<?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 */ 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() */ static $base; static $baseFull; static $maxDigit; static $msb; /** * $max10 in greatest $max10Len satisfying * $max10 = 10**$max10Len <= 2**$base. */ static $max10; /** * $max10Len in greatest $max10Len satisfying * $max10 = 10**$max10Len <= 2**$base. */ static $max10Len; 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 int|string|resource $x base-10 number or base-$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 work $versions = array(); // avoid generating errors (even with suppression) when phpinfo() is disabled (common in production systems) if (function_exists('phpinfo')) { ob_start(); @phpinfo(); $content = ob_get_contents(); ob_end_clean(); preg_match_all('#OpenSSL (Header|Library) Version(.*)#im', $content, $matches); if (!empty($matches[1])) { for ($i = 0; $i < count($matches[1]); $i++) { $fullVersion = trim(str_replace('=>', '', strip_tags($matches[2][$i]))); // Remove letter part in OpenSSL version if (!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 bytes self::$base = 31; self::$baseFull = 0x80000000; self::$maxDigit = 0x7FFFFFFF; self::$msb = 0x40000000; self::$max10 = 1000000000; self::$max10Len = 9; self::$maxDigit2 = pow(2, 62); break; //case 4: // use 64-bit floats if int size is 4 bytes default: self::$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 objects case $x instanceof \GMP: $this->value = $x; return; } $this->value = gmp_init(0); break; case self::MODE_BCMATH: $this->value = '0'; break; default: $this->value = array(); } // '0' counts as empty() but when the base is 256 '0' is equal to ord('0') or 48 // '0' is the only value like this per http://php.net/empty if (empty($x) && (abs($base) != 256 || $x !== '0')) { return; } switch ($base) { case -256: if (ord($x[0]) & 0x80) { $x = ~$x; $this->is_negative = true; } case 256: switch (MATH_BIGINTEGER_MODE) { case self::MODE_GMP: $this->value = function_exists('gmp_import') ? gmp_import($x) : gmp_init('0x' . bin2hex($x)); if ($this->is_negative) { $this->value = gmp_neg($this->value); } break; case self::MODE_BCMATH: // round $len to the nearest 4 (thanks, DavidMJ!) $len = (strlen($x) + 3) & ~3; $x = str_pad($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]*).*#s', '$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].*#s', '', $x); if (!strlen($x) || $x == '-') { $x = '0'; } 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]*).*#s', '$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 = $temp->toBytes(); if (!strlen($bytes)) { // eg. if the number we're trying to convert is -1 $bytes = chr(0); } if ($this->precision <= 0 && (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) : ''; } if (function_exists('gmp_export')) { $temp = gmp_export($this->value); } else { $temp = gmp_strval(gmp_abs($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) - 6, $start = strlen($hex) % 6; $i >= $start; $i-=6) { $bits = str_pad(decbin(hexdec(substr($hex, $i, 6))), 24, '0', STR_PAD_LEFT) . $bits; } if ($start) { // hexdec('') == 0 $bits = str_pad(decbin(hexdec(substr($hex, 0, $start))), 8 * $start, '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->bitmask = false; $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 $result; } /** * Return the size of a BigInteger in bits * * @return int */ function getLength() { if (MATH_BIGINTEGER_MODE != self::MODE_INTERNAL) { return strlen($this->toBits()); } $max = count($this->value) - 1; return $max != -1 ? $max * self::$base + intval(ceil(log($this->value[$max] + 1, 2))) : 0; } /** * Return the size of a BigInteger in bytes * * @return int */ function getLengthInBytes() { return (int) ceil($this->getLength() / 8); } /** * 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 => $x_negative ); } // subtract, if appropriate if ($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 multiplied return 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 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 $carry = self::$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; if (count($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) { $temp = self::$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 $n * @return \phpseclib\Math\BigInteger */ function _mod2($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 $temp->value; } // if (m.length >> 1) + 2 <= m.length then m is too small and n can't be reduced if ($m_length < 5) { return $this->_regularBarrett($n, $m); } // n = 2 * m.length if (($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, $temp[self::VALUE], false); // m.length + (m.length >> 1) + 1 if ($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 multiplied return 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 uncallable static $cache = array( self::VARIABLE => array(), self::DATA => array() ); if (($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**26 return $result & self::$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 that is < 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: $r = gmp_cmp($this->value, $y->value); if ($r < -1) { $r = -1; } if ($r > 1) { $r = 1; } return $r; case self::MODE_BCMATH: return bccomp($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(); $temp->value = gmp_xor(gmp_abs($this->value), gmp_abs($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->is_negative = false; $result->value = array_pad($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 $this->_normalize(new static()); } $pre_msb = decbin(ord($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 $size * @return \phpseclib\Math\BigInteger * @access private */ function _random_number_helper($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 = 145 would 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 work because, for sufficiently small ranges, the likelihood that you'd get a number within that range would be pretty small. eg. with $random's max being 255 and if your $max being 1 the probability would 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; } $length = $max->getLength(); if ($length > 8196) { user_error('Generation of random prime numbers larger than 8196 has been disabled'); } 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 == $max then no prime number exists between the specified range if ($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($initial_x)) { return false; } } } /** * Make the current number odd * * If the current number is odd it'll be unchanged. If it's even, one will be added to it. * * @see self::randomPrime() * @access private */ function _make_odd() { switch (MATH_BIGINTEGER_MODE) { case self::MODE_GMP: gmp_setbit($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 = $this->getLength(); // OpenSSL limits RSA keys to 16384 bits. The length of an RSA key is equal to the length of the modulo, which is // produced by multiplying the primes p and q by one another. The largest number two 8196 bit primes can produce is // a 16384 bit number so, basically, 8196 bit primes are the largest OpenSSL will generate and if that's the largest // that it'll generate it also stands to reason that that's the largest you'll be able to test primality on if ($length > 8196) { user_error('Primality testing is not supported for numbers larger than 8196 bits'); } if (!$t) { // see HAC 4.49 "Note (controlling the error probability)" // @codingStandardsIgnoreStart if ($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 { $t = 27; } // @codingStandardsIgnoreEnd } // ie. gmp_testbit($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 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; $r->_rshift($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 $result * @return \phpseclib\Math\BigInteger * @see self::_trim() * @access private */ function _normalize($result) { $result->precision = $this->precision; $result->bitmask = $this->bitmask; switch (MATH_BIGINTEGER_MODE) { case self::MODE_GMP: if ($this->bitmask !== false) { $flip = gmp_cmp($result->value, gmp_init(0)) < 0; if ($flip) { $result->value = gmp_neg($result->value); } $result->value = gmp_and($result->value, $result->bitmask->value); if ($flip) { $result->value = gmp_neg($result->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)) { $result->is_negative = false; 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 array $input * @param mixed $multiplier * @return array * @access private */ function _array_repeat($input, $multiplier) { return ($multiplier) ? array_fill(0, $multiplier, $input) : array(); } /** * Logical Left Shift * * Shifts binary strings $shift bits, essentially multiplying by 2**$shift. * * @param string $x (by reference) * @param int $shift * @return string * @access private */ function _base256_lshift(&$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 string $x (by referenc) * @param int $shift * @return string * @access private */ function _base256_rshift(&$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 === 31 return ($x - ($x % $y)) / $y; } }

Es la eliminación del sarro y la placa adherida a la superficie de los dientes mediante un equipo de ultrasonidos que garantiza la integridad de las piezas dentales a la vez que elimina en profundidad cualquier resto de suciedad.

A continuación se procede al pulido de los dientes mediante una fresa especial que elimina la placa bacteriana y devuelve a los dientes el aspecto sano que deben tener.

Una vez terminado todo el proceso, se mantiene al perro en observación hasta que se despierta de la anestesia, bajo la atenta supervisión de un veterinario.

¿Cada cuánto tiempo tengo que hacerle una limpieza dental a mi perro?

A partir de cierta edad, los perros pueden necesitar una limpieza dental anual o bianual. Depende de cada caso. En líneas generales, puede decirse que los perros de razas pequeñas suelen acumular más sarro y suelen necesitar una atención mayor en cuanto a higiene dental.

Riesgos de una mala higiene

Los riesgos más evidentes de una mala higiene dental en los perros son los siguientes:

Cuando la acumulación de sarro no se trata, se puede producir una inflamación y retracción de las encías que puede descalzar el diente y provocar caídas.
Mal aliento (halitosis).
Sarro perros
Puede ir a más
Las bacterias de la placa pueden trasladarse a través del torrente circulatorio a órganos vitales como el corazón ocasionando problemas de endocarditis en las válvulas. Las bacterias pueden incluso acantonarse en huesos (La osteomielitis es la infección ósea, tanto cortical como medular) provocando mucho dolor y una artritis séptica).

¿Cómo se forma el sarro?

El sarro es la calcificación de la placa dental. Los restos de alimentos, junto con las bacterias presentes en la boca, van a formar la placa bacteriana o placa dental. Si la placa no se retira, al mezclarse con la saliva y los minerales presentes en ella, reaccionará formando una costra. La placa se calcifica y se forma el sarro.

El sarro, cuando se forma, es de color blanquecino pero a medida que pasa el tiempo se va poniendo amarillo y luego marrón.

Síntomas de una pobre higiene dental
La señal más obvia de una mala salud dental canina es el mal aliento.

Sin embargo, a veces no es tan fácil de detectar
Y hay perros que no se dejan abrir la boca por su dueño. Por ejemplo…

Recientemente nos trajeron a la clínica a un perro que parpadeaba de un ojo y decía su dueño que le picaba un lado de la cara. Tenía molestias y dificultad para comer, lo que había llevado a sus dueños a comprarle comida blanda (que suele ser un poco más cara y llevar más contenido en grasa) durante medio año. Después de una exploración oftalmológica, nos dimos cuenta de que el ojo tenía una úlcera en la córnea probablemente de rascarse . Además, el canto lateral del ojo estaba inflamado. Tenía lo que en humanos llamamos flemón pero como era un perro de pelo largo, no se le notaba a simple vista. Al abrirle la boca nos llamó la atención el ver una muela llena de sarro. Le realizamos una radiografía y encontramos una fístula que llegaba hasta la parte inferior del ojo.

Le tuvimos que extraer la muela. Tras esto, el ojo se curó completamente con unos colirios y una lentilla protectora de úlcera. Afortunadamente, la úlcera no profundizó y no perforó el ojo. Ahora el perro come perfectamente a pesar de haber perdido una muela.

¿Cómo mantener la higiene dental de tu perro?
Hay varias maneras de prevenir problemas derivados de la salud dental de tu perro.

Limpiezas de dientes en casa
Es recomendable limpiar los dientes de tu perro semanal o diariamente si se puede. Existe una gran variedad de productos que se pueden utilizar:

Pastas de dientes.
Cepillos de dientes o dedales para el dedo índice, que hacen más fácil la limpieza.
Colutorios para echar en agua de bebida o directamente sobre el diente en líquido o en spray.

En la Clínica Tus Veterinarios enseñamos a nuestros clientes a tomar el hábito de limpiar los dientes de sus perros desde que son cachorros. Esto responde a nuestro compromiso con la prevención de enfermedades caninas.

Hoy en día tenemos muchos clientes que limpian los dientes todos los días a su mascota, y como resultado, se ahorran el dinero de hacer limpiezas dentales profesionales y consiguen una mejor salud de su perro.

Limpiezas dentales profesionales de perros y gatos

Recomendamos hacer una limpieza dental especializada anualmente. La realizamos con un aparato de ultrasonidos que utiliza agua para quitar el sarro. Después, procedemos a pulir los dientes con un cepillo de alta velocidad y una pasta especial. Hacemos esto para proteger el esmalte.

La frecuencia de limpiezas dentales necesaria varía mucho entre razas. En general, las razas grandes tienen buena calidad de esmalte, por lo que no necesitan hacerlo tan a menudo e incluso pueden pasarse la vida sin requerir una limpieza. Sin embargo, razas pequeñas como el Yorkshire o el Maltés, deben hacérselas todos los años desde cachorros si se quiere conservar sus piezas dentales.

Otro factor fundamental es la calidad del pienso. Algunas marcas han diseñado croquetas que limpian la superficie del diente y de la muela al masticarse.

Ultrasonido para perros

¿Se necesita anestesia para las limpiezas dentales de perros y gatos?

La limpieza dental en perros no es una técnica que pueda practicarse sin anestesia general , aunque hay veces que los propietarios no quieren anestesiar y si tiene poco sarro y el perro es muy bueno se puede intentar…… , pero no se va a poder pulir ni acceder a todas la zona de la boca …. Además los limpiadores dentales van a irrigar agua y hay riesgo de aspiración a vías respiratorias si no se realiza una anestesia correcta con intubación traqueal . En resumen , sin anestesia no se va hacer una correcta limpieza dental.

Tampoco sirve la sedación ya que necesitamos que el animal esté totalmente quieto, y el veterinario tenga un acceso completo a todas sus piezas dentales y encías.

Alimentos para la limpieza dental

Hay que tener cierto cuidado a la hora de comprar determinados alimentos porque no todos son saludables. Algunos tienen demasiado contenido graso, que en exceso puede causar problemas cardiovasculares y obesidad.

Los mejores alimentos para los dientes son aquellos que están elaborados por empresas farmacéuticas y llevan componentes químicos con tratamientos específicos para el diente del perro. Esto implica no solo limpieza a través de la acción mecánica de morder sino también un tratamiento antibacteriano para prevenir el sarro.

Conclusión

Si eres como la mayoría de dueños, por falta de tiempo , es probable que no estés prestando la suficiente atención a la limpieza dental de tu perro. Por eso te animamos a que comiences a limpiar los dientes de tu perro y consideres atender a su higiene bucal con frecuencia.

Estas simples medidas pueden conllevar a que tu perro tenga una vida más larga y mucho más saludable.

Si te resulta imposible introducir un cepillo de dientes a tu perro en la boca, pásate con él por clínica Tus Veterinarios y te explicamos cómo hacerlo.

Necesitas hacer una limpieza dental profesional a tu mascota?
Llámanos al 622575274 o contacta con nosotros

¿Cada cuánto tiempo tengo que hacerle una limpieza dental a mi perro?

Riesgos de una mala higiene

¿Cómo se forma el sarro?

Limpiezas dentales profesionales de perros y gatos

Ultrasonido para perros

Alimentos para la limpieza dental

Conclusión

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