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<?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 <terrafrost@php.net>
* @copyright MMVI Jim Wigginton
* @license http://www.opensource.org/licenses/mit-license.html MIT License
* @version $Id$
* @link http://pear.php.net/package/Math_BigInteger
*/
/**#@+
* Reduction constants
*
* @access private
* @see Math_BigInteger::_reduce()
*/
/**
* @see Math_BigInteger::_montgomery()
* @see Math_BigInteger::_prepMontgomery()
*/
define('MATH_BIGINTEGER_MONTGOMERY', 0);
/**
* @see Math_BigInteger::_barrett()
*/
define('MATH_BIGINTEGER_BARRETT', 1);
/**
* @see Math_BigInteger::_mod2()
*/
define('MATH_BIGINTEGER_POWEROF2', 2);
/**
* @see Math_BigInteger::_remainder()
*/
define('MATH_BIGINTEGER_CLASSIC', 3);
/**
* @see Math_BigInteger::__clone()
*/
define('MATH_BIGINTEGER_NONE', 4);
/**#@-*/
/**#@+
* Array constants
*
* Rather than create a thousands and thousands of new Math_BigInteger objects in repeated function calls to add() and
* multiply() or whatever, we'll just work directly on arrays, taking them in as parameters and returning them.
*
* @access private
*/
/**
* $result[MATH_BIGINTEGER_VALUE] contains the value.
*/
define('MATH_BIGINTEGER_VALUE', 0);
/**
* $result[MATH_BIGINTEGER_SIGN] contains the sign.
*/
define('MATH_BIGINTEGER_SIGN', 1);
/**#@-*/
/**#@+
* @access private
* @see Math_BigInteger::_montgomery()
* @see Math_BigInteger::_barrett()
*/
/**
* Cache constants
*
* $cache[MATH_BIGINTEGER_VARIABLE] tells us whether or not the cached data is still valid.
*/
define('MATH_BIGINTEGER_VARIABLE', 0);
/**
* $cache[MATH_BIGINTEGER_DATA] contains the cached data.
*/
define('MATH_BIGINTEGER_DATA', 1);
/**#@-*/
/**#@+
* Mode constants.
*
* @access private
* @see Math_BigInteger::Math_BigInteger()
*/
/**
* To use the pure-PHP implementation
*/
define('MATH_BIGINTEGER_MODE_INTERNAL', 1);
/**
* To use the BCMath library
*
* (if enabled; otherwise, the internal implementation will be used)
*/
define('MATH_BIGINTEGER_MODE_BCMATH', 2);
/**
* To use the GMP library
*
* (if present; otherwise, either the BCMath or the internal implementation will be used)
*/
define('MATH_BIGINTEGER_MODE_GMP', 3);
/**#@-*/
/**
* The largest digit that may be used in addition / subtraction
*
* (we do pow(2, 52) instead of using 4503599627370496, directly, because some PHP installations
* will truncate 4503599627370496)
*
* @access private
*/
define('MATH_BIGINTEGER_MAX_DIGIT52', pow(2, 52));
/**
* Karatsuba Cutoff
*
* At what point do we switch between Karatsuba multiplication and schoolbook long multiplication?
*
* @access private
*/
define('MATH_BIGINTEGER_KARATSUBA_CUTOFF', 25);
/**
* Pure-PHP arbitrary precision integer arithmetic library. Supports base-2, base-10, base-16, and base-256
* numbers.
*
* @author Jim Wigginton <terrafrost@php.net>
* @version 1.0.0RC4
* @access public
* @package Math_BigInteger
*/
class Math_BigInteger {
/**
* Holds the BigInteger's value.
*
* @var Array
* @access private
*/
var $value;
/**
* Holds the BigInteger's magnitude.
*
* @var Boolean
* @access private
*/
var $is_negative = false;
/**
* Random number generator function
*
* @see setRandomGenerator()
* @access private
*/
var $generator = 'mt_rand';
/**
* Precision
*
* @see setPrecision()
* @access private
*/
var $precision = -1;
/**
* Precision Bitmask
*
* @see setPrecision()
* @access private
*/
var $bitmask = false;
/**
* Mode independant value used for serialization.
*
* If the bcmath or gmp extensions are installed $this->value will be a non-serializable resource, hence the need for
* a variable that'll be serializable regardless of whether or not extensions are being used. Unlike $this->value,
* however, $this->hex is only calculated when $this->__sleep() is called.
*
* @see __sleep()
* @see __wakeup()
* @var String
* @access private
*/
var $hex;
/**
* Converts base-2, base-10, base-16, and binary strings (eg. base-256) to BigIntegers.
*
* If the second parameter - $base - is negative, then it will be assumed that the number's are encoded using
* two's compliment. The sole exception to this is -10, which is treated the same as 10 is.
*
* Here's an example:
* <code>
* <?php
* include('Math/BigInteger.php');
*
* $a = new Math_BigInteger('0x32', 16); // 50 in base-16
*
* echo $a->toString(); // outputs 50
* ?>
* </code>
*
* @param optional $x base-10 number or base-$base number if $base set.
* @param optional integer $base
* @return Math_BigInteger
* @access public
*/
function Math_BigInteger($x = 0, $base = 10)
{
if ( !defined('MATH_BIGINTEGER_MODE') ) {
switch (true) {
case extension_loaded('gmp'):
define('MATH_BIGINTEGER_MODE', MATH_BIGINTEGER_MODE_GMP);
break;
case extension_loaded('bcmath'):
define('MATH_BIGINTEGER_MODE', MATH_BIGINTEGER_MODE_BCMATH);
break;
default:
define('MATH_BIGINTEGER_MODE', MATH_BIGINTEGER_MODE_INTERNAL);
}
}
switch ( MATH_BIGINTEGER_MODE ) {
case MATH_BIGINTEGER_MODE_GMP:
if (is_resource($x) && get_resource_type($x) == 'GMP integer') {
$this->value = $x;
return;
}
$this->value = gmp_init(0);
break;
case MATH_BIGINTEGER_MODE_BCMATH:
$this->value = '0';
break;
default:
$this->value = array();
}
// '0' counts as empty() but when the base is 256 '0' is equal to ord('0') or 48
// '0' is the only value like this per http://php.net/empty
if (empty($x) && (abs($base) != 256 || $x !== '0')) {
return;
}
switch ($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', 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 $this->_normalize($n->subtract($temp));
}
extract($this->extendedGCD($n));
if (!$gcd->equals($one)) {
return false;
}
$x = $x->compare($zero) < 0 ? $x->add($n) : $x;
return $this->compare($zero) < 0 ? $this->_normalize($n->subtract($x)) : $this->_normalize($x);
}
/**
* Calculates the greatest common divisor and Bézout's identity.
*
* Say you have 693 and 609. The GCD is 21. Bézout's identity states that there exist integers x and y such that
* 693*x + 609*y == 21. In point of fact, there are actually an infinite number of x and y combinations and which
* combination is returned is dependant upon which mode is in use. See
* {@link http://en.wikipedia.org/wiki/B%C3%A9zout%27s_identity Bézout's identity - Wikipedia} for more information.
*
* Here's an example:
* <code>
* <?php
* include('Math/BigInteger.php');
*
* $a = new Math_BigInteger(693);
* $b = new Math_BigInteger(609);
*
* extract($a->extendedGCD($b));
*
* echo $gcd->toString() . "\r\n"; // outputs 21
* echo $a->toString() * $x->toString() + $b->toString() * $y->toString(); // outputs 21
* ?>
* </code>
*
* @param Math_BigInteger $n
* @return Math_BigInteger
* @access public
* @internal Calculates the GCD using the binary xGCD algorithim described in
* {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=19 HAC 14.61}. As the text above 14.61 notes,
* the more traditional algorithim requires "relatively costly multiple-precision divisions".
*/
function extendedGCD($n)
{
switch ( MATH_BIGINTEGER_MODE ) {
case MATH_BIGINTEGER_MODE_GMP:
extract(gmp_gcdext($this->value, $n->value));
return array(
'gcd' => $this->_normalize(new Math_BigInteger($g)),
'x' => $this->_normalize(new Math_BigInteger($s)),
'y' => $this->_normalize(new Math_BigInteger($t))
);
case MATH_BIGINTEGER_MODE_BCMATH:
// it might be faster to use the binary xGCD algorithim here, as well, but (1) that algorithim works
// best when the base is a power of 2 and (2) i don't think it'd make much difference, anyway. as is,
// the basic extended euclidean algorithim is what we're using.
$u = $this->value;
$v = $n->value;
$a = '1';
$b = '0';
$c = '0';
$d = '1';
while (bccomp($v, '0', 0) != 0) {
$q = bcdiv($u, $v, 0);
$temp = $u;
$u = $v;
$v = bcsub($temp, bcmul($v, $q, 0), 0);
$temp = $a;
$a = $c;
$c = bcsub($temp, bcmul($a, $q, 0), 0);
$temp = $b;
$b = $d;
$d = bcsub($temp, bcmul($b, $q, 0), 0);
}
return array(
'gcd' => $this->_normalize(new Math_BigInteger($u)),
'x' => $this->_normalize(new Math_BigInteger($a)),
'y' => $this->_normalize(new Math_BigInteger($b))
);
}
$y = $n->copy();
$x = $this->copy();
$g = new Math_BigInteger();
$g->value = array(1);
while ( !(($x->value[0] & 1)|| ($y->value[0] & 1)) ) {
$x->_rshift(1);
$y->_rshift(1);
$g->_lshift(1);
}
$u = $x->copy();
$v = $y->copy();
$a = new Math_BigInteger();
$b = new Math_BigInteger();
$c = new Math_BigInteger();
$d = new Math_BigInteger();
$a->value = $d->value = $g->value = array(1);
$b->value = $c->value = array();
while ( !empty($u->value) ) {
while ( !($u->value[0] & 1) ) {
$u->_rshift(1);
if ( (!empty($a->value) && ($a->value[0] & 1)) || (!empty($b->value) && ($b->value[0] & 1)) ) {
$a = $a->add($y);
$b = $b->subtract($x);
}
$a->_rshift(1);
$b->_rshift(1);
}
while ( !($v->value[0] & 1) ) {
$v->_rshift(1);
if ( (!empty($d->value) && ($d->value[0] & 1)) || (!empty($c->value) && ($c->value[0] & 1)) ) {
$c = $c->add($y);
$d = $d->subtract($x);
}
$c->_rshift(1);
$d->_rshift(1);
}
if ($u->compare($v) >= 0) {
$u = $u->subtract($v);
$a = $a->subtract($c);
$b = $b->subtract($d);
} else {
$v = $v->subtract($u);
$c = $c->subtract($a);
$d = $d->subtract($b);
}
}
return array(
'gcd' => $this->_normalize($g->multiply($v)),
'x' => $this->_normalize($c),
'y' => $this->_normalize($d)
);
}
/**
* Calculates the greatest common divisor
*
* Say you have 693 and 609. The GCD is 21.
*
* Here's an example:
* <code>
* <?php
* include('Math/BigInteger.php');
*
* $a = new Math_BigInteger(693);
* $b = new Math_BigInteger(609);
*
* $gcd = a->extendedGCD($b);
*
* echo $gcd->toString() . "\r\n"; // outputs 21
* ?>
* </code>
*
* @param Math_BigInteger $n
* @return Math_BigInteger
* @access public
*/
function gcd($n)
{
extract($this->extendedGCD($n));
return $gcd;
}
/**
* Absolute value.
*
* @return Math_BigInteger
* @access public
*/
function abs()
{
$temp = new Math_BigInteger();
switch ( MATH_BIGINTEGER_MODE ) {
case MATH_BIGINTEGER_MODE_GMP:
$temp->value = gmp_abs($this->value);
break;
case MATH_BIGINTEGER_MODE_BCMATH:
$temp->value = (bccomp($this->value, '0', 0) < 0) ? substr($this->value, 1) : $this->value;
break;
default:
$temp->value = $this->value;
}
return $temp;
}
/**
* Compares two numbers.
*
* Although one might think !$x->compare($y) means $x != $y, it, in fact, means the opposite. The reason for this is
* demonstrated thusly:
*
* $x > $y: $x->compare($y) > 0
* $x < $y: $x->compare($y) < 0
* $x == $y: $x->compare($y) == 0
*
* Note how the same comparison operator is used. If you want to test for equality, use $x->equals($y).
*
* @param Math_BigInteger $x
* @return Integer < 0 if $this is less than $x; > 0 if $this is greater than $x, and 0 if they are equal.
* @access public
* @see equals()
* @internal Could return $this->subtract($x), but that's not as fast as what we do do.
*/
function compare($y)
{
switch ( MATH_BIGINTEGER_MODE ) {
case MATH_BIGINTEGER_MODE_GMP:
return gmp_cmp($this->value, $y->value);
case MATH_BIGINTEGER_MODE_BCMATH:
return bccomp($this->value, $y->value, 0);
}
return $this->_compare($this->value, $this->is_negative, $y->value, $y->is_negative);
}
/**
* Compares two numbers.
*
* @param Array $x_value
* @param Boolean $x_negative
* @param Array $y_value
* @param Boolean $y_negative
* @return Integer
* @see compare()
* @access private
*/
function _compare($x_value, $x_negative, $y_value, $y_negative)
{
if ( $x_negative != $y_negative ) {
return ( !$x_negative && $y_negative ) ? 1 : -1;
}
$result = $x_negative ? -1 : 1;
if ( count($x_value) != count($y_value) ) {
return ( count($x_value) > count($y_value) ) ? $result : -$result;
}
$size = max(count($x_value), count($y_value));
$x_value = array_pad($x_value, $size, 0);
$y_value = array_pad($y_value, $size, 0);
for ($i = count($x_value) - 1; $i >= 0; --$i) {
if ($x_value[$i] != $y_value[$i]) {
return ( $x_value[$i] > $y_value[$i] ) ? $result : -$result;
}
}
return 0;
}
/**
* Tests the equality of two numbers.
*
* If you need to see if one number is greater than or less than another number, use Math_BigInteger::compare()
*
* @param Math_BigInteger $x
* @return Boolean
* @access public
* @see compare()
*/
function equals($x)
{
switch ( MATH_BIGINTEGER_MODE ) {
case MATH_BIGINTEGER_MODE_GMP:
return gmp_cmp($this->value, $x->value) == 0;
default:
return $this->value === $x->value && $this->is_negative == $x->is_negative;
}
}
/**
* Set Precision
*
* Some bitwise operations give different results depending on the precision being used. Examples include left
* shift, not, and rotates.
*
* @param Math_BigInteger $x
* @access public
* @return Math_BigInteger
*/
function setPrecision($bits)
{
$this->precision = $bits;
if ( MATH_BIGINTEGER_MODE != MATH_BIGINTEGER_MODE_BCMATH ) {
$this->bitmask = new Math_BigInteger(chr((1 << ($bits & 0x7)) - 1) . str_repeat(chr(0xFF), $bits >> 3), 256);
} else {
$this->bitmask = new Math_BigInteger(bcpow('2', $bits, 0));
}
$temp = $this->_normalize($this);
$this->value = $temp->value;
}
/**
* Logical And
*
* @param Math_BigInteger $x
* @access public
* @internal Implemented per a request by Lluis Pamies i Juarez <lluis _a_ pamies.cat>
* @return Math_BigInteger
*/
function bitwise_and($x)
{
switch ( MATH_BIGINTEGER_MODE ) {
case MATH_BIGINTEGER_MODE_GMP:
$temp = new Math_BigInteger();
$temp->value = gmp_and($this->value, $x->value);
return $this->_normalize($temp);
case MATH_BIGINTEGER_MODE_BCMATH:
$left = $this->toBytes();
$right = $x->toBytes();
$length = max(strlen($left), strlen($right));
$left = str_pad($left, $length, chr(0), STR_PAD_LEFT);
$right = str_pad($right, $length, chr(0), STR_PAD_LEFT);
return $this->_normalize(new Math_BigInteger($left & $right, 256));
}
$result = $this->copy();
$length = min(count($x->value), count($this->value));
$result->value = array_slice($result->value, 0, $length);
for ($i = 0; $i < $length; ++$i) {
$result->value[$i] = $result->value[$i] & $x->value[$i];
}
return $this->_normalize($result);
}
/**
* Logical Or
*
* @param Math_BigInteger $x
* @access public
* @internal Implemented per a request by Lluis Pamies i Juarez <lluis _a_ pamies.cat>
* @return Math_BigInteger
*/
function bitwise_or($x)
{
switch ( MATH_BIGINTEGER_MODE ) {
case MATH_BIGINTEGER_MODE_GMP:
$temp = new Math_BigInteger();
$temp->value = gmp_or($this->value, $x->value);
return $this->_normalize($temp);
case MATH_BIGINTEGER_MODE_BCMATH:
$left = $this->toBytes();
$right = $x->toBytes();
$length = max(strlen($left), strlen($right));
$left = str_pad($left, $length, chr(0), STR_PAD_LEFT);
$right = str_pad($right, $length, chr(0), STR_PAD_LEFT);
return $this->_normalize(new Math_BigInteger($left | $right, 256));
}
$length = max(count($this->value), count($x->value));
$result = $this->copy();
$result->value = array_pad($result->value, 0, $length);
$x->value = array_pad($x->value, 0, $length);
for ($i = 0; $i < $length; ++$i) {
$result->value[$i] = $this->value[$i] | $x->value[$i];
}
return $this->_normalize($result);
}
/**
* Logical Exclusive-Or
*
* @param Math_BigInteger $x
* @access public
* @internal Implemented per a request by Lluis Pamies i Juarez <lluis _a_ pamies.cat>
* @return Math_BigInteger
*/
function bitwise_xor($x)
{
switch ( MATH_BIGINTEGER_MODE ) {
case MATH_BIGINTEGER_MODE_GMP:
$temp = new Math_BigInteger();
$temp->value = gmp_xor($this->value, $x->value);
return $this->_normalize($temp);
case MATH_BIGINTEGER_MODE_BCMATH:
$left = $this->toBytes();
$right = $x->toBytes();
$length = max(strlen($left), strlen($right));
$left = str_pad($left, $length, chr(0), STR_PAD_LEFT);
$right = str_pad($right, $length, chr(0), STR_PAD_LEFT);
return $this->_normalize(new Math_BigInteger($left ^ $right, 256));
}
$length = max(count($this->value), count($x->value));
$result = $this->copy();
$result->value = array_pad($result->value, 0, $length);
$x->value = array_pad($x->value, 0, $length);
for ($i = 0; $i < $length; ++$i) {
$result->value[$i] = $this->value[$i] ^ $x->value[$i];
}
return $this->_normalize($result);
}
/**
* Logical Not
*
* @access public
* @internal Implemented per a request by Lluis Pamies i Juarez <lluis _a_ pamies.cat>
* @return Math_BigInteger
*/
function bitwise_not()
{
// calculuate "not" without regard to $this->precision
// (will always result in a smaller number. ie. ~1 isn't 1111 1110 - it's 0)
$temp = $this->toBytes();
$pre_msb = decbin(ord($temp[0]));
$temp = ~$temp;
$msb = decbin(ord($temp[0]));
if (strlen($msb) == 8) {
$msb = substr($msb, strpos($msb, '0'));
}
$temp[0] = chr(bindec($msb));
// see if we need to add extra leading 1's
$current_bits = strlen($pre_msb) + 8 * strlen($temp) - 8;
$new_bits = $this->precision - $current_bits;
if ($new_bits <= 0) {
return $this->_normalize(new Math_BigInteger($temp, 256));
}
// generate as many leading 1's as we need to.
$leading_ones = chr((1 << ($new_bits & 0x7)) - 1) . str_repeat(chr(0xFF), $new_bits >> 3);
$this->_base256_lshift($leading_ones, $current_bits);
$temp = str_pad($temp, ceil($this->bits / 8), chr(0), STR_PAD_LEFT);
return $this->_normalize(new Math_BigInteger($leading_ones | $temp, 256));
}
/**
* Logical Right Shift
*
* Shifts BigInteger's by $shift bits, effectively dividing by 2**$shift.
*
* @param Integer $shift
* @return Math_BigInteger
* @access public
* @internal The only version that yields any speed increases is the internal version.
*/
function bitwise_rightShift($shift)
{
$temp = new Math_BigInteger();
switch ( MATH_BIGINTEGER_MODE ) {
case MATH_BIGINTEGER_MODE_GMP:
static $two;
if (!isset($two)) {
$two = gmp_init('2');
}
$temp->value = gmp_div_q($this->value, gmp_pow($two, $shift));
break;
case MATH_BIGINTEGER_MODE_BCMATH:
$temp->value = bcdiv($this->value, bcpow('2', $shift, 0), 0);
break;
default: // could just replace _lshift with this, but then all _lshift() calls would need to be rewritten
// and I don't want to do that...
$temp->value = $this->value;
$temp->_rshift($shift);
}
return $this->_normalize($temp);
}
/**
* Logical Left Shift
*
* Shifts BigInteger's by $shift bits, effectively multiplying by 2**$shift.
*
* @param Integer $shift
* @return Math_BigInteger
* @access public
* @internal The only version that yields any speed increases is the internal version.
*/
function bitwise_leftShift($shift)
{
$temp = new Math_BigInteger();
switch ( MATH_BIGINTEGER_MODE ) {
case MATH_BIGINTEGER_MODE_GMP:
static $two;
if (!isset($two)) {
$two = gmp_init('2');
}
$temp->value = gmp_mul($this->value, gmp_pow($two, $shift));
break;
case MATH_BIGINTEGER_MODE_BCMATH:
$temp->value = bcmul($this->value, bcpow('2', $shift, 0), 0);
break;
default: // could just replace _rshift with this, but then all _lshift() calls would need to be rewritten
// and I don't want to do that...
$temp->value = $this->value;
$temp->_lshift($shift);
}
return $this->_normalize($temp);
}
/**
* Logical Left Rotate
*
* Instead of the top x bits being dropped they're appended to the shifted bit string.
*
* @param Integer $shift
* @return Math_BigInteger
* @access public
*/
function bitwise_leftRotate($shift)
{
$bits = $this->toBytes();
if ($this->precision > 0) {
$precision = $this->precision;
if ( MATH_BIGINTEGER_MODE == MATH_BIGINTEGER_MODE_BCMATH ) {
$mask = $this->bitmask->subtract(new Math_BigInteger(1));
$mask = $mask->toBytes();
} else {
$mask = $this->bitmask->toBytes();
}
} else {
$temp = ord($bits[0]);
for ($i = 0; $temp >> $i; ++$i);
$precision = 8 * strlen($bits) - 8 + $i;
$mask = chr((1 << ($precision & 0x7)) - 1) . str_repeat(chr(0xFF), $precision >> 3);
}
if ($shift < 0) {
$shift+= $precision;
}
$shift%= $precision;
if (!$shift) {
return $this->copy();
}
$left = $this->bitwise_leftShift($shift);
$left = $left->bitwise_and(new Math_BigInteger($mask, 256));
$right = $this->bitwise_rightShift($precision - $shift);
$result = MATH_BIGINTEGER_MODE != MATH_BIGINTEGER_MODE_BCMATH ? $left->bitwise_or($right) : $left->add($right);
return $this->_normalize($result);
}
/**
* Logical Right Rotate
*
* Instead of the bottom x bits being dropped they're prepended to the shifted bit string.
*
* @param Integer $shift
* @return Math_BigInteger
* @access public
*/
function bitwise_rightRotate($shift)
{
return $this->bitwise_leftRotate(-$shift);
}
/**
* Set random number generator function
*
* $generator should be the name of a random generating function whose first parameter is the minimum
* value and whose second parameter is the maximum value. If this function needs to be seeded, it should
* be seeded prior to calling Math_BigInteger::random() or Math_BigInteger::randomPrime()
*
* If the random generating function is not explicitly set, it'll be assumed to be mt_rand().
*
* @see random()
* @see randomPrime()
* @param optional String $generator
* @access public
*/
function setRandomGenerator($generator)
{
$this->generator = $generator;
}
/**
* Generate a random number
*
* @param optional Integer $min
* @param optional Integer $max
* @return Math_BigInteger
* @access public
*/
function random($min = false, $max = false)
{
if ($min === false) {
$min = new Math_BigInteger(0);
}
if ($max === false) {
$max = new Math_BigInteger(0x7FFFFFFF);
}
$compare = $max->compare($min);
if (!$compare) {
return $this->_normalize($min);
} else if ($compare < 0) {
// if $min is bigger then $max, swap $min and $max
$temp = $max;
$max = $min;
$min = $temp;
}
$generator = $this->generator;
$max = $max->subtract($min);
$max = ltrim($max->toBytes(), chr(0));
$size = strlen($max) - 1;
$random = '';
$bytes = $size & 1;
for ($i = 0; $i < $bytes; ++$i) {
$random.= chr($generator(0, 255));
}
$blocks = $size >> 1;
for ($i = 0; $i < $blocks; ++$i) {
// mt_rand(-2147483648, 0x7FFFFFFF) always produces -2147483648 on some systems
$random.= pack('n', $generator(0, 0xFFFF));
}
$temp = new Math_BigInteger($random, 256);
if ($temp->compare(new Math_BigInteger(substr($max, 1), 256)) > 0) {
$random = chr($generator(0, ord($max[0]) - 1)) . $random;
} else {
$random = chr($generator(0, ord($max[0]) )) . $random;
}
$random = new Math_BigInteger($random, 256);
return $this->_normalize($random->add($min));
}
/**
* Generate a random prime number.
*
* If there's not a prime within the given range, false will be returned. If more than $timeout seconds have elapsed,
* give up and return false.
*
* @param optional Integer $min
* @param optional Integer $max
* @param optional Integer $timeout
* @return Math_BigInteger
* @access public
* @internal See {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap4.pdf#page=15 HAC 4.44}.
*/
function randomPrime($min = false, $max = false, $timeout = false)
{
$compare = $max->compare($min);
if (!$compare) {
return $min;
} else if ($compare < 0) {
// if $min is bigger then $max, swap $min and $max
$temp = $max;
$max = $min;
$min = $temp;
}
// gmp_nextprime() requires PHP 5 >= 5.2.0 per <http://php.net/gmp-nextprime>.
if ( MATH_BIGINTEGER_MODE == MATH_BIGINTEGER_MODE_GMP && function_exists('gmp_nextprime') ) {
// we don't rely on Math_BigInteger::random()'s min / max when gmp_nextprime() is being used since this function
// does its own checks on $max / $min when gmp_nextprime() is used. When gmp_nextprime() is not used, however,
// the same $max / $min checks are not performed.
if ($min === false) {
$min = new Math_BigInteger(0);
}
if ($max === false) {
$max = new Math_BigInteger(0x7FFFFFFF);
}
$x = $this->random($min, $max);
$x->value = gmp_nextprime($x->value);
if ($x->compare($max) <= 0) {
return $x;
}
$x->value = gmp_nextprime($min->value);
if ($x->compare($max) <= 0) {
return $x;
}
return false;
}
static $one, $two;
if (!isset($one)) {
$one = new Math_BigInteger(1);
$two = new Math_BigInteger(2);
}
$start = time();
$x = $this->random($min, $max);
if ($x->equals($two)) {
return $x;
}
$x->_make_odd();
if ($x->compare($max) > 0) {
// if $x > $max then $max is even and if $min == $max then no prime number exists between the specified range
if ($min->equals($max)) {
return false;
}
$x = $min->copy();
$x->_make_odd();
}
$initial_x = $x->copy();
while (true) {
if ($timeout !== false && time() - $start > $timeout) {
return false;
}
if ($x->isPrime()) {
return $x;
}
$x = $x->add($two);
if ($x->compare($max) > 0) {
$x = $min->copy();
if ($x->equals($two)) {
return $x;
}
$x->_make_odd();
}
if ($x->equals($initial_x)) {
return false;
}
}
}
/**
* Make the current number odd
*
* If the current number is odd it'll be unchanged. If it's even, one will be added to it.
*
* @see randomPrime()
* @access private
*/
function _make_odd()
{
switch ( MATH_BIGINTEGER_MODE ) {
case MATH_BIGINTEGER_MODE_GMP:
gmp_setbit($this->value, 0);
break;
case MATH_BIGINTEGER_MODE_BCMATH:
if ($this->value[strlen($this->value) - 1] % 2 == 0) {
$this->value = bcadd($this->value, '1');
}
break;
default:
$this->value[0] |= 1;
}
}
/**
* Checks a numer to see if it's prime
*
* Assuming the $t parameter is not set, this function has an error rate of 2**-80. The main motivation for the
* $t parameter is distributability. Math_BigInteger::randomPrime() can be distributed accross multiple pageloads
* on a website instead of just one.
*
* @param optional Integer $t
* @return Boolean
* @access public
* @internal Uses the
* {@link http://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test Miller-Rabin primality test}. See
* {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap4.pdf#page=8 HAC 4.24}.