diff options
Diffstat (limited to 'library/phpsec/Math/BigInteger.php')
| -rw-r--r-- | library/phpsec/Math/BigInteger.php | 3545 |
1 files changed, 0 insertions, 3545 deletions
diff --git a/library/phpsec/Math/BigInteger.php b/library/phpsec/Math/BigInteger.php deleted file mode 100644 index 5b3a4fc8b..000000000 --- a/library/phpsec/Math/BigInteger.php +++ /dev/null @@ -1,3545 +0,0 @@ -<?php
-/* vim: set expandtab tabstop=4 shiftwidth=4 softtabstop=4: */
-
-/**
- * Pure-PHP arbitrary precision integer arithmetic library.
- *
- * Supports base-2, base-10, base-16, and base-256 numbers. Uses the GMP or BCMath extensions, if available,
- * and an internal implementation, otherwise.
- *
- * PHP versions 4 and 5
- *
- * {@internal (all DocBlock comments regarding implementation - such as the one that follows - refer to the
- * {@link MATH_BIGINTEGER_MODE_INTERNAL MATH_BIGINTEGER_MODE_INTERNAL} mode)
- *
- * Math_BigInteger uses base-2**26 to perform operations such as multiplication and division and
- * base-2**52 (ie. two base 2**26 digits) to perform addition and subtraction. Because the largest possible
- * value when multiplying two base-2**26 numbers together is a base-2**52 number, double precision floating
- * point numbers - numbers that should be supported on most hardware and whose significand is 53 bits - are
- * used. As a consequence, bitwise operators such as >> and << cannot be used, nor can the modulo operator %,
- * which only supports integers. Although this fact will slow this library down, the fact that such a high
- * base is being used should more than compensate.
- *
- * When PHP version 6 is officially released, we'll be able to use 64-bit integers. This should, once again,
- * allow bitwise operators, and will increase the maximum possible base to 2**31 (or 2**62 for addition /
- * subtraction).
- *
- * Numbers are stored in {@link http://en.wikipedia.org/wiki/Endianness little endian} format. ie.
- * (new Math_BigInteger(pow(2, 26)))->value = array(0, 1)
- *
- * Useful resources are as follows:
- *
- * - {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf Handbook of Applied Cryptography (HAC)}
- * - {@link http://math.libtomcrypt.com/files/tommath.pdf Multi-Precision Math (MPM)}
- * - Java's BigInteger classes. See /j2se/src/share/classes/java/math in jdk-1_5_0-src-jrl.zip
- *
- * Here's an example of how to use this library:
- * <code>
- * <?php
- * include('Math/BigInteger.php');
- *
- * $a = new Math_BigInteger(2);
- * $b = new Math_BigInteger(3);
- *
- * $c = $a->add($b);
- *
- * echo $c->toString(); // outputs 5
- * ?>
- * </code>
- *
- * LICENSE: This library is free software; you can redistribute it and/or
- * modify it under the terms of the GNU Lesser General Public
- * License as published by the Free Software Foundation; either
- * version 2.1 of the License, or (at your option) any later version.
- *
- * This library is distributed in the hope that it will be useful,
- * but WITHOUT ANY WARRANTY; without even the implied warranty of
- * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
- * Lesser General Public License for more details.
- *
- * You should have received a copy of the GNU Lesser General Public
- * License along with this library; if not, write to the Free Software
- * Foundation, Inc., 59 Temple Place, Suite 330, Boston,
- * MA 02111-1307 USA
- *
- * @category Math
- * @package Math_BigInteger
- * @author Jim Wigginton <terrafrost@php.net>
- * @copyright MMVI Jim Wigginton
- * @license http://www.gnu.org/licenses/lgpl.txt
- * @version $Id: BigInteger.php,v 1.33 2010/03/22 22:32:03 terrafrost Exp $
- * @link http://pear.php.net/package/Math_BigInteger
- */
-
-/**#@+
- * Reduction constants
- *
- * @access private
- * @see Math_BigInteger::_reduce()
- */
-/**
- * @see Math_BigInteger::_montgomery()
- * @see Math_BigInteger::_prepMontgomery()
- */
-define('MATH_BIGINTEGER_MONTGOMERY', 0);
-/**
- * @see Math_BigInteger::_barrett()
- */
-define('MATH_BIGINTEGER_BARRETT', 1);
-/**
- * @see Math_BigInteger::_mod2()
- */
-define('MATH_BIGINTEGER_POWEROF2', 2);
-/**
- * @see Math_BigInteger::_remainder()
- */
-define('MATH_BIGINTEGER_CLASSIC', 3);
-/**
- * @see Math_BigInteger::__clone()
- */
-define('MATH_BIGINTEGER_NONE', 4);
-/**#@-*/
-
-/**#@+
- * Array constants
- *
- * Rather than create a thousands and thousands of new Math_BigInteger objects in repeated function calls to add() and
- * multiply() or whatever, we'll just work directly on arrays, taking them in as parameters and returning them.
- *
- * @access private
- */
-/**
- * $result[MATH_BIGINTEGER_VALUE] contains the value.
- */
-define('MATH_BIGINTEGER_VALUE', 0);
-/**
- * $result[MATH_BIGINTEGER_SIGN] contains the sign.
- */
-define('MATH_BIGINTEGER_SIGN', 1);
-/**#@-*/
-
-/**#@+
- * @access private
- * @see Math_BigInteger::_montgomery()
- * @see Math_BigInteger::_barrett()
- */
-/**
- * Cache constants
- *
- * $cache[MATH_BIGINTEGER_VARIABLE] tells us whether or not the cached data is still valid.
- */
-define('MATH_BIGINTEGER_VARIABLE', 0);
-/**
- * $cache[MATH_BIGINTEGER_DATA] contains the cached data.
- */
-define('MATH_BIGINTEGER_DATA', 1);
-/**#@-*/
-
-/**#@+
- * Mode constants.
- *
- * @access private
- * @see Math_BigInteger::Math_BigInteger()
- */
-/**
- * To use the pure-PHP implementation
- */
-define('MATH_BIGINTEGER_MODE_INTERNAL', 1);
-/**
- * To use the BCMath library
- *
- * (if enabled; otherwise, the internal implementation will be used)
- */
-define('MATH_BIGINTEGER_MODE_BCMATH', 2);
-/**
- * To use the GMP library
- *
- * (if present; otherwise, either the BCMath or the internal implementation will be used)
- */
-define('MATH_BIGINTEGER_MODE_GMP', 3);
-/**#@-*/
-
-/**
- * The largest digit that may be used in addition / subtraction
- *
- * (we do pow(2, 52) instead of using 4503599627370496, directly, because some PHP installations
- * will truncate 4503599627370496)
- *
- * @access private
- */
-define('MATH_BIGINTEGER_MAX_DIGIT52', pow(2, 52));
-
-/**
- * Karatsuba Cutoff
- *
- * At what point do we switch between Karatsuba multiplication and schoolbook long multiplication?
- *
- * @access private
- */
-define('MATH_BIGINTEGER_KARATSUBA_CUTOFF', 25);
-
-/**
- * Pure-PHP arbitrary precision integer arithmetic library. Supports base-2, base-10, base-16, and base-256
- * numbers.
- *
- * @author Jim Wigginton <terrafrost@php.net>
- * @version 1.0.0RC4
- * @access public
- * @package Math_BigInteger
- */
-class Math_BigInteger {
- /**
- * Holds the BigInteger's value.
- *
- * @var Array
- * @access private
- */
- var $value;
-
- /**
- * Holds the BigInteger's magnitude.
- *
- * @var Boolean
- * @access private
- */
- var $is_negative = false;
-
- /**
- * Random number generator function
- *
- * @see setRandomGenerator()
- * @access private
- */
- var $generator = 'mt_rand';
-
- /**
- * Precision
- *
- * @see setPrecision()
- * @access private
- */
- var $precision = -1;
-
- /**
- * Precision Bitmask
- *
- * @see setPrecision()
- * @access private
- */
- var $bitmask = false;
-
- /**
- * Mode independant value used for serialization.
- *
- * If the bcmath or gmp extensions are installed $this->value will be a non-serializable resource, hence the need for
- * a variable that'll be serializable regardless of whether or not extensions are being used. Unlike $this->value,
- * however, $this->hex is only calculated when $this->__sleep() is called.
- *
- * @see __sleep()
- * @see __wakeup()
- * @var String
- * @access private
- */
- var $hex;
-
- /**
- * Converts base-2, base-10, base-16, and binary strings (eg. base-256) to BigIntegers.
- *
- * If the second parameter - $base - is negative, then it will be assumed that the number's are encoded using
- * two's compliment. The sole exception to this is -10, which is treated the same as 10 is.
- *
- * Here's an example:
- * <code>
- * <?php
- * include('Math/BigInteger.php');
- *
- * $a = new Math_BigInteger('0x32', 16); // 50 in base-16
- *
- * echo $a->toString(); // outputs 50
- * ?>
- * </code>
- *
- * @param optional $x base-10 number or base-$base number if $base set.
- * @param optional integer $base
- * @return Math_BigInteger
- * @access public
- */
- function Math_BigInteger($x = 0, $base = 10)
- {
- if ( !defined('MATH_BIGINTEGER_MODE') ) {
- switch (true) {
- case extension_loaded('gmp'):
- define('MATH_BIGINTEGER_MODE', MATH_BIGINTEGER_MODE_GMP);
- break;
- case extension_loaded('bcmath'):
- define('MATH_BIGINTEGER_MODE', MATH_BIGINTEGER_MODE_BCMATH);
- break;
- default:
- define('MATH_BIGINTEGER_MODE', MATH_BIGINTEGER_MODE_INTERNAL);
- }
- }
-
- switch ( MATH_BIGINTEGER_MODE ) {
- case MATH_BIGINTEGER_MODE_GMP:
- if (is_resource($x) && get_resource_type($x) == 'GMP integer') {
- $this->value = $x;
- return;
- }
- $this->value = gmp_init(0);
- break;
- case MATH_BIGINTEGER_MODE_BCMATH:
- $this->value = '0';
- break;
- default:
- $this->value = array();
- }
-
- if (empty($x)) {
- return;
- }
-
- switch ($base) {
- case -256:
- if (ord($x[0]) & 0x80) {
- $x = ~$x;
- $this->is_negative = true;
- }
- case 256:
- switch ( MATH_BIGINTEGER_MODE ) {
- case MATH_BIGINTEGER_MODE_GMP:
- $sign = $this->is_negative ? '-' : '';
- $this->value = gmp_init($sign . '0x' . bin2hex($x));
- break;
- case MATH_BIGINTEGER_MODE_BCMATH:
- // round $len to the nearest 4 (thanks, DavidMJ!)
- $len = (strlen($x) + 3) & 0xFFFFFFFC;
-
- $x = str_pad($x, $len, chr(0), STR_PAD_LEFT);
-
- for ($i = 0; $i < $len; $i+= 4) {
- $this->value = bcmul($this->value, '4294967296', 0); // 4294967296 == 2**32
- $this->value = bcadd($this->value, 0x1000000 * ord($x[$i]) + ((ord($x[$i + 1]) << 16) | (ord($x[$i + 2]) << 8) | ord($x[$i + 3])), 0);
- }
-
- if ($this->is_negative) {
- $this->value = '-' . $this->value;
- }
-
- break;
- // converts a base-2**8 (big endian / msb) number to base-2**26 (little endian / lsb)
- default:
- while (strlen($x)) {
- $this->value[] = $this->_bytes2int($this->_base256_rshift($x, 26));
- }
- }
-
- if ($this->is_negative) {
- if (MATH_BIGINTEGER_MODE != MATH_BIGINTEGER_MODE_INTERNAL) {
- $this->is_negative = false;
- }
- $temp = $this->add(new Math_BigInteger('-1'));
- $this->value = $temp->value;
- }
- break;
- case 16:
- case -16:
- if ($base > 0 && $x[0] == '-') {
- $this->is_negative = true;
- $x = substr($x, 1);
- }
-
- $x = preg_replace('#^(?:0x)?([A-Fa-f0-9]*).*#', '$1', $x);
-
- $is_negative = false;
- if ($base < 0 && hexdec($x[0]) >= 8) {
- $this->is_negative = $is_negative = true;
- $x = bin2hex(~pack('H*', $x));
- }
-
- switch ( MATH_BIGINTEGER_MODE ) {
- case MATH_BIGINTEGER_MODE_GMP:
- $temp = $this->is_negative ? '-0x' . $x : '0x' . $x;
- $this->value = gmp_init($temp);
- $this->is_negative = false;
- break;
- case MATH_BIGINTEGER_MODE_BCMATH:
- $x = ( strlen($x) & 1 ) ? '0' . $x : $x;
- $temp = new Math_BigInteger(pack('H*', $x), 256);
- $this->value = $this->is_negative ? '-' . $temp->value : $temp->value;
- $this->is_negative = false;
- break;
- default:
- $x = ( strlen($x) & 1 ) ? '0' . $x : $x;
- $temp = new Math_BigInteger(pack('H*', $x), 256);
- $this->value = $temp->value;
- }
-
- if ($is_negative) {
- $temp = $this->add(new Math_BigInteger('-1'));
- $this->value = $temp->value;
- }
- break;
- case 10:
- case -10:
- $x = preg_replace('#^(-?[0-9]*).*#', '$1', $x);
-
- switch ( MATH_BIGINTEGER_MODE ) {
- case MATH_BIGINTEGER_MODE_GMP:
- $this->value = gmp_init($x);
- break;
- case MATH_BIGINTEGER_MODE_BCMATH:
- // explicitly casting $x to a string is necessary, here, since doing $x[0] on -1 yields different
- // results then doing it on '-1' does (modInverse does $x[0])
- $this->value = (string) $x;
- break;
- default:
- $temp = new Math_BigInteger();
-
- // array(10000000) is 10**7 in base-2**26. 10**7 is the closest to 2**26 we can get without passing it.
- $multiplier = new Math_BigInteger();
- $multiplier->value = array(10000000);
-
- if ($x[0] == '-') {
- $this->is_negative = true;
- $x = substr($x, 1);
- }
-
- $x = str_pad($x, strlen($x) + (6 * strlen($x)) % 7, 0, STR_PAD_LEFT);
-
- while (strlen($x)) {
- $temp = $temp->multiply($multiplier);
- $temp = $temp->add(new Math_BigInteger($this->_int2bytes(substr($x, 0, 7)), 256));
- $x = substr($x, 7);
- }
-
- $this->value = $temp->value;
- }
- break;
- case 2: // base-2 support originally implemented by Lluis Pamies - thanks!
- case -2:
- if ($base > 0 && $x[0] == '-') {
- $this->is_negative = true;
- $x = substr($x, 1);
- }
-
- $x = preg_replace('#^([01]*).*#', '$1', $x);
- $x = str_pad($x, strlen($x) + (3 * strlen($x)) % 4, 0, STR_PAD_LEFT);
-
- $str = '0x';
- while (strlen($x)) {
- $part = substr($x, 0, 4);
- $str.= dechex(bindec($part));
- $x = substr($x, 4);
- }
-
- if ($this->is_negative) {
- $str = '-' . $str;
- }
-
- $temp = new Math_BigInteger($str, 8 * $base); // ie. either -16 or +16
- $this->value = $temp->value;
- $this->is_negative = $temp->is_negative;
-
- break;
- default:
- // base not supported, so we'll let $this == 0
- }
- }
-
- /**
- * Converts a BigInteger to a byte string (eg. base-256).
- *
- * Negative numbers are saved as positive numbers, unless $twos_compliment is set to true, at which point, they're
- * saved as two's compliment.
- *
- * Here's an example:
- * <code>
- * <?php
- * include('Math/BigInteger.php');
- *
- * $a = new Math_BigInteger('65');
- *
- * echo $a->toBytes(); // outputs chr(65)
- * ?>
- * </code>
- *
- * @param Boolean $twos_compliment
- * @return String
- * @access public
- * @internal Converts a base-2**26 number to base-2**8
- */
- function toBytes($twos_compliment = false)
- {
- if ($twos_compliment) {
- $comparison = $this->compare(new Math_BigInteger());
- if ($comparison == 0) {
- return $this->precision > 0 ? str_repeat(chr(0), ($this->precision + 1) >> 3) : '';
- }
-
- $temp = $comparison < 0 ? $this->add(new Math_BigInteger(1)) : $this->copy();
- $bytes = $temp->toBytes();
-
- if (empty($bytes)) { // eg. if the number we're trying to convert is -1
- $bytes = chr(0);
- }
-
- if (ord($bytes[0]) & 0x80) {
- $bytes = chr(0) . $bytes;
- }
-
- return $comparison < 0 ? ~$bytes : $bytes;
- }
-
- switch ( MATH_BIGINTEGER_MODE ) {
- case MATH_BIGINTEGER_MODE_GMP:
- if (gmp_cmp($this->value, gmp_init(0)) == 0) {
- return $this->precision > 0 ? str_repeat(chr(0), ($this->precision + 1) >> 3) : '';
- }
-
- $temp = gmp_strval(gmp_abs($this->value), 16);
- $temp = ( strlen($temp) & 1 ) ? '0' . $temp : $temp;
- $temp = pack('H*', $temp);
-
- return $this->precision > 0 ?
- substr(str_pad($temp, $this->precision >> 3, chr(0), STR_PAD_LEFT), -($this->precision >> 3)) :
- ltrim($temp, chr(0));
- case MATH_BIGINTEGER_MODE_BCMATH:
- if ($this->value === '0') {
- return $this->precision > 0 ? str_repeat(chr(0), ($this->precision + 1) >> 3) : '';
- }
-
- $value = '';
- $current = $this->value;
-
- if ($current[0] == '-') {
- $current = substr($current, 1);
- }
-
- while (bccomp($current, '0', 0) > 0) {
- $temp = bcmod($current, '16777216');
- $value = chr($temp >> 16) . chr($temp >> 8) . chr($temp) . $value;
- $current = bcdiv($current, '16777216', 0);
- }
-
- return $this->precision > 0 ?
- substr(str_pad($value, $this->precision >> 3, chr(0), STR_PAD_LEFT), -($this->precision >> 3)) :
- ltrim($value, chr(0));
- }
-
- if (!count($this->value)) {
- return $this->precision > 0 ? str_repeat(chr(0), ($this->precision + 1) >> 3) : '';
- }
- $result = $this->_int2bytes($this->value[count($this->value) - 1]);
-
- $temp = $this->copy();
-
- for ($i = count($temp->value) - 2; $i >= 0; --$i) {
- $temp->_base256_lshift($result, 26);
- $result = $result | str_pad($temp->_int2bytes($temp->value[$i]), strlen($result), chr(0), STR_PAD_LEFT);
- }
-
- return $this->precision > 0 ?
- str_pad(substr($result, -(($this->precision + 7) >> 3)), ($this->precision + 7) >> 3, chr(0), STR_PAD_LEFT) :
- $result;
- }
-
- /**
- * Converts a BigInteger to a hex string (eg. base-16)).
- *
- * Negative numbers are saved as positive numbers, unless $twos_compliment is set to true, at which point, they're
- * saved as two's compliment.
- *
- * Here's an example:
- * <code>
- * <?php
- * include('Math/BigInteger.php');
- *
- * $a = new Math_BigInteger('65');
- *
- * echo $a->toHex(); // outputs '41'
- * ?>
- * </code>
- *
- * @param Boolean $twos_compliment
- * @return String
- * @access public
- * @internal Converts a base-2**26 number to base-2**8
- */
- function toHex($twos_compliment = false)
- {
- return bin2hex($this->toBytes($twos_compliment));
- }
-
- /**
- * Converts a BigInteger to a bit string (eg. base-2).
- *
- * Negative numbers are saved as positive numbers, unless $twos_compliment is set to true, at which point, they're
- * saved as two's compliment.
- *
- * Here's an example:
- * <code>
- * <?php
- * include('Math/BigInteger.php');
- *
- * $a = new Math_BigInteger('65');
- *
- * echo $a->toBits(); // outputs '1000001'
- * ?>
- * </code>
- *
- * @param Boolean $twos_compliment
- * @return String
- * @access public
- * @internal Converts a base-2**26 number to base-2**2
- */
- function toBits($twos_compliment = false)
- {
- $hex = $this->toHex($twos_compliment);
- $bits = '';
- for ($i = 0; $i < strlen($hex); $i+=8) {
- $bits.= str_pad(decbin(hexdec(substr($hex, $i, 8))), 32, '0', STR_PAD_LEFT);
- }
- return $this->precision > 0 ? substr($bits, -$this->precision) : ltrim($bits, '0');
- }
-
- /**
- * Converts a BigInteger to a base-10 number.
- *
- * Here's an example:
- * <code>
- * <?php
- * include('Math/BigInteger.php');
- *
- * $a = new Math_BigInteger('50');
- *
- * echo $a->toString(); // outputs 50
- * ?>
- * </code>
- *
- * @return String
- * @access public
- * @internal Converts a base-2**26 number to base-10**7 (which is pretty much base-10)
- */
- function toString()
- {
- switch ( MATH_BIGINTEGER_MODE ) {
- case MATH_BIGINTEGER_MODE_GMP:
- return gmp_strval($this->value);
- case MATH_BIGINTEGER_MODE_BCMATH:
- if ($this->value === '0') {
- return '0';
- }
-
- return ltrim($this->value, '0');
- }
-
- if (!count($this->value)) {
- return '0';
- }
-
- $temp = $this->copy();
- $temp->is_negative = false;
-
- $divisor = new Math_BigInteger();
- $divisor->value = array(10000000); // eg. 10**7
- $result = '';
- while (count($temp->value)) {
- list($temp, $mod) = $temp->divide($divisor);
- $result = str_pad(isset($mod->value[0]) ? $mod->value[0] : '', 7, '0', STR_PAD_LEFT) . $result;
- }
- $result = ltrim($result, '0');
- if (empty($result)) {
- $result = '0';
- }
-
- if ($this->is_negative) {
- $result = '-' . $result;
- }
-
- return $result;
- }
-
- /**
- * Copy an object
- *
- * PHP5 passes objects by reference while PHP4 passes by value. As such, we need a function to guarantee
- * that all objects are passed by value, when appropriate. More information can be found here:
- *
- * {@link http://php.net/language.oop5.basic#51624}
- *
- * @access public
- * @see __clone()
- * @return Math_BigInteger
- */
- function copy()
- {
- $temp = new Math_BigInteger();
- $temp->value = $this->value;
- $temp->is_negative = $this->is_negative;
- $temp->generator = $this->generator;
- $temp->precision = $this->precision;
- $temp->bitmask = $this->bitmask;
- return $temp;
- }
-
- /**
- * __toString() magic method
- *
- * Will be called, automatically, if you're supporting just PHP5. If you're supporting PHP4, you'll need to call
- * toString().
- *
- * @access public
- * @internal Implemented per a suggestion by Techie-Michael - thanks!
- */
- function __toString()
- {
- return $this->toString();
- }
-
- /**
- * __clone() magic method
- *
- * Although you can call Math_BigInteger::__toString() directly in PHP5, you cannot call Math_BigInteger::__clone()
- * directly in PHP5. You can in PHP4 since it's not a magic method, but in PHP5, you have to call it by using the PHP5
- * only syntax of $y = clone $x. As such, if you're trying to write an application that works on both PHP4 and PHP5,
- * call Math_BigInteger::copy(), instead.
- *
- * @access public
- * @see copy()
- * @return Math_BigInteger
- */
- function __clone()
- {
- return $this->copy();
- }
-
- /**
- * __sleep() magic method
- *
- * Will be called, automatically, when serialize() is called on a Math_BigInteger object.
- *
- * @see __wakeup()
- * @access public
- */
- function __sleep()
- {
- $this->hex = $this->toHex(true);
- $vars = array('hex');
- if ($this->generator != 'mt_rand') {
- $vars[] = 'generator';
- }
- if ($this->precision > 0) {
- $vars[] = 'precision';
- }
- return $vars;
-
- }
-
- /**
- * __wakeup() magic method
- *
- * Will be called, automatically, when unserialize() is called on a Math_BigInteger object.
- *
- * @see __sleep()
- * @access public
- */
- function __wakeup()
- {
- $temp = new Math_BigInteger($this->hex, -16);
- $this->value = $temp->value;
- $this->is_negative = $temp->is_negative;
- $this->setRandomGenerator($this->generator);
- if ($this->precision > 0) {
- // recalculate $this->bitmask
- $this->setPrecision($this->precision);
- }
- }
-
- /**
- * Adds two BigIntegers.
- *
- * Here's an example:
- * <code>
- * <?php
- * include('Math/BigInteger.php');
- *
- * $a = new Math_BigInteger('10');
- * $b = new Math_BigInteger('20');
- *
- * $c = $a->add($b);
- *
- * echo $c->toString(); // outputs 30
- * ?>
- * </code>
- *
- * @param Math_BigInteger $y
- * @return Math_BigInteger
- * @access public
- * @internal Performs base-2**52 addition
- */
- function add($y)
- {
- switch ( MATH_BIGINTEGER_MODE ) {
- case MATH_BIGINTEGER_MODE_GMP:
- $temp = new Math_BigInteger();
- $temp->value = gmp_add($this->value, $y->value);
-
- return $this->_normalize($temp);
- case MATH_BIGINTEGER_MODE_BCMATH:
- $temp = new Math_BigInteger();
- $temp->value = bcadd($this->value, $y->value, 0);
-
- return $this->_normalize($temp);
- }
-
- $temp = $this->_add($this->value, $this->is_negative, $y->value, $y->is_negative);
-
- $result = new Math_BigInteger();
- $result->value = $temp[MATH_BIGINTEGER_VALUE];
- $result->is_negative = $temp[MATH_BIGINTEGER_SIGN];
-
- return $this->_normalize($result);
- }
-
- /**
- * Performs addition.
- *
- * @param Array $x_value
- * @param Boolean $x_negative
- * @param Array $y_value
- * @param Boolean $y_negative
- * @return Array
- * @access private
- */
- function _add($x_value, $x_negative, $y_value, $y_negative)
- {
- $x_size = count($x_value);
- $y_size = count($y_value);
-
- if ($x_size == 0) {
- return array(
- MATH_BIGINTEGER_VALUE => $y_value,
- MATH_BIGINTEGER_SIGN => $y_negative
- );
- } else if ($y_size == 0) {
- return array(
- MATH_BIGINTEGER_VALUE => $x_value,
- MATH_BIGINTEGER_SIGN => $x_negative
- );
- }
-
- // subtract, if appropriate
- if ( $x_negative != $y_negative ) {
- if ( $x_value == $y_value ) {
- return array(
- MATH_BIGINTEGER_VALUE => array(),
- MATH_BIGINTEGER_SIGN => false
- );
- }
-
- $temp = $this->_subtract($x_value, false, $y_value, false);
- $temp[MATH_BIGINTEGER_SIGN] = $this->_compare($x_value, false, $y_value, false) > 0 ?
- $x_negative : $y_negative;
-
- return $temp;
- }
-
- if ($x_size < $y_size) {
- $size = $x_size;
- $value = $y_value;
- } else {
- $size = $y_size;
- $value = $x_value;
- }
-
- $value[] = 0; // just in case the carry adds an extra digit
-
- $carry = 0;
- for ($i = 0, $j = 1; $j < $size; $i+=2, $j+=2) {
- $sum = $x_value[$j] * 0x4000000 + $x_value[$i] + $y_value[$j] * 0x4000000 + $y_value[$i] + $carry;
- $carry = $sum >= MATH_BIGINTEGER_MAX_DIGIT52; // eg. floor($sum / 2**52); only possible values (in any base) are 0 and 1
- $sum = $carry ? $sum - MATH_BIGINTEGER_MAX_DIGIT52 : $sum;
-
- $temp = (int) ($sum / 0x4000000);
-
- $value[$i] = (int) ($sum - 0x4000000 * $temp); // eg. a faster alternative to fmod($sum, 0x4000000)
- $value[$j] = $temp;
- }
-
- if ($j == $size) { // ie. if $y_size is odd
- $sum = $x_value[$i] + $y_value[$i] + $carry;
- $carry = $sum >= 0x4000000;
- $value[$i] = $carry ? $sum - 0x4000000 : $sum;
- ++$i; // ie. let $i = $j since we've just done $value[$i]
- }
-
- if ($carry) {
- for (; $value[$i] == 0x3FFFFFF; ++$i) {
- $value[$i] = 0;
- }
- ++$value[$i];
- }
-
- return array(
- MATH_BIGINTEGER_VALUE => $this->_trim($value),
- MATH_BIGINTEGER_SIGN => $x_negative
- );
- }
-
- /**
- * Subtracts two BigIntegers.
- *
- * Here's an example:
- * <code>
- * <?php
- * include('Math/BigInteger.php');
- *
- * $a = new Math_BigInteger('10');
- * $b = new Math_BigInteger('20');
- *
- * $c = $a->subtract($b);
- *
- * echo $c->toString(); // outputs -10
- * ?>
- * </code>
- *
- * @param Math_BigInteger $y
- * @return Math_BigInteger
- * @access public
- * @internal Performs base-2**52 subtraction
- */
- function subtract($y)
- {
- switch ( MATH_BIGINTEGER_MODE ) {
- case MATH_BIGINTEGER_MODE_GMP:
- $temp = new Math_BigInteger();
- $temp->value = gmp_sub($this->value, $y->value);
-
- return $this->_normalize($temp);
- case MATH_BIGINTEGER_MODE_BCMATH:
- $temp = new Math_BigInteger();
- $temp->value = bcsub($this->value, $y->value, 0);
-
- return $this->_normalize($temp);
- }
-
- $temp = $this->_subtract($this->value, $this->is_negative, $y->value, $y->is_negative);
-
- $result = new Math_BigInteger();
- $result->value = $temp[MATH_BIGINTEGER_VALUE];
- $result->is_negative = $temp[MATH_BIGINTEGER_SIGN];
-
- return $this->_normalize($result);
- }
-
- /**
- * Performs subtraction.
- *
- * @param Array $x_value
- * @param Boolean $x_negative
- * @param Array $y_value
- * @param Boolean $y_negative
- * @return Array
- * @access private
- */
- function _subtract($x_value, $x_negative, $y_value, $y_negative)
- {
- $x_size = count($x_value);
- $y_size = count($y_value);
-
- if ($x_size == 0) {
- return array(
- MATH_BIGINTEGER_VALUE => $y_value,
- MATH_BIGINTEGER_SIGN => !$y_negative
- );
- } else if ($y_size == 0) {
- return array(
- MATH_BIGINTEGER_VALUE => $x_value,
- MATH_BIGINTEGER_SIGN => $x_negative
- );
- }
-
- // add, if appropriate (ie. -$x - +$y or +$x - -$y)
- if ( $x_negative != $y_negative ) {
- $temp = $this->_add($x_value, false, $y_value, false);
- $temp[MATH_BIGINTEGER_SIGN] = $x_negative;
-
- return $temp;
- }
-
- $diff = $this->_compare($x_value, $x_negative, $y_value, $y_negative);
-
- if ( !$diff ) {
- return array(
- MATH_BIGINTEGER_VALUE => array(),
- MATH_BIGINTEGER_SIGN => false
- );
- }
-
- // switch $x and $y around, if appropriate.
- if ( (!$x_negative && $diff < 0) || ($x_negative && $diff > 0) ) {
- $temp = $x_value;
- $x_value = $y_value;
- $y_value = $temp;
-
- $x_negative = !$x_negative;
-
- $x_size = count($x_value);
- $y_size = count($y_value);
- }
-
- // at this point, $x_value should be at least as big as - if not bigger than - $y_value
-
- $carry = 0;
- for ($i = 0, $j = 1; $j < $y_size; $i+=2, $j+=2) {
- $sum = $x_value[$j] * 0x4000000 + $x_value[$i] - $y_value[$j] * 0x4000000 - $y_value[$i] - $carry;
- $carry = $sum < 0; // eg. floor($sum / 2**52); only possible values (in any base) are 0 and 1
- $sum = $carry ? $sum + MATH_BIGINTEGER_MAX_DIGIT52 : $sum;
-
- $temp = (int) ($sum / 0x4000000);
-
- $x_value[$i] = (int) ($sum - 0x4000000 * $temp);
- $x_value[$j] = $temp;
- }
-
- if ($j == $y_size) { // ie. if $y_size is odd
- $sum = $x_value[$i] - $y_value[$i] - $carry;
- $carry = $sum < 0;
- $x_value[$i] = $carry ? $sum + 0x4000000 : $sum;
- ++$i;
- }
-
- if ($carry) {
- for (; !$x_value[$i]; ++$i) {
- $x_value[$i] = 0x3FFFFFF;
- }
- --$x_value[$i];
- }
-
- return array(
- MATH_BIGINTEGER_VALUE => $this->_trim($x_value),
- MATH_BIGINTEGER_SIGN => $x_negative
- );
- }
-
- /**
- * Multiplies two BigIntegers
- *
- * Here's an example:
- * <code>
- * <?php
- * include('Math/BigInteger.php');
- *
- * $a = new Math_BigInteger('10');
- * $b = new Math_BigInteger('20');
- *
- * $c = $a->multiply($b);
- *
- * echo $c->toString(); // outputs 200
- * ?>
- * </code>
- *
- * @param Math_BigInteger $x
- * @return Math_BigInteger
- * @access public
- */
- function multiply($x)
- {
- switch ( MATH_BIGINTEGER_MODE ) {
- case MATH_BIGINTEGER_MODE_GMP:
- $temp = new Math_BigInteger();
- $temp->value = gmp_mul($this->value, $x->value);
-
- return $this->_normalize($temp);
- case MATH_BIGINTEGER_MODE_BCMATH:
- $temp = new Math_BigInteger();
- $temp->value = bcmul($this->value, $x->value, 0);
-
- return $this->_normalize($temp);
- }
-
- $temp = $this->_multiply($this->value, $this->is_negative, $x->value, $x->is_negative);
-
- $product = new Math_BigInteger();
- $product->value = $temp[MATH_BIGINTEGER_VALUE];
- $product->is_negative = $temp[MATH_BIGINTEGER_SIGN];
-
- return $this->_normalize($product);
- }
-
- /**
- * Performs multiplication.
- *
- * @param Array $x_value
- * @param Boolean $x_negative
- * @param Array $y_value
- * @param Boolean $y_negative
- * @return Array
- * @access private
- */
- function _multiply($x_value, $x_negative, $y_value, $y_negative)
- {
- //if ( $x_value == $y_value ) {
- // return array(
- // MATH_BIGINTEGER_VALUE => $this->_square($x_value),
- // MATH_BIGINTEGER_SIGN => $x_sign != $y_value
- // );
- //}
-
- $x_length = count($x_value);
- $y_length = count($y_value);
-
- if ( !$x_length || !$y_length ) { // a 0 is being multiplied
- return array(
- MATH_BIGINTEGER_VALUE => array(),
- MATH_BIGINTEGER_SIGN => false
- );
- }
-
- return array(
- MATH_BIGINTEGER_VALUE => min($x_length, $y_length) < 2 * MATH_BIGINTEGER_KARATSUBA_CUTOFF ?
- $this->_trim($this->_regularMultiply($x_value, $y_value)) :
- $this->_trim($this->_karatsuba($x_value, $y_value)),
- MATH_BIGINTEGER_SIGN => $x_negative != $y_negative
- );
- }
-
- /**
- * Performs long multiplication on two BigIntegers
- *
- * Modeled after 'multiply' in MutableBigInteger.java.
- *
- * @param Array $x_value
- * @param Array $y_value
- * @return Array
- * @access private
- */
- function _regularMultiply($x_value, $y_value)
- {
- $x_length = count($x_value);
- $y_length = count($y_value);
-
- if ( !$x_length || !$y_length ) { // a 0 is being multiplied
- return array();
- }
-
- if ( $x_length < $y_length ) {
- $temp = $x_value;
- $x_value = $y_value;
- $y_value = $temp;
-
- $x_length = count($x_value);
- $y_length = count($y_value);
- }
-
- $product_value = $this->_array_repeat(0, $x_length + $y_length);
-
- // the following for loop could be removed if the for loop following it
- // (the one with nested for loops) initially set $i to 0, but
- // doing so would also make the result in one set of unnecessary adds,
- // since on the outermost loops first pass, $product->value[$k] is going
- // to always be 0
-
- $carry = 0;
-
- for ($j = 0; $j < $x_length; ++$j) { // ie. $i = 0
- $temp = $x_value[$j] * $y_value[0] + $carry; // $product_value[$k] == 0
- $carry = (int) ($temp / 0x4000000);
- $product_value[$j] = (int) ($temp - 0x4000000 * $carry);
- }
-
- $product_value[$j] = $carry;
-
- // the above for loop is what the previous comment was talking about. the
- // following for loop is the "one with nested for loops"
- for ($i = 1; $i < $y_length; ++$i) {
- $carry = 0;
-
- for ($j = 0, $k = $i; $j < $x_length; ++$j, ++$k) {
- $temp = $product_value[$k] + $x_value[$j] * $y_value[$i] + $carry;
- $carry = (int) ($temp / 0x4000000);
- $product_value[$k] = (int) ($temp - 0x4000000 * $carry);
- }
-
- $product_value[$k] = $carry;
- }
-
- return $product_value;
- }
-
- /**
- * Performs Karatsuba multiplication on two BigIntegers
- *
- * See {@link http://en.wikipedia.org/wiki/Karatsuba_algorithm Karatsuba algorithm} and
- * {@link http://math.libtomcrypt.com/files/tommath.pdf#page=120 MPM 5.2.3}.
- *
- * @param Array $x_value
- * @param Array $y_value
- * @return Array
- * @access private
- */
- function _karatsuba($x_value, $y_value)
- {
- $m = min(count($x_value) >> 1, count($y_value) >> 1);
-
- if ($m < MATH_BIGINTEGER_KARATSUBA_CUTOFF) {
- return $this->_regularMultiply($x_value, $y_value);
- }
-
- $x1 = array_slice($x_value, $m);
- $x0 = array_slice($x_value, 0, $m);
- $y1 = array_slice($y_value, $m);
- $y0 = array_slice($y_value, 0, $m);
-
- $z2 = $this->_karatsuba($x1, $y1);
- $z0 = $this->_karatsuba($x0, $y0);
-
- $z1 = $this->_add($x1, false, $x0, false);
- $temp = $this->_add($y1, false, $y0, false);
- $z1 = $this->_karatsuba($z1[MATH_BIGINTEGER_VALUE], $temp[MATH_BIGINTEGER_VALUE]);
- $temp = $this->_add($z2, false, $z0, false);
- $z1 = $this->_subtract($z1, false, $temp[MATH_BIGINTEGER_VALUE], false);
-
- $z2 = array_merge(array_fill(0, 2 * $m, 0), $z2);
- $z1[MATH_BIGINTEGER_VALUE] = array_merge(array_fill(0, $m, 0), $z1[MATH_BIGINTEGER_VALUE]);
-
- $xy = $this->_add($z2, false, $z1[MATH_BIGINTEGER_VALUE], $z1[MATH_BIGINTEGER_SIGN]);
- $xy = $this->_add($xy[MATH_BIGINTEGER_VALUE], $xy[MATH_BIGINTEGER_SIGN], $z0, false);
-
- return $xy[MATH_BIGINTEGER_VALUE];
- }
-
- /**
- * Performs squaring
- *
- * @param Array $x
- * @return Array
- * @access private
- */
- function _square($x = false)
- {
- return count($x) < 2 * MATH_BIGINTEGER_KARATSUBA_CUTOFF ?
- $this->_trim($this->_baseSquare($x)) :
- $this->_trim($this->_karatsubaSquare($x));
- }
-
- /**
- * Performs traditional squaring on two BigIntegers
- *
- * Squaring can be done faster than multiplying a number by itself can be. See
- * {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=7 HAC 14.2.4} /
- * {@link http://math.libtomcrypt.com/files/tommath.pdf#page=141 MPM 5.3} for more information.
- *
- * @param Array $value
- * @return Array
- * @access private
- */
- function _baseSquare($value)
- {
- if ( empty($value) ) {
- return array();
- }
- $square_value = $this->_array_repeat(0, 2 * count($value));
-
- for ($i = 0, $max_index = count($value) - 1; $i <= $max_index; ++$i) {
- $i2 = $i << 1;
-
- $temp = $square_value[$i2] + $value[$i] * $value[$i];
- $carry = (int) ($temp / 0x4000000);
- $square_value[$i2] = (int) ($temp - 0x4000000 * $carry);
-
- // note how we start from $i+1 instead of 0 as we do in multiplication.
- for ($j = $i + 1, $k = $i2 + 1; $j <= $max_index; ++$j, ++$k) {
- $temp = $square_value[$k] + 2 * $value[$j] * $value[$i] + $carry;
- $carry = (int) ($temp / 0x4000000);
- $square_value[$k] = (int) ($temp - 0x4000000 * $carry);
- }
-
- // the following line can yield values larger 2**15. at this point, PHP should switch
- // over to floats.
- $square_value[$i + $max_index + 1] = $carry;
- }
-
- return $square_value;
- }
-
- /**
- * Performs Karatsuba "squaring" on two BigIntegers
- *
- * See {@link http://en.wikipedia.org/wiki/Karatsuba_algorithm Karatsuba algorithm} and
- * {@link http://math.libtomcrypt.com/files/tommath.pdf#page=151 MPM 5.3.4}.
- *
- * @param Array $value
- * @return Array
- * @access private
- */
- function _karatsubaSquare($value)
- {
- $m = count($value) >> 1;
-
- if ($m < MATH_BIGINTEGER_KARATSUBA_CUTOFF) {
- return $this->_baseSquare($value);
- }
-
- $x1 = array_slice($value, $m);
- $x0 = array_slice($value, 0, $m);
-
- $z2 = $this->_karatsubaSquare($x1);
- $z0 = $this->_karatsubaSquare($x0);
-
- $z1 = $this->_add($x1, false, $x0, false);
- $z1 = $this->_karatsubaSquare($z1[MATH_BIGINTEGER_VALUE]);
- $temp = $this->_add($z2, false, $z0, false);
- $z1 = $this->_subtract($z1, false, $temp[MATH_BIGINTEGER_VALUE], false);
-
- $z2 = array_merge(array_fill(0, 2 * $m, 0), $z2);
- $z1[MATH_BIGINTEGER_VALUE] = array_merge(array_fill(0, $m, 0), $z1[MATH_BIGINTEGER_VALUE]);
-
- $xx = $this->_add($z2, false, $z1[MATH_BIGINTEGER_VALUE], $z1[MATH_BIGINTEGER_SIGN]);
- $xx = $this->_add($xx[MATH_BIGINTEGER_VALUE], $xx[MATH_BIGINTEGER_SIGN], $z0, false);
-
- return $xx[MATH_BIGINTEGER_VALUE];
- }
-
- /**
- * Divides two BigIntegers.
- *
- * Returns an array whose first element contains the quotient and whose second element contains the
- * "common residue". If the remainder would be positive, the "common residue" and the remainder are the
- * same. If the remainder would be negative, the "common residue" is equal to the sum of the remainder
- * and the divisor (basically, the "common residue" is the first positive modulo).
- *
- * Here's an example:
- * <code>
- * <?php
- * include('Math/BigInteger.php');
- *
- * $a = new Math_BigInteger('10');
- * $b = new Math_BigInteger('20');
- *
- * list($quotient, $remainder) = $a->divide($b);
- *
- * echo $quotient->toString(); // outputs 0
- * echo "\r\n";
- * echo $remainder->toString(); // outputs 10
- * ?>
- * </code>
- *
- * @param Math_BigInteger $y
- * @return Array
- * @access public
- * @internal This function is based off of {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=9 HAC 14.20}.
- */
- function divide($y)
- {
- switch ( MATH_BIGINTEGER_MODE ) {
- case MATH_BIGINTEGER_MODE_GMP:
- $quotient = new Math_BigInteger();
- $remainder = new Math_BigInteger();
-
- list($quotient->value, $remainder->value) = gmp_div_qr($this->value, $y->value);
-
- if (gmp_sign($remainder->value) < 0) {
- $remainder->value = gmp_add($remainder->value, gmp_abs($y->value));
- }
-
- return array($this->_normalize($quotient), $this->_normalize($remainder));
- case MATH_BIGINTEGER_MODE_BCMATH:
- $quotient = new Math_BigInteger();
- $remainder = new Math_BigInteger();
-
- $quotient->value = bcdiv($this->value, $y->value, 0);
- $remainder->value = bcmod($this->value, $y->value);
-
- if ($remainder->value[0] == '-') {
- $remainder->value = bcadd($remainder->value, $y->value[0] == '-' ? substr($y->value, 1) : $y->value, 0);
- }
-
- return array($this->_normalize($quotient), $this->_normalize($remainder));
- }
-
- if (count($y->value) == 1) {
- list($q, $r) = $this->_divide_digit($this->value, $y->value[0]);
- $quotient = new Math_BigInteger();
- $remainder = new Math_BigInteger();
- $quotient->value = $q;
- $remainder->value = array($r);
- $quotient->is_negative = $this->is_negative != $y->is_negative;
- return array($this->_normalize($quotient), $this->_normalize($remainder));
- }
-
- static $zero;
- if ( !isset($zero) ) {
- $zero = new Math_BigInteger();
- }
-
- $x = $this->copy();
- $y = $y->copy();
-
- $x_sign = $x->is_negative;
- $y_sign = $y->is_negative;
-
- $x->is_negative = $y->is_negative = false;
-
- $diff = $x->compare($y);
-
- if ( !$diff ) {
- $temp = new Math_BigInteger();
- $temp->value = array(1);
- $temp->is_negative = $x_sign != $y_sign;
- return array($this->_normalize($temp), $this->_normalize(new Math_BigInteger()));
- }
-
- if ( $diff < 0 ) {
- // if $x is negative, "add" $y.
- if ( $x_sign ) {
- $x = $y->subtract($x);
- }
- return array($this->_normalize(new Math_BigInteger()), $this->_normalize($x));
- }
-
- // normalize $x and $y as described in HAC 14.23 / 14.24
- $msb = $y->value[count($y->value) - 1];
- for ($shift = 0; !($msb & 0x2000000); ++$shift) {
- $msb <<= 1;
- }
- $x->_lshift($shift);
- $y->_lshift($shift);
- $y_value = &$y->value;
-
- $x_max = count($x->value) - 1;
- $y_max = count($y->value) - 1;
-
- $quotient = new Math_BigInteger();
- $quotient_value = &$quotient->value;
- $quotient_value = $this->_array_repeat(0, $x_max - $y_max + 1);
-
- static $temp, $lhs, $rhs;
- if (!isset($temp)) {
- $temp = new Math_BigInteger();
- $lhs = new Math_BigInteger();
- $rhs = new Math_BigInteger();
- }
- $temp_value = &$temp->value;
- $rhs_value = &$rhs->value;
-
- // $temp = $y << ($x_max - $y_max-1) in base 2**26
- $temp_value = array_merge($this->_array_repeat(0, $x_max - $y_max), $y_value);
-
- while ( $x->compare($temp) >= 0 ) {
- // calculate the "common residue"
- ++$quotient_value[$x_max - $y_max];
- $x = $x->subtract($temp);
- $x_max = count($x->value) - 1;
- }
-
- for ($i = $x_max; $i >= $y_max + 1; --$i) {
- $x_value = &$x->value;
- $x_window = array(
- isset($x_value[$i]) ? $x_value[$i] : 0,
- isset($x_value[$i - 1]) ? $x_value[$i - 1] : 0,
- isset($x_value[$i - 2]) ? $x_value[$i - 2] : 0
- );
- $y_window = array(
- $y_value[$y_max],
- ( $y_max > 0 ) ? $y_value[$y_max - 1] : 0
- );
-
- $q_index = $i - $y_max - 1;
- if ($x_window[0] == $y_window[0]) {
- $quotient_value[$q_index] = 0x3FFFFFF;
- } else {
- $quotient_value[$q_index] = (int) (
- ($x_window[0] * 0x4000000 + $x_window[1])
- /
- $y_window[0]
- );
- }
-
- $temp_value = array($y_window[1], $y_window[0]);
-
- $lhs->value = array($quotient_value[$q_index]);
- $lhs = $lhs->multiply($temp);
-
- $rhs_value = array($x_window[2], $x_window[1], $x_window[0]);
-
- while ( $lhs->compare($rhs) > 0 ) {
- --$quotient_value[$q_index];
-
- $lhs->value = array($quotient_value[$q_index]);
- $lhs = $lhs->multiply($temp);
- }
-
- $adjust = $this->_array_repeat(0, $q_index);
- $temp_value = array($quotient_value[$q_index]);
- $temp = $temp->multiply($y);
- $temp_value = &$temp->value;
- $temp_value = array_merge($adjust, $temp_value);
-
- $x = $x->subtract($temp);
-
- if ($x->compare($zero) < 0) {
- $temp_value = array_merge($adjust, $y_value);
- $x = $x->add($temp);
-
- --$quotient_value[$q_index];
- }
-
- $x_max = count($x_value) - 1;
- }
-
- // unnormalize the remainder
- $x->_rshift($shift);
-
- $quotient->is_negative = $x_sign != $y_sign;
-
- // calculate the "common residue", if appropriate
- if ( $x_sign ) {
- $y->_rshift($shift);
- $x = $y->subtract($x);
- }
-
- return array($this->_normalize($quotient), $this->_normalize($x));
- }
-
- /**
- * Divides a BigInteger by a regular integer
- *
- * abc / x = a00 / x + b0 / x + c / x
- *
- * @param Array $dividend
- * @param Array $divisor
- * @return Array
- * @access private
- */
- function _divide_digit($dividend, $divisor)
- {
- $carry = 0;
- $result = array();
-
- for ($i = count($dividend) - 1; $i >= 0; --$i) {
- $temp = 0x4000000 * $carry + $dividend[$i];
- $result[$i] = (int) ($temp / $divisor);
- $carry = (int) ($temp - $divisor * $result[$i]);
- }
-
- return array($result, $carry);
- }
-
- /**
- * Performs modular exponentiation.
- *
- * Here's an example:
- * <code>
- * <?php
- * include('Math/BigInteger.php');
- *
- * $a = new Math_BigInteger('10');
- * $b = new Math_BigInteger('20');
- * $c = new Math_BigInteger('30');
- *
- * $c = $a->modPow($b, $c);
- *
- * echo $c->toString(); // outputs 10
- * ?>
- * </code>
- *
- * @param Math_BigInteger $e
- * @param Math_BigInteger $n
- * @return Math_BigInteger
- * @access public
- * @internal The most naive approach to modular exponentiation has very unreasonable requirements, and
- * and although the approach involving repeated squaring does vastly better, it, too, is impractical
- * for our purposes. The reason being that division - by far the most complicated and time-consuming
- * of the basic operations (eg. +,-,*,/) - occurs multiple times within it.
- *
- * Modular reductions resolve this issue. Although an individual modular reduction takes more time
- * then an individual division, when performed in succession (with the same modulo), they're a lot faster.
- *
- * The two most commonly used modular reductions are Barrett and Montgomery reduction. Montgomery reduction,
- * although faster, only works when the gcd of the modulo and of the base being used is 1. In RSA, when the
- * base is a power of two, the modulo - a product of two primes - is always going to have a gcd of 1 (because
- * the product of two odd numbers is odd), but what about when RSA isn't used?
- *
- * In contrast, Barrett reduction has no such constraint. As such, some bigint implementations perform a
- * Barrett reduction after every operation in the modpow function. Others perform Barrett reductions when the
- * modulo is even and Montgomery reductions when the modulo is odd. BigInteger.java's modPow method, however,
- * uses a trick involving the Chinese Remainder Theorem to factor the even modulo into two numbers - one odd and
- * the other, a power of two - and recombine them, later. This is the method that this modPow function uses.
- * {@link http://islab.oregonstate.edu/papers/j34monex.pdf Montgomery Reduction with Even Modulus} elaborates.
- */
- function modPow($e, $n)
- {
- $n = $this->bitmask !== false && $this->bitmask->compare($n) < 0 ? $this->bitmask : $n->abs();
-
- if ($e->compare(new Math_BigInteger()) < 0) {
- $e = $e->abs();
-
- $temp = $this->modInverse($n);
- if ($temp === false) {
- return false;
- }
-
- return $this->_normalize($temp->modPow($e, $n));
- }
-
- switch ( MATH_BIGINTEGER_MODE ) {
- case MATH_BIGINTEGER_MODE_GMP:
- $temp = new Math_BigInteger();
- $temp->value = gmp_powm($this->value, $e->value, $n->value);
-
- return $this->_normalize($temp);
- case MATH_BIGINTEGER_MODE_BCMATH:
- $temp = new Math_BigInteger();
- $temp->value = bcpowmod($this->value, $e->value, $n->value, 0);
-
- return $this->_normalize($temp);
- }
-
- if ( empty($e->value) ) {
- $temp = new Math_BigInteger();
- $temp->value = array(1);
- return $this->_normalize($temp);
- }
-
- if ( $e->value == array(1) ) {
- list(, $temp) = $this->divide($n);
- return $this->_normalize($temp);
- }
-
- if ( $e->value == array(2) ) {
- $temp = new Math_BigInteger();
- $temp->value = $this->_square($this->value);
- list(, $temp) = $temp->divide($n);
- return $this->_normalize($temp);
- }
-
- return $this->_normalize($this->_slidingWindow($e, $n, MATH_BIGINTEGER_BARRETT));
-
- // is the modulo odd?
- if ( $n->value[0] & 1 ) {
- return $this->_normalize($this->_slidingWindow($e, $n, MATH_BIGINTEGER_MONTGOMERY));
- }
- // if it's not, it's even
-
- // find the lowest set bit (eg. the max pow of 2 that divides $n)
- for ($i = 0; $i < count($n->value); ++$i) {
- if ( $n->value[$i] ) {
- $temp = decbin($n->value[$i]);
- $j = strlen($temp) - strrpos($temp, '1') - 1;
- $j+= 26 * $i;
- break;
- }
- }
- // at this point, 2^$j * $n/(2^$j) == $n
-
- $mod1 = $n->copy();
- $mod1->_rshift($j);
- $mod2 = new Math_BigInteger();
- $mod2->value = array(1);
- $mod2->_lshift($j);
-
- $part1 = ( $mod1->value != array(1) ) ? $this->_slidingWindow($e, $mod1, MATH_BIGINTEGER_MONTGOMERY) : new Math_BigInteger();
- $part2 = $this->_slidingWindow($e, $mod2, MATH_BIGINTEGER_POWEROF2);
-
- $y1 = $mod2->modInverse($mod1);
- $y2 = $mod1->modInverse($mod2);
-
- $result = $part1->multiply($mod2);
- $result = $result->multiply($y1);
-
- $temp = $part2->multiply($mod1);
- $temp = $temp->multiply($y2);
-
- $result = $result->add($temp);
- list(, $result) = $result->divide($n);
-
- return $this->_normalize($result);
- }
-
- /**
- * Performs modular exponentiation.
- *
- * Alias for Math_BigInteger::modPow()
- *
- * @param Math_BigInteger $e
- * @param Math_BigInteger $n
- * @return Math_BigInteger
- * @access public
- */
- function powMod($e, $n)
- {
- return $this->modPow($e, $n);
- }
-
- /**
- * Sliding Window k-ary Modular Exponentiation
- *
- * Based on {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=27 HAC 14.85} /
- * {@link http://math.libtomcrypt.com/files/tommath.pdf#page=210 MPM 7.7}. In a departure from those algorithims,
- * however, this function performs a modular reduction after every multiplication and squaring operation.
- * As such, this function has the same preconditions that the reductions being used do.
- *
- * @param Math_BigInteger $e
- * @param Math_BigInteger $n
- * @param Integer $mode
- * @return Math_BigInteger
- * @access private
- */
- function _slidingWindow($e, $n, $mode)
- {
- static $window_ranges = array(7, 25, 81, 241, 673, 1793); // from BigInteger.java's oddModPow function
- //static $window_ranges = array(0, 7, 36, 140, 450, 1303, 3529); // from MPM 7.3.1
-
- $e_value = $e->value;
- $e_length = count($e_value) - 1;
- $e_bits = decbin($e_value[$e_length]);
- for ($i = $e_length - 1; $i >= 0; --$i) {
- $e_bits.= str_pad(decbin($e_value[$i]), 26, '0', STR_PAD_LEFT);
- }
-
- $e_length = strlen($e_bits);
-
- // calculate the appropriate window size.
- // $window_size == 3 if $window_ranges is between 25 and 81, for example.
- for ($i = 0, $window_size = 1; $e_length > $window_ranges[$i] && $i < count($window_ranges); ++$window_size, ++$i);
-
- $n_value = $n->value;
-
- // precompute $this^0 through $this^$window_size
- $powers = array();
- $powers[1] = $this->_prepareReduce($this->value, $n_value, $mode);
- $powers[2] = $this->_squareReduce($powers[1], $n_value, $mode);
-
- // we do every other number since substr($e_bits, $i, $j+1) (see below) is supposed to end
- // in a 1. ie. it's supposed to be odd.
- $temp = 1 << ($window_size - 1);
- for ($i = 1; $i < $temp; ++$i) {
- $i2 = $i << 1;
- $powers[$i2 + 1] = $this->_multiplyReduce($powers[$i2 - 1], $powers[2], $n_value, $mode);
- }
-
- $result = array(1);
- $result = $this->_prepareReduce($result, $n_value, $mode);
-
- for ($i = 0; $i < $e_length; ) {
- if ( !$e_bits[$i] ) {
- $result = $this->_squareReduce($result, $n_value, $mode);
- ++$i;
- } else {
- for ($j = $window_size - 1; $j > 0; --$j) {
- if ( !empty($e_bits[$i + $j]) ) {
- break;
- }
- }
-
- for ($k = 0; $k <= $j; ++$k) {// eg. the length of substr($e_bits, $i, $j+1)
- $result = $this->_squareReduce($result, $n_value, $mode);
- }
-
- $result = $this->_multiplyReduce($result, $powers[bindec(substr($e_bits, $i, $j + 1))], $n_value, $mode);
-
- $i+=$j + 1;
- }
- }
-
- $temp = new Math_BigInteger();
- $temp->value = $this->_reduce($result, $n_value, $mode);
-
- return $temp;
- }
-
- /**
- * Modular reduction
- *
- * For most $modes this will return the remainder.
- *
- * @see _slidingWindow()
- * @access private
- * @param Array $x
- * @param Array $n
- * @param Integer $mode
- * @return Array
- */
- function _reduce($x, $n, $mode)
- {
- switch ($mode) {
- case MATH_BIGINTEGER_MONTGOMERY:
- return $this->_montgomery($x, $n);
- case MATH_BIGINTEGER_BARRETT:
- return $this->_barrett($x, $n);
- case MATH_BIGINTEGER_POWEROF2:
- $lhs = new Math_BigInteger();
- $lhs->value = $x;
- $rhs = new Math_BigInteger();
- $rhs->value = $n;
- return $x->_mod2($n);
- case MATH_BIGINTEGER_CLASSIC:
- $lhs = new Math_BigInteger();
- $lhs->value = $x;
- $rhs = new Math_BigInteger();
- $rhs->value = $n;
- list(, $temp) = $lhs->divide($rhs);
- return $temp->value;
- case MATH_BIGINTEGER_NONE:
- return $x;
- default:
- // an invalid $mode was provided
- }
- }
-
- /**
- * Modular reduction preperation
- *
- * @see _slidingWindow()
- * @access private
- * @param Array $x
- * @param Array $n
- * @param Integer $mode
- * @return Array
- */
- function _prepareReduce($x, $n, $mode)
- {
- if ($mode == MATH_BIGINTEGER_MONTGOMERY) {
- return $this->_prepMontgomery($x, $n);
- }
- return $this->_reduce($x, $n, $mode);
- }
-
- /**
- * Modular multiply
- *
- * @see _slidingWindow()
- * @access private
- * @param Array $x
- * @param Array $y
- * @param Array $n
- * @param Integer $mode
- * @return Array
- */
- function _multiplyReduce($x, $y, $n, $mode)
- {
- if ($mode == MATH_BIGINTEGER_MONTGOMERY) {
- return $this->_montgomeryMultiply($x, $y, $n);
- }
- $temp = $this->_multiply($x, false, $y, false);
- return $this->_reduce($temp[MATH_BIGINTEGER_VALUE], $n, $mode);
- }
-
- /**
- * Modular square
- *
- * @see _slidingWindow()
- * @access private
- * @param Array $x
- * @param Array $n
- * @param Integer $mode
- * @return Array
- */
- function _squareReduce($x, $n, $mode)
- {
- if ($mode == MATH_BIGINTEGER_MONTGOMERY) {
- return $this->_montgomeryMultiply($x, $x, $n);
- }
- return $this->_reduce($this->_square($x), $n, $mode);
- }
-
- /**
- * Modulos for Powers of Two
- *
- * Calculates $x%$n, where $n = 2**$e, for some $e. Since this is basically the same as doing $x & ($n-1),
- * we'll just use this function as a wrapper for doing that.
- *
- * @see _slidingWindow()
- * @access private
- * @param Math_BigInteger
- * @return Math_BigInteger
- */
- function _mod2($n)
- {
- $temp = new Math_BigInteger();
- $temp->value = array(1);
- return $this->bitwise_and($n->subtract($temp));
- }
-
- /**
- * Barrett Modular Reduction
- *
- * See {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=14 HAC 14.3.3} /
- * {@link http://math.libtomcrypt.com/files/tommath.pdf#page=165 MPM 6.2.5} for more information. Modified slightly,
- * so as not to require negative numbers (initially, this script didn't support negative numbers).
- *
- * Employs "folding", as described at
- * {@link http://www.cosic.esat.kuleuven.be/publications/thesis-149.pdf#page=66 thesis-149.pdf#page=66}. To quote from
- * 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."
- *
- * Unfortunately, the "Barrett Reduction with Folding" algorithm described in thesis-149.pdf is not, as written, all that
- * usable on account of (1) its not using reasonable radix points as discussed in
- * {@link http://math.libtomcrypt.com/files/tommath.pdf#page=162 MPM 6.2.2} and (2) the fact that, even with reasonable
- * radix points, it only works when there are an even number of digits in the denominator. The reason for (2) is that
- * (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
- * comments for details.
- *
- * @see _slidingWindow()
- * @access private
- * @param Array $n
- * @param Array $m
- * @return Array
- */
- function _barrett($n, $m)
- {
- static $cache = array(
- MATH_BIGINTEGER_VARIABLE => array(),
- MATH_BIGINTEGER_DATA => array()
- );
-
- $m_length = count($m);
-
- // if ($this->_compare($n, $this->_square($m)) >= 0) {
- if (count($n) > 2 * $m_length) {
- $lhs = new Math_BigInteger();
- $rhs = new Math_BigInteger();
- $lhs->value = $n;
- $rhs->value = $m;
- list(, $temp) = $lhs->divide($rhs);
- return $temp->value;
- }
-
- // if (m.length >> 1) + 2 <= m.length then m is too small and n can't be reduced
- if ($m_length < 5) {
- return $this->_regularBarrett($n, $m);
- }
-
- // n = 2 * m.length
-
- if ( ($key = array_search($m, $cache[MATH_BIGINTEGER_VARIABLE])) === false ) {
- $key = count($cache[MATH_BIGINTEGER_VARIABLE]);
- $cache[MATH_BIGINTEGER_VARIABLE][] = $m;
-
- $lhs = new Math_BigInteger();
- $lhs_value = &$lhs->value;
- $lhs_value = $this->_array_repeat(0, $m_length + ($m_length >> 1));
- $lhs_value[] = 1;
- $rhs = new Math_BigInteger();
- $rhs->value = $m;
-
- list($u, $m1) = $lhs->divide($rhs);
- $u = $u->value;
- $m1 = $m1->value;
-
- $cache[MATH_BIGINTEGER_DATA][] = array(
- 'u' => $u, // m.length >> 1 (technically (m.length >> 1) + 1)
- 'm1'=> $m1 // m.length
- );
- } else {
- extract($cache[MATH_BIGINTEGER_DATA][$key]);
- }
-
- $cutoff = $m_length + ($m_length >> 1);
- $lsd = array_slice($n, 0, $cutoff); // m.length + (m.length >> 1)
- $msd = array_slice($n, $cutoff); // m.length >> 1
- $lsd = $this->_trim($lsd);
- $temp = $this->_multiply($msd, false, $m1, false);
- $n = $this->_add($lsd, false, $temp[MATH_BIGINTEGER_VALUE], false); // m.length + (m.length >> 1) + 1
-
- if ($m_length & 1) {
- return $this->_regularBarrett($n[MATH_BIGINTEGER_VALUE], $m);
- }
-
- // (m.length + (m.length >> 1) + 1) - (m.length - 1) == (m.length >> 1) + 2
- $temp = array_slice($n[MATH_BIGINTEGER_VALUE], $m_length - 1);
- // if even: ((m.length >> 1) + 2) + (m.length >> 1) == m.length + 2
- // if odd: ((m.length >> 1) + 2) + (m.length >> 1) == (m.length - 1) + 2 == m.length + 1
- $temp = $this->_multiply($temp, false, $u, false);
- // if even: (m.length + 2) - ((m.length >> 1) + 1) = m.length - (m.length >> 1) + 1
- // if odd: (m.length + 1) - ((m.length >> 1) + 1) = m.length - (m.length >> 1)
- $temp = array_slice($temp[MATH_BIGINTEGER_VALUE], ($m_length >> 1) + 1);
- // if even: (m.length - (m.length >> 1) + 1) + m.length = 2 * m.length - (m.length >> 1) + 1
- // if odd: (m.length - (m.length >> 1)) + m.length = 2 * m.length - (m.length >> 1)
- $temp = $this->_multiply($temp, false, $m, false);
-
- // at this point, if m had an odd number of digits, we'd be subtracting a 2 * m.length - (m.length >> 1) digit
- // number from a m.length + (m.length >> 1) + 1 digit number. ie. there'd be an extra digit and the while loop
- // following this comment would loop a lot (hence our calling _regularBarrett() in that situation).
-
- $result = $this->_subtract($n[MATH_BIGINTEGER_VALUE], false, $temp[MATH_BIGINTEGER_VALUE], false);
-
- while ($this->_compare($result[MATH_BIGINTEGER_VALUE], $result[MATH_BIGINTEGER_SIGN], $m, false) >= 0) {
- $result = $this->_subtract($result[MATH_BIGINTEGER_VALUE], $result[MATH_BIGINTEGER_SIGN], $m, false);
- }
-
- return $result[MATH_BIGINTEGER_VALUE];
- }
-
- /**
- * (Regular) Barrett Modular Reduction
- *
- * For numbers with more than four digits Math_BigInteger::_barrett() is faster. The difference between that and this
- * is that this function does not fold the denominator into a smaller form.
- *
- * @see _slidingWindow()
- * @access private
- * @param Array $x
- * @param Array $n
- * @return Array
- */
- function _regularBarrett($x, $n)
- {
- static $cache = array(
- MATH_BIGINTEGER_VARIABLE => array(),
- MATH_BIGINTEGER_DATA => array()
- );
-
- $n_length = count($n);
-
- if (count($x) > 2 * $n_length) {
- $lhs = new Math_BigInteger();
- $rhs = new Math_BigInteger();
- $lhs->value = $x;
- $rhs->value = $n;
- list(, $temp) = $lhs->divide($rhs);
- return $temp->value;
- }
-
- if ( ($key = array_search($n, $cache[MATH_BIGINTEGER_VARIABLE])) === false ) {
- $key = count($cache[MATH_BIGINTEGER_VARIABLE]);
- $cache[MATH_BIGINTEGER_VARIABLE][] = $n;
- $lhs = new Math_BigInteger();
- $lhs_value = &$lhs->value;
- $lhs_value = $this->_array_repeat(0, 2 * $n_length);
- $lhs_value[] = 1;
- $rhs = new Math_BigInteger();
- $rhs->value = $n;
- list($temp, ) = $lhs->divide($rhs); // m.length
- $cache[MATH_BIGINTEGER_DATA][] = $temp->value;
- }
-
- // 2 * m.length - (m.length - 1) = m.length + 1
- $temp = array_slice($x, $n_length - 1);
- // (m.length + 1) + m.length = 2 * m.length + 1
- $temp = $this->_multiply($temp, false, $cache[MATH_BIGINTEGER_DATA][$key], false);
- // (2 * m.length + 1) - (m.length - 1) = m.length + 2
- $temp = array_slice($temp[MATH_BIGINTEGER_VALUE], $n_length + 1);
-
- // m.length + 1
- $result = array_slice($x, 0, $n_length + 1);
- // m.length + 1
- $temp = $this->_multiplyLower($temp, false, $n, false, $n_length + 1);
- // $temp == array_slice($temp->_multiply($temp, false, $n, false)->value, 0, $n_length + 1)
-
- if ($this->_compare($result, false, $temp[MATH_BIGINTEGER_VALUE], $temp[MATH_BIGINTEGER_SIGN]) < 0) {
- $corrector_value = $this->_array_repeat(0, $n_length + 1);
- $corrector_value[] = 1;
- $result = $this->_add($result, false, $corrector, false);
- $result = $result[MATH_BIGINTEGER_VALUE];
- }
-
- // at this point, we're subtracting a number with m.length + 1 digits from another number with m.length + 1 digits
- $result = $this->_subtract($result, false, $temp[MATH_BIGINTEGER_VALUE], $temp[MATH_BIGINTEGER_SIGN]);
- while ($this->_compare($result[MATH_BIGINTEGER_VALUE], $result[MATH_BIGINTEGER_SIGN], $n, false) > 0) {
- $result = $this->_subtract($result[MATH_BIGINTEGER_VALUE], $result[MATH_BIGINTEGER_SIGN], $n, false);
- }
-
- return $result[MATH_BIGINTEGER_VALUE];
- }
-
- /**
- * Performs long multiplication up to $stop digits
- *
- * If you're going to be doing array_slice($product->value, 0, $stop), some cycles can be saved.
- *
- * @see _regularBarrett()
- * @param Array $x_value
- * @param Boolean $x_negative
- * @param Array $y_value
- * @param Boolean $y_negative
- * @return Array
- * @access private
- */
- function _multiplyLower($x_value, $x_negative, $y_value, $y_negative, $stop)
- {
- $x_length = count($x_value);
- $y_length = count($y_value);
-
- if ( !$x_length || !$y_length ) { // a 0 is being multiplied
- return array(
- MATH_BIGINTEGER_VALUE => array(),
- MATH_BIGINTEGER_SIGN => false
- );
- }
-
- if ( $x_length < $y_length ) {
- $temp = $x_value;
- $x_value = $y_value;
- $y_value = $temp;
-
- $x_length = count($x_value);
- $y_length = count($y_value);
- }
-
- $product_value = $this->_array_repeat(0, $x_length + $y_length);
-
- // the following for loop could be removed if the for loop following it
- // (the one with nested for loops) initially set $i to 0, but
- // doing so would also make the result in one set of unnecessary adds,
- // since on the outermost loops first pass, $product->value[$k] is going
- // to always be 0
-
- $carry = 0;
-
- for ($j = 0; $j < $x_length; ++$j) { // ie. $i = 0, $k = $i
- $temp = $x_value[$j] * $y_value[0] + $carry; // $product_value[$k] == 0
- $carry = (int) ($temp / 0x4000000);
- $product_value[$j] = (int) ($temp - 0x4000000 * $carry);
- }
-
- if ($j < $stop) {
- $product_value[$j] = $carry;
- }
-
- // the above for loop is what the previous comment was talking about. the
- // following for loop is the "one with nested for loops"
-
- for ($i = 1; $i < $y_length; ++$i) {
- $carry = 0;
-
- for ($j = 0, $k = $i; $j < $x_length && $k < $stop; ++$j, ++$k) {
- $temp = $product_value[$k] + $x_value[$j] * $y_value[$i] + $carry;
- $carry = (int) ($temp / 0x4000000);
- $product_value[$k] = (int) ($temp - 0x4000000 * $carry);
- }
-
- if ($k < $stop) {
- $product_value[$k] = $carry;
- }
- }
-
- return array(
- MATH_BIGINTEGER_VALUE => $this->_trim($product_value),
- MATH_BIGINTEGER_SIGN => $x_negative != $y_negative
- );
- }
-
- /**
- * Montgomery Modular Reduction
- *
- * ($x->_prepMontgomery($n))->_montgomery($n) yields $x % $n.
- * {@link http://math.libtomcrypt.com/files/tommath.pdf#page=170 MPM 6.3} provides insights on how this can be
- * improved upon (basically, by using the comba method). gcd($n, 2) must be equal to one for this function
- * to work correctly.
- *
- * @see _prepMontgomery()
- * @see _slidingWindow()
- * @access private
- * @param Array $x
- * @param Array $n
- * @return Array
- */
- function _montgomery($x, $n)
- {
- static $cache = array(
- MATH_BIGINTEGER_VARIABLE => array(),
- MATH_BIGINTEGER_DATA => array()
- );
-
- if ( ($key = array_search($n, $cache[MATH_BIGINTEGER_VARIABLE])) === false ) {
- $key = count($cache[MATH_BIGINTEGER_VARIABLE]);
- $cache[MATH_BIGINTEGER_VARIABLE][] = $x;
- $cache[MATH_BIGINTEGER_DATA][] = $this->_modInverse67108864($n);
- }
-
- $k = count($n);
-
- $result = array(MATH_BIGINTEGER_VALUE => $x);
-
- for ($i = 0; $i < $k; ++$i) {
- $temp = $result[MATH_BIGINTEGER_VALUE][$i] * $cache[MATH_BIGINTEGER_DATA][$key];
- $temp = (int) ($temp - 0x4000000 * ((int) ($temp / 0x4000000)));
- $temp = $this->_regularMultiply(array($temp), $n);
- $temp = array_merge($this->_array_repeat(0, $i), $temp);
- $result = $this->_add($result[MATH_BIGINTEGER_VALUE], false, $temp, false);
- }
-
- $result[MATH_BIGINTEGER_VALUE] = array_slice($result[MATH_BIGINTEGER_VALUE], $k);
-
- if ($this->_compare($result, false, $n, false) >= 0) {
- $result = $this->_subtract($result[MATH_BIGINTEGER_VALUE], false, $n, false);
- }
-
- return $result[MATH_BIGINTEGER_VALUE];
- }
-
- /**
- * Montgomery Multiply
- *
- * Interleaves the montgomery reduction and long multiplication algorithms together as described in
- * {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=13 HAC 14.36}
- *
- * @see _prepMontgomery()
- * @see _montgomery()
- * @access private
- * @param Array $x
- * @param Array $y
- * @param Array $m
- * @return Array
- */
- function _montgomeryMultiply($x, $y, $m)
- {
- $temp = $this->_multiply($x, false, $y, false);
- return $this->_montgomery($temp[MATH_BIGINTEGER_VALUE], $m);
-
- static $cache = array(
- MATH_BIGINTEGER_VARIABLE => array(),
- MATH_BIGINTEGER_DATA => array()
- );
-
- if ( ($key = array_search($m, $cache[MATH_BIGINTEGER_VARIABLE])) === false ) {
- $key = count($cache[MATH_BIGINTEGER_VARIABLE]);
- $cache[MATH_BIGINTEGER_VARIABLE][] = $m;
- $cache[MATH_BIGINTEGER_DATA][] = $this->_modInverse67108864($m);
- }
-
- $n = max(count($x), count($y), count($m));
- $x = array_pad($x, $n, 0);
- $y = array_pad($y, $n, 0);
- $m = array_pad($m, $n, 0);
- $a = array(MATH_BIGINTEGER_VALUE => $this->_array_repeat(0, $n + 1));
- for ($i = 0; $i < $n; ++$i) {
- $temp = $a[MATH_BIGINTEGER_VALUE][0] + $x[$i] * $y[0];
- $temp = (int) ($temp - 0x4000000 * ((int) ($temp / 0x4000000)));
- $temp = $temp * $cache[MATH_BIGINTEGER_DATA][$key];
- $temp = (int) ($temp - 0x4000000 * ((int) ($temp / 0x4000000)));
- $temp = $this->_add($this->_regularMultiply(array($x[$i]), $y), false, $this->_regularMultiply(array($temp), $m), false);
- $a = $this->_add($a[MATH_BIGINTEGER_VALUE], false, $temp[MATH_BIGINTEGER_VALUE], false);
- $a[MATH_BIGINTEGER_VALUE] = array_slice($a[MATH_BIGINTEGER_VALUE], 1);
- }
- if ($this->_compare($a[MATH_BIGINTEGER_VALUE], false, $m, false) >= 0) {
- $a = $this->_subtract($a[MATH_BIGINTEGER_VALUE], false, $m, false);
- }
- return $a[MATH_BIGINTEGER_VALUE];
- }
-
- /**
- * Prepare a number for use in Montgomery Modular Reductions
- *
- * @see _montgomery()
- * @see _slidingWindow()
- * @access private
- * @param Array $x
- * @param Array $n
- * @return Array
- */
- function _prepMontgomery($x, $n)
- {
- $lhs = new Math_BigInteger();
- $lhs->value = array_merge($this->_array_repeat(0, count($n)), $x);
- $rhs = new Math_BigInteger();
- $rhs->value = $n;
-
- list(, $temp) = $lhs->divide($rhs);
- return $temp->value;
- }
-
- /**
- * Modular Inverse of a number mod 2**26 (eg. 67108864)
- *
- * Based off of the bnpInvDigit function implemented and justified in the following URL:
- *
- * {@link http://www-cs-students.stanford.edu/~tjw/jsbn/jsbn.js}
- *
- * The following URL provides more info:
- *
- * {@link http://groups.google.com/group/sci.crypt/msg/7a137205c1be7d85}
- *
- * As for why we do all the bitmasking... strange things can happen when converting from floats to ints. For
- * instance, on some computers, var_dump((int) -4294967297) yields int(-1) and on others, it yields
- * int(-2147483648). To avoid problems stemming from this, we use bitmasks to guarantee that ints aren't
- * auto-converted to floats. The outermost bitmask is present because without it, there's no guarantee that
- * the "residue" returned would be the so-called "common residue". We use fmod, in the last step, because the
- * maximum possible $x is 26 bits and the maximum $result is 16 bits. Thus, we have to be able to handle up to
- * 40 bits, which only 64-bit floating points will support.
- *
- * Thanks to Pedro Gimeno Fortea for input!
- *
- * @see _montgomery()
- * @access private
- * @param Array $x
- * @return Integer
- */
- function _modInverse67108864($x) // 2**26 == 67108864
- {
- $x = -$x[0];
- $result = $x & 0x3; // x**-1 mod 2**2
- $result = ($result * (2 - $x * $result)) & 0xF; // x**-1 mod 2**4
- $result = ($result * (2 - ($x & 0xFF) * $result)) & 0xFF; // x**-1 mod 2**8
- $result = ($result * ((2 - ($x & 0xFFFF) * $result) & 0xFFFF)) & 0xFFFF; // x**-1 mod 2**16
- $result = fmod($result * (2 - fmod($x * $result, 0x4000000)), 0x4000000); // x**-1 mod 2**26
- return $result & 0x3FFFFFF;
- }
-
- /**
- * Calculates modular inverses.
- *
- * Say you have (30 mod 17 * x mod 17) mod 17 == 1. x can be found using modular inverses.
- *
- * Here's an example:
- * <code>
- * <?php
- * include('Math/BigInteger.php');
- *
- * $a = new Math_BigInteger(30);
- * $b = new Math_BigInteger(17);
- *
- * $c = $a->modInverse($b);
- * echo $c->toString(); // outputs 4
- *
- * echo "\r\n";
- *
- * $d = $a->multiply($c);
- * list(, $d) = $d->divide($b);
- * echo $d; // outputs 1 (as per the definition of modular inverse)
- * ?>
- * </code>
- *
- * @param Math_BigInteger $n
- * @return mixed false, if no modular inverse exists, Math_BigInteger, otherwise.
- * @access public
- * @internal See {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=21 HAC 14.64} for more information.
- */
- function modInverse($n)
- {
- switch ( MATH_BIGINTEGER_MODE ) {
- case MATH_BIGINTEGER_MODE_GMP:
- $temp = new Math_BigInteger();
- $temp->value = gmp_invert($this->value, $n->value);
-
- return ( $temp->value === false ) ? false : $this->_normalize($temp);
- }
-
- static $zero, $one;
- if (!isset($zero)) {
- $zero = new Math_BigInteger();
- $one = new Math_BigInteger(1);
- }
-
- // $x mod $n == $x mod -$n.
- $n = $n->abs();
-
- if ($this->compare($zero) < 0) {
- $temp = $this->abs();
- $temp = $temp->modInverse($n);
- return $negated === false ? false : $this->_normalize($n->subtract($temp));
- }
-
- extract($this->extendedGCD($n));
-
- if (!$gcd->equals($one)) {
- return false;
- }
-
- $x = $x->compare($zero) < 0 ? $x->add($n) : $x;
-
- return $this->compare($zero) < 0 ? $this->_normalize($n->subtract($x)) : $this->_normalize($x);
- }
-
- /**
- * Calculates the greatest common divisor and Bézout's identity.
- *
- * Say you have 693 and 609. The GCD is 21. Bézout's identity states that there exist integers x and y such that
- * 693*x + 609*y == 21. In point of fact, there are actually an infinite number of x and y combinations and which
- * combination is returned is dependant upon which mode is in use. See
- * {@link http://en.wikipedia.org/wiki/B%C3%A9zout%27s_identity Bézout's identity - Wikipedia} for more information.
- *
- * Here's an example:
- * <code>
- * <?php
- * include('Math/BigInteger.php');
- *
- * $a = new Math_BigInteger(693);
- * $b = new Math_BigInteger(609);
- *
- * extract($a->extendedGCD($b));
- *
- * echo $gcd->toString() . "\r\n"; // outputs 21
- * echo $a->toString() * $x->toString() + $b->toString() * $y->toString(); // outputs 21
- * ?>
- * </code>
- *
- * @param Math_BigInteger $n
- * @return Math_BigInteger
- * @access public
- * @internal Calculates the GCD using the binary xGCD algorithim described in
- * {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=19 HAC 14.61}. As the text above 14.61 notes,
- * the more traditional algorithim requires "relatively costly multiple-precision divisions".
- */
- function extendedGCD($n)
- {
- switch ( MATH_BIGINTEGER_MODE ) {
- case MATH_BIGINTEGER_MODE_GMP:
- extract(gmp_gcdext($this->value, $n->value));
-
- return array(
- 'gcd' => $this->_normalize(new Math_BigInteger($g)),
- 'x' => $this->_normalize(new Math_BigInteger($s)),
- 'y' => $this->_normalize(new Math_BigInteger($t))
- );
- case MATH_BIGINTEGER_MODE_BCMATH:
- // it might be faster to use the binary xGCD algorithim here, as well, but (1) that algorithim works
- // best when the base is a power of 2 and (2) i don't think it'd make much difference, anyway. as is,
- // the basic extended euclidean algorithim is what we're using.
-
- $u = $this->value;
- $v = $n->value;
-
- $a = '1';
- $b = '0';
- $c = '0';
- $d = '1';
-
- while (bccomp($v, '0', 0) != 0) {
- $q = bcdiv($u, $v, 0);
-
- $temp = $u;
- $u = $v;
- $v = bcsub($temp, bcmul($v, $q, 0), 0);
-
- $temp = $a;
- $a = $c;
- $c = bcsub($temp, bcmul($a, $q, 0), 0);
-
- $temp = $b;
- $b = $d;
- $d = bcsub($temp, bcmul($b, $q, 0), 0);
- }
-
- return array(
- 'gcd' => $this->_normalize(new Math_BigInteger($u)),
- 'x' => $this->_normalize(new Math_BigInteger($a)),
- 'y' => $this->_normalize(new Math_BigInteger($b))
- );
- }
-
- $y = $n->copy();
- $x = $this->copy();
- $g = new Math_BigInteger();
- $g->value = array(1);
-
- while ( !(($x->value[0] & 1)|| ($y->value[0] & 1)) ) {
- $x->_rshift(1);
- $y->_rshift(1);
- $g->_lshift(1);
- }
-
- $u = $x->copy();
- $v = $y->copy();
-
- $a = new Math_BigInteger();
- $b = new Math_BigInteger();
- $c = new Math_BigInteger();
- $d = new Math_BigInteger();
-
- $a->value = $d->value = $g->value = array(1);
- $b->value = $c->value = array();
-
- while ( !empty($u->value) ) {
- while ( !($u->value[0] & 1) ) {
- $u->_rshift(1);
- if ( (!empty($a->value) && ($a->value[0] & 1)) || (!empty($b->value) && ($b->value[0] & 1)) ) {
- $a = $a->add($y);
- $b = $b->subtract($x);
- }
- $a->_rshift(1);
- $b->_rshift(1);
- }
-
- while ( !($v->value[0] & 1) ) {
- $v->_rshift(1);
- if ( (!empty($d->value) && ($d->value[0] & 1)) || (!empty($c->value) && ($c->value[0] & 1)) ) {
- $c = $c->add($y);
- $d = $d->subtract($x);
- }
- $c->_rshift(1);
- $d->_rshift(1);
- }
-
- if ($u->compare($v) >= 0) {
- $u = $u->subtract($v);
- $a = $a->subtract($c);
- $b = $b->subtract($d);
- } else {
- $v = $v->subtract($u);
- $c = $c->subtract($a);
- $d = $d->subtract($b);
- }
- }
-
- return array(
- 'gcd' => $this->_normalize($g->multiply($v)),
- 'x' => $this->_normalize($c),
- 'y' => $this->_normalize($d)
- );
- }
-
- /**
- * Calculates the greatest common divisor
- *
- * Say you have 693 and 609. The GCD is 21.
- *
- * Here's an example:
- * <code>
- * <?php
- * include('Math/BigInteger.php');
- *
- * $a = new Math_BigInteger(693);
- * $b = new Math_BigInteger(609);
- *
- * $gcd = a->extendedGCD($b);
- *
- * echo $gcd->toString() . "\r\n"; // outputs 21
- * ?>
- * </code>
- *
- * @param Math_BigInteger $n
- * @return Math_BigInteger
- * @access public
- */
- function gcd($n)
- {
- extract($this->extendedGCD($n));
- return $gcd;
- }
-
- /**
- * Absolute value.
- *
- * @return Math_BigInteger
- * @access public
- */
- function abs()
- {
- $temp = new Math_BigInteger();
-
- switch ( MATH_BIGINTEGER_MODE ) {
- case MATH_BIGINTEGER_MODE_GMP:
- $temp->value = gmp_abs($this->value);
- break;
- case MATH_BIGINTEGER_MODE_BCMATH:
- $temp->value = (bccomp($this->value, '0', 0) < 0) ? substr($this->value, 1) : $this->value;
- break;
- default:
- $temp->value = $this->value;
- }
-
- return $temp;
- }
-
- /**
- * Compares two numbers.
- *
- * Although one might think !$x->compare($y) means $x != $y, it, in fact, means the opposite. The reason for this is
- * demonstrated thusly:
- *
- * $x > $y: $x->compare($y) > 0
- * $x < $y: $x->compare($y) < 0
- * $x == $y: $x->compare($y) == 0
- *
- * Note how the same comparison operator is used. If you want to test for equality, use $x->equals($y).
- *
- * @param Math_BigInteger $x
- * @return Integer < 0 if $this is less than $x; > 0 if $this is greater than $x, and 0 if they are equal.
- * @access public
- * @see equals()
- * @internal Could return $this->subtract($x), but that's not as fast as what we do do.
- */
- function compare($y)
- {
- switch ( MATH_BIGINTEGER_MODE ) {
- case MATH_BIGINTEGER_MODE_GMP:
- return gmp_cmp($this->value, $y->value);
- case MATH_BIGINTEGER_MODE_BCMATH:
- return bccomp($this->value, $y->value, 0);
- }
-
- return $this->_compare($this->value, $this->is_negative, $y->value, $y->is_negative);
- }
-
- /**
- * Compares two numbers.
- *
- * @param Array $x_value
- * @param Boolean $x_negative
- * @param Array $y_value
- * @param Boolean $y_negative
- * @return Integer
- * @see compare()
- * @access private
- */
- function _compare($x_value, $x_negative, $y_value, $y_negative)
- {
- if ( $x_negative != $y_negative ) {
- return ( !$x_negative && $y_negative ) ? 1 : -1;
- }
-
- $result = $x_negative ? -1 : 1;
-
- if ( count($x_value) != count($y_value) ) {
- return ( count($x_value) > count($y_value) ) ? $result : -$result;
- }
- $size = max(count($x_value), count($y_value));
-
- $x_value = array_pad($x_value, $size, 0);
- $y_value = array_pad($y_value, $size, 0);
-
- for ($i = count($x_value) - 1; $i >= 0; --$i) {
- if ($x_value[$i] != $y_value[$i]) {
- return ( $x_value[$i] > $y_value[$i] ) ? $result : -$result;
- }
- }
-
- return 0;
- }
-
- /**
- * Tests the equality of two numbers.
- *
- * If you need to see if one number is greater than or less than another number, use Math_BigInteger::compare()
- *
- * @param Math_BigInteger $x
- * @return Boolean
- * @access public
- * @see compare()
- */
- function equals($x)
- {
- switch ( MATH_BIGINTEGER_MODE ) {
- case MATH_BIGINTEGER_MODE_GMP:
- return gmp_cmp($this->value, $x->value) == 0;
- default:
- return $this->value === $x->value && $this->is_negative == $x->is_negative;
- }
- }
-
- /**
- * Set Precision
- *
- * Some bitwise operations give different results depending on the precision being used. Examples include left
- * shift, not, and rotates.
- *
- * @param Math_BigInteger $x
- * @access public
- * @return Math_BigInteger
- */
- function setPrecision($bits)
- {
- $this->precision = $bits;
- if ( MATH_BIGINTEGER_MODE != MATH_BIGINTEGER_MODE_BCMATH ) {
- $this->bitmask = new Math_BigInteger(chr((1 << ($bits & 0x7)) - 1) . str_repeat(chr(0xFF), $bits >> 3), 256);
- } else {
- $this->bitmask = new Math_BigInteger(bcpow('2', $bits, 0));
- }
-
- $temp = $this->_normalize($this);
- $this->value = $temp->value;
- }
-
- /**
- * Logical And
- *
- * @param Math_BigInteger $x
- * @access public
- * @internal Implemented per a request by Lluis Pamies i Juarez <lluis _a_ pamies.cat>
- * @return Math_BigInteger
- */
- function bitwise_and($x)
- {
- switch ( MATH_BIGINTEGER_MODE ) {
- case MATH_BIGINTEGER_MODE_GMP:
- $temp = new Math_BigInteger();
- $temp->value = gmp_and($this->value, $x->value);
-
- return $this->_normalize($temp);
- case MATH_BIGINTEGER_MODE_BCMATH:
- $left = $this->toBytes();
- $right = $x->toBytes();
-
- $length = max(strlen($left), strlen($right));
-
- $left = str_pad($left, $length, chr(0), STR_PAD_LEFT);
- $right = str_pad($right, $length, chr(0), STR_PAD_LEFT);
-
- return $this->_normalize(new Math_BigInteger($left & $right, 256));
- }
-
- $result = $this->copy();
-
- $length = min(count($x->value), count($this->value));
-
- $result->value = array_slice($result->value, 0, $length);
-
- for ($i = 0; $i < $length; ++$i) {
- $result->value[$i] = $result->value[$i] & $x->value[$i];
- }
-
- return $this->_normalize($result);
- }
-
- /**
- * Logical Or
- *
- * @param Math_BigInteger $x
- * @access public
- * @internal Implemented per a request by Lluis Pamies i Juarez <lluis _a_ pamies.cat>
- * @return Math_BigInteger
- */
- function bitwise_or($x)
- {
- switch ( MATH_BIGINTEGER_MODE ) {
- case MATH_BIGINTEGER_MODE_GMP:
- $temp = new Math_BigInteger();
- $temp->value = gmp_or($this->value, $x->value);
-
- return $this->_normalize($temp);
- case MATH_BIGINTEGER_MODE_BCMATH:
- $left = $this->toBytes();
- $right = $x->toBytes();
-
- $length = max(strlen($left), strlen($right));
-
- $left = str_pad($left, $length, chr(0), STR_PAD_LEFT);
- $right = str_pad($right, $length, chr(0), STR_PAD_LEFT);
-
- return $this->_normalize(new Math_BigInteger($left | $right, 256));
- }
-
- $length = max(count($this->value), count($x->value));
- $result = $this->copy();
- $result->value = array_pad($result->value, 0, $length);
- $x->value = array_pad($x->value, 0, $length);
-
- for ($i = 0; $i < $length; ++$i) {
- $result->value[$i] = $this->value[$i] | $x->value[$i];
- }
-
- return $this->_normalize($result);
- }
-
- /**
- * Logical Exclusive-Or
- *
- * @param Math_BigInteger $x
- * @access public
- * @internal Implemented per a request by Lluis Pamies i Juarez <lluis _a_ pamies.cat>
- * @return Math_BigInteger
- */
- function bitwise_xor($x)
- {
- switch ( MATH_BIGINTEGER_MODE ) {
- case MATH_BIGINTEGER_MODE_GMP:
- $temp = new Math_BigInteger();
- $temp->value = gmp_xor($this->value, $x->value);
-
- return $this->_normalize($temp);
- case MATH_BIGINTEGER_MODE_BCMATH:
- $left = $this->toBytes();
- $right = $x->toBytes();
-
- $length = max(strlen($left), strlen($right));
-
- $left = str_pad($left, $length, chr(0), STR_PAD_LEFT);
- $right = str_pad($right, $length, chr(0), STR_PAD_LEFT);
-
- return $this->_normalize(new Math_BigInteger($left ^ $right, 256));
- }
-
- $length = max(count($this->value), count($x->value));
- $result = $this->copy();
- $result->value = array_pad($result->value, 0, $length);
- $x->value = array_pad($x->value, 0, $length);
-
- for ($i = 0; $i < $length; ++$i) {
- $result->value[$i] = $this->value[$i] ^ $x->value[$i];
- }
-
- return $this->_normalize($result);
- }
-
- /**
- * Logical Not
- *
- * @access public
- * @internal Implemented per a request by Lluis Pamies i Juarez <lluis _a_ pamies.cat>
- * @return Math_BigInteger
- */
- function bitwise_not()
- {
- // calculuate "not" without regard to $this->precision
- // (will always result in a smaller number. ie. ~1 isn't 1111 1110 - it's 0)
- $temp = $this->toBytes();
- $pre_msb = decbin(ord($temp[0]));
- $temp = ~$temp;
- $msb = decbin(ord($temp[0]));
- if (strlen($msb) == 8) {
- $msb = substr($msb, strpos($msb, '0'));
- }
- $temp[0] = chr(bindec($msb));
-
- // see if we need to add extra leading 1's
- $current_bits = strlen($pre_msb) + 8 * strlen($temp) - 8;
- $new_bits = $this->precision - $current_bits;
- if ($new_bits <= 0) {
- return $this->_normalize(new Math_BigInteger($temp, 256));
- }
-
- // generate as many leading 1's as we need to.
- $leading_ones = chr((1 << ($new_bits & 0x7)) - 1) . str_repeat(chr(0xFF), $new_bits >> 3);
- $this->_base256_lshift($leading_ones, $current_bits);
-
- $temp = str_pad($temp, ceil($this->bits / 8), chr(0), STR_PAD_LEFT);
-
- return $this->_normalize(new Math_BigInteger($leading_ones | $temp, 256));
- }
-
- /**
- * Logical Right Shift
- *
- * Shifts BigInteger's by $shift bits, effectively dividing by 2**$shift.
- *
- * @param Integer $shift
- * @return Math_BigInteger
- * @access public
- * @internal The only version that yields any speed increases is the internal version.
- */
- function bitwise_rightShift($shift)
- {
- $temp = new Math_BigInteger();
-
- switch ( MATH_BIGINTEGER_MODE ) {
- case MATH_BIGINTEGER_MODE_GMP:
- static $two;
-
- if (!isset($two)) {
- $two = gmp_init('2');
- }
-
- $temp->value = gmp_div_q($this->value, gmp_pow($two, $shift));
-
- break;
- case MATH_BIGINTEGER_MODE_BCMATH:
- $temp->value = bcdiv($this->value, bcpow('2', $shift, 0), 0);
-
- break;
- default: // could just replace _lshift with this, but then all _lshift() calls would need to be rewritten
- // and I don't want to do that...
- $temp->value = $this->value;
- $temp->_rshift($shift);
- }
-
- return $this->_normalize($temp);
- }
-
- /**
- * Logical Left Shift
- *
- * Shifts BigInteger's by $shift bits, effectively multiplying by 2**$shift.
- *
- * @param Integer $shift
- * @return Math_BigInteger
- * @access public
- * @internal The only version that yields any speed increases is the internal version.
- */
- function bitwise_leftShift($shift)
- {
- $temp = new Math_BigInteger();
-
- switch ( MATH_BIGINTEGER_MODE ) {
- case MATH_BIGINTEGER_MODE_GMP:
- static $two;
-
- if (!isset($two)) {
- $two = gmp_init('2');
- }
-
- $temp->value = gmp_mul($this->value, gmp_pow($two, $shift));
-
- break;
- case MATH_BIGINTEGER_MODE_BCMATH:
- $temp->value = bcmul($this->value, bcpow('2', $shift, 0), 0);
-
- break;
- default: // could just replace _rshift with this, but then all _lshift() calls would need to be rewritten
- // and I don't want to do that...
- $temp->value = $this->value;
- $temp->_lshift($shift);
- }
-
- return $this->_normalize($temp);
- }
-
- /**
- * Logical Left Rotate
- *
- * Instead of the top x bits being dropped they're appended to the shifted bit string.
- *
- * @param Integer $shift
- * @return Math_BigInteger
- * @access public
- */
- function bitwise_leftRotate($shift)
- {
- $bits = $this->toBytes();
-
- if ($this->precision > 0) {
- $precision = $this->precision;
- if ( MATH_BIGINTEGER_MODE == MATH_BIGINTEGER_MODE_BCMATH ) {
- $mask = $this->bitmask->subtract(new Math_BigInteger(1));
- $mask = $mask->toBytes();
- } else {
- $mask = $this->bitmask->toBytes();
- }
- } else {
- $temp = ord($bits[0]);
- for ($i = 0; $temp >> $i; ++$i);
- $precision = 8 * strlen($bits) - 8 + $i;
- $mask = chr((1 << ($precision & 0x7)) - 1) . str_repeat(chr(0xFF), $precision >> 3);
- }
-
- if ($shift < 0) {
- $shift+= $precision;
- }
- $shift%= $precision;
-
- if (!$shift) {
- return $this->copy();
- }
-
- $left = $this->bitwise_leftShift($shift);
- $left = $left->bitwise_and(new Math_BigInteger($mask, 256));
- $right = $this->bitwise_rightShift($precision - $shift);
- $result = MATH_BIGINTEGER_MODE != MATH_BIGINTEGER_MODE_BCMATH ? $left->bitwise_or($right) : $left->add($right);
- return $this->_normalize($result);
- }
-
- /**
- * Logical Right Rotate
- *
- * Instead of the bottom x bits being dropped they're prepended to the shifted bit string.
- *
- * @param Integer $shift
- * @return Math_BigInteger
- * @access public
- */
- function bitwise_rightRotate($shift)
- {
- return $this->bitwise_leftRotate(-$shift);
- }
-
- /**
- * Set random number generator function
- *
- * $generator should be the name of a random generating function whose first parameter is the minimum
- * value and whose second parameter is the maximum value. If this function needs to be seeded, it should
- * be seeded prior to calling Math_BigInteger::random() or Math_BigInteger::randomPrime()
- *
- * If the random generating function is not explicitly set, it'll be assumed to be mt_rand().
- *
- * @see random()
- * @see randomPrime()
- * @param optional String $generator
- * @access public
- */
- function setRandomGenerator($generator)
- {
- $this->generator = $generator;
- }
-
- /**
- * Generate a random number
- *
- * @param optional Integer $min
- * @param optional Integer $max
- * @return Math_BigInteger
- * @access public
- */
- function random($min = false, $max = false)
- {
- if ($min === false) {
- $min = new Math_BigInteger(0);
- }
-
- if ($max === false) {
- $max = new Math_BigInteger(0x7FFFFFFF);
- }
-
- $compare = $max->compare($min);
-
- if (!$compare) {
- return $this->_normalize($min);
- } else if ($compare < 0) {
- // if $min is bigger then $max, swap $min and $max
- $temp = $max;
- $max = $min;
- $min = $temp;
- }
-
- $generator = $this->generator;
-
- $max = $max->subtract($min);
- $max = ltrim($max->toBytes(), chr(0));
- $size = strlen($max) - 1;
- $random = '';
-
- $bytes = $size & 1;
- for ($i = 0; $i < $bytes; ++$i) {
- $random.= chr($generator(0, 255));
- }
-
- $blocks = $size >> 1;
- for ($i = 0; $i < $blocks; ++$i) {
- // mt_rand(-2147483648, 0x7FFFFFFF) always produces -2147483648 on some systems
- $random.= pack('n', $generator(0, 0xFFFF));
- }
-
- $temp = new Math_BigInteger($random, 256);
- if ($temp->compare(new Math_BigInteger(substr($max, 1), 256)) > 0) {
- $random = chr($generator(0, ord($max[0]) - 1)) . $random;
- } else {
- $random = chr($generator(0, ord($max[0]) )) . $random;
- }
-
- $random = new Math_BigInteger($random, 256);
-
- return $this->_normalize($random->add($min));
- }
-
- /**
- * Generate a random prime number.
- *
- * If there's not a prime within the given range, false will be returned. If more than $timeout seconds have elapsed,
- * give up and return false.
- *
- * @param optional Integer $min
- * @param optional Integer $max
- * @param optional Integer $timeout
- * @return Math_BigInteger
- * @access public
- * @internal See {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap4.pdf#page=15 HAC 4.44}.
- */
- function randomPrime($min = false, $max = false, $timeout = false)
- {
- $compare = $max->compare($min);
-
- if (!$compare) {
- return $min;
- } else if ($compare < 0) {
- // if $min is bigger then $max, swap $min and $max
- $temp = $max;
- $max = $min;
- $min = $temp;
- }
-
- // gmp_nextprime() requires PHP 5 >= 5.2.0 per <http://php.net/gmp-nextprime>.
- if ( MATH_BIGINTEGER_MODE == MATH_BIGINTEGER_MODE_GMP && function_exists('gmp_nextprime') ) {
- // we don't rely on Math_BigInteger::random()'s min / max when gmp_nextprime() is being used since this function
- // does its own checks on $max / $min when gmp_nextprime() is used. When gmp_nextprime() is not used, however,
- // the same $max / $min checks are not performed.
- if ($min === false) {
- $min = new Math_BigInteger(0);
- }
-
- if ($max === false) {
- $max = new Math_BigInteger(0x7FFFFFFF);
- }
-
- $x = $this->random($min, $max);
-
- $x->value = gmp_nextprime($x->value);
-
- if ($x->compare($max) <= 0) {
- return $x;
- }
-
- $x->value = gmp_nextprime($min->value);
-
- if ($x->compare($max) <= 0) {
- return $x;
- }
-
- return false;
- }
-
- static $one, $two;
- if (!isset($one)) {
- $one = new Math_BigInteger(1);
- $two = new Math_BigInteger(2);
- }
-
- $start = time();
-
- $x = $this->random($min, $max);
- if ($x->equals($two)) {
- return $x;
- }
-
- $x->_make_odd();
- if ($x->compare($max) > 0) {
- // if $x > $max then $max is even and if $min == $max then no prime number exists between the specified range
- if ($min->equals($max)) {
- return false;
- }
- $x = $min->copy();
- $x->_make_odd();
- }
-
- $initial_x = $x->copy();
-
- while (true) {
- if ($timeout !== false && time() - $start > $timeout) {
- return false;
- }
-
- if ($x->isPrime()) {
- return $x;
- }
-
- $x = $x->add($two);
-
- if ($x->compare($max) > 0) {
- $x = $min->copy();
- if ($x->equals($two)) {
- return $x;
- }
- $x->_make_odd();
- }
-
- if ($x->equals($initial_x)) {
- return false;
- }
- }
- }
-
- /**
- * Make the current number odd
- *
- * If the current number is odd it'll be unchanged. If it's even, one will be added to it.
- *
- * @see randomPrime()
- * @access private
- */
- function _make_odd()
- {
- switch ( MATH_BIGINTEGER_MODE ) {
- case MATH_BIGINTEGER_MODE_GMP:
- gmp_setbit($this->value, 0);
- break;
- case MATH_BIGINTEGER_MODE_BCMATH:
- if ($this->value[strlen($this->value) - 1] % 2 == 0) {
- $this->value = bcadd($this->value, '1');
- }
- break;
- default:
- $this->value[0] |= 1;
- }
- }
-
- /**
- * Checks a numer to see if it's prime
- *
- * Assuming the $t parameter is not set, this function has an error rate of 2**-80. The main motivation for the
- * $t parameter is distributability. Math_BigInteger::randomPrime() can be distributed accross multiple pageloads
- * on a website instead of just one.
- *
- * @param optional Integer $t
- * @return Boolean
- * @access public
- * @internal Uses the
- * {@link http://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test Miller-Rabin primality test}. See
- * {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap4.pdf#page=8 HAC 4.24}.
- */
- function isPrime($t = false)
- {
- $length = strlen($this->toBytes());
-
- if (!$t) {
- // see HAC 4.49 "Note (controlling the error probability)"
- if ($length >= 163) { $t = 2; } // floor(1300 / 8)
- else if ($length >= 106) { $t = 3; } // floor( 850 / 8)
- else if ($length >= 81 ) { $t = 4; } // floor( 650 / 8)
- else if ($length >= 68 ) { $t = 5; } // floor( 550 / 8)
- else if ($length >= 56 ) { $t = 6; } // floor( 450 / 8)
- else if ($length >= 50 ) { $t = 7; } // floor( 400 / 8)
- else if ($length >= 43 ) { $t = 8; } // floor( 350 / 8)
- else if ($length >= 37 ) { $t = 9; } // floor( 300 / 8)
- else if ($length >= 31 ) { $t = 12; } // floor( 250 / 8)
- else if ($length >= 25 ) { $t = 15; } // floor( 200 / 8)
- else if ($length >= 18 ) { $t = 18; } // floor( 150 / 8)
- else { $t = 27; }
- }
-
- // ie. gmp_testbit($this, 0)
- // ie. isEven() or !isOdd()
- switch ( MATH_BIGINTEGER_MODE ) {
- case MATH_BIGINTEGER_MODE_GMP:
- return gmp_prob_prime($this->value, $t) != 0;
- case MATH_BIGINTEGER_MODE_BCMATH:
- if ($this->value === '2') {
- return true;
- }
- if ($this->value[strlen($this->value) - 1] % 2 == 0) {
- return false;
- }
- break;
- default:
- if ($this->value == array(2)) {
- return true;
- }
- if (~$this->value[0] & 1) {
- return false;
- }
- }
-
- static $primes, $zero, $one, $two;
-
- if (!isset($primes)) {
- $primes = array(
- 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59,
- 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137,
- 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227,
- 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313,
- 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419,
- 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509,
- 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617,
- 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727,
- 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829,
- 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947,
- 953, 967, 971, 977, 983, 991, 997
- );
-
- if ( MATH_BIGINTEGER_MODE != MATH_BIGINTEGER_MODE_INTERNAL ) {
- for ($i = 0; $i < count($primes); ++$i) {
- $primes[$i] = new Math_BigInteger($primes[$i]);
- }
- }
-
- $zero = new Math_BigInteger();
- $one = new Math_BigInteger(1);
- $two = new Math_BigInteger(2);
- }
-
- if ($this->equals($one)) {
- return false;
- }
-
- // see HAC 4.4.1 "Random search for probable primes"
- if ( MATH_BIGINTEGER_MODE != MATH_BIGINTEGER_MODE_INTERNAL ) {
- foreach ($primes as $prime) {
- list(, $r) = $this->divide($prime);
- if ($r->equals($zero)) {
- return $this->equals($prime);
- }
- }
- } else {
- $value = $this->value;
- foreach ($primes as $prime) {
- list(, $r) = $this->_divide_digit($value, $prime);
- if (!$r) {
- return count($value) == 1 && $value[0] == $prime;
- }
- }
- }
-
- $n = $this->copy();
- $n_1 = $n->subtract($one);
- $n_2 = $n->subtract($two);
-
- $r = $n_1->copy();
- $r_value = $r->value;
- // ie. $s = gmp_scan1($n, 0) and $r = gmp_div_q($n, gmp_pow(gmp_init('2'), $s));
- if ( MATH_BIGINTEGER_MODE == MATH_BIGINTEGER_MODE_BCMATH ) {
- $s = 0;
- // if $n was 1, $r would be 0 and this would be an infinite loop, hence our $this->equals($one) check earlier
- while ($r->value[strlen($r->value) - 1] % 2 == 0) {
- $r->value = bcdiv($r->value, '2', 0);
- ++$s;
- }
- } else {
- for ($i = 0, $r_length = count($r_value); $i < $r_length; ++$i) {
- $temp = ~$r_value[$i] & 0xFFFFFF;
- for ($j = 1; ($temp >> $j) & 1; ++$j);
- if ($j != 25) {
- break;
- }
- }
- $s = 26 * $i + $j - 1;
- $r->_rshift($s);
- }
-
- for ($i = 0; $i < $t; ++$i) {
- $a = $this->random($two, $n_2);
- $y = $a->modPow($r, $n);
-
- if (!$y->equals($one) && !$y->equals($n_1)) {
- for ($j = 1; $j < $s && !$y->equals($n_1); ++$j) {
- $y = $y->modPow($two, $n);
- if ($y->equals($one)) {
- return false;
- }
- }
-
- if (!$y->equals($n_1)) {
- return false;
- }
- }
- }
- return true;
- }
-
- /**
- * Logical Left Shift
- *
- * Shifts BigInteger's by $shift bits.
- *
- * @param Integer $shift
- * @access private
- */
- function _lshift($shift)
- {
- if ( $shift == 0 ) {
- return;
- }
-
- $num_digits = (int) ($shift / 26);
- $shift %= 26;
- $shift = 1 << $shift;
-
- $carry = 0;
-
- for ($i = 0; $i < count($this->value); ++$i) {
- $temp = $this->value[$i] * $shift + $carry;
- $carry = (int) ($temp / 0x4000000);
- $this->value[$i] = (int) ($temp - $carry * 0x4000000);
- }
-
- if ( $carry ) {
- $this->value[] = $carry;
- }
-
- while ($num_digits--) {
- array_unshift($this->value, 0);
- }
- }
-
- /**
- * Logical Right Shift
- *
- * Shifts BigInteger's by $shift bits.
- *
- * @param Integer $shift
- * @access private
- */
- function _rshift($shift)
- {
- if ($shift == 0) {
- return;
- }
-
- $num_digits = (int) ($shift / 26);
- $shift %= 26;
- $carry_shift = 26 - $shift;
- $carry_mask = (1 << $shift) - 1;
-
- if ( $num_digits ) {
- $this->value = array_slice($this->value, $num_digits);
- }
-
- $carry = 0;
-
- for ($i = count($this->value) - 1; $i >= 0; --$i) {
- $temp = $this->value[$i] >> $shift | $carry;
- $carry = ($this->value[$i] & $carry_mask) << $carry_shift;
- $this->value[$i] = $temp;
- }
-
- $this->value = $this->_trim($this->value);
- }
-
- /**
- * Normalize
- *
- * Removes leading zeros and truncates (if necessary) to maintain the appropriate precision
- *
- * @param Math_BigInteger
- * @return Math_BigInteger
- * @see _trim()
- * @access private
- */
- function _normalize($result)
- {
- $result->precision = $this->precision;
- $result->bitmask = $this->bitmask;
-
- switch ( MATH_BIGINTEGER_MODE ) {
- case MATH_BIGINTEGER_MODE_GMP:
- if (!empty($result->bitmask->value)) {
- $result->value = gmp_and($result->value, $result->bitmask->value);
- }
-
- return $result;
- case MATH_BIGINTEGER_MODE_BCMATH:
- if (!empty($result->bitmask->value)) {
- $result->value = bcmod($result->value, $result->bitmask->value);
- }
-
- return $result;
- }
-
- $value = &$result->value;
-
- if ( !count($value) ) {
- return $result;
- }
-
- $value = $this->_trim($value);
-
- if (!empty($result->bitmask->value)) {
- $length = min(count($value), count($this->bitmask->value));
- $value = array_slice($value, 0, $length);
-
- for ($i = 0; $i < $length; ++$i) {
- $value[$i] = $value[$i] & $this->bitmask->value[$i];
- }
- }
-
- return $result;
- }
-
- /**
- * Trim
- *
- * Removes leading zeros
- *
- * @return Math_BigInteger
- * @access private
- */
- function _trim($value)
- {
- for ($i = count($value) - 1; $i >= 0; --$i) {
- if ( $value[$i] ) {
- break;
- }
- unset($value[$i]);
- }
-
- return $value;
- }
-
- /**
- * Array Repeat
- *
- * @param $input Array
- * @param $multiplier mixed
- * @return Array
- * @access private
- */
- function _array_repeat($input, $multiplier)
- {
- return ($multiplier) ? array_fill(0, $multiplier, $input) : array();
- }
-
- /**
- * Logical Left Shift
- *
- * Shifts binary strings $shift bits, essentially multiplying by 2**$shift.
- *
- * @param $x String
- * @param $shift Integer
- * @return String
- * @access private
- */
- function _base256_lshift(&$x, $shift)
- {
- if ($shift == 0) {
- return;
- }
-
- $num_bytes = $shift >> 3; // eg. floor($shift/8)
- $shift &= 7; // eg. $shift % 8
-
- $carry = 0;
- for ($i = strlen($x) - 1; $i >= 0; --$i) {
- $temp = ord($x[$i]) << $shift | $carry;
- $x[$i] = chr($temp);
- $carry = $temp >> 8;
- }
- $carry = ($carry != 0) ? chr($carry) : '';
- $x = $carry . $x . str_repeat(chr(0), $num_bytes);
- }
-
- /**
- * Logical Right Shift
- *
- * Shifts binary strings $shift bits, essentially dividing by 2**$shift and returning the remainder.
- *
- * @param $x String
- * @param $shift Integer
- * @return String
- * @access private
- */
- function _base256_rshift(&$x, $shift)
- {
- if ($shift == 0) {
- $x = ltrim($x, chr(0));
- return '';
- }
-
- $num_bytes = $shift >> 3; // eg. floor($shift/8)
- $shift &= 7; // eg. $shift % 8
-
- $remainder = '';
- if ($num_bytes) {
- $start = $num_bytes > strlen($x) ? -strlen($x) : -$num_bytes;
- $remainder = substr($x, $start);
- $x = substr($x, 0, -$num_bytes);
- }
-
- $carry = 0;
- $carry_shift = 8 - $shift;
- for ($i = 0; $i < strlen($x); ++$i) {
- $temp = (ord($x[$i]) >> $shift) | $carry;
- $carry = (ord($x[$i]) << $carry_shift) & 0xFF;
- $x[$i] = chr($temp);
- }
- $x = ltrim($x, chr(0));
-
- $remainder = chr($carry >> $carry_shift) . $remainder;
-
- return ltrim($remainder, chr(0));
- }
-
- // one quirk about how the following functions are implemented is that PHP defines N to be an unsigned long
- // at 32-bits, while java's longs are 64-bits.
-
- /**
- * Converts 32-bit integers to bytes.
- *
- * @param Integer $x
- * @return String
- * @access private
- */
- function _int2bytes($x)
- {
- return ltrim(pack('N', $x), chr(0));
- }
-
- /**
- * Converts bytes to 32-bit integers
- *
- * @param String $x
- * @return Integer
- * @access private
- */
- function _bytes2int($x)
- {
- $temp = unpack('Nint', str_pad($x, 4, chr(0), STR_PAD_LEFT));
- return $temp['int'];
- }
-}
\ No newline at end of file |