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Diffstat (limited to 'plugins/OStatus/extlib/Math')
| -rw-r--r-- | plugins/OStatus/extlib/Math/BigInteger.php | 3060 |
1 files changed, 3060 insertions, 0 deletions
diff --git a/plugins/OStatus/extlib/Math/BigInteger.php b/plugins/OStatus/extlib/Math/BigInteger.php new file mode 100644 index 000000000..ce0e08354 --- /dev/null +++ b/plugins/OStatus/extlib/Math/BigInteger.php @@ -0,0 +1,3060 @@ +<?php
+/* vim: set expandtab tabstop=4 shiftwidth=4 softtabstop=4: */
+
+/**
+ * Pure-PHP arbitrary precision integer arithmetic library.
+ *
+ * Supports base-2, base-10, base-16, and base-256 numbers. Uses the GMP or BCMath extensions, if available,
+ * and an internal implementation, otherwise.
+ *
+ * PHP versions 4 and 5
+ *
+ * {@internal (all DocBlock comments regarding implementation - such as the one that follows - refer to the
+ * {@link MATH_BIGINTEGER_MODE_INTERNAL MATH_BIGINTEGER_MODE_INTERNAL} mode)
+ *
+ * Math_BigInteger uses base-2**26 to perform operations such as multiplication and division and
+ * base-2**52 (ie. two base 2**26 digits) to perform addition and subtraction. Because the largest possible
+ * value when multiplying two base-2**26 numbers together is a base-2**52 number, double precision floating
+ * point numbers - numbers that should be supported on most hardware and whose significand is 53 bits - are
+ * used. As a consequence, bitwise operators such as >> and << cannot be used, nor can the modulo operator %,
+ * which only supports integers. Although this fact will slow this library down, the fact that such a high
+ * base is being used should more than compensate.
+ *
+ * When PHP version 6 is officially released, we'll be able to use 64-bit integers. This should, once again,
+ * allow bitwise operators, and will increase the maximum possible base to 2**31 (or 2**62 for addition /
+ * subtraction).
+ *
+ * Useful resources are as follows:
+ *
+ * - {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf Handbook of Applied Cryptography (HAC)}
+ * - {@link http://math.libtomcrypt.com/files/tommath.pdf Multi-Precision Math (MPM)}
+ * - Java's BigInteger classes. See /j2se/src/share/classes/java/math in jdk-1_5_0-src-jrl.zip
+ *
+ * One idea for optimization is to use the comba method to reduce the number of operations performed.
+ * MPM uses this quite extensively. The following URL elaborates:
+ *
+ * {@link http://www.everything2.com/index.pl?node_id=1736418}}}
+ *
+ * Here's 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 1.18 2009/12/04 19:12:18 terrafrost Exp $
+ * @link http://pear.php.net/package/Math_BigInteger
+ */
+
+/**#@+
+ * @access private
+ * @see Math_BigInteger::_slidingWindow()
+ */
+/**
+ * @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);
+/**#@-*/
+
+/**#@+
+ * @access private
+ * @see Math_BigInteger::_montgomery()
+ * @see Math_BigInteger::_barrett()
+ */
+/**
+ * $cache[MATH_BIGINTEGER_VARIABLE] tells us whether or not the cached data is still valid.
+ */
+define('MATH_BIGINTEGER_VARIABLE', 0);
+/**
+ * $cache[MATH_BIGINTEGER_DATA] contains the cached data.
+ */
+define('MATH_BIGINTEGER_DATA', 1);
+/**#@-*/
+
+/**#@+
+ * @access private
+ * @see Math_BigInteger::Math_BigInteger()
+ */
+/**
+ * To use the pure-PHP implementation
+ */
+define('MATH_BIGINTEGER_MODE_INTERNAL', 1);
+/**
+ * To use the BCMath library
+ *
+ * (if enabled; otherwise, the internal implementation will be used)
+ */
+define('MATH_BIGINTEGER_MODE_BCMATH', 2);
+/**
+ * To use the GMP library
+ *
+ * (if present; otherwise, either the BCMath or the internal implementation will be used)
+ */
+define('MATH_BIGINTEGER_MODE_GMP', 3);
+/**#@-*/
+
+/**
+ * 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', 15);
+
+/**
+ * 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 1.0.0RC3
+ * @access public
+ * @package Math_BigInteger
+ */
+class Math_BigInteger {
+ /**
+ * Holds the BigInteger's value.
+ *
+ * @var Array
+ * @access private
+ */
+ var $value;
+
+ /**
+ * Holds the BigInteger's magnitude.
+ *
+ * @var Boolean
+ * @access private
+ */
+ var $is_negative = false;
+
+ /**
+ * 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;
+
+ /**
+ * 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:
+ $this->value = array();
+ }
+
+ if ($x === 0) {
+ return;
+ }
+
+ switch ($base) {
+ case -256:
+ if (ord($x[0]) & 0x80) {
+ $x = ~$x;
+ $this->is_negative = true;
+ }
+ case 256:
+ switch ( MATH_BIGINTEGER_MODE ) {
+ case MATH_BIGINTEGER_MODE_GMP:
+ $sign = $this->is_negative ? '-' : '';
+ $this->value = gmp_init($sign . '0x' . bin2hex($x));
+ break;
+ case MATH_BIGINTEGER_MODE_BCMATH:
+ // round $len to the nearest 4 (thanks, DavidMJ!)
+ $len = (strlen($x) + 3) & 0xFFFFFFFC;
+
+ $x = str_pad($x, $len, chr(0), STR_PAD_LEFT);
+
+ for ($i = 0; $i < $len; $i+= 4) {
+ $this->value = bcmul($this->value, '4294967296'); // 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])));
+ }
+
+ 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);
+ }
+
+ // we don't do four bytes at a time because then numbers larger than 1<<31 would be negative
+ // two's complimented numbers, which would break chr.
+ while (bccomp($current, '0') > 0) {
+ $temp = bcmod($current, 0x1000000);
+ $value = chr($temp >> 16) . chr($temp >> 8) . chr($temp) . $value;
+ $current = bcdiv($current, 0x1000000);
+ }
+
+ return $this->precision > 0 ?
+ substr(str_pad($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 $this->precision > 0 ?
+ substr(str_pad($result, $this->precision >> 3, chr(0), STR_PAD_LEFT), -($this->precision >> 3)) :
+ $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);
+ $result = str_pad($mod->value[0], 7, '0', STR_PAD_LEFT) . $result;
+ }
+ $result = ltrim($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();
+ }
+
+ /**
+ * 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);
+
+ return $this->_normalize($temp);
+ }
+
+ $this_size = count($this->value);
+ $y_size = count($y->value);
+
+ if ($this_size == 0) {
+ return $y->copy();
+ } else if ($y_size == 0) {
+ return $this->copy();
+ }
+
+ // subtract, if appropriate
+ if ( $this->is_negative != $y->is_negative ) {
+ // is $y the negative number?
+ $y_negative = $this->compare($y) > 0;
+
+ $temp = $this->copy();
+ $y = $y->copy();
+ $temp->is_negative = $y->is_negative = false;
+
+ $diff = $temp->compare($y);
+ if ( !$diff ) {
+ $temp = new Math_BigInteger();
+ return $this->_normalize($temp);
+ }
+
+ $temp = $temp->subtract($y);
+
+ $temp->is_negative = ($diff > 0) ? !$y_negative : $y_negative;
+
+ return $this->_normalize($temp);
+ }
+
+ $result = new Math_BigInteger();
+ $carry = 0;
+
+ $size = max($this_size, $y_size);
+ $size+= $size & 1; // rounds $size to the nearest 2.
+
+ $x = array_pad($this->value, $size, 0);
+ $y = array_pad($y->value, $size, 0);
+
+ for ($i = 0; $i < $size - 1; $i+=2) {
+ $sum = $x[$i + 1] * 0x4000000 + $x[$i] + $y[$i + 1] * 0x4000000 + $y[$i] + $carry;
+ $carry = $sum >= MATH_BIGINTEGER_MAX_DIGIT52; // eg. floor($sum / 2**52); only possible values (in any base) are 0 and 1
+ $sum = $carry ? $sum - MATH_BIGINTEGER_MAX_DIGIT52 : $sum;
+
+ $temp = floor($sum / 0x4000000);
+
+ $result->value[] = $sum - 0x4000000 * $temp; // eg. a faster alternative to fmod($sum, 0x4000000)
+ $result->value[] = $temp;
+ }
+
+ if ($carry) {
+ $result->value[] = (int) $carry;
+ }
+
+ $result->is_negative = $this->is_negative;
+
+ return $this->_normalize($result);
+ }
+
+ /**
+ * 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);
+
+ return $this->_normalize($temp);
+ }
+
+ $this_size = count($this->value);
+ $y_size = count($y->value);
+
+ if ($this_size == 0) {
+ $temp = $y->copy();
+ $temp->is_negative = !$temp->is_negative;
+ return $temp;
+ } else if ($y_size == 0) {
+ return $this->copy();
+ }
+
+ // add, if appropriate (ie. -$x - +$y or +$x - -$y)
+ if ( $this->is_negative != $y->is_negative ) {
+ $is_negative = $y->compare($this) > 0;
+
+ $temp = $this->copy();
+ $y = $y->copy();
+ $temp->is_negative = $y->is_negative = false;
+
+ $temp = $temp->add($y);
+
+ $temp->is_negative = $is_negative;
+
+ return $this->_normalize($temp);
+ }
+
+ $diff = $this->compare($y);
+
+ if ( !$diff ) {
+ $temp = new Math_BigInteger();
+ return $this->_normalize($temp);
+ }
+
+ // switch $this and $y around, if appropriate.
+ if ( (!$this->is_negative && $diff < 0) || ($this->is_negative && $diff > 0) ) {
+ $is_negative = $y->is_negative;
+
+ $temp = $this->copy();
+ $y = $y->copy();
+ $temp->is_negative = $y->is_negative = false;
+
+ $temp = $y->subtract($temp);
+ $temp->is_negative = !$is_negative;
+
+ return $this->_normalize($temp);
+ }
+
+ $result = new Math_BigInteger();
+ $carry = 0;
+
+ $size = max($this_size, $y_size);
+ $size+= $size % 2;
+
+ $x = array_pad($this->value, $size, 0);
+ $y = array_pad($y->value, $size, 0);
+
+ for ($i = 0; $i < $size - 1; $i+=2) {
+ $sum = $x[$i + 1] * 0x4000000 + $x[$i] - $y[$i + 1] * 0x4000000 - $y[$i] + $carry;
+ $carry = $sum < 0 ? -1 : 0; // eg. floor($sum / 2**52); only possible values (in any base) are 0 and 1
+ $sum = $carry ? $sum + MATH_BIGINTEGER_MAX_DIGIT52 : $sum;
+
+ $temp = floor($sum / 0x4000000);
+
+ $result->value[] = $sum - 0x4000000 * $temp;
+ $result->value[] = $temp;
+ }
+
+ // $carry shouldn't be anything other than zero, at this point, since we already made sure that $this
+ // was bigger than $y.
+
+ $result->is_negative = $this->is_negative;
+
+ return $this->_normalize($result);
+ }
+
+ /**
+ * 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);
+
+ return $this->_normalize($temp);
+ }
+
+ static $cutoff = false;
+ if ($cutoff === false) {
+ $cutoff = 2 * MATH_BIGINTEGER_KARATSUBA_CUTOFF;
+ }
+
+ if ( $this->equals($x) ) {
+ return $this->_square();
+ }
+
+ $this_length = count($this->value);
+ $x_length = count($x->value);
+
+ if ( !$this_length || !$x_length ) { // a 0 is being multiplied
+ $temp = new Math_BigInteger();
+ return $this->_normalize($temp);
+ }
+
+ $product = min($this_length, $x_length) < $cutoff ? $this->_multiply($x) : $this->_karatsuba($x);
+
+ $product->is_negative = $this->is_negative != $x->is_negative;
+
+ return $this->_normalize($product);
+ }
+
+ /**
+ * 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 _barrett()
+ * @param Math_BigInteger $x
+ * @return Math_BigInteger
+ * @access private
+ */
+ function _multiplyLower($x, $stop)
+ {
+ $this_length = count($this->value);
+ $x_length = count($x->value);
+
+ if ( !$this_length || !$x_length ) { // a 0 is being multiplied
+ return new Math_BigInteger();
+ }
+
+ if ( $this_length < $x_length ) {
+ return $x->_multiplyLower($this, $stop);
+ }
+
+ $product = new Math_BigInteger();
+ $product->value = $this->_array_repeat(0, $this_length + $x_length);
+
+ // the following for loop could be removed if the for loop following it
+ // (the one with nested for loops) initially set $i to 0, but
+ // doing so would also make the result in one set of unnecessary adds,
+ // since on the outermost loops first pass, $product->value[$k] is going
+ // to always be 0
+
+ $carry = 0;
+
+ for ($j = 0; $j < $this_length; $j++) { // ie. $i = 0, $k = $i
+ $temp = $this->value[$j] * $x->value[0] + $carry; // $product->value[$k] == 0
+ $carry = floor($temp / 0x4000000);
+ $product->value[$j] = $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 < $x_length; $i++) {
+ $carry = 0;
+
+ for ($j = 0, $k = $i; $j < $this_length && $k < $stop; $j++, $k++) {
+ $temp = $product->value[$k] + $this->value[$j] * $x->value[$i] + $carry;
+ $carry = floor($temp / 0x4000000);
+ $product->value[$k] = $temp - 0x4000000 * $carry;
+ }
+
+ if ($k < $stop) {
+ $product->value[$k] = $carry;
+ }
+ }
+
+ $product->is_negative = $this->is_negative != $x->is_negative;
+
+ return $product;
+ }
+
+ /**
+ * Performs long multiplication on two BigIntegers
+ *
+ * Modeled after 'multiply' in MutableBigInteger.java.
+ *
+ * @param Math_BigInteger $x
+ * @return Math_BigInteger
+ * @access private
+ */
+ function _multiply($x)
+ {
+ $this_length = count($this->value);
+ $x_length = count($x->value);
+
+ if ( !$this_length || !$x_length ) { // a 0 is being multiplied
+ return new Math_BigInteger();
+ }
+
+ if ( $this_length < $x_length ) {
+ return $x->_multiply($this);
+ }
+
+ $product = new Math_BigInteger();
+ $product->value = $this->_array_repeat(0, $this_length + $x_length);
+
+ // the following for loop could be removed if the for loop following it
+ // (the one with nested for loops) initially set $i to 0, but
+ // doing so would also make the result in one set of unnecessary adds,
+ // since on the outermost loops first pass, $product->value[$k] is going
+ // to always be 0
+
+ $carry = 0;
+
+ for ($j = 0; $j < $this_length; $j++) { // ie. $i = 0
+ $temp = $this->value[$j] * $x->value[0] + $carry; // $product->value[$k] == 0
+ $carry = floor($temp / 0x4000000);
+ $product->value[$j] = $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 < $x_length; $i++) {
+ $carry = 0;
+
+ for ($j = 0, $k = $i; $j < $this_length; $j++, $k++) {
+ $temp = $product->value[$k] + $this->value[$j] * $x->value[$i] + $carry;
+ $carry = floor($temp / 0x4000000);
+ $product->value[$k] = $temp - 0x4000000 * $carry;
+ }
+
+ $product->value[$k] = $carry;
+ }
+
+ $product->is_negative = $this->is_negative != $x->is_negative;
+
+ return $this->_normalize($product);
+ }
+
+ /**
+ * 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 Math_BigInteger $y
+ * @return Math_BigInteger
+ * @access private
+ */
+ function _karatsuba($y)
+ {
+ $x = $this->copy();
+
+ $m = min(count($x->value) >> 1, count($y->value) >> 1);
+
+ if ($m < MATH_BIGINTEGER_KARATSUBA_CUTOFF) {
+ return $x->_multiply($y);
+ }
+
+ $x1 = new Math_BigInteger();
+ $x0 = new Math_BigInteger();
+ $y1 = new Math_BigInteger();
+ $y0 = new Math_BigInteger();
+
+ $x1->value = array_slice($x->value, $m);
+ $x0->value = array_slice($x->value, 0, $m);
+ $y1->value = array_slice($y->value, $m);
+ $y0->value = array_slice($y->value, 0, $m);
+
+ $z2 = $x1->_karatsuba($y1);
+ $z0 = $x0->_karatsuba($y0);
+
+ $z1 = $x1->add($x0);
+ $z1 = $z1->_karatsuba($y1->add($y0));
+ $z1 = $z1->subtract($z2->add($z0));
+
+ $z2->value = array_merge(array_fill(0, 2 * $m, 0), $z2->value);
+ $z1->value = array_merge(array_fill(0, $m, 0), $z1->value);
+
+ $xy = $z2->add($z1);
+ $xy = $xy->add($z0);
+
+ return $xy;
+ }
+
+ /**
+ * Squares a BigInteger
+ *
+ * @return Math_BigInteger
+ * @access private
+ */
+ function _square()
+ {
+ static $cutoff = false;
+ if ($cutoff === false) {
+ $cutoff = 2 * MATH_BIGINTEGER_KARATSUBA_CUTOFF;
+ }
+
+ return count($this->value) < $cutoff ? $this->_baseSquare() : $this->_karatsubaSquare();
+ }
+
+ /**
+ * Performs traditional squaring on two BigIntegers
+ *
+ * Squaring can be done faster than multiplying a number by itself can be. See
+ * {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=7 HAC 14.2.4} /
+ * {@link http://math.libtomcrypt.com/files/tommath.pdf#page=141 MPM 5.3} for more information.
+ *
+ * @return Math_BigInteger
+ * @access private
+ */
+ function _baseSquare()
+ {
+ if ( empty($this->value) ) {
+ return new Math_BigInteger();
+ }
+
+ $square = new Math_BigInteger();
+ $square->value = $this->_array_repeat(0, 2 * count($this->value));
+
+ for ($i = 0, $max_index = count($this->value) - 1; $i <= $max_index; $i++) {
+ $i2 = 2 * $i;
+
+ $temp = $square->value[$i2] + $this->value[$i] * $this->value[$i];
+ $carry = floor($temp / 0x4000000);
+ $square->value[$i2] = $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 * $this->value[$j] * $this->value[$i] + $carry;
+ $carry = floor($temp / 0x4000000);
+ $square->value[$k] = $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;
+ }
+
+ /**
+ * 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 Math_BigInteger $y
+ * @return Math_BigInteger
+ * @access private
+ */
+ function _karatsubaSquare()
+ {
+ $m = count($this->value) >> 1;
+
+ if ($m < MATH_BIGINTEGER_KARATSUBA_CUTOFF) {
+ return $this->_square();
+ }
+
+ $x1 = new Math_BigInteger();
+ $x0 = new Math_BigInteger();
+
+ $x1->value = array_slice($this->value, $m);
+ $x0->value = array_slice($this->value, 0, $m);
+
+ $z2 = $x1->_karatsubaSquare();
+ $z0 = $x0->_karatsubaSquare();
+
+ $z1 = $x1->add($x0);
+ $z1 = $z1->_karatsubaSquare();
+ $z1 = $z1->subtract($z2->add($z0));
+
+ $z2->value = array_merge(array_fill(0, 2 * $m, 0), $z2->value);
+ $z1->value = array_merge(array_fill(0, $m, 0), $z1->value);
+
+ $xx = $z2->add($z1);
+ $xx = $xx->add($z0);
+
+ return $xx;
+ }
+
+ /**
+ * 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);
+ $remainder->value = bcmod($this->value, $y->value);
+
+ if ($remainder->value[0] == '-') {
+ $remainder->value = bcadd($remainder->value, $y->value[0] == '-' ? substr($y->value, 1) : $y->value);
+ }
+
+ return array($this->_normalize($quotient), $this->_normalize($remainder));
+ }
+
+ if (count($y->value) == 1) {
+ $temp = $this->_divide_digit($y->value[0]);
+ $temp[0]->is_negative = $this->is_negative != $y->is_negative;
+ return array($this->_normalize($temp[0]), $this->_normalize($temp[1]));
+ }
+
+ 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);
+
+ $x_max = count($x->value) - 1;
+ $y_max = count($y->value) - 1;
+
+ $quotient = new Math_BigInteger();
+ $quotient->value = $this->_array_repeat(0, $x_max - $y_max + 1);
+
+ // $temp = $y << ($x_max - $y_max-1) in base 2**26
+ $temp = new Math_BigInteger();
+ $temp->value = array_merge($this->_array_repeat(0, $x_max - $y_max), $y->value);
+
+ while ( $x->compare($temp) >= 0 ) {
+ // calculate the "common residue"
+ $quotient->value[$x_max - $y_max]++;
+ $x = $x->subtract($temp);
+ $x_max = count($x->value) - 1;
+ }
+
+ for ($i = $x_max; $i >= $y_max + 1; $i--) {
+ $x_value = array(
+ $x->value[$i],
+ ( $i > 0 ) ? $x->value[$i - 1] : 0,
+ ( $i > 1 ) ? $x->value[$i - 2] : 0
+ );
+ $y_value = array(
+ $y->value[$y_max],
+ ( $y_max > 0 ) ? $y->value[$y_max - 1] : 0
+ );
+
+ $q_index = $i - $y_max - 1;
+ if ($x_value[0] == $y_value[0]) {
+ $quotient->value[$q_index] = 0x3FFFFFF;
+ } else {
+ $quotient->value[$q_index] = floor(
+ ($x_value[0] * 0x4000000 + $x_value[1])
+ /
+ $y_value[0]
+ );
+ }
+
+ $temp = new Math_BigInteger();
+ $temp->value = array($y_value[1], $y_value[0]);
+
+ $lhs = new Math_BigInteger();
+ $lhs->value = array($quotient->value[$q_index]);
+ $lhs = $lhs->multiply($temp);
+
+ $rhs = new Math_BigInteger();
+ $rhs->value = array($x_value[2], $x_value[1], $x_value[0]);
+
+ while ( $lhs->compare($rhs) > 0 ) {
+ $quotient->value[$q_index]--;
+
+ $lhs = new Math_BigInteger();
+ $lhs->value = array($quotient->value[$q_index]);
+ $lhs = $lhs->multiply($temp);
+ }
+
+ $adjust = $this->_array_repeat(0, $q_index);
+ $temp = new Math_BigInteger();
+ $temp->value = array($quotient->value[$q_index]);
+ $temp = $temp->multiply($y);
+ $temp->value = array_merge($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 Math_BigInteger $divisor
+ * @return Array
+ * @access public
+ */
+ function _divide_digit($divisor)
+ {
+ $carry = 0;
+ $result = new Math_BigInteger();
+
+ for ($i = count($this->value) - 1; $i >= 0; $i--) {
+ $temp = 0x4000000 * $carry + $this->value[$i];
+ $result->value[$i] = floor($temp / $divisor);
+ $carry = fmod($temp, $divisor);
+ }
+
+ $remainder = new Math_BigInteger();
+ $remainder->value = array($carry);
+
+ return array($result, $remainder);
+ }
+
+ /**
+ * 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);
+
+ 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 = $this->_square();
+ 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
+
+ $e_length = count($e->value) - 1;
+ $e_bits = decbin($e->value[$e_length]);
+ for ($i = $e_length - 1; $i >= 0; $i--) {
+ $e_bits.= str_pad(decbin($e->value[$i]), 26, '0', STR_PAD_LEFT);
+ }
+
+ $e_length = strlen($e_bits);
+
+ // calculate the appropriate window size.
+ // $window_size == 3 if $window_ranges is between 25 and 81, for example.
+ for ($i = 0, $window_size = 1; $e_length > $window_ranges[$i] && $i < count($window_ranges); $window_size++, $i++);
+ switch ($mode) {
+ case MATH_BIGINTEGER_MONTGOMERY:
+ $reduce = '_montgomery';
+ $prep = '_prepMontgomery';
+ break;
+ case MATH_BIGINTEGER_BARRETT:
+ $reduce = '_barrett';
+ $prep = '_barrett';
+ break;
+ case MATH_BIGINTEGER_POWEROF2:
+ $reduce = '_mod2';
+ $prep = '_mod2';
+ break;
+ case MATH_BIGINTEGER_CLASSIC:
+ $reduce = '_remainder';
+ $prep = '_remainder';
+ break;
+ case MATH_BIGINTEGER_NONE:
+ // ie. do no modular reduction. useful if you want to just do pow as opposed to modPow.
+ $reduce = 'copy';
+ $prep = 'copy';
+ break;
+ default:
+ // an invalid $mode was provided
+ }
+
+ // precompute $this^0 through $this^$window_size
+ $powers = array();
+ $powers[1] = $this->$prep($n);
+ $powers[2] = $powers[1]->_square();
+ $powers[2] = $powers[2]->$reduce($n);
+
+ // we do every other number since substr($e_bits, $i, $j+1) (see below) is supposed to end
+ // in a 1. ie. it's supposed to be odd.
+ $temp = 1 << ($window_size - 1);
+ for ($i = 1; $i < $temp; $i++) {
+ $powers[2 * $i + 1] = $powers[2 * $i - 1]->multiply($powers[2]);
+ $powers[2 * $i + 1] = $powers[2 * $i + 1]->$reduce($n);
+ }
+
+ $result = new Math_BigInteger();
+ $result->value = array(1);
+ $result = $result->$prep($n);
+
+ for ($i = 0; $i < $e_length; ) {
+ if ( !$e_bits[$i] ) {
+ $result = $result->_square();
+ $result = $result->$reduce($n);
+ $i++;
+ } else {
+ for ($j = $window_size - 1; $j > 0; $j--) {
+ if ( !empty($e_bits[$i + $j]) ) {
+ break;
+ }
+ }
+
+ for ($k = 0; $k <= $j; $k++) {// eg. the length of substr($e_bits, $i, $j+1)
+ $result = $result->_square();
+ $result = $result->$reduce($n);
+ }
+
+ $result = $result->multiply($powers[bindec(substr($e_bits, $i, $j + 1))]);
+ $result = $result->$reduce($n);
+
+ $i+=$j + 1;
+ }
+ }
+
+ $result = $result->$reduce($n);
+
+ return $result;
+ }
+
+ /**
+ * Remainder
+ *
+ * A wrapper for the divide function.
+ *
+ * @see divide()
+ * @see _slidingWindow()
+ * @access private
+ * @param Math_BigInteger
+ * @return Math_BigInteger
+ */
+ function _remainder($n)
+ {
+ list(, $temp) = $this->divide($n);
+ return $temp;
+ }
+
+ /**
+ * Modulos for Powers of Two
+ *
+ * Calculates $x%$n, where $n = 2**$e, for some $e. Since this is basically the same as doing $x & ($n-1),
+ * we'll just use this function as a wrapper for doing that.
+ *
+ * @see _slidingWindow()
+ * @access private
+ * @param Math_BigInteger
+ * @return Math_BigInteger
+ */
+ function _mod2($n)
+ {
+ $temp = new Math_BigInteger();
+ $temp->value = array(1);
+ return $this->bitwise_and($n->subtract($temp));
+ }
+
+ /**
+ * Barrett Modular Reduction
+ *
+ * See {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=14 HAC 14.3.3} /
+ * {@link http://math.libtomcrypt.com/files/tommath.pdf#page=165 MPM 6.2.5} for more information. Modified slightly,
+ * so as not to require negative numbers (initially, this script didn't support negative numbers).
+ *
+ * @see _slidingWindow()
+ * @access private
+ * @param Math_BigInteger
+ * @return Math_BigInteger
+ */
+ function _barrett($n)
+ {
+ static $cache = array(
+ MATH_BIGINTEGER_VARIABLE => array(),
+ MATH_BIGINTEGER_DATA => array()
+ );
+
+ $n_length = count($n->value);
+
+ if (count($this->value) > 2 * $n_length) {
+ list(, $temp) = $this->divide($n);
+ return $temp;
+ }
+
+ if ( ($key = array_search($n->value, $cache[MATH_BIGINTEGER_VARIABLE])) === false ) {
+ $key = count($cache[MATH_BIGINTEGER_VARIABLE]);
+ $cache[MATH_BIGINTEGER_VARIABLE][] = $n->value;
+ $temp = new Math_BigInteger();
+ $temp->value = $this->_array_repeat(0, 2 * $n_length);
+ $temp->value[] = 1;
+ list($cache[MATH_BIGINTEGER_DATA][], ) = $temp->divide($n);
+ }
+
+ $temp = new Math_BigInteger();
+ $temp->value = array_slice($this->value, $n_length - 1);
+ $temp = $temp->multiply($cache[MATH_BIGINTEGER_DATA][$key]);
+ $temp->value = array_slice($temp->value, $n_length + 1);
+
+ $result = new Math_BigInteger();
+ $result->value = array_slice($this->value, 0, $n_length + 1);
+ $temp = $temp->_multiplyLower($n, $n_length + 1);
+ // $temp->value == array_slice($temp->multiply($n)->value, 0, $n_length + 1)
+
+ if ($result->compare($temp) < 0) {
+ $corrector = new Math_BigInteger();
+ $corrector->value = $this->_array_repeat(0, $n_length + 1);
+ $corrector->value[] = 1;
+ $result = $result->add($corrector);
+ }
+
+ $result = $result->subtract($temp);
+ while ($result->compare($n) > 0) {
+ $result = $result->subtract($n);
+ }
+
+ return $result;
+ }
+
+ /**
+ * Montgomery Modular Reduction
+ *
+ * ($this->_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($n, 2) must be equal to one for this function
+ * to work correctly.
+ *
+ * @see _prepMontgomery()
+ * @see _slidingWindow()
+ * @access private
+ * @param Math_BigInteger
+ * @return Math_BigInteger
+ */
+ function _montgomery($n)
+ {
+ static $cache = array(
+ MATH_BIGINTEGER_VARIABLE => array(),
+ MATH_BIGINTEGER_DATA => array()
+ );
+
+ if ( ($key = array_search($n->value, $cache[MATH_BIGINTEGER_VARIABLE])) === false ) {
+ $key = count($cache[MATH_BIGINTEGER_VARIABLE]);
+ $cache[MATH_BIGINTEGER_VARIABLE][] = $n->value;
+ $cache[MATH_BIGINTEGER_DATA][] = $n->_modInverse67108864();
+ }
+
+ $k = count($n->value);
+
+ $result = $this->copy();
+
+ for ($i = 0; $i < $k; $i++) {
+ $temp = new Math_BigInteger();
+ $temp->value = array(
+ ($result->value[$i] * $cache[MATH_BIGINTEGER_DATA][$key]) & 0x3FFFFFF
+ );
+
+ $temp = $temp->multiply($n);
+ $temp->value = array_merge($this->_array_repeat(0, $i), $temp->value);
+ $result = $result->add($temp);
+ }
+
+ $result->value = array_slice($result->value, $k);
+
+ if ($result->compare($n) >= 0) {
+ $result = $result->subtract($n);
+ }
+
+ return $result;
+ }
+
+ /**
+ * Prepare a number for use in Montgomery Modular Reductions
+ *
+ * @see _montgomery()
+ * @see _slidingWindow()
+ * @access private
+ * @param Math_BigInteger
+ * @return Math_BigInteger
+ */
+ function _prepMontgomery($n)
+ {
+ $k = count($n->value);
+
+ $temp = new Math_BigInteger();
+ $temp->value = array_merge($this->_array_repeat(0, $k), $this->value);
+
+ list(, $temp) = $temp->divide($n);
+ return $temp;
+ }
+
+ /**
+ * 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
+ * @return Integer
+ */
+ function _modInverse67108864() // 2**26 == 67108864
+ {
+ $x = -$this->value[0];
+ $result = $x & 0x3; // x**-1 mod 2**2
+ $result = ($result * (2 - $x * $result)) & 0xF; // x**-1 mod 2**4
+ $result = ($result * (2 - ($x & 0xFF) * $result)) & 0xFF; // x**-1 mod 2**8
+ $result = ($result * ((2 - ($x & 0xFFFF) * $result) & 0xFFFF)) & 0xFFFF; // x**-1 mod 2**16
+ $result = fmod($result * (2 - fmod($x * $result, 0x4000000)), 0x4000000); // x**-1 mod 2**26
+ return $result & 0x3FFFFFF;
+ }
+
+ /**
+ * Calculates modular inverses.
+ *
+ * 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) {
+ $q = bcdiv($u, $v);
+
+ $temp = $u;
+ $u = $v;
+ $v = bcsub($temp, bcmul($v, $q));
+
+ $temp = $a;
+ $a = $c;
+ $c = bcsub($temp, bcmul($a, $q));
+
+ $temp = $b;
+ $b = $d;
+ $d = bcsub($temp, bcmul($b, $q));
+ }
+
+ 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 = $g->value = array(1);
+
+ while ( !empty($u->value) ) {
+ while ( !($u->value[0] & 1) ) {
+ $u->_rshift(1);
+ if ( ($a->value[0] & 1) || ($b->value[0] & 1) ) {
+ $a = $a->add($y);
+ $b = $b->subtract($x);
+ }
+ $a->_rshift(1);
+ $b->_rshift(1);
+ }
+
+ while ( !($v->value[0] & 1) ) {
+ $v->_rshift(1);
+ if ( ($c->value[0] & 1) || ($d->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) ? 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 $this->sub($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);
+ }
+
+ $x = $this->_normalize($this->copy());
+ $y = $this->_normalize($y);
+
+ if ( $x->is_negative != $y->is_negative ) {
+ return ( !$x->is_negative && $y->is_negative ) ? 1 : -1;
+ }
+
+ $result = $x->is_negative ? -1 : 1;
+
+ if ( count($x->value) != count($y->value) ) {
+ return ( count($x->value) > count($y->value) ) ? $result : -$result;
+ }
+
+ 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));
+ }
+ }
+
+ /**
+ * 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 $two;
+
+ if (empty($two)) {
+ $two = gmp_init('2');
+ }
+
+ $temp->value = gmp_div_q($this->value, gmp_pow($two, $shift));
+
+ break;
+ case MATH_BIGINTEGER_MODE_BCMATH:
+ $temp->value = bcdiv($this->value, bcpow('2', $shift));
+
+ break;
+ default: // could just replace _lshift with this, but then all _lshift() calls would need to be rewritten
+ // and I don't want to do that...
+ $temp->value = $this->value;
+ $temp->_rshift($shift);
+ }
+
+ return $this->_normalize($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 $two;
+
+ if (empty($two)) {
+ $two = gmp_init('2');
+ }
+
+ $temp->value = gmp_mul($this->value, gmp_pow($two, $shift));
+
+ break;
+ case MATH_BIGINTEGER_MODE_BCMATH:
+ $temp->value = bcmul($this->value, bcpow('2', $shift));
+
+ break;
+ default: // could just replace _rshift with this, but then all _lshift() calls would need to be rewritten
+ // and I don't want to do that...
+ $temp->value = $this->value;
+ $temp->_lshift($shift);
+ }
+
+ return $this->_normalize($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)
+ {
+ // 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);
+ }
+
+ $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;
+ }
+
+ $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;
+ }
+
+ $repeat1 = $repeat2 = array();
+
+ $one = new Math_BigInteger(1);
+ $two = new Math_BigInteger(2);
+
+ $start = time();
+
+ do {
+ if ($timeout !== false && time() - $start > $timeout) {
+ return false;
+ }
+
+ $x = $this->random($min, $max);
+ if ($x->equals($two)) {
+ return $x;
+ }
+
+ // make the number odd
+ switch ( MATH_BIGINTEGER_MODE ) {
+ case MATH_BIGINTEGER_MODE_GMP:
+ gmp_setbit($x->value, 0);
+ break;
+ case MATH_BIGINTEGER_MODE_BCMATH:
+ if ($x->value[strlen($x->value) - 1] % 2 == 0) {
+ $x = $x->add($one);
+ }
+ break;
+ default:
+ $x->value[0] |= 1;
+ }
+
+ // if we've seen this number twice before, assume there are no prime numbers within the given range
+ if (in_array($x->value, $repeat1)) {
+ if (in_array($x->value, $repeat2)) {
+ return false;
+ } else {
+ $repeat2[] = $x->value;
+ }
+ } else {
+ $repeat1[] = $x->value;
+ }
+ } while (!$x->isPrime());
+
+ return $x;
+ }
+
+ /**
+ * Checks a numer to see if it's prime
+ *
+ * Assuming the $t parameter is not set, this functoin 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)) {
+ $primes = array(
+ 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59,
+ 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137,
+ 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227,
+ 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313,
+ 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419,
+ 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509,
+ 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617,
+ 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727,
+ 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829,
+ 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947,
+ 953, 967, 971, 977, 983, 991, 997
+ );
+
+ for ($i = 0; $i < count($primes); $i++) {
+ $primes[$i] = new Math_BigInteger($primes[$i]);
+ }
+
+ $zero = new Math_BigInteger();
+ $one = new Math_BigInteger(1);
+ $two = new Math_BigInteger(2);
+ }
+
+ // see HAC 4.4.1 "Random search for probable primes"
+ for ($i = 0; $i < count($primes); $i++) {
+ list(, $r) = $this->divide($primes[$i]);
+ if ($r->equals($zero)) {
+ return false;
+ }
+ }
+
+ $n = $this->copy();
+ $n_1 = $n->subtract($one);
+ $n_2 = $n->subtract($two);
+
+ $r = $n_1->copy();
+ // 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 ) {
+ $s = 0;
+ while ($r->value[strlen($r->value) - 1] % 2 == 0) {
+ $r->value = bcdiv($r->value, 2);
+ $s++;
+ }
+ } else {
+ for ($i = 0; $i < count($r->value); $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 = new Math_BigInteger();
+ $a = $a->random($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;
+ }
+
+ $num_digits = floor($shift / 26);
+ $shift %= 26;
+ $shift = 1 << $shift;
+
+ $carry = 0;
+
+ for ($i = 0; $i < count($this->value); $i++) {
+ $temp = $this->value[$i] * $shift + $carry;
+ $carry = floor($temp / 0x4000000);
+ $this->value[$i] = $temp - $carry * 0x4000000;
+ }
+
+ if ( $carry ) {
+ $this->value[] = $carry;
+ }
+
+ while ($num_digits--) {
+ array_unshift($this->value, 0);
+ }
+ }
+
+ /**
+ * Logical Right Shift
+ *
+ * Shifts BigInteger's by $shift bits.
+ *
+ * @param Integer $shift
+ * @access private
+ */
+ function _rshift($shift)
+ {
+ if ($shift == 0) {
+ return;
+ }
+
+ $num_digits = floor($shift / 26);
+ $shift %= 26;
+ $carry_shift = 26 - $shift;
+ $carry_mask = (1 << $shift) - 1;
+
+ if ( $num_digits ) {
+ $this->value = array_slice($this->value, $num_digits);
+ }
+
+ $carry = 0;
+
+ for ($i = count($this->value) - 1; $i >= 0; $i--) {
+ $temp = $this->value[$i] >> $shift | $carry;
+ $carry = ($this->value[$i] & $carry_mask) << $carry_shift;
+ $this->value[$i] = $temp;
+ }
+ }
+
+ /**
+ * Normalize
+ *
+ * Deletes leading zeros and truncates (if necessary) to maintain the appropriate precision
+ *
+ * @return Math_BigInteger
+ * @access private
+ */
+ function _normalize($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 $result;
+ }
+
+ if ( !count($result->value) ) {
+ return $result;
+ }
+
+ for ($i = count($result->value) - 1; $i >= 0; $i--) {
+ if ( $result->value[$i] ) {
+ break;
+ }
+ unset($result->value[$i]);
+ }
+
+ if (!empty($result->bitmask->value)) {
+ $length = min(count($result->value), count($this->bitmask->value));
+ $result->value = array_slice($result->value, 0, $length);
+
+ for ($i = 0; $i < $length; $i++) {
+ $result->value[$i] = $result->value[$i] & $this->bitmask->value[$i];
+ }
+ }
+
+ return $result;
+ }
+
+ /**
+ * 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 |