mirror of
https://github.com/danog/tgseclib.git
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367ddebf80
git-svn-id: http://phpseclib.svn.sourceforge.net/svnroot/phpseclib/trunk@3 21d32557-59b3-4da0-833f-c5933fad653e
2174 lines
70 KiB
PHP
2174 lines
70 KiB
PHP
<?php
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/* vim: set expandtab tabstop=4 shiftwidth=4 softtabstop=4: */
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/**
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* Pure-PHP arbitrary precision integer arithmetic library.
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*
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* Supports base-2, base-10, base-16, and base-256 numbers. Uses the GMP or BCMath extensions, if available,
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* and an internal implementation, otherwise.
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*
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* PHP versions 4 and 5
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*
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* {@internal (all DocBlock comments regarding implementation - such as the one that follows - refer to the
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* {@link MATH_BIGINTEGER_MODE_INTERNAL MATH_BIGINTEGER_MODE_INTERNAL} mode)
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*
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* Math_BigInteger uses base-2**26 to perform operations such as multiplication and division and
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* base-2**52 (ie. two base 2**26 digits) to perform addition and subtraction. Because the largest possible
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* value when multiplying two base-2**26 numbers together is a base-2**52 number, double precision floating
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* point numbers - numbers that should be supported on most hardware and whose significand is 53 bits - are
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* used. As a consequence, bitwise operators such as >> and << cannot be used, nor can the modulo operator %,
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* which only supports integers. Although this fact will slow this library down, the fact that such a high
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* base is being used should more than compensate.
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*
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* When PHP version 6 is officially released, we'll be able to use 64-bit integers. This should, once again,
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* allow bitwise operators, and will increase the maximum possible base to 2**31 (or 2**62 for addition /
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* subtraction).
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*
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* Useful resources are as follows:
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*
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* - {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf Handbook of Applied Cryptography (HAC)}
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* - {@link http://math.libtomcrypt.com/files/tommath.pdf Multi-Precision Math (MPM)}
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* - Java's BigInteger classes. See /j2se/src/share/classes/java/math in jdk-1_5_0-src-jrl.zip
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*
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* One idea for optimization is to use the comba method to reduce the number of operations performed.
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* MPM uses this quite extensively. The following URL elaborates:
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*
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* {@link http://www.everything2.com/index.pl?node_id=1736418}}}
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*
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* Here's a quick 'n dirty example of how to use this library:
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* <code>
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* <?php
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* include('Math/BigInteger.php');
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*
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* $a = new Math_BigInteger(2);
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* $b = new Math_BigInteger(3);
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*
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* $c = $a->add($b);
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*
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* echo $c->toString(); // outputs 5
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* ?>
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* </code>
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*
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* LICENSE: This library is free software; you can redistribute it and/or
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* modify it under the terms of the GNU Lesser General Public
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* License as published by the Free Software Foundation; either
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* version 2.1 of the License, or (at your option) any later version.
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*
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* This library is distributed in the hope that it will be useful,
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* but WITHOUT ANY WARRANTY; without even the implied warranty of
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* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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* Lesser General Public License for more details.
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*
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* You should have received a copy of the GNU Lesser General Public
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* License along with this library; if not, write to the Free Software
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* Foundation, Inc., 59 Temple Place, Suite 330, Boston,
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* MA 02111-1307 USA
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*
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* @category Math
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* @package Math_BigInteger
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* @author Jim Wigginton <terrafrost@php.net>
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* @copyright MMVI Jim Wigginton
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* @license http://www.gnu.org/licenses/lgpl.txt
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* @version $Id: BigInteger.php,v 1.1 2007-07-02 04:19:47 terrafrost Exp $
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* @link http://pear.php.net/package/Math_BigInteger
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*/
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/**
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* Include PHP_Compat module bcpowmod since that function does not exist in PHP4:
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* {@link http://pear.php.net/package/PHP_Compat/}
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* {@link http://php.net/function.bcpowmod}
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*/
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require_once 'PHP/Compat/Function/bcpowmod.php';
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/**
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* Include PHP_Compat module array_fill since that function requires PHP4.2.0+:
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* {@link http://pear.php.net/package/PHP_Compat/}
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* {@link http://php.net/function.array_fill}
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*/
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require_once 'PHP/Compat/Function/array_fill.php';
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/**#@+
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* @access private
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* @see Math_BigInteger::_slidingWindow()
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*/
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/**
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* @see Math_BigInteger::_montgomery()
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* @see Math_BigInteger::_undoMontgomery()
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*/
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define('MATH_BIGINTEGER_MONTGOMERY', 0);
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/**
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* @see Math_BigInteger::_barrett()
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*/
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define('MATH_BIGINTEGER_BARRETT', 1);
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/**
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* @see Math_BigInteger::_mod2()
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*/
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define('MATH_BIGINTEGER_POWEROF2', 2);
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/**
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* @see Math_BigInteger::_remainder()
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*/
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define('MATH_BIGINTEGER_CLASSIC', 3);
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/**
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* @see Math_BigInteger::_copy()
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*/
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define('MATH_BIGINTEGER_NONE', 4);
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/**#@-*/
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/**#@+
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* @access private
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* @see Math_BigInteger::_montgomery()
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* @see Math_BigInteger::_barrett()
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*/
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/**
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* $cache[MATH_BIGINTEGER_VARIABLE] tells us whether or not the cached data is still valid.
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*/
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define('MATH_BIGINTEGER_VARIABLE', 0);
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/**
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* $cache[MATH_BIGINTEGER_DATA] contains the cached data.
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*/
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define('MATH_BIGINTEGER_DATA', 1);
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/**#@-*/
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/**#@+
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* @access private
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* @see Math_BigInteger::Math_BigInteger()
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*/
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/**
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* To use the pure-PHP implementation
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*/
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define('MATH_BIGINTEGER_MODE_INTERNAL', 1);
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/**
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* To use the BCMath library
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*
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* (if enabled; otherwise, the internal implementation will be used)
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*/
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define('MATH_BIGINTEGER_MODE_BCMATH', 2);
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/**
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* To use the GMP library
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*
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* (if present; otherwise, either the BCMath or the internal implementation will be used)
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*/
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define('MATH_BIGINTEGER_MODE_GMP', 3);
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/**#@-*/
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/**
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* Pure-PHP arbitrary precission integer arithmetic library. Supports base-2, base-10, base-16, and base-256
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* numbers.
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*
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* @author Jim Wigginton <terrafrost@php.net>
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* @version 1.0.0RC3
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* @access public
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* @package Math_BigInteger
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*/
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class Math_BigInteger {
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/**
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* Holds the BigInteger's value.
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*
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* @var Array
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* @access private
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*/
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var $value;
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/**
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* Holds the BigInteger's magnitude.
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*
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* @var Boolean
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* @access private
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*/
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var $is_negative = false;
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/**
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* Converts base-2, base-10, base-16, and binary strings (eg. base-256) to BigIntegers.
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*
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* If the second parameter - $base - is negative, then it will be assumed that the number's are encoded using
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* two's compliment. The sole exception to this is -10, which is treated the same as 10 is.
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*
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* Here's a quick 'n dirty example:
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* <code>
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* <?php
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* include('Math/BigInteger.php');
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*
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* $a = new Math_BigInteger('0x32', 16); // 50 in base-16
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*
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* echo $a->toString(); // outputs 50
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* ?>
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* </code>
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*
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* @param optional $x base-10 number or base-$base number if $base set.
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* @param optional integer $base
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* @return Math_BigInteger
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* @access public
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*/
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function Math_BigInteger($x = 0, $base = 10)
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{
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if ( !defined('MATH_BIGINTEGER_MODE') ) {
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switch (true) {
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case extension_loaded('gmp'):
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define('MATH_BIGINTEGER_MODE', MATH_BIGINTEGER_MODE_GMP);
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break;
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case extension_loaded('bcmath'):
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define('MATH_BIGINTEGER_MODE', MATH_BIGINTEGER_MODE_BCMATH);
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break;
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default:
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define('MATH_BIGINTEGER_MODE', MATH_BIGINTEGER_MODE_INTERNAL);
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}
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}
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switch ( MATH_BIGINTEGER_MODE ) {
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case MATH_BIGINTEGER_MODE_GMP:
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$this->value = gmp_init(0);
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break;
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case MATH_BIGINTEGER_MODE_BCMATH:
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$this->value = '0';
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break;
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default:
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$this->value = array();
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}
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if ($x === 0) {
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return;
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}
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switch ($base) {
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case -256:
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if (ord($x[0]) & 0x80) {
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$x = ~$x;
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$this->is_negative = true;
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}
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case 256:
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switch ( MATH_BIGINTEGER_MODE ) {
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case MATH_BIGINTEGER_MODE_GMP:
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$temp = unpack('H*hex', $x);
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$sign = $this->is_negative ? '-' : '';
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$this->value = gmp_init($sign . '0x' . $temp['hex']);
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break;
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case MATH_BIGINTEGER_MODE_BCMATH:
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// round $len to the nearest 4 (thanks, DavidMJ!)
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$len = (strlen($x) + 3) & 0xFFFFFFFC;
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$x = str_pad($x, $len, chr(0), STR_PAD_LEFT);
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for ($i = 0; $i < $len; $i+= 4) {
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$this->value = bcmul($this->value, '4294967296'); // 4294967296 == 2**32
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$this->value = bcadd($this->value, 0x1000000 * ord($x[$i]) + ((ord($x[$i + 1]) << 16) | (ord($x[$i + 2]) << 8) | ord($x[$i + 3])));
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}
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if ($this->is_negative) {
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$this->value = '-' . $this->value;
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}
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break;
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// converts a base-2**8 (big endian / msb) number to base-2**26 (little endian / lsb)
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case MATH_BIGINTEGER_MODE_INTERNAL:
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while (strlen($x)) {
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$this->value[] = $this->_bytes2int($this->_base256_rshift($x, 26));
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}
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}
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if ($this->is_negative) {
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if (MATH_BIGINTEGER_MODE != MATH_BIGINTEGER_MODE_INTERNAL) {
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$this->is_negative = false;
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}
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$temp = $this->add(new Math_BigInteger('-1'));
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$this->value = $temp->value;
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}
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break;
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case 16:
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case -16:
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if ($base > 0 && $x[0] == '-') {
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$this->is_negative = true;
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$x = substr($x, 1);
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}
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$x = preg_replace('#^(?:0x)?([A-Fa-f0-9]*).*#', '$1', $x);
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$is_negative = false;
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if ($base < 0 && hexdec($x[0]) >= 8) {
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$this->is_negative = $is_negative = true;
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$x = bin2hex(~pack('H*', $x));
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}
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switch ( MATH_BIGINTEGER_MODE ) {
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case MATH_BIGINTEGER_MODE_GMP:
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$temp = $this->is_negative ? '-0x' . $x : '0x' . $x;
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$this->value = gmp_init($temp);
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$this->is_negative = false;
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break;
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case MATH_BIGINTEGER_MODE_BCMATH:
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$x = ( strlen($x) & 1 ) ? '0' . $x : $x;
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$temp = new Math_BigInteger(pack('H*', $x), 256);
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$this->value = $this->is_negative ? '-' . $temp->value : $temp->value;
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$this->is_negative = false;
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break;
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case MATH_BIGINTEGER_MODE_INTERNAL:
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$x = ( strlen($x) & 1 ) ? '0' . $x : $x;
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$temp = new Math_BigInteger(pack('H*', $x), 256);
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$this->value = $temp->value;
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}
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if ($is_negative) {
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$temp = $this->add(new Math_BigInteger('-1'));
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$this->value = $temp->value;
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}
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break;
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case 10:
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case -10:
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$x = preg_replace('#^(-?[0-9]*).*#', '$1', $x);
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switch ( MATH_BIGINTEGER_MODE ) {
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case MATH_BIGINTEGER_MODE_GMP:
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$this->value = gmp_init($x);
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break;
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case MATH_BIGINTEGER_MODE_BCMATH:
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// explicitly casting $x to a string is necessary, here, since doing $x[0] on -1 yields different
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// results then doing it on '-1' does (modInverse does $x[0])
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$this->value = (string) $x;
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break;
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case MATH_BIGINTEGER_MODE_INTERNAL:
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$temp = new Math_BigInteger();
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// array(10000000) is 10**7 in base-2**26. 10**7 is the closest to 2**26 we can get without passing it.
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$multiplier = new Math_BigInteger();
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$multiplier->value = array(10000000);
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if ($x[0] == '-') {
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$this->is_negative = true;
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$x = substr($x, 1);
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}
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$x = str_pad($x, strlen($x) + (6 * strlen($x)) % 7, 0, STR_PAD_LEFT);
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while (strlen($x)) {
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$temp = $temp->multiply($multiplier);
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$temp = $temp->add(new Math_BigInteger($this->_int2bytes(substr($x, 0, 7)), 256));
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$x = substr($x, 7);
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}
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$this->value = $temp->value;
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}
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break;
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case 2: // base-2 support originally implemented by Lluis Pamies - thanks!
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case -2:
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if ($base > 0 && $x[0] == '-') {
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$this->is_negative = true;
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$x = substr($x, 1);
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}
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$x = preg_replace('#^([01]*).*#', '$1', $x);
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$x = str_pad($x, strlen($x) + (3 * strlen($x)) % 4, 0, STR_PAD_LEFT);
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$str = '0x';
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while (strlen($x)) {
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$part = substr($x, 0, 4);
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$str.= dechex(bindec($part));
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$x = substr($x, 4);
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}
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if ($this->is_negative) {
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$str = '-' . $str;
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}
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$temp = new Math_BigInteger($str, 8 * $base); // ie. either -16 or +16
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$this->value = $temp->value;
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$this->is_negative = $temp->is_negative;
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break;
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default:
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// base not supported, so we'll let $this == 0
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}
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}
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/**
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* Converts a BigInteger to a byte string (eg. base-256).
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*
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* Negative numbers are saved as positive numbers, unless $twos_compliment is set to true, at which point, they're
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* saved as two's compliment.
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*
|
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* Here's a quick 'n dirty example:
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* <code>
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* <?php
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* include('Math/BigInteger.php');
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*
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* $a = new Math_BigInteger('65');
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*
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* echo $a->toBytes(); // outputs chr(65)
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* ?>
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* </code>
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*
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* @param Boolean $twos_compliment
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* @return String
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* @access public
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* @internal Converts a base-2**26 number to base-2**8
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*/
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function toBytes($twos_compliment = false)
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{
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if ($twos_compliment) {
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$comparison = $this->compare(new Math_BigInteger());
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if ($comparison == 0) {
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return '';
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}
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$temp = $comparison < 0 ? $this->add(new Math_BigInteger(1)) : $this->_copy();
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$bytes = $temp->toBytes();
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if (empty($bytes)) { // eg. if the number we're trying to convert is -1
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$bytes = chr(0);
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}
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if (ord($bytes[0]) & 0x80) {
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$bytes = chr(0) . $bytes;
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}
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return $comparison < 0 ? ~$bytes : $bytes;
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}
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|
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switch ( MATH_BIGINTEGER_MODE ) {
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case MATH_BIGINTEGER_MODE_GMP:
|
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if (gmp_cmp($this->value, gmp_init(0)) == 0) {
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return '';
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}
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$temp = gmp_strval(gmp_abs($this->value), 16);
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$temp = ( strlen($temp) & 1 ) ? '0' . $temp : $temp;
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return ltrim(pack('H*', $temp), chr(0));
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case MATH_BIGINTEGER_MODE_BCMATH:
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if ($this->value === '0') {
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return '';
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}
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|
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$value = '';
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$current = $this->value;
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|
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if ($current[0] == '-') {
|
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$current = substr($current, 1);
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}
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// we don't do four bytes at a time because then numbers larger than 1<<31 would be negative
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// two's complimented numbers, which would break chr.
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while (bccomp($current, '0') > 0) {
|
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$temp = bcmod($current, 0x1000000);
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$value = chr($temp >> 16) . chr($temp >> 8) . chr($temp) . $value;
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$current = bcdiv($current, 0x1000000);
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}
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return ltrim($value, chr(0));
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}
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if (!count($this->value)) {
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return '';
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}
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$result = $this->_int2bytes($this->value[count($this->value) - 1]);
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$temp = $this->_copy();
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for ($i = count($temp->value) - 2; $i >= 0; $i--) {
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$temp->_base256_lshift($result, 26);
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$result = $result | str_pad($temp->_int2bytes($temp->value[$i]), strlen($result), chr(0), STR_PAD_LEFT);
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}
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return $result;
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}
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|
|
|
/**
|
|
* 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 $a->toString(); // outputs 50
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* ?>
|
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* </code>
|
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*
|
|
* @return String
|
|
* @access public
|
|
* @internal Converts a base-2**26 number to base-10**7 (which is pretty much base-10)
|
|
*/
|
|
function toString()
|
|
{
|
|
switch ( MATH_BIGINTEGER_MODE ) {
|
|
case MATH_BIGINTEGER_MODE_GMP:
|
|
return gmp_strval($this->value);
|
|
case MATH_BIGINTEGER_MODE_BCMATH:
|
|
if ($this->value === '0') {
|
|
return '0';
|
|
}
|
|
|
|
return ltrim($this->value, '0');
|
|
}
|
|
|
|
if (!count($this->value)) {
|
|
return '0';
|
|
}
|
|
|
|
$temp = $this->_copy();
|
|
$temp->is_negative = false;
|
|
|
|
$divisor = new Math_BigInteger();
|
|
$divisor->value = array(10000000); // eg. 10**7
|
|
while (count($temp->value)) {
|
|
list($temp, $mod) = $temp->divide($divisor);
|
|
$result = str_pad($this->_bytes2int($mod->toBytes()), 7, '0', STR_PAD_LEFT) . $result;
|
|
}
|
|
$result = ltrim($result, '0');
|
|
|
|
if ($this->is_negative) {
|
|
$result = '-' . $result;
|
|
}
|
|
|
|
return $result;
|
|
}
|
|
|
|
/**
|
|
* 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 & 1; // rounds $size to the nearest 2.
|
|
|
|
$x = array_pad($this->value, $size,0);
|
|
$y = array_pad($y->value, $size, 0);
|
|
|
|
for ($i = 0; $i < $size - 1; $i+=2) {
|
|
$sum = $x[$i + 1] * 0x4000000 + $x[$i] + $y[$i + 1] * 0x4000000 + $y[$i] + $carry;
|
|
$carry = $sum >= 4503599627370496; // eg. floor($sum / 2**52); only possible values (in any base) are 0 and 1
|
|
$sum = $carry ? $sum - 4503599627370496 : $sum;
|
|
|
|
$temp = floor($sum / 0x4000000);
|
|
|
|
$result->value[] = $sum - 0x4000000 * $temp; // eg. a faster alternative to fmod($sum, 0x4000000)
|
|
$result->value[] = $temp;
|
|
}
|
|
|
|
if ($carry) {
|
|
$result->value[] = $carry;
|
|
}
|
|
|
|
$result->is_negative = $this->is_negative;
|
|
|
|
return $result->_normalize();
|
|
}
|
|
|
|
/**
|
|
* Subtracts two BigIntegers.
|
|
*
|
|
* 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 (basically, the "common residue" is the first positive modulo).
|
|
*
|
|
* 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:
|
|
$temp = new Math_BigInteger();
|
|
$temp->value = bcpowmod($this->value, $e->value, $n->value);
|
|
|
|
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++) {
|
|
$temp = new Math_BigInteger();
|
|
$temp->value = array(
|
|
($result->value[$i] * $cache[MATH_BIGINTEGER_DATA]) & 0x3FFFFFF
|
|
);
|
|
$temp = $temp->multiply($n);
|
|
$temp->value = array_merge($this->_array_repeat(0, $i), $temp->value);
|
|
$result = $result->add($temp);
|
|
}
|
|
|
|
$result->value = array_slice($result->value, $n_length);
|
|
|
|
if ($result->compare($n) >= 0) {
|
|
$result = $result->subtract($n);
|
|
}
|
|
|
|
return $result->_normalize();
|
|
}
|
|
|
|
/**
|
|
* Undo Montgomery Modular Reduction
|
|
*
|
|
* @see _montgomery()
|
|
* @see _slidingWindow()
|
|
* @access private
|
|
* @param Math_BigInteger
|
|
* @return Math_BigInteger
|
|
*/
|
|
function _undoMontgomery($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 be faster to use the binary xGCD algorithim here, as well, but (1) that algorithim works
|
|
// best when the base is a power of 2 and (2) i don't think it'd make much difference, anyway. as is,
|
|
// the basic extended euclidean algorithim is what we're using.
|
|
|
|
// if $x is less than 0, the first character of $x is a '-', so we'll remove it. we can do this because
|
|
// $x mod $n == $x mod -$n.
|
|
$n = (bccomp($n->value, '0') < 0) ? substr($n->value, 1) : $n->value;
|
|
|
|
if (bccomp($this->value,'0') < 0) {
|
|
$negated_this = new Math_BigInteger();
|
|
$negated_this->value = substr($this->value, 1);
|
|
|
|
$temp = $negated_this->modInverse(new Math_BigInteger($n));
|
|
|
|
if ($temp === false) {
|
|
return false;
|
|
}
|
|
|
|
$temp->value = bcsub($n, $temp->value);
|
|
|
|
return $temp;
|
|
}
|
|
|
|
$u = $this->value;
|
|
$v = $n;
|
|
|
|
$a = '1';
|
|
$c = '0';
|
|
|
|
while (true) {
|
|
$q = bcdiv($u, $v);
|
|
$temp = $u;
|
|
$u = $v;
|
|
$v = bcsub($temp, bcmul($v, $q));
|
|
|
|
if (bccomp($v, '0') == 0) {
|
|
break;
|
|
}
|
|
|
|
$temp = $a;
|
|
$a = $c;
|
|
$c = bcsub($temp, bcmul($c, $q));
|
|
}
|
|
|
|
$temp = new Math_BigInteger();
|
|
$temp->value = (bccomp($c, '0') < 0) ? bcadd($c, $n) : $c;
|
|
|
|
// $u contains the gcd of $this and $n
|
|
return (bccomp($u,'1') == 0) ? $temp : false;
|
|
}
|
|
|
|
// if $this and $n are even, return false.
|
|
if ( !($this->value[0]&1) && !($n->value[0]&1) ) {
|
|
return false;
|
|
}
|
|
|
|
$n = $n->_copy();
|
|
$n->is_negative = false;
|
|
|
|
if ($this->compare(new Math_BigInteger()) < 0) {
|
|
// is_negative is currently true. since we need it to be false, we'll just set it to false, temporarily,
|
|
// and reset it as true, later.
|
|
$this->is_negative = false;
|
|
|
|
$temp = $this->modInverse($n);
|
|
|
|
if ($temp === false) {
|
|
return false;
|
|
}
|
|
|
|
$temp = $n->subtract($temp);
|
|
|
|
$this->is_negative = true;
|
|
|
|
return $temp;
|
|
}
|
|
|
|
$u = $n->_copy();
|
|
$x = $this;
|
|
//list(, $x) = $this->divide($n);
|
|
$v = $x->_copy();
|
|
|
|
$a = new Math_BigInteger();
|
|
$b = new Math_BigInteger();
|
|
$c = new Math_BigInteger();
|
|
$d = new Math_BigInteger();
|
|
|
|
$a->value = $d->value = array(1);
|
|
|
|
while ( !empty($u->value) ) {
|
|
while ( !($u->value[0] & 1) ) {
|
|
$u->_rshift(1);
|
|
if ( ($a->value[0] & 1) || ($b->value[0] & 1) ) {
|
|
$a = $a->add($x);
|
|
$b = $b->subtract($n);
|
|
}
|
|
$a->_rshift(1);
|
|
$b->_rshift(1);
|
|
}
|
|
|
|
while ( !($v->value[0] & 1) ) {
|
|
$v->_rshift(1);
|
|
if ( ($c->value[0] & 1) || ($d->value[0] & 1) ) {
|
|
$c = $c->add($x);
|
|
$d = $d->subtract($n);
|
|
}
|
|
$c->_rshift(1);
|
|
$d->_rshift(1);
|
|
}
|
|
|
|
if ($u->compare($v) >= 0) {
|
|
$u = $u->subtract($v);
|
|
$a = $a->subtract($c);
|
|
$b = $b->subtract($d);
|
|
} else {
|
|
$v = $v->subtract($u);
|
|
$c = $c->subtract($a);
|
|
$d = $d->subtract($b);
|
|
}
|
|
|
|
$u->_normalize();
|
|
}
|
|
|
|
// at this point, $v == gcd($this, $n). if it's not equal to 1, no modular inverse exists.
|
|
if ( $v->value != array(1) ) {
|
|
return false;
|
|
}
|
|
|
|
$d = ($d->compare(new Math_BigInteger()) < 0) ? $d->add($n) : $d;
|
|
|
|
return ($this->is_negative) ? $n->subtract($d) : $d;
|
|
}
|
|
|
|
/**
|
|
* Absolute value.
|
|
*
|
|
* @return Math_BigInteger
|
|
* @access public
|
|
*/
|
|
function abs()
|
|
{
|
|
$temp = new Math_BigInteger();
|
|
|
|
switch ( MATH_BIGINTEGER_MODE ) {
|
|
case MATH_BIGINTEGER_MODE_GMP:
|
|
$temp->value = gmp_abs($this->value);
|
|
break;
|
|
case MATH_BIGINTEGER_MODE_BCMATH:
|
|
$temp->value = (bccomp($this->value, '0') < 0) ? substr($this->value, 1) : $this->value;
|
|
break;
|
|
default:
|
|
$temp->value = $this->value;
|
|
}
|
|
|
|
return $temp;
|
|
}
|
|
|
|
/**
|
|
* Compares two numbers.
|
|
*
|
|
* @param Math_BigInteger $x
|
|
* @return Integer < 0 if $this is less than $x; > 0 if $this is greater than $x, and 0 if they are equal.
|
|
* @access public
|
|
* @internal Could return $this->sub($x), but that's not as fast as what we do do.
|
|
*/
|
|
function compare($x)
|
|
{
|
|
switch ( MATH_BIGINTEGER_MODE ) {
|
|
case MATH_BIGINTEGER_MODE_GMP:
|
|
return gmp_cmp($this->value, $x->value);
|
|
case MATH_BIGINTEGER_MODE_BCMATH:
|
|
return bccomp($this->value, $x->value);
|
|
}
|
|
|
|
$this->_normalize();
|
|
$x->_normalize();
|
|
|
|
if ( $this->is_negative != $x->is_negative ) {
|
|
return ( !$this->is_negative && $x->is_negative ) ? 1 : -1;
|
|
}
|
|
|
|
$result = $this->is_negative ? -1 : 1;
|
|
|
|
if ( count($this->value) != count($x->value) ) {
|
|
return ( count($this->value) > count($x->value) ) ? $result : -$result;
|
|
}
|
|
|
|
for ($i = count($this->value) - 1; $i >= 0; $i--) {
|
|
if ($this->value[$i] != $x->value[$i]) {
|
|
return ( $this->value[$i] > $x->value[$i] ) ? $result : -$result;
|
|
}
|
|
}
|
|
|
|
return 0;
|
|
}
|
|
|
|
/**
|
|
* Returns a copy of $this
|
|
*
|
|
* PHP5 passes objects by reference while PHP4 passes by value. As such, we need a function to guarantee
|
|
* that all objects are passed by value, when appropriate. More information can be found here:
|
|
*
|
|
* {@link http://www.php.net/manual/en/language.oop5.basic.php#51624}
|
|
*
|
|
* @access private
|
|
* @return Math_BigInteger
|
|
*/
|
|
function _copy()
|
|
{
|
|
$temp = new Math_BigInteger();
|
|
$temp->value = $this->value;
|
|
$temp->is_negative = $this->is_negative;
|
|
return $temp;
|
|
}
|
|
|
|
/**
|
|
* Logical And
|
|
*
|
|
* @param Math_BigInteger $x
|
|
* @access public
|
|
* @internal Implemented per a request by Lluis Pamies i Juarez <lluis _a_ pamies.cat>
|
|
* @return Math_BigInteger
|
|
*/
|
|
function bitwise_and($x)
|
|
{
|
|
switch ( MATH_BIGINTEGER_MODE ) {
|
|
case MATH_BIGINTEGER_MODE_GMP:
|
|
$temp = new Math_BigInteger();
|
|
$temp->value = gmp_and($this->value, $x->value);
|
|
|
|
return $temp;
|
|
case MATH_BIGINTEGER_MODE_BCMATH:
|
|
return new Math_BigInteger($this->toBytes() & $x->toBytes(), 256);
|
|
}
|
|
|
|
$result = new Math_BigInteger();
|
|
|
|
$x_length = count($x->value);
|
|
for ($i = 0; $i < $x_length; $i++) {
|
|
$result->value[] = $this->value[$i] & $x->value[$i];
|
|
}
|
|
|
|
return $result->_normalize();
|
|
}
|
|
|
|
/**
|
|
* Logical Or
|
|
*
|
|
* @param Math_BigInteger $x
|
|
* @access public
|
|
* @internal Implemented per a request by Lluis Pamies i Juarez <lluis _a_ pamies.cat>
|
|
* @return Math_BigInteger
|
|
*/
|
|
function bitwise_or($x)
|
|
{
|
|
switch ( MATH_BIGINTEGER_MODE ) {
|
|
case MATH_BIGINTEGER_MODE_GMP:
|
|
$temp = new Math_BigInteger();
|
|
$temp->value = gmp_or($this->value, $x->value);
|
|
|
|
return $temp;
|
|
case MATH_BIGINTEGER_MODE_BCMATH:
|
|
return new Math_BigInteger($this->toBytes() | $x->toBytes(), 256);
|
|
}
|
|
|
|
$result = new Math_BigInteger();
|
|
|
|
$x_length = count($x->value);
|
|
for ($i = 0; $i < $x_length; $i++) {
|
|
$result->value[] = $this->value[$i] | $x->value[$i];
|
|
}
|
|
|
|
return $result->_normalize();
|
|
}
|
|
|
|
/**
|
|
* Logical Exclusive-Or
|
|
*
|
|
* @param Math_BigInteger $x
|
|
* @access public
|
|
* @internal Implemented per a request by Lluis Pamies i Juarez <lluis _a_ pamies.cat>
|
|
* @return Math_BigInteger
|
|
*/
|
|
function bitwise_xor($x)
|
|
{
|
|
switch ( MATH_BIGINTEGER_MODE ) {
|
|
case MATH_BIGINTEGER_MODE_GMP:
|
|
$temp = new Math_BigInteger();
|
|
$temp->value = gmp_xor($this->value, $x->value);
|
|
|
|
return $temp;
|
|
case MATH_BIGINTEGER_MODE_BCMATH:
|
|
return new Math_BigInteger($this->toBytes() ^ $x->toBytes(), 256);
|
|
}
|
|
|
|
$result = new Math_BigInteger();
|
|
|
|
$x_length = count($x->value);
|
|
for ($i = 0; $i < $x_length; $i++) {
|
|
$result->value[] = $this->value[$i] ^ $x->value[$i];
|
|
}
|
|
|
|
return $result->_normalize();
|
|
}
|
|
|
|
/**
|
|
* Logical Not
|
|
*
|
|
* Although integers can be converted to and from various bases with relative ease, there is one piece
|
|
* of information that is lost during such conversions. The number of leading zeros that number had
|
|
* or should have in any given base. Per that, if you convert 1 from decimal to binary, there's no
|
|
* way to know just how many leading zero's there should be. In truth, there could be any number.
|
|
*
|
|
* Normally, the number of leading zero's is unimportant. When doing "not", however, it is. The "not"
|
|
* of 1 on an 8-bit representation of 1 is 1111 1110. The "not" of 1 on a 16-bit representation of 1 is
|
|
* 1111 1111 1111 1110. When doing it on a number that's preceeded by an infinite number of zero's, it's
|
|
* infinite.
|
|
*
|
|
* This function assumes that there are no leading zero's - that the bit-representation being used is
|
|
* equal to the minimum number of required bits, unless otherwise specified in the optional parameter,
|
|
* where the optional parameter represents the bit-representation being used. If the specified
|
|
* bit-representation is smaller than the minimum number of bits required to represent the number, the
|
|
* latter will be used as the bit-representation.
|
|
*
|
|
* @param $bits Integer
|
|
* @access public
|
|
* @internal Implemented per a request by Lluis Pamies i Juarez <lluis _a_ pamies.cat>
|
|
* @return Math_BigInteger
|
|
*/
|
|
function bitwise_not($bits = -1)
|
|
{
|
|
// calculuate "not" without regard to $bits
|
|
$temp = ~$this->toBytes();
|
|
$msb = decbin(ord($temp[0]));
|
|
$msb = substr($msb, strpos($msb, '0'));
|
|
$temp[0] = chr(bindec($msb));
|
|
|
|
// see if we need to add extra leading 1's
|
|
$current_bits = strlen($msb) + 8 * strlen($temp) - 8;
|
|
$new_bits = $bits - $current_bits;
|
|
if ($new_bits <= 0) {
|
|
return new Math_BigInteger($temp, 256);
|
|
}
|
|
|
|
// generate as many leading 1's as we need to.
|
|
$leading_ones = chr((1 << ($new_bits & 0x7)) - 1) . str_repeat(chr(0xFF), $new_bits >> 3);
|
|
$this->_base256_lshift($leading_ones, $current_bits);
|
|
|
|
$temp = str_pad($temp, ceil($bits / 8), chr(0), STR_PAD_LEFT);
|
|
|
|
return new Math_BigInteger($leading_ones | $temp, 256);
|
|
}
|
|
|
|
/**
|
|
* Logical Right Shift
|
|
*
|
|
* Shifts BigInteger's by $shift bits, effectively dividing by 2**$shift.
|
|
*
|
|
* @param Integer $shift
|
|
* @return Math_BigInteger
|
|
* @access public
|
|
* @internal The only version that yields any speed increases is the internal version.
|
|
*/
|
|
function bitwise_rightShift($shift)
|
|
{
|
|
$temp = new Math_BigInteger();
|
|
|
|
switch ( MATH_BIGINTEGER_MODE ) {
|
|
case MATH_BIGINTEGER_MODE_GMP:
|
|
static $two;
|
|
|
|
if (empty($two)) {
|
|
$two = gmp_init('2');
|
|
}
|
|
|
|
$temp->value = gmp_div_q($this->value, gmp_pow($two, $shift));
|
|
|
|
break;
|
|
case MATH_BIGINTEGER_MODE_BCMATH:
|
|
|
|
$temp->value = bcdiv($this->value, bcpow('2', $shift));
|
|
|
|
break;
|
|
default: // could just replace _lshift with this, but then all _lshift() calls would need to be rewritten
|
|
// and I don't want to do that...
|
|
$temp->value = $this->value;
|
|
$temp->_rshift($shift);
|
|
}
|
|
|
|
return $temp;
|
|
}
|
|
|
|
/**
|
|
* Logical Left Shift
|
|
*
|
|
* Shifts BigInteger's by $shift bits, effectively multiplying by 2**$shift.
|
|
*
|
|
* @param Integer $shift
|
|
* @return Math_BigInteger
|
|
* @access public
|
|
* @internal The only version that yields any speed increases is the internal version.
|
|
*/
|
|
function bitwise_leftShift($shift)
|
|
{
|
|
$temp = new Math_BigInteger();
|
|
|
|
switch ( MATH_BIGINTEGER_MODE ) {
|
|
case MATH_BIGINTEGER_MODE_GMP:
|
|
static $two;
|
|
|
|
if (empty($two)) {
|
|
$two = gmp_init('2');
|
|
}
|
|
|
|
$temp->value = gmp_mul($this->value, gmp_pow($two, $shift));
|
|
|
|
break;
|
|
case MATH_BIGINTEGER_MODE_BCMATH:
|
|
$temp->value = bcmul($this->value, bcpow('2', $shift));
|
|
|
|
break;
|
|
default: // could just replace _rshift with this, but then all _lshift() calls would need to be rewritten
|
|
// and I don't want to do that...
|
|
$temp->value = $this->value;
|
|
$temp->_lshift($shift);
|
|
}
|
|
|
|
return $temp;
|
|
}
|
|
|
|
/**
|
|
* Generate a random number
|
|
*
|
|
* $generator should be the name of a random number generating function whose first parameter is the minimum
|
|
* value and whose second parameter is the maximum value. If this function needs to be seeded, it should be
|
|
* done before this function is called.
|
|
*
|
|
* @param optional Integer $min
|
|
* @param optional Integer $max
|
|
* @param optional String $generator
|
|
* @return Math_BigInteger
|
|
* @access public
|
|
*/
|
|
function random($min = false, $max = false, $generator = 'mt_rand')
|
|
{
|
|
if ($min === false) {
|
|
$min = new Math_BigInteger(0);
|
|
}
|
|
|
|
if ($max === false) {
|
|
$max = new Math_BigInteger(0x7FFFFFFF);
|
|
}
|
|
|
|
$compare = $max->compare($min);
|
|
|
|
if (!$compare) {
|
|
return $min;
|
|
} else if ($compare < 0) {
|
|
// if $min is bigger then $max, swap $min and $max
|
|
$temp = $max;
|
|
$max = $min;
|
|
$min = $temp;
|
|
}
|
|
|
|
$max = $max->subtract($min);
|
|
$max = ltrim($max->toBytes(), chr(0));
|
|
$size = strlen($max) - 1;
|
|
|
|
$bytes = $size & 3;
|
|
for ($i = 0; $i < $bytes; $i++) {
|
|
$random.= chr($generator(0, 255));
|
|
}
|
|
|
|
$blocks = $size >> 2;
|
|
for ($i = 0; $i < $blocks; $i++) {
|
|
$random.= pack('N', $generator(-2147483648, 0x7FFFFFFF));
|
|
}
|
|
|
|
$temp = new Math_BigInteger($random, 256);
|
|
if ($temp->compare(new Math_BigInteger(substr($max, 1), 256)) > 0) {
|
|
$random = chr($generator(0, ord($max[0]) - 1)) . $random;
|
|
} else {
|
|
$random = chr($generator(0, ord($max[0]) )) . $random;
|
|
}
|
|
|
|
$random = new Math_BigInteger($random, 256);
|
|
|
|
return $random->add($min);
|
|
}
|
|
|
|
/**
|
|
* Logical Left Shift
|
|
*
|
|
* Shifts BigInteger's by $shift bits.
|
|
*
|
|
* @param Integer $shift
|
|
* @access private
|
|
*/
|
|
function _lshift($shift)
|
|
{
|
|
if ( $shift == 0 ) {
|
|
return;
|
|
}
|
|
|
|
$num_digits = floor($shift / 26);
|
|
$shift %= 26;
|
|
$shift = 1 << $shift;
|
|
|
|
$carry = 0;
|
|
|
|
for ($i = 0; $i < count($this->value); $i++) {
|
|
$temp = $this->value[$i] * $shift + $carry;
|
|
$carry = floor($temp / 0x4000000);
|
|
$this->value[$i] = $temp - $carry * 0x4000000;
|
|
}
|
|
|
|
if ( $carry ) {
|
|
$this->value[] = $carry;
|
|
}
|
|
|
|
while ($num_digits--) {
|
|
array_unshift($this->value, 0);
|
|
}
|
|
}
|
|
|
|
/**
|
|
* Logical Right Shift
|
|
*
|
|
* Shifts BigInteger's by $shift bits.
|
|
*
|
|
* @param Integer $shift
|
|
* @access private
|
|
*/
|
|
function _rshift($shift)
|
|
{
|
|
if ($shift == 0) {
|
|
$this->_normalize();
|
|
}
|
|
|
|
$num_digits = floor($shift / 26);
|
|
$shift %= 26;
|
|
$carry_shift = 26 - $shift;
|
|
$carry_mask = (1 << $shift) - 1;
|
|
|
|
if ( $num_digits ) {
|
|
$this->value = array_slice($this->value, $num_digits);
|
|
}
|
|
|
|
$carry = 0;
|
|
|
|
for ($i = count($this->value) - 1; $i >= 0; $i--) {
|
|
$temp = $this->value[$i] >> $shift | $carry;
|
|
$carry = ($this->value[$i] & $carry_mask) << $carry_shift;
|
|
$this->value[$i] = $temp;
|
|
}
|
|
|
|
$this->_normalize();
|
|
}
|
|
|
|
/**
|
|
* Normalize
|
|
*
|
|
* Deletes leading zeros.
|
|
*
|
|
* @see divide()
|
|
* @return Math_BigInteger
|
|
* @access private
|
|
*/
|
|
function _normalize()
|
|
{
|
|
if ( !count($this->value) ) {
|
|
return $this;
|
|
}
|
|
|
|
for ($i=count($this->value) - 1; $i >= 0; $i--) {
|
|
if ( $this->value[$i] ) {
|
|
break;
|
|
}
|
|
unset($this->value[$i]);
|
|
}
|
|
|
|
return $this;
|
|
}
|
|
|
|
/**
|
|
* Array Repeat
|
|
*
|
|
* @param $input Array
|
|
* @param $multiplier mixed
|
|
* @return Array
|
|
* @access private
|
|
*/
|
|
function _array_repeat($input, $multiplier)
|
|
{
|
|
return ($multiplier) ? array_fill(0, $multiplier, $input) : array();
|
|
}
|
|
|
|
/**
|
|
* Logical Left Shift
|
|
*
|
|
* Shifts binary strings $shift bits, essentially multiplying by 2**$shift.
|
|
*
|
|
* @param $x String
|
|
* @param $shift Integer
|
|
* @return String
|
|
* @access private
|
|
*/
|
|
function _base256_lshift(&$x, $shift)
|
|
{
|
|
if ($shift == 0) {
|
|
return;
|
|
}
|
|
|
|
$num_bytes = $shift >> 3; // eg. floor($shift/8)
|
|
$shift &= 7; // eg. $shift % 8
|
|
|
|
$carry = 0;
|
|
for ($i = strlen($x) - 1; $i >= 0; $i--) {
|
|
$temp = ord($x{$i}) << $shift | $carry;
|
|
$x{$i} = chr($temp);
|
|
$carry = $temp >> 8;
|
|
}
|
|
$carry = ($carry != 0) ? chr($carry) : '';
|
|
$x = $carry . $x . str_repeat(chr(0), $num_bytes);
|
|
}
|
|
|
|
/**
|
|
* Logical Right Shift
|
|
*
|
|
* Shifts binary strings $shift bits, essentially dividing by 2**$shift and returning the remainder.
|
|
*
|
|
* @param $x String
|
|
* @param $shift Integer
|
|
* @return String
|
|
* @access private
|
|
*/
|
|
function _base256_rshift(&$x, $shift)
|
|
{
|
|
if ($shift == 0) {
|
|
$x = ltrim($x, chr(0));
|
|
return '';
|
|
}
|
|
|
|
$num_bytes = $shift >> 3; // eg. floor($shift/8)
|
|
$shift &= 7; // eg. $shift % 8
|
|
|
|
$remainder = '';
|
|
if ($num_bytes) {
|
|
$start = $num_bytes > strlen($x) ? -strlen($x) : -$num_bytes;
|
|
$remainder = substr($x, $start);
|
|
$x = substr($x, 0, -$num_bytes);
|
|
}
|
|
|
|
$carry = 0;
|
|
$carry_shift = 8-$shift;
|
|
for ($i = 0; $i < strlen($x); $i++) {
|
|
$temp = (ord($x{$i}) >> $shift) | $carry;
|
|
$carry = (ord($x{$i}) << $carry_shift) & 0xFF;
|
|
$x{$i} = chr($temp);
|
|
}
|
|
$x = ltrim($x, chr(0));
|
|
|
|
$remainder = chr($carry >> $carry_shift) . $remainder;
|
|
|
|
return ltrim($remainder, chr(0));
|
|
}
|
|
|
|
// one quirk about how the following functions are implemented is that PHP defines N to be an unsigned long
|
|
// at 32-bits, while java's longs are 64-bits.
|
|
|
|
/**
|
|
* Converts 32-bit integers to bytes.
|
|
*
|
|
* @param Integer $x
|
|
* @return String
|
|
* @access private
|
|
*/
|
|
function _int2bytes($x)
|
|
{
|
|
return ltrim(pack('N', $x), chr(0));
|
|
}
|
|
|
|
/**
|
|
* Converts bytes to 32-bit integers
|
|
*
|
|
* @param String $x
|
|
* @return Integer
|
|
* @access private
|
|
*/
|
|
function _bytes2int($x)
|
|
{
|
|
$temp = unpack('Nint', str_pad($x, 4, chr(0), STR_PAD_LEFT));
|
|
return $temp['int'];
|
|
}
|
|
}
|
|
|
|
// vim: ts=4:sw=4:et:
|
|
// vim6: fdl=1:
|
|
?>
|