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<?php/*** 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 version 5** {@internal (all DocBlock comments regarding implementation - such as the one that follows - refer to the* {@link self::MODE_INTERNAL self::MODE_INTERNAL} mode)** 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.** Numbers are stored in {@link http://en.wikipedia.org/wiki/Endianness little endian} format. ie.* (new \phpseclib\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* $a = new \phpseclib\Math\BigInteger(2);* $b = new \phpseclib\Math\BigInteger(3);** $c = $a->add($b);** echo $c->toString(); // outputs 5* ?>* </code>** @category Math* @package BigInteger* @author Jim Wigginton <terrafrost@php.net>* @copyright 2006 Jim Wigginton* @license http://www.opensource.org/licenses/mit-license.html MIT License* @link http://pear.php.net/package/Math_BigInteger*/namespace phpseclib\Math;use phpseclib\Crypt\Random;/*** Pure-PHP arbitrary precision integer arithmetic library. Supports base-2, base-10, base-16, and base-256* numbers.** @package BigInteger* @author Jim Wigginton <terrafrost@php.net>* @access public*/class BigInteger{/**#@+* Reduction constants** @access private* @see BigInteger::_reduce()*//*** @see BigInteger::_montgomery()* @see BigInteger::_prepMontgomery()*/const MONTGOMERY = 0;/*** @see BigInteger::_barrett()*/const BARRETT = 1;/*** @see BigInteger::_mod2()*/const POWEROF2 = 2;/*** @see BigInteger::_remainder()*/const CLASSIC = 3;/*** @see BigInteger::__clone()*/const NONE = 4;/**#@-*//**#@+* Array constants** Rather than create a thousands and thousands of new 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[self::VALUE] contains the value.*/const VALUE = 0;/*** $result[self::SIGN] contains the sign.*/const SIGN = 1;/**#@-*//**#@+* @access private* @see BigInteger::_montgomery()* @see BigInteger::_barrett()*//*** Cache constants** $cache[self::VARIABLE] tells us whether or not the cached data is still valid.*/const VARIABLE = 0;/*** $cache[self::DATA] contains the cached data.*/const DATA = 1;/**#@-*//**#@+* Mode constants.** @access private* @see BigInteger::__construct()*//*** To use the pure-PHP implementation*/const MODE_INTERNAL = 1;/*** To use the BCMath library** (if enabled; otherwise, the internal implementation will be used)*/const MODE_BCMATH = 2;/*** To use the GMP library** (if present; otherwise, either the BCMath or the internal implementation will be used)*/const MODE_GMP = 3;/**#@-*//*** Karatsuba Cutoff** At what point do we switch between Karatsuba multiplication and schoolbook long multiplication?** @access private*/const KARATSUBA_CUTOFF = 25;/**#@+* Static properties used by the pure-PHP implementation.** @see __construct()*/protected static $base;protected static $baseFull;protected static $maxDigit;protected static $msb;/*** $max10 in greatest $max10Len satisfying* $max10 = 10**$max10Len <= 2**$base.*/protected static $max10;/*** $max10Len in greatest $max10Len satisfying* $max10 = 10**$max10Len <= 2**$base.*/protected static $max10Len;protected static $maxDigit2;/**#@-*//*** Holds the BigInteger's value.** @var array* @access private*/var $value;/*** Holds the BigInteger's magnitude.** @var bool* @access private*/var $is_negative = false;/*** Precision** @see self::setPrecision()* @access private*/var $precision = -1;/*** Precision Bitmask** @see self::setPrecision()* @access private*/var $bitmask = false;/*** Mode independent 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 self::__sleep()* @see self::__wakeup()* @var string* @access private*/var $hex;/*** Converts base-2, base-10, base-16, and binary strings (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* $a = new \phpseclib\Math\BigInteger('0x32', 16); // 50 in base-16** echo $a->toString(); // outputs 50* ?>* </code>** @param $x base-10 number or base-$base number if $base set.* @param int $base* @return \phpseclib\Math\BigInteger* @access public*/function __construct($x = 0, $base = 10){if (!defined('MATH_BIGINTEGER_MODE')) {switch (true) {case extension_loaded('gmp'):define('MATH_BIGINTEGER_MODE', self::MODE_GMP);break;case extension_loaded('bcmath'):define('MATH_BIGINTEGER_MODE', self::MODE_BCMATH);break;default:define('MATH_BIGINTEGER_MODE', self::MODE_INTERNAL);}}if (extension_loaded('openssl') && !defined('MATH_BIGINTEGER_OPENSSL_DISABLE') && !defined('MATH_BIGINTEGER_OPENSSL_ENABLED')) {// some versions of XAMPP have mismatched versions of OpenSSL which causes it not to workob_start();@phpinfo();$content = ob_get_contents();ob_end_clean();preg_match_all('#OpenSSL (Header|Library) Version(.*)#im', $content, $matches);$versions = array();if (!empty($matches[1])) {for ($i = 0; $i < count($matches[1]); $i++) {$fullVersion = trim(str_replace('=>', '', strip_tags($matches[2][$i])));// Remove letter part in OpenSSL versionif (!preg_match('/(\d+\.\d+\.\d+)/i', $fullVersion, $m)) {$versions[$matches[1][$i]] = $fullVersion;} else {$versions[$matches[1][$i]] = $m[0];}}}// it doesn't appear that OpenSSL versions were reported upon until PHP 5.3+switch (true) {case !isset($versions['Header']):case !isset($versions['Library']):case $versions['Header'] == $versions['Library']:case version_compare($versions['Header'], '1.0.0') >= 0 && version_compare($versions['Library'], '1.0.0') >= 0:define('MATH_BIGINTEGER_OPENSSL_ENABLED', true);break;default:define('MATH_BIGINTEGER_OPENSSL_DISABLE', true);}}if (!defined('PHP_INT_SIZE')) {define('PHP_INT_SIZE', 4);}if (empty(self::$base) && MATH_BIGINTEGER_MODE == self::MODE_INTERNAL) {switch (PHP_INT_SIZE) {case 8: // use 64-bit integers if int size is 8 bytesself::$base = 31;self::$baseFull = 0x80000000;self::$maxDigit = 0x7FFFFFFF;self::$msb = 0x40000000;self::$max10 = 1000000000;self::$max10Len = 9;self::$maxDigit2 = pow(2, 62);break;//case 4: // use 64-bit floats if int size is 4 bytesdefault:self::$base = 26;self::$baseFull = 0x4000000;self::$maxDigit = 0x3FFFFFF;self::$msb = 0x2000000;self::$max10 = 10000000;self::$max10Len = 7;self::$maxDigit2 = pow(2, 52); // pow() prevents truncation}}switch (MATH_BIGINTEGER_MODE) {case self::MODE_GMP:switch (true) {case is_resource($x) && get_resource_type($x) == 'GMP integer':// PHP 5.6 switched GMP from using resources to objectscase $x instanceof \GMP:$this->value = $x;return;}$this->value = gmp_init(0);break;case self::MODE_BCMATH:$this->value = '0';break;default:$this->value = array();}// '0' counts as empty() but when the base is 256 '0' is equal to ord('0') or 48// '0' is the only value like this per http://php.net/emptyif (empty($x) && (abs($base) != 256 || $x !== '0')) {return;}switch ($base) {case -256:if (ord($x[0]) & 0x80) {$x = ~$x;$this->is_negative = true;}case 256:switch (MATH_BIGINTEGER_MODE) {case self::MODE_GMP:$sign = $this->is_negative ? '-' : '';$this->value = gmp_init($sign . '0x' . bin2hex($x));break;case self::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($x, self::$base));}}if ($this->is_negative) {if (MATH_BIGINTEGER_MODE != self::MODE_INTERNAL) {$this->is_negative = false;}$temp = $this->add(new static('-1'));$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*', $x));}switch (MATH_BIGINTEGER_MODE) {case self::MODE_GMP:$temp = $this->is_negative ? '-0x' . $x : '0x' . $x;$this->value = gmp_init($temp);$this->is_negative = false;break;case self::MODE_BCMATH:$x = (strlen($x) & 1) ? '0' . $x : $x;$temp = new static(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 static(pack('H*', $x), 256);$this->value = $temp->value;}if ($is_negative) {$temp = $this->add(new static('-1'));$this->value = $temp->value;}break;case 10:case -10:// (?<!^)(?:-).*: find any -'s that aren't at the beginning and then any characters that follow that// (?<=^|-)0*: find any 0's that are preceded by the start of the string or by a - (ie. octals)// [^-0-9].*: find any non-numeric characters and then any characters that follow that$x = preg_replace('#(?<!^)(?:-).*|(?<=^|-)0*|[^-0-9].*#', '', $x);switch (MATH_BIGINTEGER_MODE) {case self::MODE_GMP:$this->value = gmp_init($x);break;case self::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 = $x === '-' ? '0' : (string) $x;break;default:$temp = new static();$multiplier = new static();$multiplier->value = array(self::$max10);if ($x[0] == '-') {$this->is_negative = true;$x = substr($x, 1);}$x = str_pad($x, strlen($x) + ((self::$max10Len - 1) * strlen($x)) % self::$max10Len, 0, STR_PAD_LEFT);while (strlen($x)) {$temp = $temp->multiply($multiplier);$temp = $temp->add(new static($this->_int2bytes(substr($x, 0, self::$max10Len)), 256));$x = substr($x, self::$max10Len);}$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 static($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* $a = new \phpseclib\Math\BigInteger('65');** echo $a->toBytes(); // outputs chr(65)* ?>* </code>** @param bool $twos_compliment* @return string* @access public* @internal Converts a base-2**26 number to base-2**8*/function toBytes($twos_compliment = false){if ($twos_compliment) {$comparison = $this->compare(new static());if ($comparison == 0) {return $this->precision > 0 ? str_repeat(chr(0), ($this->precision + 1) >> 3) : '';}$temp = $comparison < 0 ? $this->add(new static(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 : $bytes;}switch (MATH_BIGINTEGER_MODE) {case self::MODE_GMP:if (gmp_cmp($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($temp, chr(0));case self::MODE_BCMATH:if ($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, self::$base);$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* $a = new \phpseclib\Math\BigInteger('65');** echo $a->toHex(); // outputs '41'* ?>* </code>** @param bool $twos_compliment* @return string* @access public* @internal Converts a base-2**26 number to base-2**8*/function toHex($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* $a = new \phpseclib\Math\BigInteger('65');** echo $a->toBits(); // outputs '1000001'* ?>* </code>** @param bool $twos_compliment* @return string* @access public* @internal Converts a base-2**26 number to base-2**2*/function toBits($twos_compliment = false){$hex = $this->toHex($twos_compliment);$bits = '';for ($i = strlen($hex) - 8, $start = strlen($hex) & 7; $i >= $start; $i-=8) {$bits = str_pad(decbin(hexdec(substr($hex, $i, 8))), 32, '0', STR_PAD_LEFT) . $bits;}if ($start) { // hexdec('') == 0$bits = str_pad(decbin(hexdec(substr($hex, 0, $start))), 8, '0', STR_PAD_LEFT) . $bits;}$result = $this->precision > 0 ? substr($bits, -$this->precision) : ltrim($bits, '0');if ($twos_compliment && $this->compare(new static()) > 0 && $this->precision <= 0) {return '0' . $result;}return $result;}/*** Converts a BigInteger to a base-10 number.** Here's an example:* <code>* <?php* $a = new \phpseclib\Math\BigInteger('50');** echo $a->toString(); // outputs 50* ?>* </code>** @return string* @access public* @internal Converts a base-2**26 number to base-10**7 (which is pretty much base-10)*/function toString(){switch (MATH_BIGINTEGER_MODE) {case self::MODE_GMP:return gmp_strval($this->value);case self::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;$divisor = new static();$divisor->value = array(self::$max10);$result = '';while (count($temp->value)) {list($temp, $mod) = $temp->divide($divisor);$result = str_pad(isset($mod->value[0]) ? $mod->value[0] : '', self::$max10Len, '0', STR_PAD_LEFT) . $result;}$result = ltrim($result, '0');if (empty($result)) {$result = '0';}if ($this->is_negative) {$result = '-' . $result;}return $result;}/*** Copy an object** PHP5 passes objects by reference while PHP4 passes by value. As such, we need a function to guarantee* that all objects are passed by value, when appropriate. More information can be found here:** {@link http://php.net/language.oop5.basic#51624}** @access public* @see self::__clone()* @return \phpseclib\Math\BigInteger*/function copy(){$temp = new static();$temp->value = $this->value;$temp->is_negative = $this->is_negative;$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 BigInteger::__toString() directly in PHP5, you cannot call 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 BigInteger::copy(), instead.** @access public* @see self::copy()* @return \phpseclib\Math\BigInteger*/function __clone(){return $this->copy();}/*** __sleep() magic method** Will be called, automatically, when serialize() is called on a BigInteger object.** @see self::__wakeup()* @access public*/function __sleep(){$this->hex = $this->toHex(true);$vars = array('hex');if ($this->precision > 0) {$vars[] = 'precision';}return $vars;}/*** __wakeup() magic method** Will be called, automatically, when unserialize() is called on a BigInteger object.** @see self::__sleep()* @access public*/function __wakeup(){$temp = new static($this->hex, -16);$this->value = $temp->value;$this->is_negative = $temp->is_negative;if ($this->precision > 0) {// recalculate $this->bitmask$this->setPrecision($this->precision);}}/*** __debugInfo() magic method** Will be called, automatically, when print_r() or var_dump() are called** @access public*/function __debugInfo(){$opts = array();switch (MATH_BIGINTEGER_MODE) {case self::MODE_GMP:$engine = 'gmp';break;case self::MODE_BCMATH:$engine = 'bcmath';break;case self::MODE_INTERNAL:$engine = 'internal';$opts[] = PHP_INT_SIZE == 8 ? '64-bit' : '32-bit';}if (MATH_BIGINTEGER_MODE != self::MODE_GMP && defined('MATH_BIGINTEGER_OPENSSL_ENABLED')) {$opts[] = 'OpenSSL';}if (!empty($opts)) {$engine.= ' (' . implode($opts, ', ') . ')';}return array('value' => '0x' . $this->toHex(true),'engine' => $engine);}/*** Adds two BigIntegers.** Here's an example:* <code>* <?php* $a = new \phpseclib\Math\BigInteger('10');* $b = new \phpseclib\Math\BigInteger('20');** $c = $a->add($b);** echo $c->toString(); // outputs 30* ?>* </code>** @param \phpseclib\Math\BigInteger $y* @return \phpseclib\Math\BigInteger* @access public* @internal Performs base-2**52 addition*/function add($y){switch (MATH_BIGINTEGER_MODE) {case self::MODE_GMP:$temp = new static();$temp->value = gmp_add($this->value, $y->value);return $this->_normalize($temp);case self::MODE_BCMATH:$temp = new static();$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 static();$result->value = $temp[self::VALUE];$result->is_negative = $temp[self::SIGN];return $this->_normalize($result);}/*** Performs addition.** @param array $x_value* @param bool $x_negative* @param array $y_value* @param bool $y_negative* @return array* @access private*/function _add($x_value, $x_negative, $y_value, $y_negative){$x_size = count($x_value);$y_size = count($y_value);if ($x_size == 0) {return array(self::VALUE => $y_value,self::SIGN => $y_negative);} elseif ($y_size == 0) {return array(self::VALUE => $x_value,self::SIGN => $x_negative);}// subtract, if appropriateif ($x_negative != $y_negative) {if ($x_value == $y_value) {return array(self::VALUE => array(),self::SIGN => false);}$temp = $this->_subtract($x_value, false, $y_value, false);$temp[self::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[count($value)] = 0; // just in case the carry adds an extra digit$carry = 0;for ($i = 0, $j = 1; $j < $size; $i+=2, $j+=2) {$sum = $x_value[$j] * self::$baseFull + $x_value[$i] + $y_value[$j] * self::$baseFull + $y_value[$i] + $carry;$carry = $sum >= self::$maxDigit2; // eg. floor($sum / 2**52); only possible values (in any base) are 0 and 1$sum = $carry ? $sum - self::$maxDigit2 : $sum;$temp = self::$base === 26 ? intval($sum / 0x4000000) : ($sum >> 31);$value[$i] = (int) ($sum - self::$baseFull * $temp); // eg. a faster alternative to fmod($sum, 0x4000000)$value[$j] = $temp;}if ($j == $size) { // ie. if $y_size is odd$sum = $x_value[$i] + $y_value[$i] + $carry;$carry = $sum >= self::$baseFull;$value[$i] = $carry ? $sum - self::$baseFull : $sum;++$i; // ie. let $i = $j since we've just done $value[$i]}if ($carry) {for (; $value[$i] == self::$maxDigit; ++$i) {$value[$i] = 0;}++$value[$i];}return array(self::VALUE => $this->_trim($value),self::SIGN => $x_negative);}/*** Subtracts two BigIntegers.** Here's an example:* <code>* <?php* $a = new \phpseclib\Math\BigInteger('10');* $b = new \phpseclib\Math\BigInteger('20');** $c = $a->subtract($b);** echo $c->toString(); // outputs -10* ?>* </code>** @param \phpseclib\Math\BigInteger $y* @return \phpseclib\Math\BigInteger* @access public* @internal Performs base-2**52 subtraction*/function subtract($y){switch (MATH_BIGINTEGER_MODE) {case self::MODE_GMP:$temp = new static();$temp->value = gmp_sub($this->value, $y->value);return $this->_normalize($temp);case self::MODE_BCMATH:$temp = new static();$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 static();$result->value = $temp[self::VALUE];$result->is_negative = $temp[self::SIGN];return $this->_normalize($result);}/*** Performs subtraction.** @param array $x_value* @param bool $x_negative* @param array $y_value* @param bool $y_negative* @return array* @access private*/function _subtract($x_value, $x_negative, $y_value, $y_negative){$x_size = count($x_value);$y_size = count($y_value);if ($x_size == 0) {return array(self::VALUE => $y_value,self::SIGN => !$y_negative);} elseif ($y_size == 0) {return array(self::VALUE => $x_value,self::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[self::SIGN] = $x_negative;return $temp;}$diff = $this->_compare($x_value, $x_negative, $y_value, $y_negative);if (!$diff) {return array(self::VALUE => array(),self::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] * self::$baseFull + $x_value[$i] - $y_value[$j] * self::$baseFull - $y_value[$i] - $carry;$carry = $sum < 0; // eg. floor($sum / 2**52); only possible values (in any base) are 0 and 1$sum = $carry ? $sum + self::$maxDigit2 : $sum;$temp = self::$base === 26 ? intval($sum / 0x4000000) : ($sum >> 31);$x_value[$i] = (int) ($sum - self::$baseFull * $temp);$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 ? $sum + self::$baseFull : $sum;++$i;}if ($carry) {for (; !$x_value[$i]; ++$i) {$x_value[$i] = self::$maxDigit;}--$x_value[$i];}return array(self::VALUE => $this->_trim($x_value),self::SIGN => $x_negative);}/*** Multiplies two BigIntegers** Here's an example:* <code>* <?php* $a = new \phpseclib\Math\BigInteger('10');* $b = new \phpseclib\Math\BigInteger('20');** $c = $a->multiply($b);** echo $c->toString(); // outputs 200* ?>* </code>** @param \phpseclib\Math\BigInteger $x* @return \phpseclib\Math\BigInteger* @access public*/function multiply($x){switch (MATH_BIGINTEGER_MODE) {case self::MODE_GMP:$temp = new static();$temp->value = gmp_mul($this->value, $x->value);return $this->_normalize($temp);case self::MODE_BCMATH:$temp = new static();$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 static();$product->value = $temp[self::VALUE];$product->is_negative = $temp[self::SIGN];return $this->_normalize($product);}/*** Performs multiplication.** @param array $x_value* @param bool $x_negative* @param array $y_value* @param bool $y_negative* @return array* @access private*/function _multiply($x_value, $x_negative, $y_value, $y_negative){//if ( $x_value == $y_value ) {// return array(// self::VALUE => $this->_square($x_value),// self::SIGN => $x_sign != $y_value// );//}$x_length = count($x_value);$y_length = count($y_value);if (!$x_length || !$y_length) { // a 0 is being multipliedreturn array(self::VALUE => array(),self::SIGN => false);}return array(self::VALUE => min($x_length, $y_length) < 2 * self::KARATSUBA_CUTOFF ?$this->_trim($this->_regularMultiply($x_value, $y_value)) :$this->_trim($this->_karatsuba($x_value, $y_value)),self::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 || !$y_length) { // a 0 is being multipliedreturn array();}if ($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$carry = self::$base === 26 ? intval($temp / 0x4000000) : ($temp >> 31);$product_value[$j] = (int) ($temp - self::$baseFull * $carry);}$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 = self::$base === 26 ? intval($temp / 0x4000000) : ($temp >> 31);$product_value[$k] = (int) ($temp - self::$baseFull * $carry);}$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 < self::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[self::VALUE], $temp[self::VALUE]);$temp = $this->_add($z2, false, $z0, false);$z1 = $this->_subtract($z1, false, $temp[self::VALUE], false);$z2 = array_merge(array_fill(0, 2 * $m, 0), $z2);$z1[self::VALUE] = array_merge(array_fill(0, $m, 0), $z1[self::VALUE]);$xy = $this->_add($z2, false, $z1[self::VALUE], $z1[self::SIGN]);$xy = $this->_add($xy[self::VALUE], $xy[self::SIGN], $z0, false);return $xy[self::VALUE];}/*** Performs squaring** @param array $x* @return array* @access private*/function _square($x = false){return count($x) < 2 * self::KARATSUBA_CUTOFF ?$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];$carry = self::$base === 26 ? intval($temp / 0x4000000) : ($temp >> 31);$square_value[$i2] = (int) ($temp - self::$baseFull * $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 = self::$base === 26 ? intval($temp / 0x4000000) : ($temp >> 31);$square_value[$k] = (int) ($temp - self::$baseFull * $carry);}// the following line can yield values larger 2**15. at this point, PHP should switch// over to floats.$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 < self::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[self::VALUE]);$temp = $this->_add($z2, false, $z0, false);$z1 = $this->_subtract($z1, false, $temp[self::VALUE], false);$z2 = array_merge(array_fill(0, 2 * $m, 0), $z2);$z1[self::VALUE] = array_merge(array_fill(0, $m, 0), $z1[self::VALUE]);$xx = $this->_add($z2, false, $z1[self::VALUE], $z1[self::SIGN]);$xx = $this->_add($xx[self::VALUE], $xx[self::SIGN], $z0, false);return $xx[self::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* $a = new \phpseclib\Math\BigInteger('10');* $b = new \phpseclib\Math\BigInteger('20');** list($quotient, $remainder) = $a->divide($b);** echo $quotient->toString(); // outputs 0* echo "\r\n";* echo $remainder->toString(); // outputs 10* ?>* </code>** @param \phpseclib\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 self::MODE_GMP:$quotient = new static();$remainder = new static();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 self::MODE_BCMATH:$quotient = new static();$remainder = new static();$quotient->value = bcdiv($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 static();$remainder = new static();$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)) {$zero = new static();}$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 static();$temp->value = array(1);$temp->is_negative = $x_sign != $y_sign;return array($this->_normalize($temp), $this->_normalize(new static()));}if ($diff < 0) {// if $x is negative, "add" $y.if ($x_sign) {$x = $y->subtract($x);}return array($this->_normalize(new static()), $this->_normalize($x));}// normalize $x and $y as described in HAC 14.23 / 14.24$msb = $y->value[count($y->value) - 1];for ($shift = 0; !($msb & self::$msb); ++$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 static();$quotient_value = &$quotient->value;$quotient_value = $this->_array_repeat(0, $x_max - $y_max + 1);static $temp, $lhs, $rhs;if (!isset($temp)) {$temp = new static();$lhs = new static();$rhs = new static();}$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[$q_index] = self::$maxDigit;} else {$quotient_value[$q_index] = $this->_safe_divide($x_window[0] * self::$baseFull + $x_window[1],$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 != $y_sign;// calculate the "common residue", if appropriateif ($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) {$temp = self::$baseFull * $carry + $dividend[$i];$result[$i] = $this->_safe_divide($temp, $divisor);$carry = (int) ($temp - $divisor * $result[$i]);}return array($result, $carry);}/*** Performs modular exponentiation.** Here's an example:* <code>* <?php* $a = new \phpseclib\Math\BigInteger('10');* $b = new \phpseclib\Math\BigInteger('20');* $c = new \phpseclib\Math\BigInteger('30');** $c = $a->modPow($b, $c);** echo $c->toString(); // outputs 10* ?>* </code>** @param \phpseclib\Math\BigInteger $e* @param \phpseclib\Math\BigInteger $n* @return \phpseclib\Math\BigInteger* @access public* @internal The most naive approach to modular exponentiation has very unreasonable requirements, and* and although the approach involving repeated squaring does vastly better, it, too, is impractical* for our purposes. The reason being that division - by far the most complicated and time-consuming* of the basic operations (eg. +,-,*,/) - occurs multiple times within it.** Modular reductions resolve this issue. Although an individual modular reduction takes more time* then an individual division, when performed in succession (with the same modulo), they're a lot faster.** The two most commonly used modular reductions are Barrett and Montgomery reduction. Montgomery reduction,* although faster, only works when the gcd of the modulo and of the base being used is 1. In RSA, when the* base is a power of two, the modulo - a product of two primes - is always going to have a gcd of 1 (because* the product of two odd numbers is odd), but what about when RSA isn't used?** In contrast, Barrett reduction has no such constraint. As such, some bigint implementations perform a* Barrett reduction after every operation in the modpow function. Others perform Barrett reductions when the* modulo is even and Montgomery reductions when the modulo is odd. BigInteger.java's modPow method, however,* uses a trick involving the Chinese Remainder Theorem to factor the even modulo into two numbers - one odd and* the other, a power of two - and recombine them, later. This is the method that this modPow function uses.* {@link http://islab.oregonstate.edu/papers/j34monex.pdf Montgomery Reduction with Even Modulus} elaborates.*/function modPow($e, $n){$n = $this->bitmask !== false && $this->bitmask->compare($n) < 0 ? $this->bitmask : $n->abs();if ($e->compare(new static()) < 0) {$e = $e->abs();$temp = $this->modInverse($n);if ($temp === false) {return false;}return $this->_normalize($temp->modPow($e, $n));}if (MATH_BIGINTEGER_MODE == self::MODE_GMP) {$temp = new static();$temp->value = gmp_powm($this->value, $e->value, $n->value);return $this->_normalize($temp);}if ($this->compare(new static()) < 0 || $this->compare($n) > 0) {list(, $temp) = $this->divide($n);return $temp->modPow($e, $n);}if (defined('MATH_BIGINTEGER_OPENSSL_ENABLED')) {$components = array('modulus' => $n->toBytes(true),'publicExponent' => $e->toBytes(true));$components = array('modulus' => pack('Ca*a*', 2, $this->_encodeASN1Length(strlen($components['modulus'])), $components['modulus']),'publicExponent' => pack('Ca*a*', 2, $this->_encodeASN1Length(strlen($components['publicExponent'])), $components['publicExponent']));$RSAPublicKey = pack('Ca*a*a*',48,$this->_encodeASN1Length(strlen($components['modulus']) + strlen($components['publicExponent'])),$components['modulus'],$components['publicExponent']);$rsaOID = pack('H*', '300d06092a864886f70d0101010500'); // hex version of MA0GCSqGSIb3DQEBAQUA$RSAPublicKey = chr(0) . $RSAPublicKey;$RSAPublicKey = chr(3) . $this->_encodeASN1Length(strlen($RSAPublicKey)) . $RSAPublicKey;$encapsulated = pack('Ca*a*',48,$this->_encodeASN1Length(strlen($rsaOID . $RSAPublicKey)),$rsaOID . $RSAPublicKey);$RSAPublicKey = "-----BEGIN PUBLIC KEY-----\r\n" .chunk_split(base64_encode($encapsulated)) .'-----END PUBLIC KEY-----';$plaintext = str_pad($this->toBytes(), strlen($n->toBytes(true)) - 1, "\0", STR_PAD_LEFT);if (openssl_public_encrypt($plaintext, $result, $RSAPublicKey, OPENSSL_NO_PADDING)) {return new static($result, 256);}}if (MATH_BIGINTEGER_MODE == self::MODE_BCMATH) {$temp = new static();$temp->value = bcpowmod($this->value, $e->value, $n->value, 0);return $this->_normalize($temp);}if (empty($e->value)) {$temp = new static();$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 static();$temp->value = $this->_square($this->value);list(, $temp) = $temp->divide($n);return $this->_normalize($temp);}return $this->_normalize($this->_slidingWindow($e, $n, self::BARRETT));// the following code, although not callable, can be run independently of the above code// although the above code performed better in my benchmarks the following could might// perform better under different circumstances. in lieu of deleting it it's just been// made uncallable// is the modulo odd?if ($n->value[0] & 1) {return $this->_normalize($this->_slidingWindow($e, $n, self::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 static();$mod2->value = array(1);$mod2->_lshift($j);$part1 = ($mod1->value != array(1)) ? $this->_slidingWindow($e, $mod1, self::MONTGOMERY) : new static();$part2 = $this->_slidingWindow($e, $mod2, self::POWEROF2);$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 modPow().** @param \phpseclib\Math\BigInteger $e* @param \phpseclib\Math\BigInteger $n* @return \phpseclib\Math\BigInteger* @access public*/function powMod($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 \phpseclib\Math\BigInteger $e* @param \phpseclib\Math\BigInteger $n* @param int $mode* @return \phpseclib\Math\BigInteger* @access private*/function _slidingWindow($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[$i]), self::$base, '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; $i < count($window_ranges) && $e_length > $window_ranges[$i]; ++$window_size, ++$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 + $j])) {break;}}// eg. the length of substr($e_bits, $i, $j + 1)for ($k = 0; $k <= $j; ++$k) {$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 static();$temp->value = $this->_reduce($result, $n_value, $mode);return $temp;}/*** Modular reduction** For most $modes this will return the remainder.** @see self::_slidingWindow()* @access private* @param array $x* @param array $n* @param int $mode* @return array*/function _reduce($x, $n, $mode){switch ($mode) {case self::MONTGOMERY:return $this->_montgomery($x, $n);case self::BARRETT:return $this->_barrett($x, $n);case self::POWEROF2:$lhs = new static();$lhs->value = $x;$rhs = new static();$rhs->value = $n;return $x->_mod2($n);case self::CLASSIC:$lhs = new static();$lhs->value = $x;$rhs = new static();$rhs->value = $n;list(, $temp) = $lhs->divide($rhs);return $temp->value;case self::NONE:return $x;default:// an invalid $mode was provided}}/*** Modular reduction preperation** @see self::_slidingWindow()* @access private* @param array $x* @param array $n* @param int $mode* @return array*/function _prepareReduce($x, $n, $mode){if ($mode == self::MONTGOMERY) {return $this->_prepMontgomery($x, $n);}return $this->_reduce($x, $n, $mode);}/*** Modular multiply** @see self::_slidingWindow()* @access private* @param array $x* @param array $y* @param array $n* @param int $mode* @return array*/function _multiplyReduce($x, $y, $n, $mode){if ($mode == self::MONTGOMERY) {return $this->_montgomeryMultiply($x, $y, $n);}$temp = $this->_multiply($x, false, $y, false);return $this->_reduce($temp[self::VALUE], $n, $mode);}/*** Modular square** @see self::_slidingWindow()* @access private* @param array $x* @param array $n* @param int $mode* @return array*/function _squareReduce($x, $n, $mode){if ($mode == self::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 self::_slidingWindow()* @access private* @param \phpseclib\Math\BigInteger* @return \phpseclib\Math\BigInteger*/function _mod2($n){$temp = new static();$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 self::_slidingWindow()* @access private* @param array $n* @param array $m* @return array*/function _barrett($n, $m){static $cache = array(self::VARIABLE => array(),self::DATA => array());$m_length = count($m);// if ($this->_compare($n, $this->_square($m)) >= 0) {if (count($n) > 2 * $m_length) {$lhs = new static();$rhs = new static();$lhs->value = $n;$rhs->value = $m;list(, $temp) = $lhs->divide($rhs);return $temp->value;}// if (m.length >> 1) + 2 <= m.length then m is too small and n can't be reducedif ($m_length < 5) {return $this->_regularBarrett($n, $m);}// n = 2 * m.lengthif (($key = array_search($m, $cache[self::VARIABLE])) === false) {$key = count($cache[self::VARIABLE]);$cache[self::VARIABLE][] = $m;$lhs = new static();$lhs_value = &$lhs->value;$lhs_value = $this->_array_repeat(0, $m_length + ($m_length >> 1));$lhs_value[] = 1;$rhs = new static();$rhs->value = $m;list($u, $m1) = $lhs->divide($rhs);$u = $u->value;$m1 = $m1->value;$cache[self::DATA][] = array('u' => $u, // m.length >> 1 (technically (m.length >> 1) + 1)'m1'=> $m1 // m.length);} else {extract($cache[self::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, $temp[self::VALUE], false); // m.length + (m.length >> 1) + 1if ($m_length & 1) {return $this->_regularBarrett($n[self::VALUE], $m);}// (m.length + (m.length >> 1) + 1) - (m.length - 1) == (m.length >> 1) + 2$temp = array_slice($n[self::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($temp[self::VALUE], ($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[self::VALUE], false, $temp[self::VALUE], false);while ($this->_compare($result[self::VALUE], $result[self::SIGN], $m, false) >= 0) {$result = $this->_subtract($result[self::VALUE], $result[self::SIGN], $m, false);}return $result[self::VALUE];}/*** (Regular) Barrett Modular Reduction** For numbers with more than four digits BigInteger::_barrett() is faster. The difference between that and this* is that this function does not fold the denominator into a smaller form.** @see self::_slidingWindow()* @access private* @param array $x* @param array $n* @return array*/function _regularBarrett($x, $n){static $cache = array(self::VARIABLE => array(),self::DATA => array());$n_length = count($n);if (count($x) > 2 * $n_length) {$lhs = new static();$rhs = new static();$lhs->value = $x;$rhs->value = $n;list(, $temp) = $lhs->divide($rhs);return $temp->value;}if (($key = array_search($n, $cache[self::VARIABLE])) === false) {$key = count($cache[self::VARIABLE]);$cache[self::VARIABLE][] = $n;$lhs = new static();$lhs_value = &$lhs->value;$lhs_value = $this->_array_repeat(0, 2 * $n_length);$lhs_value[] = 1;$rhs = new static();$rhs->value = $n;list($temp, ) = $lhs->divide($rhs); // m.length$cache[self::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, $cache[self::DATA][$key], false);// (2 * m.length + 1) - (m.length - 1) = m.length + 2$temp = array_slice($temp[self::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[self::VALUE], $temp[self::SIGN]) < 0) {$corrector_value = $this->_array_repeat(0, $n_length + 1);$corrector_value[count($corrector_value)] = 1;$result = $this->_add($result, false, $corrector_value, false);$result = $result[self::VALUE];}// at this point, we're subtracting a number with m.length + 1 digits from another number with m.length + 1 digits$result = $this->_subtract($result, false, $temp[self::VALUE], $temp[self::SIGN]);while ($this->_compare($result[self::VALUE], $result[self::SIGN], $n, false) > 0) {$result = $this->_subtract($result[self::VALUE], $result[self::SIGN], $n, false);}return $result[self::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 self::_regularBarrett()* @param array $x_value* @param bool $x_negative* @param array $y_value* @param bool $y_negative* @param int $stop* @return array* @access private*/function _multiplyLower($x_value, $x_negative, $y_value, $y_negative, $stop){$x_length = count($x_value);$y_length = count($y_value);if (!$x_length || !$y_length) { // a 0 is being multipliedreturn array(self::VALUE => array(),self::SIGN => false);}if ($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$carry = self::$base === 26 ? intval($temp / 0x4000000) : ($temp >> 31);$product_value[$j] = (int) ($temp - self::$baseFull * $carry);}if ($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 = self::$base === 26 ? intval($temp / 0x4000000) : ($temp >> 31);$product_value[$k] = (int) ($temp - self::$baseFull * $carry);}if ($k < $stop) {$product_value[$k] = $carry;}}return array(self::VALUE => $this->_trim($product_value),self::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 self::_prepMontgomery()* @see self::_slidingWindow()* @access private* @param array $x* @param array $n* @return array*/function _montgomery($x, $n){static $cache = array(self::VARIABLE => array(),self::DATA => array());if (($key = array_search($n, $cache[self::VARIABLE])) === false) {$key = count($cache[self::VARIABLE]);$cache[self::VARIABLE][] = $x;$cache[self::DATA][] = $this->_modInverse67108864($n);}$k = count($n);$result = array(self::VALUE => $x);for ($i = 0; $i < $k; ++$i) {$temp = $result[self::VALUE][$i] * $cache[self::DATA][$key];$temp = $temp - self::$baseFull * (self::$base === 26 ? intval($temp / 0x4000000) : ($temp >> 31));$temp = $this->_regularMultiply(array($temp), $n);$temp = array_merge($this->_array_repeat(0, $i), $temp);$result = $this->_add($result[self::VALUE], false, $temp, false);}$result[self::VALUE] = array_slice($result[self::VALUE], $k);if ($this->_compare($result, false, $n, false) >= 0) {$result = $this->_subtract($result[self::VALUE], false, $n, false);}return $result[self::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 self::_prepMontgomery()* @see self::_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[self::VALUE], $m);// the following code, although not callable, can be run independently of the above code// although the above code performed better in my benchmarks the following could might// perform better under different circumstances. in lieu of deleting it it's just been// made uncallablestatic $cache = array(self::VARIABLE => array(),self::DATA => array());if (($key = array_search($m, $cache[self::VARIABLE])) === false) {$key = count($cache[self::VARIABLE]);$cache[self::VARIABLE][] = $m;$cache[self::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(self::VALUE => $this->_array_repeat(0, $n + 1));for ($i = 0; $i < $n; ++$i) {$temp = $a[self::VALUE][0] + $x[$i] * $y[0];$temp = $temp - self::$baseFull * (self::$base === 26 ? intval($temp / 0x4000000) : ($temp >> 31));$temp = $temp * $cache[self::DATA][$key];$temp = $temp - self::$baseFull * (self::$base === 26 ? intval($temp / 0x4000000) : ($temp >> 31));$temp = $this->_add($this->_regularMultiply(array($x[$i]), $y), false, $this->_regularMultiply(array($temp), $m), false);$a = $this->_add($a[self::VALUE], false, $temp[self::VALUE], false);$a[self::VALUE] = array_slice($a[self::VALUE], 1);}if ($this->_compare($a[self::VALUE], false, $m, false) >= 0) {$a = $this->_subtract($a[self::VALUE], false, $m, false);}return $a[self::VALUE];}/*** Prepare a number for use in Montgomery Modular Reductions** @see self::_montgomery()* @see self::_slidingWindow()* @access private* @param array $x* @param array $n* @return array*/function _prepMontgomery($x, $n){$lhs = new static();$lhs->value = array_merge($this->_array_repeat(0, count($n)), $x);$rhs = new static();$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 self::_montgomery()* @access private* @param array $x* @return int*/function _modInverse67108864($x) // 2**26 == 67,108,864{$x = -$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 * $result, self::$baseFull)), self::$baseFull); // x**-1 mod 2**26return $result & self::$maxDigit;}/*** 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* $a = new \phpseclib\Math\BigInteger(30);* $b = new \phpseclib\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 \phpseclib\Math\BigInteger $n* @return \phpseclib\Math\BigInteger|false* @access public* @internal See {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=21 HAC 14.64} for more information.*/function modInverse($n){switch (MATH_BIGINTEGER_MODE) {case self::MODE_GMP:$temp = new static();$temp->value = gmp_invert($this->value, $n->value);return ($temp->value === false) ? false : $this->_normalize($temp);}static $zero, $one;if (!isset($zero)) {$zero = new static();$one = new static(1);}// $x mod -$n == $x mod $n.$n = $n->abs();if ($this->compare($zero) < 0) {$temp = $this->abs();$temp = $temp->modInverse($n);return $this->_normalize($n->subtract($temp));}extract($this->extendedGCD($n));if (!$gcd->equals($one)) {return false;}$x = $x->compare($zero) < 0 ? $x->add($n) : $x;return $this->compare($zero) < 0 ? $this->_normalize($n->subtract($x)) : $this->_normalize($x);}/*** Calculates the greatest common divisor and Bezout's identity.** Say you have 693 and 609. The GCD is 21. Bezout'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 dependent upon which mode is in use. See* {@link http://en.wikipedia.org/wiki/B%C3%A9zout%27s_identity Bezout's identity - Wikipedia} for more information.** Here's an example:* <code>* <?php* $a = new \phpseclib\Math\BigInteger(693);* $b = new \phpseclib\Math\BigInteger(609);** extract($a->extendedGCD($b));** echo $gcd->toString() . "\r\n"; // outputs 21* echo $a->toString() * $x->toString() + $b->toString() * $y->toString(); // outputs 21* ?>* </code>** @param \phpseclib\Math\BigInteger $n* @return \phpseclib\Math\BigInteger* @access public* @internal Calculates the GCD using the binary xGCD algorithim described in* {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=19 HAC 14.61}. As the text above 14.61 notes,* the more traditional algorithim requires "relatively costly multiple-precision divisions".*/function extendedGCD($n){switch (MATH_BIGINTEGER_MODE) {case self::MODE_GMP:extract(gmp_gcdext($this->value, $n->value));return array('gcd' => $this->_normalize(new static($g)),'x' => $this->_normalize(new static($s)),'y' => $this->_normalize(new static($t)));case self::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' => $this->_normalize(new static($u)),'x' => $this->_normalize(new static($a)),'y' => $this->_normalize(new static($b)));}$y = $n->copy();$x = $this->copy();$g = new static();$g->value = array(1);while (!(($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 static();$b = new static();$c = new static();$d = new static();$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* $a = new \phpseclib\Math\BigInteger(693);* $b = new \phpseclib\Math\BigInteger(609);** $gcd = a->extendedGCD($b);** echo $gcd->toString() . "\r\n"; // outputs 21* ?>* </code>** @param \phpseclib\Math\BigInteger $n* @return \phpseclib\Math\BigInteger* @access public*/function gcd($n){extract($this->extendedGCD($n));return $gcd;}/*** Absolute value.** @return \phpseclib\Math\BigInteger* @access public*/function abs(){$temp = new static();switch (MATH_BIGINTEGER_MODE) {case self::MODE_GMP:$temp->value = gmp_abs($this->value);break;case self::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 \phpseclib\Math\BigInteger $y* @return int < 0 if $this is less than $y; > 0 if $this is greater than $y, and 0 if they are equal.* @access public* @see self::equals()* @internal Could return $this->subtract($x), but that's not as fast as what we do do.*/function compare($y){switch (MATH_BIGINTEGER_MODE) {case self::MODE_GMP:return gmp_cmp($this->value, $y->value);case self::MODE_BCMATH:return bccomp($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 bool $x_negative* @param array $y_value* @param bool $y_negative* @return int* @see self::compare()* @access private*/function _compare($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 BigInteger::compare()** @param \phpseclib\Math\BigInteger $x* @return bool* @access public* @see self::compare()*/function equals($x){switch (MATH_BIGINTEGER_MODE) {case self::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 int $bits* @access public*/function setPrecision($bits){$this->precision = $bits;if (MATH_BIGINTEGER_MODE != self::MODE_BCMATH) {$this->bitmask = new static(chr((1 << ($bits & 0x7)) - 1) . str_repeat(chr(0xFF), $bits >> 3), 256);} else {$this->bitmask = new static(bcpow('2', $bits, 0));}$temp = $this->_normalize($this);$this->value = $temp->value;}/*** Logical And** @param \phpseclib\Math\BigInteger $x* @access public* @internal Implemented per a request by Lluis Pamies i Juarez <lluis _a_ pamies.cat>* @return \phpseclib\Math\BigInteger*/function bitwise_and($x){switch (MATH_BIGINTEGER_MODE) {case self::MODE_GMP:$temp = new static();$temp->value = gmp_and($this->value, $x->value);return $this->_normalize($temp);case self::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 $this->_normalize(new static($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]&= $x->value[$i];}return $this->_normalize($result);}/*** Logical Or** @param \phpseclib\Math\BigInteger $x* @access public* @internal Implemented per a request by Lluis Pamies i Juarez <lluis _a_ pamies.cat>* @return \phpseclib\Math\BigInteger*/function bitwise_or($x){switch (MATH_BIGINTEGER_MODE) {case self::MODE_GMP:$temp = new static();$temp->value = gmp_or($this->value, $x->value);return $this->_normalize($temp);case self::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 $this->_normalize(new static($left | $right, 256));}$length = max(count($this->value), count($x->value));$result = $this->copy();$result->value = array_pad($result->value, $length, 0);$x->value = array_pad($x->value, $length, 0);for ($i = 0; $i < $length; ++$i) {$result->value[$i]|= $x->value[$i];}return $this->_normalize($result);}/*** Logical Exclusive-Or** @param \phpseclib\Math\BigInteger $x* @access public* @internal Implemented per a request by Lluis Pamies i Juarez <lluis _a_ pamies.cat>* @return \phpseclib\Math\BigInteger*/function bitwise_xor($x){switch (MATH_BIGINTEGER_MODE) {case self::MODE_GMP:$temp = new static();$temp->value = gmp_xor($this->value, $x->value);return $this->_normalize($temp);case self::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 $this->_normalize(new static($left ^ $right, 256));}$length = max(count($this->value), count($x->value));$result = $this->copy();$result->value = array_pad($result->value, $length, 0);$x->value = array_pad($x->value, $length, 0);for ($i = 0; $i < $length; ++$i) {$result->value[$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 \phpseclib\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 = $this->toBytes();if ($temp == '') {return '';}$pre_msb = decbin(ord($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 $this->_normalize(new static($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($temp, strlen($leading_ones), chr(0), STR_PAD_LEFT);return $this->_normalize(new static($leading_ones | $temp, 256));}/*** Logical Right Shift** Shifts BigInteger's by $shift bits, effectively dividing by 2**$shift.** @param int $shift* @return \phpseclib\Math\BigInteger* @access public* @internal The only version that yields any speed increases is the internal version.*/function bitwise_rightShift($shift){$temp = new static();switch (MATH_BIGINTEGER_MODE) {case self::MODE_GMP:static $two;if (!isset($two)) {$two = gmp_init('2');}$temp->value = gmp_div_q($this->value, gmp_pow($two, $shift));break;case self::MODE_BCMATH:$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 int $shift* @return \phpseclib\Math\BigInteger* @access public* @internal The only version that yields any speed increases is the internal version.*/function bitwise_leftShift($shift){$temp = new static();switch (MATH_BIGINTEGER_MODE) {case self::MODE_GMP:static $two;if (!isset($two)) {$two = gmp_init('2');}$temp->value = gmp_mul($this->value, gmp_pow($two, $shift));break;case self::MODE_BCMATH:$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 int $shift* @return \phpseclib\Math\BigInteger* @access public*/function bitwise_leftRotate($shift){$bits = $this->toBytes();if ($this->precision > 0) {$precision = $this->precision;if (MATH_BIGINTEGER_MODE == self::MODE_BCMATH) {$mask = $this->bitmask->subtract(new static(1));$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 = $left->bitwise_and(new static($mask, 256));$right = $this->bitwise_rightShift($precision - $shift);$result = MATH_BIGINTEGER_MODE != self::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 int $shift* @return \phpseclib\Math\BigInteger* @access public*/function bitwise_rightRotate($shift){return $this->bitwise_leftRotate(-$shift);}/*** Generates a random BigInteger** Byte length is equal to $length. Uses \phpseclib\Crypt\Random if it's loaded and mt_rand if it's not.** @param int $length* @return \phpseclib\Math\BigInteger* @access private*/function _random_number_helper($size){if (class_exists('\phpseclib\Crypt\Random')) {$random = Random::string($size);} else {$random = '';if ($size & 1) {$random.= chr(mt_rand(0, 255));}$blocks = $size >> 1;for ($i = 0; $i < $blocks; ++$i) {// mt_rand(-2147483648, 0x7FFFFFFF) always produces -2147483648 on some systems$random.= pack('n', mt_rand(0, 0xFFFF));}}return new static($random, 256);}/*** Generate a random number** Returns a random number between $min and $max where $min and $max* can be defined using one of the two methods:** $min->random($max)* $max->random($min)** @param \phpseclib\Math\BigInteger $arg1* @param \phpseclib\Math\BigInteger $arg2* @return \phpseclib\Math\BigInteger* @access public* @internal The API for creating random numbers used to be $a->random($min, $max), where $a was a BigInteger object.* That method is still supported for BC purposes.*/function random($arg1, $arg2 = false){if ($arg1 === false) {return false;}if ($arg2 === false) {$max = $arg1;$min = $this;} else {$min = $arg1;$max = $arg2;}$compare = $max->compare($min);if (!$compare) {return $this->_normalize($min);} elseif ($compare < 0) {// if $min is bigger then $max, swap $min and $max$temp = $max;$max = $min;$min = $temp;}static $one;if (!isset($one)) {$one = new static(1);}$max = $max->subtract($min->subtract($one));$size = strlen(ltrim($max->toBytes(), chr(0)));/*doing $random % $max doesn't work because some numbers will be more likely to occur than others.eg. if $max is 140 and $random's max is 255 then that'd mean both $random = 5 and $random = 145would produce 5 whereas the only value of random that could produce 139 would be 139. ie.not all numbers would be equally likely. some would be more likely than others.creating a whole new random number until you find one that is within the range doesn't workbecause, for sufficiently small ranges, the likelihood that you'd get a number within that rangewould be pretty small. eg. with $random's max being 255 and if your $max being 1 the probabilitywould be pretty high that $random would be greater than $max.phpseclib works around this using the technique described here:http://crypto.stackexchange.com/questions/5708/creating-a-small-number-from-a-cryptographically-secure-random-string*/$random_max = new static(chr(1) . str_repeat("\0", $size), 256);$random = $this->_random_number_helper($size);list($max_multiple) = $random_max->divide($max);$max_multiple = $max_multiple->multiply($max);while ($random->compare($max_multiple) >= 0) {$random = $random->subtract($max_multiple);$random_max = $random_max->subtract($max_multiple);$random = $random->bitwise_leftShift(8);$random = $random->add($this->_random_number_helper(1));$random_max = $random_max->bitwise_leftShift(8);list($max_multiple) = $random_max->divide($max);$max_multiple = $max_multiple->multiply($max);}list(, $random) = $random->divide($max);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 \phpseclib\Math\BigInteger $arg1* @param \phpseclib\Math\BigInteger $arg2* @param int $timeout* @return Math_BigInteger|false* @access public* @internal See {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap4.pdf#page=15 HAC 4.44}.*/function randomPrime($arg1, $arg2 = false, $timeout = false){if ($arg1 === false) {return false;}if ($arg2 === false) {$max = $arg1;$min = $this;} else {$min = $arg1;$max = $arg2;}$compare = $max->compare($min);if (!$compare) {return $min->isPrime() ? $min : false;} elseif ($compare < 0) {// if $min is bigger then $max, swap $min and $max$temp = $max;$max = $min;$min = $temp;}static $one, $two;if (!isset($one)) {$one = new static(1);$two = new static(2);}$start = time();$x = $this->random($min, $max);// gmp_nextprime() requires PHP 5 >= 5.2.0 per <http://php.net/gmp-nextprime>.if (MATH_BIGINTEGER_MODE == self::MODE_GMP && extension_loaded('gmp')) {$p = new static();$p->value = gmp_nextprime($x->value);if ($p->compare($max) <= 0) {return $p;}if (!$min->equals($x)) {$x = $x->subtract($one);}return $x->randomPrime($min, $x);}if ($x->equals($two)) {return $x;}$x->_make_odd();if ($x->compare($max) > 0) {// if $x > $max then $max is even and if $min == $max then no prime number exists between the specified rangeif ($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 self::randomPrime()* @access private*/function _make_odd(){switch (MATH_BIGINTEGER_MODE) {case self::MODE_GMP:gmp_setbit($this->value, 0);break;case self::MODE_BCMATH:if ($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. BigInteger::randomPrime() can be distributed across multiple pageloads* on a website instead of just one.** @param \phpseclib\Math\BigInteger $t* @return bool* @access public* @internal Uses the* {@link http://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test Miller-Rabin primality test}. See* {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap4.pdf#page=8 HAC 4.24}.*/function isPrime($t = false){$length = strlen($this->toBytes());if (!$t) {// see HAC 4.49 "Note (controlling the error probability)"// @codingStandardsIgnoreStartif ($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 { $t = 27; }// @codingStandardsIgnoreEnd}// ie. gmp_testbit($this, 0)// ie. isEven() or !isOdd()switch (MATH_BIGINTEGER_MODE) {case self::MODE_GMP:return gmp_prob_prime($this->value, $t) != 0;case self::MODE_BCMATH:if ($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)) {$primes = array(3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59,61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137,139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227,229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313,317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419,421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509,521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617,619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727,733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829,839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947,953, 967, 971, 977, 983, 991, 997);if (MATH_BIGINTEGER_MODE != self::MODE_INTERNAL) {for ($i = 0; $i < count($primes); ++$i) {$primes[$i] = new static($primes[$i]);}}$zero = new static();$one = new static(1);$two = new static(2);}if ($this->equals($one)) {return false;}// see HAC 4.4.1 "Random search for probable primes"if (MATH_BIGINTEGER_MODE != self::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'), $s));if (MATH_BIGINTEGER_MODE == self::MODE_BCMATH) {$s = 0;// if $n was 1, $r would be 0 and this would be an infinite loop, hence our $this->equals($one) check earlierwhile ($r->value[strlen($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 int $shift* @access private*/function _lshift($shift){if ($shift == 0) {return;}$num_digits = (int) ($shift / self::$base);$shift %= self::$base;$shift = 1 << $shift;$carry = 0;for ($i = 0; $i < count($this->value); ++$i) {$temp = $this->value[$i] * $shift + $carry;$carry = self::$base === 26 ? intval($temp / 0x4000000) : ($temp >> 31);$this->value[$i] = (int) ($temp - $carry * self::$baseFull);}if ($carry) {$this->value[count($this->value)] = $carry;}while ($num_digits--) {array_unshift($this->value, 0);}}/*** Logical Right Shift** Shifts BigInteger's by $shift bits.** @param int $shift* @access private*/function _rshift($shift){if ($shift == 0) {return;}$num_digits = (int) ($shift / self::$base);$shift %= self::$base;$carry_shift = self::$base - $shift;$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 \phpseclib\Math\BigInteger* @return \phpseclib\Math\BigInteger* @see self::_trim()* @access private*/function _normalize($result){$result->precision = $this->precision;$result->bitmask = $this->bitmask;switch (MATH_BIGINTEGER_MODE) {case self::MODE_GMP:if ($this->bitmask !== false) {$result->value = gmp_and($result->value, $result->bitmask->value);}return $result;case self::MODE_BCMATH:if (!empty($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** @param array $value* @return \phpseclib\Math\BigInteger* @access private*/function _trim($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 int $x* @return string* @access private*/function _int2bytes($x){return ltrim(pack('N', $x), chr(0));}/*** Converts bytes to 32-bit integers** @param string $x* @return int* @access private*/function _bytes2int($x){$temp = unpack('Nint', str_pad($x, 4, chr(0), STR_PAD_LEFT));return $temp['int'];}/*** DER-encode an integer** The ability to DER-encode integers is needed to create RSA public keys for use with OpenSSL** @see self::modPow()* @access private* @param int $length* @return string*/function _encodeASN1Length($length){if ($length <= 0x7F) {return chr($length);}$temp = ltrim(pack('N', $length), chr(0));return pack('Ca*', 0x80 | strlen($temp), $temp);}/*** Single digit division** Even if int64 is being used the division operator will return a float64 value* if the dividend is not evenly divisible by the divisor. Since a float64 doesn't* have the precision of int64 this is a problem so, when int64 is being used,* we'll guarantee that the dividend is divisible by first subtracting the remainder.** @access private* @param int $x* @param int $y* @return int*/function _safe_divide($x, $y){if (self::$base === 26) {return (int) ($x / $y);}// self::$base === 31return ($x - ($x % $y)) / $y;}}