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Diffstat (limited to 'plugins/OStatus/extlib/Math/BigInteger.php')
| -rw-r--r-- | plugins/OStatus/extlib/Math/BigInteger.php | 3545 |
1 files changed, 3545 insertions, 0 deletions
diff --git a/plugins/OStatus/extlib/Math/BigInteger.php b/plugins/OStatus/extlib/Math/BigInteger.php new file mode 100644 index 000000000..9733351d4 --- /dev/null +++ b/plugins/OStatus/extlib/Math/BigInteger.php @@ -0,0 +1,3545 @@ +<?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: This library is free software; you can redistribute it and/or + * modify it under the terms of the GNU Lesser General Public + * License as published by the Free Software Foundation; either + * version 2.1 of the License, or (at your option) any later version. + * + * This library is distributed in the hope that it will be useful, + * but WITHOUT ANY WARRANTY; without even the implied warranty of + * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU + * Lesser General Public License for more details. + * + * You should have received a copy of the GNU Lesser General Public + * License along with this library; if not, write to the Free Software + * Foundation, Inc., 59 Temple Place, Suite 330, Boston, + * MA 02111-1307 USA + * + * @category Math + * @package Math_BigInteger + * @author Jim Wigginton <terrafrost@php.net> + * @copyright MMVI Jim Wigginton + * @license http://www.gnu.org/licenses/lgpl.txt + * @version $Id: BigInteger.php,v 1.31 2010/03/01 17:28:19 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 <terrafrost@php.net> + * @version 1.0.0RC4 + * @access public + * @package Math_BigInteger + */ +class Math_BigInteger { + /** + * Holds the BigInteger's value. + * + * @var Array + * @access private + */ + var $value; + + /** + * Holds the BigInteger's magnitude. + * + * @var Boolean + * @access private + */ + var $is_negative = false; + + /** + * Random number generator function + * + * @see setRandomGenerator() + * @access private + */ + var $generator = 'mt_rand'; + + /** + * Precision + * + * @see setPrecision() + * @access private + */ + var $precision = -1; + + /** + * Precision Bitmask + * + * @see setPrecision() + * @access private + */ + var $bitmask = false; + + /** + * Mode independant value used for serialization. + * + * If the bcmath or gmp extensions are installed $this->value will be a non-serializable resource, hence the need for + * a variable that'll be serializable regardless of whether or not extensions are being used. Unlike $this->value, + * however, $this->hex is only calculated when $this->__sleep() is called. + * + * @see __sleep() + * @see __wakeup() + * @var String + * @access private + */ + var $hex; + + /** + * Converts base-2, base-10, base-16, and binary strings (eg. base-256) to BigIntegers. + * + * If the second parameter - $base - is negative, then it will be assumed that the number's are encoded using + * two's compliment. The sole exception to this is -10, which is treated the same as 10 is. + * + * Here's an example: + * <code> + * <?php + * include('Math/BigInteger.php'); + * + * $a = new Math_BigInteger('0x32', 16); // 50 in base-16 + * + * echo $a->toString(); // outputs 50 + * ?> + * </code> + * + * @param optional $x base-10 number or base-$base number if $base set. + * @param optional integer $base + * @return Math_BigInteger + * @access public + */ + function Math_BigInteger($x = 0, $base = 10) + { + if ( !defined('MATH_BIGINTEGER_MODE') ) { + switch (true) { + case extension_loaded('gmp'): + define('MATH_BIGINTEGER_MODE', MATH_BIGINTEGER_MODE_GMP); + break; + case extension_loaded('bcmath'): + define('MATH_BIGINTEGER_MODE', MATH_BIGINTEGER_MODE_BCMATH); + break; + default: + define('MATH_BIGINTEGER_MODE', MATH_BIGINTEGER_MODE_INTERNAL); + } + } + + switch ( MATH_BIGINTEGER_MODE ) { + case MATH_BIGINTEGER_MODE_GMP: + if (is_resource($x) && get_resource_type($x) == 'GMP integer') { + $this->value = $x; + return; + } + $this->value = gmp_init(0); + break; + case MATH_BIGINTEGER_MODE_BCMATH: + $this->value = '0'; + break; + default: + $this->value = array(); + } + + if ($x === 0) { + return; + } + + switch ($base) { + case -256: + if (ord($x[0]) & 0x80) { + $x = ~$x; + $this->is_negative = true; + } + case 256: + switch ( MATH_BIGINTEGER_MODE ) { + case MATH_BIGINTEGER_MODE_GMP: + $sign = $this->is_negative ? '-' : ''; + $this->value = gmp_init($sign . '0x' . bin2hex($x)); + break; + case MATH_BIGINTEGER_MODE_BCMATH: + // round $len to the nearest 4 (thanks, DavidMJ!) + $len = (strlen($x) + 3) & 0xFFFFFFFC; + + $x = str_pad($x, $len, chr(0), STR_PAD_LEFT); + + for ($i = 0; $i < $len; $i+= 4) { + $this->value = bcmul($this->value, '4294967296', 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; $i < strlen($hex); $i+=8) { + $bits.= str_pad(decbin(hexdec(substr($hex, $i, 8))), 32, '0', STR_PAD_LEFT); + } + return $this->precision > 0 ? substr($bits, -$this->precision) : ltrim($bits, '0'); + } + + /** + * Converts a BigInteger to a base-10 number. + * + * Here's an example: + * <code> + * <?php + * include('Math/BigInteger.php'); + * + * $a = new Math_BigInteger('50'); + * + * echo $a->toString(); // outputs 50 + * ?> + * </code> + * + * @return String + * @access public + * @internal Converts a base-2**26 number to base-10**7 (which is pretty much base-10) + */ + function toString() + { + switch ( MATH_BIGINTEGER_MODE ) { + case MATH_BIGINTEGER_MODE_GMP: + return gmp_strval($this->value); + case MATH_BIGINTEGER_MODE_BCMATH: + if ($this->value === '0') { + return '0'; + } + + return ltrim($this->value, '0'); + } + + if (!count($this->value)) { + return '0'; + } + + $temp = $this->copy(); + $temp->is_negative = false; + + $divisor = new Math_BigInteger(); + $divisor->value = array(10000000); // eg. 10**7 + $result = ''; + while (count($temp->value)) { + list($temp, $mod) = $temp->divide($divisor); + $result = str_pad(isset($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: + $temp->value = $this->value; + } + + return $temp; + } + + /** + * Compares two numbers. + * + * Although one might think !$x->compare($y) means $x != $y, it, in fact, means the opposite. The reason for this is + * demonstrated thusly: + * + * $x > $y: $x->compare($y) > 0 + * $x < $y: $x->compare($y) < 0 + * $x == $y: $x->compare($y) == 0 + * + * Note how the same comparison operator is used. If you want to test for equality, use $x->equals($y). + * + * @param Math_BigInteger $x + * @return Integer < 0 if $this is less than $x; > 0 if $this is greater than $x, and 0 if they are equal. + * @access public + * @see equals() + * @internal Could return $this->subtract($x), but that's not as fast as what we do do. + */ + function compare($y) + { + switch ( MATH_BIGINTEGER_MODE ) { + case MATH_BIGINTEGER_MODE_GMP: + return gmp_cmp($this->value, $y->value); + case MATH_BIGINTEGER_MODE_BCMATH: + return bccomp($this->value, $y->value, 0); + } + + return $this->_compare($this->value, $this->is_negative, $y->value, $y->is_negative); + } + + /** + * Compares two numbers. + * + * @param Array $x_value + * @param Boolean $x_negative + * @param Array $y_value + * @param Boolean $y_negative + * @return Integer + * @see compare() + * @access private + */ + function _compare($x_value, $x_negative, $y_value, $y_negative) + { + if ( $x_negative != $y_negative ) { + return ( !$x_negative && $y_negative ) ? 1 : -1; + } + + $result = $x_negative ? -1 : 1; + + if ( count($x_value) != count($y_value) ) { + return ( count($x_value) > count($y_value) ) ? $result : -$result; + } + $size = max(count($x_value), count($y_value)); + + $x_value = array_pad($x_value, $size, 0); + $y_value = array_pad($y_value, $size, 0); + + for ($i = count($x_value) - 1; $i >= 0; --$i) { + if ($x_value[$i] != $y_value[$i]) { + return ( $x_value[$i] > $y_value[$i] ) ? $result : -$result; + } + } + + return 0; + } + + /** + * Tests the equality of two numbers. + * + * If you need to see if one number is greater than or less than another number, use Math_BigInteger::compare() + * + * @param Math_BigInteger $x + * @return Boolean + * @access public + * @see compare() + */ + function equals($x) + { + switch ( MATH_BIGINTEGER_MODE ) { + case MATH_BIGINTEGER_MODE_GMP: + return gmp_cmp($this->value, $x->value) == 0; + default: + return $this->value === $x->value && $this->is_negative == $x->is_negative; + } + } + + /** + * Set Precision + * + * Some bitwise operations give different results depending on the precision being used. Examples include left + * shift, not, and rotates. + * + * @param Math_BigInteger $x + * @access public + * @return Math_BigInteger + */ + function setPrecision($bits) + { + $this->precision = $bits; + if ( MATH_BIGINTEGER_MODE != MATH_BIGINTEGER_MODE_BCMATH ) { + $this->bitmask = new Math_BigInteger(chr((1 << ($bits & 0x7)) - 1) . str_repeat(chr(0xFF), $bits >> 3), 256); + } else { + $this->bitmask = new Math_BigInteger(bcpow('2', $bits, 0)); + } + + $temp = $this->_normalize($this); + $this->value = $temp->value; + } + + /** + * Logical And + * + * @param Math_BigInteger $x + * @access public + * @internal Implemented per a request by Lluis Pamies i Juarez <lluis _a_ pamies.cat> + * @return Math_BigInteger + */ + function bitwise_and($x) + { + switch ( MATH_BIGINTEGER_MODE ) { + case MATH_BIGINTEGER_MODE_GMP: + $temp = new Math_BigInteger(); + $temp->value = gmp_and($this->value, $x->value); + + return $this->_normalize($temp); + case MATH_BIGINTEGER_MODE_BCMATH: + $left = $this->toBytes(); + $right = $x->toBytes(); + + $length = max(strlen($left), strlen($right)); + + $left = str_pad($left, $length, chr(0), STR_PAD_LEFT); + $right = str_pad($right, $length, chr(0), STR_PAD_LEFT); + + return $this->_normalize(new Math_BigInteger($left & $right, 256)); + } + + $result = $this->copy(); + + $length = min(count($x->value), count($this->value)); + + $result->value = array_slice($result->value, 0, $length); + + for ($i = 0; $i < $length; ++$i) { + $result->value[$i] = $result->value[$i] & $x->value[$i]; + } + + return $this->_normalize($result); + } + + /** + * Logical Or + * + * @param Math_BigInteger $x + * @access public + * @internal Implemented per a request by Lluis Pamies i Juarez <lluis _a_ pamies.cat> + * @return Math_BigInteger + */ + function bitwise_or($x) + { + switch ( MATH_BIGINTEGER_MODE ) { + case MATH_BIGINTEGER_MODE_GMP: + $temp = new Math_BigInteger(); + $temp->value = gmp_or($this->value, $x->value); + + return $this->_normalize($temp); + case MATH_BIGINTEGER_MODE_BCMATH: + $left = $this->toBytes(); + $right = $x->toBytes(); + + $length = max(strlen($left), strlen($right)); + + $left = str_pad($left, $length, chr(0), STR_PAD_LEFT); + $right = str_pad($right, $length, chr(0), STR_PAD_LEFT); + + return $this->_normalize(new Math_BigInteger($left | $right, 256)); + } + + $length = max(count($this->value), count($x->value)); + $result = $this->copy(); + $result->value = array_pad($result->value, 0, $length); + $x->value = array_pad($x->value, 0, $length); + + for ($i = 0; $i < $length; ++$i) { + $result->value[$i] = $this->value[$i] | $x->value[$i]; + } + + return $this->_normalize($result); + } + + /** + * Logical Exclusive-Or + * + * @param Math_BigInteger $x + * @access public + * @internal Implemented per a request by Lluis Pamies i Juarez <lluis _a_ pamies.cat> + * @return Math_BigInteger + */ + function bitwise_xor($x) + { + switch ( MATH_BIGINTEGER_MODE ) { + case MATH_BIGINTEGER_MODE_GMP: + $temp = new Math_BigInteger(); + $temp->value = gmp_xor($this->value, $x->value); + + return $this->_normalize($temp); + case MATH_BIGINTEGER_MODE_BCMATH: + $left = $this->toBytes(); + $right = $x->toBytes(); + + $length = max(strlen($left), strlen($right)); + + $left = str_pad($left, $length, chr(0), STR_PAD_LEFT); + $right = str_pad($right, $length, chr(0), STR_PAD_LEFT); + + return $this->_normalize(new Math_BigInteger($left ^ $right, 256)); + } + + $length = max(count($this->value), count($x->value)); + $result = $this->copy(); + $result->value = array_pad($result->value, 0, $length); + $x->value = array_pad($x->value, 0, $length); + + for ($i = 0; $i < $length; ++$i) { + $result->value[$i] = $this->value[$i] ^ $x->value[$i]; + } + + return $this->_normalize($result); + } + + /** + * Logical Not + * + * @access public + * @internal Implemented per a request by Lluis Pamies i Juarez <lluis _a_ pamies.cat> + * @return Math_BigInteger + */ + function bitwise_not() + { + // calculuate "not" without regard to $this->precision + // (will always result in a smaller number. ie. ~1 isn't 1111 1110 - it's 0) + $temp = $this->toBytes(); + $pre_msb = decbin(ord($temp[0])); + $temp = ~$temp; + $msb = decbin(ord($temp[0])); + if (strlen($msb) == 8) { + $msb = substr($msb, strpos($msb, '0')); + } + $temp[0] = chr(bindec($msb)); + + // see if we need to add extra leading 1's + $current_bits = strlen($pre_msb) + 8 * strlen($temp) - 8; + $new_bits = $this->precision - $current_bits; + if ($new_bits <= 0) { + return $this->_normalize(new Math_BigInteger($temp, 256)); + } + + // generate as many leading 1's as we need to. + $leading_ones = chr((1 << ($new_bits & 0x7)) - 1) . str_repeat(chr(0xFF), $new_bits >> 3); + $this->_base256_lshift($leading_ones, $current_bits); + + $temp = str_pad($temp, ceil($this->bits / 8), chr(0), STR_PAD_LEFT); + + return $this->_normalize(new Math_BigInteger($leading_ones | $temp, 256)); + } + + /** + * Logical Right Shift + * + * Shifts BigInteger's by $shift bits, effectively dividing by 2**$shift. + * + * @param Integer $shift + * @return Math_BigInteger + * @access public + * @internal The only version that yields any speed increases is the internal version. + */ + function bitwise_rightShift($shift) + { + $temp = new Math_BigInteger(); + + switch ( MATH_BIGINTEGER_MODE ) { + case MATH_BIGINTEGER_MODE_GMP: + static $two; + + if (!isset($two)) { + $two = gmp_init('2'); + } + + $temp->value = gmp_div_q($this->value, gmp_pow($two, $shift)); + + break; + case MATH_BIGINTEGER_MODE_BCMATH: + $temp->value = bcdiv($this->value, bcpow('2', $shift, 0), 0); + + break; + default: // could just replace _lshift with this, but then all _lshift() calls would need to be rewritten + // and I don't want to do that... + $temp->value = $this->value; + $temp->_rshift($shift); + } + + return $this->_normalize($temp); + } + + /** + * Logical Left Shift + * + * Shifts BigInteger's by $shift bits, effectively multiplying by 2**$shift. + * + * @param Integer $shift + * @return Math_BigInteger + * @access public + * @internal The only version that yields any speed increases is the internal version. + */ + function bitwise_leftShift($shift) + { + $temp = new Math_BigInteger(); + + switch ( MATH_BIGINTEGER_MODE ) { + case MATH_BIGINTEGER_MODE_GMP: + static $two; + + if (!isset($two)) { + $two = gmp_init('2'); + } + + $temp->value = gmp_mul($this->value, gmp_pow($two, $shift)); + + break; + case MATH_BIGINTEGER_MODE_BCMATH: + $temp->value = bcmul($this->value, bcpow('2', $shift, 0), 0); + + break; + default: // could just replace _rshift with this, but then all _lshift() calls would need to be rewritten + // and I don't want to do that... + $temp->value = $this->value; + $temp->_lshift($shift); + } + + return $this->_normalize($temp); + } + + /** + * Logical Left Rotate + * + * Instead of the top x bits being dropped they're appended to the shifted bit string. + * + * @param Integer $shift + * @return Math_BigInteger + * @access public + */ + function bitwise_leftRotate($shift) + { + $bits = $this->toBytes(); + + if ($this->precision > 0) { + $precision = $this->precision; + if ( MATH_BIGINTEGER_MODE == MATH_BIGINTEGER_MODE_BCMATH ) { + $mask = $this->bitmask->subtract(new Math_BigInteger(1)); + $mask = $mask->toBytes(); + } else { + $mask = $this->bitmask->toBytes(); + } + } else { + $temp = ord($bits[0]); + for ($i = 0; $temp >> $i; ++$i); + $precision = 8 * strlen($bits) - 8 + $i; + $mask = chr((1 << ($precision & 0x7)) - 1) . str_repeat(chr(0xFF), $precision >> 3); + } + + if ($shift < 0) { + $shift+= $precision; + } + $shift%= $precision; + + if (!$shift) { + return $this->copy(); + } + + $left = $this->bitwise_leftShift($shift); + $left = $left->bitwise_and(new Math_BigInteger($mask, 256)); + $right = $this->bitwise_rightShift($precision - $shift); + $result = MATH_BIGINTEGER_MODE != MATH_BIGINTEGER_MODE_BCMATH ? $left->bitwise_or($right) : $left->add($right); + return $this->_normalize($result); + } + + /** + * Logical Right Rotate + * + * Instead of the bottom x bits being dropped they're prepended to the shifted bit string. + * + * @param Integer $shift + * @return Math_BigInteger + * @access public + */ + function bitwise_rightRotate($shift) + { + return $this->bitwise_leftRotate(-$shift); + } + + /** + * Set random number generator function + * + * $generator should be the name of a random generating function whose first parameter is the minimum + * value and whose second parameter is the maximum value. If this function needs to be seeded, it should + * be seeded prior to calling Math_BigInteger::random() or Math_BigInteger::randomPrime() + * + * If the random generating function is not explicitly set, it'll be assumed to be mt_rand(). + * + * @see random() + * @see randomPrime() + * @param optional String $generator + * @access public + */ + function setRandomGenerator($generator) + { + $this->generator = $generator; + } + + /** + * Generate a random number + * + * @param optional Integer $min + * @param optional Integer $max + * @return Math_BigInteger + * @access public + */ + function random($min = false, $max = false) + { + if ($min === false) { + $min = new Math_BigInteger(0); + } + + if ($max === false) { + $max = new Math_BigInteger(0x7FFFFFFF); + } + + $compare = $max->compare($min); + + if (!$compare) { + return $this->_normalize($min); + } else if ($compare < 0) { + // if $min is bigger then $max, swap $min and $max + $temp = $max; + $max = $min; + $min = $temp; + } + + $generator = $this->generator; + + $max = $max->subtract($min); + $max = ltrim($max->toBytes(), chr(0)); + $size = strlen($max) - 1; + $random = ''; + + $bytes = $size & 1; + for ($i = 0; $i < $bytes; ++$i) { + $random.= chr($generator(0, 255)); + } + + $blocks = $size >> 1; + for ($i = 0; $i < $blocks; ++$i) { + // mt_rand(-2147483648, 0x7FFFFFFF) always produces -2147483648 on some systems + $random.= pack('n', $generator(0, 0xFFFF)); + } + + $temp = new Math_BigInteger($random, 256); + if ($temp->compare(new Math_BigInteger(substr($max, 1), 256)) > 0) { + $random = chr($generator(0, ord($max[0]) - 1)) . $random; + } else { + $random = chr($generator(0, ord($max[0]) )) . $random; + } + + $random = new Math_BigInteger($random, 256); + + return $this->_normalize($random->add($min)); + } + + /** + * Generate a random prime number. + * + * If there's not a prime within the given range, false will be returned. If more than $timeout seconds have elapsed, + * give up and return false. + * + * @param optional Integer $min + * @param optional Integer $max + * @param optional Integer $timeout + * @return Math_BigInteger + * @access public + * @internal See {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap4.pdf#page=15 HAC 4.44}. + */ + function randomPrime($min = false, $max = false, $timeout = false) + { + $compare = $max->compare($min); + + if (!$compare) { + return $min; + } else if ($compare < 0) { + // if $min is bigger then $max, swap $min and $max + $temp = $max; + $max = $min; + $min = $temp; + } + + // gmp_nextprime() requires PHP 5 >= 5.2.0 per <http://php.net/gmp-nextprime>. + if ( MATH_BIGINTEGER_MODE == MATH_BIGINTEGER_MODE_GMP && function_exists('gmp_nextprime') ) { + // we don't rely on Math_BigInteger::random()'s min / max when gmp_nextprime() is being used since this function + // does its own checks on $max / $min when gmp_nextprime() is used. When gmp_nextprime() is not used, however, + // the same $max / $min checks are not performed. + if ($min === false) { + $min = new Math_BigInteger(0); + } + + if ($max === false) { + $max = new Math_BigInteger(0x7FFFFFFF); + } + + $x = $this->random($min, $max); + + $x->value = gmp_nextprime($x->value); + + if ($x->compare($max) <= 0) { + return $x; + } + + $x->value = gmp_nextprime($min->value); + + if ($x->compare($max) <= 0) { + return $x; + } + + return false; + } + + static $one, $two; + if (!isset($one)) { + $one = new Math_BigInteger(1); + $two = new Math_BigInteger(2); + } + + $start = time(); + + $x = $this->random($min, $max); + if ($x->equals($two)) { + return $x; + } + + $x->_make_odd(); + if ($x->compare($max) > 0) { + // if $x > $max then $max is even and if $min == $max then no prime number exists between the specified range + if ($min->equals($max)) { + return false; + } + $x = $min->copy(); + $x->_make_odd(); + } + + $initial_x = $x->copy(); + + while (true) { + if ($timeout !== false && time() - $start > $timeout) { + return false; + } + + if ($x->isPrime()) { + return $x; + } + + $x = $x->add($two); + + if ($x->compare($max) > 0) { + $x = $min->copy(); + if ($x->equals($two)) { + return $x; + } + $x->_make_odd(); + } + + if ($x->equals($initial_x)) { + return false; + } + } + } + + /** + * Make the current number odd + * + * If the current number is odd it'll be unchanged. If it's even, one will be added to it. + * + * @see randomPrime() + * @access private + */ + function _make_odd() + { + switch ( MATH_BIGINTEGER_MODE ) { + case MATH_BIGINTEGER_MODE_GMP: + gmp_setbit($this->value, 0); + break; + case MATH_BIGINTEGER_MODE_BCMATH: + if ($this->value[strlen($this->value) - 1] % 2 == 0) { + $this->value = bcadd($this->value, '1'); + } + break; + default: + $this->value[0] |= 1; + } + } + + /** + * Checks a numer to see if it's prime + * + * Assuming the $t parameter is not set, this function has an error rate of 2**-80. The main motivation for the + * $t parameter is distributability. Math_BigInteger::randomPrime() can be distributed accross multiple pageloads + * on a website instead of just one. + * + * @param optional Integer $t + * @return Boolean + * @access public + * @internal Uses the + * {@link http://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test Miller-Rabin primality test}. See + * {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap4.pdf#page=8 HAC 4.24}. + */ + function isPrime($t = false) + { + $length = strlen($this->toBytes()); + + if (!$t) { + // see HAC 4.49 "Note (controlling the error probability)" + if ($length >= 163) { $t = 2; } // floor(1300 / 8) + else if ($length >= 106) { $t = 3; } // floor( 850 / 8) + else if ($length >= 81 ) { $t = 4; } // floor( 650 / 8) + else if ($length >= 68 ) { $t = 5; } // floor( 550 / 8) + else if ($length >= 56 ) { $t = 6; } // floor( 450 / 8) + else if ($length >= 50 ) { $t = 7; } // floor( 400 / 8) + else if ($length >= 43 ) { $t = 8; } // floor( 350 / 8) + else if ($length >= 37 ) { $t = 9; } // floor( 300 / 8) + else if ($length >= 31 ) { $t = 12; } // floor( 250 / 8) + else if ($length >= 25 ) { $t = 15; } // floor( 200 / 8) + else if ($length >= 18 ) { $t = 18; } // floor( 150 / 8) + else { $t = 27; } + } + + // ie. gmp_testbit($this, 0) + // ie. isEven() or !isOdd() + switch ( MATH_BIGINTEGER_MODE ) { + case MATH_BIGINTEGER_MODE_GMP: + return gmp_prob_prime($this->value, $t) != 0; + case MATH_BIGINTEGER_MODE_BCMATH: + if ($this->value === '2') { + return true; + } + if ($this->value[strlen($this->value) - 1] % 2 == 0) { + return false; + } + break; + default: + if ($this->value == array(2)) { + return true; + } + if (~$this->value[0] & 1) { + return false; + } + } + + static $primes, $zero, $one, $two; + + if (!isset($primes)) { + $primes = array( + 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, + 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, + 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, + 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, + 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, + 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, + 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, + 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, + 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, + 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, + 953, 967, 971, 977, 983, 991, 997 + ); + + if ( MATH_BIGINTEGER_MODE != MATH_BIGINTEGER_MODE_INTERNAL ) { + for ($i = 0; $i < count($primes); ++$i) { + $primes[$i] = new Math_BigInteger($primes[$i]); + } + } + + $zero = new Math_BigInteger(); + $one = new Math_BigInteger(1); + $two = new Math_BigInteger(2); + } + + if ($this->equals($one)) { + return false; + } + + // see HAC 4.4.1 "Random search for probable primes" + if ( MATH_BIGINTEGER_MODE != MATH_BIGINTEGER_MODE_INTERNAL ) { + foreach ($primes as $prime) { + list(, $r) = $this->divide($prime); + if ($r->equals($zero)) { + return $this->equals($prime); + } + } + } else { + $value = $this->value; + foreach ($primes as $prime) { + list(, $r) = $this->_divide_digit($value, $prime); + if (!$r) { + return count($value) == 1 && $value[0] == $prime; + } + } + } + + $n = $this->copy(); + $n_1 = $n->subtract($one); + $n_2 = $n->subtract($two); + + $r = $n_1->copy(); + $r_value = $r->value; + // ie. $s = gmp_scan1($n, 0) and $r = gmp_div_q($n, gmp_pow(gmp_init('2'), $s)); + if ( MATH_BIGINTEGER_MODE == MATH_BIGINTEGER_MODE_BCMATH ) { + $s = 0; + // if $n was 1, $r would be 0 and this would be an infinite loop, hence our $this->equals($one) check earlier + while ($r->value[strlen($r->value) - 1] % 2 == 0) { + $r->value = bcdiv($r->value, '2', 0); + ++$s; + } + } else { + for ($i = 0, $r_length = count($r_value); $i < $r_length; ++$i) { + $temp = ~$r_value[$i] & 0xFFFFFF; + for ($j = 1; ($temp >> $j) & 1; ++$j); + if ($j != 25) { + break; + } + } + $s = 26 * $i + $j - 1; + $r->_rshift($s); + } + + for ($i = 0; $i < $t; ++$i) { + $a = $this->random($two, $n_2); + $y = $a->modPow($r, $n); + + if (!$y->equals($one) && !$y->equals($n_1)) { + for ($j = 1; $j < $s && !$y->equals($n_1); ++$j) { + $y = $y->modPow($two, $n); + if ($y->equals($one)) { + return false; + } + } + + if (!$y->equals($n_1)) { + return false; + } + } + } + return true; + } + + /** + * Logical Left Shift + * + * Shifts BigInteger's by $shift bits. + * + * @param Integer $shift + * @access private + */ + function _lshift($shift) + { + if ( $shift == 0 ) { + return; + } + + $num_digits = (int) ($shift / 26); + $shift %= 26; + $shift = 1 << $shift; + + $carry = 0; + + for ($i = 0; $i < count($this->value); ++$i) { + $temp = $this->value[$i] * $shift + $carry; + $carry = (int) ($temp / 0x4000000); + $this->value[$i] = (int) ($temp - $carry * 0x4000000); + } + + if ( $carry ) { + $this->value[] = $carry; + } + + while ($num_digits--) { + array_unshift($this->value, 0); + } + } + + /** + * Logical Right Shift + * + * Shifts BigInteger's by $shift bits. + * + * @param Integer $shift + * @access private + */ + function _rshift($shift) + { + if ($shift == 0) { + return; + } + + $num_digits = (int) ($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->value = $this->_trim($this->value); + } + + /** + * Normalize + * + * Removes leading zeros and truncates (if necessary) to maintain the appropriate precision + * + * @param Math_BigInteger + * @return Math_BigInteger + * @see _trim() + * @access private + */ + function _normalize($result) + { + $result->precision = $this->precision; + $result->bitmask = $this->bitmask; + + switch ( MATH_BIGINTEGER_MODE ) { + case MATH_BIGINTEGER_MODE_GMP: + if (!empty($result->bitmask->value)) { + $result->value = gmp_and($result->value, $result->bitmask->value); + } + + return $result; + case MATH_BIGINTEGER_MODE_BCMATH: + if (!empty($result->bitmask->value)) { + $result->value = bcmod($result->value, $result->bitmask->value); + } + + return $result; + } + + $value = &$result->value; + + if ( !count($value) ) { + return $result; + } + + $value = $this->_trim($value); + + if (!empty($result->bitmask->value)) { + $length = min(count($value), count($this->bitmask->value)); + $value = array_slice($value, 0, $length); + + for ($i = 0; $i < $length; ++$i) { + $value[$i] = $value[$i] & $this->bitmask->value[$i]; + } + } + + return $result; + } + + /** + * Trim + * + * Removes leading zeros + * + * @return Math_BigInteger + * @access private + */ + function _trim($value) + { + for ($i = count($value) - 1; $i >= 0; --$i) { + if ( $value[$i] ) { + break; + } + unset($value[$i]); + } + + return $value; + } + + /** + * 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']; + } +}
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