($n->value);



        for ($i = 0; $i < $n_length; $i++) {

            $digit = $result->value[$i] * $cache[MATH_BIGINTEGER_DATA];

            $temp = new Math_BigInteger();

            $temp->value = array(

                $digit - floor($digit / 0x4000000) * 0x4000000

            );

            $temp = $temp->multiply($n);

            $temp->value = array_merge($this->_array_repeat(0, $i), $temp->value);

            $result = $result->add($temp);

        }



        $result->value = array_slice($result->value, $n_length);



        if ($result->compare($n) >= 0) {

            $result = $result->subtract($n);

        }



        return $result->_normalize();

    }



    /**

     * Undo Montgomery Modular Reduction

     *

     * @see _montgomery()

     * @see _slidingWindow()

     * @access private

     * @param Math_BigInteger

     * @return Math_BigInteger

     */

    function _undoMontgomery($n)

    {

        $temp = new Math_BigInteger();

        $temp->value = array_merge($this->_array_repeat(0, count($n->value)), $this->value);

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

        return $temp->_normalize();

    }



    /**

     * Modular Inverse of a number mod 2**26 (eg. 67108864)

     *

     * Based off of the bnpInvDigit function implemented and justified in the following URL:

     *

     * {@link http://www-cs-students.stanford.edu/~tjw/jsbn/jsbn.js}

     *

     * The following URL provides more info:

     *

     * {@link http://groups.google.com/group/sci.crypt/msg/7a137205c1be7d85}

     *

     * As for why we do all the bitmasking...  strange things can happen when converting from flots to ints. For

     * instance, on some computers, var_dump((int) -4294967297) yields int(-1) and on others, it yields 

     * int(-2147483648).  To avoid problems stemming from this, we use bitmasks to guarntee that ints aren't

     * auto-converted to floats.  The outermost bitmask is present because without it, there's no guarantee that

     * the "residue" returned would be the so-called "common residue".  We use fmod, in the last step, because the

     * maximum possible $x is 26 bits and the maximum $result is 16 bits.  Thus, we have to be able to handle up to

     * 40 bits, which only 64-bit floating points will support.

     *

     * Thanks to Pedro Gimeno Fortea for input!

     *

     * @see _montgomery()

     * @access private

     * @return Integer

     */

    function _modInverse67108864() // 2**26 == 67108864

    {

        $x = -$this->value[0];

        $result = $x & 0x3; // x**-1 mod 2**2

        $result = ($result * (2 - $x * $result)) & 0xF; // x**-1 mod 2**4

        $result = ($result * (2 - ($x & 0xFF) * $result))  & 0xFF; // x**-1 mod 2**8

        $result = ($result * ((2 - ($x & 0xFFFF) * $result) & 0xFFFF)) & 0xFFFF; // x**-1 mod 2**16

        $result = fmod($result * (2 - fmod($x * $result, 0x4000000)), 0x4000000); // x**-1 mod 2**26

        return $result & 0x3FFFFFF;

    }



    /**

     * Calculates modular inverses.

     *

     * Here's a quick 'n dirty example:

     * <code>

     * <?php

     *    include('Math/BigInteger.php');

     *

     *    $a = new Math_BigInteger(30);

     *    $b = new Math_BigInteger(17);

     *

     *    $c = $a->modInverse($b);

     *

     *    echo $c->toString(); // outputs 4

     * ?>

     * </code>

     *

     * @param Math_BigInteger $n

     * @return mixed false, if no modular inverse exists, Math_BigInteger, otherwise.

     * @access public

     * @internal Calculates the modular inverse of $this mod $n using the binary xGCD algorithim described in

     *    {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=19 HAC 14.61}.  As the text above 14.61 notes,

     *    the more traditional algorithim requires "relatively costly multiple-precision divisions".  See

     *    {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=21 HAC 14.64} for more information.

     */

    function modInverse($n)

    {

        switch ( MATH_BIGINTEGER_MODE ) {

            case MATH_BIGINTEGER_MODE_GMP:

                $temp = new Math_BigInteger();

                $temp->value = gmp_invert($this->value, $n->value);



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

            case MATH_BIGINTEGER_MODE_BCMATH:

                // it might b…