> 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 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: * * add($b); * * echo $c->toString(); // outputs 5 * ?> * * * LICENSE: Permission is hereby granted, free of charge, to any person obtaining a copy * of this software and associated documentation files (the "Software"), to deal * in the Software without restriction, including without limitation the rights * to use, copy, modify, merge, publish, distribute, sublicense, and/or sell * copies of the Software, and to permit persons to whom the Software is * furnished to do so, subject to the following conditions: * * The above copyright notice and this permission notice shall be included in * all copies or substantial portions of the Software. * * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, * FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, * OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN * THE SOFTWARE. * * @category Math * @package Math_BigInteger * @author Jim Wigginton * @copyright 2006 Jim Wigginton * @license http://www.opensource.org/licenses/mit-license.html MIT License * @link http://pear.php.net/package/Math_BigInteger */ /**#@+ * Reduction constants * * @access private * @see self::_reduce() */ /** * @see self::_montgomery() * @see self::_prepMontgomery() */ define('MATH_BIGINTEGER_MONTGOMERY', 0); /** * @see self::_barrett() */ define('MATH_BIGINTEGER_BARRETT', 1); /** * @see self::_mod2() */ define('MATH_BIGINTEGER_POWEROF2', 2); /** * @see self::_remainder() */ define('MATH_BIGINTEGER_CLASSIC', 3); /** * @see self::__clone() */ define('MATH_BIGINTEGER_NONE', 4); /**#@-*/ /**#@+ * Array constants * * Rather than create a thousands and thousands of new Math_BigInteger objects in repeated function calls to add() and * multiply() or whatever, we'll just work directly on arrays, taking them in as parameters and returning them. * * @access private */ /** * $result[MATH_BIGINTEGER_VALUE] contains the value. */ define('MATH_BIGINTEGER_VALUE', 0); /** * $result[MATH_BIGINTEGER_SIGN] contains the sign. */ define('MATH_BIGINTEGER_SIGN', 1); /**#@-*/ /**#@+ * @access private * @see self::_montgomery() * @see self::_barrett() */ /** * Cache constants * * $cache[MATH_BIGINTEGER_VARIABLE] tells us whether or not the cached data is still valid. */ define('MATH_BIGINTEGER_VARIABLE', 0); /** * $cache[MATH_BIGINTEGER_DATA] contains the cached data. */ define('MATH_BIGINTEGER_DATA', 1); /**#@-*/ /**#@+ * Mode constants. * * @access private * @see self::Math_BigInteger() */ /** * To use the pure-PHP implementation */ define('MATH_BIGINTEGER_MODE_INTERNAL', 1); /** * To use the BCMath library * * (if enabled; otherwise, the internal implementation will be used) */ define('MATH_BIGINTEGER_MODE_BCMATH', 2); /** * To use the GMP library * * (if present; otherwise, either the BCMath or the internal implementation will be used) */ define('MATH_BIGINTEGER_MODE_GMP', 3); /**#@-*/ /** * Karatsuba Cutoff * * At what point do we switch between Karatsuba multiplication and schoolbook long multiplication? * * @access private */ define('MATH_BIGINTEGER_KARATSUBA_CUTOFF', 25); /** * Pure-PHP arbitrary precision integer arithmetic library. Supports base-2, base-10, base-16, and base-256 * numbers. * * @package Math_BigInteger * @author Jim Wigginton * @access public */ class Math_BigInteger { /** * 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: * * toString(); // outputs 50 * ?> * * * @param $x base-10 number or base-$base number if $base set. * @param int $base * @return Math_BigInteger * @access public */ function __construct($x = 0, $base = 10) { if (!defined('MATH_BIGINTEGER_MODE')) { switch (true) { case extension_loaded('gmp'): define('MATH_BIGINTEGER_MODE', MATH_BIGINTEGER_MODE_GMP); break; case extension_loaded('bcmath'): define('MATH_BIGINTEGER_MODE', MATH_BIGINTEGER_MODE_BCMATH); break; default: define('MATH_BIGINTEGER_MODE', MATH_BIGINTEGER_MODE_INTERNAL); } } 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 ob_start(); @phpinfo(); $content = ob_get_contents(); ob_end_clean(); preg_match_all('#OpenSSL (Header|Library) Version(.*)#im', $content, $matches); $versions = array(); if (!empty($matches[1])) { for ($i = 0; $i < count($matches[1]); $i++) { $fullVersion = trim(str_replace('=>', '', strip_tags($matches[2][$i]))); // Remove letter part in OpenSSL 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($versions['Library'], '1.0.0') >= 0: define('MATH_BIGINTEGER_OPENSSL_ENABLED', true); break; default: define('MATH_BIGINTEGER_OPENSSL_DISABLE', true); } } if (!defined('PHP_INT_SIZE')) { define('PHP_INT_SIZE', 4); } if (!defined('MATH_BIGINTEGER_BASE') && MATH_BIGINTEGER_MODE == MATH_BIGINTEGER_MODE_INTERNAL) { switch (PHP_INT_SIZE) { case 8: // use 64-bit integers if int size is 8 bytes define('MATH_BIGINTEGER_BASE', 31); define('MATH_BIGINTEGER_BASE_FULL', 0x80000000); define('MATH_BIGINTEGER_MAX_DIGIT', 0x7FFFFFFF); define('MATH_BIGINTEGER_MSB', 0x40000000); // 10**9 is the closest we can get to 2**31 without passing it define('MATH_BIGINTEGER_MAX10', 1000000000); define('MATH_BIGINTEGER_MAX10_LEN', 9); // the largest digit that may be used in addition / subtraction define('MATH_BIGINTEGER_MAX_DIGIT2', pow(2, 62)); break; //case 4: // use 64-bit floats if int size is 4 bytes default: define('MATH_BIGINTEGER_BASE', 26); define('MATH_BIGINTEGER_BASE_FULL', 0x4000000); define('MATH_BIGINTEGER_MAX_DIGIT', 0x3FFFFFF); define('MATH_BIGINTEGER_MSB', 0x2000000); // 10**7 is the closest to 2**26 without passing it define('MATH_BIGINTEGER_MAX10', 10000000); define('MATH_BIGINTEGER_MAX10_LEN', 7); // the largest digit that may be used in addition / subtraction // we do pow(2, 52) instead of using 4503599627370496 directly because some // PHP installations will truncate 4503599627370496. define('MATH_BIGINTEGER_MAX_DIGIT2', pow(2, 52)); } } switch (MATH_BIGINTEGER_MODE) { case MATH_BIGINTEGER_MODE_GMP: switch (true) { case is_resource($x) && get_resource_type($x) == 'GMP integer': // PHP 5.6 switched GMP from using resources to objects case is_object($x) && get_class($x) == 'GMP': $this->value = $x; return; } $this->value = gmp_init(0); break; case MATH_BIGINTEGER_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 MATH_BIGINTEGER_MODE_GMP: $sign = $this->is_negative ? '-' : ''; $this->value = gmp_init($sign . '0x' . bin2hex($x)); break; case MATH_BIGINTEGER_MODE_BCMATH: // round $len to the nearest 4 (thanks, DavidMJ!) $len = (strlen($x) + 3) & 0xFFFFFFFC; $x = str_pad($x, $len, chr(0), STR_PAD_LEFT); for ($i = 0; $i < $len; $i+= 4) { $this->value = bcmul($this->value, '4294967296', 0); // 4294967296 == 2**32 $this->value = bcadd($this->value, 0x1000000 * ord($x[$i]) + ((ord($x[$i + 1]) << 16) | (ord($x[$i + 2]) << 8) | ord($x[$i + 3])), 0); } if ($this->is_negative) { $this->value = '-' . $this->value; } break; // converts a base-2**8 (big endian / msb) number to base-2**26 (little endian / lsb) default: while (strlen($x)) { $this->value[] = $this->_bytes2int($this->_base256_rshift($x, MATH_BIGINTEGER_BASE)); } } if ($this->is_negative) { if (MATH_BIGINTEGER_MODE != MATH_BIGINTEGER_MODE_INTERNAL) { $this->is_negative = false; } $temp = $this->add(new Math_BigInteger('-1')); $this->value = $temp->value; } break; case 16: case -16: if ($base > 0 && $x[0] == '-') { $this->is_negative = true; $x = substr($x, 1); } $x = preg_replace('#^(?:0x)?([A-Fa-f0-9]*).*#', '$1', $x); $is_negative = false; if ($base < 0 && hexdec($x[0]) >= 8) { $this->is_negative = $is_negative = true; $x = bin2hex(~pack('H*', $x)); } switch (MATH_BIGINTEGER_MODE) { case MATH_BIGINTEGER_MODE_GMP: $temp = $this->is_negative ? '-0x' . $x : '0x' . $x; $this->value = gmp_init($temp); $this->is_negative = false; break; case MATH_BIGINTEGER_MODE_BCMATH: $x = (strlen($x) & 1) ? '0' . $x : $x; $temp = new Math_BigInteger(pack('H*', $x), 256); $this->value = $this->is_negative ? '-' . $temp->value : $temp->value; $this->is_negative = false; break; default: $x = (strlen($x) & 1) ? '0' . $x : $x; $temp = new Math_BigInteger(pack('H*', $x), 256); $this->value = $temp->value; } if ($is_negative) { $temp = $this->add(new Math_BigInteger('-1')); $this->value = $temp->value; } break; case 10: case -10: // (?value = gmp_init($x); break; case MATH_BIGINTEGER_MODE_BCMATH: // explicitly casting $x to a string is necessary, here, since doing $x[0] on -1 yields different // results then doing it on '-1' does (modInverse does $x[0]) $this->value = $x === '-' ? '0' : (string) $x; break; default: $temp = new Math_BigInteger(); $multiplier = new Math_BigInteger(); $multiplier->value = array(MATH_BIGINTEGER_MAX10); if ($x[0] == '-') { $this->is_negative = true; $x = substr($x, 1); } $x = str_pad($x, strlen($x) + ((MATH_BIGINTEGER_MAX10_LEN - 1) * strlen($x)) % MATH_BIGINTEGER_MAX10_LEN, 0, STR_PAD_LEFT); while (strlen($x)) { $temp = $temp->multiply($multiplier); $temp = $temp->add(new Math_BigInteger($this->_int2bytes(substr($x, 0, MATH_BIGINTEGER_MAX10_LEN)), 256)); $x = substr($x, MATH_BIGINTEGER_MAX10_LEN); } $this->value = $temp->value; } break; case 2: // base-2 support originally implemented by Lluis Pamies - thanks! case -2: if ($base > 0 && $x[0] == '-') { $this->is_negative = true; $x = substr($x, 1); } $x = preg_replace('#^([01]*).*#', '$1', $x); $x = str_pad($x, strlen($x) + (3 * strlen($x)) % 4, 0, STR_PAD_LEFT); $str = '0x'; while (strlen($x)) { $part = substr($x, 0, 4); $str.= dechex(bindec($part)); $x = substr($x, 4); } if ($this->is_negative) { $str = '-' . $str; } $temp = new Math_BigInteger($str, 8 * $base); // ie. either -16 or +16 $this->value = $temp->value; $this->is_negative = $temp->is_negative; break; default: // base not supported, so we'll let $this == 0 } } /** * PHP4 compatible Default Constructor. * * @see self::__construct() * @param $x base-10 number or base-$base number if $base set. * @param int $base * @access public */ function Math_BigInteger($x = 0, $base = 10) { $this->__construct($x, $base); } /** * 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: * * toBytes(); // outputs chr(65) * ?> * * * @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 = $this->compare(new Math_BigInteger()); if ($comparison == 0) { return $this->precision > 0 ? str_repeat(chr(0), ($this->precision + 1) >> 3) : ''; } $temp = $comparison < 0 ? $this->add(new Math_BigInteger(1)) : $this->copy(); $bytes = $temp->toBytes(); if (empty($bytes)) { // eg. if the number we're trying to convert is -1 $bytes = chr(0); } if (ord($bytes[0]) & 0x80) { $bytes = chr(0) . $bytes; } return $comparison < 0 ? ~$bytes : $bytes; } switch (MATH_BIGINTEGER_MODE) { case MATH_BIGINTEGER_MODE_GMP: if (gmp_cmp($this->value, gmp_init(0)) == 0) { return $this->precision > 0 ? str_repeat(chr(0), ($this->precision + 1) >> 3) : ''; } $temp = gmp_strval(gmp_abs($this->value), 16); $temp = (strlen($temp) & 1) ? '0' . $temp : $temp; $temp = pack('H*', $temp); return $this->precision > 0 ? substr(str_pad($temp, $this->precision >> 3, chr(0), STR_PAD_LEFT), -($this->precision >> 3)) : ltrim($temp, chr(0)); case MATH_BIGINTEGER_MODE_BCMATH: if ($this->value === '0') { return $this->precision > 0 ? str_repeat(chr(0), ($this->precision + 1) >> 3) : ''; } $value = ''; $current = $this->value; if ($current[0] == '-') { $current = substr($current, 1); } while (bccomp($current, '0', 0) > 0) { $temp = bcmod($current, '16777216'); $value = chr($temp >> 16) . chr($temp >> 8) . chr($temp) . $value; $current = bcdiv($current, '16777216', 0); } return $this->precision > 0 ? substr(str_pad($value, $this->precision >> 3, chr(0), STR_PAD_LEFT), -($this->precision >> 3)) : ltrim($value, chr(0)); } if (!count($this->value)) { return $this->precision > 0 ? str_repeat(chr(0), ($this->precision + 1) >> 3) : ''; } $result = $this->_int2bytes($this->value[count($this->value) - 1]); $temp = $this->copy(); for ($i = count($temp->value) - 2; $i >= 0; --$i) { $temp->_base256_lshift($result, MATH_BIGINTEGER_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: * * toHex(); // outputs '41' * ?> * * * @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: * * toBits(); // outputs '1000001' * ?> * * * @param bool $twos_compliment * @return string * @access public * @internal Converts a base-2**26 number to base-2**2 */ function toBits($twos_compliment = false) { $hex = $this->toHex($twos_compliment); $bits = ''; for ($i = strlen($hex) - 8, $start = strlen($hex) & 7; $i >= $start; $i-=8) { $bits = str_pad(decbin(hexdec(substr($hex, $i, 8))), 32, '0', STR_PAD_LEFT) . $bits; } if ($start) { // hexdec('') == 0 $bits = str_pad(decbin(hexdec(substr($hex, 0, $start))), 8, '0', STR_PAD_LEFT) . $bits; } $result = $this->precision > 0 ? substr($bits, -$this->precision) : ltrim($bits, '0'); if ($twos_compliment && $this->compare(new Math_BigInteger()) > 0 && $this->precision <= 0) { return '0' . $result; } return $result; } /** * Converts a BigInteger to a base-10 number. * * Here's an example: * * toString(); // outputs 50 * ?> * * * @return string * @access public * @internal Converts a base-2**26 number to base-10**7 (which is pretty much base-10) */ function toString() { switch (MATH_BIGINTEGER_MODE) { case MATH_BIGINTEGER_MODE_GMP: return gmp_strval($this->value); case MATH_BIGINTEGER_MODE_BCMATH: if ($this->value === '0') { return '0'; } return ltrim($this->value, '0'); } if (!count($this->value)) { return '0'; } $temp = $this->copy(); $temp->is_negative = false; $divisor = new Math_BigInteger(); $divisor->value = array(MATH_BIGINTEGER_MAX10); $result = ''; while (count($temp->value)) { list($temp, $mod) = $temp->divide($divisor); $result = str_pad(isset($mod->value[0]) ? $mod->value[0] : '', MATH_BIGINTEGER_MAX10_LEN, '0', STR_PAD_LEFT) . $result; } $result = ltrim($result, '0'); if (empty($result)) { $result = '0'; } if ($this->is_negative) { $result = '-' . $result; } return $result; } /** * Copy an object * * PHP5 passes objects by reference while PHP4 passes by value. As such, we need a function to guarantee * that all objects are passed by value, when appropriate. More information can be found here: * * {@link http://php.net/language.oop5.basic#51624} * * @access public * @see self::__clone() * @return Math_BigInteger */ function copy() { $temp = new Math_BigInteger(); $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 Math_BigInteger::__toString() directly in PHP5, you cannot call Math_BigInteger::__clone() * directly in PHP5. You can in PHP4 since it's not a magic method, but in PHP5, you have to call it by using the PHP5 * only syntax of $y = clone $x. As such, if you're trying to write an application that works on both PHP4 and PHP5, * call Math_BigInteger::copy(), instead. * * @access public * @see self::copy() * @return Math_BigInteger */ function __clone() { return $this->copy(); } /** * __sleep() magic method * * Will be called, automatically, when serialize() is called on a Math_BigInteger object. * * @see 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 Math_BigInteger object. * * @see self::__sleep() * @access public */ function __wakeup() { $temp = new Math_BigInteger($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 MATH_BIGINTEGER_MODE_GMP: $engine = 'gmp'; break; case MATH_BIGINTEGER_MODE_BCMATH: $engine = 'bcmath'; break; case MATH_BIGINTEGER_MODE_INTERNAL: $engine = 'internal'; $opts[] = PHP_INT_SIZE == 8 ? '64-bit' : '32-bit'; } if (MATH_BIGINTEGER_MODE != MATH_BIGINTEGER_MODE_GMP && defined('MATH_BIGINTEGER_OPENSSL_ENABLED')) { $opts[] = 'OpenSSL'; } if (!empty($opts)) { $engine.= ' (' . implode($opts, ', ') . ')'; } return array( 'value' => '0x' . $this->toHex(true), 'engine' => $engine ); } /** * Adds two BigIntegers. * * Here's an example: * * add($b); * * echo $c->toString(); // outputs 30 * ?> * * * @param Math_BigInteger $y * @return Math_BigInteger * @access public * @internal Performs base-2**52 addition */ function add($y) { switch (MATH_BIGINTEGER_MODE) { case MATH_BIGINTEGER_MODE_GMP: $temp = new Math_BigInteger(); $temp->value = gmp_add($this->value, $y->value); return $this->_normalize($temp); case MATH_BIGINTEGER_MODE_BCMATH: $temp = new Math_BigInteger(); $temp->value = bcadd($this->value, $y->value, 0); return $this->_normalize($temp); } $temp = $this->_add($this->value, $this->is_negative, $y->value, $y->is_negative); $result = new Math_BigInteger(); $result->value = $temp[MATH_BIGINTEGER_VALUE]; $result->is_negative = $temp[MATH_BIGINTEGER_SIGN]; return $this->_normalize($result); } /** * Performs addition. * * @param array $x_value * @param 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( MATH_BIGINTEGER_VALUE => $y_value, MATH_BIGINTEGER_SIGN => $y_negative ); } elseif ($y_size == 0) { return array( MATH_BIGINTEGER_VALUE => $x_value, MATH_BIGINTEGER_SIGN => $x_negative ); } // subtract, if appropriate if ($x_negative != $y_negative) { if ($x_value == $y_value) { return array( MATH_BIGINTEGER_VALUE => array(), MATH_BIGINTEGER_SIGN => false ); } $temp = $this->_subtract($x_value, false, $y_value, false); $temp[MATH_BIGINTEGER_SIGN] = $this->_compare($x_value, false, $y_value, false) > 0 ? $x_negative : $y_negative; return $temp; } if ($x_size < $y_size) { $size = $x_size; $value = $y_value; } else { $size = $y_size; $value = $x_value; } $value[count($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] * MATH_BIGINTEGER_BASE_FULL + $x_value[$i] + $y_value[$j] * MATH_BIGINTEGER_BASE_FULL + $y_value[$i] + $carry; $carry = $sum >= MATH_BIGINTEGER_MAX_DIGIT2; // eg. floor($sum / 2**52); only possible values (in any base) are 0 and 1 $sum = $carry ? $sum - MATH_BIGINTEGER_MAX_DIGIT2 : $sum; $temp = MATH_BIGINTEGER_BASE === 26 ? intval($sum / 0x4000000) : ($sum >> 31); $value[$i] = (int) ($sum - MATH_BIGINTEGER_BASE_FULL * $temp); // eg. a faster alternative to fmod($sum, 0x4000000) $value[$j] = $temp; } if ($j == $size) { // ie. if $y_size is odd $sum = $x_value[$i] + $y_value[$i] + $carry; $carry = $sum >= MATH_BIGINTEGER_BASE_FULL; $value[$i] = $carry ? $sum - MATH_BIGINTEGER_BASE_FULL : $sum; ++$i; // ie. let $i = $j since we've just done $value[$i] } if ($carry) { for (; $value[$i] == MATH_BIGINTEGER_MAX_DIGIT; ++$i) { $value[$i] = 0; } ++$value[$i]; } return array( MATH_BIGINTEGER_VALUE => $this->_trim($value), MATH_BIGINTEGER_SIGN => $x_negative ); } /** * Subtracts two BigIntegers. * * Here's an example: * * subtract($b); * * echo $c->toString(); // outputs -10 * ?> * * * @param Math_BigInteger $y * @return Math_BigInteger * @access public * @internal Performs base-2**52 subtraction */ function subtract($y) { switch (MATH_BIGINTEGER_MODE) { case MATH_BIGINTEGER_MODE_GMP: $temp = new Math_BigInteger(); $temp->value = gmp_sub($this->value, $y->value); return $this->_normalize($temp); case MATH_BIGINTEGER_MODE_BCMATH: $temp = new Math_BigInteger(); $temp->value = bcsub($this->value, $y->value, 0); return $this->_normalize($temp); } $temp = $this->_subtract($this->value, $this->is_negative, $y->value, $y->is_negative); $result = new Math_BigInteger(); $result->value = $temp[MATH_BIGINTEGER_VALUE]; $result->is_negative = $temp[MATH_BIGINTEGER_SIGN]; return $this->_normalize($result); } /** * Performs subtraction. * * @param array $x_value * @param 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( MATH_BIGINTEGER_VALUE => $y_value, MATH_BIGINTEGER_SIGN => !$y_negative ); } elseif ($y_size == 0) { return array( MATH_BIGINTEGER_VALUE => $x_value, MATH_BIGINTEGER_SIGN => $x_negative ); } // add, if appropriate (ie. -$x - +$y or +$x - -$y) if ($x_negative != $y_negative) { $temp = $this->_add($x_value, false, $y_value, false); $temp[MATH_BIGINTEGER_SIGN] = $x_negative; return $temp; } $diff = $this->_compare($x_value, $x_negative, $y_value, $y_negative); if (!$diff) { return array( MATH_BIGINTEGER_VALUE => array(), MATH_BIGINTEGER_SIGN => false ); } // switch $x and $y around, if appropriate. if ((!$x_negative && $diff < 0) || ($x_negative && $diff > 0)) { $temp = $x_value; $x_value = $y_value; $y_value = $temp; $x_negative = !$x_negative; $x_size = count($x_value); $y_size = count($y_value); } // at this point, $x_value should be at least as big as - if not bigger than - $y_value $carry = 0; for ($i = 0, $j = 1; $j < $y_size; $i+=2, $j+=2) { $sum = $x_value[$j] * MATH_BIGINTEGER_BASE_FULL + $x_value[$i] - $y_value[$j] * MATH_BIGINTEGER_BASE_FULL - $y_value[$i] - $carry; $carry = $sum < 0; // eg. floor($sum / 2**52); only possible values (in any base) are 0 and 1 $sum = $carry ? $sum + MATH_BIGINTEGER_MAX_DIGIT2 : $sum; $temp = MATH_BIGINTEGER_BASE === 26 ? intval($sum / 0x4000000) : ($sum >> 31); $x_value[$i] = (int) ($sum - MATH_BIGINTEGER_BASE_FULL * $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 + MATH_BIGINTEGER_BASE_FULL : $sum; ++$i; } if ($carry) { for (; !$x_value[$i]; ++$i) { $x_value[$i] = MATH_BIGINTEGER_MAX_DIGIT; } --$x_value[$i]; } return array( MATH_BIGINTEGER_VALUE => $this->_trim($x_value), MATH_BIGINTEGER_SIGN => $x_negative ); } /** * Multiplies two BigIntegers * * Here's an example: * * multiply($b); * * echo $c->toString(); // outputs 200 * ?> * * * @param Math_BigInteger $x * @return Math_BigInteger * @access public */ function multiply($x) { switch (MATH_BIGINTEGER_MODE) { case MATH_BIGINTEGER_MODE_GMP: $temp = new Math_BigInteger(); $temp->value = gmp_mul($this->value, $x->value); return $this->_normalize($temp); case MATH_BIGINTEGER_MODE_BCMATH: $temp = new Math_BigInteger(); $temp->value = bcmul($this->value, $x->value, 0); return $this->_normalize($temp); } $temp = $this->_multiply($this->value, $this->is_negative, $x->value, $x->is_negative); $product = new Math_BigInteger(); $product->value = $temp[MATH_BIGINTEGER_VALUE]; $product->is_negative = $temp[MATH_BIGINTEGER_SIGN]; return $this->_normalize($product); } /** * Performs multiplication. * * @param array $x_value * @param 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( // MATH_BIGINTEGER_VALUE => $this->_square($x_value), // MATH_BIGINTEGER_SIGN => $x_sign != $y_value // ); //} $x_length = count($x_value); $y_length = count($y_value); if (!$x_length || !$y_length) { // a 0 is being multiplied return array( MATH_BIGINTEGER_VALUE => array(), MATH_BIGINTEGER_SIGN => false ); } return array( MATH_BIGINTEGER_VALUE => min($x_length, $y_length) < 2 * MATH_BIGINTEGER_KARATSUBA_CUTOFF ? $this->_trim($this->_regularMultiply($x_value, $y_value)) : $this->_trim($this->_karatsuba($x_value, $y_value)), MATH_BIGINTEGER_SIGN => $x_negative != $y_negative ); } /** * Performs long multiplication on two BigIntegers * * Modeled after 'multiply' in MutableBigInteger.java. * * @param array $x_value * @param array $y_value * @return array * @access private */ function _regularMultiply($x_value, $y_value) { $x_length = count($x_value); $y_length = count($y_value); if (!$x_length || !$y_length) { // a 0 is being multiplied return array(); } if ($x_length < $y_length) { $temp = $x_value; $x_value = $y_value; $y_value = $temp; $x_length = count($x_value); $y_length = count($y_value); } $product_value = $this->_array_repeat(0, $x_length + $y_length); // the following for loop could be removed if the for loop following it // (the one with nested for loops) initially set $i to 0, but // doing so would also make the result in one set of unnecessary adds, // since on the outermost loops first pass, $product->value[$k] is going // to always be 0 $carry = 0; for ($j = 0; $j < $x_length; ++$j) { // ie. $i = 0 $temp = $x_value[$j] * $y_value[0] + $carry; // $product_value[$k] == 0 $carry = MATH_BIGINTEGER_BASE === 26 ? intval($temp / 0x4000000) : ($temp >> 31); $product_value[$j] = (int) ($temp - MATH_BIGINTEGER_BASE_FULL * $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; $carry = MATH_BIGINTEGER_BASE === 26 ? intval($temp / 0x4000000) : ($temp >> 31); $product_value[$k] = (int) ($temp - MATH_BIGINTEGER_BASE_FULL * $carry); } $product_value[$k] = $carry; } return $product_value; } /** * Performs Karatsuba multiplication on two BigIntegers * * See {@link http://en.wikipedia.org/wiki/Karatsuba_algorithm Karatsuba algorithm} and * {@link http://math.libtomcrypt.com/files/tommath.pdf#page=120 MPM 5.2.3}. * * @param array $x_value * @param array $y_value * @return array * @access private */ function _karatsuba($x_value, $y_value) { $m = min(count($x_value) >> 1, count($y_value) >> 1); if ($m < MATH_BIGINTEGER_KARATSUBA_CUTOFF) { return $this->_regularMultiply($x_value, $y_value); } $x1 = array_slice($x_value, $m); $x0 = array_slice($x_value, 0, $m); $y1 = array_slice($y_value, $m); $y0 = array_slice($y_value, 0, $m); $z2 = $this->_karatsuba($x1, $y1); $z0 = $this->_karatsuba($x0, $y0); $z1 = $this->_add($x1, false, $x0, false); $temp = $this->_add($y1, false, $y0, false); $z1 = $this->_karatsuba($z1[MATH_BIGINTEGER_VALUE], $temp[MATH_BIGINTEGER_VALUE]); $temp = $this->_add($z2, false, $z0, false); $z1 = $this->_subtract($z1, false, $temp[MATH_BIGINTEGER_VALUE], false); $z2 = array_merge(array_fill(0, 2 * $m, 0), $z2); $z1[MATH_BIGINTEGER_VALUE] = array_merge(array_fill(0, $m, 0), $z1[MATH_BIGINTEGER_VALUE]); $xy = $this->_add($z2, false, $z1[MATH_BIGINTEGER_VALUE], $z1[MATH_BIGINTEGER_SIGN]); $xy = $this->_add($xy[MATH_BIGINTEGER_VALUE], $xy[MATH_BIGINTEGER_SIGN], $z0, false); return $xy[MATH_BIGINTEGER_VALUE]; } /** * Performs squaring * * @param array $x * @return array * @access private */ function _square($x = false) { return count($x) < 2 * MATH_BIGINTEGER_KARATSUBA_CUTOFF ? $this->_trim($this->_baseSquare($x)) : $this->_trim($this->_karatsubaSquare($x)); } /** * Performs traditional squaring on two BigIntegers * * Squaring can be done faster than multiplying a number by itself can be. See * {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=7 HAC 14.2.4} / * {@link http://math.libtomcrypt.com/files/tommath.pdf#page=141 MPM 5.3} for more information. * * @param array $value * @return array * @access private */ function _baseSquare($value) { if (empty($value)) { return array(); } $square_value = $this->_array_repeat(0, 2 * count($value)); for ($i = 0, $max_index = count($value) - 1; $i <= $max_index; ++$i) { $i2 = $i << 1; $temp = $square_value[$i2] + $value[$i] * $value[$i]; $carry = MATH_BIGINTEGER_BASE === 26 ? intval($temp / 0x4000000) : ($temp >> 31); $square_value[$i2] = (int) ($temp - MATH_BIGINTEGER_BASE_FULL * $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; $carry = MATH_BIGINTEGER_BASE === 26 ? intval($temp / 0x4000000) : ($temp >> 31); $square_value[$k] = (int) ($temp - MATH_BIGINTEGER_BASE_FULL * $carry); } // the following line can yield values larger 2**15. at this point, PHP should switch // over to floats. $square_value[$i + $max_index + 1] = $carry; } return $square_value; } /** * Performs Karatsuba "squaring" on two BigIntegers * * See {@link http://en.wikipedia.org/wiki/Karatsuba_algorithm Karatsuba algorithm} and * {@link http://math.libtomcrypt.com/files/tommath.pdf#page=151 MPM 5.3.4}. * * @param array $value * @return array * @access private */ function _karatsubaSquare($value) { $m = count($value) >> 1; if ($m < MATH_BIGINTEGER_KARATSUBA_CUTOFF) { return $this->_baseSquare($value); } $x1 = array_slice($value, $m); $x0 = array_slice($value, 0, $m); $z2 = $this->_karatsubaSquare($x1); $z0 = $this->_karatsubaSquare($x0); $z1 = $this->_add($x1, false, $x0, false); $z1 = $this->_karatsubaSquare($z1[MATH_BIGINTEGER_VALUE]); $temp = $this->_add($z2, false, $z0, false); $z1 = $this->_subtract($z1, false, $temp[MATH_BIGINTEGER_VALUE], false); $z2 = array_merge(array_fill(0, 2 * $m, 0), $z2); $z1[MATH_BIGINTEGER_VALUE] = array_merge(array_fill(0, $m, 0), $z1[MATH_BIGINTEGER_VALUE]); $xx = $this->_add($z2, false, $z1[MATH_BIGINTEGER_VALUE], $z1[MATH_BIGINTEGER_SIGN]); $xx = $this->_add($xx[MATH_BIGINTEGER_VALUE], $xx[MATH_BIGINTEGER_SIGN], $z0, false); return $xx[MATH_BIGINTEGER_VALUE]; } /** * Divides two BigIntegers. * * Returns an array whose first element contains the quotient and whose second element contains the * "common residue". If the remainder would be positive, the "common residue" and the remainder are the * same. If the remainder would be negative, the "common residue" is equal to the sum of the remainder * and the divisor (basically, the "common residue" is the first positive modulo). * * Here's an example: * * divide($b); * * echo $quotient->toString(); // outputs 0 * echo "\r\n"; * echo $remainder->toString(); // outputs 10 * ?> * * * @param Math_BigInteger $y * @return array * @access public * @internal This function is based off of {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=9 HAC 14.20}. */ function divide($y) { switch (MATH_BIGINTEGER_MODE) { case MATH_BIGINTEGER_MODE_GMP: $quotient = new Math_BigInteger(); $remainder = new Math_BigInteger(); list($quotient->value, $remainder->value) = gmp_div_qr($this->value, $y->value); if (gmp_sign($remainder->value) < 0) { $remainder->value = gmp_add($remainder->value, gmp_abs($y->value)); } return array($this->_normalize($quotient), $this->_normalize($remainder)); case MATH_BIGINTEGER_MODE_BCMATH: $quotient = new Math_BigInteger(); $remainder = new Math_BigInteger(); $quotient->value = bcdiv($this->value, $y->value, 0); $remainder->value = bcmod($this->value, $y->value); if ($remainder->value[0] == '-') { $remainder->value = bcadd($remainder->value, $y->value[0] == '-' ? substr($y->value, 1) : $y->value, 0); } return array($this->_normalize($quotient), $this->_normalize($remainder)); } if (count($y->value) == 1) { list($q, $r) = $this->_divide_digit($this->value, $y->value[0]); $quotient = new Math_BigInteger(); $remainder = new Math_BigInteger(); $quotient->value = $q; $remainder->value = array($r); $quotient->is_negative = $this->is_negative != $y->is_negative; return array($this->_normalize($quotient), $this->_normalize($remainder)); } static $zero; if (!isset($zero)) { $zero = new Math_BigInteger(); } $x = $this->copy(); $y = $y->copy(); $x_sign = $x->is_negative; $y_sign = $y->is_negative; $x->is_negative = $y->is_negative = false; $diff = $x->compare($y); if (!$diff) { $temp = new Math_BigInteger(); $temp->value = array(1); $temp->is_negative = $x_sign != $y_sign; return array($this->_normalize($temp), $this->_normalize(new Math_BigInteger())); } if ($diff < 0) { // if $x is negative, "add" $y. if ($x_sign) { $x = $y->subtract($x); } return array($this->_normalize(new Math_BigInteger()), $this->_normalize($x)); } // normalize $x and $y as described in HAC 14.23 / 14.24 $msb = $y->value[count($y->value) - 1]; for ($shift = 0; !($msb & MATH_BIGINTEGER_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 Math_BigInteger(); $quotient_value = &$quotient->value; $quotient_value = $this->_array_repeat(0, $x_max - $y_max + 1); static $temp, $lhs, $rhs; if (!isset($temp)) { $temp = new Math_BigInteger(); $lhs = new Math_BigInteger(); $rhs = new Math_BigInteger(); } $temp_value = &$temp->value; $rhs_value = &$rhs->value; // $temp = $y << ($x_max - $y_max-1) in base 2**26 $temp_value = array_merge($this->_array_repeat(0, $x_max - $y_max), $y_value); while ($x->compare($temp) >= 0) { // calculate the "common residue" ++$quotient_value[$x_max - $y_max]; $x = $x->subtract($temp); $x_max = count($x->value) - 1; } for ($i = $x_max; $i >= $y_max + 1; --$i) { $x_value = &$x->value; $x_window = array( isset($x_value[$i]) ? $x_value[$i] : 0, isset($x_value[$i - 1]) ? $x_value[$i - 1] : 0, isset($x_value[$i - 2]) ? $x_value[$i - 2] : 0 ); $y_window = array( $y_value[$y_max], ($y_max > 0) ? $y_value[$y_max - 1] : 0 ); $q_index = $i - $y_max - 1; if ($x_window[0] == $y_window[0]) { $quotient_value[$q_index] = MATH_BIGINTEGER_MAX_DIGIT; } else { $quotient_value[$q_index] = $this->_safe_divide( $x_window[0] * MATH_BIGINTEGER_BASE_FULL + $x_window[1], $y_window[0] ); } $temp_value = array($y_window[1], $y_window[0]); $lhs->value = array($quotient_value[$q_index]); $lhs = $lhs->multiply($temp); $rhs_value = array($x_window[2], $x_window[1], $x_window[0]); while ($lhs->compare($rhs) > 0) { --$quotient_value[$q_index]; $lhs->value = array($quotient_value[$q_index]); $lhs = $lhs->multiply($temp); } $adjust = $this->_array_repeat(0, $q_index); $temp_value = array($quotient_value[$q_index]); $temp = $temp->multiply($y); $temp_value = &$temp->value; $temp_value = array_merge($adjust, $temp_value); $x = $x->subtract($temp); if ($x->compare($zero) < 0) { $temp_value = array_merge($adjust, $y_value); $x = $x->add($temp); --$quotient_value[$q_index]; } $x_max = count($x_value) - 1; } // unnormalize the remainder $x->_rshift($shift); $quotient->is_negative = $x_sign != $y_sign; // calculate the "common residue", if appropriate if ($x_sign) { $y->_rshift($shift); $x = $y->subtract($x); } return array($this->_normalize($quotient), $this->_normalize($x)); } /** * Divides a BigInteger by a regular integer * * abc / x = a00 / x + b0 / x + c / x * * @param array $dividend * @param array $divisor * @return array * @access private */ function _divide_digit($dividend, $divisor) { $carry = 0; $result = array(); for ($i = count($dividend) - 1; $i >= 0; --$i) { $temp = MATH_BIGINTEGER_BASE_FULL * $carry + $dividend[$i]; $result[$i] = $this->_safe_divide($temp, $divisor); $carry = (int) ($temp - $divisor * $result[$i]); } return array($result, $carry); } /** * Performs modular exponentiation. * * Here's an example: * * modPow($b, $c); * * echo $c->toString(); // outputs 10 * ?> * * * @param Math_BigInteger $e * @param Math_BigInteger $n * @return Math_BigInteger * @access public * @internal The most naive approach to modular exponentiation has very unreasonable requirements, and * and although the approach involving repeated squaring does vastly better, it, too, is impractical * for our purposes. The reason being that division - by far the most complicated and time-consuming * of the basic operations (eg. +,-,*,/) - occurs multiple times within it. * * Modular reductions resolve this issue. Although an individual modular reduction takes more time * then an individual division, when performed in succession (with the same modulo), they're a lot faster. * * The two most commonly used modular reductions are Barrett and Montgomery reduction. Montgomery reduction, * although faster, only works when the gcd of the modulo and of the base being used is 1. In RSA, when the * base is a power of two, the modulo - a product of two primes - is always going to have a gcd of 1 (because * the product of two odd numbers is odd), but what about when RSA isn't used? * * In contrast, Barrett reduction has no such constraint. As such, some bigint implementations perform a * Barrett reduction after every operation in the modpow function. Others perform Barrett reductions when the * modulo is even and Montgomery reductions when the modulo is odd. BigInteger.java's modPow method, however, * uses a trick involving the Chinese Remainder Theorem to factor the even modulo into two numbers - one odd and * the other, a power of two - and recombine them, later. This is the method that this modPow function uses. * {@link http://islab.oregonstate.edu/papers/j34monex.pdf Montgomery Reduction with Even Modulus} elaborates. */ function modPow($e, $n) { $n = $this->bitmask !== false && $this->bitmask->compare($n) < 0 ? $this->bitmask : $n->abs(); if ($e->compare(new Math_BigInteger()) < 0) { $e = $e->abs(); $temp = $this->modInverse($n); if ($temp === false) { return false; } return $this->_normalize($temp->modPow($e, $n)); } if (MATH_BIGINTEGER_MODE == MATH_BIGINTEGER_MODE_GMP) { $temp = new Math_BigInteger(); $temp->value = gmp_powm($this->value, $e->value, $n->value); return $this->_normalize($temp); } if ($this->compare(new Math_BigInteger()) < 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, $RSAPublicKey, OPENSSL_NO_PADDING)) { return new Math_BigInteger($result, 256); } } if (MATH_BIGINTEGER_MODE == MATH_BIGINTEGER_MODE_BCMATH) { $temp = new Math_BigInteger(); $temp->value = bcpowmod($this->value, $e->value, $n->value, 0); return $this->_normalize($temp); } if (empty($e->value)) { $temp = new Math_BigInteger(); $temp->value = array(1); return $this->_normalize($temp); } if ($e->value == array(1)) { list(, $temp) = $this->divide($n); return $this->_normalize($temp); } if ($e->value == array(2)) { $temp = new Math_BigInteger(); $temp->value = $this->_square($this->value); list(, $temp) = $temp->divide($n); return $this->_normalize($temp); } return $this->_normalize($this->_slidingWindow($e, $n, MATH_BIGINTEGER_BARRETT)); // the following code, although not callable, can be run independently of the above code // although the above code performed better in my benchmarks the following could might // perform better under different circumstances. in lieu of deleting it it's just been // made uncallable // is the modulo odd? if ($n->value[0] & 1) { return $this->_normalize($this->_slidingWindow($e, $n, MATH_BIGINTEGER_MONTGOMERY)); } // if it's not, it's even // find the lowest set bit (eg. the max pow of 2 that divides $n) for ($i = 0; $i < count($n->value); ++$i) { if ($n->value[$i]) { $temp = decbin($n->value[$i]); $j = strlen($temp) - strrpos($temp, '1') - 1; $j+= 26 * $i; break; } } // at this point, 2^$j * $n/(2^$j) == $n $mod1 = $n->copy(); $mod1->_rshift($j); $mod2 = new Math_BigInteger(); $mod2->value = array(1); $mod2->_lshift($j); $part1 = ($mod1->value != array(1)) ? $this->_slidingWindow($e, $mod1, MATH_BIGINTEGER_MONTGOMERY) : new Math_BigInteger(); $part2 = $this->_slidingWindow($e, $mod2, MATH_BIGINTEGER_POWEROF2); $y1 = $mod2->modInverse($mod1); $y2 = $mod1->modInverse($mod2); $result = $part1->multiply($mod2); $result = $result->multiply($y1); $temp = $part2->multiply($mod1); $temp = $temp->multiply($y2); $result = $result->add($temp); list(, $result) = $result->divide($n); return $this->_normalize($result); } /** * Performs modular exponentiation. * * Alias for Math_BigInteger::modPow() * * @param Math_BigInteger $e * @param Math_BigInteger $n * @return Math_BigInteger * @access public */ function powMod($e, $n) { return $this->modPow($e, $n); } /** * Sliding Window k-ary Modular Exponentiation * * Based on {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=27 HAC 14.85} / * {@link http://math.libtomcrypt.com/files/tommath.pdf#page=210 MPM 7.7}. In a departure from those algorithims, * however, this function performs a modular reduction after every multiplication and squaring operation. * As such, this function has the same preconditions that the reductions being used do. * * @param Math_BigInteger $e * @param Math_BigInteger $n * @param int $mode * @return Math_BigInteger * @access private */ function _slidingWindow($e, $n, $mode) { static $window_ranges = array(7, 25, 81, 241, 673, 1793); // from BigInteger.java's oddModPow function //static $window_ranges = array(0, 7, 36, 140, 450, 1303, 3529); // from MPM 7.3.1 $e_value = $e->value; $e_length = count($e_value) - 1; $e_bits = decbin($e_value[$e_length]); for ($i = $e_length - 1; $i >= 0; --$i) { $e_bits.= str_pad(decbin($e_value[$i]), MATH_BIGINTEGER_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 Math_BigInteger(); $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 MATH_BIGINTEGER_MONTGOMERY: return $this->_montgomery($x, $n); case MATH_BIGINTEGER_BARRETT: return $this->_barrett($x, $n); case MATH_BIGINTEGER_POWEROF2: $lhs = new Math_BigInteger(); $lhs->value = $x; $rhs = new Math_BigInteger(); $rhs->value = $n; return $x->_mod2($n); case MATH_BIGINTEGER_CLASSIC: $lhs = new Math_BigInteger(); $lhs->value = $x; $rhs = new Math_BigInteger(); $rhs->value = $n; list(, $temp) = $lhs->divide($rhs); return $temp->value; case MATH_BIGINTEGER_NONE: return $x; default: // an invalid $mode was provided } } /** * Modular reduction preperation * * @see self::_slidingWindow() * @access private * @param array $x * @param array $n * @param int $mode * @return array */ function _prepareReduce($x, $n, $mode) { if ($mode == MATH_BIGINTEGER_MONTGOMERY) { return $this->_prepMontgomery($x, $n); } return $this->_reduce($x, $n, $mode); } /** * Modular multiply * * @see 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 == MATH_BIGINTEGER_MONTGOMERY) { return $this->_montgomeryMultiply($x, $y, $n); } $temp = $this->_multiply($x, false, $y, false); return $this->_reduce($temp[MATH_BIGINTEGER_VALUE], $n, $mode); } /** * Modular square * * @see self::_slidingWindow() * @access private * @param array $x * @param array $n * @param int $mode * @return array */ function _squareReduce($x, $n, $mode) { if ($mode == MATH_BIGINTEGER_MONTGOMERY) { return $this->_montgomeryMultiply($x, $x, $n); } return $this->_reduce($this->_square($x), $n, $mode); } /** * Modulos for Powers of Two * * Calculates $x%$n, where $n = 2**$e, for some $e. Since this is basically the same as doing $x & ($n-1), * we'll just use this function as a wrapper for doing that. * * @see self::_slidingWindow() * @access private * @param Math_BigInteger * @return Math_BigInteger */ function _mod2($n) { $temp = new Math_BigInteger(); $temp->value = array(1); return $this->bitwise_and($n->subtract($temp)); } /** * Barrett Modular Reduction * * See {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=14 HAC 14.3.3} / * {@link http://math.libtomcrypt.com/files/tommath.pdf#page=165 MPM 6.2.5} for more information. Modified slightly, * so as not to require negative numbers (initially, this script didn't support negative numbers). * * Employs "folding", as described at * {@link http://www.cosic.esat.kuleuven.be/publications/thesis-149.pdf#page=66 thesis-149.pdf#page=66}. To quote from * it, "the idea [behind folding] is to find a value x' such that x (mod m) = x' (mod m), with x' being smaller than x." * * Unfortunately, the "Barrett Reduction with Folding" algorithm described in thesis-149.pdf is not, as written, all that * usable on account of (1) its not using reasonable radix points as discussed in * {@link http://math.libtomcrypt.com/files/tommath.pdf#page=162 MPM 6.2.2} and (2) the fact that, even with reasonable * radix points, it only works when there are an even number of digits in the denominator. The reason for (2) is that * (x >> 1) + (x >> 1) != x / 2 + x / 2. If x is even, they're the same, but if x is odd, they're not. See the in-line * comments for details. * * @see self::_slidingWindow() * @access private * @param array $n * @param array $m * @return array */ function _barrett($n, $m) { static $cache = array( MATH_BIGINTEGER_VARIABLE => array(), MATH_BIGINTEGER_DATA => array() ); $m_length = count($m); // if ($this->_compare($n, $this->_square($m)) >= 0) { if (count($n) > 2 * $m_length) { $lhs = new Math_BigInteger(); $rhs = new Math_BigInteger(); $lhs->value = $n; $rhs->value = $m; list(, $temp) = $lhs->divide($rhs); return $temp->value; } // if (m.length >> 1) + 2 <= m.length then m is too small and n can't be reduced if ($m_length < 5) { return $this->_regularBarrett($n, $m); } // n = 2 * m.length if (($key = array_search($m, $cache[MATH_BIGINTEGER_VARIABLE])) === false) { $key = count($cache[MATH_BIGINTEGER_VARIABLE]); $cache[MATH_BIGINTEGER_VARIABLE][] = $m; $lhs = new Math_BigInteger(); $lhs_value = &$lhs->value; $lhs_value = $this->_array_repeat(0, $m_length + ($m_length >> 1)); $lhs_value[] = 1; $rhs = new Math_BigInteger(); $rhs->value = $m; list($u, $m1) = $lhs->divide($rhs); $u = $u->value; $m1 = $m1->value; $cache[MATH_BIGINTEGER_DATA][] = array( 'u' => $u, // m.length >> 1 (technically (m.length >> 1) + 1) 'm1'=> $m1 // m.length ); } else { extract($cache[MATH_BIGINTEGER_DATA][$key]); } $cutoff = $m_length + ($m_length >> 1); $lsd = array_slice($n, 0, $cutoff); // m.length + (m.length >> 1) $msd = array_slice($n, $cutoff); // m.length >> 1 $lsd = $this->_trim($lsd); $temp = $this->_multiply($msd, false, $m1, false); $n = $this->_add($lsd, false, $temp[MATH_BIGINTEGER_VALUE], false); // m.length + (m.length >> 1) + 1 if ($m_length & 1) { return $this->_regularBarrett($n[MATH_BIGINTEGER_VALUE], $m); } // (m.length + (m.length >> 1) + 1) - (m.length - 1) == (m.length >> 1) + 2 $temp = array_slice($n[MATH_BIGINTEGER_VALUE], $m_length - 1); // if even: ((m.length >> 1) + 2) + (m.length >> 1) == m.length + 2 // if odd: ((m.length >> 1) + 2) + (m.length >> 1) == (m.length - 1) + 2 == m.length + 1 $temp = $this->_multiply($temp, false, $u, false); // if even: (m.length + 2) - ((m.length >> 1) + 1) = m.length - (m.length >> 1) + 1 // if odd: (m.length + 1) - ((m.length >> 1) + 1) = m.length - (m.length >> 1) $temp = array_slice($temp[MATH_BIGINTEGER_VALUE], ($m_length >> 1) + 1); // if even: (m.length - (m.length >> 1) + 1) + m.length = 2 * m.length - (m.length >> 1) + 1 // if odd: (m.length - (m.length >> 1)) + m.length = 2 * m.length - (m.length >> 1) $temp = $this->_multiply($temp, false, $m, false); // at this point, if m had an odd number of digits, we'd be subtracting a 2 * m.length - (m.length >> 1) digit // number from a m.length + (m.length >> 1) + 1 digit number. ie. there'd be an extra digit and the while loop // following this comment would loop a lot (hence our calling _regularBarrett() in that situation). $result = $this->_subtract($n[MATH_BIGINTEGER_VALUE], false, $temp[MATH_BIGINTEGER_VALUE], false); while ($this->_compare($result[MATH_BIGINTEGER_VALUE], $result[MATH_BIGINTEGER_SIGN], $m, false) >= 0) { $result = $this->_subtract($result[MATH_BIGINTEGER_VALUE], $result[MATH_BIGINTEGER_SIGN], $m, false); } return $result[MATH_BIGINTEGER_VALUE]; } /** * (Regular) Barrett Modular Reduction * * For numbers with more than four digits Math_BigInteger::_barrett() is faster. The difference between that and this * is that this function does not fold the denominator into a smaller form. * * @see self::_slidingWindow() * @access private * @param array $x * @param array $n * @return array */ function _regularBarrett($x, $n) { static $cache = array( MATH_BIGINTEGER_VARIABLE => array(), MATH_BIGINTEGER_DATA => array() ); $n_length = count($n); if (count($x) > 2 * $n_length) { $lhs = new Math_BigInteger(); $rhs = new Math_BigInteger(); $lhs->value = $x; $rhs->value = $n; list(, $temp) = $lhs->divide($rhs); return $temp->value; } if (($key = array_search($n, $cache[MATH_BIGINTEGER_VARIABLE])) === false) { $key = count($cache[MATH_BIGINTEGER_VARIABLE]); $cache[MATH_BIGINTEGER_VARIABLE][] = $n; $lhs = new Math_BigInteger(); $lhs_value = &$lhs->value; $lhs_value = $this->_array_repeat(0, 2 * $n_length); $lhs_value[] = 1; $rhs = new Math_BigInteger(); $rhs->value = $n; list($temp, ) = $lhs->divide($rhs); // m.length $cache[MATH_BIGINTEGER_DATA][] = $temp->value; } // 2 * m.length - (m.length - 1) = m.length + 1 $temp = array_slice($x, $n_length - 1); // (m.length + 1) + m.length = 2 * m.length + 1 $temp = $this->_multiply($temp, false, $cache[MATH_BIGINTEGER_DATA][$key], false); // (2 * m.length + 1) - (m.length - 1) = m.length + 2 $temp = array_slice($temp[MATH_BIGINTEGER_VALUE], $n_length + 1); // m.length + 1 $result = array_slice($x, 0, $n_length + 1); // m.length + 1 $temp = $this->_multiplyLower($temp, false, $n, false, $n_length + 1); // $temp == array_slice($temp->_multiply($temp, false, $n, false)->value, 0, $n_length + 1) if ($this->_compare($result, false, $temp[MATH_BIGINTEGER_VALUE], $temp[MATH_BIGINTEGER_SIGN]) < 0) { $corrector_value = $this->_array_repeat(0, $n_length + 1); $corrector_value[count($corrector_value)] = 1; $result = $this->_add($result, false, $corrector_value, false); $result = $result[MATH_BIGINTEGER_VALUE]; } // at this point, we're subtracting a number with m.length + 1 digits from another number with m.length + 1 digits $result = $this->_subtract($result, false, $temp[MATH_BIGINTEGER_VALUE], $temp[MATH_BIGINTEGER_SIGN]); while ($this->_compare($result[MATH_BIGINTEGER_VALUE], $result[MATH_BIGINTEGER_SIGN], $n, false) > 0) { $result = $this->_subtract($result[MATH_BIGINTEGER_VALUE], $result[MATH_BIGINTEGER_SIGN], $n, false); } return $result[MATH_BIGINTEGER_VALUE]; } /** * Performs long multiplication up to $stop digits * * If you're going to be doing array_slice($product->value, 0, $stop), some cycles can be saved. * * @see 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( MATH_BIGINTEGER_VALUE => array(), MATH_BIGINTEGER_SIGN => false ); } if ($x_length < $y_length) { $temp = $x_value; $x_value = $y_value; $y_value = $temp; $x_length = count($x_value); $y_length = count($y_value); } $product_value = $this->_array_repeat(0, $x_length + $y_length); // the following for loop could be removed if the for loop following it // (the one with nested for loops) initially set $i to 0, but // doing so would also make the result in one set of unnecessary adds, // since on the outermost loops first pass, $product->value[$k] is going // to always be 0 $carry = 0; for ($j = 0; $j < $x_length; ++$j) { // ie. $i = 0, $k = $i $temp = $x_value[$j] * $y_value[0] + $carry; // $product_value[$k] == 0 $carry = MATH_BIGINTEGER_BASE === 26 ? intval($temp / 0x4000000) : ($temp >> 31); $product_value[$j] = (int) ($temp - MATH_BIGINTEGER_BASE_FULL * $carry); } if ($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; $carry = MATH_BIGINTEGER_BASE === 26 ? intval($temp / 0x4000000) : ($temp >> 31); $product_value[$k] = (int) ($temp - MATH_BIGINTEGER_BASE_FULL * $carry); } if ($k < $stop) { $product_value[$k] = $carry; } } return array( MATH_BIGINTEGER_VALUE => $this->_trim($product_value), MATH_BIGINTEGER_SIGN => $x_negative != $y_negative ); } /** * Montgomery Modular Reduction * * ($x->_prepMontgomery($n))->_montgomery($n) yields $x % $n. * {@link http://math.libtomcrypt.com/files/tommath.pdf#page=170 MPM 6.3} provides insights on how this can be * improved upon (basically, by using the comba method). gcd($n, 2) must be equal to one for this function * to work correctly. * * @see self::_prepMontgomery() * @see self::_slidingWindow() * @access private * @param array $x * @param array $n * @return array */ function _montgomery($x, $n) { static $cache = array( MATH_BIGINTEGER_VARIABLE => array(), MATH_BIGINTEGER_DATA => array() ); if (($key = array_search($n, $cache[MATH_BIGINTEGER_VARIABLE])) === false) { $key = count($cache[MATH_BIGINTEGER_VARIABLE]); $cache[MATH_BIGINTEGER_VARIABLE][] = $x; $cache[MATH_BIGINTEGER_DATA][] = $this->_modInverse67108864($n); } $k = count($n); $result = array(MATH_BIGINTEGER_VALUE => $x); for ($i = 0; $i < $k; ++$i) { $temp = $result[MATH_BIGINTEGER_VALUE][$i] * $cache[MATH_BIGINTEGER_DATA][$key]; $temp = $temp - MATH_BIGINTEGER_BASE_FULL * (MATH_BIGINTEGER_BASE === 26 ? intval($temp / 0x4000000) : ($temp >> 31)); $temp = $this->_regularMultiply(array($temp), $n); $temp = array_merge($this->_array_repeat(0, $i), $temp); $result = $this->_add($result[MATH_BIGINTEGER_VALUE], false, $temp, false); } $result[MATH_BIGINTEGER_VALUE] = array_slice($result[MATH_BIGINTEGER_VALUE], $k); if ($this->_compare($result, false, $n, false) >= 0) { $result = $this->_subtract($result[MATH_BIGINTEGER_VALUE], false, $n, false); } return $result[MATH_BIGINTEGER_VALUE]; } /** * Montgomery Multiply * * Interleaves the montgomery reduction and long multiplication algorithms together as described in * {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=13 HAC 14.36} * * @see 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[MATH_BIGINTEGER_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 $cache = array( MATH_BIGINTEGER_VARIABLE => array(), MATH_BIGINTEGER_DATA => array() ); if (($key = array_search($m, $cache[MATH_BIGINTEGER_VARIABLE])) === false) { $key = count($cache[MATH_BIGINTEGER_VARIABLE]); $cache[MATH_BIGINTEGER_VARIABLE][] = $m; $cache[MATH_BIGINTEGER_DATA][] = $this->_modInverse67108864($m); } $n = max(count($x), count($y), count($m)); $x = array_pad($x, $n, 0); $y = array_pad($y, $n, 0); $m = array_pad($m, $n, 0); $a = array(MATH_BIGINTEGER_VALUE => $this->_array_repeat(0, $n + 1)); for ($i = 0; $i < $n; ++$i) { $temp = $a[MATH_BIGINTEGER_VALUE][0] + $x[$i] * $y[0]; $temp = $temp - MATH_BIGINTEGER_BASE_FULL * (MATH_BIGINTEGER_BASE === 26 ? intval($temp / 0x4000000) : ($temp >> 31)); $temp = $temp * $cache[MATH_BIGINTEGER_DATA][$key]; $temp = $temp - MATH_BIGINTEGER_BASE_FULL * (MATH_BIGINTEGER_BASE === 26 ? intval($temp / 0x4000000) : ($temp >> 31)); $temp = $this->_add($this->_regularMultiply(array($x[$i]), $y), false, $this->_regularMultiply(array($temp), $m), false); $a = $this->_add($a[MATH_BIGINTEGER_VALUE], false, $temp[MATH_BIGINTEGER_VALUE], false); $a[MATH_BIGINTEGER_VALUE] = array_slice($a[MATH_BIGINTEGER_VALUE], 1); } if ($this->_compare($a[MATH_BIGINTEGER_VALUE], false, $m, false) >= 0) { $a = $this->_subtract($a[MATH_BIGINTEGER_VALUE], false, $m, false); } return $a[MATH_BIGINTEGER_VALUE]; } /** * Prepare a number for use in Montgomery Modular Reductions * * @see self::_montgomery() * @see self::_slidingWindow() * @access private * @param array $x * @param array $n * @return array */ function _prepMontgomery($x, $n) { $lhs = new Math_BigInteger(); $lhs->value = array_merge($this->_array_repeat(0, count($n)), $x); $rhs = new Math_BigInteger(); $rhs->value = $n; list(, $temp) = $lhs->divide($rhs); return $temp->value; } /** * Modular Inverse of a number mod 2**26 (eg. 67108864) * * Based off of the bnpInvDigit function implemented and justified in the following URL: * * {@link http://www-cs-students.stanford.edu/~tjw/jsbn/jsbn.js} * * The following URL provides more info: * * {@link http://groups.google.com/group/sci.crypt/msg/7a137205c1be7d85} * * As for why we do all the bitmasking... strange things can happen when converting from floats to ints. For * instance, on some computers, var_dump((int) -4294967297) yields int(-1) and on others, it yields * int(-2147483648). To avoid problems stemming from this, we use bitmasks to guarantee that ints aren't * auto-converted to floats. The outermost bitmask is present because without it, there's no guarantee that * the "residue" returned would be the so-called "common residue". We use fmod, in the last step, because the * maximum possible $x is 26 bits and the maximum $result is 16 bits. Thus, we have to be able to handle up to * 40 bits, which only 64-bit floating points will support. * * Thanks to Pedro Gimeno Fortea for input! * * @see 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, MATH_BIGINTEGER_BASE_FULL)), MATH_BIGINTEGER_BASE_FULL); // x**-1 mod 2**26 return $result & MATH_BIGINTEGER_MAX_DIGIT; } /** * Calculates modular inverses. * * Say you have (30 mod 17 * x mod 17) mod 17 == 1. x can be found using modular inverses. * * Here's an example: * * modInverse($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) * ?> * * * @param Math_BigInteger $n * @return Math_BigInteger|false * @access public * @internal See {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=21 HAC 14.64} for more information. */ function modInverse($n) { switch (MATH_BIGINTEGER_MODE) { case MATH_BIGINTEGER_MODE_GMP: $temp = new Math_BigInteger(); $temp->value = gmp_invert($this->value, $n->value); return ($temp->value === false) ? false : $this->_normalize($temp); } static $zero, $one; if (!isset($zero)) { $zero = new Math_BigInteger(); $one = new Math_BigInteger(1); } // $x mod -$n == $x mod $n. $n = $n->abs(); if ($this->compare($zero) < 0) { $temp = $this->abs(); $temp = $temp->modInverse($n); return $this->_normalize($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: * * extendedGCD($b)); * * echo $gcd->toString() . "\r\n"; // outputs 21 * echo $a->toString() * $x->toString() + $b->toString() * $y->toString(); // outputs 21 * ?> * * * @param Math_BigInteger $n * @return Math_BigInteger * @access public * @internal Calculates the GCD using the binary xGCD algorithim described in * {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=19 HAC 14.61}. As the text above 14.61 notes, * the more traditional algorithim requires "relatively costly multiple-precision divisions". */ function extendedGCD($n) { switch (MATH_BIGINTEGER_MODE) { case MATH_BIGINTEGER_MODE_GMP: extract(gmp_gcdext($this->value, $n->value)); return array( 'gcd' => $this->_normalize(new Math_BigInteger($g)), 'x' => $this->_normalize(new Math_BigInteger($s)), 'y' => $this->_normalize(new Math_BigInteger($t)) ); case MATH_BIGINTEGER_MODE_BCMATH: // it might be faster to use the binary xGCD algorithim here, as well, but (1) that algorithim works // best when the base is a power of 2 and (2) i don't think it'd make much difference, anyway. as is, // the basic extended euclidean algorithim is what we're using. $u = $this->value; $v = $n->value; $a = '1'; $b = '0'; $c = '0'; $d = '1'; while (bccomp($v, '0', 0) != 0) { $q = bcdiv($u, $v, 0); $temp = $u; $u = $v; $v = bcsub($temp, bcmul($v, $q, 0), 0); $temp = $a; $a = $c; $c = bcsub($temp, bcmul($a, $q, 0), 0); $temp = $b; $b = $d; $d = bcsub($temp, bcmul($b, $q, 0), 0); } return array( 'gcd' => $this->_normalize(new Math_BigInteger($u)), 'x' => $this->_normalize(new Math_BigInteger($a)), 'y' => $this->_normalize(new Math_BigInteger($b)) ); } $y = $n->copy(); $x = $this->copy(); $g = new Math_BigInteger(); $g->value = array(1); while (!(($x->value[0] & 1)|| ($y->value[0] & 1))) { $x->_rshift(1); $y->_rshift(1); $g->_lshift(1); } $u = $x->copy(); $v = $y->copy(); $a = new Math_BigInteger(); $b = new Math_BigInteger(); $c = new Math_BigInteger(); $d = new Math_BigInteger(); $a->value = $d->value = $g->value = array(1); $b->value = $c->value = array(); while (!empty($u->value)) { while (!($u->value[0] & 1)) { $u->_rshift(1); if ((!empty($a->value) && ($a->value[0] & 1)) || (!empty($b->value) && ($b->value[0] & 1))) { $a = $a->add($y); $b = $b->subtract($x); } $a->_rshift(1); $b->_rshift(1); } while (!($v->value[0] & 1)) { $v->_rshift(1); if ((!empty($d->value) && ($d->value[0] & 1)) || (!empty($c->value) && ($c->value[0] & 1))) { $c = $c->add($y); $d = $d->subtract($x); } $c->_rshift(1); $d->_rshift(1); } if ($u->compare($v) >= 0) { $u = $u->subtract($v); $a = $a->subtract($c); $b = $b->subtract($d); } else { $v = $v->subtract($u); $c = $c->subtract($a); $d = $d->subtract($b); } } return array( 'gcd' => $this->_normalize($g->multiply($v)), 'x' => $this->_normalize($c), 'y' => $this->_normalize($d) ); } /** * Calculates the greatest common divisor * * Say you have 693 and 609. The GCD is 21. * * Here's an example: * * extendedGCD($b); * * echo $gcd->toString() . "\r\n"; // outputs 21 * ?> * * * @param Math_BigInteger $n * @return Math_BigInteger * @access public */ function gcd($n) { extract($this->extendedGCD($n)); return $gcd; } /** * Absolute value. * * @return Math_BigInteger * @access public */ function abs() { $temp = new Math_BigInteger(); switch (MATH_BIGINTEGER_MODE) { case MATH_BIGINTEGER_MODE_GMP: $temp->value = gmp_abs($this->value); break; case MATH_BIGINTEGER_MODE_BCMATH: $temp->value = (bccomp($this->value, '0', 0) < 0) ? substr($this->value, 1) : $this->value; break; default: $temp->value = $this->value; } return $temp; } /** * Compares two numbers. * * Although one might think !$x->compare($y) means $x != $y, it, in fact, means the opposite. The reason for this is * demonstrated thusly: * * $x > $y: $x->compare($y) > 0 * $x < $y: $x->compare($y) < 0 * $x == $y: $x->compare($y) == 0 * * Note how the same comparison operator is used. If you want to test for equality, use $x->equals($y). * * @param Math_BigInteger $y * @return int < 0 if $this is less than $y; > 0 if $this is greater than $y, and 0 if they are equal. * @access public * @see self::equals() * @internal Could return $this->subtract($x), but that's not as fast as what we do do. */ function compare($y) { switch (MATH_BIGINTEGER_MODE) { case MATH_BIGINTEGER_MODE_GMP: return gmp_cmp($this->value, $y->value); case MATH_BIGINTEGER_MODE_BCMATH: return bccomp($this->value, $y->value, 0); } return $this->_compare($this->value, $this->is_negative, $y->value, $y->is_negative); } /** * Compares two numbers. * * @param array $x_value * @param 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 Math_BigInteger::compare() * * @param Math_BigInteger $x * @return bool * @access public * @see self::compare() */ function equals($x) { switch (MATH_BIGINTEGER_MODE) { case MATH_BIGINTEGER_MODE_GMP: return gmp_cmp($this->value, $x->value) == 0; default: return $this->value === $x->value && $this->is_negative == $x->is_negative; } } /** * Set Precision * * Some bitwise operations give different results depending on the precision being used. Examples include left * shift, not, and rotates. * * @param int $bits * @access public */ function setPrecision($bits) { $this->precision = $bits; if (MATH_BIGINTEGER_MODE != MATH_BIGINTEGER_MODE_BCMATH) { $this->bitmask = new Math_BigInteger(chr((1 << ($bits & 0x7)) - 1) . str_repeat(chr(0xFF), $bits >> 3), 256); } else { $this->bitmask = new Math_BigInteger(bcpow('2', $bits, 0)); } $temp = $this->_normalize($this); $this->value = $temp->value; } /** * Logical And * * @param Math_BigInteger $x * @access public * @internal Implemented per a request by Lluis Pamies i Juarez * @return Math_BigInteger */ function bitwise_and($x) { switch (MATH_BIGINTEGER_MODE) { case MATH_BIGINTEGER_MODE_GMP: $temp = new Math_BigInteger(); $temp->value = gmp_and($this->value, $x->value); return $this->_normalize($temp); case MATH_BIGINTEGER_MODE_BCMATH: $left = $this->toBytes(); $right = $x->toBytes(); $length = max(strlen($left), strlen($right)); $left = str_pad($left, $length, chr(0), STR_PAD_LEFT); $right = str_pad($right, $length, chr(0), STR_PAD_LEFT); return $this->_normalize(new Math_BigInteger($left & $right, 256)); } $result = $this->copy(); $length = min(count($x->value), count($this->value)); $result->value = array_slice($result->value, 0, $length); for ($i = 0; $i < $length; ++$i) { $result->value[$i]&= $x->value[$i]; } return $this->_normalize($result); } /** * Logical Or * * @param Math_BigInteger $x * @access public * @internal Implemented per a request by Lluis Pamies i Juarez * @return Math_BigInteger */ function bitwise_or($x) { switch (MATH_BIGINTEGER_MODE) { case MATH_BIGINTEGER_MODE_GMP: $temp = new Math_BigInteger(); $temp->value = gmp_or($this->value, $x->value); return $this->_normalize($temp); case MATH_BIGINTEGER_MODE_BCMATH: $left = $this->toBytes(); $right = $x->toBytes(); $length = max(strlen($left), strlen($right)); $left = str_pad($left, $length, chr(0), STR_PAD_LEFT); $right = str_pad($right, $length, chr(0), STR_PAD_LEFT); return $this->_normalize(new Math_BigInteger($left | $right, 256)); } $length = max(count($this->value), count($x->value)); $result = $this->copy(); $result->value = array_pad($result->value, $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 Math_BigInteger $x * @access public * @internal Implemented per a request by Lluis Pamies i Juarez * @return Math_BigInteger */ function bitwise_xor($x) { switch (MATH_BIGINTEGER_MODE) { case MATH_BIGINTEGER_MODE_GMP: $temp = new Math_BigInteger(); $temp->value = gmp_xor($this->value, $x->value); return $this->_normalize($temp); case MATH_BIGINTEGER_MODE_BCMATH: $left = $this->toBytes(); $right = $x->toBytes(); $length = max(strlen($left), strlen($right)); $left = str_pad($left, $length, chr(0), STR_PAD_LEFT); $right = str_pad($right, $length, chr(0), STR_PAD_LEFT); return $this->_normalize(new Math_BigInteger($left ^ $right, 256)); } $length = max(count($this->value), count($x->value)); $result = $this->copy(); $result->value = array_pad($result->value, $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 * @return Math_BigInteger */ function bitwise_not() { // calculuate "not" without regard to $this->precision // (will always result in a smaller number. ie. ~1 isn't 1111 1110 - it's 0) $temp = $this->toBytes(); if ($temp == '') { return ''; } $pre_msb = decbin(ord($temp[0])); $temp = ~$temp; $msb = decbin(ord($temp[0])); if (strlen($msb) == 8) { $msb = substr($msb, strpos($msb, '0')); } $temp[0] = chr(bindec($msb)); // see if we need to add extra leading 1's $current_bits = strlen($pre_msb) + 8 * strlen($temp) - 8; $new_bits = $this->precision - $current_bits; if ($new_bits <= 0) { return $this->_normalize(new Math_BigInteger($temp, 256)); } // generate as many leading 1's as we need to. $leading_ones = chr((1 << ($new_bits & 0x7)) - 1) . str_repeat(chr(0xFF), $new_bits >> 3); $this->_base256_lshift($leading_ones, $current_bits); $temp = str_pad($temp, strlen($leading_ones), chr(0), STR_PAD_LEFT); return $this->_normalize(new Math_BigInteger($leading_ones | $temp, 256)); } /** * Logical Right Shift * * Shifts BigInteger's by $shift bits, effectively dividing by 2**$shift. * * @param int $shift * @return Math_BigInteger * @access public * @internal The only version that yields any speed increases is the internal version. */ function bitwise_rightShift($shift) { $temp = new Math_BigInteger(); switch (MATH_BIGINTEGER_MODE) { case MATH_BIGINTEGER_MODE_GMP: static $two; if (!isset($two)) { $two = gmp_init('2'); } $temp->value = gmp_div_q($this->value, gmp_pow($two, $shift)); break; case MATH_BIGINTEGER_MODE_BCMATH: $temp->value = bcdiv($this->value, bcpow('2', $shift, 0), 0); break; default: // could just replace _lshift with this, but then all _lshift() calls would need to be rewritten // and I don't want to do that... $temp->value = $this->value; $temp->_rshift($shift); } return $this->_normalize($temp); } /** * Logical Left Shift * * Shifts BigInteger's by $shift bits, effectively multiplying by 2**$shift. * * @param int $shift * @return Math_BigInteger * @access public * @internal The only version that yields any speed increases is the internal version. */ function bitwise_leftShift($shift) { $temp = new Math_BigInteger(); switch (MATH_BIGINTEGER_MODE) { case MATH_BIGINTEGER_MODE_GMP: static $two; if (!isset($two)) { $two = gmp_init('2'); } $temp->value = gmp_mul($this->value, gmp_pow($two, $shift)); break; case MATH_BIGINTEGER_MODE_BCMATH: $temp->value = bcmul($this->value, bcpow('2', $shift, 0), 0); break; default: // could just replace _rshift with this, but then all _lshift() calls would need to be rewritten // and I don't want to do that... $temp->value = $this->value; $temp->_lshift($shift); } return $this->_normalize($temp); } /** * Logical Left Rotate * * Instead of the top x bits being dropped they're appended to the shifted bit string. * * @param int $shift * @return Math_BigInteger * @access public */ function bitwise_leftRotate($shift) { $bits = $this->toBytes(); if ($this->precision > 0) { $precision = $this->precision; if (MATH_BIGINTEGER_MODE == MATH_BIGINTEGER_MODE_BCMATH) { $mask = $this->bitmask->subtract(new Math_BigInteger(1)); $mask = $mask->toBytes(); } else { $mask = $this->bitmask->toBytes(); } } else { $temp = ord($bits[0]); for ($i = 0; $temp >> $i; ++$i) { } $precision = 8 * strlen($bits) - 8 + $i; $mask = chr((1 << ($precision & 0x7)) - 1) . str_repeat(chr(0xFF), $precision >> 3); } if ($shift < 0) { $shift+= $precision; } $shift%= $precision; if (!$shift) { return $this->copy(); } $left = $this->bitwise_leftShift($shift); $left = $left->bitwise_and(new Math_BigInteger($mask, 256)); $right = $this->bitwise_rightShift($precision - $shift); $result = MATH_BIGINTEGER_MODE != MATH_BIGINTEGER_MODE_BCMATH ? $left->bitwise_or($right) : $left->add($right); return $this->_normalize($result); } /** * Logical Right Rotate * * Instead of the bottom x bits being dropped they're prepended to the shifted bit string. * * @param int $shift * @return Math_BigInteger * @access public */ function bitwise_rightRotate($shift) { return $this->bitwise_leftRotate(-$shift); } /** * Set random number generator function * * This function is deprecated. * * @param string $generator * @access public */ function setRandomGenerator($generator) { } /** * Generates a random BigInteger * * Byte length is equal to $length. Uses crypt_random if it's loaded and mt_rand if it's not. * * @param int $length * @return Math_BigInteger * @access private */ function _random_number_helper($size) { if (function_exists('crypt_random_string')) { $random = crypt_random_string($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 $random.= pack('n', mt_rand(0, 0xFFFF)); } } return new Math_BigInteger($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 Math_BigInteger $arg1 * @param Math_BigInteger $arg2 * @return Math_BigInteger * @access public * @internal The API for creating random numbers used to be $a->random($min, $max), where $a was a Math_BigInteger object. * That method is still supported for BC purposes. */ function random($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 Math_BigInteger(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 Math_BigInteger(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 Math_BigInteger $arg1 * @param Math_BigInteger $arg2 * @param int $timeout * @return Math_BigInteger|false * @access public * @internal See {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap4.pdf#page=15 HAC 4.44}. */ function randomPrime($arg1, $arg2 = false, $timeout = false) { if ($arg1 === false) { return false; } if ($arg2 === false) { $max = $arg1; $min = $this; } else { $min = $arg1; $max = $arg2; } $compare = $max->compare($min); if (!$compare) { return $min->isPrime() ? $min : false; } elseif ($compare < 0) { // if $min is bigger then $max, swap $min and $max $temp = $max; $max = $min; $min = $temp; } static $one, $two; if (!isset($one)) { $one = new Math_BigInteger(1); $two = new Math_BigInteger(2); } $start = time(); $x = $this->random($min, $max); // gmp_nextprime() requires PHP 5 >= 5.2.0 per . if (MATH_BIGINTEGER_MODE == MATH_BIGINTEGER_MODE_GMP && extension_loaded('gmp') && version_compare(PHP_VERSION, '5.2.0', '>=')) { $p = new Math_BigInteger(); $p->value = gmp_nextprime($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($initial_x)) { return false; } } } /** * Make the current number odd * * If the current number is odd it'll be unchanged. If it's even, one will be added to it. * * @see self::randomPrime() * @access private */ function _make_odd() { switch (MATH_BIGINTEGER_MODE) { case MATH_BIGINTEGER_MODE_GMP: gmp_setbit($this->value, 0); break; case MATH_BIGINTEGER_MODE_BCMATH: if ($this->value[strlen($this->value) - 1] % 2 == 0) { $this->value = bcadd($this->value, '1'); } break; default: $this->value[0] |= 1; } } /** * Checks a numer to see if it's prime * * Assuming the $t parameter is not set, this function has an error rate of 2**-80. The main motivation for the * $t parameter is distributability. Math_BigInteger::randomPrime() can be distributed across multiple pageloads * on a website instead of just one. * * @param Math_BigInteger $t * @return bool * @access public * @internal Uses the * {@link http://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test Miller-Rabin primality test}. See * {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap4.pdf#page=8 HAC 4.24}. */ function isPrime($t = false) { $length = strlen($this->toBytes()); if (!$t) { // see HAC 4.49 "Note (controlling the error probability)" // @codingStandardsIgnoreStart if ($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 MATH_BIGINTEGER_MODE_GMP: return gmp_prob_prime($this->value, $t) != 0; case MATH_BIGINTEGER_MODE_BCMATH: if ($this->value === '2') { return true; } if ($this->value[strlen($this->value) - 1] % 2 == 0) { return false; } break; default: if ($this->value == array(2)) { return true; } if (~$this->value[0] & 1) { return false; } } static $primes, $zero, $one, $two; if (!isset($primes)) { $primes = array( 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997 ); if (MATH_BIGINTEGER_MODE != MATH_BIGINTEGER_MODE_INTERNAL) { for ($i = 0; $i < count($primes); ++$i) { $primes[$i] = new Math_BigInteger($primes[$i]); } } $zero = new Math_BigInteger(); $one = new Math_BigInteger(1); $two = new Math_BigInteger(2); } if ($this->equals($one)) { return false; } // see HAC 4.4.1 "Random search for probable primes" if (MATH_BIGINTEGER_MODE != MATH_BIGINTEGER_MODE_INTERNAL) { foreach ($primes as $prime) { list(, $r) = $this->divide($prime); if ($r->equals($zero)) { return $this->equals($prime); } } } else { $value = $this->value; foreach ($primes as $prime) { list(, $r) = $this->_divide_digit($value, $prime); if (!$r) { return count($value) == 1 && $value[0] == $prime; } } } $n = $this->copy(); $n_1 = $n->subtract($one); $n_2 = $n->subtract($two); $r = $n_1->copy(); $r_value = $r->value; // ie. $s = gmp_scan1($n, 0) and $r = gmp_div_q($n, gmp_pow(gmp_init('2'), $s)); if (MATH_BIGINTEGER_MODE == MATH_BIGINTEGER_MODE_BCMATH) { $s = 0; // if $n was 1, $r would be 0 and this would be an infinite loop, hence our $this->equals($one) check earlier while ($r->value[strlen($r->value) - 1] % 2 == 0) { $r->value = bcdiv($r->value, '2', 0); ++$s; } } else { for ($i = 0, $r_length = count($r_value); $i < $r_length; ++$i) { $temp = ~$r_value[$i] & 0xFFFFFF; for ($j = 1; ($temp >> $j) & 1; ++$j) { } if ($j != 25) { break; } } $s = 26 * $i + $j - 1; $r->_rshift($s); } for ($i = 0; $i < $t; ++$i) { $a = $this->random($two, $n_2); $y = $a->modPow($r, $n); if (!$y->equals($one) && !$y->equals($n_1)) { for ($j = 1; $j < $s && !$y->equals($n_1); ++$j) { $y = $y->modPow($two, $n); if ($y->equals($one)) { return false; } } if (!$y->equals($n_1)) { return false; } } } return true; } /** * Logical Left Shift * * Shifts BigInteger's by $shift bits. * * @param int $shift * @access private */ function _lshift($shift) { if ($shift == 0) { return; } $num_digits = (int) ($shift / MATH_BIGINTEGER_BASE); $shift %= MATH_BIGINTEGER_BASE; $shift = 1 << $shift; $carry = 0; for ($i = 0; $i < count($this->value); ++$i) { $temp = $this->value[$i] * $shift + $carry; $carry = MATH_BIGINTEGER_BASE === 26 ? intval($temp / 0x4000000) : ($temp >> 31); $this->value[$i] = (int) ($temp - $carry * MATH_BIGINTEGER_BASE_FULL); } if ($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 / MATH_BIGINTEGER_BASE); $shift %= MATH_BIGINTEGER_BASE; $carry_shift = MATH_BIGINTEGER_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 Math_BigInteger * @return Math_BigInteger * @see self::_trim() * @access private */ function _normalize($result) { $result->precision = $this->precision; $result->bitmask = $this->bitmask; switch (MATH_BIGINTEGER_MODE) { case MATH_BIGINTEGER_MODE_GMP: if ($this->bitmask !== false) { $result->value = gmp_and($result->value, $result->bitmask->value); } return $result; case MATH_BIGINTEGER_MODE_BCMATH: if (!empty($result->bitmask->value)) { $result->value = bcmod($result->value, $result->bitmask->value); } return $result; } $value = &$result->value; if (!count($value)) { return $result; } $value = $this->_trim($value); if (!empty($result->bitmask->value)) { $length = min(count($value), count($this->bitmask->value)); $value = array_slice($value, 0, $length); for ($i = 0; $i < $length; ++$i) { $value[$i] = $value[$i] & $this->bitmask->value[$i]; } } return $result; } /** * Trim * * Removes leading zeros * * @param array $value * @return Math_BigInteger * @access private */ function _trim($value) { for ($i = count($value) - 1; $i >= 0; --$i) { if ($value[$i]) { break; } unset($value[$i]); } return $value; } /** * Array Repeat * * @param $input Array * @param $multiplier mixed * @return array * @access private */ function _array_repeat($input, $multiplier) { return ($multiplier) ? array_fill(0, $multiplier, $input) : array(); } /** * Logical Left Shift * * Shifts binary strings $shift bits, essentially multiplying by 2**$shift. * * @param $x String * @param $shift Integer * @return string * @access private */ function _base256_lshift(&$x, $shift) { if ($shift == 0) { return; } $num_bytes = $shift >> 3; // eg. floor($shift/8) $shift &= 7; // eg. $shift % 8 $carry = 0; for ($i = strlen($x) - 1; $i >= 0; --$i) { $temp = ord($x[$i]) << $shift | $carry; $x[$i] = chr($temp); $carry = $temp >> 8; } $carry = ($carry != 0) ? chr($carry) : ''; $x = $carry . $x . str_repeat(chr(0), $num_bytes); } /** * Logical Right Shift * * Shifts binary strings $shift bits, essentially dividing by 2**$shift and returning the remainder. * * @param $x String * @param $shift Integer * @return string * @access private */ function _base256_rshift(&$x, $shift) { if ($shift == 0) { $x = ltrim($x, chr(0)); return ''; } $num_bytes = $shift >> 3; // eg. floor($shift/8) $shift &= 7; // eg. $shift % 8 $remainder = ''; if ($num_bytes) { $start = $num_bytes > strlen($x) ? -strlen($x) : -$num_bytes; $remainder = substr($x, $start); $x = substr($x, 0, -$num_bytes); } $carry = 0; $carry_shift = 8 - $shift; for ($i = 0; $i < strlen($x); ++$i) { $temp = (ord($x[$i]) >> $shift) | $carry; $carry = (ord($x[$i]) << $carry_shift) & 0xFF; $x[$i] = chr($temp); } $x = ltrim($x, chr(0)); $remainder = chr($carry >> $carry_shift) . $remainder; return ltrim($remainder, chr(0)); } // one quirk about how the following functions are implemented is that PHP defines N to be an unsigned long // at 32-bits, while java's longs are 64-bits. /** * Converts 32-bit integers to bytes. * * @param 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, $y) { if (MATH_BIGINTEGER_BASE === 26) { return (int) ($x / $y); } // MATH_BIGINTEGER_BASE === 31 return ($x - ($x % $y)) / $y; } }