> and << cannot be used, nor can the modulo operator %,
* which only supports integers. Although this fact will slow this library down, the fact that such a high
* base is being used should more than compensate.
*
* When PHP version 6 is officially released, we'll be able to use 64-bit integers. This should, once again,
* allow bitwise operators, and will increase the maximum possible base to 2**31 (or 2**62 for addition /
* subtraction).
*
* Useful resources are as follows:
*
* - {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf Handbook of Applied Cryptography (HAC)}
* - {@link http://math.libtomcrypt.com/files/tommath.pdf Multi-Precision Math (MPM)}
* - Java's BigInteger classes. See /j2se/src/share/classes/java/math in jdk-1_5_0-src-jrl.zip
*
* One idea for optimization is to use the comba method to reduce the number of operations performed.
* MPM uses this quite extensively. The following URL elaborates:
*
* {@link http://www.everything2.com/index.pl?node_id=1736418}}}
*
* Here's a quick 'n dirty example of how to use this library:
*
* add($b);
*
* echo $c->toString(); // outputs 5
* ?>
*
*
* LICENSE: This library is free software; you can redistribute it and/or
* modify it under the terms of the GNU Lesser General Public
* License as published by the Free Software Foundation; either
* version 2.1 of the License, or (at your option) any later version.
*
* This library is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
* Lesser General Public License for more details.
*
* You should have received a copy of the GNU Lesser General Public
* License along with this library; if not, write to the Free Software
* Foundation, Inc., 59 Temple Place, Suite 330, Boston,
* MA 02111-1307 USA
*
* @category Math
* @package Math_BigInteger
* @author Jim Wigginton
* @copyright MMVI Jim Wigginton
* @license http://www.gnu.org/licenses/lgpl.txt
* @version $Id: BigInteger.php,v 1.10 2009-08-30 15:10:05 terrafrost Exp $
* @link http://pear.php.net/package/Math_BigInteger
*/
/**
* Include PHP_Compat module bcpowmod since that function does not exist in PHP4:
* {@link http://pear.php.net/package/PHP_Compat/}
* {@link http://php.net/function.bcpowmod}
*/
require_once 'PHP/Compat/Function/bcpowmod.php';
/**
* Include PHP_Compat module array_fill since that function requires PHP4.2.0+:
* {@link http://pear.php.net/package/PHP_Compat/}
* {@link http://php.net/function.array_fill}
*/
require_once 'PHP/Compat/Function/array_fill.php';
/**#@+
* @access private
* @see Math_BigInteger::_slidingWindow()
*/
/**
* @see Math_BigInteger::_montgomery()
* @see Math_BigInteger::_undoMontgomery()
*/
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::_copy()
*/
define('MATH_BIGINTEGER_NONE', 4);
/**#@-*/
/**#@+
* @access private
* @see Math_BigInteger::_montgomery()
* @see Math_BigInteger::_barrett()
*/
/**
* $cache[MATH_BIGINTEGER_VARIABLE] tells us whether or not the cached data is still valid.
*/
define('MATH_BIGINTEGER_VARIABLE', 0);
/**
* $cache[MATH_BIGINTEGER_DATA] contains the cached data.
*/
define('MATH_BIGINTEGER_DATA', 1);
/**#@-*/
/**#@+
* @access private
* @see Math_BigInteger::Math_BigInteger()
*/
/**
* To use the pure-PHP implementation
*/
define('MATH_BIGINTEGER_MODE_INTERNAL', 1);
/**
* To use the BCMath library
*
* (if enabled; otherwise, the internal implementation will be used)
*/
define('MATH_BIGINTEGER_MODE_BCMATH', 2);
/**
* To use the GMP library
*
* (if present; otherwise, either the BCMath or the internal implementation will be used)
*/
define('MATH_BIGINTEGER_MODE_GMP', 3);
/**#@-*/
/**
* Pure-PHP arbitrary precision integer arithmetic library. Supports base-2, base-10, base-16, and base-256
* numbers.
*
* @author Jim Wigginton
* @version 1.0.0RC3
* @access public
* @package Math_BigInteger
*/
class Math_BigInteger {
/**
* Holds the BigInteger's value.
*
* @var Array
* @access private
*/
var $value;
/**
* Holds the BigInteger's magnitude.
*
* @var Boolean
* @access private
*/
var $is_negative = false;
/**
* 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 a quick 'n dirty example:
*
* toString(); // outputs 50
* ?>
*
*
* @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:
$this->value = gmp_init(0);
break;
case MATH_BIGINTEGER_MODE_BCMATH:
$this->value = '0';
break;
default:
$this->value = array();
}
if ($x === 0) {
return;
}
switch ($base) {
case -256:
if (ord($x[0]) & 0x80) {
$x = ~$x;
$this->is_negative = true;
}
case 256:
switch ( MATH_BIGINTEGER_MODE ) {
case MATH_BIGINTEGER_MODE_GMP:
$temp = unpack('H*hex', $x);
$sign = $this->is_negative ? '-' : '';
$this->value = gmp_init($sign . '0x' . $temp['hex']);
break;
case MATH_BIGINTEGER_MODE_BCMATH:
// round $len to the nearest 4 (thanks, DavidMJ!)
$len = (strlen($x) + 3) & 0xFFFFFFFC;
$x = str_pad($x, $len, chr(0), STR_PAD_LEFT);
for ($i = 0; $i < $len; $i+= 4) {
$this->value = bcmul($this->value, '4294967296'); // 4294967296 == 2**32
$this->value = bcadd($this->value, 0x1000000 * ord($x[$i]) + ((ord($x[$i + 1]) << 16) | (ord($x[$i + 2]) << 8) | ord($x[$i + 3])));
}
if ($this->is_negative) {
$this->value = '-' . $this->value;
}
break;
// converts a base-2**8 (big endian / msb) number to base-2**26 (little endian / lsb)
case MATH_BIGINTEGER_MODE_INTERNAL:
while (strlen($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;
case MATH_BIGINTEGER_MODE_INTERNAL:
$x = ( strlen($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;
case MATH_BIGINTEGER_MODE_INTERNAL:
$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 a quick 'n dirty example:
*
* toBytes(); // outputs chr(65)
* ?>
*
*
* @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 '';
}
$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 '';
}
$temp = gmp_strval(gmp_abs($this->value), 16);
$temp = ( strlen($temp) & 1 ) ? '0' . $temp : $temp;
return ltrim(pack('H*', $temp), chr(0));
case MATH_BIGINTEGER_MODE_BCMATH:
if ($this->value === '0') {
return '';
}
$value = '';
$current = $this->value;
if ($current[0] == '-') {
$current = substr($current, 1);
}
// we don't do four bytes at a time because then numbers larger than 1<<31 would be negative
// two's complimented numbers, which would break chr.
while (bccomp($current, '0') > 0) {
$temp = bcmod($current, 0x1000000);
$value = chr($temp >> 16) . chr($temp >> 8) . chr($temp) . $value;
$current = bcdiv($current, 0x1000000);
}
return ltrim($value, chr(0));
}
if (!count($this->value)) {
return '';
}
$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 $result;
}
/**
* Converts a BigInteger to a base-10 number.
*
* Here's a quick 'n dirty example:
*
* toString(); // outputs 50
* ?>
*
*
* @return String
* @access public
* @internal Converts a base-2**26 number to base-10**7 (which is pretty much base-10)
*/
function toString()
{
switch ( MATH_BIGINTEGER_MODE ) {
case MATH_BIGINTEGER_MODE_GMP:
return gmp_strval($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();
$divisor->value = array(10000000); // eg. 10**7
while (count($temp->value)) {
list($temp, $mod) = $temp->divide($divisor);
$result = str_pad($this->_bytes2int($mod->toBytes()), 7, '0', STR_PAD_LEFT) . $result;
}
$result = ltrim($result, '0');
if ($this->is_negative) {
$result = '-' . $result;
}
return $result;
}
/**
* __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 $this->toString();
}
/**
* Adds two BigIntegers.
*
* Here's a quick 'n dirty example:
*
* add($b);
*
* echo $c->toString(); // outputs 30
* ?>
*
*
* @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 $temp;
case MATH_BIGINTEGER_MODE_BCMATH:
$temp = new Math_BigInteger();
$temp->value = bcadd($this->value, $y->value);
return $temp;
}
// subtract, if appropriate
if ( $this->is_negative != $y->is_negative ) {
// is $y the negative number?
$y_negative = $this->compare($y) > 0;
$temp = $this->_copy();
$y = $y->_copy();
$temp->is_negative = $y->is_negative = false;
$diff = $temp->compare($y);
if ( !$diff ) {
return new Math_BigInteger();
}
$temp = $temp->subtract($y);
$temp->is_negative = ($diff > 0) ? !$y_negative : $y_negative;
return $temp;
}
$result = new Math_BigInteger();
$carry = 0;
$size = max(count($this->value), count($y->value));
$size+= $size & 1; // rounds $size to the nearest 2.
$x = array_pad($this->value, $size,0);
$y = array_pad($y->value, $size, 0);
for ($i = 0; $i < $size - 1; $i+=2) {
$sum = $x[$i + 1] * 0x4000000 + $x[$i] + $y[$i + 1] * 0x4000000 + $y[$i] + $carry;
$carry = $sum >= 4503599627370496; // eg. floor($sum / 2**52); only possible values (in any base) are 0 and 1
$sum = $carry ? $sum - 4503599627370496 : $sum;
$temp = floor($sum / 0x4000000);
$result->value[] = $sum - 0x4000000 * $temp; // eg. a faster alternative to fmod($sum, 0x4000000)
$result->value[] = $temp;
}
if ($carry) {
$result->value[] = $carry;
}
$result->is_negative = $this->is_negative;
return $result->_normalize();
}
/**
* Subtracts two BigIntegers.
*
* Here's a quick 'n dirty example:
*
* subtract($b);
*
* echo $c->toString(); // outputs -10
* ?>
*
*
* @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 $temp;
case MATH_BIGINTEGER_MODE_BCMATH:
$temp = new Math_BigInteger();
$temp->value = bcsub($this->value, $y->value);
return $temp;
}
// add, if appropriate
if ( $this->is_negative != $y->is_negative ) {
$is_negative = $y->compare($this) > 0;
$temp = $this->_copy();
$y = $y->_copy();
$temp->is_negative = $y->is_negative = false;
$temp = $temp->add($y);
$temp->is_negative = $is_negative;
return $temp;
}
$diff = $this->compare($y);
if ( !$diff ) {
return new Math_BigInteger();
}
// switch $this and $y around, if appropriate.
if ( (!$this->is_negative && $diff < 0) || ($this->is_negative && $diff > 0) ) {
$is_negative = $y->is_negative;
$temp = $this->_copy();
$y = $y->_copy();
$temp->is_negative = $y->is_negative = false;
$temp = $y->subtract($temp);
$temp->is_negative = !$is_negative;
return $temp;
}
$result = new Math_BigInteger();
$carry = 0;
$size = max(count($this->value), count($y->value));
$size+= $size % 2;
$x = array_pad($this->value, $size, 0);
$y = array_pad($y->value, $size, 0);
for ($i = 0; $i < $size - 1; $i+=2) {
$sum = $x[$i + 1] * 0x4000000 + $x[$i] - $y[$i + 1] * 0x4000000 - $y[$i] + $carry;
$carry = $sum < 0 ? -1 : 0; // eg. floor($sum / 2**52); only possible values (in any base) are 0 and 1
$sum = $carry ? $sum + 4503599627370496 : $sum;
$temp = floor($sum / 0x4000000);
$result->value[] = $sum - 0x4000000 * $temp;
$result->value[] = $temp;
}
// $carry shouldn't be anything other than zero, at this point, since we already made sure that $this
// was bigger than $y.
$result->is_negative = $this->is_negative;
return $result->_normalize();
}
/**
* Multiplies two BigIntegers
*
* Here's a quick 'n dirty example:
*
* multiply($b);
*
* echo $c->toString(); // outputs 200
* ?>
*
*
* @param Math_BigInteger $x
* @return Math_BigInteger
* @access public
* @internal Modeled after 'multiply' in MutableBigInteger.java.
*/
function multiply($x)
{
switch ( MATH_BIGINTEGER_MODE ) {
case MATH_BIGINTEGER_MODE_GMP:
$temp = new Math_BigInteger();
$temp->value = gmp_mul($this->value, $x->value);
return $temp;
case MATH_BIGINTEGER_MODE_BCMATH:
$temp = new Math_BigInteger();
$temp->value = bcmul($this->value, $x->value);
return $temp;
}
if ( !$this->compare($x) ) {
return $this->_square();
}
$this_length = count($this->value);
$x_length = count($x->value);
if ( !$this_length || !$x_length ) { // a 0 is being multiplied
return new Math_BigInteger();
}
$product = new Math_BigInteger();
$product->value = $this->_array_repeat(0, $this_length + $x_length);
// the following for loop could be removed if the for loop following it
// (the one with nested for loops) initially set $i to 0, but
// doing so would also make the result in one set of unnecessary adds,
// since on the outermost loops first pass, $product->value[$k] is going
// to always be 0
$carry = 0;
$i = 0;
for ($j = 0, $k = $i; $j < $this_length; $j++, $k++) {
$temp = $product->value[$k] + $this->value[$j] * $x->value[$i] + $carry;
$carry = floor($temp / 0x4000000);
$product->value[$k] = $temp - 0x4000000 * $carry;
}
$product->value[$k] = $carry;
// the above for loop is what the previous comment was talking about. the
// following for loop is the "one with nested for loops"
for ($i = 1; $i < $x_length; $i++) {
$carry = 0;
for ($j = 0, $k = $i; $j < $this_length; $j++, $k++) {
$temp = $product->value[$k] + $this->value[$j] * $x->value[$i] + $carry;
$carry = floor($temp / 0x4000000);
$product->value[$k] = $temp - 0x4000000 * $carry;
}
$product->value[$k] = $carry;
}
$product->is_negative = $this->is_negative != $x->is_negative;
return $product->_normalize();
}
/**
* Squares a BigInteger
*
* Squaring can be done faster than multiplying a number by itself can be. See
* {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=7 HAC 14.2.4} /
* {@link http://math.libtomcrypt.com/files/tommath.pdf#page=141 MPM 5.3} for more information.
*
* @return Math_BigInteger
* @access private
*/
function _square()
{
if ( empty($this->value) ) {
return new Math_BigInteger();
}
$max_index = count($this->value) - 1;
$square = new Math_BigInteger();
$square->value = $this->_array_repeat(0, 2 * $max_index);
for ($i = 0; $i <= $max_index; $i++) {
$temp = $square->value[2 * $i] + $this->value[$i] * $this->value[$i];
$carry = floor($temp / 0x4000000);
$square->value[2 * $i] = $temp - 0x4000000 * $carry;
// note how we start from $i+1 instead of 0 as we do in multiplication.
for ($j = $i + 1; $j <= $max_index; $j++) {
$temp = $square->value[$i + $j] + 2 * $this->value[$j] * $this->value[$i] + $carry;
$carry = floor($temp / 0x4000000);
$square->value[$i + $j] = $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 $square->_normalize();
}
/**
* Divides two BigIntegers.
*
* Returns an array whose first element contains the quotient and whose second element contains the
* "common residue". If the remainder would be positive, the "common residue" and the remainder are the
* same. If the remainder would be negative, the "common residue" is equal to the sum of the remainder
* and the divisor (basically, the "common residue" is the first positive modulo).
*
* Here's a quick 'n dirty example:
*
* divide($b);
*
* echo $quotient->toString(); // outputs 0
* echo "\r\n";
* echo $remainder->toString(); // outputs 10
* ?>
*
*
* @param Math_BigInteger $y
* @return Array
* @access public
* @internal This function is based off of {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=9 HAC 14.20}
* with a slight variation due to the fact that this script, initially, did not support negative numbers. Now,
* it does, but I don't want to change that which already works.
*/
function divide($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($y->value));
}
return array($quotient, $remainder);
case MATH_BIGINTEGER_MODE_BCMATH:
$quotient = new Math_BigInteger();
$remainder = new Math_BigInteger();
$quotient->value = bcdiv($this->value, $y->value);
$remainder->value = bcmod($this->value, $y->value);
if ($remainder->value[0] == '-') {
$remainder->value = bcadd($remainder->value, $y->value[0] == '-' ? substr($y->value, 1) : $y->value);
}
return array($quotient, $remainder);
}
$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 != $y_sign;
return array($temp, new Math_BigInteger());
}
if ( $diff < 0 ) {
// if $x is negative, "add" $y.
if ( $x_sign ) {
$x = $y->subtract($x);
}
return array(new Math_BigInteger(), $x);
}
// normalize $x and $y as described in HAC 14.23 / 14.24
// (incidently, i haven't been able to find a definitive example showing that this
// results in worth-while speedup, but whatever)
$msb = $y->value[count($y->value) - 1];
for ($shift = 0; !($msb & 0x2000000); $shift++) {
$msb <<= 1;
}
$x->_lshift($shift);
$y->_lshift($shift);
$x_max = count($x->value) - 1;
$y_max = count($y->value) - 1;
$quotient = new Math_BigInteger();
$quotient->value = $this->_array_repeat(0, $x_max - $y_max + 1);
// $temp = $y << ($x_max - $y_max-1) in base 2**26
$temp = new Math_BigInteger();
$temp->value = array_merge($this->_array_repeat(0, $x_max - $y_max), $y->value);
while ( $x->compare($temp) >= 0 ) {
// calculate the "common residue"
$quotient->value[$x_max - $y_max]++;
$x = $x->subtract($temp);
$x_max = count($x->value) - 1;
}
for ($i = $x_max; $i >= $y_max + 1; $i--) {
$x_value = array(
$x->value[$i],
( $i > 0 ) ? $x->value[$i - 1] : 0,
( $i - 1 > 0 ) ? $x->value[$i - 2] : 0
);
$y_value = array(
$y->value[$y_max],
( $y_max > 0 ) ? $y_max - 1 : 0
);
$q_index = $i - $y_max - 1;
if ($x_value[0] == $y_value[0]) {
$quotient->value[$q_index] = 0x3FFFFFF;
} else {
$quotient->value[$q_index] = floor(
($x_value[0] * 0x4000000 + $x_value[1])
/
$y_value[0]
);
}
$temp = new Math_BigInteger();
$temp->value = array($y_value[1], $y_value[0]);
$lhs = new Math_BigInteger();
$lhs->value = array($quotient->value[$q_index]);
$lhs = $lhs->multiply($temp);
$rhs = new Math_BigInteger();
$rhs->value = array($x_value[2], $x_value[1], $x_value[0]);
while ( $lhs->compare($rhs) > 0 ) {
$quotient->value[$q_index]--;
$lhs = new Math_BigInteger();
$lhs->value = array($quotient->value[$q_index]);
$lhs = $lhs->multiply($temp);
}
$corrector = new Math_BigInteger();
$temp = new Math_BigInteger();
$corrector->value = $temp->value = $this->_array_repeat(0, $q_index);
$temp->value[] = $quotient->value[$q_index];
$temp = $temp->multiply($y);
if ( $x->compare($temp) < 0 ) {
$corrector->value[] = 1;
$x = $x->add($corrector->multiply($y));
$quotient->value[$q_index]--;
}
$x = $x->subtract($temp);
$x_max = count($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($quotient->_normalize(), $x);
}
/**
* Performs modular exponentiation.
*
* Here's a quick 'n dirty example:
*
* modPow($b, $c);
*
* echo $c->toString(); // outputs 10
* ?>
*
*
* @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 = $n->abs();
if ($e->compare(new Math_BigInteger()) < 0) {
$e = $e->abs();
$temp = $this->modInverse($n);
if ($temp === false) {
return false;
}
return $temp->modPow($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 $temp;
case MATH_BIGINTEGER_MODE_BCMATH:
$temp = new Math_BigInteger();
$temp->value = bcpowmod($this->value, $e->value, $n->value);
return $temp;
}
if ( empty($e->value) ) {
$temp = new Math_BigInteger();
$temp->value = array(1);
return $temp;
}
if ( $e->value == array(1) ) {
list(, $temp) = $this->divide($n);
return $temp;
}
if ( $e->value == array(2) ) {
$temp = $this->_square();
list(, $temp) = $temp->divide($n);
return $temp;
}
// is the modulo odd?
if ( $n->value[0] & 1 ) {
return $this->_slidingWindow($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 $result;
}
/**
* Sliding Window k-ary Modular Exponentiation
*
* Based on {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=27 HAC 14.85} /
* {@link http://math.libtomcrypt.com/files/tommath.pdf#page=210 MPM 7.7}. In a departure from those algorithims,
* however, this function performs a modular reduction after every multiplication and squaring operation.
* As such, this function has the same preconditions that the reductions being used do.
*
* @param Math_BigInteger $e
* @param Math_BigInteger $n
* @param Integer $mode
* @return Math_BigInteger
* @access private
*/
function _slidingWindow($e, $n, $mode)
{
static $window_ranges = array(7, 25, 81, 241, 673, 1793); // from BigInteger.java's oddModPow function
//static $window_ranges = array(0, 7, 36, 140, 450, 1303, 3529); // from MPM 7.3.1
$e_length = count($e->value) - 1;
$e_bits = decbin($e->value[$e_length]);
for ($i = $e_length - 1; $i >= 0; $i--) {
$e_bits.= str_pad(decbin($e->value[$i]), 26, '0', STR_PAD_LEFT);
}
$e_length = strlen($e_bits);
// calculate the appropriate window size.
// $window_size == 3 if $window_ranges is between 25 and 81, for example.
for ($i = 0, $window_size = 1; $e_length > $window_ranges[$i] && $i < count($window_ranges); $window_size++, $i++);
switch ($mode) {
case MATH_BIGINTEGER_MONTGOMERY:
$reduce = '_montgomery';
$undo = '_undoMontgomery';
break;
case MATH_BIGINTEGER_BARRETT:
$reduce = '_barrett';
$undo = '_barrett';
break;
case MATH_BIGINTEGER_POWEROF2:
$reduce = '_mod2';
$undo = '_mod2';
break;
case MATH_BIGINTEGER_CLASSIC:
$reduce = '_remainder';
$undo = '_remainder';
break;
case MATH_BIGINTEGER_NONE:
// ie. do no modular reduction. useful if you want to just do pow as opposed to modPow.
$reduce = '_copy';
$undo = '_copy';
break;
default:
// an invalid $mode was provided
}
// precompute $this^0 through $this^$window_size
$powers = array();
$powers[1] = $this->$undo($n);
$powers[2] = $powers[1]->_square();
$powers[2] = $powers[2]->$reduce($n);
// we do every other number since substr($e_bits, $i, $j+1) (see below) is supposed to end
// in a 1. ie. it's supposed to be odd.
$temp = 1 << ($window_size - 1);
for ($i = 1; $i < $temp; $i++) {
$powers[2 * $i + 1] = $powers[2 * $i - 1]->multiply($powers[2]);
$powers[2 * $i + 1] = $powers[2 * $i + 1]->$reduce($n);
}
$result = new Math_BigInteger();
$result->value = array(1);
$result = $result->$undo($n);
for ($i = 0; $i < $e_length; ) {
if ( !$e_bits[$i] ) {
$result = $result->_square();
$result = $result->$reduce($n);
$i++;
} else {
for ($j = $window_size - 1; $j >= 0; $j--) {
if ( !empty($e_bits[$i + $j]) ) {
break;
}
}
for ($k = 0; $k <= $j; $k++) {// eg. the length of substr($e_bits, $i, $j+1)
$result = $result->_square();
$result = $result->$reduce($n);
}
$result = $result->multiply($powers[bindec(substr($e_bits, $i, $j + 1))]);
$result = $result->$reduce($n);
$i+=$j + 1;
}
}
$result = $result->$reduce($n);
return $result->_normalize();
}
/**
* Remainder
*
* A wrapper for the divide function.
*
* @see divide()
* @see _slidingWindow()
* @access private
* @param Math_BigInteger
* @return Math_BigInteger
*/
function _remainder($n)
{
list(, $temp) = $this->divide($n);
return $temp;
}
/**
* Modulos for Powers of Two
*
* Calculates $x%$n, where $n = 2**$e, for some $e. Since this is basically the same as doing $x & ($n-1),
* we'll just use this function as a wrapper for doing that.
*
* @see _slidingWindow()
* @access private
* @param Math_BigInteger
* @return Math_BigInteger
*/
function _mod2($n)
{
$temp = new Math_BigInteger();
$temp->value = array(1);
return $this->bitwise_and($n->subtract($temp));
}
/**
* Barrett Modular Reduction
*
* See {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=14 HAC 14.3.3} /
* {@link http://math.libtomcrypt.com/files/tommath.pdf#page=165 MPM 6.2.5} for more information. Modified slightly,
* so as not to require negative numbers (initially, this script didn't support negative numbers).
*
* @see _slidingWindow()
* @access private
* @param Math_BigInteger
* @return Math_BigInteger
*/
function _barrett($n)
{
static $cache;
$n_length = count($n->value);
if ( !isset($cache[MATH_BIGINTEGER_VARIABLE]) || $n->compare($cache[MATH_BIGINTEGER_VARIABLE]) ) {
$cache[MATH_BIGINTEGER_VARIABLE] = $n;
$temp = new Math_BigInteger();
$temp->value = $this->_array_repeat(0, 2 * $n_length);
$temp->value[] = 1;
list($cache[MATH_BIGINTEGER_DATA], ) = $temp->divide($n);
}
$temp = new Math_BigInteger();
$temp->value = array_slice($this->value, $n_length - 1);
$temp = $temp->multiply($cache[MATH_BIGINTEGER_DATA]);
$temp->value = array_slice($temp->value, $n_length + 1);
$result = new Math_BigInteger();
$result->value = array_slice($this->value, 0, $n_length + 1);
$temp = $temp->multiply($n);
$temp->value = array_slice($temp->value, 0, $n_length + 1);
if ($result->compare($temp) < 0) {
$corrector = new Math_BigInteger();
$corrector->value = $this->_array_repeat(0, $n_length + 1);
$corrector->value[] = 1;
$result = $result->add($corrector);
}
$result = $result->subtract($temp);
while ($result->compare($n) > 0) {
$result = $result->subtract($n);
}
return $result;
}
/**
* Montgomery Modular Reduction
*
* ($this->_montgomery($n))->_undoMontgomery($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($n, 2) must be equal to one for this function
* to work correctly.
*
* @see _undoMontgomery()
* @see _slidingWindow()
* @access private
* @param Math_BigInteger
* @return Math_BigInteger
*/
function _montgomery($n)
{
static $cache;
if ( !isset($cache[MATH_BIGINTEGER_VARIABLE]) || $n->compare($cache[MATH_BIGINTEGER_VARIABLE]) ) {
$cache[MATH_BIGINTEGER_VARIABLE] = $n;
$cache[MATH_BIGINTEGER_DATA] = $n->_modInverse67108864();
}
$result = $this->_copy();
$n_length = count($n->value);
for ($i = 0; $i < $n_length; $i++) {
$temp = new Math_BigInteger();
$temp->value = array(
($result->value[$i] * $cache[MATH_BIGINTEGER_DATA]) & 0x3FFFFFF
);
$temp = $temp->multiply($n);
$temp->value = array_merge($this->_array_repeat(0, $i), $temp->value);
$result = $result->add($temp);
}
$result->value = array_slice($result->value, $n_length);
if ($result->compare($n) >= 0) {
$result = $result->subtract($n);
}
return $result->_normalize();
}
/**
* Undo Montgomery Modular Reduction
*
* @see _montgomery()
* @see _slidingWindow()
* @access private
* @param Math_BigInteger
* @return Math_BigInteger
*/
function _undoMontgomery($n)
{
$temp = new Math_BigInteger();
$temp->value = array_merge($this->_array_repeat(0, count($n->value)), $this->value);
list(, $temp) = $temp->divide($n);
return $temp->_normalize();
}
/**
* Modular Inverse of a number mod 2**26 (eg. 67108864)
*
* Based off of the bnpInvDigit function implemented and justified in the following URL:
*
* {@link http://www-cs-students.stanford.edu/~tjw/jsbn/jsbn.js}
*
* The following URL provides more info:
*
* {@link http://groups.google.com/group/sci.crypt/msg/7a137205c1be7d85}
*
* As for why we do all the bitmasking... strange things can happen when converting from floats to ints. For
* instance, on some computers, var_dump((int) -4294967297) yields int(-1) and on others, it yields
* int(-2147483648). To avoid problems stemming from this, we use bitmasks to guarantee that ints aren't
* auto-converted to floats. The outermost bitmask is present because without it, there's no guarantee that
* the "residue" returned would be the so-called "common residue". We use fmod, in the last step, because the
* maximum possible $x is 26 bits and the maximum $result is 16 bits. Thus, we have to be able to handle up to
* 40 bits, which only 64-bit floating points will support.
*
* Thanks to Pedro Gimeno Fortea for input!
*
* @see _montgomery()
* @access private
* @return Integer
*/
function _modInverse67108864() // 2**26 == 67108864
{
$x = -$this->value[0];
$result = $x & 0x3; // x**-1 mod 2**2
$result = ($result * (2 - $x * $result)) & 0xF; // x**-1 mod 2**4
$result = ($result * (2 - ($x & 0xFF) * $result)) & 0xFF; // x**-1 mod 2**8
$result = ($result * ((2 - ($x & 0xFFFF) * $result) & 0xFFFF)) & 0xFFFF; // x**-1 mod 2**16
$result = fmod($result * (2 - fmod($x * $result, 0x4000000)), 0x4000000); // x**-1 mod 2**26
return $result & 0x3FFFFFF;
}
/**
* Calculates modular inverses.
*
* Here's a quick 'n dirty example:
*
* modInverse($b);
*
* echo $c->toString(); // outputs 4
* ?>
*
*
* @param Math_BigInteger $n
* @return mixed false, if no modular inverse exists, Math_BigInteger, otherwise.
* @access public
* @internal Calculates the modular inverse of $this mod $n using the binary xGCD algorithim described in
* {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=19 HAC 14.61}. As the text above 14.61 notes,
* the more traditional algorithim requires "relatively costly multiple-precision divisions". See
* {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=21 HAC 14.64} for more information.
*/
function modInverse($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 : $temp;
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.
// if $x is less than 0, the first character of $x is a '-', so we'll remove it. we can do this because
// $x mod $n == $x mod -$n.
$n = (bccomp($n->value, '0') < 0) ? substr($n->value, 1) : $n->value;
if (bccomp($this->value,'0') < 0) {
$negated_this = new Math_BigInteger();
$negated_this->value = substr($this->value, 1);
$temp = $negated_this->modInverse(new Math_BigInteger($n));
if ($temp === false) {
return false;
}
$temp->value = bcsub($n, $temp->value);
return $temp;
}
$u = $this->value;
$v = $n;
$a = '1';
$c = '0';
while (true) {
$q = bcdiv($u, $v);
$temp = $u;
$u = $v;
$v = bcsub($temp, bcmul($v, $q));
if (bccomp($v, '0') == 0) {
break;
}
$temp = $a;
$a = $c;
$c = bcsub($temp, bcmul($c, $q));
}
$temp = new Math_BigInteger();
$temp->value = (bccomp($c, '0') < 0) ? bcadd($c, $n) : $c;
// $u contains the gcd of $this and $n
return (bccomp($u,'1') == 0) ? $temp : false;
}
// if $this and $n are even, return false.
if ( !($this->value[0]&1) && !($n->value[0]&1) ) {
return false;
}
$n = $n->_copy();
$n->is_negative = false;
if ($this->compare(new Math_BigInteger()) < 0) {
// is_negative is currently true. since we need it to be false, we'll just set it to false, temporarily,
// and reset it as true, later.
$this->is_negative = false;
$temp = $this->modInverse($n);
if ($temp === false) {
return false;
}
$temp = $n->subtract($temp);
$this->is_negative = true;
return $temp;
}
$u = $n->_copy();
$x = $this;
//list(, $x) = $this->divide($n);
$v = $x->_copy();
$a = new Math_BigInteger();
$b = new Math_BigInteger();
$c = new Math_BigInteger();
$d = new Math_BigInteger();
$a->value = $d->value = array(1);
while ( !empty($u->value) ) {
while ( !($u->value[0] & 1) ) {
$u->_rshift(1);
if ( ($a->value[0] & 1) || ($b->value[0] & 1) ) {
$a = $a->add($x);
$b = $b->subtract($n);
}
$a->_rshift(1);
$b->_rshift(1);
}
while ( !($v->value[0] & 1) ) {
$v->_rshift(1);
if ( ($c->value[0] & 1) || ($d->value[0] & 1) ) {
$c = $c->add($x);
$d = $d->subtract($n);
}
$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);
}
$u->_normalize();
}
// at this point, $v == gcd($this, $n). if it's not equal to 1, no modular inverse exists.
if ( $v->value != array(1) ) {
return false;
}
$d = ($d->compare(new Math_BigInteger()) < 0) ? $d->add($n) : $d;
return ($this->is_negative) ? $n->subtract($d) : $d;
}
/**
* Absolute value.
*
* @return Math_BigInteger
* @access public
*/
function abs()
{
$temp = new Math_BigInteger();
switch ( MATH_BIGINTEGER_MODE ) {
case MATH_BIGINTEGER_MODE_GMP:
$temp->value = gmp_abs($this->value);
break;
case MATH_BIGINTEGER_MODE_BCMATH:
$temp->value = (bccomp($this->value, '0') < 0) ? substr($this->value, 1) : $this->value;
break;
default:
$temp->value = $this->value;
}
return $temp;
}
/**
* Compares two numbers.
*
* @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
* @internal Could return $this->sub($x), but that's not as fast as what we do do.
*/
function compare($x)
{
switch ( MATH_BIGINTEGER_MODE ) {
case MATH_BIGINTEGER_MODE_GMP:
return gmp_cmp($this->value, $x->value);
case MATH_BIGINTEGER_MODE_BCMATH:
return bccomp($this->value, $x->value);
}
$this->_normalize();
$x->_normalize();
if ( $this->is_negative != $x->is_negative ) {
return ( !$this->is_negative && $x->is_negative ) ? 1 : -1;
}
$result = $this->is_negative ? -1 : 1;
if ( count($this->value) != count($x->value) ) {
return ( count($this->value) > count($x->value) ) ? $result : -$result;
}
for ($i = count($this->value) - 1; $i >= 0; $i--) {
if ($this->value[$i] != $x->value[$i]) {
return ( $this->value[$i] > $x->value[$i] ) ? $result : -$result;
}
}
return 0;
}
/**
* Returns a copy of $this
*
* 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://www.php.net/manual/en/language.oop5.basic.php#51624}
*
* @access private
* @return Math_BigInteger
*/
function _copy()
{
$temp = new Math_BigInteger();
$temp->value = $this->value;
$temp->is_negative = $this->is_negative;
return $temp;
}
/**
* Logical And
*
* @param Math_BigInteger $x
* @access public
* @internal Implemented per a request by Lluis Pamies i Juarez
* @return Math_BigInteger
*/
function bitwise_and($x)
{
switch ( MATH_BIGINTEGER_MODE ) {
case MATH_BIGINTEGER_MODE_GMP:
$temp = new Math_BigInteger();
$temp->value = gmp_and($this->value, $x->value);
return $temp;
case MATH_BIGINTEGER_MODE_BCMATH:
return new Math_BigInteger($this->toBytes() & $x->toBytes(), 256);
}
$result = new Math_BigInteger();
$x_length = count($x->value);
for ($i = 0; $i < $x_length; $i++) {
$result->value[] = $this->value[$i] & $x->value[$i];
}
return $result->_normalize();
}
/**
* Logical Or
*
* @param Math_BigInteger $x
* @access public
* @internal Implemented per a request by Lluis Pamies i Juarez
* @return Math_BigInteger
*/
function bitwise_or($x)
{
switch ( MATH_BIGINTEGER_MODE ) {
case MATH_BIGINTEGER_MODE_GMP:
$temp = new Math_BigInteger();
$temp->value = gmp_or($this->value, $x->value);
return $temp;
case MATH_BIGINTEGER_MODE_BCMATH:
return new Math_BigInteger($this->toBytes() | $x->toBytes(), 256);
}
$result = $this->_copy();
$x_length = count($x->value);
for ($i = 0; $i < $x_length; $i++) {
$result->value[$i] = $this->value[$i] | $x->value[$i];
}
return $result->_normalize();
}
/**
* Logical Exclusive-Or
*
* @param Math_BigInteger $x
* @access public
* @internal Implemented per a request by Lluis Pamies i Juarez
* @return Math_BigInteger
*/
function bitwise_xor($x)
{
switch ( MATH_BIGINTEGER_MODE ) {
case MATH_BIGINTEGER_MODE_GMP:
$temp = new Math_BigInteger();
$temp->value = gmp_xor($this->value, $x->value);
return $temp;
case MATH_BIGINTEGER_MODE_BCMATH:
return new Math_BigInteger($this->toBytes() ^ $x->toBytes(), 256);
}
$result = $this->_copy();
$x_length = count($x->value);
for ($i = 0; $i < $x_length; $i++) {
$result->value[$i] = $this->value[$i] ^ $x->value[$i];
}
return $result->_normalize();
}
/**
* Logical Not
*
* Although integers can be converted to and from various bases with relative ease, there is one piece
* of information that is lost during such conversions. The number of leading zeros that number had
* or should have in any given base. Per that, if you convert 1 from decimal to binary, there's no
* way to know just how many leading zero's there should be. In truth, there could be any number.
*
* Normally, the number of leading zero's is unimportant. When doing "not", however, it is. The "not"
* of 1 on an 8-bit representation of 1 is 1111 1110. The "not" of 1 on a 16-bit representation of 1 is
* 1111 1111 1111 1110. When doing it on a number that's preceeded by an infinite number of zero's, it's
* infinite.
*
* This function assumes that there are no leading zero's - that the bit-representation being used is
* equal to the minimum number of required bits, unless otherwise specified in the optional parameter,
* where the optional parameter represents the bit-representation being used. If the specified
* bit-representation is smaller than the minimum number of bits required to represent the number, the
* latter will be used as the bit-representation.
*
* @param $bits Integer
* @access public
* @internal Implemented per a request by Lluis Pamies i Juarez
* @return Math_BigInteger
*/
function bitwise_not($bits = -1)
{
// calculuate "not" without regard to $bits
$temp = ~$this->toBytes();
$msb = decbin(ord($temp[0]));
$msb = substr($msb, strpos($msb, '0'));
$temp[0] = chr(bindec($msb));
// see if we need to add extra leading 1's
$current_bits = strlen($msb) + 8 * strlen($temp) - 8;
$new_bits = $bits - $current_bits;
if ($new_bits <= 0) {
return 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($bits / 8), chr(0), STR_PAD_LEFT);
return new Math_BigInteger($leading_ones | $temp, 256);
}
/**
* Logical Right Shift
*
* Shifts BigInteger's by $shift bits, effectively dividing by 2**$shift.
*
* @param Integer $shift
* @return Math_BigInteger
* @access public
* @internal The only version that yields any speed increases is the internal version.
*/
function bitwise_rightShift($shift)
{
$temp = new Math_BigInteger();
switch ( MATH_BIGINTEGER_MODE ) {
case MATH_BIGINTEGER_MODE_GMP:
static $two;
if (empty($two)) {
$two = gmp_init('2');
}
$temp->value = gmp_div_q($this->value, gmp_pow($two, $shift));
break;
case MATH_BIGINTEGER_MODE_BCMATH:
$temp->value = bcdiv($this->value, bcpow('2', $shift));
break;
default: // could just replace _lshift with this, but then all _lshift() calls would need to be rewritten
// and I don't want to do that...
$temp->value = $this->value;
$temp->_rshift($shift);
}
return $temp;
}
/**
* Logical Left Shift
*
* Shifts BigInteger's by $shift bits, effectively multiplying by 2**$shift.
*
* @param Integer $shift
* @return Math_BigInteger
* @access public
* @internal The only version that yields any speed increases is the internal version.
*/
function bitwise_leftShift($shift)
{
$temp = new Math_BigInteger();
switch ( MATH_BIGINTEGER_MODE ) {
case MATH_BIGINTEGER_MODE_GMP:
static $two;
if (empty($two)) {
$two = gmp_init('2');
}
$temp->value = gmp_mul($this->value, gmp_pow($two, $shift));
break;
case MATH_BIGINTEGER_MODE_BCMATH:
$temp->value = bcmul($this->value, bcpow('2', $shift));
break;
default: // could just replace _rshift with this, but then all _lshift() calls would need to be rewritten
// and I don't want to do that...
$temp->value = $this->value;
$temp->_lshift($shift);
}
return $temp;
}
/**
* Generate a random number
*
* $generator should be the name of a random number 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
* done before this function is called.
*
* @param optional Integer $min
* @param optional Integer $max
* @param optional String $generator
* @return Math_BigInteger
* @access public
*/
function random($min = false, $max = false, $generator = 'mt_rand')
{
if ($min === false) {
$min = new Math_BigInteger(0);
}
if ($max === false) {
$max = new Math_BigInteger(0x7FFFFFFF);
}
$compare = $max->compare($min);
if (!$compare) {
return $min;
} else if ($compare < 0) {
// if $min is bigger then $max, swap $min and $max
$temp = $max;
$max = $min;
$min = $temp;
}
$max = $max->subtract($min);
$max = ltrim($max->toBytes(), chr(0));
$size = strlen($max) - 1;
$random = '';
$bytes = $size & 3;
for ($i = 0; $i < $bytes; $i++) {
$random.= chr($generator(0, 255));
}
$blocks = $size >> 2;
for ($i = 0; $i < $blocks; $i++) {
$random.= pack('N', $generator(-2147483648, 0x7FFFFFFF));
}
$temp = new Math_BigInteger($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 $random->add($min);
}
/**
* Logical Left Shift
*
* Shifts BigInteger's by $shift bits.
*
* @param Integer $shift
* @access private
*/
function _lshift($shift)
{
if ( $shift == 0 ) {
return;
}
$num_digits = floor($shift / 26);
$shift %= 26;
$shift = 1 << $shift;
$carry = 0;
for ($i = 0; $i < count($this->value); $i++) {
$temp = $this->value[$i] * $shift + $carry;
$carry = floor($temp / 0x4000000);
$this->value[$i] = $temp - $carry * 0x4000000;
}
if ( $carry ) {
$this->value[] = $carry;
}
while ($num_digits--) {
array_unshift($this->value, 0);
}
}
/**
* Logical Right Shift
*
* Shifts BigInteger's by $shift bits.
*
* @param Integer $shift
* @access private
*/
function _rshift($shift)
{
if ($shift == 0) {
$this->_normalize();
}
$num_digits = floor($shift / 26);
$shift %= 26;
$carry_shift = 26 - $shift;
$carry_mask = (1 << $shift) - 1;
if ( $num_digits ) {
$this->value = array_slice($this->value, $num_digits);
}
$carry = 0;
for ($i = count($this->value) - 1; $i >= 0; $i--) {
$temp = $this->value[$i] >> $shift | $carry;
$carry = ($this->value[$i] & $carry_mask) << $carry_shift;
$this->value[$i] = $temp;
}
$this->_normalize();
}
/**
* Normalize
*
* Deletes leading zeros.
*
* @see divide()
* @return Math_BigInteger
* @access private
*/
function _normalize()
{
if ( !count($this->value) ) {
return $this;
}
for ($i=count($this->value) - 1; $i >= 0; $i--) {
if ( $this->value[$i] ) {
break;
}
unset($this->value[$i]);
}
return $this;
}
/**
* Array Repeat
*
* @param $input Array
* @param $multiplier mixed
* @return Array
* @access private
*/
function _array_repeat($input, $multiplier)
{
return ($multiplier) ? array_fill(0, $multiplier, $input) : array();
}
/**
* Logical Left Shift
*
* Shifts binary strings $shift bits, essentially multiplying by 2**$shift.
*
* @param $x String
* @param $shift Integer
* @return String
* @access private
*/
function _base256_lshift(&$x, $shift)
{
if ($shift == 0) {
return;
}
$num_bytes = $shift >> 3; // eg. floor($shift/8)
$shift &= 7; // eg. $shift % 8
$carry = 0;
for ($i = strlen($x) - 1; $i >= 0; $i--) {
$temp = ord($x[$i]) << $shift | $carry;
$x[$i] = chr($temp);
$carry = $temp >> 8;
}
$carry = ($carry != 0) ? chr($carry) : '';
$x = $carry . $x . str_repeat(chr(0), $num_bytes);
}
/**
* Logical Right Shift
*
* Shifts binary strings $shift bits, essentially dividing by 2**$shift and returning the remainder.
*
* @param $x String
* @param $shift Integer
* @return String
* @access private
*/
function _base256_rshift(&$x, $shift)
{
if ($shift == 0) {
$x = ltrim($x, chr(0));
return '';
}
$num_bytes = $shift >> 3; // eg. floor($shift/8)
$shift &= 7; // eg. $shift % 8
$remainder = '';
if ($num_bytes) {
$start = $num_bytes > strlen($x) ? -strlen($x) : -$num_bytes;
$remainder = substr($x, $start);
$x = substr($x, 0, -$num_bytes);
}
$carry = 0;
$carry_shift = 8 - $shift;
for ($i = 0; $i < strlen($x); $i++) {
$temp = (ord($x[$i]) >> $shift) | $carry;
$carry = (ord($x[$i]) << $carry_shift) & 0xFF;
$x[$i] = chr($temp);
}
$x = ltrim($x, chr(0));
$remainder = chr($carry >> $carry_shift) . $remainder;
return ltrim($remainder, chr(0));
}
// one quirk about how the following functions are implemented is that PHP defines N to be an unsigned long
// at 32-bits, while java's longs are 64-bits.
/**
* Converts 32-bit integers to bytes.
*
* @param Integer $x
* @return String
* @access private
*/
function _int2bytes($x)
{
return ltrim(pack('N', $x), chr(0));
}
/**
* Converts bytes to 32-bit integers
*
* @param String $x
* @return Integer
* @access private
*/
function _bytes2int($x)
{
$temp = unpack('Nint', str_pad($x, 4, chr(0), STR_PAD_LEFT));
return $temp['int'];
}
}