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Posted to commits@vcl.apache.org by jf...@apache.org on 2011/06/16 22:18:37 UTC
svn commit: r1136647 [6/11] - in /incubator/vcl/trunk/web/.ht-inc/phpseclib:
./ Crypt/ Math/ Net/ PHP/ PHP/Compat/ PHP/Compat/Function/
Added: incubator/vcl/trunk/web/.ht-inc/phpseclib/Math/BigInteger.php
URL: http://svn.apache.org/viewvc/incubator/vcl/trunk/web/.ht-inc/phpseclib/Math/BigInteger.php?rev=1136647&view=auto
==============================================================================
--- incubator/vcl/trunk/web/.ht-inc/phpseclib/Math/BigInteger.php (added)
+++ incubator/vcl/trunk/web/.ht-inc/phpseclib/Math/BigInteger.php Thu Jun 16 20:18:36 2011
@@ -0,0 +1,3551 @@
+<?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: Permission is hereby granted, free of charge, to any person obtaining a copy
+ * of this software and associated documentation files (the "Software"), to deal
+ * in the Software without restriction, including without limitation the rights
+ * to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
+ * copies of the Software, and to permit persons to whom the Software is
+ * furnished to do so, subject to the following conditions:
+ *
+ * The above copyright notice and this permission notice shall be included in
+ * all copies or substantial portions of the Software.
+ *
+ * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+ * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+ * FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
+ * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
+ * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
+ * OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
+ * THE SOFTWARE.
+ *
+ * @category Math
+ * @package Math_BigInteger
+ * @author Jim Wigginton <te...@php.net>
+ * @copyright MMVI Jim Wigginton
+ * @license http://www.opensource.org/licenses/mit-license.html MIT License
+ * @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 <te...@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, $end = strlen($hex) & 0xFFFFFFF8; $i < $end; $i+=8) {
+ $bits.= str_pad(decbin(hexdec(substr($hex, $i, 8))), 32, '0', STR_PAD_LEFT);
+ }
+ if ($end != strlen($hex)) { // hexdec('') == 0
+ $bits.= str_pad(decbin(hexdec(substr($hex, $end))), strlen($hex) & 7, '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;
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