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<?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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* Numbers are stored in {@link http://en.wikipedia.org/wiki/Endianness little endian} format. ie.
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* (new Math_BigInteger(pow(2, 26)))->value = array(0, 1)
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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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* Here's an 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 1.33 2010/03/22 22:32:03 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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* Reduction constants
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*
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* @access private
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* @see Math_BigInteger::_reduce()
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*/
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/**
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* @see Math_BigInteger::_montgomery()
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* @see Math_BigInteger::_prepMontgomery()
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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::__clone()
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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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* Array constants
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*
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* Rather than create a thousands and thousands of new Math_BigInteger objects in repeated function calls to add() and
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* multiply() or whatever, we'll just work directly on arrays, taking them in as parameters and returning them.
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*
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* @access private
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*/
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/**
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* $result[MATH_BIGINTEGER_VALUE] contains the value.
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*/
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define('MATH_BIGINTEGER_VALUE', 0);
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/**
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* $result[MATH_BIGINTEGER_SIGN] contains the sign.
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*/
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define('MATH_BIGINTEGER_SIGN', 1);
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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 constants
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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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* Mode constants.
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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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* The largest digit that may be used in addition / subtraction
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*
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* (we do pow(2, 52) instead of using 4503599627370496, directly, because some PHP installations
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* will truncate 4503599627370496)
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*
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* @access private
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*/
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define('MATH_BIGINTEGER_MAX_DIGIT52', pow(2, 52));
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/**
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* Karatsuba Cutoff
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*
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* At what point do we switch between Karatsuba multiplication and schoolbook long multiplication?
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*
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* @access private
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*/
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define('MATH_BIGINTEGER_KARATSUBA_CUTOFF', 25);
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/**
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* Pure-PHP arbitrary precision 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 1.0.0RC4
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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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* Random number generator function
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*
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* @see setRandomGenerator()
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* @access private
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*/
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var $generator = 'mt_rand';
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/**
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* Precision
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*
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* @see setPrecision()
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* @access private
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*/
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var $precision = -1;
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/**
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* Precision Bitmask
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*
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* @see setPrecision()
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* @access private
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*/
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var $bitmask = false;
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/**
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* Mode independant value used for serialization.
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*
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* If the bcmath or gmp extensions are installed $this->value will be a non-serializable resource, hence the need for
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* a variable that'll be serializable regardless of whether or not extensions are being used. Unlike $this->value,
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* however, $this->hex is only calculated when $this->__sleep() is called.
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*
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* @see __sleep()
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* @see __wakeup()
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* @var String
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* @access private
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*/
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var $hex;
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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 an 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, $base = 10)
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{
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if ( !defined('MATH_BIGINTEGER_MODE') ) {
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switch (true) {
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case extension_loaded('gmp'):
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define('MATH_BIGINTEGER_MODE', MATH_BIGINTEGER_MODE_GMP);
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break;
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case extension_loaded('bcmath'):
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define('MATH_BIGINTEGER_MODE', MATH_BIGINTEGER_MODE_BCMATH);
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break;
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default:
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define('MATH_BIGINTEGER_MODE', MATH_BIGINTEGER_MODE_INTERNAL);
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}
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}
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switch ( MATH_BIGINTEGER_MODE ) {
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case MATH_BIGINTEGER_MODE_GMP:
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if (is_resource($x) && get_resource_type($x) == 'GMP integer') {
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$this->value = $x;
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return;
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}
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$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 (empty($x)) {
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return;
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}
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switch ($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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$sign = $this->is_negative ? '-' : '';
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$this->value = gmp_init($sign . '0x' . bin2hex($x));
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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($this->value, '4294967296', 0); // 4294967296 == 2**32
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$this->value = bcadd($this->value, 0x1000000 * ord($x[$i]) + ((ord($x[$i + 1]) << 16) | (ord($x[$i + 2]) << 8) | ord($x[$i + 3])), 0);
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}
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if ($this->is_negative) {
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$this->value = '-' . $this->value;
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}
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break;
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// converts a base-2**8 (big endian / msb) number to base-2**26 (little endian / lsb)
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default:
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while (strlen($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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|
355 |
$this->is_negative = $is_negative = true;
|
|
|
356 |
$x = bin2hex(~pack('H*', $x));
|
|
|
357 |
}
|
|
|
358 |
|
|
|
359 |
switch ( MATH_BIGINTEGER_MODE ) {
|
|
|
360 |
case MATH_BIGINTEGER_MODE_GMP:
|
|
|
361 |
$temp = $this->is_negative ? '-0x' . $x : '0x' . $x;
|
|
|
362 |
$this->value = gmp_init($temp);
|
|
|
363 |
$this->is_negative = false;
|
|
|
364 |
break;
|
|
|
365 |
case MATH_BIGINTEGER_MODE_BCMATH:
|
|
|
366 |
$x = ( strlen($x) & 1 ) ? '0' . $x : $x;
|
|
|
367 |
$temp = new Math_BigInteger(pack('H*', $x), 256);
|
|
|
368 |
$this->value = $this->is_negative ? '-' . $temp->value : $temp->value;
|
|
|
369 |
$this->is_negative = false;
|
|
|
370 |
break;
|
|
|
371 |
default:
|
|
|
372 |
$x = ( strlen($x) & 1 ) ? '0' . $x : $x;
|
|
|
373 |
$temp = new Math_BigInteger(pack('H*', $x), 256);
|
|
|
374 |
$this->value = $temp->value;
|
|
|
375 |
}
|
|
|
376 |
|
|
|
377 |
if ($is_negative) {
|
|
|
378 |
$temp = $this->add(new Math_BigInteger('-1'));
|
|
|
379 |
$this->value = $temp->value;
|
|
|
380 |
}
|
|
|
381 |
break;
|
|
|
382 |
case 10:
|
|
|
383 |
case -10:
|
|
|
384 |
$x = preg_replace('#^(-?[0-9]*).*#', '$1', $x);
|
|
|
385 |
|
|
|
386 |
switch ( MATH_BIGINTEGER_MODE ) {
|
|
|
387 |
case MATH_BIGINTEGER_MODE_GMP:
|
|
|
388 |
$this->value = gmp_init($x);
|
|
|
389 |
break;
|
|
|
390 |
case MATH_BIGINTEGER_MODE_BCMATH:
|
|
|
391 |
// explicitly casting $x to a string is necessary, here, since doing $x[0] on -1 yields different
|
|
|
392 |
// results then doing it on '-1' does (modInverse does $x[0])
|
|
|
393 |
$this->value = (string) $x;
|
|
|
394 |
break;
|
|
|
395 |
default:
|
|
|
396 |
$temp = new Math_BigInteger();
|
|
|
397 |
|
|
|
398 |
// array(10000000) is 10**7 in base-2**26. 10**7 is the closest to 2**26 we can get without passing it.
|
|
|
399 |
$multiplier = new Math_BigInteger();
|
|
|
400 |
$multiplier->value = array(10000000);
|
|
|
401 |
|
|
|
402 |
if ($x[0] == '-') {
|
|
|
403 |
$this->is_negative = true;
|
|
|
404 |
$x = substr($x, 1);
|
|
|
405 |
}
|
|
|
406 |
|
|
|
407 |
$x = str_pad($x, strlen($x) + (6 * strlen($x)) % 7, 0, STR_PAD_LEFT);
|
|
|
408 |
|
|
|
409 |
while (strlen($x)) {
|
|
|
410 |
$temp = $temp->multiply($multiplier);
|
|
|
411 |
$temp = $temp->add(new Math_BigInteger($this->_int2bytes(substr($x, 0, 7)), 256));
|
|
|
412 |
$x = substr($x, 7);
|
|
|
413 |
}
|
|
|
414 |
|
|
|
415 |
$this->value = $temp->value;
|
|
|
416 |
}
|
|
|
417 |
break;
|
|
|
418 |
case 2: // base-2 support originally implemented by Lluis Pamies - thanks!
|
|
|
419 |
case -2:
|
|
|
420 |
if ($base > 0 && $x[0] == '-') {
|
|
|
421 |
$this->is_negative = true;
|
|
|
422 |
$x = substr($x, 1);
|
|
|
423 |
}
|
|
|
424 |
|
|
|
425 |
$x = preg_replace('#^([01]*).*#', '$1', $x);
|
|
|
426 |
$x = str_pad($x, strlen($x) + (3 * strlen($x)) % 4, 0, STR_PAD_LEFT);
|
|
|
427 |
|
|
|
428 |
$str = '0x';
|
|
|
429 |
while (strlen($x)) {
|
|
|
430 |
$part = substr($x, 0, 4);
|
|
|
431 |
$str.= dechex(bindec($part));
|
|
|
432 |
$x = substr($x, 4);
|
|
|
433 |
}
|
|
|
434 |
|
|
|
435 |
if ($this->is_negative) {
|
|
|
436 |
$str = '-' . $str;
|
|
|
437 |
}
|
|
|
438 |
|
|
|
439 |
$temp = new Math_BigInteger($str, 8 * $base); // ie. either -16 or +16
|
|
|
440 |
$this->value = $temp->value;
|
|
|
441 |
$this->is_negative = $temp->is_negative;
|
|
|
442 |
|
|
|
443 |
break;
|
|
|
444 |
default:
|
|
|
445 |
// base not supported, so we'll let $this == 0
|
|
|
446 |
}
|
|
|
447 |
}
|
|
|
448 |
|
|
|
449 |
/**
|
|
|
450 |
* Converts a BigInteger to a byte string (eg. base-256).
|
|
|
451 |
*
|
|
|
452 |
* Negative numbers are saved as positive numbers, unless $twos_compliment is set to true, at which point, they're
|
|
|
453 |
* saved as two's compliment.
|
|
|
454 |
*
|
|
|
455 |
* Here's an example:
|
|
|
456 |
* <code>
|
|
|
457 |
* <?php
|
|
|
458 |
* include('Math/BigInteger.php');
|
|
|
459 |
*
|
|
|
460 |
* $a = new Math_BigInteger('65');
|
|
|
461 |
*
|
|
|
462 |
* echo $a->toBytes(); // outputs chr(65)
|
|
|
463 |
* ?>
|
|
|
464 |
* </code>
|
|
|
465 |
*
|
|
|
466 |
* @param Boolean $twos_compliment
|
|
|
467 |
* @return String
|
|
|
468 |
* @access public
|
|
|
469 |
* @internal Converts a base-2**26 number to base-2**8
|
|
|
470 |
*/
|
|
|
471 |
function toBytes($twos_compliment = false)
|
|
|
472 |
{
|
|
|
473 |
if ($twos_compliment) {
|
|
|
474 |
$comparison = $this->compare(new Math_BigInteger());
|
|
|
475 |
if ($comparison == 0) {
|
|
|
476 |
return $this->precision > 0 ? str_repeat(chr(0), ($this->precision + 1) >> 3) : '';
|
|
|
477 |
}
|
|
|
478 |
|
|
|
479 |
$temp = $comparison < 0 ? $this->add(new Math_BigInteger(1)) : $this->copy();
|
|
|
480 |
$bytes = $temp->toBytes();
|
|
|
481 |
|
|
|
482 |
if (empty($bytes)) { // eg. if the number we're trying to convert is -1
|
|
|
483 |
$bytes = chr(0);
|
|
|
484 |
}
|
|
|
485 |
|
|
|
486 |
if (ord($bytes[0]) & 0x80) {
|
|
|
487 |
$bytes = chr(0) . $bytes;
|
|
|
488 |
}
|
|
|
489 |
|
|
|
490 |
return $comparison < 0 ? ~$bytes : $bytes;
|
|
|
491 |
}
|
|
|
492 |
|
|
|
493 |
switch ( MATH_BIGINTEGER_MODE ) {
|
|
|
494 |
case MATH_BIGINTEGER_MODE_GMP:
|
|
|
495 |
if (gmp_cmp($this->value, gmp_init(0)) == 0) {
|
|
|
496 |
return $this->precision > 0 ? str_repeat(chr(0), ($this->precision + 1) >> 3) : '';
|
|
|
497 |
}
|
|
|
498 |
|
|
|
499 |
$temp = gmp_strval(gmp_abs($this->value), 16);
|
|
|
500 |
$temp = ( strlen($temp) & 1 ) ? '0' . $temp : $temp;
|
|
|
501 |
$temp = pack('H*', $temp);
|
|
|
502 |
|
|
|
503 |
return $this->precision > 0 ?
|
|
|
504 |
substr(str_pad($temp, $this->precision >> 3, chr(0), STR_PAD_LEFT), -($this->precision >> 3)) :
|
|
|
505 |
ltrim($temp, chr(0));
|
|
|
506 |
case MATH_BIGINTEGER_MODE_BCMATH:
|
|
|
507 |
if ($this->value === '0') {
|
|
|
508 |
return $this->precision > 0 ? str_repeat(chr(0), ($this->precision + 1) >> 3) : '';
|
|
|
509 |
}
|
|
|
510 |
|
|
|
511 |
$value = '';
|
|
|
512 |
$current = $this->value;
|
|
|
513 |
|
|
|
514 |
if ($current[0] == '-') {
|
|
|
515 |
$current = substr($current, 1);
|
|
|
516 |
}
|
|
|
517 |
|
|
|
518 |
while (bccomp($current, '0', 0) > 0) {
|
|
|
519 |
$temp = bcmod($current, '16777216');
|
|
|
520 |
$value = chr($temp >> 16) . chr($temp >> 8) . chr($temp) . $value;
|
|
|
521 |
$current = bcdiv($current, '16777216', 0);
|
|
|
522 |
}
|
|
|
523 |
|
|
|
524 |
return $this->precision > 0 ?
|
|
|
525 |
substr(str_pad($value, $this->precision >> 3, chr(0), STR_PAD_LEFT), -($this->precision >> 3)) :
|
|
|
526 |
ltrim($value, chr(0));
|
|
|
527 |
}
|
|
|
528 |
|
|
|
529 |
if (!count($this->value)) {
|
|
|
530 |
return $this->precision > 0 ? str_repeat(chr(0), ($this->precision + 1) >> 3) : '';
|
|
|
531 |
}
|
|
|
532 |
$result = $this->_int2bytes($this->value[count($this->value) - 1]);
|
|
|
533 |
|
|
|
534 |
$temp = $this->copy();
|
|
|
535 |
|
|
|
536 |
for ($i = count($temp->value) - 2; $i >= 0; --$i) {
|
|
|
537 |
$temp->_base256_lshift($result, 26);
|
|
|
538 |
$result = $result | str_pad($temp->_int2bytes($temp->value[$i]), strlen($result), chr(0), STR_PAD_LEFT);
|
|
|
539 |
}
|
|
|
540 |
|
|
|
541 |
return $this->precision > 0 ?
|
|
|
542 |
str_pad(substr($result, -(($this->precision + 7) >> 3)), ($this->precision + 7) >> 3, chr(0), STR_PAD_LEFT) :
|
|
|
543 |
$result;
|
|
|
544 |
}
|
|
|
545 |
|
|
|
546 |
/**
|
|
|
547 |
* Converts a BigInteger to a hex string (eg. base-16)).
|
|
|
548 |
*
|
|
|
549 |
* Negative numbers are saved as positive numbers, unless $twos_compliment is set to true, at which point, they're
|
|
|
550 |
* saved as two's compliment.
|
|
|
551 |
*
|
|
|
552 |
* Here's an example:
|
|
|
553 |
* <code>
|
|
|
554 |
* <?php
|
|
|
555 |
* include('Math/BigInteger.php');
|
|
|
556 |
*
|
|
|
557 |
* $a = new Math_BigInteger('65');
|
|
|
558 |
*
|
|
|
559 |
* echo $a->toHex(); // outputs '41'
|
|
|
560 |
* ?>
|
|
|
561 |
* </code>
|
|
|
562 |
*
|
|
|
563 |
* @param Boolean $twos_compliment
|
|
|
564 |
* @return String
|
|
|
565 |
* @access public
|
|
|
566 |
* @internal Converts a base-2**26 number to base-2**8
|
|
|
567 |
*/
|
|
|
568 |
function toHex($twos_compliment = false)
|
|
|
569 |
{
|
|
|
570 |
return bin2hex($this->toBytes($twos_compliment));
|
|
|
571 |
}
|
|
|
572 |
|
|
|
573 |
/**
|
|
|
574 |
* Converts a BigInteger to a bit string (eg. base-2).
|
|
|
575 |
*
|
|
|
576 |
* Negative numbers are saved as positive numbers, unless $twos_compliment is set to true, at which point, they're
|
|
|
577 |
* saved as two's compliment.
|
|
|
578 |
*
|
|
|
579 |
* Here's an example:
|
|
|
580 |
* <code>
|
|
|
581 |
* <?php
|
|
|
582 |
* include('Math/BigInteger.php');
|
|
|
583 |
*
|
|
|
584 |
* $a = new Math_BigInteger('65');
|
|
|
585 |
*
|
|
|
586 |
* echo $a->toBits(); // outputs '1000001'
|
|
|
587 |
* ?>
|
|
|
588 |
* </code>
|
|
|
589 |
*
|
|
|
590 |
* @param Boolean $twos_compliment
|
|
|
591 |
* @return String
|
|
|
592 |
* @access public
|
|
|
593 |
* @internal Converts a base-2**26 number to base-2**2
|
|
|
594 |
*/
|
|
|
595 |
function toBits($twos_compliment = false)
|
|
|
596 |
{
|
|
|
597 |
$hex = $this->toHex($twos_compliment);
|
|
|
598 |
$bits = '';
|
|
|
599 |
for ($i = 0; $i < strlen($hex); $i+=8) {
|
|
|
600 |
$bits.= str_pad(decbin(hexdec(substr($hex, $i, 8))), 32, '0', STR_PAD_LEFT);
|
|
|
601 |
}
|
|
|
602 |
return $this->precision > 0 ? substr($bits, -$this->precision) : ltrim($bits, '0');
|
|
|
603 |
}
|
|
|
604 |
|
|
|
605 |
/**
|
|
|
606 |
* Converts a BigInteger to a base-10 number.
|
|
|
607 |
*
|
|
|
608 |
* Here's an example:
|
|
|
609 |
* <code>
|
|
|
610 |
* <?php
|
|
|
611 |
* include('Math/BigInteger.php');
|
|
|
612 |
*
|
|
|
613 |
* $a = new Math_BigInteger('50');
|
|
|
614 |
*
|
|
|
615 |
* echo $a->toString(); // outputs 50
|
|
|
616 |
* ?>
|
|
|
617 |
* </code>
|
|
|
618 |
*
|
|
|
619 |
* @return String
|
|
|
620 |
* @access public
|
|
|
621 |
* @internal Converts a base-2**26 number to base-10**7 (which is pretty much base-10)
|
|
|
622 |
*/
|
|
|
623 |
function toString()
|
|
|
624 |
{
|
|
|
625 |
switch ( MATH_BIGINTEGER_MODE ) {
|
|
|
626 |
case MATH_BIGINTEGER_MODE_GMP:
|
|
|
627 |
return gmp_strval($this->value);
|
|
|
628 |
case MATH_BIGINTEGER_MODE_BCMATH:
|
|
|
629 |
if ($this->value === '0') {
|
|
|
630 |
return '0';
|
|
|
631 |
}
|
|
|
632 |
|
|
|
633 |
return ltrim($this->value, '0');
|
|
|
634 |
}
|
|
|
635 |
|
|
|
636 |
if (!count($this->value)) {
|
|
|
637 |
return '0';
|
|
|
638 |
}
|
|
|
639 |
|
|
|
640 |
$temp = $this->copy();
|
|
|
641 |
$temp->is_negative = false;
|
|
|
642 |
|
|
|
643 |
$divisor = new Math_BigInteger();
|
|
|
644 |
$divisor->value = array(10000000); // eg. 10**7
|
|
|
645 |
$result = '';
|
|
|
646 |
while (count($temp->value)) {
|
|
|
647 |
list($temp, $mod) = $temp->divide($divisor);
|
|
|
648 |
$result = str_pad(isset($mod->value[0]) ? $mod->value[0] : '', 7, '0', STR_PAD_LEFT) . $result;
|
|
|
649 |
}
|
|
|
650 |
$result = ltrim($result, '0');
|
|
|
651 |
if (empty($result)) {
|
|
|
652 |
$result = '0';
|
|
|
653 |
}
|
|
|
654 |
|
|
|
655 |
if ($this->is_negative) {
|
|
|
656 |
$result = '-' . $result;
|
|
|
657 |
}
|
|
|
658 |
|
|
|
659 |
return $result;
|
|
|
660 |
}
|
|
|
661 |
|
|
|
662 |
/**
|
|
|
663 |
* Copy an object
|
|
|
664 |
*
|
|
|
665 |
* PHP5 passes objects by reference while PHP4 passes by value. As such, we need a function to guarantee
|
|
|
666 |
* that all objects are passed by value, when appropriate. More information can be found here:
|
|
|
667 |
*
|
|
|
668 |
* {@link http://php.net/language.oop5.basic#51624}
|
|
|
669 |
*
|
|
|
670 |
* @access public
|
|
|
671 |
* @see __clone()
|
|
|
672 |
* @return Math_BigInteger
|
|
|
673 |
*/
|
|
|
674 |
function copy()
|
|
|
675 |
{
|
|
|
676 |
$temp = new Math_BigInteger();
|
|
|
677 |
$temp->value = $this->value;
|
|
|
678 |
$temp->is_negative = $this->is_negative;
|
|
|
679 |
$temp->generator = $this->generator;
|
|
|
680 |
$temp->precision = $this->precision;
|
|
|
681 |
$temp->bitmask = $this->bitmask;
|
|
|
682 |
return $temp;
|
|
|
683 |
}
|
|
|
684 |
|
|
|
685 |
/**
|
|
|
686 |
* __toString() magic method
|
|
|
687 |
*
|
|
|
688 |
* Will be called, automatically, if you're supporting just PHP5. If you're supporting PHP4, you'll need to call
|
|
|
689 |
* toString().
|
|
|
690 |
*
|
|
|
691 |
* @access public
|
|
|
692 |
* @internal Implemented per a suggestion by Techie-Michael - thanks!
|
|
|
693 |
*/
|
|
|
694 |
function __toString()
|
|
|
695 |
{
|
|
|
696 |
return $this->toString();
|
|
|
697 |
}
|
|
|
698 |
|
|
|
699 |
/**
|
|
|
700 |
* __clone() magic method
|
|
|
701 |
*
|
|
|
702 |
* Although you can call Math_BigInteger::__toString() directly in PHP5, you cannot call Math_BigInteger::__clone()
|
|
|
703 |
* directly in PHP5. You can in PHP4 since it's not a magic method, but in PHP5, you have to call it by using the PHP5
|
|
|
704 |
* only syntax of $y = clone $x. As such, if you're trying to write an application that works on both PHP4 and PHP5,
|
|
|
705 |
* call Math_BigInteger::copy(), instead.
|
|
|
706 |
*
|
|
|
707 |
* @access public
|
|
|
708 |
* @see copy()
|
|
|
709 |
* @return Math_BigInteger
|
|
|
710 |
*/
|
|
|
711 |
function __clone()
|
|
|
712 |
{
|
|
|
713 |
return $this->copy();
|
|
|
714 |
}
|
|
|
715 |
|
|
|
716 |
/**
|
|
|
717 |
* __sleep() magic method
|
|
|
718 |
*
|
|
|
719 |
* Will be called, automatically, when serialize() is called on a Math_BigInteger object.
|
|
|
720 |
*
|
|
|
721 |
* @see __wakeup()
|
|
|
722 |
* @access public
|
|
|
723 |
*/
|
|
|
724 |
function __sleep()
|
|
|
725 |
{
|
|
|
726 |
$this->hex = $this->toHex(true);
|
|
|
727 |
$vars = array('hex');
|
|
|
728 |
if ($this->generator != 'mt_rand') {
|
|
|
729 |
$vars[] = 'generator';
|
|
|
730 |
}
|
|
|
731 |
if ($this->precision > 0) {
|
|
|
732 |
$vars[] = 'precision';
|
|
|
733 |
}
|
|
|
734 |
return $vars;
|
|
|
735 |
|
|
|
736 |
}
|
|
|
737 |
|
|
|
738 |
/**
|
|
|
739 |
* __wakeup() magic method
|
|
|
740 |
*
|
|
|
741 |
* Will be called, automatically, when unserialize() is called on a Math_BigInteger object.
|
|
|
742 |
*
|
|
|
743 |
* @see __sleep()
|
|
|
744 |
* @access public
|
|
|
745 |
*/
|
|
|
746 |
function __wakeup()
|
|
|
747 |
{
|
|
|
748 |
$temp = new Math_BigInteger($this->hex, -16);
|
|
|
749 |
$this->value = $temp->value;
|
|
|
750 |
$this->is_negative = $temp->is_negative;
|
|
|
751 |
$this->setRandomGenerator($this->generator);
|
|
|
752 |
if ($this->precision > 0) {
|
|
|
753 |
// recalculate $this->bitmask
|
|
|
754 |
$this->setPrecision($this->precision);
|
|
|
755 |
}
|
|
|
756 |
}
|
|
|
757 |
|
|
|
758 |
/**
|
|
|
759 |
* Adds two BigIntegers.
|
|
|
760 |
*
|
|
|
761 |
* Here's an example:
|
|
|
762 |
* <code>
|
|
|
763 |
* <?php
|
|
|
764 |
* include('Math/BigInteger.php');
|
|
|
765 |
*
|
|
|
766 |
* $a = new Math_BigInteger('10');
|
|
|
767 |
* $b = new Math_BigInteger('20');
|
|
|
768 |
*
|
|
|
769 |
* $c = $a->add($b);
|
|
|
770 |
*
|
|
|
771 |
* echo $c->toString(); // outputs 30
|
|
|
772 |
* ?>
|
|
|
773 |
* </code>
|
|
|
774 |
*
|
|
|
775 |
* @param Math_BigInteger $y
|
|
|
776 |
* @return Math_BigInteger
|
|
|
777 |
* @access public
|
|
|
778 |
* @internal Performs base-2**52 addition
|
|
|
779 |
*/
|
|
|
780 |
function add($y)
|
|
|
781 |
{
|
|
|
782 |
switch ( MATH_BIGINTEGER_MODE ) {
|
|
|
783 |
case MATH_BIGINTEGER_MODE_GMP:
|
|
|
784 |
$temp = new Math_BigInteger();
|
|
|
785 |
$temp->value = gmp_add($this->value, $y->value);
|
|
|
786 |
|
|
|
787 |
return $this->_normalize($temp);
|
|
|
788 |
case MATH_BIGINTEGER_MODE_BCMATH:
|
|
|
789 |
$temp = new Math_BigInteger();
|
|
|
790 |
$temp->value = bcadd($this->value, $y->value, 0);
|
|
|
791 |
|
|
|
792 |
return $this->_normalize($temp);
|
|
|
793 |
}
|
|
|
794 |
|
|
|
795 |
$temp = $this->_add($this->value, $this->is_negative, $y->value, $y->is_negative);
|
|
|
796 |
|
|
|
797 |
$result = new Math_BigInteger();
|
|
|
798 |
$result->value = $temp[MATH_BIGINTEGER_VALUE];
|
|
|
799 |
$result->is_negative = $temp[MATH_BIGINTEGER_SIGN];
|
|
|
800 |
|
|
|
801 |
return $this->_normalize($result);
|
|
|
802 |
}
|
|
|
803 |
|
|
|
804 |
/**
|
|
|
805 |
* Performs addition.
|
|
|
806 |
*
|
|
|
807 |
* @param Array $x_value
|
|
|
808 |
* @param Boolean $x_negative
|
|
|
809 |
* @param Array $y_value
|
|
|
810 |
* @param Boolean $y_negative
|
|
|
811 |
* @return Array
|
|
|
812 |
* @access private
|
|
|
813 |
*/
|
|
|
814 |
function _add($x_value, $x_negative, $y_value, $y_negative)
|
|
|
815 |
{
|
|
|
816 |
$x_size = count($x_value);
|
|
|
817 |
$y_size = count($y_value);
|
|
|
818 |
|
|
|
819 |
if ($x_size == 0) {
|
|
|
820 |
return array(
|
|
|
821 |
MATH_BIGINTEGER_VALUE => $y_value,
|
|
|
822 |
MATH_BIGINTEGER_SIGN => $y_negative
|
|
|
823 |
);
|
|
|
824 |
} else if ($y_size == 0) {
|
|
|
825 |
return array(
|
|
|
826 |
MATH_BIGINTEGER_VALUE => $x_value,
|
|
|
827 |
MATH_BIGINTEGER_SIGN => $x_negative
|
|
|
828 |
);
|
|
|
829 |
}
|
|
|
830 |
|
|
|
831 |
// subtract, if appropriate
|
|
|
832 |
if ( $x_negative != $y_negative ) {
|
|
|
833 |
if ( $x_value == $y_value ) {
|
|
|
834 |
return array(
|
|
|
835 |
MATH_BIGINTEGER_VALUE => array(),
|
|
|
836 |
MATH_BIGINTEGER_SIGN => false
|
|
|
837 |
);
|
|
|
838 |
}
|
|
|
839 |
|
|
|
840 |
$temp = $this->_subtract($x_value, false, $y_value, false);
|
|
|
841 |
$temp[MATH_BIGINTEGER_SIGN] = $this->_compare($x_value, false, $y_value, false) > 0 ?
|
|
|
842 |
$x_negative : $y_negative;
|
|
|
843 |
|
|
|
844 |
return $temp;
|
|
|
845 |
}
|
|
|
846 |
|
|
|
847 |
if ($x_size < $y_size) {
|
|
|
848 |
$size = $x_size;
|
|
|
849 |
$value = $y_value;
|
|
|
850 |
} else {
|
|
|
851 |
$size = $y_size;
|
|
|
852 |
$value = $x_value;
|
|
|
853 |
}
|
|
|
854 |
|
|
|
855 |
$value[] = 0; // just in case the carry adds an extra digit
|
|
|
856 |
|
|
|
857 |
$carry = 0;
|
|
|
858 |
for ($i = 0, $j = 1; $j < $size; $i+=2, $j+=2) {
|
|
|
859 |
$sum = $x_value[$j] * 0x4000000 + $x_value[$i] + $y_value[$j] * 0x4000000 + $y_value[$i] + $carry;
|
|
|
860 |
$carry = $sum >= MATH_BIGINTEGER_MAX_DIGIT52; // eg. floor($sum / 2**52); only possible values (in any base) are 0 and 1
|
|
|
861 |
$sum = $carry ? $sum - MATH_BIGINTEGER_MAX_DIGIT52 : $sum;
|
|
|
862 |
|
|
|
863 |
$temp = (int) ($sum / 0x4000000);
|
|
|
864 |
|
|
|
865 |
$value[$i] = (int) ($sum - 0x4000000 * $temp); // eg. a faster alternative to fmod($sum, 0x4000000)
|
|
|
866 |
$value[$j] = $temp;
|
|
|
867 |
}
|
|
|
868 |
|
|
|
869 |
if ($j == $size) { // ie. if $y_size is odd
|
|
|
870 |
$sum = $x_value[$i] + $y_value[$i] + $carry;
|
|
|
871 |
$carry = $sum >= 0x4000000;
|
|
|
872 |
$value[$i] = $carry ? $sum - 0x4000000 : $sum;
|
|
|
873 |
++$i; // ie. let $i = $j since we've just done $value[$i]
|
|
|
874 |
}
|
|
|
875 |
|
|
|
876 |
if ($carry) {
|
|
|
877 |
for (; $value[$i] == 0x3FFFFFF; ++$i) {
|
|
|
878 |
$value[$i] = 0;
|
|
|
879 |
}
|
|
|
880 |
++$value[$i];
|
|
|
881 |
}
|
|
|
882 |
|
|
|
883 |
return array(
|
|
|
884 |
MATH_BIGINTEGER_VALUE => $this->_trim($value),
|
|
|
885 |
MATH_BIGINTEGER_SIGN => $x_negative
|
|
|
886 |
);
|
|
|
887 |
}
|
|
|
888 |
|
|
|
889 |
/**
|
|
|
890 |
* Subtracts two BigIntegers.
|
|
|
891 |
*
|
|
|
892 |
* Here's an example:
|
|
|
893 |
* <code>
|
|
|
894 |
* <?php
|
|
|
895 |
* include('Math/BigInteger.php');
|
|
|
896 |
*
|
|
|
897 |
* $a = new Math_BigInteger('10');
|
|
|
898 |
* $b = new Math_BigInteger('20');
|
|
|
899 |
*
|
|
|
900 |
* $c = $a->subtract($b);
|
|
|
901 |
*
|
|
|
902 |
* echo $c->toString(); // outputs -10
|
|
|
903 |
* ?>
|
|
|
904 |
* </code>
|
|
|
905 |
*
|
|
|
906 |
* @param Math_BigInteger $y
|
|
|
907 |
* @return Math_BigInteger
|
|
|
908 |
* @access public
|
|
|
909 |
* @internal Performs base-2**52 subtraction
|
|
|
910 |
*/
|
|
|
911 |
function subtract($y)
|
|
|
912 |
{
|
|
|
913 |
switch ( MATH_BIGINTEGER_MODE ) {
|
|
|
914 |
case MATH_BIGINTEGER_MODE_GMP:
|
|
|
915 |
$temp = new Math_BigInteger();
|
|
|
916 |
$temp->value = gmp_sub($this->value, $y->value);
|
|
|
917 |
|
|
|
918 |
return $this->_normalize($temp);
|
|
|
919 |
case MATH_BIGINTEGER_MODE_BCMATH:
|
|
|
920 |
$temp = new Math_BigInteger();
|
|
|
921 |
$temp->value = bcsub($this->value, $y->value, 0);
|
|
|
922 |
|
|
|
923 |
return $this->_normalize($temp);
|
|
|
924 |
}
|
|
|
925 |
|
|
|
926 |
$temp = $this->_subtract($this->value, $this->is_negative, $y->value, $y->is_negative);
|
|
|
927 |
|
|
|
928 |
$result = new Math_BigInteger();
|
|
|
929 |
$result->value = $temp[MATH_BIGINTEGER_VALUE];
|
|
|
930 |
$result->is_negative = $temp[MATH_BIGINTEGER_SIGN];
|
|
|
931 |
|
|
|
932 |
return $this->_normalize($result);
|
|
|
933 |
}
|
|
|
934 |
|
|
|
935 |
/**
|
|
|
936 |
* Performs subtraction.
|
|
|
937 |
*
|
|
|
938 |
* @param Array $x_value
|
|
|
939 |
* @param Boolean $x_negative
|
|
|
940 |
* @param Array $y_value
|
|
|
941 |
* @param Boolean $y_negative
|
|
|
942 |
* @return Array
|
|
|
943 |
* @access private
|
|
|
944 |
*/
|
|
|
945 |
function _subtract($x_value, $x_negative, $y_value, $y_negative)
|
|
|
946 |
{
|
|
|
947 |
$x_size = count($x_value);
|
|
|
948 |
$y_size = count($y_value);
|
|
|
949 |
|
|
|
950 |
if ($x_size == 0) {
|
|
|
951 |
return array(
|
|
|
952 |
MATH_BIGINTEGER_VALUE => $y_value,
|
|
|
953 |
MATH_BIGINTEGER_SIGN => !$y_negative
|
|
|
954 |
);
|
|
|
955 |
} else if ($y_size == 0) {
|
|
|
956 |
return array(
|
|
|
957 |
MATH_BIGINTEGER_VALUE => $x_value,
|
|
|
958 |
MATH_BIGINTEGER_SIGN => $x_negative
|
|
|
959 |
);
|
|
|
960 |
}
|
|
|
961 |
|
|
|
962 |
// add, if appropriate (ie. -$x - +$y or +$x - -$y)
|
|
|
963 |
if ( $x_negative != $y_negative ) {
|
|
|
964 |
$temp = $this->_add($x_value, false, $y_value, false);
|
|
|
965 |
$temp[MATH_BIGINTEGER_SIGN] = $x_negative;
|
|
|
966 |
|
|
|
967 |
return $temp;
|
|
|
968 |
}
|
|
|
969 |
|
|
|
970 |
$diff = $this->_compare($x_value, $x_negative, $y_value, $y_negative);
|
|
|
971 |
|
|
|
972 |
if ( !$diff ) {
|
|
|
973 |
return array(
|
|
|
974 |
MATH_BIGINTEGER_VALUE => array(),
|
|
|
975 |
MATH_BIGINTEGER_SIGN => false
|
|
|
976 |
);
|
|
|
977 |
}
|
|
|
978 |
|
|
|
979 |
// switch $x and $y around, if appropriate.
|
|
|
980 |
if ( (!$x_negative && $diff < 0) || ($x_negative && $diff > 0) ) {
|
|
|
981 |
$temp = $x_value;
|
|
|
982 |
$x_value = $y_value;
|
|
|
983 |
$y_value = $temp;
|
|
|
984 |
|
|
|
985 |
$x_negative = !$x_negative;
|
|
|
986 |
|
|
|
987 |
$x_size = count($x_value);
|
|
|
988 |
$y_size = count($y_value);
|
|
|
989 |
}
|
|
|
990 |
|
|
|
991 |
// at this point, $x_value should be at least as big as - if not bigger than - $y_value
|
|
|
992 |
|
|
|
993 |
$carry = 0;
|
|
|
994 |
for ($i = 0, $j = 1; $j < $y_size; $i+=2, $j+=2) {
|
|
|
995 |
$sum = $x_value[$j] * 0x4000000 + $x_value[$i] - $y_value[$j] * 0x4000000 - $y_value[$i] - $carry;
|
|
|
996 |
$carry = $sum < 0; // eg. floor($sum / 2**52); only possible values (in any base) are 0 and 1
|
|
|
997 |
$sum = $carry ? $sum + MATH_BIGINTEGER_MAX_DIGIT52 : $sum;
|
|
|
998 |
|
|
|
999 |
$temp = (int) ($sum / 0x4000000);
|
|
|
1000 |
|
|
|
1001 |
$x_value[$i] = (int) ($sum - 0x4000000 * $temp);
|
|
|
1002 |
$x_value[$j] = $temp;
|
|
|
1003 |
}
|
|
|
1004 |
|
|
|
1005 |
if ($j == $y_size) { // ie. if $y_size is odd
|
|
|
1006 |
$sum = $x_value[$i] - $y_value[$i] - $carry;
|
|
|
1007 |
$carry = $sum < 0;
|
|
|
1008 |
$x_value[$i] = $carry ? $sum + 0x4000000 : $sum;
|
|
|
1009 |
++$i;
|
|
|
1010 |
}
|
|
|
1011 |
|
|
|
1012 |
if ($carry) {
|
|
|
1013 |
for (; !$x_value[$i]; ++$i) {
|
|
|
1014 |
$x_value[$i] = 0x3FFFFFF;
|
|
|
1015 |
}
|
|
|
1016 |
--$x_value[$i];
|
|
|
1017 |
}
|
|
|
1018 |
|
|
|
1019 |
return array(
|
|
|
1020 |
MATH_BIGINTEGER_VALUE => $this->_trim($x_value),
|
|
|
1021 |
MATH_BIGINTEGER_SIGN => $x_negative
|
|
|
1022 |
);
|
|
|
1023 |
}
|
|
|
1024 |
|
|
|
1025 |
/**
|
|
|
1026 |
* Multiplies two BigIntegers
|
|
|
1027 |
*
|
|
|
1028 |
* Here's an example:
|
|
|
1029 |
* <code>
|
|
|
1030 |
* <?php
|
|
|
1031 |
* include('Math/BigInteger.php');
|
|
|
1032 |
*
|
|
|
1033 |
* $a = new Math_BigInteger('10');
|
|
|
1034 |
* $b = new Math_BigInteger('20');
|
|
|
1035 |
*
|
|
|
1036 |
* $c = $a->multiply($b);
|
|
|
1037 |
*
|
|
|
1038 |
* echo $c->toString(); // outputs 200
|
|
|
1039 |
* ?>
|
|
|
1040 |
* </code>
|
|
|
1041 |
*
|
|
|
1042 |
* @param Math_BigInteger $x
|
|
|
1043 |
* @return Math_BigInteger
|
|
|
1044 |
* @access public
|
|
|
1045 |
*/
|
|
|
1046 |
function multiply($x)
|
|
|
1047 |
{
|
|
|
1048 |
switch ( MATH_BIGINTEGER_MODE ) {
|
|
|
1049 |
case MATH_BIGINTEGER_MODE_GMP:
|
|
|
1050 |
$temp = new Math_BigInteger();
|
|
|
1051 |
$temp->value = gmp_mul($this->value, $x->value);
|
|
|
1052 |
|
|
|
1053 |
return $this->_normalize($temp);
|
|
|
1054 |
case MATH_BIGINTEGER_MODE_BCMATH:
|
|
|
1055 |
$temp = new Math_BigInteger();
|
|
|
1056 |
$temp->value = bcmul($this->value, $x->value, 0);
|
|
|
1057 |
|
|
|
1058 |
return $this->_normalize($temp);
|
|
|
1059 |
}
|
|
|
1060 |
|
|
|
1061 |
$temp = $this->_multiply($this->value, $this->is_negative, $x->value, $x->is_negative);
|
|
|
1062 |
|
|
|
1063 |
$product = new Math_BigInteger();
|
|
|
1064 |
$product->value = $temp[MATH_BIGINTEGER_VALUE];
|
|
|
1065 |
$product->is_negative = $temp[MATH_BIGINTEGER_SIGN];
|
|
|
1066 |
|
|
|
1067 |
return $this->_normalize($product);
|
|
|
1068 |
}
|
|
|
1069 |
|
|
|
1070 |
/**
|
|
|
1071 |
* Performs multiplication.
|
|
|
1072 |
*
|
|
|
1073 |
* @param Array $x_value
|
|
|
1074 |
* @param Boolean $x_negative
|
|
|
1075 |
* @param Array $y_value
|
|
|
1076 |
* @param Boolean $y_negative
|
|
|
1077 |
* @return Array
|
|
|
1078 |
* @access private
|
|
|
1079 |
*/
|
|
|
1080 |
function _multiply($x_value, $x_negative, $y_value, $y_negative)
|
|
|
1081 |
{
|
|
|
1082 |
//if ( $x_value == $y_value ) {
|
|
|
1083 |
// return array(
|
|
|
1084 |
// MATH_BIGINTEGER_VALUE => $this->_square($x_value),
|
|
|
1085 |
// MATH_BIGINTEGER_SIGN => $x_sign != $y_value
|
|
|
1086 |
// );
|
|
|
1087 |
//}
|
|
|
1088 |
|
|
|
1089 |
$x_length = count($x_value);
|
|
|
1090 |
$y_length = count($y_value);
|
|
|
1091 |
|
|
|
1092 |
if ( !$x_length || !$y_length ) { // a 0 is being multiplied
|
|
|
1093 |
return array(
|
|
|
1094 |
MATH_BIGINTEGER_VALUE => array(),
|
|
|
1095 |
MATH_BIGINTEGER_SIGN => false
|
|
|
1096 |
);
|
|
|
1097 |
}
|
|
|
1098 |
|
|
|
1099 |
return array(
|
|
|
1100 |
MATH_BIGINTEGER_VALUE => min($x_length, $y_length) < 2 * MATH_BIGINTEGER_KARATSUBA_CUTOFF ?
|
|
|
1101 |
$this->_trim($this->_regularMultiply($x_value, $y_value)) :
|
|
|
1102 |
$this->_trim($this->_karatsuba($x_value, $y_value)),
|
|
|
1103 |
MATH_BIGINTEGER_SIGN => $x_negative != $y_negative
|
|
|
1104 |
);
|
|
|
1105 |
}
|
|
|
1106 |
|
|
|
1107 |
/**
|
|
|
1108 |
* Performs long multiplication on two BigIntegers
|
|
|
1109 |
*
|
|
|
1110 |
* Modeled after 'multiply' in MutableBigInteger.java.
|
|
|
1111 |
*
|
|
|
1112 |
* @param Array $x_value
|
|
|
1113 |
* @param Array $y_value
|
|
|
1114 |
* @return Array
|
|
|
1115 |
* @access private
|
|
|
1116 |
*/
|
|
|
1117 |
function _regularMultiply($x_value, $y_value)
|
|
|
1118 |
{
|
|
|
1119 |
$x_length = count($x_value);
|
|
|
1120 |
$y_length = count($y_value);
|
|
|
1121 |
|
|
|
1122 |
if ( !$x_length || !$y_length ) { // a 0 is being multiplied
|
|
|
1123 |
return array();
|
|
|
1124 |
}
|
|
|
1125 |
|
|
|
1126 |
if ( $x_length < $y_length ) {
|
|
|
1127 |
$temp = $x_value;
|
|
|
1128 |
$x_value = $y_value;
|
|
|
1129 |
$y_value = $temp;
|
|
|
1130 |
|
|
|
1131 |
$x_length = count($x_value);
|
|
|
1132 |
$y_length = count($y_value);
|
|
|
1133 |
}
|
|
|
1134 |
|
|
|
1135 |
$product_value = $this->_array_repeat(0, $x_length + $y_length);
|
|
|
1136 |
|
|
|
1137 |
// the following for loop could be removed if the for loop following it
|
|
|
1138 |
// (the one with nested for loops) initially set $i to 0, but
|
|
|
1139 |
// doing so would also make the result in one set of unnecessary adds,
|
|
|
1140 |
// since on the outermost loops first pass, $product->value[$k] is going
|
|
|
1141 |
// to always be 0
|
|
|
1142 |
|
|
|
1143 |
$carry = 0;
|
|
|
1144 |
|
|
|
1145 |
for ($j = 0; $j < $x_length; ++$j) { // ie. $i = 0
|
|
|
1146 |
$temp = $x_value[$j] * $y_value[0] + $carry; // $product_value[$k] == 0
|
|
|
1147 |
$carry = (int) ($temp / 0x4000000);
|
|
|
1148 |
$product_value[$j] = (int) ($temp - 0x4000000 * $carry);
|
|
|
1149 |
}
|
|
|
1150 |
|
|
|
1151 |
$product_value[$j] = $carry;
|
|
|
1152 |
|
|
|
1153 |
// the above for loop is what the previous comment was talking about. the
|
|
|
1154 |
// following for loop is the "one with nested for loops"
|
|
|
1155 |
for ($i = 1; $i < $y_length; ++$i) {
|
|
|
1156 |
$carry = 0;
|
|
|
1157 |
|
|
|
1158 |
for ($j = 0, $k = $i; $j < $x_length; ++$j, ++$k) {
|
|
|
1159 |
$temp = $product_value[$k] + $x_value[$j] * $y_value[$i] + $carry;
|
|
|
1160 |
$carry = (int) ($temp / 0x4000000);
|
|
|
1161 |
$product_value[$k] = (int) ($temp - 0x4000000 * $carry);
|
|
|
1162 |
}
|
|
|
1163 |
|
|
|
1164 |
$product_value[$k] = $carry;
|
|
|
1165 |
}
|
|
|
1166 |
|
|
|
1167 |
return $product_value;
|
|
|
1168 |
}
|
|
|
1169 |
|
|
|
1170 |
/**
|
|
|
1171 |
* Performs Karatsuba multiplication on two BigIntegers
|
|
|
1172 |
*
|
|
|
1173 |
* See {@link http://en.wikipedia.org/wiki/Karatsuba_algorithm Karatsuba algorithm} and
|
|
|
1174 |
* {@link http://math.libtomcrypt.com/files/tommath.pdf#page=120 MPM 5.2.3}.
|
|
|
1175 |
*
|
|
|
1176 |
* @param Array $x_value
|
|
|
1177 |
* @param Array $y_value
|
|
|
1178 |
* @return Array
|
|
|
1179 |
* @access private
|
|
|
1180 |
*/
|
|
|
1181 |
function _karatsuba($x_value, $y_value)
|
|
|
1182 |
{
|
|
|
1183 |
$m = min(count($x_value) >> 1, count($y_value) >> 1);
|
|
|
1184 |
|
|
|
1185 |
if ($m < MATH_BIGINTEGER_KARATSUBA_CUTOFF) {
|
|
|
1186 |
return $this->_regularMultiply($x_value, $y_value);
|
|
|
1187 |
}
|
|
|
1188 |
|
|
|
1189 |
$x1 = array_slice($x_value, $m);
|
|
|
1190 |
$x0 = array_slice($x_value, 0, $m);
|
|
|
1191 |
$y1 = array_slice($y_value, $m);
|
|
|
1192 |
$y0 = array_slice($y_value, 0, $m);
|
|
|
1193 |
|
|
|
1194 |
$z2 = $this->_karatsuba($x1, $y1);
|
|
|
1195 |
$z0 = $this->_karatsuba($x0, $y0);
|
|
|
1196 |
|
|
|
1197 |
$z1 = $this->_add($x1, false, $x0, false);
|
|
|
1198 |
$temp = $this->_add($y1, false, $y0, false);
|
|
|
1199 |
$z1 = $this->_karatsuba($z1[MATH_BIGINTEGER_VALUE], $temp[MATH_BIGINTEGER_VALUE]);
|
|
|
1200 |
$temp = $this->_add($z2, false, $z0, false);
|
|
|
1201 |
$z1 = $this->_subtract($z1, false, $temp[MATH_BIGINTEGER_VALUE], false);
|
|
|
1202 |
|
|
|
1203 |
$z2 = array_merge(array_fill(0, 2 * $m, 0), $z2);
|
|
|
1204 |
$z1[MATH_BIGINTEGER_VALUE] = array_merge(array_fill(0, $m, 0), $z1[MATH_BIGINTEGER_VALUE]);
|
|
|
1205 |
|
|
|
1206 |
$xy = $this->_add($z2, false, $z1[MATH_BIGINTEGER_VALUE], $z1[MATH_BIGINTEGER_SIGN]);
|
|
|
1207 |
$xy = $this->_add($xy[MATH_BIGINTEGER_VALUE], $xy[MATH_BIGINTEGER_SIGN], $z0, false);
|
|
|
1208 |
|
|
|
1209 |
return $xy[MATH_BIGINTEGER_VALUE];
|
|
|
1210 |
}
|
|
|
1211 |
|
|
|
1212 |
/**
|
|
|
1213 |
* Performs squaring
|
|
|
1214 |
*
|
|
|
1215 |
* @param Array $x
|
|
|
1216 |
* @return Array
|
|
|
1217 |
* @access private
|
|
|
1218 |
*/
|
|
|
1219 |
function _square($x = false)
|
|
|
1220 |
{
|
|
|
1221 |
return count($x) < 2 * MATH_BIGINTEGER_KARATSUBA_CUTOFF ?
|
|
|
1222 |
$this->_trim($this->_baseSquare($x)) :
|
|
|
1223 |
$this->_trim($this->_karatsubaSquare($x));
|
|
|
1224 |
}
|
|
|
1225 |
|
|
|
1226 |
/**
|
|
|
1227 |
* Performs traditional squaring on two BigIntegers
|
|
|
1228 |
*
|
|
|
1229 |
* Squaring can be done faster than multiplying a number by itself can be. See
|
|
|
1230 |
* {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=7 HAC 14.2.4} /
|
|
|
1231 |
* {@link http://math.libtomcrypt.com/files/tommath.pdf#page=141 MPM 5.3} for more information.
|
|
|
1232 |
*
|
|
|
1233 |
* @param Array $value
|
|
|
1234 |
* @return Array
|
|
|
1235 |
* @access private
|
|
|
1236 |
*/
|
|
|
1237 |
function _baseSquare($value)
|
|
|
1238 |
{
|
|
|
1239 |
if ( empty($value) ) {
|
|
|
1240 |
return array();
|
|
|
1241 |
}
|
|
|
1242 |
$square_value = $this->_array_repeat(0, 2 * count($value));
|
|
|
1243 |
|
|
|
1244 |
for ($i = 0, $max_index = count($value) - 1; $i <= $max_index; ++$i) {
|
|
|
1245 |
$i2 = $i << 1;
|
|
|
1246 |
|
|
|
1247 |
$temp = $square_value[$i2] + $value[$i] * $value[$i];
|
|
|
1248 |
$carry = (int) ($temp / 0x4000000);
|
|
|
1249 |
$square_value[$i2] = (int) ($temp - 0x4000000 * $carry);
|
|
|
1250 |
|
|
|
1251 |
// note how we start from $i+1 instead of 0 as we do in multiplication.
|
|
|
1252 |
for ($j = $i + 1, $k = $i2 + 1; $j <= $max_index; ++$j, ++$k) {
|
|
|
1253 |
$temp = $square_value[$k] + 2 * $value[$j] * $value[$i] + $carry;
|
|
|
1254 |
$carry = (int) ($temp / 0x4000000);
|
|
|
1255 |
$square_value[$k] = (int) ($temp - 0x4000000 * $carry);
|
|
|
1256 |
}
|
|
|
1257 |
|
|
|
1258 |
// the following line can yield values larger 2**15. at this point, PHP should switch
|
|
|
1259 |
// over to floats.
|
|
|
1260 |
$square_value[$i + $max_index + 1] = $carry;
|
|
|
1261 |
}
|
|
|
1262 |
|
|
|
1263 |
return $square_value;
|
|
|
1264 |
}
|
|
|
1265 |
|
|
|
1266 |
/**
|
|
|
1267 |
* Performs Karatsuba "squaring" on two BigIntegers
|
|
|
1268 |
*
|
|
|
1269 |
* See {@link http://en.wikipedia.org/wiki/Karatsuba_algorithm Karatsuba algorithm} and
|
|
|
1270 |
* {@link http://math.libtomcrypt.com/files/tommath.pdf#page=151 MPM 5.3.4}.
|
|
|
1271 |
*
|
|
|
1272 |
* @param Array $value
|
|
|
1273 |
* @return Array
|
|
|
1274 |
* @access private
|
|
|
1275 |
*/
|
|
|
1276 |
function _karatsubaSquare($value)
|
|
|
1277 |
{
|
|
|
1278 |
$m = count($value) >> 1;
|
|
|
1279 |
|
|
|
1280 |
if ($m < MATH_BIGINTEGER_KARATSUBA_CUTOFF) {
|
|
|
1281 |
return $this->_baseSquare($value);
|
|
|
1282 |
}
|
|
|
1283 |
|
|
|
1284 |
$x1 = array_slice($value, $m);
|
|
|
1285 |
$x0 = array_slice($value, 0, $m);
|
|
|
1286 |
|
|
|
1287 |
$z2 = $this->_karatsubaSquare($x1);
|
|
|
1288 |
$z0 = $this->_karatsubaSquare($x0);
|
|
|
1289 |
|
|
|
1290 |
$z1 = $this->_add($x1, false, $x0, false);
|
|
|
1291 |
$z1 = $this->_karatsubaSquare($z1[MATH_BIGINTEGER_VALUE]);
|
|
|
1292 |
$temp = $this->_add($z2, false, $z0, false);
|
|
|
1293 |
$z1 = $this->_subtract($z1, false, $temp[MATH_BIGINTEGER_VALUE], false);
|
|
|
1294 |
|
|
|
1295 |
$z2 = array_merge(array_fill(0, 2 * $m, 0), $z2);
|
|
|
1296 |
$z1[MATH_BIGINTEGER_VALUE] = array_merge(array_fill(0, $m, 0), $z1[MATH_BIGINTEGER_VALUE]);
|
|
|
1297 |
|
|
|
1298 |
$xx = $this->_add($z2, false, $z1[MATH_BIGINTEGER_VALUE], $z1[MATH_BIGINTEGER_SIGN]);
|
|
|
1299 |
$xx = $this->_add($xx[MATH_BIGINTEGER_VALUE], $xx[MATH_BIGINTEGER_SIGN], $z0, false);
|
|
|
1300 |
|
|
|
1301 |
return $xx[MATH_BIGINTEGER_VALUE];
|
|
|
1302 |
}
|
|
|
1303 |
|
|
|
1304 |
/**
|
|
|
1305 |
* Divides two BigIntegers.
|
|
|
1306 |
*
|
|
|
1307 |
* Returns an array whose first element contains the quotient and whose second element contains the
|
|
|
1308 |
* "common residue". If the remainder would be positive, the "common residue" and the remainder are the
|
|
|
1309 |
* same. If the remainder would be negative, the "common residue" is equal to the sum of the remainder
|
|
|
1310 |
* and the divisor (basically, the "common residue" is the first positive modulo).
|
|
|
1311 |
*
|
|
|
1312 |
* Here's an example:
|
|
|
1313 |
* <code>
|
|
|
1314 |
* <?php
|
|
|
1315 |
* include('Math/BigInteger.php');
|
|
|
1316 |
*
|
|
|
1317 |
* $a = new Math_BigInteger('10');
|
|
|
1318 |
* $b = new Math_BigInteger('20');
|
|
|
1319 |
*
|
|
|
1320 |
* list($quotient, $remainder) = $a->divide($b);
|
|
|
1321 |
*
|
|
|
1322 |
* echo $quotient->toString(); // outputs 0
|
|
|
1323 |
* echo "\r\n";
|
|
|
1324 |
* echo $remainder->toString(); // outputs 10
|
|
|
1325 |
* ?>
|
|
|
1326 |
* </code>
|
|
|
1327 |
*
|
|
|
1328 |
* @param Math_BigInteger $y
|
|
|
1329 |
* @return Array
|
|
|
1330 |
* @access public
|
|
|
1331 |
* @internal This function is based off of {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=9 HAC 14.20}.
|
|
|
1332 |
*/
|
|
|
1333 |
function divide($y)
|
|
|
1334 |
{
|
|
|
1335 |
switch ( MATH_BIGINTEGER_MODE ) {
|
|
|
1336 |
case MATH_BIGINTEGER_MODE_GMP:
|
|
|
1337 |
$quotient = new Math_BigInteger();
|
|
|
1338 |
$remainder = new Math_BigInteger();
|
|
|
1339 |
|
|
|
1340 |
list($quotient->value, $remainder->value) = gmp_div_qr($this->value, $y->value);
|
|
|
1341 |
|
|
|
1342 |
if (gmp_sign($remainder->value) < 0) {
|
|
|
1343 |
$remainder->value = gmp_add($remainder->value, gmp_abs($y->value));
|
|
|
1344 |
}
|
|
|
1345 |
|
|
|
1346 |
return array($this->_normalize($quotient), $this->_normalize($remainder));
|
|
|
1347 |
case MATH_BIGINTEGER_MODE_BCMATH:
|
|
|
1348 |
$quotient = new Math_BigInteger();
|
|
|
1349 |
$remainder = new Math_BigInteger();
|
|
|
1350 |
|
|
|
1351 |
$quotient->value = bcdiv($this->value, $y->value, 0);
|
|
|
1352 |
$remainder->value = bcmod($this->value, $y->value);
|
|
|
1353 |
|
|
|
1354 |
if ($remainder->value[0] == '-') {
|
|
|
1355 |
$remainder->value = bcadd($remainder->value, $y->value[0] == '-' ? substr($y->value, 1) : $y->value, 0);
|
|
|
1356 |
}
|
|
|
1357 |
|
|
|
1358 |
return array($this->_normalize($quotient), $this->_normalize($remainder));
|
|
|
1359 |
}
|
|
|
1360 |
|
|
|
1361 |
if (count($y->value) == 1) {
|
|
|
1362 |
list($q, $r) = $this->_divide_digit($this->value, $y->value[0]);
|
|
|
1363 |
$quotient = new Math_BigInteger();
|
|
|
1364 |
$remainder = new Math_BigInteger();
|
|
|
1365 |
$quotient->value = $q;
|
|
|
1366 |
$remainder->value = array($r);
|
|
|
1367 |
$quotient->is_negative = $this->is_negative != $y->is_negative;
|
|
|
1368 |
return array($this->_normalize($quotient), $this->_normalize($remainder));
|
|
|
1369 |
}
|
|
|
1370 |
|
|
|
1371 |
static $zero;
|
|
|
1372 |
if ( !isset($zero) ) {
|
|
|
1373 |
$zero = new Math_BigInteger();
|
|
|
1374 |
}
|
|
|
1375 |
|
|
|
1376 |
$x = $this->copy();
|
|
|
1377 |
$y = $y->copy();
|
|
|
1378 |
|
|
|
1379 |
$x_sign = $x->is_negative;
|
|
|
1380 |
$y_sign = $y->is_negative;
|
|
|
1381 |
|
|
|
1382 |
$x->is_negative = $y->is_negative = false;
|
|
|
1383 |
|
|
|
1384 |
$diff = $x->compare($y);
|
|
|
1385 |
|
|
|
1386 |
if ( !$diff ) {
|
|
|
1387 |
$temp = new Math_BigInteger();
|
|
|
1388 |
$temp->value = array(1);
|
|
|
1389 |
$temp->is_negative = $x_sign != $y_sign;
|
|
|
1390 |
return array($this->_normalize($temp), $this->_normalize(new Math_BigInteger()));
|
|
|
1391 |
}
|
|
|
1392 |
|
|
|
1393 |
if ( $diff < 0 ) {
|
|
|
1394 |
// if $x is negative, "add" $y.
|
|
|
1395 |
if ( $x_sign ) {
|
|
|
1396 |
$x = $y->subtract($x);
|
|
|
1397 |
}
|
|
|
1398 |
return array($this->_normalize(new Math_BigInteger()), $this->_normalize($x));
|
|
|
1399 |
}
|
|
|
1400 |
|
|
|
1401 |
// normalize $x and $y as described in HAC 14.23 / 14.24
|
|
|
1402 |
$msb = $y->value[count($y->value) - 1];
|
|
|
1403 |
for ($shift = 0; !($msb & 0x2000000); ++$shift) {
|
|
|
1404 |
$msb <<= 1;
|
|
|
1405 |
}
|
|
|
1406 |
$x->_lshift($shift);
|
|
|
1407 |
$y->_lshift($shift);
|
|
|
1408 |
$y_value = &$y->value;
|
|
|
1409 |
|
|
|
1410 |
$x_max = count($x->value) - 1;
|
|
|
1411 |
$y_max = count($y->value) - 1;
|
|
|
1412 |
|
|
|
1413 |
$quotient = new Math_BigInteger();
|
|
|
1414 |
$quotient_value = &$quotient->value;
|
|
|
1415 |
$quotient_value = $this->_array_repeat(0, $x_max - $y_max + 1);
|
|
|
1416 |
|
|
|
1417 |
static $temp, $lhs, $rhs;
|
|
|
1418 |
if (!isset($temp)) {
|
|
|
1419 |
$temp = new Math_BigInteger();
|
|
|
1420 |
$lhs = new Math_BigInteger();
|
|
|
1421 |
$rhs = new Math_BigInteger();
|
|
|
1422 |
}
|
|
|
1423 |
$temp_value = &$temp->value;
|
|
|
1424 |
$rhs_value = &$rhs->value;
|
|
|
1425 |
|
|
|
1426 |
// $temp = $y << ($x_max - $y_max-1) in base 2**26
|
|
|
1427 |
$temp_value = array_merge($this->_array_repeat(0, $x_max - $y_max), $y_value);
|
|
|
1428 |
|
|
|
1429 |
while ( $x->compare($temp) >= 0 ) {
|
|
|
1430 |
// calculate the "common residue"
|
|
|
1431 |
++$quotient_value[$x_max - $y_max];
|
|
|
1432 |
$x = $x->subtract($temp);
|
|
|
1433 |
$x_max = count($x->value) - 1;
|
|
|
1434 |
}
|
|
|
1435 |
|
|
|
1436 |
for ($i = $x_max; $i >= $y_max + 1; --$i) {
|
|
|
1437 |
$x_value = &$x->value;
|
|
|
1438 |
$x_window = array(
|
|
|
1439 |
isset($x_value[$i]) ? $x_value[$i] : 0,
|
|
|
1440 |
isset($x_value[$i - 1]) ? $x_value[$i - 1] : 0,
|
|
|
1441 |
isset($x_value[$i - 2]) ? $x_value[$i - 2] : 0
|
|
|
1442 |
);
|
|
|
1443 |
$y_window = array(
|
|
|
1444 |
$y_value[$y_max],
|
|
|
1445 |
( $y_max > 0 ) ? $y_value[$y_max - 1] : 0
|
|
|
1446 |
);
|
|
|
1447 |
|
|
|
1448 |
$q_index = $i - $y_max - 1;
|
|
|
1449 |
if ($x_window[0] == $y_window[0]) {
|
|
|
1450 |
$quotient_value[$q_index] = 0x3FFFFFF;
|
|
|
1451 |
} else {
|
|
|
1452 |
$quotient_value[$q_index] = (int) (
|
|
|
1453 |
($x_window[0] * 0x4000000 + $x_window[1])
|
|
|
1454 |
/
|
|
|
1455 |
$y_window[0]
|
|
|
1456 |
);
|
|
|
1457 |
}
|
|
|
1458 |
|
|
|
1459 |
$temp_value = array($y_window[1], $y_window[0]);
|
|
|
1460 |
|
|
|
1461 |
$lhs->value = array($quotient_value[$q_index]);
|
|
|
1462 |
$lhs = $lhs->multiply($temp);
|
|
|
1463 |
|
|
|
1464 |
$rhs_value = array($x_window[2], $x_window[1], $x_window[0]);
|
|
|
1465 |
|
|
|
1466 |
while ( $lhs->compare($rhs) > 0 ) {
|
|
|
1467 |
--$quotient_value[$q_index];
|
|
|
1468 |
|
|
|
1469 |
$lhs->value = array($quotient_value[$q_index]);
|
|
|
1470 |
$lhs = $lhs->multiply($temp);
|
|
|
1471 |
}
|
|
|
1472 |
|
|
|
1473 |
$adjust = $this->_array_repeat(0, $q_index);
|
|
|
1474 |
$temp_value = array($quotient_value[$q_index]);
|
|
|
1475 |
$temp = $temp->multiply($y);
|
|
|
1476 |
$temp_value = &$temp->value;
|
|
|
1477 |
$temp_value = array_merge($adjust, $temp_value);
|
|
|
1478 |
|
|
|
1479 |
$x = $x->subtract($temp);
|
|
|
1480 |
|
|
|
1481 |
if ($x->compare($zero) < 0) {
|
|
|
1482 |
$temp_value = array_merge($adjust, $y_value);
|
|
|
1483 |
$x = $x->add($temp);
|
|
|
1484 |
|
|
|
1485 |
--$quotient_value[$q_index];
|
|
|
1486 |
}
|
|
|
1487 |
|
|
|
1488 |
$x_max = count($x_value) - 1;
|
|
|
1489 |
}
|
|
|
1490 |
|
|
|
1491 |
// unnormalize the remainder
|
|
|
1492 |
$x->_rshift($shift);
|
|
|
1493 |
|
|
|
1494 |
$quotient->is_negative = $x_sign != $y_sign;
|
|
|
1495 |
|
|
|
1496 |
// calculate the "common residue", if appropriate
|
|
|
1497 |
if ( $x_sign ) {
|
|
|
1498 |
$y->_rshift($shift);
|
|
|
1499 |
$x = $y->subtract($x);
|
|
|
1500 |
}
|
|
|
1501 |
|
|
|
1502 |
return array($this->_normalize($quotient), $this->_normalize($x));
|
|
|
1503 |
}
|
|
|
1504 |
|
|
|
1505 |
/**
|
|
|
1506 |
* Divides a BigInteger by a regular integer
|
|
|
1507 |
*
|
|
|
1508 |
* abc / x = a00 / x + b0 / x + c / x
|
|
|
1509 |
*
|
|
|
1510 |
* @param Array $dividend
|
|
|
1511 |
* @param Array $divisor
|
|
|
1512 |
* @return Array
|
|
|
1513 |
* @access private
|
|
|
1514 |
*/
|
|
|
1515 |
function _divide_digit($dividend, $divisor)
|
|
|
1516 |
{
|
|
|
1517 |
$carry = 0;
|
|
|
1518 |
$result = array();
|
|
|
1519 |
|
|
|
1520 |
for ($i = count($dividend) - 1; $i >= 0; --$i) {
|
|
|
1521 |
$temp = 0x4000000 * $carry + $dividend[$i];
|
|
|
1522 |
$result[$i] = (int) ($temp / $divisor);
|
|
|
1523 |
$carry = (int) ($temp - $divisor * $result[$i]);
|
|
|
1524 |
}
|
|
|
1525 |
|
|
|
1526 |
return array($result, $carry);
|
|
|
1527 |
}
|
|
|
1528 |
|
|
|
1529 |
/**
|
|
|
1530 |
* Performs modular exponentiation.
|
|
|
1531 |
*
|
|
|
1532 |
* Here's an example:
|
|
|
1533 |
* <code>
|
|
|
1534 |
* <?php
|
|
|
1535 |
* include('Math/BigInteger.php');
|
|
|
1536 |
*
|
|
|
1537 |
* $a = new Math_BigInteger('10');
|
|
|
1538 |
* $b = new Math_BigInteger('20');
|
|
|
1539 |
* $c = new Math_BigInteger('30');
|
|
|
1540 |
*
|
|
|
1541 |
* $c = $a->modPow($b, $c);
|
|
|
1542 |
*
|
|
|
1543 |
* echo $c->toString(); // outputs 10
|
|
|
1544 |
* ?>
|
|
|
1545 |
* </code>
|
|
|
1546 |
*
|
|
|
1547 |
* @param Math_BigInteger $e
|
|
|
1548 |
* @param Math_BigInteger $n
|
|
|
1549 |
* @return Math_BigInteger
|
|
|
1550 |
* @access public
|
|
|
1551 |
* @internal The most naive approach to modular exponentiation has very unreasonable requirements, and
|
|
|
1552 |
* and although the approach involving repeated squaring does vastly better, it, too, is impractical
|
|
|
1553 |
* for our purposes. The reason being that division - by far the most complicated and time-consuming
|
|
|
1554 |
* of the basic operations (eg. +,-,*,/) - occurs multiple times within it.
|
|
|
1555 |
*
|
|
|
1556 |
* Modular reductions resolve this issue. Although an individual modular reduction takes more time
|
|
|
1557 |
* then an individual division, when performed in succession (with the same modulo), they're a lot faster.
|
|
|
1558 |
*
|
|
|
1559 |
* The two most commonly used modular reductions are Barrett and Montgomery reduction. Montgomery reduction,
|
|
|
1560 |
* although faster, only works when the gcd of the modulo and of the base being used is 1. In RSA, when the
|
|
|
1561 |
* base is a power of two, the modulo - a product of two primes - is always going to have a gcd of 1 (because
|
|
|
1562 |
* the product of two odd numbers is odd), but what about when RSA isn't used?
|
|
|
1563 |
*
|
|
|
1564 |
* In contrast, Barrett reduction has no such constraint. As such, some bigint implementations perform a
|
|
|
1565 |
* Barrett reduction after every operation in the modpow function. Others perform Barrett reductions when the
|
|
|
1566 |
* modulo is even and Montgomery reductions when the modulo is odd. BigInteger.java's modPow method, however,
|
|
|
1567 |
* uses a trick involving the Chinese Remainder Theorem to factor the even modulo into two numbers - one odd and
|
|
|
1568 |
* the other, a power of two - and recombine them, later. This is the method that this modPow function uses.
|
|
|
1569 |
* {@link http://islab.oregonstate.edu/papers/j34monex.pdf Montgomery Reduction with Even Modulus} elaborates.
|
|
|
1570 |
*/
|
|
|
1571 |
function modPow($e, $n)
|
|
|
1572 |
{
|
|
|
1573 |
$n = $this->bitmask !== false && $this->bitmask->compare($n) < 0 ? $this->bitmask : $n->abs();
|
|
|
1574 |
|
|
|
1575 |
if ($e->compare(new Math_BigInteger()) < 0) {
|
|
|
1576 |
$e = $e->abs();
|
|
|
1577 |
|
|
|
1578 |
$temp = $this->modInverse($n);
|
|
|
1579 |
if ($temp === false) {
|
|
|
1580 |
return false;
|
|
|
1581 |
}
|
|
|
1582 |
|
|
|
1583 |
return $this->_normalize($temp->modPow($e, $n));
|
|
|
1584 |
}
|
|
|
1585 |
|
|
|
1586 |
switch ( MATH_BIGINTEGER_MODE ) {
|
|
|
1587 |
case MATH_BIGINTEGER_MODE_GMP:
|
|
|
1588 |
$temp = new Math_BigInteger();
|
|
|
1589 |
$temp->value = gmp_powm($this->value, $e->value, $n->value);
|
|
|
1590 |
|
|
|
1591 |
return $this->_normalize($temp);
|
|
|
1592 |
case MATH_BIGINTEGER_MODE_BCMATH:
|
|
|
1593 |
$temp = new Math_BigInteger();
|
|
|
1594 |
$temp->value = bcpowmod($this->value, $e->value, $n->value, 0);
|
|
|
1595 |
|
|
|
1596 |
return $this->_normalize($temp);
|
|
|
1597 |
}
|
|
|
1598 |
|
|
|
1599 |
if ( empty($e->value) ) {
|
|
|
1600 |
$temp = new Math_BigInteger();
|
|
|
1601 |
$temp->value = array(1);
|
|
|
1602 |
return $this->_normalize($temp);
|
|
|
1603 |
}
|
|
|
1604 |
|
|
|
1605 |
if ( $e->value == array(1) ) {
|
|
|
1606 |
list(, $temp) = $this->divide($n);
|
|
|
1607 |
return $this->_normalize($temp);
|
|
|
1608 |
}
|
|
|
1609 |
|
|
|
1610 |
if ( $e->value == array(2) ) {
|
|
|
1611 |
$temp = new Math_BigInteger();
|
|
|
1612 |
$temp->value = $this->_square($this->value);
|
|
|
1613 |
list(, $temp) = $temp->divide($n);
|
|
|
1614 |
return $this->_normalize($temp);
|
|
|
1615 |
}
|
|
|
1616 |
|
|
|
1617 |
return $this->_normalize($this->_slidingWindow($e, $n, MATH_BIGINTEGER_BARRETT));
|
|
|
1618 |
|
|
|
1619 |
// is the modulo odd?
|
|
|
1620 |
if ( $n->value[0] & 1 ) {
|
|
|
1621 |
return $this->_normalize($this->_slidingWindow($e, $n, MATH_BIGINTEGER_MONTGOMERY));
|
|
|
1622 |
}
|
|
|
1623 |
// if it's not, it's even
|
|
|
1624 |
|
|
|
1625 |
// find the lowest set bit (eg. the max pow of 2 that divides $n)
|
|
|
1626 |
for ($i = 0; $i < count($n->value); ++$i) {
|
|
|
1627 |
if ( $n->value[$i] ) {
|
|
|
1628 |
$temp = decbin($n->value[$i]);
|
|
|
1629 |
$j = strlen($temp) - strrpos($temp, '1') - 1;
|
|
|
1630 |
$j+= 26 * $i;
|
|
|
1631 |
break;
|
|
|
1632 |
}
|
|
|
1633 |
}
|
|
|
1634 |
// at this point, 2^$j * $n/(2^$j) == $n
|
|
|
1635 |
|
|
|
1636 |
$mod1 = $n->copy();
|
|
|
1637 |
$mod1->_rshift($j);
|
|
|
1638 |
$mod2 = new Math_BigInteger();
|
|
|
1639 |
$mod2->value = array(1);
|
|
|
1640 |
$mod2->_lshift($j);
|
|
|
1641 |
|
|
|
1642 |
$part1 = ( $mod1->value != array(1) ) ? $this->_slidingWindow($e, $mod1, MATH_BIGINTEGER_MONTGOMERY) : new Math_BigInteger();
|
|
|
1643 |
$part2 = $this->_slidingWindow($e, $mod2, MATH_BIGINTEGER_POWEROF2);
|
|
|
1644 |
|
|
|
1645 |
$y1 = $mod2->modInverse($mod1);
|
|
|
1646 |
$y2 = $mod1->modInverse($mod2);
|
|
|
1647 |
|
|
|
1648 |
$result = $part1->multiply($mod2);
|
|
|
1649 |
$result = $result->multiply($y1);
|
|
|
1650 |
|
|
|
1651 |
$temp = $part2->multiply($mod1);
|
|
|
1652 |
$temp = $temp->multiply($y2);
|
|
|
1653 |
|
|
|
1654 |
$result = $result->add($temp);
|
|
|
1655 |
list(, $result) = $result->divide($n);
|
|
|
1656 |
|
|
|
1657 |
return $this->_normalize($result);
|
|
|
1658 |
}
|
|
|
1659 |
|
|
|
1660 |
/**
|
|
|
1661 |
* Performs modular exponentiation.
|
|
|
1662 |
*
|
|
|
1663 |
* Alias for Math_BigInteger::modPow()
|
|
|
1664 |
*
|
|
|
1665 |
* @param Math_BigInteger $e
|
|
|
1666 |
* @param Math_BigInteger $n
|
|
|
1667 |
* @return Math_BigInteger
|
|
|
1668 |
* @access public
|
|
|
1669 |
*/
|
|
|
1670 |
function powMod($e, $n)
|
|
|
1671 |
{
|
|
|
1672 |
return $this->modPow($e, $n);
|
|
|
1673 |
}
|
|
|
1674 |
|
|
|
1675 |
/**
|
|
|
1676 |
* Sliding Window k-ary Modular Exponentiation
|
|
|
1677 |
*
|
|
|
1678 |
* Based on {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=27 HAC 14.85} /
|
|
|
1679 |
* {@link http://math.libtomcrypt.com/files/tommath.pdf#page=210 MPM 7.7}. In a departure from those algorithims,
|
|
|
1680 |
* however, this function performs a modular reduction after every multiplication and squaring operation.
|
|
|
1681 |
* As such, this function has the same preconditions that the reductions being used do.
|
|
|
1682 |
*
|
|
|
1683 |
* @param Math_BigInteger $e
|
|
|
1684 |
* @param Math_BigInteger $n
|
|
|
1685 |
* @param Integer $mode
|
|
|
1686 |
* @return Math_BigInteger
|
|
|
1687 |
* @access private
|
|
|
1688 |
*/
|
|
|
1689 |
function _slidingWindow($e, $n, $mode)
|
|
|
1690 |
{
|
|
|
1691 |
static $window_ranges = array(7, 25, 81, 241, 673, 1793); // from BigInteger.java's oddModPow function
|
|
|
1692 |
//static $window_ranges = array(0, 7, 36, 140, 450, 1303, 3529); // from MPM 7.3.1
|
|
|
1693 |
|
|
|
1694 |
$e_value = $e->value;
|
|
|
1695 |
$e_length = count($e_value) - 1;
|
|
|
1696 |
$e_bits = decbin($e_value[$e_length]);
|
|
|
1697 |
for ($i = $e_length - 1; $i >= 0; --$i) {
|
|
|
1698 |
$e_bits.= str_pad(decbin($e_value[$i]), 26, '0', STR_PAD_LEFT);
|
|
|
1699 |
}
|
|
|
1700 |
|
|
|
1701 |
$e_length = strlen($e_bits);
|
|
|
1702 |
|
|
|
1703 |
// calculate the appropriate window size.
|
|
|
1704 |
// $window_size == 3 if $window_ranges is between 25 and 81, for example.
|
|
|
1705 |
for ($i = 0, $window_size = 1; $e_length > $window_ranges[$i] && $i < count($window_ranges); ++$window_size, ++$i);
|
|
|
1706 |
|
|
|
1707 |
$n_value = $n->value;
|
|
|
1708 |
|
|
|
1709 |
// precompute $this^0 through $this^$window_size
|
|
|
1710 |
$powers = array();
|
|
|
1711 |
$powers[1] = $this->_prepareReduce($this->value, $n_value, $mode);
|
|
|
1712 |
$powers[2] = $this->_squareReduce($powers[1], $n_value, $mode);
|
|
|
1713 |
|
|
|
1714 |
// we do every other number since substr($e_bits, $i, $j+1) (see below) is supposed to end
|
|
|
1715 |
// in a 1. ie. it's supposed to be odd.
|
|
|
1716 |
$temp = 1 << ($window_size - 1);
|
|
|
1717 |
for ($i = 1; $i < $temp; ++$i) {
|
|
|
1718 |
$i2 = $i << 1;
|
|
|
1719 |
$powers[$i2 + 1] = $this->_multiplyReduce($powers[$i2 - 1], $powers[2], $n_value, $mode);
|
|
|
1720 |
}
|
|
|
1721 |
|
|
|
1722 |
$result = array(1);
|
|
|
1723 |
$result = $this->_prepareReduce($result, $n_value, $mode);
|
|
|
1724 |
|
|
|
1725 |
for ($i = 0; $i < $e_length; ) {
|
|
|
1726 |
if ( !$e_bits[$i] ) {
|
|
|
1727 |
$result = $this->_squareReduce($result, $n_value, $mode);
|
|
|
1728 |
++$i;
|
|
|
1729 |
} else {
|
|
|
1730 |
for ($j = $window_size - 1; $j > 0; --$j) {
|
|
|
1731 |
if ( !empty($e_bits[$i + $j]) ) {
|
|
|
1732 |
break;
|
|
|
1733 |
}
|
|
|
1734 |
}
|
|
|
1735 |
|
|
|
1736 |
for ($k = 0; $k <= $j; ++$k) {// eg. the length of substr($e_bits, $i, $j+1)
|
|
|
1737 |
$result = $this->_squareReduce($result, $n_value, $mode);
|
|
|
1738 |
}
|
|
|
1739 |
|
|
|
1740 |
$result = $this->_multiplyReduce($result, $powers[bindec(substr($e_bits, $i, $j + 1))], $n_value, $mode);
|
|
|
1741 |
|
|
|
1742 |
$i+=$j + 1;
|
|
|
1743 |
}
|
|
|
1744 |
}
|
|
|
1745 |
|
|
|
1746 |
$temp = new Math_BigInteger();
|
|
|
1747 |
$temp->value = $this->_reduce($result, $n_value, $mode);
|
|
|
1748 |
|
|
|
1749 |
return $temp;
|
|
|
1750 |
}
|
|
|
1751 |
|
|
|
1752 |
/**
|
|
|
1753 |
* Modular reduction
|
|
|
1754 |
*
|
|
|
1755 |
* For most $modes this will return the remainder.
|
|
|
1756 |
*
|
|
|
1757 |
* @see _slidingWindow()
|
|
|
1758 |
* @access private
|
|
|
1759 |
* @param Array $x
|
|
|
1760 |
* @param Array $n
|
|
|
1761 |
* @param Integer $mode
|
|
|
1762 |
* @return Array
|
|
|
1763 |
*/
|
|
|
1764 |
function _reduce($x, $n, $mode)
|
|
|
1765 |
{
|
|
|
1766 |
switch ($mode) {
|
|
|
1767 |
case MATH_BIGINTEGER_MONTGOMERY:
|
|
|
1768 |
return $this->_montgomery($x, $n);
|
|
|
1769 |
case MATH_BIGINTEGER_BARRETT:
|
|
|
1770 |
return $this->_barrett($x, $n);
|
|
|
1771 |
case MATH_BIGINTEGER_POWEROF2:
|
|
|
1772 |
$lhs = new Math_BigInteger();
|
|
|
1773 |
$lhs->value = $x;
|
|
|
1774 |
$rhs = new Math_BigInteger();
|
|
|
1775 |
$rhs->value = $n;
|
|
|
1776 |
return $x->_mod2($n);
|
|
|
1777 |
case MATH_BIGINTEGER_CLASSIC:
|
|
|
1778 |
$lhs = new Math_BigInteger();
|
|
|
1779 |
$lhs->value = $x;
|
|
|
1780 |
$rhs = new Math_BigInteger();
|
|
|
1781 |
$rhs->value = $n;
|
|
|
1782 |
list(, $temp) = $lhs->divide($rhs);
|
|
|
1783 |
return $temp->value;
|
|
|
1784 |
case MATH_BIGINTEGER_NONE:
|
|
|
1785 |
return $x;
|
|
|
1786 |
default:
|
|
|
1787 |
// an invalid $mode was provided
|
|
|
1788 |
}
|
|
|
1789 |
}
|
|
|
1790 |
|
|
|
1791 |
/**
|
|
|
1792 |
* Modular reduction preperation
|
|
|
1793 |
*
|
|
|
1794 |
* @see _slidingWindow()
|
|
|
1795 |
* @access private
|
|
|
1796 |
* @param Array $x
|
|
|
1797 |
* @param Array $n
|
|
|
1798 |
* @param Integer $mode
|
|
|
1799 |
* @return Array
|
|
|
1800 |
*/
|
|
|
1801 |
function _prepareReduce($x, $n, $mode)
|
|
|
1802 |
{
|
|
|
1803 |
if ($mode == MATH_BIGINTEGER_MONTGOMERY) {
|
|
|
1804 |
return $this->_prepMontgomery($x, $n);
|
|
|
1805 |
}
|
|
|
1806 |
return $this->_reduce($x, $n, $mode);
|
|
|
1807 |
}
|
|
|
1808 |
|
|
|
1809 |
/**
|
|
|
1810 |
* Modular multiply
|
|
|
1811 |
*
|
|
|
1812 |
* @see _slidingWindow()
|
|
|
1813 |
* @access private
|
|
|
1814 |
* @param Array $x
|
|
|
1815 |
* @param Array $y
|
|
|
1816 |
* @param Array $n
|
|
|
1817 |
* @param Integer $mode
|
|
|
1818 |
* @return Array
|
|
|
1819 |
*/
|
|
|
1820 |
function _multiplyReduce($x, $y, $n, $mode)
|
|
|
1821 |
{
|
|
|
1822 |
if ($mode == MATH_BIGINTEGER_MONTGOMERY) {
|
|
|
1823 |
return $this->_montgomeryMultiply($x, $y, $n);
|
|
|
1824 |
}
|
|
|
1825 |
$temp = $this->_multiply($x, false, $y, false);
|
|
|
1826 |
return $this->_reduce($temp[MATH_BIGINTEGER_VALUE], $n, $mode);
|
|
|
1827 |
}
|
|
|
1828 |
|
|
|
1829 |
/**
|
|
|
1830 |
* Modular square
|
|
|
1831 |
*
|
|
|
1832 |
* @see _slidingWindow()
|
|
|
1833 |
* @access private
|
|
|
1834 |
* @param Array $x
|
|
|
1835 |
* @param Array $n
|
|
|
1836 |
* @param Integer $mode
|
|
|
1837 |
* @return Array
|
|
|
1838 |
*/
|
|
|
1839 |
function _squareReduce($x, $n, $mode)
|
|
|
1840 |
{
|
|
|
1841 |
if ($mode == MATH_BIGINTEGER_MONTGOMERY) {
|
|
|
1842 |
return $this->_montgomeryMultiply($x, $x, $n);
|
|
|
1843 |
}
|
|
|
1844 |
return $this->_reduce($this->_square($x), $n, $mode);
|
|
|
1845 |
}
|
|
|
1846 |
|
|
|
1847 |
/**
|
|
|
1848 |
* Modulos for Powers of Two
|
|
|
1849 |
*
|
|
|
1850 |
* Calculates $x%$n, where $n = 2**$e, for some $e. Since this is basically the same as doing $x & ($n-1),
|
|
|
1851 |
* we'll just use this function as a wrapper for doing that.
|
|
|
1852 |
*
|
|
|
1853 |
* @see _slidingWindow()
|
|
|
1854 |
* @access private
|
|
|
1855 |
* @param Math_BigInteger
|
|
|
1856 |
* @return Math_BigInteger
|
|
|
1857 |
*/
|
|
|
1858 |
function _mod2($n)
|
|
|
1859 |
{
|
|
|
1860 |
$temp = new Math_BigInteger();
|
|
|
1861 |
$temp->value = array(1);
|
|
|
1862 |
return $this->bitwise_and($n->subtract($temp));
|
|
|
1863 |
}
|
|
|
1864 |
|
|
|
1865 |
/**
|
|
|
1866 |
* Barrett Modular Reduction
|
|
|
1867 |
*
|
|
|
1868 |
* See {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=14 HAC 14.3.3} /
|
|
|
1869 |
* {@link http://math.libtomcrypt.com/files/tommath.pdf#page=165 MPM 6.2.5} for more information. Modified slightly,
|
|
|
1870 |
* so as not to require negative numbers (initially, this script didn't support negative numbers).
|
|
|
1871 |
*
|
|
|
1872 |
* Employs "folding", as described at
|
|
|
1873 |
* {@link http://www.cosic.esat.kuleuven.be/publications/thesis-149.pdf#page=66 thesis-149.pdf#page=66}. To quote from
|
|
|
1874 |
* it, "the idea [behind folding] is to find a value x' such that x (mod m) = x' (mod m), with x' being smaller than x."
|
|
|
1875 |
*
|
|
|
1876 |
* Unfortunately, the "Barrett Reduction with Folding" algorithm described in thesis-149.pdf is not, as written, all that
|
|
|
1877 |
* usable on account of (1) its not using reasonable radix points as discussed in
|
|
|
1878 |
* {@link http://math.libtomcrypt.com/files/tommath.pdf#page=162 MPM 6.2.2} and (2) the fact that, even with reasonable
|
|
|
1879 |
* radix points, it only works when there are an even number of digits in the denominator. The reason for (2) is that
|
|
|
1880 |
* (x >> 1) + (x >> 1) != x / 2 + x / 2. If x is even, they're the same, but if x is odd, they're not. See the in-line
|
|
|
1881 |
* comments for details.
|
|
|
1882 |
*
|
|
|
1883 |
* @see _slidingWindow()
|
|
|
1884 |
* @access private
|
|
|
1885 |
* @param Array $n
|
|
|
1886 |
* @param Array $m
|
|
|
1887 |
* @return Array
|
|
|
1888 |
*/
|
|
|
1889 |
function _barrett($n, $m)
|
|
|
1890 |
{
|
|
|
1891 |
static $cache = array(
|
|
|
1892 |
MATH_BIGINTEGER_VARIABLE => array(),
|
|
|
1893 |
MATH_BIGINTEGER_DATA => array()
|
|
|
1894 |
);
|
|
|
1895 |
|
|
|
1896 |
$m_length = count($m);
|
|
|
1897 |
|
|
|
1898 |
// if ($this->_compare($n, $this->_square($m)) >= 0) {
|
|
|
1899 |
if (count($n) > 2 * $m_length) {
|
|
|
1900 |
$lhs = new Math_BigInteger();
|
|
|
1901 |
$rhs = new Math_BigInteger();
|
|
|
1902 |
$lhs->value = $n;
|
|
|
1903 |
$rhs->value = $m;
|
|
|
1904 |
list(, $temp) = $lhs->divide($rhs);
|
|
|
1905 |
return $temp->value;
|
|
|
1906 |
}
|
|
|
1907 |
|
|
|
1908 |
// if (m.length >> 1) + 2 <= m.length then m is too small and n can't be reduced
|
|
|
1909 |
if ($m_length < 5) {
|
|
|
1910 |
return $this->_regularBarrett($n, $m);
|
|
|
1911 |
}
|
|
|
1912 |
|
|
|
1913 |
// n = 2 * m.length
|
|
|
1914 |
|
|
|
1915 |
if ( ($key = array_search($m, $cache[MATH_BIGINTEGER_VARIABLE])) === false ) {
|
|
|
1916 |
$key = count($cache[MATH_BIGINTEGER_VARIABLE]);
|
|
|
1917 |
$cache[MATH_BIGINTEGER_VARIABLE][] = $m;
|
|
|
1918 |
|
|
|
1919 |
$lhs = new Math_BigInteger();
|
|
|
1920 |
$lhs_value = &$lhs->value;
|
|
|
1921 |
$lhs_value = $this->_array_repeat(0, $m_length + ($m_length >> 1));
|
|
|
1922 |
$lhs_value[] = 1;
|
|
|
1923 |
$rhs = new Math_BigInteger();
|
|
|
1924 |
$rhs->value = $m;
|
|
|
1925 |
|
|
|
1926 |
list($u, $m1) = $lhs->divide($rhs);
|
|
|
1927 |
$u = $u->value;
|
|
|
1928 |
$m1 = $m1->value;
|
|
|
1929 |
|
|
|
1930 |
$cache[MATH_BIGINTEGER_DATA][] = array(
|
|
|
1931 |
'u' => $u, // m.length >> 1 (technically (m.length >> 1) + 1)
|
|
|
1932 |
'm1'=> $m1 // m.length
|
|
|
1933 |
);
|
|
|
1934 |
} else {
|
|
|
1935 |
extract($cache[MATH_BIGINTEGER_DATA][$key]);
|
|
|
1936 |
}
|
|
|
1937 |
|
|
|
1938 |
$cutoff = $m_length + ($m_length >> 1);
|
|
|
1939 |
$lsd = array_slice($n, 0, $cutoff); // m.length + (m.length >> 1)
|
|
|
1940 |
$msd = array_slice($n, $cutoff); // m.length >> 1
|
|
|
1941 |
$lsd = $this->_trim($lsd);
|
|
|
1942 |
$temp = $this->_multiply($msd, false, $m1, false);
|
|
|
1943 |
$n = $this->_add($lsd, false, $temp[MATH_BIGINTEGER_VALUE], false); // m.length + (m.length >> 1) + 1
|
|
|
1944 |
|
|
|
1945 |
if ($m_length & 1) {
|
|
|
1946 |
return $this->_regularBarrett($n[MATH_BIGINTEGER_VALUE], $m);
|
|
|
1947 |
}
|
|
|
1948 |
|
|
|
1949 |
// (m.length + (m.length >> 1) + 1) - (m.length - 1) == (m.length >> 1) + 2
|
|
|
1950 |
$temp = array_slice($n[MATH_BIGINTEGER_VALUE], $m_length - 1);
|
|
|
1951 |
// if even: ((m.length >> 1) + 2) + (m.length >> 1) == m.length + 2
|
|
|
1952 |
// if odd: ((m.length >> 1) + 2) + (m.length >> 1) == (m.length - 1) + 2 == m.length + 1
|
|
|
1953 |
$temp = $this->_multiply($temp, false, $u, false);
|
|
|
1954 |
// if even: (m.length + 2) - ((m.length >> 1) + 1) = m.length - (m.length >> 1) + 1
|
|
|
1955 |
// if odd: (m.length + 1) - ((m.length >> 1) + 1) = m.length - (m.length >> 1)
|
|
|
1956 |
$temp = array_slice($temp[MATH_BIGINTEGER_VALUE], ($m_length >> 1) + 1);
|
|
|
1957 |
// if even: (m.length - (m.length >> 1) + 1) + m.length = 2 * m.length - (m.length >> 1) + 1
|
|
|
1958 |
// if odd: (m.length - (m.length >> 1)) + m.length = 2 * m.length - (m.length >> 1)
|
|
|
1959 |
$temp = $this->_multiply($temp, false, $m, false);
|
|
|
1960 |
|
|
|
1961 |
// at this point, if m had an odd number of digits, we'd be subtracting a 2 * m.length - (m.length >> 1) digit
|
|
|
1962 |
// number from a m.length + (m.length >> 1) + 1 digit number. ie. there'd be an extra digit and the while loop
|
|
|
1963 |
// following this comment would loop a lot (hence our calling _regularBarrett() in that situation).
|
|
|
1964 |
|
|
|
1965 |
$result = $this->_subtract($n[MATH_BIGINTEGER_VALUE], false, $temp[MATH_BIGINTEGER_VALUE], false);
|
|
|
1966 |
|
|
|
1967 |
while ($this->_compare($result[MATH_BIGINTEGER_VALUE], $result[MATH_BIGINTEGER_SIGN], $m, false) >= 0) {
|
|
|
1968 |
$result = $this->_subtract($result[MATH_BIGINTEGER_VALUE], $result[MATH_BIGINTEGER_SIGN], $m, false);
|
|
|
1969 |
}
|
|
|
1970 |
|
|
|
1971 |
return $result[MATH_BIGINTEGER_VALUE];
|
|
|
1972 |
}
|
|
|
1973 |
|
|
|
1974 |
/**
|
|
|
1975 |
* (Regular) Barrett Modular Reduction
|
|
|
1976 |
*
|
|
|
1977 |
* For numbers with more than four digits Math_BigInteger::_barrett() is faster. The difference between that and this
|
|
|
1978 |
* is that this function does not fold the denominator into a smaller form.
|
|
|
1979 |
*
|
|
|
1980 |
* @see _slidingWindow()
|
|
|
1981 |
* @access private
|
|
|
1982 |
* @param Array $x
|
|
|
1983 |
* @param Array $n
|
|
|
1984 |
* @return Array
|
|
|
1985 |
*/
|
|
|
1986 |
function _regularBarrett($x, $n)
|
|
|
1987 |
{
|
|
|
1988 |
static $cache = array(
|
|
|
1989 |
MATH_BIGINTEGER_VARIABLE => array(),
|
|
|
1990 |
MATH_BIGINTEGER_DATA => array()
|
|
|
1991 |
);
|
|
|
1992 |
|
|
|
1993 |
$n_length = count($n);
|
|
|
1994 |
|
|
|
1995 |
if (count($x) > 2 * $n_length) {
|
|
|
1996 |
$lhs = new Math_BigInteger();
|
|
|
1997 |
$rhs = new Math_BigInteger();
|
|
|
1998 |
$lhs->value = $x;
|
|
|
1999 |
$rhs->value = $n;
|
|
|
2000 |
list(, $temp) = $lhs->divide($rhs);
|
|
|
2001 |
return $temp->value;
|
|
|
2002 |
}
|
|
|
2003 |
|
|
|
2004 |
if ( ($key = array_search($n, $cache[MATH_BIGINTEGER_VARIABLE])) === false ) {
|
|
|
2005 |
$key = count($cache[MATH_BIGINTEGER_VARIABLE]);
|
|
|
2006 |
$cache[MATH_BIGINTEGER_VARIABLE][] = $n;
|
|
|
2007 |
$lhs = new Math_BigInteger();
|
|
|
2008 |
$lhs_value = &$lhs->value;
|
|
|
2009 |
$lhs_value = $this->_array_repeat(0, 2 * $n_length);
|
|
|
2010 |
$lhs_value[] = 1;
|
|
|
2011 |
$rhs = new Math_BigInteger();
|
|
|
2012 |
$rhs->value = $n;
|
|
|
2013 |
list($temp, ) = $lhs->divide($rhs); // m.length
|
|
|
2014 |
$cache[MATH_BIGINTEGER_DATA][] = $temp->value;
|
|
|
2015 |
}
|
|
|
2016 |
|
|
|
2017 |
// 2 * m.length - (m.length - 1) = m.length + 1
|
|
|
2018 |
$temp = array_slice($x, $n_length - 1);
|
|
|
2019 |
// (m.length + 1) + m.length = 2 * m.length + 1
|
|
|
2020 |
$temp = $this->_multiply($temp, false, $cache[MATH_BIGINTEGER_DATA][$key], false);
|
|
|
2021 |
// (2 * m.length + 1) - (m.length - 1) = m.length + 2
|
|
|
2022 |
$temp = array_slice($temp[MATH_BIGINTEGER_VALUE], $n_length + 1);
|
|
|
2023 |
|
|
|
2024 |
// m.length + 1
|
|
|
2025 |
$result = array_slice($x, 0, $n_length + 1);
|
|
|
2026 |
// m.length + 1
|
|
|
2027 |
$temp = $this->_multiplyLower($temp, false, $n, false, $n_length + 1);
|
|
|
2028 |
// $temp == array_slice($temp->_multiply($temp, false, $n, false)->value, 0, $n_length + 1)
|
|
|
2029 |
|
|
|
2030 |
if ($this->_compare($result, false, $temp[MATH_BIGINTEGER_VALUE], $temp[MATH_BIGINTEGER_SIGN]) < 0) {
|
|
|
2031 |
$corrector_value = $this->_array_repeat(0, $n_length + 1);
|
|
|
2032 |
$corrector_value[] = 1;
|
|
|
2033 |
$result = $this->_add($result, false, $corrector, false);
|
|
|
2034 |
$result = $result[MATH_BIGINTEGER_VALUE];
|
|
|
2035 |
}
|
|
|
2036 |
|
|
|
2037 |
// at this point, we're subtracting a number with m.length + 1 digits from another number with m.length + 1 digits
|
|
|
2038 |
$result = $this->_subtract($result, false, $temp[MATH_BIGINTEGER_VALUE], $temp[MATH_BIGINTEGER_SIGN]);
|
|
|
2039 |
while ($this->_compare($result[MATH_BIGINTEGER_VALUE], $result[MATH_BIGINTEGER_SIGN], $n, false) > 0) {
|
|
|
2040 |
$result = $this->_subtract($result[MATH_BIGINTEGER_VALUE], $result[MATH_BIGINTEGER_SIGN], $n, false);
|
|
|
2041 |
}
|
|
|
2042 |
|
|
|
2043 |
return $result[MATH_BIGINTEGER_VALUE];
|
|
|
2044 |
}
|
|
|
2045 |
|
|
|
2046 |
/**
|
|
|
2047 |
* Performs long multiplication up to $stop digits
|
|
|
2048 |
*
|
|
|
2049 |
* If you're going to be doing array_slice($product->value, 0, $stop), some cycles can be saved.
|
|
|
2050 |
*
|
|
|
2051 |
* @see _regularBarrett()
|
|
|
2052 |
* @param Array $x_value
|
|
|
2053 |
* @param Boolean $x_negative
|
|
|
2054 |
* @param Array $y_value
|
|
|
2055 |
* @param Boolean $y_negative
|
|
|
2056 |
* @return Array
|
|
|
2057 |
* @access private
|
|
|
2058 |
*/
|
|
|
2059 |
function _multiplyLower($x_value, $x_negative, $y_value, $y_negative, $stop)
|
|
|
2060 |
{
|
|
|
2061 |
$x_length = count($x_value);
|
|
|
2062 |
$y_length = count($y_value);
|
|
|
2063 |
|
|
|
2064 |
if ( !$x_length || !$y_length ) { // a 0 is being multiplied
|
|
|
2065 |
return array(
|
|
|
2066 |
MATH_BIGINTEGER_VALUE => array(),
|
|
|
2067 |
MATH_BIGINTEGER_SIGN => false
|
|
|
2068 |
);
|
|
|
2069 |
}
|
|
|
2070 |
|
|
|
2071 |
if ( $x_length < $y_length ) {
|
|
|
2072 |
$temp = $x_value;
|
|
|
2073 |
$x_value = $y_value;
|
|
|
2074 |
$y_value = $temp;
|
|
|
2075 |
|
|
|
2076 |
$x_length = count($x_value);
|
|
|
2077 |
$y_length = count($y_value);
|
|
|
2078 |
}
|
|
|
2079 |
|
|
|
2080 |
$product_value = $this->_array_repeat(0, $x_length + $y_length);
|
|
|
2081 |
|
|
|
2082 |
// the following for loop could be removed if the for loop following it
|
|
|
2083 |
// (the one with nested for loops) initially set $i to 0, but
|
|
|
2084 |
// doing so would also make the result in one set of unnecessary adds,
|
|
|
2085 |
// since on the outermost loops first pass, $product->value[$k] is going
|
|
|
2086 |
// to always be 0
|
|
|
2087 |
|
|
|
2088 |
$carry = 0;
|
|
|
2089 |
|
|
|
2090 |
for ($j = 0; $j < $x_length; ++$j) { // ie. $i = 0, $k = $i
|
|
|
2091 |
$temp = $x_value[$j] * $y_value[0] + $carry; // $product_value[$k] == 0
|
|
|
2092 |
$carry = (int) ($temp / 0x4000000);
|
|
|
2093 |
$product_value[$j] = (int) ($temp - 0x4000000 * $carry);
|
|
|
2094 |
}
|
|
|
2095 |
|
|
|
2096 |
if ($j < $stop) {
|
|
|
2097 |
$product_value[$j] = $carry;
|
|
|
2098 |
}
|
|
|
2099 |
|
|
|
2100 |
// the above for loop is what the previous comment was talking about. the
|
|
|
2101 |
// following for loop is the "one with nested for loops"
|
|
|
2102 |
|
|
|
2103 |
for ($i = 1; $i < $y_length; ++$i) {
|
|
|
2104 |
$carry = 0;
|
|
|
2105 |
|
|
|
2106 |
for ($j = 0, $k = $i; $j < $x_length && $k < $stop; ++$j, ++$k) {
|
|
|
2107 |
$temp = $product_value[$k] + $x_value[$j] * $y_value[$i] + $carry;
|
|
|
2108 |
$carry = (int) ($temp / 0x4000000);
|
|
|
2109 |
$product_value[$k] = (int) ($temp - 0x4000000 * $carry);
|
|
|
2110 |
}
|
|
|
2111 |
|
|
|
2112 |
if ($k < $stop) {
|
|
|
2113 |
$product_value[$k] = $carry;
|
|
|
2114 |
}
|
|
|
2115 |
}
|
|
|
2116 |
|
|
|
2117 |
return array(
|
|
|
2118 |
MATH_BIGINTEGER_VALUE => $this->_trim($product_value),
|
|
|
2119 |
MATH_BIGINTEGER_SIGN => $x_negative != $y_negative
|
|
|
2120 |
);
|
|
|
2121 |
}
|
|
|
2122 |
|
|
|
2123 |
/**
|
|
|
2124 |
* Montgomery Modular Reduction
|
|
|
2125 |
*
|
|
|
2126 |
* ($x->_prepMontgomery($n))->_montgomery($n) yields $x % $n.
|
|
|
2127 |
* {@link http://math.libtomcrypt.com/files/tommath.pdf#page=170 MPM 6.3} provides insights on how this can be
|
|
|
2128 |
* improved upon (basically, by using the comba method). gcd($n, 2) must be equal to one for this function
|
|
|
2129 |
* to work correctly.
|
|
|
2130 |
*
|
|
|
2131 |
* @see _prepMontgomery()
|
|
|
2132 |
* @see _slidingWindow()
|
|
|
2133 |
* @access private
|
|
|
2134 |
* @param Array $x
|
|
|
2135 |
* @param Array $n
|
|
|
2136 |
* @return Array
|
|
|
2137 |
*/
|
|
|
2138 |
function _montgomery($x, $n)
|
|
|
2139 |
{
|
|
|
2140 |
static $cache = array(
|
|
|
2141 |
MATH_BIGINTEGER_VARIABLE => array(),
|
|
|
2142 |
MATH_BIGINTEGER_DATA => array()
|
|
|
2143 |
);
|
|
|
2144 |
|
|
|
2145 |
if ( ($key = array_search($n, $cache[MATH_BIGINTEGER_VARIABLE])) === false ) {
|
|
|
2146 |
$key = count($cache[MATH_BIGINTEGER_VARIABLE]);
|
|
|
2147 |
$cache[MATH_BIGINTEGER_VARIABLE][] = $x;
|
|
|
2148 |
$cache[MATH_BIGINTEGER_DATA][] = $this->_modInverse67108864($n);
|
|
|
2149 |
}
|
|
|
2150 |
|
|
|
2151 |
$k = count($n);
|
|
|
2152 |
|
|
|
2153 |
$result = array(MATH_BIGINTEGER_VALUE => $x);
|
|
|
2154 |
|
|
|
2155 |
for ($i = 0; $i < $k; ++$i) {
|
|
|
2156 |
$temp = $result[MATH_BIGINTEGER_VALUE][$i] * $cache[MATH_BIGINTEGER_DATA][$key];
|
|
|
2157 |
$temp = (int) ($temp - 0x4000000 * ((int) ($temp / 0x4000000)));
|
|
|
2158 |
$temp = $this->_regularMultiply(array($temp), $n);
|
|
|
2159 |
$temp = array_merge($this->_array_repeat(0, $i), $temp);
|
|
|
2160 |
$result = $this->_add($result[MATH_BIGINTEGER_VALUE], false, $temp, false);
|
|
|
2161 |
}
|
|
|
2162 |
|
|
|
2163 |
$result[MATH_BIGINTEGER_VALUE] = array_slice($result[MATH_BIGINTEGER_VALUE], $k);
|
|
|
2164 |
|
|
|
2165 |
if ($this->_compare($result, false, $n, false) >= 0) {
|
|
|
2166 |
$result = $this->_subtract($result[MATH_BIGINTEGER_VALUE], false, $n, false);
|
|
|
2167 |
}
|
|
|
2168 |
|
|
|
2169 |
return $result[MATH_BIGINTEGER_VALUE];
|
|
|
2170 |
}
|
|
|
2171 |
|
|
|
2172 |
/**
|
|
|
2173 |
* Montgomery Multiply
|
|
|
2174 |
*
|
|
|
2175 |
* Interleaves the montgomery reduction and long multiplication algorithms together as described in
|
|
|
2176 |
* {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=13 HAC 14.36}
|
|
|
2177 |
*
|
|
|
2178 |
* @see _prepMontgomery()
|
|
|
2179 |
* @see _montgomery()
|
|
|
2180 |
* @access private
|
|
|
2181 |
* @param Array $x
|
|
|
2182 |
* @param Array $y
|
|
|
2183 |
* @param Array $m
|
|
|
2184 |
* @return Array
|
|
|
2185 |
*/
|
|
|
2186 |
function _montgomeryMultiply($x, $y, $m)
|
|
|
2187 |
{
|
|
|
2188 |
$temp = $this->_multiply($x, false, $y, false);
|
|
|
2189 |
return $this->_montgomery($temp[MATH_BIGINTEGER_VALUE], $m);
|
|
|
2190 |
|
|
|
2191 |
static $cache = array(
|
|
|
2192 |
MATH_BIGINTEGER_VARIABLE => array(),
|
|
|
2193 |
MATH_BIGINTEGER_DATA => array()
|
|
|
2194 |
);
|
|
|
2195 |
|
|
|
2196 |
if ( ($key = array_search($m, $cache[MATH_BIGINTEGER_VARIABLE])) === false ) {
|
|
|
2197 |
$key = count($cache[MATH_BIGINTEGER_VARIABLE]);
|
|
|
2198 |
$cache[MATH_BIGINTEGER_VARIABLE][] = $m;
|
|
|
2199 |
$cache[MATH_BIGINTEGER_DATA][] = $this->_modInverse67108864($m);
|
|
|
2200 |
}
|
|
|
2201 |
|
|
|
2202 |
$n = max(count($x), count($y), count($m));
|
|
|
2203 |
$x = array_pad($x, $n, 0);
|
|
|
2204 |
$y = array_pad($y, $n, 0);
|
|
|
2205 |
$m = array_pad($m, $n, 0);
|
|
|
2206 |
$a = array(MATH_BIGINTEGER_VALUE => $this->_array_repeat(0, $n + 1));
|
|
|
2207 |
for ($i = 0; $i < $n; ++$i) {
|
|
|
2208 |
$temp = $a[MATH_BIGINTEGER_VALUE][0] + $x[$i] * $y[0];
|
|
|
2209 |
$temp = (int) ($temp - 0x4000000 * ((int) ($temp / 0x4000000)));
|
|
|
2210 |
$temp = $temp * $cache[MATH_BIGINTEGER_DATA][$key];
|
|
|
2211 |
$temp = (int) ($temp - 0x4000000 * ((int) ($temp / 0x4000000)));
|
|
|
2212 |
$temp = $this->_add($this->_regularMultiply(array($x[$i]), $y), false, $this->_regularMultiply(array($temp), $m), false);
|
|
|
2213 |
$a = $this->_add($a[MATH_BIGINTEGER_VALUE], false, $temp[MATH_BIGINTEGER_VALUE], false);
|
|
|
2214 |
$a[MATH_BIGINTEGER_VALUE] = array_slice($a[MATH_BIGINTEGER_VALUE], 1);
|
|
|
2215 |
}
|
|
|
2216 |
if ($this->_compare($a[MATH_BIGINTEGER_VALUE], false, $m, false) >= 0) {
|
|
|
2217 |
$a = $this->_subtract($a[MATH_BIGINTEGER_VALUE], false, $m, false);
|
|
|
2218 |
}
|
|
|
2219 |
return $a[MATH_BIGINTEGER_VALUE];
|
|
|
2220 |
}
|
|
|
2221 |
|
|
|
2222 |
/**
|
|
|
2223 |
* Prepare a number for use in Montgomery Modular Reductions
|
|
|
2224 |
*
|
|
|
2225 |
* @see _montgomery()
|
|
|
2226 |
* @see _slidingWindow()
|
|
|
2227 |
* @access private
|
|
|
2228 |
* @param Array $x
|
|
|
2229 |
* @param Array $n
|
|
|
2230 |
* @return Array
|
|
|
2231 |
*/
|
|
|
2232 |
function _prepMontgomery($x, $n)
|
|
|
2233 |
{
|
|
|
2234 |
$lhs = new Math_BigInteger();
|
|
|
2235 |
$lhs->value = array_merge($this->_array_repeat(0, count($n)), $x);
|
|
|
2236 |
$rhs = new Math_BigInteger();
|
|
|
2237 |
$rhs->value = $n;
|
|
|
2238 |
|
|
|
2239 |
list(, $temp) = $lhs->divide($rhs);
|
|
|
2240 |
return $temp->value;
|
|
|
2241 |
}
|
|
|
2242 |
|
|
|
2243 |
/**
|
|
|
2244 |
* Modular Inverse of a number mod 2**26 (eg. 67108864)
|
|
|
2245 |
*
|
|
|
2246 |
* Based off of the bnpInvDigit function implemented and justified in the following URL:
|
|
|
2247 |
*
|
|
|
2248 |
* {@link http://www-cs-students.stanford.edu/~tjw/jsbn/jsbn.js}
|
|
|
2249 |
*
|
|
|
2250 |
* The following URL provides more info:
|
|
|
2251 |
*
|
|
|
2252 |
* {@link http://groups.google.com/group/sci.crypt/msg/7a137205c1be7d85}
|
|
|
2253 |
*
|
|
|
2254 |
* As for why we do all the bitmasking... strange things can happen when converting from floats to ints. For
|
|
|
2255 |
* instance, on some computers, var_dump((int) -4294967297) yields int(-1) and on others, it yields
|
|
|
2256 |
* int(-2147483648). To avoid problems stemming from this, we use bitmasks to guarantee that ints aren't
|
|
|
2257 |
* auto-converted to floats. The outermost bitmask is present because without it, there's no guarantee that
|
|
|
2258 |
* the "residue" returned would be the so-called "common residue". We use fmod, in the last step, because the
|
|
|
2259 |
* maximum possible $x is 26 bits and the maximum $result is 16 bits. Thus, we have to be able to handle up to
|
|
|
2260 |
* 40 bits, which only 64-bit floating points will support.
|
|
|
2261 |
*
|
|
|
2262 |
* Thanks to Pedro Gimeno Fortea for input!
|
|
|
2263 |
*
|
|
|
2264 |
* @see _montgomery()
|
|
|
2265 |
* @access private
|
|
|
2266 |
* @param Array $x
|
|
|
2267 |
* @return Integer
|
|
|
2268 |
*/
|
|
|
2269 |
function _modInverse67108864($x) // 2**26 == 67108864
|
|
|
2270 |
{
|
|
|
2271 |
$x = -$x[0];
|
|
|
2272 |
$result = $x & 0x3; // x**-1 mod 2**2
|
|
|
2273 |
$result = ($result * (2 - $x * $result)) & 0xF; // x**-1 mod 2**4
|
|
|
2274 |
$result = ($result * (2 - ($x & 0xFF) * $result)) & 0xFF; // x**-1 mod 2**8
|
|
|
2275 |
$result = ($result * ((2 - ($x & 0xFFFF) * $result) & 0xFFFF)) & 0xFFFF; // x**-1 mod 2**16
|
|
|
2276 |
$result = fmod($result * (2 - fmod($x * $result, 0x4000000)), 0x4000000); // x**-1 mod 2**26
|
|
|
2277 |
return $result & 0x3FFFFFF;
|
|
|
2278 |
}
|
|
|
2279 |
|
|
|
2280 |
/**
|
|
|
2281 |
* Calculates modular inverses.
|
|
|
2282 |
*
|
|
|
2283 |
* Say you have (30 mod 17 * x mod 17) mod 17 == 1. x can be found using modular inverses.
|
|
|
2284 |
*
|
|
|
2285 |
* Here's an example:
|
|
|
2286 |
* <code>
|
|
|
2287 |
* <?php
|
|
|
2288 |
* include('Math/BigInteger.php');
|
|
|
2289 |
*
|
|
|
2290 |
* $a = new Math_BigInteger(30);
|
|
|
2291 |
* $b = new Math_BigInteger(17);
|
|
|
2292 |
*
|
|
|
2293 |
* $c = $a->modInverse($b);
|
|
|
2294 |
* echo $c->toString(); // outputs 4
|
|
|
2295 |
*
|
|
|
2296 |
* echo "\r\n";
|
|
|
2297 |
*
|
|
|
2298 |
* $d = $a->multiply($c);
|
|
|
2299 |
* list(, $d) = $d->divide($b);
|
|
|
2300 |
* echo $d; // outputs 1 (as per the definition of modular inverse)
|
|
|
2301 |
* ?>
|
|
|
2302 |
* </code>
|
|
|
2303 |
*
|
|
|
2304 |
* @param Math_BigInteger $n
|
|
|
2305 |
* @return mixed false, if no modular inverse exists, Math_BigInteger, otherwise.
|
|
|
2306 |
* @access public
|
|
|
2307 |
* @internal See {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=21 HAC 14.64} for more information.
|
|
|
2308 |
*/
|
|
|
2309 |
function modInverse($n)
|
|
|
2310 |
{
|
|
|
2311 |
switch ( MATH_BIGINTEGER_MODE ) {
|
|
|
2312 |
case MATH_BIGINTEGER_MODE_GMP:
|
|
|
2313 |
$temp = new Math_BigInteger();
|
|
|
2314 |
$temp->value = gmp_invert($this->value, $n->value);
|
|
|
2315 |
|
|
|
2316 |
return ( $temp->value === false ) ? false : $this->_normalize($temp);
|
|
|
2317 |
}
|
|
|
2318 |
|
|
|
2319 |
static $zero, $one;
|
|
|
2320 |
if (!isset($zero)) {
|
|
|
2321 |
$zero = new Math_BigInteger();
|
|
|
2322 |
$one = new Math_BigInteger(1);
|
|
|
2323 |
}
|
|
|
2324 |
|
|
|
2325 |
// $x mod $n == $x mod -$n.
|
|
|
2326 |
$n = $n->abs();
|
|
|
2327 |
|
|
|
2328 |
if ($this->compare($zero) < 0) {
|
|
|
2329 |
$temp = $this->abs();
|
|
|
2330 |
$temp = $temp->modInverse($n);
|
|
|
2331 |
return $negated === false ? false : $this->_normalize($n->subtract($temp));
|
|
|
2332 |
}
|
|
|
2333 |
|
|
|
2334 |
extract($this->extendedGCD($n));
|
|
|
2335 |
|
|
|
2336 |
if (!$gcd->equals($one)) {
|
|
|
2337 |
return false;
|
|
|
2338 |
}
|
|
|
2339 |
|
|
|
2340 |
$x = $x->compare($zero) < 0 ? $x->add($n) : $x;
|
|
|
2341 |
|
|
|
2342 |
return $this->compare($zero) < 0 ? $this->_normalize($n->subtract($x)) : $this->_normalize($x);
|
|
|
2343 |
}
|
|
|
2344 |
|
|
|
2345 |
/**
|
|
|
2346 |
* Calculates the greatest common divisor and Bézout's identity.
|
|
|
2347 |
*
|
|
|
2348 |
* Say you have 693 and 609. The GCD is 21. Bézout's identity states that there exist integers x and y such that
|
|
|
2349 |
* 693*x + 609*y == 21. In point of fact, there are actually an infinite number of x and y combinations and which
|
|
|
2350 |
* combination is returned is dependant upon which mode is in use. See
|
|
|
2351 |
* {@link http://en.wikipedia.org/wiki/B%C3%A9zout%27s_identity Bézout's identity - Wikipedia} for more information.
|
|
|
2352 |
*
|
|
|
2353 |
* Here's an example:
|
|
|
2354 |
* <code>
|
|
|
2355 |
* <?php
|
|
|
2356 |
* include('Math/BigInteger.php');
|
|
|
2357 |
*
|
|
|
2358 |
* $a = new Math_BigInteger(693);
|
|
|
2359 |
* $b = new Math_BigInteger(609);
|
|
|
2360 |
*
|
|
|
2361 |
* extract($a->extendedGCD($b));
|
|
|
2362 |
*
|
|
|
2363 |
* echo $gcd->toString() . "\r\n"; // outputs 21
|
|
|
2364 |
* echo $a->toString() * $x->toString() + $b->toString() * $y->toString(); // outputs 21
|
|
|
2365 |
* ?>
|
|
|
2366 |
* </code>
|
|
|
2367 |
*
|
|
|
2368 |
* @param Math_BigInteger $n
|
|
|
2369 |
* @return Math_BigInteger
|
|
|
2370 |
* @access public
|
|
|
2371 |
* @internal Calculates the GCD using the binary xGCD algorithim described in
|
|
|
2372 |
* {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=19 HAC 14.61}. As the text above 14.61 notes,
|
|
|
2373 |
* the more traditional algorithim requires "relatively costly multiple-precision divisions".
|
|
|
2374 |
*/
|
|
|
2375 |
function extendedGCD($n)
|
|
|
2376 |
{
|
|
|
2377 |
switch ( MATH_BIGINTEGER_MODE ) {
|
|
|
2378 |
case MATH_BIGINTEGER_MODE_GMP:
|
|
|
2379 |
extract(gmp_gcdext($this->value, $n->value));
|
|
|
2380 |
|
|
|
2381 |
return array(
|
|
|
2382 |
'gcd' => $this->_normalize(new Math_BigInteger($g)),
|
|
|
2383 |
'x' => $this->_normalize(new Math_BigInteger($s)),
|
|
|
2384 |
'y' => $this->_normalize(new Math_BigInteger($t))
|
|
|
2385 |
);
|
|
|
2386 |
case MATH_BIGINTEGER_MODE_BCMATH:
|
|
|
2387 |
// it might be faster to use the binary xGCD algorithim here, as well, but (1) that algorithim works
|
|
|
2388 |
// best when the base is a power of 2 and (2) i don't think it'd make much difference, anyway. as is,
|
|
|
2389 |
// the basic extended euclidean algorithim is what we're using.
|
|
|
2390 |
|
|
|
2391 |
$u = $this->value;
|
|
|
2392 |
$v = $n->value;
|
|
|
2393 |
|
|
|
2394 |
$a = '1';
|
|
|
2395 |
$b = '0';
|
|
|
2396 |
$c = '0';
|
|
|
2397 |
$d = '1';
|
|
|
2398 |
|
|
|
2399 |
while (bccomp($v, '0', 0) != 0) {
|
|
|
2400 |
$q = bcdiv($u, $v, 0);
|
|
|
2401 |
|
|
|
2402 |
$temp = $u;
|
|
|
2403 |
$u = $v;
|
|
|
2404 |
$v = bcsub($temp, bcmul($v, $q, 0), 0);
|
|
|
2405 |
|
|
|
2406 |
$temp = $a;
|
|
|
2407 |
$a = $c;
|
|
|
2408 |
$c = bcsub($temp, bcmul($a, $q, 0), 0);
|
|
|
2409 |
|
|
|
2410 |
$temp = $b;
|
|
|
2411 |
$b = $d;
|
|
|
2412 |
$d = bcsub($temp, bcmul($b, $q, 0), 0);
|
|
|
2413 |
}
|
|
|
2414 |
|
|
|
2415 |
return array(
|
|
|
2416 |
'gcd' => $this->_normalize(new Math_BigInteger($u)),
|
|
|
2417 |
'x' => $this->_normalize(new Math_BigInteger($a)),
|
|
|
2418 |
'y' => $this->_normalize(new Math_BigInteger($b))
|
|
|
2419 |
);
|
|
|
2420 |
}
|
|
|
2421 |
|
|
|
2422 |
$y = $n->copy();
|
|
|
2423 |
$x = $this->copy();
|
|
|
2424 |
$g = new Math_BigInteger();
|
|
|
2425 |
$g->value = array(1);
|
|
|
2426 |
|
|
|
2427 |
while ( !(($x->value[0] & 1)|| ($y->value[0] & 1)) ) {
|
|
|
2428 |
$x->_rshift(1);
|
|
|
2429 |
$y->_rshift(1);
|
|
|
2430 |
$g->_lshift(1);
|
|
|
2431 |
}
|
|
|
2432 |
|
|
|
2433 |
$u = $x->copy();
|
|
|
2434 |
$v = $y->copy();
|
|
|
2435 |
|
|
|
2436 |
$a = new Math_BigInteger();
|
|
|
2437 |
$b = new Math_BigInteger();
|
|
|
2438 |
$c = new Math_BigInteger();
|
|
|
2439 |
$d = new Math_BigInteger();
|
|
|
2440 |
|
|
|
2441 |
$a->value = $d->value = $g->value = array(1);
|
|
|
2442 |
$b->value = $c->value = array();
|
|
|
2443 |
|
|
|
2444 |
while ( !empty($u->value) ) {
|
|
|
2445 |
while ( !($u->value[0] & 1) ) {
|
|
|
2446 |
$u->_rshift(1);
|
|
|
2447 |
if ( (!empty($a->value) && ($a->value[0] & 1)) || (!empty($b->value) && ($b->value[0] & 1)) ) {
|
|
|
2448 |
$a = $a->add($y);
|
|
|
2449 |
$b = $b->subtract($x);
|
|
|
2450 |
}
|
|
|
2451 |
$a->_rshift(1);
|
|
|
2452 |
$b->_rshift(1);
|
|
|
2453 |
}
|
|
|
2454 |
|
|
|
2455 |
while ( !($v->value[0] & 1) ) {
|
|
|
2456 |
$v->_rshift(1);
|
|
|
2457 |
if ( (!empty($d->value) && ($d->value[0] & 1)) || (!empty($c->value) && ($c->value[0] & 1)) ) {
|
|
|
2458 |
$c = $c->add($y);
|
|
|
2459 |
$d = $d->subtract($x);
|
|
|
2460 |
}
|
|
|
2461 |
$c->_rshift(1);
|
|
|
2462 |
$d->_rshift(1);
|
|
|
2463 |
}
|
|
|
2464 |
|
|
|
2465 |
if ($u->compare($v) >= 0) {
|
|
|
2466 |
$u = $u->subtract($v);
|
|
|
2467 |
$a = $a->subtract($c);
|
|
|
2468 |
$b = $b->subtract($d);
|
|
|
2469 |
} else {
|
|
|
2470 |
$v = $v->subtract($u);
|
|
|
2471 |
$c = $c->subtract($a);
|
|
|
2472 |
$d = $d->subtract($b);
|
|
|
2473 |
}
|
|
|
2474 |
}
|
|
|
2475 |
|
|
|
2476 |
return array(
|
|
|
2477 |
'gcd' => $this->_normalize($g->multiply($v)),
|
|
|
2478 |
'x' => $this->_normalize($c),
|
|
|
2479 |
'y' => $this->_normalize($d)
|
|
|
2480 |
);
|
|
|
2481 |
}
|
|
|
2482 |
|
|
|
2483 |
/**
|
|
|
2484 |
* Calculates the greatest common divisor
|
|
|
2485 |
*
|
|
|
2486 |
* Say you have 693 and 609. The GCD is 21.
|
|
|
2487 |
*
|
|
|
2488 |
* Here's an example:
|
|
|
2489 |
* <code>
|
|
|
2490 |
* <?php
|
|
|
2491 |
* include('Math/BigInteger.php');
|
|
|
2492 |
*
|
|
|
2493 |
* $a = new Math_BigInteger(693);
|
|
|
2494 |
* $b = new Math_BigInteger(609);
|
|
|
2495 |
*
|
|
|
2496 |
* $gcd = a->extendedGCD($b);
|
|
|
2497 |
*
|
|
|
2498 |
* echo $gcd->toString() . "\r\n"; // outputs 21
|
|
|
2499 |
* ?>
|
|
|
2500 |
* </code>
|
|
|
2501 |
*
|
|
|
2502 |
* @param Math_BigInteger $n
|
|
|
2503 |
* @return Math_BigInteger
|
|
|
2504 |
* @access public
|
|
|
2505 |
*/
|
|
|
2506 |
function gcd($n)
|
|
|
2507 |
{
|
|
|
2508 |
extract($this->extendedGCD($n));
|
|
|
2509 |
return $gcd;
|
|
|
2510 |
}
|
|
|
2511 |
|
|
|
2512 |
/**
|
|
|
2513 |
* Absolute value.
|
|
|
2514 |
*
|
|
|
2515 |
* @return Math_BigInteger
|
|
|
2516 |
* @access public
|
|
|
2517 |
*/
|
|
|
2518 |
function abs()
|
|
|
2519 |
{
|
|
|
2520 |
$temp = new Math_BigInteger();
|
|
|
2521 |
|
|
|
2522 |
switch ( MATH_BIGINTEGER_MODE ) {
|
|
|
2523 |
case MATH_BIGINTEGER_MODE_GMP:
|
|
|
2524 |
$temp->value = gmp_abs($this->value);
|
|
|
2525 |
break;
|
|
|
2526 |
case MATH_BIGINTEGER_MODE_BCMATH:
|
|
|
2527 |
$temp->value = (bccomp($this->value, '0', 0) < 0) ? substr($this->value, 1) : $this->value;
|
|
|
2528 |
break;
|
|
|
2529 |
default:
|
|
|
2530 |
$temp->value = $this->value;
|
|
|
2531 |
}
|
|
|
2532 |
|
|
|
2533 |
return $temp;
|
|
|
2534 |
}
|
|
|
2535 |
|
|
|
2536 |
/**
|
|
|
2537 |
* Compares two numbers.
|
|
|
2538 |
*
|
|
|
2539 |
* Although one might think !$x->compare($y) means $x != $y, it, in fact, means the opposite. The reason for this is
|
|
|
2540 |
* demonstrated thusly:
|
|
|
2541 |
*
|
|
|
2542 |
* $x > $y: $x->compare($y) > 0
|
|
|
2543 |
* $x < $y: $x->compare($y) < 0
|
|
|
2544 |
* $x == $y: $x->compare($y) == 0
|
|
|
2545 |
*
|
|
|
2546 |
* Note how the same comparison operator is used. If you want to test for equality, use $x->equals($y).
|
|
|
2547 |
*
|
|
|
2548 |
* @param Math_BigInteger $x
|
|
|
2549 |
* @return Integer < 0 if $this is less than $x; > 0 if $this is greater than $x, and 0 if they are equal.
|
|
|
2550 |
* @access public
|
|
|
2551 |
* @see equals()
|
|
|
2552 |
* @internal Could return $this->subtract($x), but that's not as fast as what we do do.
|
|
|
2553 |
*/
|
|
|
2554 |
function compare($y)
|
|
|
2555 |
{
|
|
|
2556 |
switch ( MATH_BIGINTEGER_MODE ) {
|
|
|
2557 |
case MATH_BIGINTEGER_MODE_GMP:
|
|
|
2558 |
return gmp_cmp($this->value, $y->value);
|
|
|
2559 |
case MATH_BIGINTEGER_MODE_BCMATH:
|
|
|
2560 |
return bccomp($this->value, $y->value, 0);
|
|
|
2561 |
}
|
|
|
2562 |
|
|
|
2563 |
return $this->_compare($this->value, $this->is_negative, $y->value, $y->is_negative);
|
|
|
2564 |
}
|
|
|
2565 |
|
|
|
2566 |
/**
|
|
|
2567 |
* Compares two numbers.
|
|
|
2568 |
*
|
|
|
2569 |
* @param Array $x_value
|
|
|
2570 |
* @param Boolean $x_negative
|
|
|
2571 |
* @param Array $y_value
|
|
|
2572 |
* @param Boolean $y_negative
|
|
|
2573 |
* @return Integer
|
|
|
2574 |
* @see compare()
|
|
|
2575 |
* @access private
|
|
|
2576 |
*/
|
|
|
2577 |
function _compare($x_value, $x_negative, $y_value, $y_negative)
|
|
|
2578 |
{
|
|
|
2579 |
if ( $x_negative != $y_negative ) {
|
|
|
2580 |
return ( !$x_negative && $y_negative ) ? 1 : -1;
|
|
|
2581 |
}
|
|
|
2582 |
|
|
|
2583 |
$result = $x_negative ? -1 : 1;
|
|
|
2584 |
|
|
|
2585 |
if ( count($x_value) != count($y_value) ) {
|
|
|
2586 |
return ( count($x_value) > count($y_value) ) ? $result : -$result;
|
|
|
2587 |
}
|
|
|
2588 |
$size = max(count($x_value), count($y_value));
|
|
|
2589 |
|
|
|
2590 |
$x_value = array_pad($x_value, $size, 0);
|
|
|
2591 |
$y_value = array_pad($y_value, $size, 0);
|
|
|
2592 |
|
|
|
2593 |
for ($i = count($x_value) - 1; $i >= 0; --$i) {
|
|
|
2594 |
if ($x_value[$i] != $y_value[$i]) {
|
|
|
2595 |
return ( $x_value[$i] > $y_value[$i] ) ? $result : -$result;
|
|
|
2596 |
}
|
|
|
2597 |
}
|
|
|
2598 |
|
|
|
2599 |
return 0;
|
|
|
2600 |
}
|
|
|
2601 |
|
|
|
2602 |
/**
|
|
|
2603 |
* Tests the equality of two numbers.
|
|
|
2604 |
*
|
|
|
2605 |
* If you need to see if one number is greater than or less than another number, use Math_BigInteger::compare()
|
|
|
2606 |
*
|
|
|
2607 |
* @param Math_BigInteger $x
|
|
|
2608 |
* @return Boolean
|
|
|
2609 |
* @access public
|
|
|
2610 |
* @see compare()
|
|
|
2611 |
*/
|
|
|
2612 |
function equals($x)
|
|
|
2613 |
{
|
|
|
2614 |
switch ( MATH_BIGINTEGER_MODE ) {
|
|
|
2615 |
case MATH_BIGINTEGER_MODE_GMP:
|
|
|
2616 |
return gmp_cmp($this->value, $x->value) == 0;
|
|
|
2617 |
default:
|
|
|
2618 |
return $this->value === $x->value && $this->is_negative == $x->is_negative;
|
|
|
2619 |
}
|
|
|
2620 |
}
|
|
|
2621 |
|
|
|
2622 |
/**
|
|
|
2623 |
* Set Precision
|
|
|
2624 |
*
|
|
|
2625 |
* Some bitwise operations give different results depending on the precision being used. Examples include left
|
|
|
2626 |
* shift, not, and rotates.
|
|
|
2627 |
*
|
|
|
2628 |
* @param Math_BigInteger $x
|
|
|
2629 |
* @access public
|
|
|
2630 |
* @return Math_BigInteger
|
|
|
2631 |
*/
|
|
|
2632 |
function setPrecision($bits)
|
|
|
2633 |
{
|
|
|
2634 |
$this->precision = $bits;
|
|
|
2635 |
if ( MATH_BIGINTEGER_MODE != MATH_BIGINTEGER_MODE_BCMATH ) {
|
|
|
2636 |
$this->bitmask = new Math_BigInteger(chr((1 << ($bits & 0x7)) - 1) . str_repeat(chr(0xFF), $bits >> 3), 256);
|
|
|
2637 |
} else {
|
|
|
2638 |
$this->bitmask = new Math_BigInteger(bcpow('2', $bits, 0));
|
|
|
2639 |
}
|
|
|
2640 |
|
|
|
2641 |
$temp = $this->_normalize($this);
|
|
|
2642 |
$this->value = $temp->value;
|
|
|
2643 |
}
|
|
|
2644 |
|
|
|
2645 |
/**
|
|
|
2646 |
* Logical And
|
|
|
2647 |
*
|
|
|
2648 |
* @param Math_BigInteger $x
|
|
|
2649 |
* @access public
|
|
|
2650 |
* @internal Implemented per a request by Lluis Pamies i Juarez <lluis _a_ pamies.cat>
|
|
|
2651 |
* @return Math_BigInteger
|
|
|
2652 |
*/
|
|
|
2653 |
function bitwise_and($x)
|
|
|
2654 |
{
|
|
|
2655 |
switch ( MATH_BIGINTEGER_MODE ) {
|
|
|
2656 |
case MATH_BIGINTEGER_MODE_GMP:
|
|
|
2657 |
$temp = new Math_BigInteger();
|
|
|
2658 |
$temp->value = gmp_and($this->value, $x->value);
|
|
|
2659 |
|
|
|
2660 |
return $this->_normalize($temp);
|
|
|
2661 |
case MATH_BIGINTEGER_MODE_BCMATH:
|
|
|
2662 |
$left = $this->toBytes();
|
|
|
2663 |
$right = $x->toBytes();
|
|
|
2664 |
|
|
|
2665 |
$length = max(strlen($left), strlen($right));
|
|
|
2666 |
|
|
|
2667 |
$left = str_pad($left, $length, chr(0), STR_PAD_LEFT);
|
|
|
2668 |
$right = str_pad($right, $length, chr(0), STR_PAD_LEFT);
|
|
|
2669 |
|
|
|
2670 |
return $this->_normalize(new Math_BigInteger($left & $right, 256));
|
|
|
2671 |
}
|
|
|
2672 |
|
|
|
2673 |
$result = $this->copy();
|
|
|
2674 |
|
|
|
2675 |
$length = min(count($x->value), count($this->value));
|
|
|
2676 |
|
|
|
2677 |
$result->value = array_slice($result->value, 0, $length);
|
|
|
2678 |
|
|
|
2679 |
for ($i = 0; $i < $length; ++$i) {
|
|
|
2680 |
$result->value[$i] = $result->value[$i] & $x->value[$i];
|
|
|
2681 |
}
|
|
|
2682 |
|
|
|
2683 |
return $this->_normalize($result);
|
|
|
2684 |
}
|
|
|
2685 |
|
|
|
2686 |
/**
|
|
|
2687 |
* Logical Or
|
|
|
2688 |
*
|
|
|
2689 |
* @param Math_BigInteger $x
|
|
|
2690 |
* @access public
|
|
|
2691 |
* @internal Implemented per a request by Lluis Pamies i Juarez <lluis _a_ pamies.cat>
|
|
|
2692 |
* @return Math_BigInteger
|
|
|
2693 |
*/
|
|
|
2694 |
function bitwise_or($x)
|
|
|
2695 |
{
|
|
|
2696 |
switch ( MATH_BIGINTEGER_MODE ) {
|
|
|
2697 |
case MATH_BIGINTEGER_MODE_GMP:
|
|
|
2698 |
$temp = new Math_BigInteger();
|
|
|
2699 |
$temp->value = gmp_or($this->value, $x->value);
|
|
|
2700 |
|
|
|
2701 |
return $this->_normalize($temp);
|
|
|
2702 |
case MATH_BIGINTEGER_MODE_BCMATH:
|
|
|
2703 |
$left = $this->toBytes();
|
|
|
2704 |
$right = $x->toBytes();
|
|
|
2705 |
|
|
|
2706 |
$length = max(strlen($left), strlen($right));
|
|
|
2707 |
|
|
|
2708 |
$left = str_pad($left, $length, chr(0), STR_PAD_LEFT);
|
|
|
2709 |
$right = str_pad($right, $length, chr(0), STR_PAD_LEFT);
|
|
|
2710 |
|
|
|
2711 |
return $this->_normalize(new Math_BigInteger($left | $right, 256));
|
|
|
2712 |
}
|
|
|
2713 |
|
|
|
2714 |
$length = max(count($this->value), count($x->value));
|
|
|
2715 |
$result = $this->copy();
|
|
|
2716 |
$result->value = array_pad($result->value, 0, $length);
|
|
|
2717 |
$x->value = array_pad($x->value, 0, $length);
|
|
|
2718 |
|
|
|
2719 |
for ($i = 0; $i < $length; ++$i) {
|
|
|
2720 |
$result->value[$i] = $this->value[$i] | $x->value[$i];
|
|
|
2721 |
}
|
|
|
2722 |
|
|
|
2723 |
return $this->_normalize($result);
|
|
|
2724 |
}
|
|
|
2725 |
|
|
|
2726 |
/**
|
|
|
2727 |
* Logical Exclusive-Or
|
|
|
2728 |
*
|
|
|
2729 |
* @param Math_BigInteger $x
|
|
|
2730 |
* @access public
|
|
|
2731 |
* @internal Implemented per a request by Lluis Pamies i Juarez <lluis _a_ pamies.cat>
|
|
|
2732 |
* @return Math_BigInteger
|
|
|
2733 |
*/
|
|
|
2734 |
function bitwise_xor($x)
|
|
|
2735 |
{
|
|
|
2736 |
switch ( MATH_BIGINTEGER_MODE ) {
|
|
|
2737 |
case MATH_BIGINTEGER_MODE_GMP:
|
|
|
2738 |
$temp = new Math_BigInteger();
|
|
|
2739 |
$temp->value = gmp_xor($this->value, $x->value);
|
|
|
2740 |
|
|
|
2741 |
return $this->_normalize($temp);
|
|
|
2742 |
case MATH_BIGINTEGER_MODE_BCMATH:
|
|
|
2743 |
$left = $this->toBytes();
|
|
|
2744 |
$right = $x->toBytes();
|
|
|
2745 |
|
|
|
2746 |
$length = max(strlen($left), strlen($right));
|
|
|
2747 |
|
|
|
2748 |
$left = str_pad($left, $length, chr(0), STR_PAD_LEFT);
|
|
|
2749 |
$right = str_pad($right, $length, chr(0), STR_PAD_LEFT);
|
|
|
2750 |
|
|
|
2751 |
return $this->_normalize(new Math_BigInteger($left ^ $right, 256));
|
|
|
2752 |
}
|
|
|
2753 |
|
|
|
2754 |
$length = max(count($this->value), count($x->value));
|
|
|
2755 |
$result = $this->copy();
|
|
|
2756 |
$result->value = array_pad($result->value, 0, $length);
|
|
|
2757 |
$x->value = array_pad($x->value, 0, $length);
|
|
|
2758 |
|
|
|
2759 |
for ($i = 0; $i < $length; ++$i) {
|
|
|
2760 |
$result->value[$i] = $this->value[$i] ^ $x->value[$i];
|
|
|
2761 |
}
|
|
|
2762 |
|
|
|
2763 |
return $this->_normalize($result);
|
|
|
2764 |
}
|
|
|
2765 |
|
|
|
2766 |
/**
|
|
|
2767 |
* Logical Not
|
|
|
2768 |
*
|
|
|
2769 |
* @access public
|
|
|
2770 |
* @internal Implemented per a request by Lluis Pamies i Juarez <lluis _a_ pamies.cat>
|
|
|
2771 |
* @return Math_BigInteger
|
|
|
2772 |
*/
|
|
|
2773 |
function bitwise_not()
|
|
|
2774 |
{
|
|
|
2775 |
// calculuate "not" without regard to $this->precision
|
|
|
2776 |
// (will always result in a smaller number. ie. ~1 isn't 1111 1110 - it's 0)
|
|
|
2777 |
$temp = $this->toBytes();
|
|
|
2778 |
$pre_msb = decbin(ord($temp[0]));
|
|
|
2779 |
$temp = ~$temp;
|
|
|
2780 |
$msb = decbin(ord($temp[0]));
|
|
|
2781 |
if (strlen($msb) == 8) {
|
|
|
2782 |
$msb = substr($msb, strpos($msb, '0'));
|
|
|
2783 |
}
|
|
|
2784 |
$temp[0] = chr(bindec($msb));
|
|
|
2785 |
|
|
|
2786 |
// see if we need to add extra leading 1's
|
|
|
2787 |
$current_bits = strlen($pre_msb) + 8 * strlen($temp) - 8;
|
|
|
2788 |
$new_bits = $this->precision - $current_bits;
|
|
|
2789 |
if ($new_bits <= 0) {
|
|
|
2790 |
return $this->_normalize(new Math_BigInteger($temp, 256));
|
|
|
2791 |
}
|
|
|
2792 |
|
|
|
2793 |
// generate as many leading 1's as we need to.
|
|
|
2794 |
$leading_ones = chr((1 << ($new_bits & 0x7)) - 1) . str_repeat(chr(0xFF), $new_bits >> 3);
|
|
|
2795 |
$this->_base256_lshift($leading_ones, $current_bits);
|
|
|
2796 |
|
|
|
2797 |
$temp = str_pad($temp, ceil($this->bits / 8), chr(0), STR_PAD_LEFT);
|
|
|
2798 |
|
|
|
2799 |
return $this->_normalize(new Math_BigInteger($leading_ones | $temp, 256));
|
|
|
2800 |
}
|
|
|
2801 |
|
|
|
2802 |
/**
|
|
|
2803 |
* Logical Right Shift
|
|
|
2804 |
*
|
|
|
2805 |
* Shifts BigInteger's by $shift bits, effectively dividing by 2**$shift.
|
|
|
2806 |
*
|
|
|
2807 |
* @param Integer $shift
|
|
|
2808 |
* @return Math_BigInteger
|
|
|
2809 |
* @access public
|
|
|
2810 |
* @internal The only version that yields any speed increases is the internal version.
|
|
|
2811 |
*/
|
|
|
2812 |
function bitwise_rightShift($shift)
|
|
|
2813 |
{
|
|
|
2814 |
$temp = new Math_BigInteger();
|
|
|
2815 |
|
|
|
2816 |
switch ( MATH_BIGINTEGER_MODE ) {
|
|
|
2817 |
case MATH_BIGINTEGER_MODE_GMP:
|
|
|
2818 |
static $two;
|
|
|
2819 |
|
|
|
2820 |
if (!isset($two)) {
|
|
|
2821 |
$two = gmp_init('2');
|
|
|
2822 |
}
|
|
|
2823 |
|
|
|
2824 |
$temp->value = gmp_div_q($this->value, gmp_pow($two, $shift));
|
|
|
2825 |
|
|
|
2826 |
break;
|
|
|
2827 |
case MATH_BIGINTEGER_MODE_BCMATH:
|
|
|
2828 |
$temp->value = bcdiv($this->value, bcpow('2', $shift, 0), 0);
|
|
|
2829 |
|
|
|
2830 |
break;
|
|
|
2831 |
default: // could just replace _lshift with this, but then all _lshift() calls would need to be rewritten
|
|
|
2832 |
// and I don't want to do that...
|
|
|
2833 |
$temp->value = $this->value;
|
|
|
2834 |
$temp->_rshift($shift);
|
|
|
2835 |
}
|
|
|
2836 |
|
|
|
2837 |
return $this->_normalize($temp);
|
|
|
2838 |
}
|
|
|
2839 |
|
|
|
2840 |
/**
|
|
|
2841 |
* Logical Left Shift
|
|
|
2842 |
*
|
|
|
2843 |
* Shifts BigInteger's by $shift bits, effectively multiplying by 2**$shift.
|
|
|
2844 |
*
|
|
|
2845 |
* @param Integer $shift
|
|
|
2846 |
* @return Math_BigInteger
|
|
|
2847 |
* @access public
|
|
|
2848 |
* @internal The only version that yields any speed increases is the internal version.
|
|
|
2849 |
*/
|
|
|
2850 |
function bitwise_leftShift($shift)
|
|
|
2851 |
{
|
|
|
2852 |
$temp = new Math_BigInteger();
|
|
|
2853 |
|
|
|
2854 |
switch ( MATH_BIGINTEGER_MODE ) {
|
|
|
2855 |
case MATH_BIGINTEGER_MODE_GMP:
|
|
|
2856 |
static $two;
|
|
|
2857 |
|
|
|
2858 |
if (!isset($two)) {
|
|
|
2859 |
$two = gmp_init('2');
|
|
|
2860 |
}
|
|
|
2861 |
|
|
|
2862 |
$temp->value = gmp_mul($this->value, gmp_pow($two, $shift));
|
|
|
2863 |
|
|
|
2864 |
break;
|
|
|
2865 |
case MATH_BIGINTEGER_MODE_BCMATH:
|
|
|
2866 |
$temp->value = bcmul($this->value, bcpow('2', $shift, 0), 0);
|
|
|
2867 |
|
|
|
2868 |
break;
|
|
|
2869 |
default: // could just replace _rshift with this, but then all _lshift() calls would need to be rewritten
|
|
|
2870 |
// and I don't want to do that...
|
|
|
2871 |
$temp->value = $this->value;
|
|
|
2872 |
$temp->_lshift($shift);
|
|
|
2873 |
}
|
|
|
2874 |
|
|
|
2875 |
return $this->_normalize($temp);
|
|
|
2876 |
}
|
|
|
2877 |
|
|
|
2878 |
/**
|
|
|
2879 |
* Logical Left Rotate
|
|
|
2880 |
*
|
|
|
2881 |
* Instead of the top x bits being dropped they're appended to the shifted bit string.
|
|
|
2882 |
*
|
|
|
2883 |
* @param Integer $shift
|
|
|
2884 |
* @return Math_BigInteger
|
|
|
2885 |
* @access public
|
|
|
2886 |
*/
|
|
|
2887 |
function bitwise_leftRotate($shift)
|
|
|
2888 |
{
|
|
|
2889 |
$bits = $this->toBytes();
|
|
|
2890 |
|
|
|
2891 |
if ($this->precision > 0) {
|
|
|
2892 |
$precision = $this->precision;
|
|
|
2893 |
if ( MATH_BIGINTEGER_MODE == MATH_BIGINTEGER_MODE_BCMATH ) {
|
|
|
2894 |
$mask = $this->bitmask->subtract(new Math_BigInteger(1));
|
|
|
2895 |
$mask = $mask->toBytes();
|
|
|
2896 |
} else {
|
|
|
2897 |
$mask = $this->bitmask->toBytes();
|
|
|
2898 |
}
|
|
|
2899 |
} else {
|
|
|
2900 |
$temp = ord($bits[0]);
|
|
|
2901 |
for ($i = 0; $temp >> $i; ++$i);
|
|
|
2902 |
$precision = 8 * strlen($bits) - 8 + $i;
|
|
|
2903 |
$mask = chr((1 << ($precision & 0x7)) - 1) . str_repeat(chr(0xFF), $precision >> 3);
|
|
|
2904 |
}
|
|
|
2905 |
|
|
|
2906 |
if ($shift < 0) {
|
|
|
2907 |
$shift+= $precision;
|
|
|
2908 |
}
|
|
|
2909 |
$shift%= $precision;
|
|
|
2910 |
|
|
|
2911 |
if (!$shift) {
|
|
|
2912 |
return $this->copy();
|
|
|
2913 |
}
|
|
|
2914 |
|
|
|
2915 |
$left = $this->bitwise_leftShift($shift);
|
|
|
2916 |
$left = $left->bitwise_and(new Math_BigInteger($mask, 256));
|
|
|
2917 |
$right = $this->bitwise_rightShift($precision - $shift);
|
|
|
2918 |
$result = MATH_BIGINTEGER_MODE != MATH_BIGINTEGER_MODE_BCMATH ? $left->bitwise_or($right) : $left->add($right);
|
|
|
2919 |
return $this->_normalize($result);
|
|
|
2920 |
}
|
|
|
2921 |
|
|
|
2922 |
/**
|
|
|
2923 |
* Logical Right Rotate
|
|
|
2924 |
*
|
|
|
2925 |
* Instead of the bottom x bits being dropped they're prepended to the shifted bit string.
|
|
|
2926 |
*
|
|
|
2927 |
* @param Integer $shift
|
|
|
2928 |
* @return Math_BigInteger
|
|
|
2929 |
* @access public
|
|
|
2930 |
*/
|
|
|
2931 |
function bitwise_rightRotate($shift)
|
|
|
2932 |
{
|
|
|
2933 |
return $this->bitwise_leftRotate(-$shift);
|
|
|
2934 |
}
|
|
|
2935 |
|
|
|
2936 |
/**
|
|
|
2937 |
* Set random number generator function
|
|
|
2938 |
*
|
|
|
2939 |
* $generator should be the name of a random generating function whose first parameter is the minimum
|
|
|
2940 |
* value and whose second parameter is the maximum value. If this function needs to be seeded, it should
|
|
|
2941 |
* be seeded prior to calling Math_BigInteger::random() or Math_BigInteger::randomPrime()
|
|
|
2942 |
*
|
|
|
2943 |
* If the random generating function is not explicitly set, it'll be assumed to be mt_rand().
|
|
|
2944 |
*
|
|
|
2945 |
* @see random()
|
|
|
2946 |
* @see randomPrime()
|
|
|
2947 |
* @param optional String $generator
|
|
|
2948 |
* @access public
|
|
|
2949 |
*/
|
|
|
2950 |
function setRandomGenerator($generator)
|
|
|
2951 |
{
|
|
|
2952 |
$this->generator = $generator;
|
|
|
2953 |
}
|
|
|
2954 |
|
|
|
2955 |
/**
|
|
|
2956 |
* Generate a random number
|
|
|
2957 |
*
|
|
|
2958 |
* @param optional Integer $min
|
|
|
2959 |
* @param optional Integer $max
|
|
|
2960 |
* @return Math_BigInteger
|
|
|
2961 |
* @access public
|
|
|
2962 |
*/
|
|
|
2963 |
function random($min = false, $max = false)
|
|
|
2964 |
{
|
|
|
2965 |
if ($min === false) {
|
|
|
2966 |
$min = new Math_BigInteger(0);
|
|
|
2967 |
}
|
|
|
2968 |
|
|
|
2969 |
if ($max === false) {
|
|
|
2970 |
$max = new Math_BigInteger(0x7FFFFFFF);
|
|
|
2971 |
}
|
|
|
2972 |
|
|
|
2973 |
$compare = $max->compare($min);
|
|
|
2974 |
|
|
|
2975 |
if (!$compare) {
|
|
|
2976 |
return $this->_normalize($min);
|
|
|
2977 |
} else if ($compare < 0) {
|
|
|
2978 |
// if $min is bigger then $max, swap $min and $max
|
|
|
2979 |
$temp = $max;
|
|
|
2980 |
$max = $min;
|
|
|
2981 |
$min = $temp;
|
|
|
2982 |
}
|
|
|
2983 |
|
|
|
2984 |
$generator = $this->generator;
|
|
|
2985 |
|
|
|
2986 |
$max = $max->subtract($min);
|
|
|
2987 |
$max = ltrim($max->toBytes(), chr(0));
|
|
|
2988 |
$size = strlen($max) - 1;
|
|
|
2989 |
$random = '';
|
|
|
2990 |
|
|
|
2991 |
$bytes = $size & 1;
|
|
|
2992 |
for ($i = 0; $i < $bytes; ++$i) {
|
|
|
2993 |
$random.= chr($generator(0, 255));
|
|
|
2994 |
}
|
|
|
2995 |
|
|
|
2996 |
$blocks = $size >> 1;
|
|
|
2997 |
for ($i = 0; $i < $blocks; ++$i) {
|
|
|
2998 |
// mt_rand(-2147483648, 0x7FFFFFFF) always produces -2147483648 on some systems
|
|
|
2999 |
$random.= pack('n', $generator(0, 0xFFFF));
|
|
|
3000 |
}
|
|
|
3001 |
|
|
|
3002 |
$temp = new Math_BigInteger($random, 256);
|
|
|
3003 |
if ($temp->compare(new Math_BigInteger(substr($max, 1), 256)) > 0) {
|
|
|
3004 |
$random = chr($generator(0, ord($max[0]) - 1)) . $random;
|
|
|
3005 |
} else {
|
|
|
3006 |
$random = chr($generator(0, ord($max[0]) )) . $random;
|
|
|
3007 |
}
|
|
|
3008 |
|
|
|
3009 |
$random = new Math_BigInteger($random, 256);
|
|
|
3010 |
|
|
|
3011 |
return $this->_normalize($random->add($min));
|
|
|
3012 |
}
|
|
|
3013 |
|
|
|
3014 |
/**
|
|
|
3015 |
* Generate a random prime number.
|
|
|
3016 |
*
|
|
|
3017 |
* If there's not a prime within the given range, false will be returned. If more than $timeout seconds have elapsed,
|
|
|
3018 |
* give up and return false.
|
|
|
3019 |
*
|
|
|
3020 |
* @param optional Integer $min
|
|
|
3021 |
* @param optional Integer $max
|
|
|
3022 |
* @param optional Integer $timeout
|
|
|
3023 |
* @return Math_BigInteger
|
|
|
3024 |
* @access public
|
|
|
3025 |
* @internal See {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap4.pdf#page=15 HAC 4.44}.
|
|
|
3026 |
*/
|
|
|
3027 |
function randomPrime($min = false, $max = false, $timeout = false)
|
|
|
3028 |
{
|
|
|
3029 |
$compare = $max->compare($min);
|
|
|
3030 |
|
|
|
3031 |
if (!$compare) {
|
|
|
3032 |
return $min;
|
|
|
3033 |
} else if ($compare < 0) {
|
|
|
3034 |
// if $min is bigger then $max, swap $min and $max
|
|
|
3035 |
$temp = $max;
|
|
|
3036 |
$max = $min;
|
|
|
3037 |
$min = $temp;
|
|
|
3038 |
}
|
|
|
3039 |
|
|
|
3040 |
// gmp_nextprime() requires PHP 5 >= 5.2.0 per <http://php.net/gmp-nextprime>.
|
|
|
3041 |
if ( MATH_BIGINTEGER_MODE == MATH_BIGINTEGER_MODE_GMP && function_exists('gmp_nextprime') ) {
|
|
|
3042 |
// we don't rely on Math_BigInteger::random()'s min / max when gmp_nextprime() is being used since this function
|
|
|
3043 |
// does its own checks on $max / $min when gmp_nextprime() is used. When gmp_nextprime() is not used, however,
|
|
|
3044 |
// the same $max / $min checks are not performed.
|
|
|
3045 |
if ($min === false) {
|
|
|
3046 |
$min = new Math_BigInteger(0);
|
|
|
3047 |
}
|
|
|
3048 |
|
|
|
3049 |
if ($max === false) {
|
|
|
3050 |
$max = new Math_BigInteger(0x7FFFFFFF);
|
|
|
3051 |
}
|
|
|
3052 |
|
|
|
3053 |
$x = $this->random($min, $max);
|
|
|
3054 |
|
|
|
3055 |
$x->value = gmp_nextprime($x->value);
|
|
|
3056 |
|
|
|
3057 |
if ($x->compare($max) <= 0) {
|
|
|
3058 |
return $x;
|
|
|
3059 |
}
|
|
|
3060 |
|
|
|
3061 |
$x->value = gmp_nextprime($min->value);
|
|
|
3062 |
|
|
|
3063 |
if ($x->compare($max) <= 0) {
|
|
|
3064 |
return $x;
|
|
|
3065 |
}
|
|
|
3066 |
|
|
|
3067 |
return false;
|
|
|
3068 |
}
|
|
|
3069 |
|
|
|
3070 |
static $one, $two;
|
|
|
3071 |
if (!isset($one)) {
|
|
|
3072 |
$one = new Math_BigInteger(1);
|
|
|
3073 |
$two = new Math_BigInteger(2);
|
|
|
3074 |
}
|
|
|
3075 |
|
|
|
3076 |
$start = time();
|
|
|
3077 |
|
|
|
3078 |
$x = $this->random($min, $max);
|
|
|
3079 |
if ($x->equals($two)) {
|
|
|
3080 |
return $x;
|
|
|
3081 |
}
|
|
|
3082 |
|
|
|
3083 |
$x->_make_odd();
|
|
|
3084 |
if ($x->compare($max) > 0) {
|
|
|
3085 |
// if $x > $max then $max is even and if $min == $max then no prime number exists between the specified range
|
|
|
3086 |
if ($min->equals($max)) {
|
|
|
3087 |
return false;
|
|
|
3088 |
}
|
|
|
3089 |
$x = $min->copy();
|
|
|
3090 |
$x->_make_odd();
|
|
|
3091 |
}
|
|
|
3092 |
|
|
|
3093 |
$initial_x = $x->copy();
|
|
|
3094 |
|
|
|
3095 |
while (true) {
|
|
|
3096 |
if ($timeout !== false && time() - $start > $timeout) {
|
|
|
3097 |
return false;
|
|
|
3098 |
}
|
|
|
3099 |
|
|
|
3100 |
if ($x->isPrime()) {
|
|
|
3101 |
return $x;
|
|
|
3102 |
}
|
|
|
3103 |
|
|
|
3104 |
$x = $x->add($two);
|
|
|
3105 |
|
|
|
3106 |
if ($x->compare($max) > 0) {
|
|
|
3107 |
$x = $min->copy();
|
|
|
3108 |
if ($x->equals($two)) {
|
|
|
3109 |
return $x;
|
|
|
3110 |
}
|
|
|
3111 |
$x->_make_odd();
|
|
|
3112 |
}
|
|
|
3113 |
|
|
|
3114 |
if ($x->equals($initial_x)) {
|
|
|
3115 |
return false;
|
|
|
3116 |
}
|
|
|
3117 |
}
|
|
|
3118 |
}
|
|
|
3119 |
|
|
|
3120 |
/**
|
|
|
3121 |
* Make the current number odd
|
|
|
3122 |
*
|
|
|
3123 |
* If the current number is odd it'll be unchanged. If it's even, one will be added to it.
|
|
|
3124 |
*
|
|
|
3125 |
* @see randomPrime()
|
|
|
3126 |
* @access private
|
|
|
3127 |
*/
|
|
|
3128 |
function _make_odd()
|
|
|
3129 |
{
|
|
|
3130 |
switch ( MATH_BIGINTEGER_MODE ) {
|
|
|
3131 |
case MATH_BIGINTEGER_MODE_GMP:
|
|
|
3132 |
gmp_setbit($this->value, 0);
|
|
|
3133 |
break;
|
|
|
3134 |
case MATH_BIGINTEGER_MODE_BCMATH:
|
|
|
3135 |
if ($this->value[strlen($this->value) - 1] % 2 == 0) {
|
|
|
3136 |
$this->value = bcadd($this->value, '1');
|
|
|
3137 |
}
|
|
|
3138 |
break;
|
|
|
3139 |
default:
|
|
|
3140 |
$this->value[0] |= 1;
|
|
|
3141 |
}
|
|
|
3142 |
}
|
|
|
3143 |
|
|
|
3144 |
/**
|
|
|
3145 |
* Checks a numer to see if it's prime
|
|
|
3146 |
*
|
|
|
3147 |
* Assuming the $t parameter is not set, this function has an error rate of 2**-80. The main motivation for the
|
|
|
3148 |
* $t parameter is distributability. Math_BigInteger::randomPrime() can be distributed accross multiple pageloads
|
|
|
3149 |
* on a website instead of just one.
|
|
|
3150 |
*
|
|
|
3151 |
* @param optional Integer $t
|
|
|
3152 |
* @return Boolean
|
|
|
3153 |
* @access public
|
|
|
3154 |
* @internal Uses the
|
|
|
3155 |
* {@link http://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test Miller-Rabin primality test}. See
|
|
|
3156 |
* {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap4.pdf#page=8 HAC 4.24}.
|
|
|
3157 |
*/
|
|
|
3158 |
function isPrime($t = false)
|
|
|
3159 |
{
|
|
|
3160 |
$length = strlen($this->toBytes());
|
|
|
3161 |
|
|
|
3162 |
if (!$t) {
|
|
|
3163 |
// see HAC 4.49 "Note (controlling the error probability)"
|
|
|
3164 |
if ($length >= 163) { $t = 2; } // floor(1300 / 8)
|
|
|
3165 |
else if ($length >= 106) { $t = 3; } // floor( 850 / 8)
|
|
|
3166 |
else if ($length >= 81 ) { $t = 4; } // floor( 650 / 8)
|
|
|
3167 |
else if ($length >= 68 ) { $t = 5; } // floor( 550 / 8)
|
|
|
3168 |
else if ($length >= 56 ) { $t = 6; } // floor( 450 / 8)
|
|
|
3169 |
else if ($length >= 50 ) { $t = 7; } // floor( 400 / 8)
|
|
|
3170 |
else if ($length >= 43 ) { $t = 8; } // floor( 350 / 8)
|
|
|
3171 |
else if ($length >= 37 ) { $t = 9; } // floor( 300 / 8)
|
|
|
3172 |
else if ($length >= 31 ) { $t = 12; } // floor( 250 / 8)
|
|
|
3173 |
else if ($length >= 25 ) { $t = 15; } // floor( 200 / 8)
|
|
|
3174 |
else if ($length >= 18 ) { $t = 18; } // floor( 150 / 8)
|
|
|
3175 |
else { $t = 27; }
|
|
|
3176 |
}
|
|
|
3177 |
|
|
|
3178 |
// ie. gmp_testbit($this, 0)
|
|
|
3179 |
// ie. isEven() or !isOdd()
|
|
|
3180 |
switch ( MATH_BIGINTEGER_MODE ) {
|
|
|
3181 |
case MATH_BIGINTEGER_MODE_GMP:
|
|
|
3182 |
return gmp_prob_prime($this->value, $t) != 0;
|
|
|
3183 |
case MATH_BIGINTEGER_MODE_BCMATH:
|
|
|
3184 |
if ($this->value === '2') {
|
|
|
3185 |
return true;
|
|
|
3186 |
}
|
|
|
3187 |
if ($this->value[strlen($this->value) - 1] % 2 == 0) {
|
|
|
3188 |
return false;
|
|
|
3189 |
}
|
|
|
3190 |
break;
|
|
|
3191 |
default:
|
|
|
3192 |
if ($this->value == array(2)) {
|
|
|
3193 |
return true;
|
|
|
3194 |
}
|
|
|
3195 |
if (~$this->value[0] & 1) {
|
|
|
3196 |
return false;
|
|
|
3197 |
}
|
|
|
3198 |
}
|
|
|
3199 |
|
|
|
3200 |
static $primes, $zero, $one, $two;
|
|
|
3201 |
|
|
|
3202 |
if (!isset($primes)) {
|
|
|
3203 |
$primes = array(
|
|
|
3204 |
3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59,
|
|
|
3205 |
61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137,
|
|
|
3206 |
139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227,
|
|
|
3207 |
229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313,
|
|
|
3208 |
317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419,
|
|
|
3209 |
421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509,
|
|
|
3210 |
521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617,
|
|
|
3211 |
619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727,
|
|
|
3212 |
733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829,
|
|
|
3213 |
839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947,
|
|
|
3214 |
953, 967, 971, 977, 983, 991, 997
|
|
|
3215 |
);
|
|
|
3216 |
|
|
|
3217 |
if ( MATH_BIGINTEGER_MODE != MATH_BIGINTEGER_MODE_INTERNAL ) {
|
|
|
3218 |
for ($i = 0; $i < count($primes); ++$i) {
|
|
|
3219 |
$primes[$i] = new Math_BigInteger($primes[$i]);
|
|
|
3220 |
}
|
|
|
3221 |
}
|
|
|
3222 |
|
|
|
3223 |
$zero = new Math_BigInteger();
|
|
|
3224 |
$one = new Math_BigInteger(1);
|
|
|
3225 |
$two = new Math_BigInteger(2);
|
|
|
3226 |
}
|
|
|
3227 |
|
|
|
3228 |
if ($this->equals($one)) {
|
|
|
3229 |
return false;
|
|
|
3230 |
}
|
|
|
3231 |
|
|
|
3232 |
// see HAC 4.4.1 "Random search for probable primes"
|
|
|
3233 |
if ( MATH_BIGINTEGER_MODE != MATH_BIGINTEGER_MODE_INTERNAL ) {
|
|
|
3234 |
foreach ($primes as $prime) {
|
|
|
3235 |
list(, $r) = $this->divide($prime);
|
|
|
3236 |
if ($r->equals($zero)) {
|
|
|
3237 |
return $this->equals($prime);
|
|
|
3238 |
}
|
|
|
3239 |
}
|
|
|
3240 |
} else {
|
|
|
3241 |
$value = $this->value;
|
|
|
3242 |
foreach ($primes as $prime) {
|
|
|
3243 |
list(, $r) = $this->_divide_digit($value, $prime);
|
|
|
3244 |
if (!$r) {
|
|
|
3245 |
return count($value) == 1 && $value[0] == $prime;
|
|
|
3246 |
}
|
|
|
3247 |
}
|
|
|
3248 |
}
|
|
|
3249 |
|
|
|
3250 |
$n = $this->copy();
|
|
|
3251 |
$n_1 = $n->subtract($one);
|
|
|
3252 |
$n_2 = $n->subtract($two);
|
|
|
3253 |
|
|
|
3254 |
$r = $n_1->copy();
|
|
|
3255 |
$r_value = $r->value;
|
|
|
3256 |
// ie. $s = gmp_scan1($n, 0) and $r = gmp_div_q($n, gmp_pow(gmp_init('2'), $s));
|
|
|
3257 |
if ( MATH_BIGINTEGER_MODE == MATH_BIGINTEGER_MODE_BCMATH ) {
|
|
|
3258 |
$s = 0;
|
|
|
3259 |
// if $n was 1, $r would be 0 and this would be an infinite loop, hence our $this->equals($one) check earlier
|
|
|
3260 |
while ($r->value[strlen($r->value) - 1] % 2 == 0) {
|
|
|
3261 |
$r->value = bcdiv($r->value, '2', 0);
|
|
|
3262 |
++$s;
|
|
|
3263 |
}
|
|
|
3264 |
} else {
|
|
|
3265 |
for ($i = 0, $r_length = count($r_value); $i < $r_length; ++$i) {
|
|
|
3266 |
$temp = ~$r_value[$i] & 0xFFFFFF;
|
|
|
3267 |
for ($j = 1; ($temp >> $j) & 1; ++$j);
|
|
|
3268 |
if ($j != 25) {
|
|
|
3269 |
break;
|
|
|
3270 |
}
|
|
|
3271 |
}
|
|
|
3272 |
$s = 26 * $i + $j - 1;
|
|
|
3273 |
$r->_rshift($s);
|
|
|
3274 |
}
|
|
|
3275 |
|
|
|
3276 |
for ($i = 0; $i < $t; ++$i) {
|
|
|
3277 |
$a = $this->random($two, $n_2);
|
|
|
3278 |
$y = $a->modPow($r, $n);
|
|
|
3279 |
|
|
|
3280 |
if (!$y->equals($one) && !$y->equals($n_1)) {
|
|
|
3281 |
for ($j = 1; $j < $s && !$y->equals($n_1); ++$j) {
|
|
|
3282 |
$y = $y->modPow($two, $n);
|
|
|
3283 |
if ($y->equals($one)) {
|
|
|
3284 |
return false;
|
|
|
3285 |
}
|
|
|
3286 |
}
|
|
|
3287 |
|
|
|
3288 |
if (!$y->equals($n_1)) {
|
|
|
3289 |
return false;
|
|
|
3290 |
}
|
|
|
3291 |
}
|
|
|
3292 |
}
|
|
|
3293 |
return true;
|
|
|
3294 |
}
|
|
|
3295 |
|
|
|
3296 |
/**
|
|
|
3297 |
* Logical Left Shift
|
|
|
3298 |
*
|
|
|
3299 |
* Shifts BigInteger's by $shift bits.
|
|
|
3300 |
*
|
|
|
3301 |
* @param Integer $shift
|
|
|
3302 |
* @access private
|
|
|
3303 |
*/
|
|
|
3304 |
function _lshift($shift)
|
|
|
3305 |
{
|
|
|
3306 |
if ( $shift == 0 ) {
|
|
|
3307 |
return;
|
|
|
3308 |
}
|
|
|
3309 |
|
|
|
3310 |
$num_digits = (int) ($shift / 26);
|
|
|
3311 |
$shift %= 26;
|
|
|
3312 |
$shift = 1 << $shift;
|
|
|
3313 |
|
|
|
3314 |
$carry = 0;
|
|
|
3315 |
|
|
|
3316 |
for ($i = 0; $i < count($this->value); ++$i) {
|
|
|
3317 |
$temp = $this->value[$i] * $shift + $carry;
|
|
|
3318 |
$carry = (int) ($temp / 0x4000000);
|
|
|
3319 |
$this->value[$i] = (int) ($temp - $carry * 0x4000000);
|
|
|
3320 |
}
|
|
|
3321 |
|
|
|
3322 |
if ( $carry ) {
|
|
|
3323 |
$this->value[] = $carry;
|
|
|
3324 |
}
|
|
|
3325 |
|
|
|
3326 |
while ($num_digits--) {
|
|
|
3327 |
array_unshift($this->value, 0);
|
|
|
3328 |
}
|
|
|
3329 |
}
|
|
|
3330 |
|
|
|
3331 |
/**
|
|
|
3332 |
* Logical Right Shift
|
|
|
3333 |
*
|
|
|
3334 |
* Shifts BigInteger's by $shift bits.
|
|
|
3335 |
*
|
|
|
3336 |
* @param Integer $shift
|
|
|
3337 |
* @access private
|
|
|
3338 |
*/
|
|
|
3339 |
function _rshift($shift)
|
|
|
3340 |
{
|
|
|
3341 |
if ($shift == 0) {
|
|
|
3342 |
return;
|
|
|
3343 |
}
|
|
|
3344 |
|
|
|
3345 |
$num_digits = (int) ($shift / 26);
|
|
|
3346 |
$shift %= 26;
|
|
|
3347 |
$carry_shift = 26 - $shift;
|
|
|
3348 |
$carry_mask = (1 << $shift) - 1;
|
|
|
3349 |
|
|
|
3350 |
if ( $num_digits ) {
|
|
|
3351 |
$this->value = array_slice($this->value, $num_digits);
|
|
|
3352 |
}
|
|
|
3353 |
|
|
|
3354 |
$carry = 0;
|
|
|
3355 |
|
|
|
3356 |
for ($i = count($this->value) - 1; $i >= 0; --$i) {
|
|
|
3357 |
$temp = $this->value[$i] >> $shift | $carry;
|
|
|
3358 |
$carry = ($this->value[$i] & $carry_mask) << $carry_shift;
|
|
|
3359 |
$this->value[$i] = $temp;
|
|
|
3360 |
}
|
|
|
3361 |
|
|
|
3362 |
$this->value = $this->_trim($this->value);
|
|
|
3363 |
}
|
|
|
3364 |
|
|
|
3365 |
/**
|
|
|
3366 |
* Normalize
|
|
|
3367 |
*
|
|
|
3368 |
* Removes leading zeros and truncates (if necessary) to maintain the appropriate precision
|
|
|
3369 |
*
|
|
|
3370 |
* @param Math_BigInteger
|
|
|
3371 |
* @return Math_BigInteger
|
|
|
3372 |
* @see _trim()
|
|
|
3373 |
* @access private
|
|
|
3374 |
*/
|
|
|
3375 |
function _normalize($result)
|
|
|
3376 |
{
|
|
|
3377 |
$result->precision = $this->precision;
|
|
|
3378 |
$result->bitmask = $this->bitmask;
|
|
|
3379 |
|
|
|
3380 |
switch ( MATH_BIGINTEGER_MODE ) {
|
|
|
3381 |
case MATH_BIGINTEGER_MODE_GMP:
|
|
|
3382 |
if (!empty($result->bitmask->value)) {
|
|
|
3383 |
$result->value = gmp_and($result->value, $result->bitmask->value);
|
|
|
3384 |
}
|
|
|
3385 |
|
|
|
3386 |
return $result;
|
|
|
3387 |
case MATH_BIGINTEGER_MODE_BCMATH:
|
|
|
3388 |
if (!empty($result->bitmask->value)) {
|
|
|
3389 |
$result->value = bcmod($result->value, $result->bitmask->value);
|
|
|
3390 |
}
|
|
|
3391 |
|
|
|
3392 |
return $result;
|
|
|
3393 |
}
|
|
|
3394 |
|
|
|
3395 |
$value = &$result->value;
|
|
|
3396 |
|
|
|
3397 |
if ( !count($value) ) {
|
|
|
3398 |
return $result;
|
|
|
3399 |
}
|
|
|
3400 |
|
|
|
3401 |
$value = $this->_trim($value);
|
|
|
3402 |
|
|
|
3403 |
if (!empty($result->bitmask->value)) {
|
|
|
3404 |
$length = min(count($value), count($this->bitmask->value));
|
|
|
3405 |
$value = array_slice($value, 0, $length);
|
|
|
3406 |
|
|
|
3407 |
for ($i = 0; $i < $length; ++$i) {
|
|
|
3408 |
$value[$i] = $value[$i] & $this->bitmask->value[$i];
|
|
|
3409 |
}
|
|
|
3410 |
}
|
|
|
3411 |
|
|
|
3412 |
return $result;
|
|
|
3413 |
}
|
|
|
3414 |
|
|
|
3415 |
/**
|
|
|
3416 |
* Trim
|
|
|
3417 |
*
|
|
|
3418 |
* Removes leading zeros
|
|
|
3419 |
*
|
|
|
3420 |
* @return Math_BigInteger
|
|
|
3421 |
* @access private
|
|
|
3422 |
*/
|
|
|
3423 |
function _trim($value)
|
|
|
3424 |
{
|
|
|
3425 |
for ($i = count($value) - 1; $i >= 0; --$i) {
|
|
|
3426 |
if ( $value[$i] ) {
|
|
|
3427 |
break;
|
|
|
3428 |
}
|
|
|
3429 |
unset($value[$i]);
|
|
|
3430 |
}
|
|
|
3431 |
|
|
|
3432 |
return $value;
|
|
|
3433 |
}
|
|
|
3434 |
|
|
|
3435 |
/**
|
|
|
3436 |
* Array Repeat
|
|
|
3437 |
*
|
|
|
3438 |
* @param $input Array
|
|
|
3439 |
* @param $multiplier mixed
|
|
|
3440 |
* @return Array
|
|
|
3441 |
* @access private
|
|
|
3442 |
*/
|
|
|
3443 |
function _array_repeat($input, $multiplier)
|
|
|
3444 |
{
|
|
|
3445 |
return ($multiplier) ? array_fill(0, $multiplier, $input) : array();
|
|
|
3446 |
}
|
|
|
3447 |
|
|
|
3448 |
/**
|
|
|
3449 |
* Logical Left Shift
|
|
|
3450 |
*
|
|
|
3451 |
* Shifts binary strings $shift bits, essentially multiplying by 2**$shift.
|
|
|
3452 |
*
|
|
|
3453 |
* @param $x String
|
|
|
3454 |
* @param $shift Integer
|
|
|
3455 |
* @return String
|
|
|
3456 |
* @access private
|
|
|
3457 |
*/
|
|
|
3458 |
function _base256_lshift(&$x, $shift)
|
|
|
3459 |
{
|
|
|
3460 |
if ($shift == 0) {
|
|
|
3461 |
return;
|
|
|
3462 |
}
|
|
|
3463 |
|
|
|
3464 |
$num_bytes = $shift >> 3; // eg. floor($shift/8)
|
|
|
3465 |
$shift &= 7; // eg. $shift % 8
|
|
|
3466 |
|
|
|
3467 |
$carry = 0;
|
|
|
3468 |
for ($i = strlen($x) - 1; $i >= 0; --$i) {
|
|
|
3469 |
$temp = ord($x[$i]) << $shift | $carry;
|
|
|
3470 |
$x[$i] = chr($temp);
|
|
|
3471 |
$carry = $temp >> 8;
|
|
|
3472 |
}
|
|
|
3473 |
$carry = ($carry != 0) ? chr($carry) : '';
|
|
|
3474 |
$x = $carry . $x . str_repeat(chr(0), $num_bytes);
|
|
|
3475 |
}
|
|
|
3476 |
|
|
|
3477 |
/**
|
|
|
3478 |
* Logical Right Shift
|
|
|
3479 |
*
|
|
|
3480 |
* Shifts binary strings $shift bits, essentially dividing by 2**$shift and returning the remainder.
|
|
|
3481 |
*
|
|
|
3482 |
* @param $x String
|
|
|
3483 |
* @param $shift Integer
|
|
|
3484 |
* @return String
|
|
|
3485 |
* @access private
|
|
|
3486 |
*/
|
|
|
3487 |
function _base256_rshift(&$x, $shift)
|
|
|
3488 |
{
|
|
|
3489 |
if ($shift == 0) {
|
|
|
3490 |
$x = ltrim($x, chr(0));
|
|
|
3491 |
return '';
|
|
|
3492 |
}
|
|
|
3493 |
|
|
|
3494 |
$num_bytes = $shift >> 3; // eg. floor($shift/8)
|
|
|
3495 |
$shift &= 7; // eg. $shift % 8
|
|
|
3496 |
|
|
|
3497 |
$remainder = '';
|
|
|
3498 |
if ($num_bytes) {
|
|
|
3499 |
$start = $num_bytes > strlen($x) ? -strlen($x) : -$num_bytes;
|
|
|
3500 |
$remainder = substr($x, $start);
|
|
|
3501 |
$x = substr($x, 0, -$num_bytes);
|
|
|
3502 |
}
|
|
|
3503 |
|
|
|
3504 |
$carry = 0;
|
|
|
3505 |
$carry_shift = 8 - $shift;
|
|
|
3506 |
for ($i = 0; $i < strlen($x); ++$i) {
|
|
|
3507 |
$temp = (ord($x[$i]) >> $shift) | $carry;
|
|
|
3508 |
$carry = (ord($x[$i]) << $carry_shift) & 0xFF;
|
|
|
3509 |
$x[$i] = chr($temp);
|
|
|
3510 |
}
|
|
|
3511 |
$x = ltrim($x, chr(0));
|
|
|
3512 |
|
|
|
3513 |
$remainder = chr($carry >> $carry_shift) . $remainder;
|
|
|
3514 |
|
|
|
3515 |
return ltrim($remainder, chr(0));
|
|
|
3516 |
}
|
|
|
3517 |
|
|
|
3518 |
// one quirk about how the following functions are implemented is that PHP defines N to be an unsigned long
|
|
|
3519 |
// at 32-bits, while java's longs are 64-bits.
|
|
|
3520 |
|
|
|
3521 |
/**
|
|
|
3522 |
* Converts 32-bit integers to bytes.
|
|
|
3523 |
*
|
|
|
3524 |
* @param Integer $x
|
|
|
3525 |
* @return String
|
|
|
3526 |
* @access private
|
|
|
3527 |
*/
|
|
|
3528 |
function _int2bytes($x)
|
|
|
3529 |
{
|
|
|
3530 |
return ltrim(pack('N', $x), chr(0));
|
|
|
3531 |
}
|
|
|
3532 |
|
|
|
3533 |
/**
|
|
|
3534 |
* Converts bytes to 32-bit integers
|
|
|
3535 |
*
|
|
|
3536 |
* @param String $x
|
|
|
3537 |
* @return Integer
|
|
|
3538 |
* @access private
|
|
|
3539 |
*/
|
|
|
3540 |
function _bytes2int($x)
|
|
|
3541 |
{
|
|
|
3542 |
$temp = unpack('Nint', str_pad($x, 4, chr(0), STR_PAD_LEFT));
|
|
|
3543 |
return $temp['int'];
|
|
|
3544 |
}
|
|
|
3545 |
}
|