Modified: vcl/trunk/web/.ht-inc/phpseclib/Math/BigInteger.php URL: http://svn.apache.org/viewvc/vcl/trunk/web/.ht-inc/phpseclib/Math/BigInteger.php?rev=1796097&r1=1796096&r2=1796097&view=diff ============================================================================== --- vcl/trunk/web/.ht-inc/phpseclib/Math/BigInteger.php (original) +++ vcl/trunk/web/.ht-inc/phpseclib/Math/BigInteger.php Wed May 24 20:28:10 2017 @@ -1,3551 +1,3804 @@ -<?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 <[email protected]> - * @copyright MMVI Jim Wigginton - * @license http://www.opensource.org/licenses/mit-license.html MIT License - * @version $Id: BigInteger.php,v 1.33 2010/03/22 22:32:03 terrafrost Exp $ - * @link http://pear.php.net/package/Math_BigInteger - */ - -/**#@+ - * Reduction constants - * - * @access private - * @see Math_BigInteger::_reduce() - */ -/** - * @see Math_BigInteger::_montgomery() - * @see Math_BigInteger::_prepMontgomery() - */ -define('MATH_BIGINTEGER_MONTGOMERY', 0); -/** - * @see Math_BigInteger::_barrett() - */ -define('MATH_BIGINTEGER_BARRETT', 1); -/** - * @see Math_BigInteger::_mod2() - */ -define('MATH_BIGINTEGER_POWEROF2', 2); -/** - * @see Math_BigInteger::_remainder() - */ -define('MATH_BIGINTEGER_CLASSIC', 3); -/** - * @see Math_BigInteger::__clone() - */ -define('MATH_BIGINTEGER_NONE', 4); -/**#@-*/ - -/**#@+ - * Array constants - * - * Rather than create a thousands and thousands of new Math_BigInteger objects in repeated function calls to add() and - * multiply() or whatever, we'll just work directly on arrays, taking them in as parameters and returning them. - * - * @access private - */ -/** - * $result[MATH_BIGINTEGER_VALUE] contains the value. - */ -define('MATH_BIGINTEGER_VALUE', 0); -/** - * $result[MATH_BIGINTEGER_SIGN] contains the sign. - */ -define('MATH_BIGINTEGER_SIGN', 1); -/**#@-*/ - -/**#@+ - * @access private - * @see Math_BigInteger::_montgomery() - * @see Math_BigInteger::_barrett() - */ -/** - * Cache constants - * - * $cache[MATH_BIGINTEGER_VARIABLE] tells us whether or not the cached data is still valid. - */ -define('MATH_BIGINTEGER_VARIABLE', 0); -/** - * $cache[MATH_BIGINTEGER_DATA] contains the cached data. - */ -define('MATH_BIGINTEGER_DATA', 1); -/**#@-*/ - -/**#@+ - * Mode constants. - * - * @access private - * @see Math_BigInteger::Math_BigInteger() - */ -/** - * To use the pure-PHP implementation - */ -define('MATH_BIGINTEGER_MODE_INTERNAL', 1); -/** - * To use the BCMath library - * - * (if enabled; otherwise, the internal implementation will be used) - */ -define('MATH_BIGINTEGER_MODE_BCMATH', 2); -/** - * To use the GMP library - * - * (if present; otherwise, either the BCMath or the internal implementation will be used) - */ -define('MATH_BIGINTEGER_MODE_GMP', 3); -/**#@-*/ - -/** - * The largest digit that may be used in addition / subtraction - * - * (we do pow(2, 52) instead of using 4503599627370496, directly, because some PHP installations - * will truncate 4503599627370496) - * - * @access private - */ -define('MATH_BIGINTEGER_MAX_DIGIT52', pow(2, 52)); - -/** - * Karatsuba Cutoff - * - * At what point do we switch between Karatsuba multiplication and schoolbook long multiplication? - * - * @access private - */ -define('MATH_BIGINTEGER_KARATSUBA_CUTOFF', 25); - -/** - * Pure-PHP arbitrary precision integer arithmetic library. Supports base-2, base-10, base-16, and base-256 - * numbers. - * - * @author Jim Wigginton <[email protected]> - * @version 1.0.0RC4 - * @access public - * @package Math_BigInteger - */ -class Math_BigInteger { - /** - * Holds the BigInteger's value. - * - * @var Array - * @access private - */ - var $value; - - /** - * Holds the BigInteger's magnitude. - * - * @var Boolean - * @access private - */ - var $is_negative = false; - - /** - * Random number generator function - * - * @see setRandomGenerator() - * @access private - */ - var $generator = 'mt_rand'; - - /** - * Precision - * - * @see setPrecision() - * @access private - */ - var $precision = -1; - - /** - * Precision Bitmask - * - * @see setPrecision() - * @access private - */ - var $bitmask = false; - - /** - * Mode independant value used for serialization. - * - * If the bcmath or gmp extensions are installed $this->value will be a non-serializable resource, hence the need for - * a variable that'll be serializable regardless of whether or not extensions are being used. Unlike $this->value, - * however, $this->hex is only calculated when $this->__sleep() is called. - * - * @see __sleep() - * @see __wakeup() - * @var String - * @access private - */ - var $hex; - - /** - * Converts base-2, base-10, base-16, and binary strings (eg. base-256) to BigIntegers. - * - * If the second parameter - $base - is negative, then it will be assumed that the number's are encoded using - * two's compliment. The sole exception to this is -10, which is treated the same as 10 is. - * - * Here's an example: - * <code> - * <?php - * include('Math/BigInteger.php'); - * - * $a = new Math_BigInteger('0x32', 16); // 50 in base-16 - * - * echo $a->toString(); // outputs 50 - * ?> - * </code> - * - * @param optional $x base-10 number or base-$base number if $base set. - * @param optional integer $base - * @return Math_BigInteger - * @access public - */ - function Math_BigInteger($x = 0, $base = 10) - { - if ( !defined('MATH_BIGINTEGER_MODE') ) { - switch (true) { - case extension_loaded('gmp'): - define('MATH_BIGINTEGER_MODE', MATH_BIGINTEGER_MODE_GMP); - break; - case extension_loaded('bcmath'): - define('MATH_BIGINTEGER_MODE', MATH_BIGINTEGER_MODE_BCMATH); - break; - default: - define('MATH_BIGINTEGER_MODE', MATH_BIGINTEGER_MODE_INTERNAL); - } - } - - switch ( MATH_BIGINTEGER_MODE ) { - case MATH_BIGINTEGER_MODE_GMP: - if (is_resource($x) && get_resource_type($x) == 'GMP integer') { - $this->value = $x; - return; - } - $this->value = gmp_init(0); - break; - case MATH_BIGINTEGER_MODE_BCMATH: - $this->value = '0'; - break; - default: - $this->value = array(); - } - - if (empty($x)) { - return; - } - - switch ($base) { - case -256: - if (ord($x[0]) & 0x80) { - $x = ~$x; - $this->is_negative = true; - } - case 256: - switch ( MATH_BIGINTEGER_MODE ) { - case MATH_BIGINTEGER_MODE_GMP: - $sign = $this->is_negative ? '-' : ''; - $this->value = gmp_init($sign . '0x' . bin2hex($x)); - break; - case MATH_BIGINTEGER_MODE_BCMATH: - // round $len to the nearest 4 (thanks, DavidMJ!) - $len = (strlen($x) + 3) & 0xFFFFFFFC; - - $x = str_pad($x, $len, chr(0), STR_PAD_LEFT); - - for ($i = 0; $i < $len; $i+= 4) { - $this->value = bcmul($this->value, '4294967296', 0); // 4294967296 == 2**32 - $this->value = bcadd($this->value, 0x1000000 * ord($x[$i]) + ((ord($x[$i + 1]) << 16) | (ord($x[$i + 2]) << 8) | ord($x[$i + 3])), 0); - } - - if ($this->is_negative) { - $this->value = '-' . $this->value; - } - - break; - // converts a base-2**8 (big endian / msb) number to base-2**26 (little endian / lsb) - default: - while (strlen($x)) { - $this->value[] = $this->_bytes2int($this->_base256_rshift($x, 26)); - } - } - - if ($this->is_negative) { - if (MATH_BIGINTEGER_MODE != MATH_BIGINTEGER_MODE_INTERNAL) { - $this->is_negative = false; - } - $temp = $this->add(new Math_BigInteger('-1')); - $this->value = $temp->value; - } - break; - case 16: - case -16: - if ($base > 0 && $x[0] == '-') { - $this->is_negative = true; - $x = substr($x, 1); - } - - $x = preg_replace('#^(?:0x)?([A-Fa-f0-9]*).*#', '$1', $x); - - $is_negative = false; - if ($base < 0 && hexdec($x[0]) >= 8) { - $this->is_negative = $is_negative = true; - $x = bin2hex(~pack('H*', $x)); - } - - switch ( MATH_BIGINTEGER_MODE ) { - case MATH_BIGINTEGER_MODE_GMP: - $temp = $this->is_negative ? '-0x' . $x : '0x' . $x; - $this->value = gmp_init($temp); - $this->is_negative = false; - break; - case MATH_BIGINTEGER_MODE_BCMATH: - $x = ( strlen($x) & 1 ) ? '0' . $x : $x; - $temp = new Math_BigInteger(pack('H*', $x), 256); - $this->value = $this->is_negative ? '-' . $temp->value : $temp->value; - $this->is_negative = false; - break; - default: - $x = ( strlen($x) & 1 ) ? '0' . $x : $x; - $temp = new Math_BigInteger(pack('H*', $x), 256); - $this->value = $temp->value; - } - - if ($is_negative) { - $temp = $this->add(new Math_BigInteger('-1')); - $this->value = $temp->value; - } - break; - case 10: - case -10: - $x = preg_replace('#^(-?[0-9]*).*#', '$1', $x); - - switch ( MATH_BIGINTEGER_MODE ) { - case MATH_BIGINTEGER_MODE_GMP: - $this->value = gmp_init($x); - break; - case MATH_BIGINTEGER_MODE_BCMATH: - // explicitly casting $x to a string is necessary, here, since doing $x[0] on -1 yields different - // results then doing it on '-1' does (modInverse does $x[0]) - $this->value = (string) $x; - break; - default: - $temp = new Math_BigInteger(); - - // array(10000000) is 10**7 in base-2**26. 10**7 is the closest to 2**26 we can get without passing it. - $multiplier = new Math_BigInteger(); - $multiplier->value = array(10000000); - - if ($x[0] == '-') { - $this->is_negative = true; - $x = substr($x, 1); - } - - $x = str_pad($x, strlen($x) + (6 * strlen($x)) % 7, 0, STR_PAD_LEFT); - - while (strlen($x)) { - $temp = $temp->multiply($multiplier); - $temp = $temp->add(new Math_BigInteger($this->_int2bytes(substr($x, 0, 7)), 256)); - $x = substr($x, 7); - } - - $this->value = $temp->value; - } - break; - case 2: // base-2 support originally implemented by Lluis Pamies - thanks! - case -2: - if ($base > 0 && $x[0] == '-') { - $this->is_negative = true; - $x = substr($x, 1); - } - - $x = preg_replace('#^([01]*).*#', '$1', $x); - $x = str_pad($x, strlen($x) + (3 * strlen($x)) % 4, 0, STR_PAD_LEFT); - - $str = '0x'; - while (strlen($x)) { - $part = substr($x, 0, 4); - $str.= dechex(bindec($part)); - $x = substr($x, 4); - } - - if ($this->is_negative) { - $str = '-' . $str; - } - - $temp = new Math_BigInteger($str, 8 * $base); // ie. either -16 or +16 - $this->value = $temp->value; - $this->is_negative = $temp->is_negative; - - break; - default: - // base not supported, so we'll let $this == 0 - } - } - - /** - * Converts a BigInteger to a byte string (eg. base-256). - * - * Negative numbers are saved as positive numbers, unless $twos_compliment is set to true, at which point, they're - * saved as two's compliment. - * - * Here's an example: - * <code> - * <?php - * include('Math/BigInteger.php'); - * - * $a = new Math_BigInteger('65'); - * - * echo $a->toBytes(); // outputs chr(65) - * ?> - * </code> - * - * @param Boolean $twos_compliment - * @return String - * @access public - * @internal Converts a base-2**26 number to base-2**8 - */ - function toBytes($twos_compliment = false) - { - if ($twos_compliment) { - $comparison = $this->compare(new Math_BigInteger()); - if ($comparison == 0) { - return $this->precision > 0 ? str_repeat(chr(0), ($this->precision + 1) >> 3) : ''; - } - - $temp = $comparison < 0 ? $this->add(new Math_BigInteger(1)) : $this->copy(); - $bytes = $temp->toBytes(); - - if (empty($bytes)) { // eg. if the number we're trying to convert is -1 - $bytes = chr(0); - } - - if (ord($bytes[0]) & 0x80) { - $bytes = chr(0) . $bytes; - } - - return $comparison < 0 ? ~$bytes : $bytes; - } - - switch ( MATH_BIGINTEGER_MODE ) { - case MATH_BIGINTEGER_MODE_GMP: - if (gmp_cmp($this->value, gmp_init(0)) == 0) { - return $this->precision > 0 ? str_repeat(chr(0), ($this->precision + 1) >> 3) : ''; - } - - $temp = gmp_strval(gmp_abs($this->value), 16); - $temp = ( strlen($temp) & 1 ) ? '0' . $temp : $temp; - $temp = pack('H*', $temp); - - return $this->precision > 0 ? - substr(str_pad($temp, $this->precision >> 3, chr(0), STR_PAD_LEFT), -($this->precision >> 3)) : - ltrim($temp, chr(0)); - case MATH_BIGINTEGER_MODE_BCMATH: - if ($this->value === '0') { - return $this->precision > 0 ? str_repeat(chr(0), ($this->precision + 1) >> 3) : ''; - } - - $value = ''; - $current = $this->value; - - if ($current[0] == '-') { - $current = substr($current, 1); - } - - while (bccomp($current, '0', 0) > 0) { - $temp = bcmod($current, '16777216'); - $value = chr($temp >> 16) . chr($temp >> 8) . chr($temp) . $value; - $current = bcdiv($current, '16777216', 0); - } - - return $this->precision > 0 ? - substr(str_pad($value, $this->precision >> 3, chr(0), STR_PAD_LEFT), -($this->precision >> 3)) : - ltrim($value, chr(0)); - } - - if (!count($this->value)) { - return $this->precision > 0 ? str_repeat(chr(0), ($this->precision + 1) >> 3) : ''; - } - $result = $this->_int2bytes($this->value[count($this->value) - 1]); - - $temp = $this->copy(); - - for ($i = count($temp->value) - 2; $i >= 0; --$i) { - $temp->_base256_lshift($result, 26); - $result = $result | str_pad($temp->_int2bytes($temp->value[$i]), strlen($result), chr(0), STR_PAD_LEFT); - } - - return $this->precision > 0 ? - str_pad(substr($result, -(($this->precision + 7) >> 3)), ($this->precision + 7) >> 3, chr(0), STR_PAD_LEFT) : - $result; - } - - /** - * Converts a BigInteger to a hex string (eg. base-16)). - * - * Negative numbers are saved as positive numbers, unless $twos_compliment is set to true, at which point, they're - * saved as two's compliment. - * - * Here's an example: - * <code> - * <?php - * include('Math/BigInteger.php'); - * - * $a = new Math_BigInteger('65'); - * - * echo $a->toHex(); // outputs '41' - * ?> - * </code> - * - * @param Boolean $twos_compliment - * @return String - * @access public - * @internal Converts a base-2**26 number to base-2**8 - */ - function toHex($twos_compliment = false) - { - return bin2hex($this->toBytes($twos_compliment)); - } - - /** - * Converts a BigInteger to a bit string (eg. base-2). - * - * Negative numbers are saved as positive numbers, unless $twos_compliment is set to true, at which point, they're - * saved as two's compliment. - * - * Here's an example: - * <code> - * <?php - * include('Math/BigInteger.php'); - * - * $a = new Math_BigInteger('65'); - * - * echo $a->toBits(); // outputs '1000001' - * ?> - * </code> - * - * @param Boolean $twos_compliment - * @return String - * @access public - * @internal Converts a base-2**26 number to base-2**2 - */ - function toBits($twos_compliment = false) - { - $hex = $this->toHex($twos_compliment); - $bits = ''; - for ($i = 0, $end = strlen($hex) & 0xFFFFFFF8; $i < $end; $i+=8) { - $bits.= str_pad(decbin(hexdec(substr($hex, $i, 8))), 32, '0', STR_PAD_LEFT); - } - if ($end != strlen($hex)) { // hexdec('') == 0 - $bits.= str_pad(decbin(hexdec(substr($hex, $end))), strlen($hex) & 7, '0', STR_PAD_LEFT); - } - return $this->precision > 0 ? substr($bits, -$this->precision) : ltrim($bits, '0'); - } - - /** - * Converts a BigInteger to a base-10 number. - * - * Here's an example: - * <code> - * <?php - * include('Math/BigInteger.php'); - * - * $a = new Math_BigInteger('50'); - * - * echo $a->toString(); // outputs 50 - * ?> - * </code> - * - * @return String - * @access public - * @internal Converts a base-2**26 number to base-10**7 (which is pretty much base-10) - */ - function toString() - { - switch ( MATH_BIGINTEGER_MODE ) { - case MATH_BIGINTEGER_MODE_GMP: - return gmp_strval($this->value); - case MATH_BIGINTEGER_MODE_BCMATH: - if ($this->value === '0') { - return '0'; - } - - return ltrim($this->value, '0'); - } - - if (!count($this->value)) { - return '0'; - } - - $temp = $this->copy(); - $temp->is_negative = false; - - $divisor = new Math_BigInteger(); - $divisor->value = array(10000000); // eg. 10**7 - $result = ''; - while (count($temp->value)) { - list($temp, $mod) = $temp->divide($divisor); - $result = str_pad(isset($mod->value[0]) ? $mod->value[0] : '', 7, '0', STR_PAD_LEFT) . $result; - } - $result = ltrim($result, '0'); - if (empty($result)) { - $result = '0'; - } - - if ($this->is_negative) { - $result = '-' . $result; - } - - return $result; - } - - /** - * Copy an object - * - * PHP5 passes objects by reference while PHP4 passes by value. As such, we need a function to guarantee - * that all objects are passed by value, when appropriate. More information can be found here: - * - * {@link http://php.net/language.oop5.basic#51624} - * - * @access public - * @see __clone() - * @return Math_BigInteger - */ - function copy() - { - $temp = new Math_BigInteger(); - $temp->value = $this->value; - $temp->is_negative = $this->is_negative; - $temp->generator = $this->generator; - $temp->precision = $this->precision; - $temp->bitmask = $this->bitmask; - return $temp; - } - - /** - * __toString() magic method - * - * Will be called, automatically, if you're supporting just PHP5. If you're supporting PHP4, you'll need to call - * toString(). - * - * @access public - * @internal Implemented per a suggestion by Techie-Michael - thanks! - */ - function __toString() - { - return $this->toString(); - } - - /** - * __clone() magic method - * - * Although you can call Math_BigInteger::__toString() directly in PHP5, you cannot call Math_BigInteger::__clone() - * directly in PHP5. You can in PHP4 since it's not a magic method, but in PHP5, you have to call it by using the PHP5 - * only syntax of $y = clone $x. As such, if you're trying to write an application that works on both PHP4 and PHP5, - * call Math_BigInteger::copy(), instead. - * - * @access public - * @see copy() - * @return Math_BigInteger - */ - function __clone() - { - return $this->copy(); - } - - /** - * __sleep() magic method - * - * Will be called, automatically, when serialize() is called on a Math_BigInteger object. - * - * @see __wakeup() - * @access public - */ - function __sleep() - { - $this->hex = $this->toHex(true); - $vars = array('hex'); - if ($this->generator != 'mt_rand') { - $vars[] = 'generator'; - } - if ($this->precision > 0) { - $vars[] = 'precision'; - } - return $vars; - - } - - /** - * __wakeup() magic method - * - * Will be called, automatically, when unserialize() is called on a Math_BigInteger object. - * - * @see __sleep() - * @access public - */ - function __wakeup() - { - $temp = new Math_BigInteger($this->hex, -16); - $this->value = $temp->value; - $this->is_negative = $temp->is_negative; - $this->setRandomGenerator($this->generator); - if ($this->precision > 0) { - // recalculate $this->bitmask - $this->setPrecision($this->precision); - } - } - - /** - * Adds two BigIntegers. - * - * Here's an example: - * <code> - * <?php - * include('Math/BigInteger.php'); - * - * $a = new Math_BigInteger('10'); - * $b = new Math_BigInteger('20'); - * - * $c = $a->add($b); - * - * echo $c->toString(); // outputs 30 - * ?> - * </code> - * - * @param Math_BigInteger $y - * @return Math_BigInteger - * @access public - * @internal Performs base-2**52 addition - */ - function add($y) - { - switch ( MATH_BIGINTEGER_MODE ) { - case MATH_BIGINTEGER_MODE_GMP: - $temp = new Math_BigInteger(); - $temp->value = gmp_add($this->value, $y->value); - - return $this->_normalize($temp); - case MATH_BIGINTEGER_MODE_BCMATH: - $temp = new Math_BigInteger(); - $temp->value = bcadd($this->value, $y->value, 0); - - return $this->_normalize($temp); - } - - $temp = $this->_add($this->value, $this->is_negative, $y->value, $y->is_negative); - - $result = new Math_BigInteger(); - $result->value = $temp[MATH_BIGINTEGER_VALUE]; - $result->is_negative = $temp[MATH_BIGINTEGER_SIGN]; - - return $this->_normalize($result); - } - - /** - * Performs addition. - * - * @param Array $x_value - * @param Boolean $x_negative - * @param Array $y_value - * @param Boolean $y_negative - * @return Array - * @access private - */ - function _add($x_value, $x_negative, $y_value, $y_negative) - { - $x_size = count($x_value); - $y_size = count($y_value); - - if ($x_size == 0) { - return array( - MATH_BIGINTEGER_VALUE => $y_value, - MATH_BIGINTEGER_SIGN => $y_negative - ); - } else if ($y_size == 0) { - return array( - MATH_BIGINTEGER_VALUE => $x_value, - MATH_BIGINTEGER_SIGN => $x_negative - ); - } - - // subtract, if appropriate - if ( $x_negative != $y_negative ) { - if ( $x_value == $y_value ) { - return array( - MATH_BIGINTEGER_VALUE => array(), - MATH_BIGINTEGER_SIGN => false - ); - } - - $temp = $this->_subtract($x_value, false, $y_value, false); - $temp[MATH_BIGINTEGER_SIGN] = $this->_compare($x_value, false, $y_value, false) > 0 ? - $x_negative : $y_negative; - - return $temp; - } - - if ($x_size < $y_size) { - $size = $x_size; - $value = $y_value; - } else { - $size = $y_size; - $value = $x_value; - } - - $value[] = 0; // just in case the carry adds an extra digit - - $carry = 0; - for ($i = 0, $j = 1; $j < $size; $i+=2, $j+=2) { - $sum = $x_value[$j] * 0x4000000 + $x_value[$i] + $y_value[$j] * 0x4000000 + $y_value[$i] + $carry; - $carry = $sum >= MATH_BIGINTEGER_MAX_DIGIT52; // eg. floor($sum / 2**52); only possible values (in any base) are 0 and 1 - $sum = $carry ? $sum - MATH_BIGINTEGER_MAX_DIGIT52 : $sum; - - $temp = (int) ($sum / 0x4000000); - - $value[$i] = (int) ($sum - 0x4000000 * $temp); // eg. a faster alternative to fmod($sum, 0x4000000) - $value[$j] = $temp; - } - - if ($j == $size) { // ie. if $y_size is odd - $sum = $x_value[$i] + $y_value[$i] + $carry; - $carry = $sum >= 0x4000000; - $value[$i] = $carry ? $sum - 0x4000000 : $sum; - ++$i; // ie. let $i = $j since we've just done $value[$i] - } - - if ($carry) { - for (; $value[$i] == 0x3FFFFFF; ++$i) { - $value[$i] = 0; - } - ++$value[$i]; - } - - return array( - MATH_BIGINTEGER_VALUE => $this->_trim($value), - MATH_BIGINTEGER_SIGN => $x_negative - ); - } - - /** - * Subtracts two BigIntegers. - * - * Here's an example: - * <code> - * <?php - * include('Math/BigInteger.php'); - * - * $a = new Math_BigInteger('10'); - * $b = new Math_BigInteger('20'); - * - * $c = $a->subtract($b); - * - * echo $c->toString(); // outputs -10 - * ?> - * </code> - * - * @param Math_BigInteger $y - * @return Math_BigInteger - * @access public - * @internal Performs base-2**52 subtraction - */ - function subtract($y) - { - switch ( MATH_BIGINTEGER_MODE ) { - case MATH_BIGINTEGER_MODE_GMP: - $temp = new Math_BigInteger(); - $temp->value = gmp_sub($this->value, $y->value); - - return $this->_normalize($temp); - case MATH_BIGINTEGER_MODE_BCMATH: - $temp = new Math_BigInteger(); - $temp->value = bcsub($this->value, $y->value, 0); - - return $this->_normalize($temp); - } - - $temp = $this->_subtract($this->value, $this->is_negative, $y->value, $y->is_negative); - - $result = new Math_BigInteger(); - $result->value = $temp[MATH_BIGINTEGER_VALUE]; - $result->is_negative = $temp[MATH_BIGINTEGER_SIGN]; - - return $this->_normalize($result); - } - - /** - * Performs subtraction. - * - * @param Array $x_value - * @param Boolean $x_negative - * @param Array $y_value - * @param Boolean $y_negative - * @return Array - * @access private - */ - function _subtract($x_value, $x_negative, $y_value, $y_negative) - { - $x_size = count($x_value); - $y_size = count($y_value); - - if ($x_size == 0) { - return array( - MATH_BIGINTEGER_VALUE => $y_value, - MATH_BIGINTEGER_SIGN => !$y_negative - ); - } else if ($y_size == 0) { - return array( - MATH_BIGINTEGER_VALUE => $x_value, - MATH_BIGINTEGER_SIGN => $x_negative - ); - } - - // add, if appropriate (ie. -$x - +$y or +$x - -$y) - if ( $x_negative != $y_negative ) { - $temp = $this->_add($x_value, false, $y_value, false); - $temp[MATH_BIGINTEGER_SIGN] = $x_negative; - - return $temp; - } - - $diff = $this->_compare($x_value, $x_negative, $y_value, $y_negative); - - if ( !$diff ) { - return array( - MATH_BIGINTEGER_VALUE => array(), - MATH_BIGINTEGER_SIGN => false - ); - } - - // switch $x and $y around, if appropriate. - if ( (!$x_negative && $diff < 0) || ($x_negative && $diff > 0) ) { - $temp = $x_value; - $x_value = $y_value; - $y_value = $temp; - - $x_negative = !$x_negative; - - $x_size = count($x_value); - $y_size = count($y_value); - } - - // at this point, $x_value should be at least as big as - if not bigger than - $y_value - - $carry = 0; - for ($i = 0, $j = 1; $j < $y_size; $i+=2, $j+=2) { - $sum = $x_value[$j] * 0x4000000 + $x_value[$i] - $y_value[$j] * 0x4000000 - $y_value[$i] - $carry; - $carry = $sum < 0; // eg. floor($sum / 2**52); only possible values (in any base) are 0 and 1 - $sum = $carry ? $sum + MATH_BIGINTEGER_MAX_DIGIT52 : $sum; - - $temp = (int) ($sum / 0x4000000); - - $x_value[$i] = (int) ($sum - 0x4000000 * $temp); - $x_value[$j] = $temp; - } - - if ($j == $y_size) { // ie. if $y_size is odd - $sum = $x_value[$i] - $y_value[$i] - $carry; - $carry = $sum < 0; - $x_value[$i] = $carry ? $sum + 0x4000000 : $sum; - ++$i; - } - - if ($carry) { - for (; !$x_value[$i]; ++$i) { - $x_value[$i] = 0x3FFFFFF; - } - --$x_value[$i]; - } - - return array( - MATH_BIGINTEGER_VALUE => $this->_trim($x_value), - MATH_BIGINTEGER_SIGN => $x_negative - ); - } - - /** - * Multiplies two BigIntegers - * - * Here's an example: - * <code> - * <?php - * include('Math/BigInteger.php'); - * - * $a = new Math_BigInteger('10'); - * $b = new Math_BigInteger('20'); - * - * $c = $a->multiply($b); - * - * echo $c->toString(); // outputs 200 - * ?> - * </code> - * - * @param Math_BigInteger $x - * @return Math_BigInteger - * @access public - */ - function multiply($x) - { - switch ( MATH_BIGINTEGER_MODE ) { - case MATH_BIGINTEGER_MODE_GMP: - $temp = new Math_BigInteger(); - $temp->value = gmp_mul($this->value, $x->value); - - return $this->_normalize($temp); - case MATH_BIGINTEGER_MODE_BCMATH: - $temp = new Math_BigInteger(); - $temp->value = bcmul($this->value, $x->value, 0); - - return $this->_normalize($temp); - } - - $temp = $this->_multiply($this->value, $this->is_negative, $x->value, $x->is_negative); - - $product = new Math_BigInteger(); - $product->value = $temp[MATH_BIGINTEGER_VALUE]; - $product->is_negative = $temp[MATH_BIGINTEGER_SIGN]; - - return $this->_normalize($product); - } - - /** - * Performs multiplication. - * - * @param Array $x_value - * @param Boolean $x_negative - * @param Array $y_value - * @param Boolean $y_negative - * @return Array - * @access private - */ - function _multiply($x_value, $x_negative, $y_value, $y_negative) - { - //if ( $x_value == $y_value ) { - // return array( - // MATH_BIGINTEGER_VALUE => $this->_square($x_value), - // MATH_BIGINTEGER_SIGN => $x_sign != $y_value - // ); - //} - - $x_length = count($x_value); - $y_length = count($y_value); - - if ( !$x_length || !$y_length ) { // a 0 is being multiplied - return array( - MATH_BIGINTEGER_VALUE => array(), - MATH_BIGINTEGER_SIGN => false - ); - } - - return array( - MATH_BIGINTEGER_VALUE => min($x_length, $y_length) < 2 * MATH_BIGINTEGER_KARATSUBA_CUTOFF ? - $this->_trim($this->_regularMultiply($x_value, $y_value)) : - $this->_trim($this->_karatsuba($x_value, $y_value)), - MATH_BIGINTEGER_SIGN => $x_negative != $y_negative - ); - } - - /** - * Performs long multiplication on two BigIntegers - * - * Modeled after 'multiply' in MutableBigInteger.java. - * - * @param Array $x_value - * @param Array $y_value - * @return Array - * @access private - */ - function _regularMultiply($x_value, $y_value) - { - $x_length = count($x_value); - $y_length = count($y_value); - - if ( !$x_length || !$y_length ) { // a 0 is being multiplied - return array(); - } - - if ( $x_length < $y_length ) { - $temp = $x_value; - $x_value = $y_value; - $y_value = $temp; - - $x_length = count($x_value); - $y_length = count($y_value); - } - - $product_value = $this->_array_repeat(0, $x_length + $y_length); - - // the following for loop could be removed if the for loop following it - // (the one with nested for loops) initially set $i to 0, but - // doing so would also make the result in one set of unnecessary adds, - // since on the outermost loops first pass, $product->value[$k] is going - // to always be 0 - - $carry = 0; - - for ($j = 0; $j < $x_length; ++$j) { // ie. $i = 0 - $temp = $x_value[$j] * $y_value[0] + $carry; // $product_value[$k] == 0 - $carry = (int) ($temp / 0x4000000); - $product_value[$j] = (int) ($temp - 0x4000000 * $carry); - } - - $product_value[$j] = $carry; - - // the above for loop is what the previous comment was talking about. the - // following for loop is the "one with nested for loops" - for ($i = 1; $i < $y_length; ++$i) { - $carry = 0; - - for ($j = 0, $k = $i; $j < $x_length; ++$j, ++$k) { - $temp = $product_value[$k] + $x_value[$j] * $y_value[$i] + $carry; - $carry = (int) ($temp / 0x4000000); - $product_value[$k] = (int) ($temp - 0x4000000 * $carry); - } - - $product_value[$k] = $carry; - } - - return $product_value; - } - - /** - * Performs Karatsuba multiplication on two BigIntegers - * - * See {@link http://en.wikipedia.org/wiki/Karatsuba_algorithm Karatsuba algorithm} and - * {@link http://math.libtomcrypt.com/files/tommath.pdf#page=120 MPM 5.2.3}. - * - * @param Array $x_value - * @param Array $y_value - * @return Array - * @access private - */ - function _karatsuba($x_value, $y_value) - { - $m = min(count($x_value) >> 1, count($y_value) >> 1); - - if ($m < MATH_BIGINTEGER_KARATSUBA_CUTOFF) { - return $this->_regularMultiply($x_value, $y_value); - } - - $x1 = array_slice($x_value, $m); - $x0 = array_slice($x_value, 0, $m); - $y1 = array_slice($y_value, $m); - $y0 = array_slice($y_value, 0, $m); - - $z2 = $this->_karatsuba($x1, $y1); - $z0 = $this->_karatsuba($x0, $y0); - - $z1 = $this->_add($x1, false, $x0, false); - $temp = $this->_add($y1, false, $y0, false); - $z1 = $this->_karatsuba($z1[MATH_BIGINTEGER_VALUE], $temp[MATH_BIGINTEGER_VALUE]); - $temp = $this->_add($z2, false, $z0, false); - $z1 = $this->_subtract($z1, false, $temp[MATH_BIGINTEGER_VALUE], false); - - $z2 = array_merge(array_fill(0, 2 * $m, 0), $z2); - $z1[MATH_BIGINTEGER_VALUE] = array_merge(array_fill(0, $m, 0), $z1[MATH_BIGINTEGER_VALUE]); - - $xy = $this->_add($z2, false, $z1[MATH_BIGINTEGER_VALUE], $z1[MATH_BIGINTEGER_SIGN]); - $xy = $this->_add($xy[MATH_BIGINTEGER_VALUE], $xy[MATH_BIGINTEGER_SIGN], $z0, false); - - return $xy[MATH_BIGINTEGER_VALUE]; - } - - /** - * Performs squaring - * - * @param Array $x - * @return Array - * @access private - */ - function _square($x = false) - { - return count($x) < 2 * MATH_BIGINTEGER_KARATSUBA_CUTOFF ? - $this->_trim($this->_baseSquare($x)) : - $this->_trim($this->_karatsubaSquare($x)); - } - - /** - * Performs traditional squaring on two BigIntegers - * - * Squaring can be done faster than multiplying a number by itself can be. See - * {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=7 HAC 14.2.4} / - * {@link http://math.libtomcrypt.com/files/tommath.pdf#page=141 MPM 5.3} for more information. - * - * @param Array $value - * @return Array - * @access private - */ - function _baseSquare($value) - { - if ( empty($value) ) { - return array(); - } - $square_value = $this->_array_repeat(0, 2 * count($value)); - - for ($i = 0, $max_index = count($value) - 1; $i <= $max_index; ++$i) { - $i2 = $i << 1; - - $temp = $square_value[$i2] + $value[$i] * $value[$i]; - $carry = (int) ($temp / 0x4000000); - $square_value[$i2] = (int) ($temp - 0x4000000 * $carry); - - // note how we start from $i+1 instead of 0 as we do in multiplication. - for ($j = $i + 1, $k = $i2 + 1; $j <= $max_index; ++$j, ++$k) { - $temp = $square_value[$k] + 2 * $value[$j] * $value[$i] + $carry; - $carry = (int) ($temp / 0x4000000); - $square_value[$k] = (int) ($temp - 0x4000000 * $carry); - } - - // the following line can yield values larger 2**15. at this point, PHP should switch - // over to floats. - $square_value[$i + $max_index + 1] = $carry; - } - - return $square_value; - } - - /** - * Performs Karatsuba "squaring" on two BigIntegers - * - * See {@link http://en.wikipedia.org/wiki/Karatsuba_algorithm Karatsuba algorithm} and - * {@link http://math.libtomcrypt.com/files/tommath.pdf#page=151 MPM 5.3.4}. - * - * @param Array $value - * @return Array - * @access private - */ - function _karatsubaSquare($value) - { - $m = count($value) >> 1; - - if ($m < MATH_BIGINTEGER_KARATSUBA_CUTOFF) { - return $this->_baseSquare($value); - } - - $x1 = array_slice($value, $m); - $x0 = array_slice($value, 0, $m); - - $z2 = $this->_karatsubaSquare($x1); - $z0 = $this->_karatsubaSquare($x0); - - $z1 = $this->_add($x1, false, $x0, false); - $z1 = $this->_karatsubaSquare($z1[MATH_BIGINTEGER_VALUE]); - $temp = $this->_add($z2, false, $z0, false); - $z1 = $this->_subtract($z1, false, $temp[MATH_BIGINTEGER_VALUE], false); - - $z2 = array_merge(array_fill(0, 2 * $m, 0), $z2); - $z1[MATH_BIGINTEGER_VALUE] = array_merge(array_fill(0, $m, 0), $z1[MATH_BIGINTEGER_VALUE]); - - $xx = $this->_add($z2, false, $z1[MATH_BIGINTEGER_VALUE], $z1[MATH_BIGINTEGER_SIGN]); - $xx = $this->_add($xx[MATH_BIGINTEGER_VALUE], $xx[MATH_BIGINTEGER_SIGN], $z0, false); - - return $xx[MATH_BIGINTEGER_VALUE]; - } - - /** - * Divides two BigIntegers. - * - * Returns an array whose first element contains the quotient and whose second element contains the - * "common residue". If the remainder would be positive, the "common residue" and the remainder are the - * same. If the remainder would be negative, the "common residue" is equal to the sum of the remainder - * and the divisor (basically, the "common residue" is the first positive modulo). - * - * Here's an example: - * <code> - * <?php - * include('Math/BigInteger.php'); - * - * $a = new Math_BigInteger('10'); - * $b = new Math_BigInteger('20'); - * - * list($quotient, $remainder) = $a->divide($b); - * - * echo $quotient->toString(); // outputs 0 - * echo "\r\n"; - * echo $remainder->toString(); // outputs 10 - * ?> - * </code> - * - * @param Math_BigInteger $y - * @return Array - * @access public - * @internal This function is based off of {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=9 HAC 14.20}. - */ - function divide($y) - { - switch ( MATH_BIGINTEGER_MODE ) { - case MATH_BIGINTEGER_MODE_GMP: - $quotient = new Math_BigInteger(); - $remainder = new Math_BigInteger(); - - list($quotient->value, $remainder->value) = gmp_div_qr($this->value, $y->value); - - if (gmp_sign($remainder->value) < 0) { - $remainder->value = gmp_add($remainder->value, gmp_abs($y->value)); - } - - return array($this->_normalize($quotient), $this->_normalize($remainder)); - case MATH_BIGINTEGER_MODE_BCMATH: - $quotient = new Math_BigInteger(); - $remainder = new Math_BigInteger(); - - $quotient->value = bcdiv($this->value, $y->value, 0); - $remainder->value = bcmod($this->value, $y->value); - - if ($remainder->value[0] == '-') { - $remainder->value = bcadd($remainder->value, $y->value[0] == '-' ? substr($y->value, 1) : $y->value, 0); - } - - return array($this->_normalize($quotient), $this->_normalize($remainder)); - } - - if (count($y->value) == 1) { - list($q, $r) = $this->_divide_digit($this->value, $y->value[0]); - $quotient = new Math_BigInteger(); - $remainder = new Math_BigInteger(); - $quotient->value = $q; - $remainder->value = array($r); - $quotient->is_negative = $this->is_negative != $y->is_negative; - return array($this->_normalize($quotient), $this->_normalize($remainder)); - } - - static $zero; - if ( !isset($zero) ) { - $zero = new Math_BigInteger(); - } - - $x = $this->copy(); - $y = $y->copy(); - - $x_sign = $x->is_negative; - $y_sign = $y->is_negative; - - $x->is_negative = $y->is_negative = false; - - $diff = $x->compare($y); - - if ( !$diff ) { - $temp = new Math_BigInteger(); - $temp->value = array(1); - $temp->is_negative = $x_sign != $y_sign; - return array($this->_normalize($temp), $this->_normalize(new Math_BigInteger())); - } - - if ( $diff < 0 ) { - // if $x is negative, "add" $y. - if ( $x_sign ) { - $x = $y->subtract($x); - } - return array($this->_normalize(new Math_BigInteger()), $this->_normalize($x)); - } - - // normalize $x and $y as described in HAC 14.23 / 14.24 - $msb = $y->value[count($y->value) - 1]; - for ($shift = 0; !($msb & 0x2000000); ++$shift) { - $msb <<= 1; - } - $x->_lshift($shift); - $y->_lshift($shift); - $y_value = &$y->value; - - $x_max = count($x->value) - 1; - $y_max = count($y->value) - 1; - - $quotient = new Math_BigInteger(); - $quotient_value = &$quotient->value; - $quotient_value = $this->_array_repeat(0, $x_max - $y_max + 1); - - static $temp, $lhs, $rhs; - if (!isset($temp)) { - $temp = new Math_BigInteger(); - $lhs = new Math_BigInteger(); - $rhs = new Math_BigInteger(); - } - $temp_value = &$temp->value; - $rhs_value = &$rhs->value; - - // $temp = $y << ($x_max - $y_max-1) in base 2**26 - $temp_value = array_merge($this->_array_repeat(0, $x_max - $y_max), $y_value); - - while ( $x->compare($temp) >= 0 ) { - // calculate the "common residue" - ++$quotient_value[$x_max - $y_max]; - $x = $x->subtract($temp); - $x_max = count($x->value) - 1; - } - - for ($i = $x_max; $i >= $y_max + 1; --$i) { - $x_value = &$x->value; - $x_window = array( - isset($x_value[$i]) ? $x_value[$i] : 0, - isset($x_value[$i - 1]) ? $x_value[$i - 1] : 0, - isset($x_value[$i - 2]) ? $x_value[$i - 2] : 0 - ); - $y_window = array( - $y_value[$y_max], - ( $y_max > 0 ) ? $y_value[$y_max - 1] : 0 - ); - - $q_index = $i - $y_max - 1; - if ($x_window[0] == $y_window[0]) { - $quotient_value[$q_index] = 0x3FFFFFF; - } else { - $quotient_value[$q_index] = (int) ( - ($x_window[0] * 0x4000000 + $x_window[1]) - / - $y_window[0] - ); - } - - $temp_value = array($y_window[1], $y_window[0]); - - $lhs->value = array($quotient_value[$q_index]); - $lhs = $lhs->multiply($temp); - - $rhs_value = array($x_window[2], $x_window[1], $x_window[0]); - - while ( $lhs->compare($rhs) > 0 ) { - --$quotient_value[$q_index]; - - $lhs->value = array($quotient_value[$q_index]); - $lhs = $lhs->multiply($temp); - } - - $adjust = $this->_array_repeat(0, $q_index); - $temp_value = array($quotient_value[$q_index]); - $temp = $temp->multiply($y); - $temp_value = &$temp->value; - $temp_value = array_merge($adjust, $temp_value); - - $x = $x->subtract($temp); - - if ($x->compare($zero) < 0) { - $temp_value = array_merge($adjust, $y_value); - $x = $x->add($temp); - - --$quotient_value[$q_index]; - } - - $x_max = count($x_value) - 1; - } - - // unnormalize the remainder - $x->_rshift($shift); - - $quotient->is_negative = $x_sign != $y_sign; - - // calculate the "common residue", if appropriate - if ( $x_sign ) { - $y->_rshift($shift); - $x = $y->subtract($x); - } - - return array($this->_normalize($quotient), $this->_normalize($x)); - } - - /** - * Divides a BigInteger by a regular integer - * - * abc / x = a00 / x + b0 / x + c / x - * - * @param Array $dividend - * @param Array $divisor - * @return Array - * @access private - */ - function _divide_digit($dividend, $divisor) - { - $carry = 0; - $result = array(); - - for ($i = count($dividend) - 1; $i >= 0; --$i) { - $temp = 0x4000000 * $carry + $dividend[$i]; - $result[$i] = (int) ($temp / $divisor); - $carry = (int) ($temp - $divisor * $result[$i]); - } - - return array($result, $carry); - } - - /** - * Performs modular exponentiation. - * - * Here's an example: - * <code> - * <?php - * include('Math/BigInteger.php'); - * - * $a = new Math_BigInteger('10'); - * $b = new Math_BigInteger('20'); - * $c = new Math_BigInteger('30'); - * - * $c = $a->modPow($b, $c); - * - * echo $c->toString(); // outputs 10 - * ?> - * </code> - * - * @param Math_BigInteger $e - * @param Math_BigInteger $n - * @return Math_BigInteger - * @access public - * @internal The most naive approach to modular exponentiation has very unreasonable requirements, and - * and although the approach involving repeated squaring does vastly better, it, too, is impractical - * for our purposes. The reason being that division - by far the most complicated and time-consuming - * of the basic operations (eg. +,-,*,/) - occurs multiple times within it. - * - * Modular reductions resolve this issue. Although an individual modular reduction takes more time - * then an individual division, when performed in succession (with the same modulo), they're a lot faster. - * - * The two most commonly used modular reductions are Barrett and Montgomery reduction. Montgomery reduction, - * although faster, only works when the gcd of the modulo and of the base being used is 1. In RSA, when the - * base is a power of two, the modulo - a product of two primes - is always going to have a gcd of 1 (because - * the product of two odd numbers is odd), but what about when RSA isn't used? - * - * In contrast, Barrett reduction has no such constraint. As such, some bigint implementations perform a - * Barrett reduction after every operation in the modpow function. Others perform Barrett reductions when the - * modulo is even and Montgomery reductions when the modulo is odd. BigInteger.java's modPow method, however, - * uses a trick involving the Chinese Remainder Theorem to factor the even modulo into two numbers - one odd and - * the other, a power of two - and recombine them, later. This is the method that this modPow function uses. - * {@link http://islab.oregonstate.edu/papers/j34monex.pdf Montgomery Reduction with Even Modulus} elaborates. - */ - function modPow($e, $n) - { - $n = $this->bitmask !== false && $this->bitmask->compare($n) < 0 ? $this->bitmask : $n->abs(); - - if ($e->compare(new Math_BigInteger()) < 0) { - $e = $e->abs(); - - $temp = $this->modInverse($n); - if ($temp === false) { - return false; - } - - return $this->_normalize($temp->modPow($e, $n)); - } - - switch ( MATH_BIGINTEGER_MODE ) { - case MATH_BIGINTEGER_MODE_GMP: - $temp = new Math_BigInteger(); - $temp->value = gmp_powm($this->value, $e->value, $n->value); - - return $this->_normalize($temp); - case MATH_BIGINTEGER_MODE_BCMATH: - $temp = new Math_BigInteger(); - $temp->value = bcpowmod($this->value, $e->value, $n->value, 0); - - return $this->_normalize($temp); - } - - if ( empty($e->value) ) { - $temp = new Math_BigInteger(); - $temp->value = array(1); - return $this->_normalize($temp); - } - - if ( $e->value == array(1) ) { - list(, $temp) = $this->divide($n); - return $this->_normalize($temp); - } - - if ( $e->value == array(2) ) { - $temp = new Math_BigInteger(); - $temp->value = $this->_square($this->value); - list(, $temp) = $temp->divide($n); - return $this->_normalize($temp); - } - - return $this->_normalize($this->_slidingWindow($e, $n, MATH_BIGINTEGER_BARRETT)); - - // is the modulo odd? - if ( $n->value[0] & 1 ) { - return $this->_normalize($this->_slidingWindow($e, $n, MATH_BIGINTEGER_MONTGOMERY)); - } - // if it's not, it's even - - // find the lowest set bit (eg. the max pow of 2 that divides $n) - for ($i = 0; $i < count($n->value); ++$i) { - if ( $n->value[$i] ) { - $temp = decbin($n->value[$i]); - $j = strlen($temp) - strrpos($temp, '1') - 1; - $j+= 26 * $i; - break; - } - } - // at this point, 2^$j * $n/(2^$j) == $n - - $mod1 = $n->copy(); - $mod1->_rshift($j); - $mod2 = new Math_BigInteger(); - $mod2->value = array(1); - $mod2->_lshift($j); - - $part1 = ( $mod1->value != array(1) ) ? $this->_slidingWindow($e, $mod1, MATH_BIGINTEGER_MONTGOMERY) : new Math_BigInteger(); - $part2 = $this->_slidingWindow($e, $mod2, MATH_BIGINTEGER_POWEROF2); - - $y1 = $mod2->modInverse($mod1); - $y2 = $mod1->modInverse($mod2); - - $result = $part1->multiply($mod2); - $result = $result->multiply($y1); - - $temp = $part2->multiply($mod1); - $temp = $temp->multiply($y2); - - $result = $result->add($temp); - list(, $result) = $result->divide($n); - - return $this->_normalize($result); - } - - /** - * Performs modular exponentiation. - * - * Alias for Math_BigInteger::modPow() - * - * @param Math_BigInteger $e - * @param Math_BigInteger $n - * @return Math_BigInteger - * @access public - */ - function powMod($e, $n) - { - return $this->modPow($e, $n); - } - - /** - * Sliding Window k-ary Modular Exponentiation - * - * Based on {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=27 HAC 14.85} / - * {@link http://math.libtomcrypt.com/files/tommath.pdf#page=210 MPM 7.7}. In a departure from those algorithims, - * however, this function performs a modular reduction after every multiplication and squaring operation. - * As such, this function has the same preconditions that the reductions being used do. - * - * @param Math_BigInteger $e - * @param Math_BigInteger $n - * @param Integer $mode - * @return Math_BigInteger - * @access private - */ - function _slidingWindow($e, $n, $mode) - { - static $window_ranges = array(7, 25, 81, 241, 673, 1793); // from BigInteger.java's oddModPow function - //static $window_ranges = array(0, 7, 36, 140, 450, 1303, 3529); // from MPM 7.3.1 - - $e_value = $e->value; - $e_length = count($e_value) - 1; - $e_bits = decbin($e_value[$e_length]); - for ($i = $e_length - 1; $i >= 0; --$i) { - $e_bits.= str_pad(decbin($e_value[$i]), 26, '0', STR_PAD_LEFT); - } - - $e_length = strlen($e_bits); - - // calculate the appropriate window size. - // $window_size == 3 if $window_ranges is between 25 and 81, for example. - for ($i = 0, $window_size = 1; $e_length > $window_ranges[$i] && $i < count($window_ranges); ++$window_size, ++$i); - - $n_value = $n->value; - - // precompute $this^0 through $this^$window_size - $powers = array(); - $powers[1] = $this->_prepareReduce($this->value, $n_value, $mode); - $powers[2] = $this->_squareReduce($powers[1], $n_value, $mode); - - // we do every other number since substr($e_bits, $i, $j+1) (see below) is supposed to end - // in a 1. ie. it's supposed to be odd. - $temp = 1 << ($window_size - 1); - for ($i = 1; $i < $temp; ++$i) { - $i2 = $i << 1; - $powers[$i2 + 1] = $this->_multiplyReduce($powers[$i2 - 1], $powers[2], $n_value, $mode); - } - - $result = array(1); - $result = $this->_prepareReduce($result, $n_value, $mode); - - for ($i = 0; $i < $e_length; ) { - if ( !$e_bits[$i] ) { - $result = $this->_squareReduce($result, $n_value, $mode); - ++$i; - } else { - for ($j = $window_size - 1; $j > 0; --$j) { - if ( !empty($e_bits[$i + $j]) ) { - break; - } - } - - for ($k = 0; $k <= $j; ++$k) {// eg. the length of substr($e_bits, $i, $j+1) - $result = $this->_squareReduce($result, $n_value, $mode); - } - - $result = $this->_multiplyReduce($result, $powers[bindec(substr($e_bits, $i, $j + 1))], $n_value, $mode); - - $i+=$j + 1; - } - } - - $temp = new Math_BigInteger(); - $temp->value = $this->_reduce($result, $n_value, $mode); - - return $temp; - } - - /** - * Modular reduction - * - * For most $modes this will return the remainder. - * - * @see _slidingWindow() - * @access private - * @param Array $x - * @param Array $n - * @param Integer $mode - * @return Array - */ - function _reduce($x, $n, $mode) - { - switch ($mode) { - case MATH_BIGINTEGER_MONTGOMERY: - return $this->_montgomery($x, $n); - case MATH_BIGINTEGER_BARRETT: - return $this->_barrett($x, $n); - case MATH_BIGINTEGER_POWEROF2: - $lhs = new Math_BigInteger(); - $lhs->value = $x; - $rhs = new Math_BigInteger(); - $rhs->value = $n; - return $x->_mod2($n); - case MATH_BIGINTEGER_CLASSIC: - $lhs = new Math_BigInteger(); - $lhs->value = $x; - $rhs = new Math_BigInteger(); - $rhs->value = $n; - list(, $temp) = $lhs->divide($rhs); - return $temp->value; - case MATH_BIGINTEGER_NONE: - return $x; - default: - // an invalid $mode was provided - } - } - - /** - * Modular reduction preperation - * - * @see _slidingWindow() - * @access private - * @param Array $x - * @param Array $n - * @param Integer $mode - * @return Array - */ - function _prepareReduce($x, $n, $mode) - { - if ($mode == MATH_BIGINTEGER_MONTGOMERY) { - return $this->_prepMontgomery($x, $n); - } - return $this->_reduce($x, $n, $mode); - } - - /** - * Modular multiply - * - * @see _slidingWindow() - * @access private - * @param Array $x - * @param Array $y - * @param Array $n - * @param Integer $mode - * @return Array - */ - function _multiplyReduce($x, $y, $n, $mode) - { - if ($mode == MATH_BIGINTEGER_MONTGOMERY) { - return $this->_montgomeryMultiply($x, $y, $n); - } - $temp = $this->_multiply($x, false, $y, false); - return $this->_reduce($temp[MATH_BIGINTEGER_VALUE], $n, $mode); - } - - /** - * Modular square - * - * @see _slidingWindow() - * @access private - * @param Array $x - * @param Array $n - * @param Integer $mode - * @return Array - */ - function _squareReduce($x, $n, $mode) - { - if ($mode == MATH_BIGINTEGER_MONTGOMERY) { - return $this->_montgomeryMultiply($x, $x, $n); - } - return $this->_reduce($this->_square($x), $n, $mode); - } - - /** - * Modulos for Powers of Two - * - * Calculates $x%$n, where $n = 2**$e, for some $e. Since this is basically the same as doing $x & ($n-1), - * we'll just use this function as a wrapper for doing that. - * - * @see _slidingWindow() - * @access private - * @param Math_BigInteger - * @return Math_BigInteger - */ - function _mod2($n) - { - $temp = new Math_BigInteger(); - $temp->value = array(1); - return $this->bitwise_and($n->subtract($temp)); - } - - /** - * Barrett Modular Reduction - * - * See {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=14 HAC 14.3.3} / - * {@link http://math.libtomcrypt.com/files/tommath.pdf#page=165 MPM 6.2.5} for more information. Modified slightly, - * so as not to require negative numbers (initially, this script didn't support negative numbers). - * - * Employs "folding", as described at - * {@link http://www.cosic.esat.kuleuven.be/publications/thesis-149.pdf#page=66 thesis-149.pdf#page=66}. To quote from - * it, "the idea [behind folding] is to find a value x' such that x (mod m) = x' (mod m), with x' being smaller than x." - * - * Unfortunately, the "Barrett Reduction with Folding" algorithm described in thesis-149.pdf is not, as written, all that - * usable on account of (1) its not using reasonable radix points as discussed in - * {@link http://math.libtomcrypt.com/files/tommath.pdf#page=162 MPM 6.2.2} and (2) the fact that, even with reasonable - * radix points, it only works when there are an even number of digits in the denominator. The reason for (2) is that - * (x >> 1) + (x >> 1) != x / 2 + x / 2. If x is even, they're the same, but if x is odd, they're not. See the in-line - * comments for details. - * - * @see _slidingWindow() - * @access private - * @param Array $n - * @param Array $m - * @return Array - */ - function _barrett($n, $m) - { - static $cache = array( - MATH_BIGINTEGER_VARIABLE => array(), - MATH_BIGINTEGER_DATA => array() - ); - - $m_length = count($m); - - // if ($this->_compare($n, $this->_square($m)) >= 0) { - if (count($n) > 2 * $m_length) { - $lhs = new Math_BigInteger(); - $rhs = new Math_BigInteger(); - $lhs->value = $n; - $rhs->value = $m; - list(, $temp) = $lhs->divide($rhs); - return $temp->value; - } - - // if (m.length >> 1) + 2 <= m.length then m is too small and n can't be reduced - if ($m_length < 5) { - return $this->_regularBarrett($n, $m); - } - - // n = 2 * m.length - - if ( ($key = array_search($m, $cache[MATH_BIGINTEGER_VARIABLE])) === false ) { - $key = count($cache[MATH_BIGINTEGER_VARIABLE]); - $cache[MATH_BIGINTEGER_VARIABLE][] = $m; - - $lhs = new Math_BigInteger(); - $lhs_value = &$lhs->value; - $lhs_value = $this->_array_repeat(0, $m_length + ($m_length >> 1)); - $lhs_value[] = 1; - $rhs = new Math_BigInteger(); - $rhs->value = $m; - - list($u, $m1) = $lhs->divide($rhs); - $u = $u->value; - $m1 = $m1->value; - - $cache[MATH_BIGINTEGER_DATA][] = array( - 'u' => $u, // m.length >> 1 (technically (m.length >> 1) + 1) - 'm1'=> $m1 // m.length - ); - } else { - extract($cache[MATH_BIGINTEGER_DATA][$key]); - } - - $cutoff = $m_length + ($m_length >> 1); - $lsd = array_slice($n, 0, $cutoff); // m.length + (m.length >> 1) - $msd = array_slice($n, $cutoff); // m.length >> 1 - $lsd = $this->_trim($lsd); - $temp = $this->_multiply($msd, false, $m1, false); - $n = $this->_add($lsd, false, $temp[MATH_BIGINTEGER_VALUE], false); // m.length + (m.length >> 1) + 1 - - if ($m_length & 1) { - return $this->_regularBarrett($n[MATH_BIGINTEGER_VALUE], $m); - } - - // (m.length + (m.length >> 1) + 1) - (m.length - 1) == (m.length >> 1) + 2 - $temp = array_slice($n[MATH_BIGINTEGER_VALUE], $m_length - 1); - // if even: ((m.length >> 1) + 2) + (m.length >> 1) == m.length + 2 - // if odd: ((m.length >> 1) + 2) + (m.length >> 1) == (m.length - 1) + 2 == m.length + 1 - $temp = $this->_multiply($temp, false, $u, false); - // if even: (m.length + 2) - ((m.length >> 1) + 1) = m.length - (m.length >> 1) + 1 - // if odd: (m.length + 1) - ((m.length >> 1) + 1) = m.length - (m.length >> 1) - $temp = array_slice($temp[MATH_BIGINTEGER_VALUE], ($m_length >> 1) + 1); - // if even: (m.length - (m.length >> 1) + 1) + m.length = 2 * m.length - (m.length >> 1) + 1 - // if odd: (m.length - (m.length >> 1)) + m.length = 2 * m.length - (m.length >> 1) - $temp = $this->_multiply($temp, false, $m, false); - - // at this point, if m had an odd number of digits, we'd be subtracting a 2 * m.length - (m.length >> 1) digit - // number from a m.length + (m.length >> 1) + 1 digit number. ie. there'd be an extra digit and the while loop - // following this comment would loop a lot (hence our calling _regularBarrett() in that situation). - - $result = $this->_subtract($n[MATH_BIGINTEGER_VALUE], false, $temp[MATH_BIGINTEGER_VALUE], false); - - while ($this->_compare($result[MATH_BIGINTEGER_VALUE], $result[MATH_BIGINTEGER_SIGN], $m, false) >= 0) { - $result = $this->_subtract($result[MATH_BIGINTEGER_VALUE], $result[MATH_BIGINTEGER_SIGN], $m, false); - } - - return $result[MATH_BIGINTEGER_VALUE]; - } - - /** - * (Regular) Barrett Modular Reduction - * - * For numbers with more than four digits Math_BigInteger::_barrett() is faster. The difference between that and this - * is that this function does not fold the denominator into a smaller form. - * - * @see _slidingWindow() - * @access private - * @param Array $x - * @param Array $n - * @return Array - */ - function _regularBarrett($x, $n) - { - static $cache = array( - MATH_BIGINTEGER_VARIABLE => array(), - MATH_BIGINTEGER_DATA => array() - ); - - $n_length = count($n); - - if (count($x) > 2 * $n_length) { - $lhs = new Math_BigInteger(); - $rhs = new Math_BigInteger(); - $lhs->value = $x; - $rhs->value = $n; - list(, $temp) = $lhs->divide($rhs); - return $temp->value; - } - - if ( ($key = array_search($n, $cache[MATH_BIGINTEGER_VARIABLE])) === false ) { - $key = count($cache[MATH_BIGINTEGER_VARIABLE]); - $cache[MATH_BIGINTEGER_VARIABLE][] = $n; - $lhs = new Math_BigInteger(); - $lhs_value = &$lhs->value; - $lhs_value = $this->_array_repeat(0, 2 * $n_length); - $lhs_value[] = 1; - $rhs = new Math_BigInteger(); - $rhs->value = $n; - list($temp, ) = $lhs->divide($rhs); // m.length - $cache[MATH_BIGINTEGER_DATA][] = $temp->value; - } - - // 2 * m.length - (m.length - 1) = m.length + 1 - $temp = array_slice($x, $n_length - 1); - // (m.length + 1) + m.length = 2 * m.length + 1 - $temp = $this->_multiply($temp, false, $cache[MATH_BIGINTEGER_DATA][$key], false); - // (2 * m.length + 1) - (m.length - 1) = m.length + 2 - $temp = array_slice($temp[MATH_BIGINTEGER_VALUE], $n_length + 1); - - // m.length + 1 - $result = array_slice($x, 0, $n_length + 1); - // m.length + 1 - $temp = $this->_multiplyLower($temp, false, $n, false, $n_length + 1); - // $temp == array_slice($temp->_multiply($temp, false, $n, false)->value, 0, $n_length + 1) - - if ($this->_compare($result, false, $temp[MATH_BIGINTEGER_VALUE], $temp[MATH_BIGINTEGER_SIGN]) < 0) { - $corrector_value = $this->_array_repeat(0, $n_length + 1); - $corrector_value[] = 1; - $result = $this->_add($result, false, $corrector, false); - $result = $result[MATH_BIGINTEGER_VALUE]; - } - - // at this point, we're subtracting a number with m.length + 1 digits from another number with m.length + 1 digits - $result = $this->_subtract($result, false, $temp[MATH_BIGINTEGER_VALUE], $temp[MATH_BIGINTEGER_SIGN]); - while ($this->_compare($result[MATH_BIGINTEGER_VALUE], $result[MATH_BIGINTEGER_SIGN], $n, false) > 0) { - $result = $this->_subtract($result[MATH_BIGINTEGER_VALUE], $result[MATH_BIGINTEGER_SIGN], $n, false); - } - - return $result[MATH_BIGINTEGER_VALUE]; - } - - /** - * Performs long multiplication up to $stop digits - * - * If you're going to be doing array_slice($product->value, 0, $stop), some cycles can be saved. - * - * @see _regularBarrett() - * @param Array $x_value - * @param Boolean $x_negative - * @param Array $y_value - * @param Boolean $y_negative - * @return Array - * @access private - */ - function _multiplyLower($x_value, $x_negative, $y_value, $y_negative, $stop) - { - $x_length = count($x_value); - $y_length = count($y_value); - - if ( !$x_length || !$y_length ) { // a 0 is being multiplied - return array( - MATH_BIGINTEGER_VALUE => array(), - MATH_BIGINTEGER_SIGN => false - ); - } - - if ( $x_length < $y_length ) { - $temp = $x_value; - $x_value = $y_value; - $y_value = $temp; - - $x_length = count($x_value); - $y_length = count($y_value); - } - - $product_value = $this->_array_repeat(0, $x_length + $y_length); - - // the following for loop could be removed if the for loop following it - // (the one with nested for loops) initially set $i to 0, but - // doing so would also make the result in one set of unnecessary adds, - // since on the outermost loops first pass, $product->value[$k] is going - // to always be 0 - - $carry = 0; - - for ($j = 0; $j < $x_length; ++$j) { // ie. $i = 0, $k = $i - $temp = $x_value[$j] * $y_value[0] + $carry; // $product_value[$k] == 0 - $carry = (int) ($temp / 0x4000000); - $product_value[$j] = (int) ($temp - 0x4000000 * $carry); - } - - if ($j < $stop) { - $product_value[$j] = $carry; - } - - // the above for loop is what the previous comment was talking about. the - // following for loop is the "one with nested for loops" - - for ($i = 1; $i < $y_length; ++$i) { - $carry = 0; - - for ($j = 0, $k = $i; $j < $x_length && $k < $stop; ++$j, ++$k) { - $temp = $product_value[$k] + $x_value[$j] * $y_value[$i] + $carry; - $carry = (int) ($temp / 0x4000000); - $product_value[$k] = (int) ($temp - 0x4000000 * $carry); - } - - if ($k < $stop) { - $product_value[$k] = $carry; - } - } - - return array( - MATH_BIGINTEGER_VALUE => $this->_trim($product_value), - MATH_BIGINTEGER_SIGN => $x_negative != $y_negative - ); - } - - /** - * Montgomery Modular Reduction - * - * ($x->_prepMontgomery($n))->_montgomery($n) yields $x % $n. - * {@link http://math.libtomcrypt.com/files/tommath.pdf#page=170 MPM 6.3} provides insights on how this can be - * improved upon (basically, by using the comba method). gcd($n, 2) must be equal to one for this function - * to work correctly. - * - * @see _prepMontgomery() - * @see _slidingWindow() - * @access private - * @param Array $x - * @param Array $n - * @return Array - */ - function _montgomery($x, $n) - { - static $cache = array( - MATH_BIGINTEGER_VARIABLE => array(), - MATH_BIGINTEGER_DATA => array() - ); - - if ( ($key = array_search($n, $cache[MATH_BIGINTEGER_VARIABLE])) === false ) { - $key = count($cache[MATH_BIGINTEGER_VARIABLE]); - $cache[MATH_BIGINTEGER_VARIABLE][] = $x; - $cache[MATH_BIGINTEGER_DATA][] = $this->_modInverse67108864($n); - } - - $k = count($n); - - $result = array(MATH_BIGINTEGER_VALUE => $x); - - for ($i = 0; $i < $k; ++$i) { - $temp = $result[MATH_BIGINTEGER_VALUE][$i] * $cache[MATH_BIGINTEGER_DATA][$key]; - $temp = (int) ($temp - 0x4000000 * ((int) ($temp / 0x4000000))); - $temp = $this->_regularMultiply(array($temp), $n); - $temp = array_merge($this->_array_repeat(0, $i), $temp); - $result = $this->_add($result[MATH_BIGINTEGER_VALUE], false, $temp, false); - } - - $result[MATH_BIGINTEGER_VALUE] = array_slice($result[MATH_BIGINTEGER_VALUE], $k); - - if ($this->_compare($result, false, $n, false) >= 0) { - $result = $this->_subtract($result[MATH_BIGINTEGER_VALUE], false, $n, false); - } - - return $result[MATH_BIGINTEGER_VALUE]; - } - - /** - * Montgomery Multiply - * - * Interleaves the montgomery reduction and long multiplication algorithms together as described in - * {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=13 HAC 14.36} - * - * @see _prepMontgomery() - * @see _montgomery() - * @access private - * @param Array $x - * @param Array $y - * @param Array $m - * @return Array - */ - function _montgomeryMultiply($x, $y, $m) - { - $temp = $this->_multiply($x, false, $y, false); - return $this->_montgomery($temp[MATH_BIGINTEGER_VALUE], $m); - - static $cache = array( - MATH_BIGINTEGER_VARIABLE => array(), - MATH_BIGINTEGER_DATA => array() - ); - - if ( ($key = array_search($m, $cache[MATH_BIGINTEGER_VARIABLE])) === false ) { - $key = count($cache[MATH_BIGINTEGER_VARIABLE]); - $cache[MATH_BIGINTEGER_VARIABLE][] = $m; - $cache[MATH_BIGINTEGER_DATA][] = $this->_modInverse67108864($m); - } - - $n = max(count($x), count($y), count($m)); - $x = array_pad($x, $n, 0); - $y = array_pad($y, $n, 0); - $m = array_pad($m, $n, 0); - $a = array(MATH_BIGINTEGER_VALUE => $this->_array_repeat(0, $n + 1)); - for ($i = 0; $i < $n; ++$i) { - $temp = $a[MATH_BIGINTEGER_VALUE][0] + $x[$i] * $y[0]; - $temp = (int) ($temp - 0x4000000 * ((int) ($temp / 0x4000000))); - $temp = $temp * $cache[MATH_BIGINTEGER_DATA][$key]; - $temp = (int) ($temp - 0x4000000 * ((int) ($temp / 0x4000000))); - $temp = $this->_add($this->_regularMultiply(array($x[$i]), $y), false, $this->_regularMultiply(array($temp), $m), false); - $a = $this->_add($a[MATH_BIGINTEGER_VALUE], false, $temp[MATH_BIGINTEGER_VALUE], false); - $a[MATH_BIGINTEGER_VALUE] = array_slice($a[MATH_BIGINTEGER_VALUE], 1); - } - if ($this->_compare($a[MATH_BIGINTEGER_VALUE], false, $m, false) >= 0) { - $a = $this->_subtract($a[MATH_BIGINTEGER_VALUE], false, $m, false); - } - return $a[MATH_BIGINTEGER_VALUE]; - } - - /** - * Prepare a number for use in Montgomery Modular Reductions - * - * @see _montgomery() - * @see _slidingWindow() - * @access private - * @param Array $x - * @param Array $n - * @return Array - */ - function _prepMontgomery($x, $n) - { - $lhs = new Math_BigInteger(); - $lhs->value = array_merge($this->_array_repeat(0, count($n)), $x); - $rhs = new Math_BigInteger(); - $rhs->value = $n; - - list(, $temp) = $lhs->divide($rhs); - return $temp->value; - } - - /** - * Modular Inverse of a number mod 2**26 (eg. 67108864) - * - * Based off of the bnpInvDigit function implemented and justified in the following URL: - * - * {@link http://www-cs-students.stanford.edu/~tjw/jsbn/jsbn.js} - * - * The following URL provides more info: - * - * {@link http://groups.google.com/group/sci.crypt/msg/7a137205c1be7d85} - * - * As for why we do all the bitmasking... strange things can happen when converting from floats to ints. For - * instance, on some computers, var_dump((int) -4294967297) yields int(-1) and on others, it yields - * int(-2147483648). To avoid problems stemming from this, we use bitmasks to guarantee that ints aren't - * auto-converted to floats. The outermost bitmask is present because without it, there's no guarantee that - * the "residue" returned would be the so-called "common residue". We use fmod, in the last step, because the - * maximum possible $x is 26 bits and the maximum $result is 16 bits. Thus, we have to be able to handle up to - * 40 bits, which only 64-bit floating points will support. - * - * Thanks to Pedro Gimeno Fortea for input! - * - * @see _montgomery() - * @access private - * @param Array $x - * @return Integer - */ - function _modInverse67108864($x) // 2**26 == 67108864 - { - $x = -$x[0]; - $result = $x & 0x3; // x**-1 mod 2**2 - $result = ($result * (2 - $x * $result)) & 0xF; // x**-1 mod 2**4 - $result = ($result * (2 - ($x & 0xFF) * $result)) & 0xFF; // x**-1 mod 2**8 - $result = ($result * ((2 - ($x & 0xFFFF) * $result) & 0xFFFF)) & 0xFFFF; // x**-1 mod 2**16 - $result = fmod($result * (2 - fmod($x * $result, 0x4000000)), 0x4000000); // x**-1 mod 2**26 - return $result & 0x3FFFFFF; - } - - /** - * Calculates modular inverses. - * - * Say you have (30 mod 17 * x mod 17) mod 17 == 1. x can be found using modular inverses. - * - * Here's an example: - * <code> - * <?php - * include('Math/BigInteger.php'); - * - * $a = new Math_BigInteger(30); - * $b = new Math_BigInteger(17); - * - * $c = $a->modInverse($b); - * echo $c->toString(); // outputs 4 - * - * echo "\r\n"; - * - * $d = $a->multiply($c); - * list(, $d) = $d->divide($b); - * echo $d; // outputs 1 (as per the definition of modular inverse) - * ?> - * </code> - * - * @param Math_BigInteger $n - * @return mixed false, if no modular inverse exists, Math_BigInteger, otherwise. - * @access public - * @internal See {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=21 HAC 14.64} for more information. - */ - function modInverse($n) - { - switch ( MATH_BIGINTEGER_MODE ) { - case MATH_BIGINTEGER_MODE_GMP: - $temp = new Math_BigInteger(); - $temp->value = gmp_invert($this->value, $n->value); - - return ( $temp->value === false ) ? false : $this->_normalize($temp); - } - - static $zero, $one; - if (!isset($zero)) { - $zero = new Math_BigInteger(); - $one = new Math_BigInteger(1); - } - - // $x mod $n == $x mod -$n. - $n = $n->abs(); - - if ($this->compare($zero) < 0) { - $temp = $this->abs(); - $temp = $temp->modInverse($n); - return $negated === false ? false : $this->_normalize($n->subtract($temp)); - } - - extract($this->extendedGCD($n)); - - if (!$gcd->equals($one)) { - return false; - } - - $x = $x->compare($zero) < 0 ? $x->add($n) : $x; - - return $this->compare($zero) < 0 ? $this->_normalize($n->subtract($x)) : $this->_normalize($x); - } - - /** - * Calculates the greatest common divisor and Bézout's identity. - * - * Say you have 693 and 609. The GCD is 21. Bézout's identity states that there exist integers x and y such that - * 693*x + 609*y == 21. In point of fact, there are actually an infinite number of x and y combinations and which - * combination is returned is dependant upon which mode is in use. See - * {@link http://en.wikipedia.org/wiki/B%C3%A9zout%27s_identity Bézout's identity - Wikipedia} for more information. - * - * Here's an example: - * <code> - * <?php - * include('Math/BigInteger.php'); - * - * $a = new Math_BigInteger(693); - * $b = new Math_BigInteger(609); - * - * extract($a->extendedGCD($b)); - * - * echo $gcd->toString() . "\r\n"; // outputs 21 - * echo $a->toString() * $x->toString() + $b->toString() * $y->toString(); // outputs 21 - * ?> - * </code> - * - * @param Math_BigInteger $n - * @return Math_BigInteger - * @access public - * @internal Calculates the GCD using the binary xGCD algorithim described in - * {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=19 HAC 14.61}. As the text above 14.61 notes, - * the more traditional algorithim requires "relatively costly multiple-precision divisions". - */ - function extendedGCD($n) - { - switch ( MATH_BIGINTEGER_MODE ) { - case MATH_BIGINTEGER_MODE_GMP: - extract(gmp_gcdext($this->value, $n->value)); - - return array( - 'gcd' => $this->_normalize(new Math_BigInteger($g)), - 'x' => $this->_normalize(new Math_BigInteger($s)), - 'y' => $this->_normalize(new Math_BigInteger($t)) - ); - case MATH_BIGINTEGER_MODE_BCMATH: - // it might be faster to use the binary xGCD algorithim here, as well, but (1) that algorithim works - // best when the base is a power of 2 and (2) i don't think it'd make much difference, anyway. as is, - // the basic extended euclidean algorithim is what we're using. - - $u = $this->value; - $v = $n->value; - - $a = '1'; - $b = '0'; - $c = '0'; - $d = '1'; - - while (bccomp($v, '0', 0) != 0) { - $q = bcdiv($u, $v, 0); - - $temp = $u; - $u = $v; - $v = bcsub($temp, bcmul($v, $q, 0), 0); - - $temp = $a; - $a = $c; - $c = bcsub($temp, bcmul($a, $q, 0), 0); - - $temp = $b; - $b = $d; - $d = bcsub($temp, bcmul($b, $q, 0), 0); - } - - return array( - 'gcd' => $this->_normalize(new Math_BigInteger($u)), - 'x' => $this->_normalize(new Math_BigInteger($a)), - 'y' => $this->_normalize(new Math_BigInteger($b)) - ); - } - - $y = $n->copy(); - $x = $this->copy(); - $g = new Math_BigInteger(); - $g->value = array(1); - - while ( !(($x->value[0] & 1)|| ($y->value[0] & 1)) ) { - $x->_rshift(1); - $y->_rshift(1); - $g->_lshift(1); - } - - $u = $x->copy(); - $v = $y->copy(); - - $a = new Math_BigInteger(); - $b = new Math_BigInteger(); - $c = new Math_BigInteger(); - $d = new Math_BigInteger(); - - $a->value = $d->value = $g->value = array(1); - $b->value = $c->value = array(); - - while ( !empty($u->value) ) { - while ( !($u->value[0] & 1) ) { - $u->_rshift(1); - if ( (!empty($a->value) && ($a->value[0] & 1)) || (!empty($b->value) && ($b->value[0] & 1)) ) { - $a = $a->add($y); - $b = $b->subtract($x); - } - $a->_rshift(1); - $b->_rshift(1); - } - - while ( !($v->value[0] & 1) ) { - $v->_rshift(1); - if ( (!empty($d->value) && ($d->value[0] & 1)) || (!empty($c->value) && ($c->value[0] & 1)) ) { - $c = $c->add($y); - $d = $d->subtract($x); - } - $c->_rshift(1); - $d->_rshift(1); - } - - if ($u->compare($v) >= 0) { - $u = $u->subtract($v); - $a = $a->subtract($c); - $b = $b->subtract($d); - } else { - $v = $v->subtract($u); - $c = $c->subtract($a); - $d = $d->subtract($b); - } - } - - return array( - 'gcd' => $this->_normalize($g->multiply($v)), - 'x' => $this->_normalize($c), - 'y' => $this->_normalize($d) - ); - } - - /** - * Calculates the greatest common divisor - * - * Say you have 693 and 609. The GCD is 21. - * - * Here's an example: - * <code> - * <?php - * include('Math/BigInteger.php'); - * - * $a = new Math_BigInteger(693); - * $b = new Math_BigInteger(609); - * - * $gcd = a->extendedGCD($b); - * - * echo $gcd->toString() . "\r\n"; // outputs 21 - * ?> - * </code> - * - * @param Math_BigInteger $n - * @return Math_BigInteger - * @access public - */ - function gcd($n) - { - extract($this->extendedGCD($n)); - return $gcd; - } - - /** - * Absolute value. - * - * @return Math_BigInteger - * @access public - */ - function abs() - { - $temp = new Math_BigInteger(); - - switch ( MATH_BIGINTEGER_MODE ) { - case MATH_BIGINTEGER_MODE_GMP: - $temp->value = gmp_abs($this->value); - break; - case MATH_BIGINTEGER_MODE_BCMATH: - $temp->value = (bccomp($this->value, '0', 0) < 0) ? substr($this->value, 1) : $this->value; - break; - default:
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