--- /dev/null
+<?php\r
+/* vim: set expandtab tabstop=4 shiftwidth=4 softtabstop=4: */\r
+\r
+/**\r
+ * Pure-PHP arbitrary precision integer arithmetic library.\r
+ *\r
+ * Supports base-2, base-10, base-16, and base-256 numbers. Uses the GMP or BCMath extensions, if available,\r
+ * and an internal implementation, otherwise.\r
+ *\r
+ * PHP versions 4 and 5\r
+ *\r
+ * {@internal (all DocBlock comments regarding implementation - such as the one that follows - refer to the \r
+ * {@link MATH_BIGINTEGER_MODE_INTERNAL MATH_BIGINTEGER_MODE_INTERNAL} mode)\r
+ *\r
+ * Math_BigInteger uses base-2**26 to perform operations such as multiplication and division and\r
+ * base-2**52 (ie. two base 2**26 digits) to perform addition and subtraction. Because the largest possible\r
+ * value when multiplying two base-2**26 numbers together is a base-2**52 number, double precision floating\r
+ * point numbers - numbers that should be supported on most hardware and whose significand is 53 bits - are\r
+ * used. As a consequence, bitwise operators such as >> and << cannot be used, nor can the modulo operator %,\r
+ * which only supports integers. Although this fact will slow this library down, the fact that such a high\r
+ * base is being used should more than compensate.\r
+ *\r
+ * When PHP version 6 is officially released, we'll be able to use 64-bit integers. This should, once again,\r
+ * allow bitwise operators, and will increase the maximum possible base to 2**31 (or 2**62 for addition /\r
+ * subtraction).\r
+ *\r
+ * Numbers are stored in {@link http://en.wikipedia.org/wiki/Endianness little endian} format. ie.\r
+ * (new Math_BigInteger(pow(2, 26)))->value = array(0, 1)\r
+ *\r
+ * Useful resources are as follows:\r
+ *\r
+ * - {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf Handbook of Applied Cryptography (HAC)}\r
+ * - {@link http://math.libtomcrypt.com/files/tommath.pdf Multi-Precision Math (MPM)}\r
+ * - Java's BigInteger classes. See /j2se/src/share/classes/java/math in jdk-1_5_0-src-jrl.zip\r
+ *\r
+ * Here's an example of how to use this library:\r
+ * <code>\r
+ * <?php\r
+ * include('Math/BigInteger.php');\r
+ *\r
+ * $a = new Math_BigInteger(2);\r
+ * $b = new Math_BigInteger(3);\r
+ *\r
+ * $c = $a->add($b);\r
+ *\r
+ * echo $c->toString(); // outputs 5\r
+ * ?>\r
+ * </code>\r
+ *\r
+ * LICENSE: This library is free software; you can redistribute it and/or\r
+ * modify it under the terms of the GNU Lesser General Public\r
+ * License as published by the Free Software Foundation; either\r
+ * version 2.1 of the License, or (at your option) any later version.\r
+ *\r
+ * This library is distributed in the hope that it will be useful,\r
+ * but WITHOUT ANY WARRANTY; without even the implied warranty of\r
+ * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU\r
+ * Lesser General Public License for more details.\r
+ *\r
+ * You should have received a copy of the GNU Lesser General Public\r
+ * License along with this library; if not, write to the Free Software\r
+ * Foundation, Inc., 59 Temple Place, Suite 330, Boston,\r
+ * MA 02111-1307 USA\r
+ *\r
+ * @category Math\r
+ * @package Math_BigInteger\r
+ * @author Jim Wigginton <terrafrost@php.net>\r
+ * @copyright MMVI Jim Wigginton\r
+ * @license http://www.gnu.org/licenses/lgpl.txt\r
+ * @version $Id: BigInteger.php,v 1.33 2010/03/22 22:32:03 terrafrost Exp $\r
+ * @link http://pear.php.net/package/Math_BigInteger\r
+ */\r
+\r
+/**#@+\r
+ * Reduction constants\r
+ *\r
+ * @access private\r
+ * @see Math_BigInteger::_reduce()\r
+ */\r
+/**\r
+ * @see Math_BigInteger::_montgomery()\r
+ * @see Math_BigInteger::_prepMontgomery()\r
+ */\r
+define('MATH_BIGINTEGER_MONTGOMERY', 0);\r
+/**\r
+ * @see Math_BigInteger::_barrett()\r
+ */\r
+define('MATH_BIGINTEGER_BARRETT', 1);\r
+/**\r
+ * @see Math_BigInteger::_mod2()\r
+ */\r
+define('MATH_BIGINTEGER_POWEROF2', 2);\r
+/**\r
+ * @see Math_BigInteger::_remainder()\r
+ */\r
+define('MATH_BIGINTEGER_CLASSIC', 3);\r
+/**\r
+ * @see Math_BigInteger::__clone()\r
+ */\r
+define('MATH_BIGINTEGER_NONE', 4);\r
+/**#@-*/\r
+\r
+/**#@+\r
+ * Array constants\r
+ *\r
+ * Rather than create a thousands and thousands of new Math_BigInteger objects in repeated function calls to add() and\r
+ * multiply() or whatever, we'll just work directly on arrays, taking them in as parameters and returning them.\r
+ *\r
+ * @access private\r
+ */\r
+/**\r
+ * $result[MATH_BIGINTEGER_VALUE] contains the value.\r
+ */\r
+define('MATH_BIGINTEGER_VALUE', 0);\r
+/**\r
+ * $result[MATH_BIGINTEGER_SIGN] contains the sign.\r
+ */\r
+define('MATH_BIGINTEGER_SIGN', 1);\r
+/**#@-*/\r
+\r
+/**#@+\r
+ * @access private\r
+ * @see Math_BigInteger::_montgomery()\r
+ * @see Math_BigInteger::_barrett()\r
+ */\r
+/**\r
+ * Cache constants\r
+ *\r
+ * $cache[MATH_BIGINTEGER_VARIABLE] tells us whether or not the cached data is still valid.\r
+ */\r
+define('MATH_BIGINTEGER_VARIABLE', 0);\r
+/**\r
+ * $cache[MATH_BIGINTEGER_DATA] contains the cached data.\r
+ */\r
+define('MATH_BIGINTEGER_DATA', 1);\r
+/**#@-*/\r
+\r
+/**#@+\r
+ * Mode constants.\r
+ *\r
+ * @access private\r
+ * @see Math_BigInteger::Math_BigInteger()\r
+ */\r
+/**\r
+ * To use the pure-PHP implementation\r
+ */\r
+define('MATH_BIGINTEGER_MODE_INTERNAL', 1);\r
+/**\r
+ * To use the BCMath library\r
+ *\r
+ * (if enabled; otherwise, the internal implementation will be used)\r
+ */\r
+define('MATH_BIGINTEGER_MODE_BCMATH', 2);\r
+/**\r
+ * To use the GMP library\r
+ *\r
+ * (if present; otherwise, either the BCMath or the internal implementation will be used)\r
+ */\r
+define('MATH_BIGINTEGER_MODE_GMP', 3);\r
+/**#@-*/\r
+\r
+/**\r
+ * The largest digit that may be used in addition / subtraction\r
+ *\r
+ * (we do pow(2, 52) instead of using 4503599627370496, directly, because some PHP installations\r
+ * will truncate 4503599627370496)\r
+ *\r
+ * @access private\r
+ */\r
+define('MATH_BIGINTEGER_MAX_DIGIT52', pow(2, 52));\r
+\r
+/**\r
+ * Karatsuba Cutoff\r
+ *\r
+ * At what point do we switch between Karatsuba multiplication and schoolbook long multiplication?\r
+ *\r
+ * @access private\r
+ */\r
+define('MATH_BIGINTEGER_KARATSUBA_CUTOFF', 25);\r
+\r
+/**\r
+ * Pure-PHP arbitrary precision integer arithmetic library. Supports base-2, base-10, base-16, and base-256\r
+ * numbers.\r
+ *\r
+ * @author Jim Wigginton <terrafrost@php.net>\r
+ * @version 1.0.0RC4\r
+ * @access public\r
+ * @package Math_BigInteger\r
+ */\r
+class Math_BigInteger {\r
+ /**\r
+ * Holds the BigInteger's value.\r
+ *\r
+ * @var Array\r
+ * @access private\r
+ */\r
+ var $value;\r
+\r
+ /**\r
+ * Holds the BigInteger's magnitude.\r
+ *\r
+ * @var Boolean\r
+ * @access private\r
+ */\r
+ var $is_negative = false;\r
+\r
+ /**\r
+ * Random number generator function\r
+ *\r
+ * @see setRandomGenerator()\r
+ * @access private\r
+ */\r
+ var $generator = 'mt_rand';\r
+\r
+ /**\r
+ * Precision\r
+ *\r
+ * @see setPrecision()\r
+ * @access private\r
+ */\r
+ var $precision = -1;\r
+\r
+ /**\r
+ * Precision Bitmask\r
+ *\r
+ * @see setPrecision()\r
+ * @access private\r
+ */\r
+ var $bitmask = false;\r
+\r
+ /**\r
+ * Mode independant value used for serialization.\r
+ *\r
+ * If the bcmath or gmp extensions are installed $this->value will be a non-serializable resource, hence the need for \r
+ * a variable that'll be serializable regardless of whether or not extensions are being used. Unlike $this->value,\r
+ * however, $this->hex is only calculated when $this->__sleep() is called.\r
+ *\r
+ * @see __sleep()\r
+ * @see __wakeup()\r
+ * @var String\r
+ * @access private\r
+ */\r
+ var $hex;\r
+\r
+ /**\r
+ * Converts base-2, base-10, base-16, and binary strings (eg. base-256) to BigIntegers.\r
+ *\r
+ * If the second parameter - $base - is negative, then it will be assumed that the number's are encoded using\r
+ * two's compliment. The sole exception to this is -10, which is treated the same as 10 is.\r
+ *\r
+ * Here's an example:\r
+ * <code>\r
+ * <?php\r
+ * include('Math/BigInteger.php');\r
+ *\r
+ * $a = new Math_BigInteger('0x32', 16); // 50 in base-16\r
+ *\r
+ * echo $a->toString(); // outputs 50\r
+ * ?>\r
+ * </code>\r
+ *\r
+ * @param optional $x base-10 number or base-$base number if $base set.\r
+ * @param optional integer $base\r
+ * @return Math_BigInteger\r
+ * @access public\r
+ */\r
+ function Math_BigInteger($x = 0, $base = 10)\r
+ {\r
+ if ( !defined('MATH_BIGINTEGER_MODE') ) {\r
+ switch (true) {\r
+ case extension_loaded('gmp'):\r
+ define('MATH_BIGINTEGER_MODE', MATH_BIGINTEGER_MODE_GMP);\r
+ break;\r
+ case extension_loaded('bcmath'):\r
+ define('MATH_BIGINTEGER_MODE', MATH_BIGINTEGER_MODE_BCMATH);\r
+ break;\r
+ default:\r
+ define('MATH_BIGINTEGER_MODE', MATH_BIGINTEGER_MODE_INTERNAL);\r
+ }\r
+ }\r
+\r
+ switch ( MATH_BIGINTEGER_MODE ) {\r
+ case MATH_BIGINTEGER_MODE_GMP:\r
+ if (is_resource($x) && get_resource_type($x) == 'GMP integer') {\r
+ $this->value = $x;\r
+ return;\r
+ }\r
+ $this->value = gmp_init(0);\r
+ break;\r
+ case MATH_BIGINTEGER_MODE_BCMATH:\r
+ $this->value = '0';\r
+ break;\r
+ default:\r
+ $this->value = array();\r
+ }\r
+\r
+ if (empty($x)) {\r
+ return;\r
+ }\r
+\r
+ switch ($base) {\r
+ case -256:\r
+ if (ord($x[0]) & 0x80) {\r
+ $x = ~$x;\r
+ $this->is_negative = true;\r
+ }\r
+ case 256:\r
+ switch ( MATH_BIGINTEGER_MODE ) {\r
+ case MATH_BIGINTEGER_MODE_GMP:\r
+ $sign = $this->is_negative ? '-' : '';\r
+ $this->value = gmp_init($sign . '0x' . bin2hex($x));\r
+ break;\r
+ case MATH_BIGINTEGER_MODE_BCMATH:\r
+ // round $len to the nearest 4 (thanks, DavidMJ!)\r
+ $len = (strlen($x) + 3) & 0xFFFFFFFC;\r
+\r
+ $x = str_pad($x, $len, chr(0), STR_PAD_LEFT);\r
+\r
+ for ($i = 0; $i < $len; $i+= 4) {\r
+ $this->value = bcmul($this->value, '4294967296', 0); // 4294967296 == 2**32\r
+ $this->value = bcadd($this->value, 0x1000000 * ord($x[$i]) + ((ord($x[$i + 1]) << 16) | (ord($x[$i + 2]) << 8) | ord($x[$i + 3])), 0);\r
+ }\r
+\r
+ if ($this->is_negative) {\r
+ $this->value = '-' . $this->value;\r
+ }\r
+\r
+ break;\r
+ // converts a base-2**8 (big endian / msb) number to base-2**26 (little endian / lsb)\r
+ default:\r
+ while (strlen($x)) {\r
+ $this->value[] = $this->_bytes2int($this->_base256_rshift($x, 26));\r
+ }\r
+ }\r
+\r
+ if ($this->is_negative) {\r
+ if (MATH_BIGINTEGER_MODE != MATH_BIGINTEGER_MODE_INTERNAL) {\r
+ $this->is_negative = false;\r
+ }\r
+ $temp = $this->add(new Math_BigInteger('-1'));\r
+ $this->value = $temp->value;\r
+ }\r
+ break;\r
+ case 16:\r
+ case -16:\r
+ if ($base > 0 && $x[0] == '-') {\r
+ $this->is_negative = true;\r
+ $x = substr($x, 1);\r
+ }\r
+\r
+ $x = preg_replace('#^(?:0x)?([A-Fa-f0-9]*).*#', '$1', $x);\r
+\r
+ $is_negative = false;\r
+ if ($base < 0 && hexdec($x[0]) >= 8) {\r
+ $this->is_negative = $is_negative = true;\r
+ $x = bin2hex(~pack('H*', $x));\r
+ }\r
+\r
+ switch ( MATH_BIGINTEGER_MODE ) {\r
+ case MATH_BIGINTEGER_MODE_GMP:\r
+ $temp = $this->is_negative ? '-0x' . $x : '0x' . $x;\r
+ $this->value = gmp_init($temp);\r
+ $this->is_negative = false;\r
+ break;\r
+ case MATH_BIGINTEGER_MODE_BCMATH:\r
+ $x = ( strlen($x) & 1 ) ? '0' . $x : $x;\r
+ $temp = new Math_BigInteger(pack('H*', $x), 256);\r
+ $this->value = $this->is_negative ? '-' . $temp->value : $temp->value;\r
+ $this->is_negative = false;\r
+ break;\r
+ default:\r
+ $x = ( strlen($x) & 1 ) ? '0' . $x : $x;\r
+ $temp = new Math_BigInteger(pack('H*', $x), 256);\r
+ $this->value = $temp->value;\r
+ }\r
+\r
+ if ($is_negative) {\r
+ $temp = $this->add(new Math_BigInteger('-1'));\r
+ $this->value = $temp->value;\r
+ }\r
+ break;\r
+ case 10:\r
+ case -10:\r
+ $x = preg_replace('#^(-?[0-9]*).*#', '$1', $x);\r
+\r
+ switch ( MATH_BIGINTEGER_MODE ) {\r
+ case MATH_BIGINTEGER_MODE_GMP:\r
+ $this->value = gmp_init($x);\r
+ break;\r
+ case MATH_BIGINTEGER_MODE_BCMATH:\r
+ // explicitly casting $x to a string is necessary, here, since doing $x[0] on -1 yields different\r
+ // results then doing it on '-1' does (modInverse does $x[0])\r
+ $this->value = (string) $x;\r
+ break;\r
+ default:\r
+ $temp = new Math_BigInteger();\r
+\r
+ // array(10000000) is 10**7 in base-2**26. 10**7 is the closest to 2**26 we can get without passing it.\r
+ $multiplier = new Math_BigInteger();\r
+ $multiplier->value = array(10000000);\r
+\r
+ if ($x[0] == '-') {\r
+ $this->is_negative = true;\r
+ $x = substr($x, 1);\r
+ }\r
+\r
+ $x = str_pad($x, strlen($x) + (6 * strlen($x)) % 7, 0, STR_PAD_LEFT);\r
+\r
+ while (strlen($x)) {\r
+ $temp = $temp->multiply($multiplier);\r
+ $temp = $temp->add(new Math_BigInteger($this->_int2bytes(substr($x, 0, 7)), 256));\r
+ $x = substr($x, 7);\r
+ }\r
+\r
+ $this->value = $temp->value;\r
+ }\r
+ break;\r
+ case 2: // base-2 support originally implemented by Lluis Pamies - thanks!\r
+ case -2:\r
+ if ($base > 0 && $x[0] == '-') {\r
+ $this->is_negative = true;\r
+ $x = substr($x, 1);\r
+ }\r
+\r
+ $x = preg_replace('#^([01]*).*#', '$1', $x);\r
+ $x = str_pad($x, strlen($x) + (3 * strlen($x)) % 4, 0, STR_PAD_LEFT);\r
+\r
+ $str = '0x';\r
+ while (strlen($x)) {\r
+ $part = substr($x, 0, 4);\r
+ $str.= dechex(bindec($part));\r
+ $x = substr($x, 4);\r
+ }\r
+\r
+ if ($this->is_negative) {\r
+ $str = '-' . $str;\r
+ }\r
+\r
+ $temp = new Math_BigInteger($str, 8 * $base); // ie. either -16 or +16\r
+ $this->value = $temp->value;\r
+ $this->is_negative = $temp->is_negative;\r
+\r
+ break;\r
+ default:\r
+ // base not supported, so we'll let $this == 0\r
+ }\r
+ }\r
+\r
+ /**\r
+ * Converts a BigInteger to a byte string (eg. base-256).\r
+ *\r
+ * Negative numbers are saved as positive numbers, unless $twos_compliment is set to true, at which point, they're\r
+ * saved as two's compliment.\r
+ *\r
+ * Here's an example:\r
+ * <code>\r
+ * <?php\r
+ * include('Math/BigInteger.php');\r
+ *\r
+ * $a = new Math_BigInteger('65');\r
+ *\r
+ * echo $a->toBytes(); // outputs chr(65)\r
+ * ?>\r
+ * </code>\r
+ *\r
+ * @param Boolean $twos_compliment\r
+ * @return String\r
+ * @access public\r
+ * @internal Converts a base-2**26 number to base-2**8\r
+ */\r
+ function toBytes($twos_compliment = false)\r
+ {\r
+ if ($twos_compliment) {\r
+ $comparison = $this->compare(new Math_BigInteger());\r
+ if ($comparison == 0) {\r
+ return $this->precision > 0 ? str_repeat(chr(0), ($this->precision + 1) >> 3) : '';\r
+ }\r
+\r
+ $temp = $comparison < 0 ? $this->add(new Math_BigInteger(1)) : $this->copy();\r
+ $bytes = $temp->toBytes();\r
+\r
+ if (empty($bytes)) { // eg. if the number we're trying to convert is -1\r
+ $bytes = chr(0);\r
+ }\r
+\r
+ if (ord($bytes[0]) & 0x80) {\r
+ $bytes = chr(0) . $bytes;\r
+ }\r
+\r
+ return $comparison < 0 ? ~$bytes : $bytes;\r
+ }\r
+\r
+ switch ( MATH_BIGINTEGER_MODE ) {\r
+ case MATH_BIGINTEGER_MODE_GMP:\r
+ if (gmp_cmp($this->value, gmp_init(0)) == 0) {\r
+ return $this->precision > 0 ? str_repeat(chr(0), ($this->precision + 1) >> 3) : '';\r
+ }\r
+\r
+ $temp = gmp_strval(gmp_abs($this->value), 16);\r
+ $temp = ( strlen($temp) & 1 ) ? '0' . $temp : $temp;\r
+ $temp = pack('H*', $temp);\r
+\r
+ return $this->precision > 0 ?\r
+ substr(str_pad($temp, $this->precision >> 3, chr(0), STR_PAD_LEFT), -($this->precision >> 3)) :\r
+ ltrim($temp, chr(0));\r
+ case MATH_BIGINTEGER_MODE_BCMATH:\r
+ if ($this->value === '0') {\r
+ return $this->precision > 0 ? str_repeat(chr(0), ($this->precision + 1) >> 3) : '';\r
+ }\r
+\r
+ $value = '';\r
+ $current = $this->value;\r
+\r
+ if ($current[0] == '-') {\r
+ $current = substr($current, 1);\r
+ }\r
+\r
+ while (bccomp($current, '0', 0) > 0) {\r
+ $temp = bcmod($current, '16777216');\r
+ $value = chr($temp >> 16) . chr($temp >> 8) . chr($temp) . $value;\r
+ $current = bcdiv($current, '16777216', 0);\r
+ }\r
+\r
+ return $this->precision > 0 ?\r
+ substr(str_pad($value, $this->precision >> 3, chr(0), STR_PAD_LEFT), -($this->precision >> 3)) :\r
+ ltrim($value, chr(0));\r
+ }\r
+\r
+ if (!count($this->value)) {\r
+ return $this->precision > 0 ? str_repeat(chr(0), ($this->precision + 1) >> 3) : '';\r
+ }\r
+ $result = $this->_int2bytes($this->value[count($this->value) - 1]);\r
+\r
+ $temp = $this->copy();\r
+\r
+ for ($i = count($temp->value) - 2; $i >= 0; --$i) {\r
+ $temp->_base256_lshift($result, 26);\r
+ $result = $result | str_pad($temp->_int2bytes($temp->value[$i]), strlen($result), chr(0), STR_PAD_LEFT);\r
+ }\r
+\r
+ return $this->precision > 0 ?\r
+ str_pad(substr($result, -(($this->precision + 7) >> 3)), ($this->precision + 7) >> 3, chr(0), STR_PAD_LEFT) :\r
+ $result;\r
+ }\r
+\r
+ /**\r
+ * Converts a BigInteger to a hex string (eg. base-16)).\r
+ *\r
+ * Negative numbers are saved as positive numbers, unless $twos_compliment is set to true, at which point, they're\r
+ * saved as two's compliment.\r
+ *\r
+ * Here's an example:\r
+ * <code>\r
+ * <?php\r
+ * include('Math/BigInteger.php');\r
+ *\r
+ * $a = new Math_BigInteger('65');\r
+ *\r
+ * echo $a->toHex(); // outputs '41'\r
+ * ?>\r
+ * </code>\r
+ *\r
+ * @param Boolean $twos_compliment\r
+ * @return String\r
+ * @access public\r
+ * @internal Converts a base-2**26 number to base-2**8\r
+ */\r
+ function toHex($twos_compliment = false)\r
+ {\r
+ return bin2hex($this->toBytes($twos_compliment));\r
+ }\r
+\r
+ /**\r
+ * Converts a BigInteger to a bit string (eg. base-2).\r
+ *\r
+ * Negative numbers are saved as positive numbers, unless $twos_compliment is set to true, at which point, they're\r
+ * saved as two's compliment.\r
+ *\r
+ * Here's an example:\r
+ * <code>\r
+ * <?php\r
+ * include('Math/BigInteger.php');\r
+ *\r
+ * $a = new Math_BigInteger('65');\r
+ *\r
+ * echo $a->toBits(); // outputs '1000001'\r
+ * ?>\r
+ * </code>\r
+ *\r
+ * @param Boolean $twos_compliment\r
+ * @return String\r
+ * @access public\r
+ * @internal Converts a base-2**26 number to base-2**2\r
+ */\r
+ function toBits($twos_compliment = false)\r
+ {\r
+ $hex = $this->toHex($twos_compliment);\r
+ $bits = '';\r
+ for ($i = 0; $i < strlen($hex); $i+=8) {\r
+ $bits.= str_pad(decbin(hexdec(substr($hex, $i, 8))), 32, '0', STR_PAD_LEFT);\r
+ }\r
+ return $this->precision > 0 ? substr($bits, -$this->precision) : ltrim($bits, '0');\r
+ }\r
+\r
+ /**\r
+ * Converts a BigInteger to a base-10 number.\r
+ *\r
+ * Here's an example:\r
+ * <code>\r
+ * <?php\r
+ * include('Math/BigInteger.php');\r
+ *\r
+ * $a = new Math_BigInteger('50');\r
+ *\r
+ * echo $a->toString(); // outputs 50\r
+ * ?>\r
+ * </code>\r
+ *\r
+ * @return String\r
+ * @access public\r
+ * @internal Converts a base-2**26 number to base-10**7 (which is pretty much base-10)\r
+ */\r
+ function toString()\r
+ {\r
+ switch ( MATH_BIGINTEGER_MODE ) {\r
+ case MATH_BIGINTEGER_MODE_GMP:\r
+ return gmp_strval($this->value);\r
+ case MATH_BIGINTEGER_MODE_BCMATH:\r
+ if ($this->value === '0') {\r
+ return '0';\r
+ }\r
+\r
+ return ltrim($this->value, '0');\r
+ }\r
+\r
+ if (!count($this->value)) {\r
+ return '0';\r
+ }\r
+\r
+ $temp = $this->copy();\r
+ $temp->is_negative = false;\r
+\r
+ $divisor = new Math_BigInteger();\r
+ $divisor->value = array(10000000); // eg. 10**7\r
+ $result = '';\r
+ while (count($temp->value)) {\r
+ list($temp, $mod) = $temp->divide($divisor);\r
+ $result = str_pad(isset($mod->value[0]) ? $mod->value[0] : '', 7, '0', STR_PAD_LEFT) . $result;\r
+ }\r
+ $result = ltrim($result, '0');\r
+ if (empty($result)) {\r
+ $result = '0';\r
+ }\r
+\r
+ if ($this->is_negative) {\r
+ $result = '-' . $result;\r
+ }\r
+\r
+ return $result;\r
+ }\r
+\r
+ /**\r
+ * Copy an object\r
+ *\r
+ * PHP5 passes objects by reference while PHP4 passes by value. As such, we need a function to guarantee\r
+ * that all objects are passed by value, when appropriate. More information can be found here:\r
+ *\r
+ * {@link http://php.net/language.oop5.basic#51624}\r
+ *\r
+ * @access public\r
+ * @see __clone()\r
+ * @return Math_BigInteger\r
+ */\r
+ function copy()\r
+ {\r
+ $temp = new Math_BigInteger();\r
+ $temp->value = $this->value;\r
+ $temp->is_negative = $this->is_negative;\r
+ $temp->generator = $this->generator;\r
+ $temp->precision = $this->precision;\r
+ $temp->bitmask = $this->bitmask;\r
+ return $temp;\r
+ }\r
+\r
+ /**\r
+ * __toString() magic method\r
+ *\r
+ * Will be called, automatically, if you're supporting just PHP5. If you're supporting PHP4, you'll need to call\r
+ * toString().\r
+ *\r
+ * @access public\r
+ * @internal Implemented per a suggestion by Techie-Michael - thanks!\r
+ */\r
+ function __toString()\r
+ {\r
+ return $this->toString();\r
+ }\r
+\r
+ /**\r
+ * __clone() magic method\r
+ *\r
+ * Although you can call Math_BigInteger::__toString() directly in PHP5, you cannot call Math_BigInteger::__clone()\r
+ * 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\r
+ * only syntax of $y = clone $x. As such, if you're trying to write an application that works on both PHP4 and PHP5,\r
+ * call Math_BigInteger::copy(), instead.\r
+ *\r
+ * @access public\r
+ * @see copy()\r
+ * @return Math_BigInteger\r
+ */\r
+ function __clone()\r
+ {\r
+ return $this->copy();\r
+ }\r
+\r
+ /**\r
+ * __sleep() magic method\r
+ *\r
+ * Will be called, automatically, when serialize() is called on a Math_BigInteger object.\r
+ *\r
+ * @see __wakeup()\r
+ * @access public\r
+ */\r
+ function __sleep()\r
+ {\r
+ $this->hex = $this->toHex(true);\r
+ $vars = array('hex');\r
+ if ($this->generator != 'mt_rand') {\r
+ $vars[] = 'generator';\r
+ }\r
+ if ($this->precision > 0) {\r
+ $vars[] = 'precision';\r
+ }\r
+ return $vars;\r
+ \r
+ }\r
+\r
+ /**\r
+ * __wakeup() magic method\r
+ *\r
+ * Will be called, automatically, when unserialize() is called on a Math_BigInteger object.\r
+ *\r
+ * @see __sleep()\r
+ * @access public\r
+ */\r
+ function __wakeup()\r
+ {\r
+ $temp = new Math_BigInteger($this->hex, -16);\r
+ $this->value = $temp->value;\r
+ $this->is_negative = $temp->is_negative;\r
+ $this->setRandomGenerator($this->generator);\r
+ if ($this->precision > 0) {\r
+ // recalculate $this->bitmask\r
+ $this->setPrecision($this->precision);\r
+ }\r
+ }\r
+\r
+ /**\r
+ * Adds two BigIntegers.\r
+ *\r
+ * Here's an example:\r
+ * <code>\r
+ * <?php\r
+ * include('Math/BigInteger.php');\r
+ *\r
+ * $a = new Math_BigInteger('10');\r
+ * $b = new Math_BigInteger('20');\r
+ *\r
+ * $c = $a->add($b);\r
+ *\r
+ * echo $c->toString(); // outputs 30\r
+ * ?>\r
+ * </code>\r
+ *\r
+ * @param Math_BigInteger $y\r
+ * @return Math_BigInteger\r
+ * @access public\r
+ * @internal Performs base-2**52 addition\r
+ */\r
+ function add($y)\r
+ {\r
+ switch ( MATH_BIGINTEGER_MODE ) {\r
+ case MATH_BIGINTEGER_MODE_GMP:\r
+ $temp = new Math_BigInteger();\r
+ $temp->value = gmp_add($this->value, $y->value);\r
+\r
+ return $this->_normalize($temp);\r
+ case MATH_BIGINTEGER_MODE_BCMATH:\r
+ $temp = new Math_BigInteger();\r
+ $temp->value = bcadd($this->value, $y->value, 0);\r
+\r
+ return $this->_normalize($temp);\r
+ }\r
+\r
+ $temp = $this->_add($this->value, $this->is_negative, $y->value, $y->is_negative);\r
+\r
+ $result = new Math_BigInteger();\r
+ $result->value = $temp[MATH_BIGINTEGER_VALUE];\r
+ $result->is_negative = $temp[MATH_BIGINTEGER_SIGN];\r
+\r
+ return $this->_normalize($result);\r
+ }\r
+\r
+ /**\r
+ * Performs addition.\r
+ *\r
+ * @param Array $x_value\r
+ * @param Boolean $x_negative\r
+ * @param Array $y_value\r
+ * @param Boolean $y_negative\r
+ * @return Array\r
+ * @access private\r
+ */\r
+ function _add($x_value, $x_negative, $y_value, $y_negative)\r
+ {\r
+ $x_size = count($x_value);\r
+ $y_size = count($y_value);\r
+\r
+ if ($x_size == 0) {\r
+ return array(\r
+ MATH_BIGINTEGER_VALUE => $y_value,\r
+ MATH_BIGINTEGER_SIGN => $y_negative\r
+ );\r
+ } else if ($y_size == 0) {\r
+ return array(\r
+ MATH_BIGINTEGER_VALUE => $x_value,\r
+ MATH_BIGINTEGER_SIGN => $x_negative\r
+ );\r
+ }\r
+\r
+ // subtract, if appropriate\r
+ if ( $x_negative != $y_negative ) {\r
+ if ( $x_value == $y_value ) {\r
+ return array(\r
+ MATH_BIGINTEGER_VALUE => array(),\r
+ MATH_BIGINTEGER_SIGN => false\r
+ );\r
+ }\r
+\r
+ $temp = $this->_subtract($x_value, false, $y_value, false);\r
+ $temp[MATH_BIGINTEGER_SIGN] = $this->_compare($x_value, false, $y_value, false) > 0 ?\r
+ $x_negative : $y_negative;\r
+\r
+ return $temp;\r
+ }\r
+\r
+ if ($x_size < $y_size) {\r
+ $size = $x_size;\r
+ $value = $y_value;\r
+ } else {\r
+ $size = $y_size;\r
+ $value = $x_value;\r
+ }\r
+\r
+ $value[] = 0; // just in case the carry adds an extra digit\r
+\r
+ $carry = 0;\r
+ for ($i = 0, $j = 1; $j < $size; $i+=2, $j+=2) {\r
+ $sum = $x_value[$j] * 0x4000000 + $x_value[$i] + $y_value[$j] * 0x4000000 + $y_value[$i] + $carry;\r
+ $carry = $sum >= MATH_BIGINTEGER_MAX_DIGIT52; // eg. floor($sum / 2**52); only possible values (in any base) are 0 and 1\r
+ $sum = $carry ? $sum - MATH_BIGINTEGER_MAX_DIGIT52 : $sum;\r
+\r
+ $temp = (int) ($sum / 0x4000000);\r
+\r
+ $value[$i] = (int) ($sum - 0x4000000 * $temp); // eg. a faster alternative to fmod($sum, 0x4000000)\r
+ $value[$j] = $temp;\r
+ }\r
+\r
+ if ($j == $size) { // ie. if $y_size is odd\r
+ $sum = $x_value[$i] + $y_value[$i] + $carry;\r
+ $carry = $sum >= 0x4000000;\r
+ $value[$i] = $carry ? $sum - 0x4000000 : $sum;\r
+ ++$i; // ie. let $i = $j since we've just done $value[$i]\r
+ }\r
+\r
+ if ($carry) {\r
+ for (; $value[$i] == 0x3FFFFFF; ++$i) {\r
+ $value[$i] = 0;\r
+ }\r
+ ++$value[$i];\r
+ }\r
+\r
+ return array(\r
+ MATH_BIGINTEGER_VALUE => $this->_trim($value),\r
+ MATH_BIGINTEGER_SIGN => $x_negative\r
+ );\r
+ }\r
+\r
+ /**\r
+ * Subtracts two BigIntegers.\r
+ *\r
+ * Here's an example:\r
+ * <code>\r
+ * <?php\r
+ * include('Math/BigInteger.php');\r
+ *\r
+ * $a = new Math_BigInteger('10');\r
+ * $b = new Math_BigInteger('20');\r
+ *\r
+ * $c = $a->subtract($b);\r
+ *\r
+ * echo $c->toString(); // outputs -10\r
+ * ?>\r
+ * </code>\r
+ *\r
+ * @param Math_BigInteger $y\r
+ * @return Math_BigInteger\r
+ * @access public\r
+ * @internal Performs base-2**52 subtraction\r
+ */\r
+ function subtract($y)\r
+ {\r
+ switch ( MATH_BIGINTEGER_MODE ) {\r
+ case MATH_BIGINTEGER_MODE_GMP:\r
+ $temp = new Math_BigInteger();\r
+ $temp->value = gmp_sub($this->value, $y->value);\r
+\r
+ return $this->_normalize($temp);\r
+ case MATH_BIGINTEGER_MODE_BCMATH:\r
+ $temp = new Math_BigInteger();\r
+ $temp->value = bcsub($this->value, $y->value, 0);\r
+\r
+ return $this->_normalize($temp);\r
+ }\r
+\r
+ $temp = $this->_subtract($this->value, $this->is_negative, $y->value, $y->is_negative);\r
+\r
+ $result = new Math_BigInteger();\r
+ $result->value = $temp[MATH_BIGINTEGER_VALUE];\r
+ $result->is_negative = $temp[MATH_BIGINTEGER_SIGN];\r
+\r
+ return $this->_normalize($result);\r
+ }\r
+\r
+ /**\r
+ * Performs subtraction.\r
+ *\r
+ * @param Array $x_value\r
+ * @param Boolean $x_negative\r
+ * @param Array $y_value\r
+ * @param Boolean $y_negative\r
+ * @return Array\r
+ * @access private\r
+ */\r
+ function _subtract($x_value, $x_negative, $y_value, $y_negative)\r
+ {\r
+ $x_size = count($x_value);\r
+ $y_size = count($y_value);\r
+\r
+ if ($x_size == 0) {\r
+ return array(\r
+ MATH_BIGINTEGER_VALUE => $y_value,\r
+ MATH_BIGINTEGER_SIGN => !$y_negative\r
+ );\r
+ } else if ($y_size == 0) {\r
+ return array(\r
+ MATH_BIGINTEGER_VALUE => $x_value,\r
+ MATH_BIGINTEGER_SIGN => $x_negative\r
+ );\r
+ }\r
+\r
+ // add, if appropriate (ie. -$x - +$y or +$x - -$y)\r
+ if ( $x_negative != $y_negative ) {\r
+ $temp = $this->_add($x_value, false, $y_value, false);\r
+ $temp[MATH_BIGINTEGER_SIGN] = $x_negative;\r
+\r
+ return $temp;\r
+ }\r
+\r
+ $diff = $this->_compare($x_value, $x_negative, $y_value, $y_negative);\r
+\r
+ if ( !$diff ) {\r
+ return array(\r
+ MATH_BIGINTEGER_VALUE => array(),\r
+ MATH_BIGINTEGER_SIGN => false\r
+ );\r
+ }\r
+\r
+ // switch $x and $y around, if appropriate.\r
+ if ( (!$x_negative && $diff < 0) || ($x_negative && $diff > 0) ) {\r
+ $temp = $x_value;\r
+ $x_value = $y_value;\r
+ $y_value = $temp;\r
+\r
+ $x_negative = !$x_negative;\r
+\r
+ $x_size = count($x_value);\r
+ $y_size = count($y_value);\r
+ }\r
+\r
+ // at this point, $x_value should be at least as big as - if not bigger than - $y_value\r
+\r
+ $carry = 0;\r
+ for ($i = 0, $j = 1; $j < $y_size; $i+=2, $j+=2) {\r
+ $sum = $x_value[$j] * 0x4000000 + $x_value[$i] - $y_value[$j] * 0x4000000 - $y_value[$i] - $carry;\r
+ $carry = $sum < 0; // eg. floor($sum / 2**52); only possible values (in any base) are 0 and 1\r
+ $sum = $carry ? $sum + MATH_BIGINTEGER_MAX_DIGIT52 : $sum;\r
+\r
+ $temp = (int) ($sum / 0x4000000);\r
+\r
+ $x_value[$i] = (int) ($sum - 0x4000000 * $temp);\r
+ $x_value[$j] = $temp;\r
+ }\r
+\r
+ if ($j == $y_size) { // ie. if $y_size is odd\r
+ $sum = $x_value[$i] - $y_value[$i] - $carry;\r
+ $carry = $sum < 0;\r
+ $x_value[$i] = $carry ? $sum + 0x4000000 : $sum;\r
+ ++$i;\r
+ }\r
+\r
+ if ($carry) {\r
+ for (; !$x_value[$i]; ++$i) {\r
+ $x_value[$i] = 0x3FFFFFF;\r
+ }\r
+ --$x_value[$i];\r
+ }\r
+\r
+ return array(\r
+ MATH_BIGINTEGER_VALUE => $this->_trim($x_value),\r
+ MATH_BIGINTEGER_SIGN => $x_negative\r
+ );\r
+ }\r
+\r
+ /**\r
+ * Multiplies two BigIntegers\r
+ *\r
+ * Here's an example:\r
+ * <code>\r
+ * <?php\r
+ * include('Math/BigInteger.php');\r
+ *\r
+ * $a = new Math_BigInteger('10');\r
+ * $b = new Math_BigInteger('20');\r
+ *\r
+ * $c = $a->multiply($b);\r
+ *\r
+ * echo $c->toString(); // outputs 200\r
+ * ?>\r
+ * </code>\r
+ *\r
+ * @param Math_BigInteger $x\r
+ * @return Math_BigInteger\r
+ * @access public\r
+ */\r
+ function multiply($x)\r
+ {\r
+ switch ( MATH_BIGINTEGER_MODE ) {\r
+ case MATH_BIGINTEGER_MODE_GMP:\r
+ $temp = new Math_BigInteger();\r
+ $temp->value = gmp_mul($this->value, $x->value);\r
+\r
+ return $this->_normalize($temp);\r
+ case MATH_BIGINTEGER_MODE_BCMATH:\r
+ $temp = new Math_BigInteger();\r
+ $temp->value = bcmul($this->value, $x->value, 0);\r
+\r
+ return $this->_normalize($temp);\r
+ }\r
+\r
+ $temp = $this->_multiply($this->value, $this->is_negative, $x->value, $x->is_negative);\r
+\r
+ $product = new Math_BigInteger();\r
+ $product->value = $temp[MATH_BIGINTEGER_VALUE];\r
+ $product->is_negative = $temp[MATH_BIGINTEGER_SIGN];\r
+\r
+ return $this->_normalize($product);\r
+ }\r
+\r
+ /**\r
+ * Performs multiplication.\r
+ *\r
+ * @param Array $x_value\r
+ * @param Boolean $x_negative\r
+ * @param Array $y_value\r
+ * @param Boolean $y_negative\r
+ * @return Array\r
+ * @access private\r
+ */\r
+ function _multiply($x_value, $x_negative, $y_value, $y_negative)\r
+ {\r
+ //if ( $x_value == $y_value ) {\r
+ // return array(\r
+ // MATH_BIGINTEGER_VALUE => $this->_square($x_value),\r
+ // MATH_BIGINTEGER_SIGN => $x_sign != $y_value\r
+ // );\r
+ //}\r
+\r
+ $x_length = count($x_value);\r
+ $y_length = count($y_value);\r
+\r
+ if ( !$x_length || !$y_length ) { // a 0 is being multiplied\r
+ return array(\r
+ MATH_BIGINTEGER_VALUE => array(),\r
+ MATH_BIGINTEGER_SIGN => false\r
+ );\r
+ }\r
+\r
+ return array(\r
+ MATH_BIGINTEGER_VALUE => min($x_length, $y_length) < 2 * MATH_BIGINTEGER_KARATSUBA_CUTOFF ?\r
+ $this->_trim($this->_regularMultiply($x_value, $y_value)) :\r
+ $this->_trim($this->_karatsuba($x_value, $y_value)),\r
+ MATH_BIGINTEGER_SIGN => $x_negative != $y_negative\r
+ );\r
+ }\r
+\r
+ /**\r
+ * Performs long multiplication on two BigIntegers\r
+ *\r
+ * Modeled after 'multiply' in MutableBigInteger.java.\r
+ *\r
+ * @param Array $x_value\r
+ * @param Array $y_value\r
+ * @return Array\r
+ * @access private\r
+ */\r
+ function _regularMultiply($x_value, $y_value)\r
+ {\r
+ $x_length = count($x_value);\r
+ $y_length = count($y_value);\r
+\r
+ if ( !$x_length || !$y_length ) { // a 0 is being multiplied\r
+ return array();\r
+ }\r
+\r
+ if ( $x_length < $y_length ) {\r
+ $temp = $x_value;\r
+ $x_value = $y_value;\r
+ $y_value = $temp;\r
+\r
+ $x_length = count($x_value);\r
+ $y_length = count($y_value);\r
+ }\r
+\r
+ $product_value = $this->_array_repeat(0, $x_length + $y_length);\r
+\r
+ // the following for loop could be removed if the for loop following it\r
+ // (the one with nested for loops) initially set $i to 0, but\r
+ // doing so would also make the result in one set of unnecessary adds,\r
+ // since on the outermost loops first pass, $product->value[$k] is going\r
+ // to always be 0\r
+\r
+ $carry = 0;\r
+\r
+ for ($j = 0; $j < $x_length; ++$j) { // ie. $i = 0\r
+ $temp = $x_value[$j] * $y_value[0] + $carry; // $product_value[$k] == 0\r
+ $carry = (int) ($temp / 0x4000000);\r
+ $product_value[$j] = (int) ($temp - 0x4000000 * $carry);\r
+ }\r
+\r
+ $product_value[$j] = $carry;\r
+\r
+ // the above for loop is what the previous comment was talking about. the\r
+ // following for loop is the "one with nested for loops"\r
+ for ($i = 1; $i < $y_length; ++$i) {\r
+ $carry = 0;\r
+\r
+ for ($j = 0, $k = $i; $j < $x_length; ++$j, ++$k) {\r
+ $temp = $product_value[$k] + $x_value[$j] * $y_value[$i] + $carry;\r
+ $carry = (int) ($temp / 0x4000000);\r
+ $product_value[$k] = (int) ($temp - 0x4000000 * $carry);\r
+ }\r
+\r
+ $product_value[$k] = $carry;\r
+ }\r
+\r
+ return $product_value;\r
+ }\r
+\r
+ /**\r
+ * Performs Karatsuba multiplication on two BigIntegers\r
+ *\r
+ * See {@link http://en.wikipedia.org/wiki/Karatsuba_algorithm Karatsuba algorithm} and\r
+ * {@link http://math.libtomcrypt.com/files/tommath.pdf#page=120 MPM 5.2.3}.\r
+ *\r
+ * @param Array $x_value\r
+ * @param Array $y_value\r
+ * @return Array\r
+ * @access private\r
+ */\r
+ function _karatsuba($x_value, $y_value)\r
+ {\r
+ $m = min(count($x_value) >> 1, count($y_value) >> 1);\r
+\r
+ if ($m < MATH_BIGINTEGER_KARATSUBA_CUTOFF) {\r
+ return $this->_regularMultiply($x_value, $y_value);\r
+ }\r
+\r
+ $x1 = array_slice($x_value, $m);\r
+ $x0 = array_slice($x_value, 0, $m);\r
+ $y1 = array_slice($y_value, $m);\r
+ $y0 = array_slice($y_value, 0, $m);\r
+\r
+ $z2 = $this->_karatsuba($x1, $y1);\r
+ $z0 = $this->_karatsuba($x0, $y0);\r
+\r
+ $z1 = $this->_add($x1, false, $x0, false);\r
+ $temp = $this->_add($y1, false, $y0, false);\r
+ $z1 = $this->_karatsuba($z1[MATH_BIGINTEGER_VALUE], $temp[MATH_BIGINTEGER_VALUE]);\r
+ $temp = $this->_add($z2, false, $z0, false);\r
+ $z1 = $this->_subtract($z1, false, $temp[MATH_BIGINTEGER_VALUE], false);\r
+\r
+ $z2 = array_merge(array_fill(0, 2 * $m, 0), $z2);\r
+ $z1[MATH_BIGINTEGER_VALUE] = array_merge(array_fill(0, $m, 0), $z1[MATH_BIGINTEGER_VALUE]);\r
+\r
+ $xy = $this->_add($z2, false, $z1[MATH_BIGINTEGER_VALUE], $z1[MATH_BIGINTEGER_SIGN]);\r
+ $xy = $this->_add($xy[MATH_BIGINTEGER_VALUE], $xy[MATH_BIGINTEGER_SIGN], $z0, false);\r
+\r
+ return $xy[MATH_BIGINTEGER_VALUE];\r
+ }\r
+\r
+ /**\r
+ * Performs squaring\r
+ *\r
+ * @param Array $x\r
+ * @return Array\r
+ * @access private\r
+ */\r
+ function _square($x = false)\r
+ {\r
+ return count($x) < 2 * MATH_BIGINTEGER_KARATSUBA_CUTOFF ?\r
+ $this->_trim($this->_baseSquare($x)) :\r
+ $this->_trim($this->_karatsubaSquare($x));\r
+ }\r
+\r
+ /**\r
+ * Performs traditional squaring on two BigIntegers\r
+ *\r
+ * Squaring can be done faster than multiplying a number by itself can be. See\r
+ * {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=7 HAC 14.2.4} /\r
+ * {@link http://math.libtomcrypt.com/files/tommath.pdf#page=141 MPM 5.3} for more information.\r
+ *\r
+ * @param Array $value\r
+ * @return Array\r
+ * @access private\r
+ */\r
+ function _baseSquare($value)\r
+ {\r
+ if ( empty($value) ) {\r
+ return array();\r
+ }\r
+ $square_value = $this->_array_repeat(0, 2 * count($value));\r
+\r
+ for ($i = 0, $max_index = count($value) - 1; $i <= $max_index; ++$i) {\r
+ $i2 = $i << 1;\r
+\r
+ $temp = $square_value[$i2] + $value[$i] * $value[$i];\r
+ $carry = (int) ($temp / 0x4000000);\r
+ $square_value[$i2] = (int) ($temp - 0x4000000 * $carry);\r
+\r
+ // note how we start from $i+1 instead of 0 as we do in multiplication.\r
+ for ($j = $i + 1, $k = $i2 + 1; $j <= $max_index; ++$j, ++$k) {\r
+ $temp = $square_value[$k] + 2 * $value[$j] * $value[$i] + $carry;\r
+ $carry = (int) ($temp / 0x4000000);\r
+ $square_value[$k] = (int) ($temp - 0x4000000 * $carry);\r
+ }\r
+\r
+ // the following line can yield values larger 2**15. at this point, PHP should switch\r
+ // over to floats.\r
+ $square_value[$i + $max_index + 1] = $carry;\r
+ }\r
+\r
+ return $square_value;\r
+ }\r
+\r
+ /**\r
+ * Performs Karatsuba "squaring" on two BigIntegers\r
+ *\r
+ * See {@link http://en.wikipedia.org/wiki/Karatsuba_algorithm Karatsuba algorithm} and\r
+ * {@link http://math.libtomcrypt.com/files/tommath.pdf#page=151 MPM 5.3.4}.\r
+ *\r
+ * @param Array $value\r
+ * @return Array\r
+ * @access private\r
+ */\r
+ function _karatsubaSquare($value)\r
+ {\r
+ $m = count($value) >> 1;\r
+\r
+ if ($m < MATH_BIGINTEGER_KARATSUBA_CUTOFF) {\r
+ return $this->_baseSquare($value);\r
+ }\r
+\r
+ $x1 = array_slice($value, $m);\r
+ $x0 = array_slice($value, 0, $m);\r
+\r
+ $z2 = $this->_karatsubaSquare($x1);\r
+ $z0 = $this->_karatsubaSquare($x0);\r
+\r
+ $z1 = $this->_add($x1, false, $x0, false);\r
+ $z1 = $this->_karatsubaSquare($z1[MATH_BIGINTEGER_VALUE]);\r
+ $temp = $this->_add($z2, false, $z0, false);\r
+ $z1 = $this->_subtract($z1, false, $temp[MATH_BIGINTEGER_VALUE], false);\r
+\r
+ $z2 = array_merge(array_fill(0, 2 * $m, 0), $z2);\r
+ $z1[MATH_BIGINTEGER_VALUE] = array_merge(array_fill(0, $m, 0), $z1[MATH_BIGINTEGER_VALUE]);\r
+\r
+ $xx = $this->_add($z2, false, $z1[MATH_BIGINTEGER_VALUE], $z1[MATH_BIGINTEGER_SIGN]);\r
+ $xx = $this->_add($xx[MATH_BIGINTEGER_VALUE], $xx[MATH_BIGINTEGER_SIGN], $z0, false);\r
+\r
+ return $xx[MATH_BIGINTEGER_VALUE];\r
+ }\r
+\r
+ /**\r
+ * Divides two BigIntegers.\r
+ *\r
+ * Returns an array whose first element contains the quotient and whose second element contains the\r
+ * "common residue". If the remainder would be positive, the "common residue" and the remainder are the\r
+ * same. If the remainder would be negative, the "common residue" is equal to the sum of the remainder\r
+ * and the divisor (basically, the "common residue" is the first positive modulo).\r
+ *\r
+ * Here's an example:\r
+ * <code>\r
+ * <?php\r
+ * include('Math/BigInteger.php');\r
+ *\r
+ * $a = new Math_BigInteger('10');\r
+ * $b = new Math_BigInteger('20');\r
+ *\r
+ * list($quotient, $remainder) = $a->divide($b);\r
+ *\r
+ * echo $quotient->toString(); // outputs 0\r
+ * echo "\r\n";\r
+ * echo $remainder->toString(); // outputs 10\r
+ * ?>\r
+ * </code>\r
+ *\r
+ * @param Math_BigInteger $y\r
+ * @return Array\r
+ * @access public\r
+ * @internal This function is based off of {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=9 HAC 14.20}.\r
+ */\r
+ function divide($y)\r
+ {\r
+ switch ( MATH_BIGINTEGER_MODE ) {\r
+ case MATH_BIGINTEGER_MODE_GMP:\r
+ $quotient = new Math_BigInteger();\r
+ $remainder = new Math_BigInteger();\r
+\r
+ list($quotient->value, $remainder->value) = gmp_div_qr($this->value, $y->value);\r
+\r
+ if (gmp_sign($remainder->value) < 0) {\r
+ $remainder->value = gmp_add($remainder->value, gmp_abs($y->value));\r
+ }\r
+\r
+ return array($this->_normalize($quotient), $this->_normalize($remainder));\r
+ case MATH_BIGINTEGER_MODE_BCMATH:\r
+ $quotient = new Math_BigInteger();\r
+ $remainder = new Math_BigInteger();\r
+\r
+ $quotient->value = bcdiv($this->value, $y->value, 0);\r
+ $remainder->value = bcmod($this->value, $y->value);\r
+\r
+ if ($remainder->value[0] == '-') {\r
+ $remainder->value = bcadd($remainder->value, $y->value[0] == '-' ? substr($y->value, 1) : $y->value, 0);\r
+ }\r
+\r
+ return array($this->_normalize($quotient), $this->_normalize($remainder));\r
+ }\r
+\r
+ if (count($y->value) == 1) {\r
+ list($q, $r) = $this->_divide_digit($this->value, $y->value[0]);\r
+ $quotient = new Math_BigInteger();\r
+ $remainder = new Math_BigInteger();\r
+ $quotient->value = $q;\r
+ $remainder->value = array($r);\r
+ $quotient->is_negative = $this->is_negative != $y->is_negative;\r
+ return array($this->_normalize($quotient), $this->_normalize($remainder));\r
+ }\r
+\r
+ static $zero;\r
+ if ( !isset($zero) ) {\r
+ $zero = new Math_BigInteger();\r
+ }\r
+\r
+ $x = $this->copy();\r
+ $y = $y->copy();\r
+\r
+ $x_sign = $x->is_negative;\r
+ $y_sign = $y->is_negative;\r
+\r
+ $x->is_negative = $y->is_negative = false;\r
+\r
+ $diff = $x->compare($y);\r
+\r
+ if ( !$diff ) {\r
+ $temp = new Math_BigInteger();\r
+ $temp->value = array(1);\r
+ $temp->is_negative = $x_sign != $y_sign;\r
+ return array($this->_normalize($temp), $this->_normalize(new Math_BigInteger()));\r
+ }\r
+\r
+ if ( $diff < 0 ) {\r
+ // if $x is negative, "add" $y.\r
+ if ( $x_sign ) {\r
+ $x = $y->subtract($x);\r
+ }\r
+ return array($this->_normalize(new Math_BigInteger()), $this->_normalize($x));\r
+ }\r
+\r
+ // normalize $x and $y as described in HAC 14.23 / 14.24\r
+ $msb = $y->value[count($y->value) - 1];\r
+ for ($shift = 0; !($msb & 0x2000000); ++$shift) {\r
+ $msb <<= 1;\r
+ }\r
+ $x->_lshift($shift);\r
+ $y->_lshift($shift);\r
+ $y_value = &$y->value;\r
+\r
+ $x_max = count($x->value) - 1;\r
+ $y_max = count($y->value) - 1;\r
+\r
+ $quotient = new Math_BigInteger();\r
+ $quotient_value = &$quotient->value;\r
+ $quotient_value = $this->_array_repeat(0, $x_max - $y_max + 1);\r
+\r
+ static $temp, $lhs, $rhs;\r
+ if (!isset($temp)) {\r
+ $temp = new Math_BigInteger();\r
+ $lhs = new Math_BigInteger();\r
+ $rhs = new Math_BigInteger();\r
+ }\r
+ $temp_value = &$temp->value;\r
+ $rhs_value = &$rhs->value;\r
+\r
+ // $temp = $y << ($x_max - $y_max-1) in base 2**26\r
+ $temp_value = array_merge($this->_array_repeat(0, $x_max - $y_max), $y_value);\r
+\r
+ while ( $x->compare($temp) >= 0 ) {\r
+ // calculate the "common residue"\r
+ ++$quotient_value[$x_max - $y_max];\r
+ $x = $x->subtract($temp);\r
+ $x_max = count($x->value) - 1;\r
+ }\r
+\r
+ for ($i = $x_max; $i >= $y_max + 1; --$i) {\r
+ $x_value = &$x->value;\r
+ $x_window = array(\r
+ isset($x_value[$i]) ? $x_value[$i] : 0,\r
+ isset($x_value[$i - 1]) ? $x_value[$i - 1] : 0,\r
+ isset($x_value[$i - 2]) ? $x_value[$i - 2] : 0\r
+ );\r
+ $y_window = array(\r
+ $y_value[$y_max],\r
+ ( $y_max > 0 ) ? $y_value[$y_max - 1] : 0\r
+ );\r
+\r
+ $q_index = $i - $y_max - 1;\r
+ if ($x_window[0] == $y_window[0]) {\r
+ $quotient_value[$q_index] = 0x3FFFFFF;\r
+ } else {\r
+ $quotient_value[$q_index] = (int) (\r
+ ($x_window[0] * 0x4000000 + $x_window[1])\r
+ /\r
+ $y_window[0]\r
+ );\r
+ }\r
+\r
+ $temp_value = array($y_window[1], $y_window[0]);\r
+\r
+ $lhs->value = array($quotient_value[$q_index]);\r
+ $lhs = $lhs->multiply($temp);\r
+\r
+ $rhs_value = array($x_window[2], $x_window[1], $x_window[0]);\r
+\r
+ while ( $lhs->compare($rhs) > 0 ) {\r
+ --$quotient_value[$q_index];\r
+\r
+ $lhs->value = array($quotient_value[$q_index]);\r
+ $lhs = $lhs->multiply($temp);\r
+ }\r
+\r
+ $adjust = $this->_array_repeat(0, $q_index);\r
+ $temp_value = array($quotient_value[$q_index]);\r
+ $temp = $temp->multiply($y);\r
+ $temp_value = &$temp->value;\r
+ $temp_value = array_merge($adjust, $temp_value);\r
+\r
+ $x = $x->subtract($temp);\r
+\r
+ if ($x->compare($zero) < 0) {\r
+ $temp_value = array_merge($adjust, $y_value);\r
+ $x = $x->add($temp);\r
+\r
+ --$quotient_value[$q_index];\r
+ }\r
+\r
+ $x_max = count($x_value) - 1;\r
+ }\r
+\r
+ // unnormalize the remainder\r
+ $x->_rshift($shift);\r
+\r
+ $quotient->is_negative = $x_sign != $y_sign;\r
+\r
+ // calculate the "common residue", if appropriate\r
+ if ( $x_sign ) {\r
+ $y->_rshift($shift);\r
+ $x = $y->subtract($x);\r
+ }\r
+\r
+ return array($this->_normalize($quotient), $this->_normalize($x));\r
+ }\r
+\r
+ /**\r
+ * Divides a BigInteger by a regular integer\r
+ *\r
+ * abc / x = a00 / x + b0 / x + c / x\r
+ *\r
+ * @param Array $dividend\r
+ * @param Array $divisor\r
+ * @return Array\r
+ * @access private\r
+ */\r
+ function _divide_digit($dividend, $divisor)\r
+ {\r
+ $carry = 0;\r
+ $result = array();\r
+\r
+ for ($i = count($dividend) - 1; $i >= 0; --$i) {\r
+ $temp = 0x4000000 * $carry + $dividend[$i];\r
+ $result[$i] = (int) ($temp / $divisor);\r
+ $carry = (int) ($temp - $divisor * $result[$i]);\r
+ }\r
+\r
+ return array($result, $carry);\r
+ }\r
+\r
+ /**\r
+ * Performs modular exponentiation.\r
+ *\r
+ * Here's an example:\r
+ * <code>\r
+ * <?php\r
+ * include('Math/BigInteger.php');\r
+ *\r
+ * $a = new Math_BigInteger('10');\r
+ * $b = new Math_BigInteger('20');\r
+ * $c = new Math_BigInteger('30');\r
+ *\r
+ * $c = $a->modPow($b, $c);\r
+ *\r
+ * echo $c->toString(); // outputs 10\r
+ * ?>\r
+ * </code>\r
+ *\r
+ * @param Math_BigInteger $e\r
+ * @param Math_BigInteger $n\r
+ * @return Math_BigInteger\r
+ * @access public\r
+ * @internal The most naive approach to modular exponentiation has very unreasonable requirements, and\r
+ * and although the approach involving repeated squaring does vastly better, it, too, is impractical\r
+ * for our purposes. The reason being that division - by far the most complicated and time-consuming\r
+ * of the basic operations (eg. +,-,*,/) - occurs multiple times within it.\r
+ *\r
+ * Modular reductions resolve this issue. Although an individual modular reduction takes more time\r
+ * then an individual division, when performed in succession (with the same modulo), they're a lot faster.\r
+ *\r
+ * The two most commonly used modular reductions are Barrett and Montgomery reduction. Montgomery reduction,\r
+ * although faster, only works when the gcd of the modulo and of the base being used is 1. In RSA, when the\r
+ * base is a power of two, the modulo - a product of two primes - is always going to have a gcd of 1 (because\r
+ * the product of two odd numbers is odd), but what about when RSA isn't used?\r
+ *\r
+ * In contrast, Barrett reduction has no such constraint. As such, some bigint implementations perform a\r
+ * Barrett reduction after every operation in the modpow function. Others perform Barrett reductions when the\r
+ * modulo is even and Montgomery reductions when the modulo is odd. BigInteger.java's modPow method, however,\r
+ * uses a trick involving the Chinese Remainder Theorem to factor the even modulo into two numbers - one odd and\r
+ * the other, a power of two - and recombine them, later. This is the method that this modPow function uses.\r
+ * {@link http://islab.oregonstate.edu/papers/j34monex.pdf Montgomery Reduction with Even Modulus} elaborates.\r
+ */\r
+ function modPow($e, $n)\r
+ {\r
+ $n = $this->bitmask !== false && $this->bitmask->compare($n) < 0 ? $this->bitmask : $n->abs();\r
+\r
+ if ($e->compare(new Math_BigInteger()) < 0) {\r
+ $e = $e->abs();\r
+\r
+ $temp = $this->modInverse($n);\r
+ if ($temp === false) {\r
+ return false;\r
+ }\r
+\r
+ return $this->_normalize($temp->modPow($e, $n));\r
+ }\r
+\r
+ switch ( MATH_BIGINTEGER_MODE ) {\r
+ case MATH_BIGINTEGER_MODE_GMP:\r
+ $temp = new Math_BigInteger();\r
+ $temp->value = gmp_powm($this->value, $e->value, $n->value);\r
+\r
+ return $this->_normalize($temp);\r
+ case MATH_BIGINTEGER_MODE_BCMATH:\r
+ $temp = new Math_BigInteger();\r
+ $temp->value = bcpowmod($this->value, $e->value, $n->value, 0);\r
+\r
+ return $this->_normalize($temp);\r
+ }\r
+\r
+ if ( empty($e->value) ) {\r
+ $temp = new Math_BigInteger();\r
+ $temp->value = array(1);\r
+ return $this->_normalize($temp);\r
+ }\r
+\r
+ if ( $e->value == array(1) ) {\r
+ list(, $temp) = $this->divide($n);\r
+ return $this->_normalize($temp);\r
+ }\r
+\r
+ if ( $e->value == array(2) ) {\r
+ $temp = new Math_BigInteger();\r
+ $temp->value = $this->_square($this->value);\r
+ list(, $temp) = $temp->divide($n);\r
+ return $this->_normalize($temp);\r
+ }\r
+\r
+ return $this->_normalize($this->_slidingWindow($e, $n, MATH_BIGINTEGER_BARRETT));\r
+\r
+ // is the modulo odd?\r
+ if ( $n->value[0] & 1 ) {\r
+ return $this->_normalize($this->_slidingWindow($e, $n, MATH_BIGINTEGER_MONTGOMERY));\r
+ }\r
+ // if it's not, it's even\r
+\r
+ // find the lowest set bit (eg. the max pow of 2 that divides $n)\r
+ for ($i = 0; $i < count($n->value); ++$i) {\r
+ if ( $n->value[$i] ) {\r
+ $temp = decbin($n->value[$i]);\r
+ $j = strlen($temp) - strrpos($temp, '1') - 1;\r
+ $j+= 26 * $i;\r
+ break;\r
+ }\r
+ }\r
+ // at this point, 2^$j * $n/(2^$j) == $n\r
+\r
+ $mod1 = $n->copy();\r
+ $mod1->_rshift($j);\r
+ $mod2 = new Math_BigInteger();\r
+ $mod2->value = array(1);\r
+ $mod2->_lshift($j);\r
+\r
+ $part1 = ( $mod1->value != array(1) ) ? $this->_slidingWindow($e, $mod1, MATH_BIGINTEGER_MONTGOMERY) : new Math_BigInteger();\r
+ $part2 = $this->_slidingWindow($e, $mod2, MATH_BIGINTEGER_POWEROF2);\r
+\r
+ $y1 = $mod2->modInverse($mod1);\r
+ $y2 = $mod1->modInverse($mod2);\r
+\r
+ $result = $part1->multiply($mod2);\r
+ $result = $result->multiply($y1);\r
+\r
+ $temp = $part2->multiply($mod1);\r
+ $temp = $temp->multiply($y2);\r
+\r
+ $result = $result->add($temp);\r
+ list(, $result) = $result->divide($n);\r
+\r
+ return $this->_normalize($result);\r
+ }\r
+\r
+ /**\r
+ * Performs modular exponentiation.\r
+ *\r
+ * Alias for Math_BigInteger::modPow()\r
+ *\r
+ * @param Math_BigInteger $e\r
+ * @param Math_BigInteger $n\r
+ * @return Math_BigInteger\r
+ * @access public\r
+ */\r
+ function powMod($e, $n)\r
+ {\r
+ return $this->modPow($e, $n);\r
+ }\r
+\r
+ /**\r
+ * Sliding Window k-ary Modular Exponentiation\r
+ *\r
+ * Based on {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=27 HAC 14.85} /\r
+ * {@link http://math.libtomcrypt.com/files/tommath.pdf#page=210 MPM 7.7}. In a departure from those algorithims,\r
+ * however, this function performs a modular reduction after every multiplication and squaring operation.\r
+ * As such, this function has the same preconditions that the reductions being used do.\r
+ *\r
+ * @param Math_BigInteger $e\r
+ * @param Math_BigInteger $n\r
+ * @param Integer $mode\r
+ * @return Math_BigInteger\r
+ * @access private\r
+ */\r
+ function _slidingWindow($e, $n, $mode)\r
+ {\r
+ static $window_ranges = array(7, 25, 81, 241, 673, 1793); // from BigInteger.java's oddModPow function\r
+ //static $window_ranges = array(0, 7, 36, 140, 450, 1303, 3529); // from MPM 7.3.1\r
+\r
+ $e_value = $e->value;\r
+ $e_length = count($e_value) - 1;\r
+ $e_bits = decbin($e_value[$e_length]);\r
+ for ($i = $e_length - 1; $i >= 0; --$i) {\r
+ $e_bits.= str_pad(decbin($e_value[$i]), 26, '0', STR_PAD_LEFT);\r
+ }\r
+\r
+ $e_length = strlen($e_bits);\r
+\r
+ // calculate the appropriate window size.\r
+ // $window_size == 3 if $window_ranges is between 25 and 81, for example.\r
+ for ($i = 0, $window_size = 1; $e_length > $window_ranges[$i] && $i < count($window_ranges); ++$window_size, ++$i);\r
+\r
+ $n_value = $n->value;\r
+\r
+ // precompute $this^0 through $this^$window_size\r
+ $powers = array();\r
+ $powers[1] = $this->_prepareReduce($this->value, $n_value, $mode);\r
+ $powers[2] = $this->_squareReduce($powers[1], $n_value, $mode);\r
+\r
+ // we do every other number since substr($e_bits, $i, $j+1) (see below) is supposed to end\r
+ // in a 1. ie. it's supposed to be odd.\r
+ $temp = 1 << ($window_size - 1);\r
+ for ($i = 1; $i < $temp; ++$i) {\r
+ $i2 = $i << 1;\r
+ $powers[$i2 + 1] = $this->_multiplyReduce($powers[$i2 - 1], $powers[2], $n_value, $mode);\r
+ }\r
+\r
+ $result = array(1);\r
+ $result = $this->_prepareReduce($result, $n_value, $mode);\r
+\r
+ for ($i = 0; $i < $e_length; ) {\r
+ if ( !$e_bits[$i] ) {\r
+ $result = $this->_squareReduce($result, $n_value, $mode);\r
+ ++$i;\r
+ } else {\r
+ for ($j = $window_size - 1; $j > 0; --$j) {\r
+ if ( !empty($e_bits[$i + $j]) ) {\r
+ break;\r
+ }\r
+ }\r
+\r
+ for ($k = 0; $k <= $j; ++$k) {// eg. the length of substr($e_bits, $i, $j+1)\r
+ $result = $this->_squareReduce($result, $n_value, $mode);\r
+ }\r
+\r
+ $result = $this->_multiplyReduce($result, $powers[bindec(substr($e_bits, $i, $j + 1))], $n_value, $mode);\r
+\r
+ $i+=$j + 1;\r
+ }\r
+ }\r
+\r
+ $temp = new Math_BigInteger();\r
+ $temp->value = $this->_reduce($result, $n_value, $mode);\r
+\r
+ return $temp;\r
+ }\r
+\r
+ /**\r
+ * Modular reduction\r
+ *\r
+ * For most $modes this will return the remainder.\r
+ *\r
+ * @see _slidingWindow()\r
+ * @access private\r
+ * @param Array $x\r
+ * @param Array $n\r
+ * @param Integer $mode\r
+ * @return Array\r
+ */\r
+ function _reduce($x, $n, $mode)\r
+ {\r
+ switch ($mode) {\r
+ case MATH_BIGINTEGER_MONTGOMERY:\r
+ return $this->_montgomery($x, $n);\r
+ case MATH_BIGINTEGER_BARRETT:\r
+ return $this->_barrett($x, $n);\r
+ case MATH_BIGINTEGER_POWEROF2:\r
+ $lhs = new Math_BigInteger();\r
+ $lhs->value = $x;\r
+ $rhs = new Math_BigInteger();\r
+ $rhs->value = $n;\r
+ return $x->_mod2($n);\r
+ case MATH_BIGINTEGER_CLASSIC:\r
+ $lhs = new Math_BigInteger();\r
+ $lhs->value = $x;\r
+ $rhs = new Math_BigInteger();\r
+ $rhs->value = $n;\r
+ list(, $temp) = $lhs->divide($rhs);\r
+ return $temp->value;\r
+ case MATH_BIGINTEGER_NONE:\r
+ return $x;\r
+ default:\r
+ // an invalid $mode was provided\r
+ }\r
+ }\r
+\r
+ /**\r
+ * Modular reduction preperation\r
+ *\r
+ * @see _slidingWindow()\r
+ * @access private\r
+ * @param Array $x\r
+ * @param Array $n\r
+ * @param Integer $mode\r
+ * @return Array\r
+ */\r
+ function _prepareReduce($x, $n, $mode)\r
+ {\r
+ if ($mode == MATH_BIGINTEGER_MONTGOMERY) {\r
+ return $this->_prepMontgomery($x, $n);\r
+ }\r
+ return $this->_reduce($x, $n, $mode);\r
+ }\r
+\r
+ /**\r
+ * Modular multiply\r
+ *\r
+ * @see _slidingWindow()\r
+ * @access private\r
+ * @param Array $x\r
+ * @param Array $y\r
+ * @param Array $n\r
+ * @param Integer $mode\r
+ * @return Array\r
+ */\r
+ function _multiplyReduce($x, $y, $n, $mode)\r
+ {\r
+ if ($mode == MATH_BIGINTEGER_MONTGOMERY) {\r
+ return $this->_montgomeryMultiply($x, $y, $n);\r
+ }\r
+ $temp = $this->_multiply($x, false, $y, false);\r
+ return $this->_reduce($temp[MATH_BIGINTEGER_VALUE], $n, $mode);\r
+ }\r
+\r
+ /**\r
+ * Modular square\r
+ *\r
+ * @see _slidingWindow()\r
+ * @access private\r
+ * @param Array $x\r
+ * @param Array $n\r
+ * @param Integer $mode\r
+ * @return Array\r
+ */\r
+ function _squareReduce($x, $n, $mode)\r
+ {\r
+ if ($mode == MATH_BIGINTEGER_MONTGOMERY) {\r
+ return $this->_montgomeryMultiply($x, $x, $n);\r
+ }\r
+ return $this->_reduce($this->_square($x), $n, $mode);\r
+ }\r
+\r
+ /**\r
+ * Modulos for Powers of Two\r
+ *\r
+ * Calculates $x%$n, where $n = 2**$e, for some $e. Since this is basically the same as doing $x & ($n-1),\r
+ * we'll just use this function as a wrapper for doing that.\r
+ *\r
+ * @see _slidingWindow()\r
+ * @access private\r
+ * @param Math_BigInteger\r
+ * @return Math_BigInteger\r
+ */\r
+ function _mod2($n)\r
+ {\r
+ $temp = new Math_BigInteger();\r
+ $temp->value = array(1);\r
+ return $this->bitwise_and($n->subtract($temp));\r
+ }\r
+\r
+ /**\r
+ * Barrett Modular Reduction\r
+ *\r
+ * See {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=14 HAC 14.3.3} /\r
+ * {@link http://math.libtomcrypt.com/files/tommath.pdf#page=165 MPM 6.2.5} for more information. Modified slightly,\r
+ * so as not to require negative numbers (initially, this script didn't support negative numbers).\r
+ *\r
+ * Employs "folding", as described at\r
+ * {@link http://www.cosic.esat.kuleuven.be/publications/thesis-149.pdf#page=66 thesis-149.pdf#page=66}. To quote from\r
+ * 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."\r
+ *\r
+ * Unfortunately, the "Barrett Reduction with Folding" algorithm described in thesis-149.pdf is not, as written, all that\r
+ * usable on account of (1) its not using reasonable radix points as discussed in\r
+ * {@link http://math.libtomcrypt.com/files/tommath.pdf#page=162 MPM 6.2.2} and (2) the fact that, even with reasonable\r
+ * radix points, it only works when there are an even number of digits in the denominator. The reason for (2) is that\r
+ * (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\r
+ * comments for details.\r
+ *\r
+ * @see _slidingWindow()\r
+ * @access private\r
+ * @param Array $n\r
+ * @param Array $m\r
+ * @return Array\r
+ */\r
+ function _barrett($n, $m)\r
+ {\r
+ static $cache = array(\r
+ MATH_BIGINTEGER_VARIABLE => array(),\r
+ MATH_BIGINTEGER_DATA => array()\r
+ );\r
+\r
+ $m_length = count($m);\r
+\r
+ // if ($this->_compare($n, $this->_square($m)) >= 0) {\r
+ if (count($n) > 2 * $m_length) {\r
+ $lhs = new Math_BigInteger();\r
+ $rhs = new Math_BigInteger();\r
+ $lhs->value = $n;\r
+ $rhs->value = $m;\r
+ list(, $temp) = $lhs->divide($rhs);\r
+ return $temp->value;\r
+ }\r
+\r
+ // if (m.length >> 1) + 2 <= m.length then m is too small and n can't be reduced\r
+ if ($m_length < 5) {\r
+ return $this->_regularBarrett($n, $m);\r
+ }\r
+\r
+ // n = 2 * m.length\r
+\r
+ if ( ($key = array_search($m, $cache[MATH_BIGINTEGER_VARIABLE])) === false ) {\r
+ $key = count($cache[MATH_BIGINTEGER_VARIABLE]);\r
+ $cache[MATH_BIGINTEGER_VARIABLE][] = $m;\r
+\r
+ $lhs = new Math_BigInteger();\r
+ $lhs_value = &$lhs->value;\r
+ $lhs_value = $this->_array_repeat(0, $m_length + ($m_length >> 1));\r
+ $lhs_value[] = 1;\r
+ $rhs = new Math_BigInteger();\r
+ $rhs->value = $m;\r
+\r
+ list($u, $m1) = $lhs->divide($rhs);\r
+ $u = $u->value;\r
+ $m1 = $m1->value;\r
+\r
+ $cache[MATH_BIGINTEGER_DATA][] = array(\r
+ 'u' => $u, // m.length >> 1 (technically (m.length >> 1) + 1)\r
+ 'm1'=> $m1 // m.length\r
+ );\r
+ } else {\r
+ extract($cache[MATH_BIGINTEGER_DATA][$key]);\r
+ }\r
+\r
+ $cutoff = $m_length + ($m_length >> 1);\r
+ $lsd = array_slice($n, 0, $cutoff); // m.length + (m.length >> 1)\r
+ $msd = array_slice($n, $cutoff); // m.length >> 1\r
+ $lsd = $this->_trim($lsd);\r
+ $temp = $this->_multiply($msd, false, $m1, false);\r
+ $n = $this->_add($lsd, false, $temp[MATH_BIGINTEGER_VALUE], false); // m.length + (m.length >> 1) + 1\r
+\r
+ if ($m_length & 1) {\r
+ return $this->_regularBarrett($n[MATH_BIGINTEGER_VALUE], $m);\r
+ }\r
+\r
+ // (m.length + (m.length >> 1) + 1) - (m.length - 1) == (m.length >> 1) + 2\r
+ $temp = array_slice($n[MATH_BIGINTEGER_VALUE], $m_length - 1);\r
+ // if even: ((m.length >> 1) + 2) + (m.length >> 1) == m.length + 2\r
+ // if odd: ((m.length >> 1) + 2) + (m.length >> 1) == (m.length - 1) + 2 == m.length + 1\r
+ $temp = $this->_multiply($temp, false, $u, false);\r
+ // if even: (m.length + 2) - ((m.length >> 1) + 1) = m.length - (m.length >> 1) + 1\r
+ // if odd: (m.length + 1) - ((m.length >> 1) + 1) = m.length - (m.length >> 1)\r
+ $temp = array_slice($temp[MATH_BIGINTEGER_VALUE], ($m_length >> 1) + 1);\r
+ // if even: (m.length - (m.length >> 1) + 1) + m.length = 2 * m.length - (m.length >> 1) + 1\r
+ // if odd: (m.length - (m.length >> 1)) + m.length = 2 * m.length - (m.length >> 1)\r
+ $temp = $this->_multiply($temp, false, $m, false);\r
+\r
+ // at this point, if m had an odd number of digits, we'd be subtracting a 2 * m.length - (m.length >> 1) digit\r
+ // number from a m.length + (m.length >> 1) + 1 digit number. ie. there'd be an extra digit and the while loop\r
+ // following this comment would loop a lot (hence our calling _regularBarrett() in that situation).\r
+\r
+ $result = $this->_subtract($n[MATH_BIGINTEGER_VALUE], false, $temp[MATH_BIGINTEGER_VALUE], false);\r
+\r
+ while ($this->_compare($result[MATH_BIGINTEGER_VALUE], $result[MATH_BIGINTEGER_SIGN], $m, false) >= 0) {\r
+ $result = $this->_subtract($result[MATH_BIGINTEGER_VALUE], $result[MATH_BIGINTEGER_SIGN], $m, false);\r
+ }\r
+\r
+ return $result[MATH_BIGINTEGER_VALUE];\r
+ }\r
+\r
+ /**\r
+ * (Regular) Barrett Modular Reduction\r
+ *\r
+ * For numbers with more than four digits Math_BigInteger::_barrett() is faster. The difference between that and this\r
+ * is that this function does not fold the denominator into a smaller form.\r
+ *\r
+ * @see _slidingWindow()\r
+ * @access private\r
+ * @param Array $x\r
+ * @param Array $n\r
+ * @return Array\r
+ */\r
+ function _regularBarrett($x, $n)\r
+ {\r
+ static $cache = array(\r
+ MATH_BIGINTEGER_VARIABLE => array(),\r
+ MATH_BIGINTEGER_DATA => array()\r
+ );\r
+\r
+ $n_length = count($n);\r
+\r
+ if (count($x) > 2 * $n_length) {\r
+ $lhs = new Math_BigInteger();\r
+ $rhs = new Math_BigInteger();\r
+ $lhs->value = $x;\r
+ $rhs->value = $n;\r
+ list(, $temp) = $lhs->divide($rhs);\r
+ return $temp->value;\r
+ }\r
+\r
+ if ( ($key = array_search($n, $cache[MATH_BIGINTEGER_VARIABLE])) === false ) {\r
+ $key = count($cache[MATH_BIGINTEGER_VARIABLE]);\r
+ $cache[MATH_BIGINTEGER_VARIABLE][] = $n;\r
+ $lhs = new Math_BigInteger();\r
+ $lhs_value = &$lhs->value;\r
+ $lhs_value = $this->_array_repeat(0, 2 * $n_length);\r
+ $lhs_value[] = 1;\r
+ $rhs = new Math_BigInteger();\r
+ $rhs->value = $n;\r
+ list($temp, ) = $lhs->divide($rhs); // m.length\r
+ $cache[MATH_BIGINTEGER_DATA][] = $temp->value;\r
+ }\r
+\r
+ // 2 * m.length - (m.length - 1) = m.length + 1\r
+ $temp = array_slice($x, $n_length - 1);\r
+ // (m.length + 1) + m.length = 2 * m.length + 1\r
+ $temp = $this->_multiply($temp, false, $cache[MATH_BIGINTEGER_DATA][$key], false);\r
+ // (2 * m.length + 1) - (m.length - 1) = m.length + 2\r
+ $temp = array_slice($temp[MATH_BIGINTEGER_VALUE], $n_length + 1);\r
+\r
+ // m.length + 1\r
+ $result = array_slice($x, 0, $n_length + 1);\r
+ // m.length + 1\r
+ $temp = $this->_multiplyLower($temp, false, $n, false, $n_length + 1);\r
+ // $temp == array_slice($temp->_multiply($temp, false, $n, false)->value, 0, $n_length + 1)\r
+\r
+ if ($this->_compare($result, false, $temp[MATH_BIGINTEGER_VALUE], $temp[MATH_BIGINTEGER_SIGN]) < 0) {\r
+ $corrector_value = $this->_array_repeat(0, $n_length + 1);\r
+ $corrector_value[] = 1;\r
+ $result = $this->_add($result, false, $corrector, false);\r
+ $result = $result[MATH_BIGINTEGER_VALUE];\r
+ }\r
+\r
+ // at this point, we're subtracting a number with m.length + 1 digits from another number with m.length + 1 digits\r
+ $result = $this->_subtract($result, false, $temp[MATH_BIGINTEGER_VALUE], $temp[MATH_BIGINTEGER_SIGN]);\r
+ while ($this->_compare($result[MATH_BIGINTEGER_VALUE], $result[MATH_BIGINTEGER_SIGN], $n, false) > 0) {\r
+ $result = $this->_subtract($result[MATH_BIGINTEGER_VALUE], $result[MATH_BIGINTEGER_SIGN], $n, false);\r
+ }\r
+\r
+ return $result[MATH_BIGINTEGER_VALUE];\r
+ }\r
+\r
+ /**\r
+ * Performs long multiplication up to $stop digits\r
+ *\r
+ * If you're going to be doing array_slice($product->value, 0, $stop), some cycles can be saved.\r
+ *\r
+ * @see _regularBarrett()\r
+ * @param Array $x_value\r
+ * @param Boolean $x_negative\r
+ * @param Array $y_value\r
+ * @param Boolean $y_negative\r
+ * @return Array\r
+ * @access private\r
+ */\r
+ function _multiplyLower($x_value, $x_negative, $y_value, $y_negative, $stop)\r
+ {\r
+ $x_length = count($x_value);\r
+ $y_length = count($y_value);\r
+\r
+ if ( !$x_length || !$y_length ) { // a 0 is being multiplied\r
+ return array(\r
+ MATH_BIGINTEGER_VALUE => array(),\r
+ MATH_BIGINTEGER_SIGN => false\r
+ );\r
+ }\r
+\r
+ if ( $x_length < $y_length ) {\r
+ $temp = $x_value;\r
+ $x_value = $y_value;\r
+ $y_value = $temp;\r
+\r
+ $x_length = count($x_value);\r
+ $y_length = count($y_value);\r
+ }\r
+\r
+ $product_value = $this->_array_repeat(0, $x_length + $y_length);\r
+\r
+ // the following for loop could be removed if the for loop following it\r
+ // (the one with nested for loops) initially set $i to 0, but\r
+ // doing so would also make the result in one set of unnecessary adds,\r
+ // since on the outermost loops first pass, $product->value[$k] is going\r
+ // to always be 0\r
+\r
+ $carry = 0;\r
+\r
+ for ($j = 0; $j < $x_length; ++$j) { // ie. $i = 0, $k = $i\r
+ $temp = $x_value[$j] * $y_value[0] + $carry; // $product_value[$k] == 0\r
+ $carry = (int) ($temp / 0x4000000);\r
+ $product_value[$j] = (int) ($temp - 0x4000000 * $carry);\r
+ }\r
+\r
+ if ($j < $stop) {\r
+ $product_value[$j] = $carry;\r
+ }\r
+\r
+ // the above for loop is what the previous comment was talking about. the\r
+ // following for loop is the "one with nested for loops"\r
+\r
+ for ($i = 1; $i < $y_length; ++$i) {\r
+ $carry = 0;\r
+\r
+ for ($j = 0, $k = $i; $j < $x_length && $k < $stop; ++$j, ++$k) {\r
+ $temp = $product_value[$k] + $x_value[$j] * $y_value[$i] + $carry;\r
+ $carry = (int) ($temp / 0x4000000);\r
+ $product_value[$k] = (int) ($temp - 0x4000000 * $carry);\r
+ }\r
+\r
+ if ($k < $stop) {\r
+ $product_value[$k] = $carry;\r
+ }\r
+ }\r
+\r
+ return array(\r
+ MATH_BIGINTEGER_VALUE => $this->_trim($product_value),\r
+ MATH_BIGINTEGER_SIGN => $x_negative != $y_negative\r
+ );\r
+ }\r
+\r
+ /**\r
+ * Montgomery Modular Reduction\r
+ *\r
+ * ($x->_prepMontgomery($n))->_montgomery($n) yields $x % $n.\r
+ * {@link http://math.libtomcrypt.com/files/tommath.pdf#page=170 MPM 6.3} provides insights on how this can be\r
+ * improved upon (basically, by using the comba method). gcd($n, 2) must be equal to one for this function\r
+ * to work correctly.\r
+ *\r
+ * @see _prepMontgomery()\r
+ * @see _slidingWindow()\r
+ * @access private\r
+ * @param Array $x\r
+ * @param Array $n\r
+ * @return Array\r
+ */\r
+ function _montgomery($x, $n)\r
+ {\r
+ static $cache = array(\r
+ MATH_BIGINTEGER_VARIABLE => array(),\r
+ MATH_BIGINTEGER_DATA => array()\r
+ );\r
+\r
+ if ( ($key = array_search($n, $cache[MATH_BIGINTEGER_VARIABLE])) === false ) {\r
+ $key = count($cache[MATH_BIGINTEGER_VARIABLE]);\r
+ $cache[MATH_BIGINTEGER_VARIABLE][] = $x;\r
+ $cache[MATH_BIGINTEGER_DATA][] = $this->_modInverse67108864($n);\r
+ }\r
+\r
+ $k = count($n);\r
+\r
+ $result = array(MATH_BIGINTEGER_VALUE => $x);\r
+\r
+ for ($i = 0; $i < $k; ++$i) {\r
+ $temp = $result[MATH_BIGINTEGER_VALUE][$i] * $cache[MATH_BIGINTEGER_DATA][$key];\r
+ $temp = (int) ($temp - 0x4000000 * ((int) ($temp / 0x4000000)));\r
+ $temp = $this->_regularMultiply(array($temp), $n);\r
+ $temp = array_merge($this->_array_repeat(0, $i), $temp);\r
+ $result = $this->_add($result[MATH_BIGINTEGER_VALUE], false, $temp, false);\r
+ }\r
+\r
+ $result[MATH_BIGINTEGER_VALUE] = array_slice($result[MATH_BIGINTEGER_VALUE], $k);\r
+\r
+ if ($this->_compare($result, false, $n, false) >= 0) {\r
+ $result = $this->_subtract($result[MATH_BIGINTEGER_VALUE], false, $n, false);\r
+ }\r
+\r
+ return $result[MATH_BIGINTEGER_VALUE];\r
+ }\r
+\r
+ /**\r
+ * Montgomery Multiply\r
+ *\r
+ * Interleaves the montgomery reduction and long multiplication algorithms together as described in \r
+ * {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=13 HAC 14.36}\r
+ *\r
+ * @see _prepMontgomery()\r
+ * @see _montgomery()\r
+ * @access private\r
+ * @param Array $x\r
+ * @param Array $y\r
+ * @param Array $m\r
+ * @return Array\r
+ */\r
+ function _montgomeryMultiply($x, $y, $m)\r
+ {\r
+ $temp = $this->_multiply($x, false, $y, false);\r
+ return $this->_montgomery($temp[MATH_BIGINTEGER_VALUE], $m);\r
+\r
+ static $cache = array(\r
+ MATH_BIGINTEGER_VARIABLE => array(),\r
+ MATH_BIGINTEGER_DATA => array()\r
+ );\r
+\r
+ if ( ($key = array_search($m, $cache[MATH_BIGINTEGER_VARIABLE])) === false ) {\r
+ $key = count($cache[MATH_BIGINTEGER_VARIABLE]);\r
+ $cache[MATH_BIGINTEGER_VARIABLE][] = $m;\r
+ $cache[MATH_BIGINTEGER_DATA][] = $this->_modInverse67108864($m);\r
+ }\r
+\r
+ $n = max(count($x), count($y), count($m));\r
+ $x = array_pad($x, $n, 0);\r
+ $y = array_pad($y, $n, 0);\r
+ $m = array_pad($m, $n, 0);\r
+ $a = array(MATH_BIGINTEGER_VALUE => $this->_array_repeat(0, $n + 1));\r
+ for ($i = 0; $i < $n; ++$i) {\r
+ $temp = $a[MATH_BIGINTEGER_VALUE][0] + $x[$i] * $y[0];\r
+ $temp = (int) ($temp - 0x4000000 * ((int) ($temp / 0x4000000)));\r
+ $temp = $temp * $cache[MATH_BIGINTEGER_DATA][$key];\r
+ $temp = (int) ($temp - 0x4000000 * ((int) ($temp / 0x4000000)));\r
+ $temp = $this->_add($this->_regularMultiply(array($x[$i]), $y), false, $this->_regularMultiply(array($temp), $m), false);\r
+ $a = $this->_add($a[MATH_BIGINTEGER_VALUE], false, $temp[MATH_BIGINTEGER_VALUE], false);\r
+ $a[MATH_BIGINTEGER_VALUE] = array_slice($a[MATH_BIGINTEGER_VALUE], 1);\r
+ }\r
+ if ($this->_compare($a[MATH_BIGINTEGER_VALUE], false, $m, false) >= 0) {\r
+ $a = $this->_subtract($a[MATH_BIGINTEGER_VALUE], false, $m, false);\r
+ }\r
+ return $a[MATH_BIGINTEGER_VALUE];\r
+ }\r
+\r
+ /**\r
+ * Prepare a number for use in Montgomery Modular Reductions\r
+ *\r
+ * @see _montgomery()\r
+ * @see _slidingWindow()\r
+ * @access private\r
+ * @param Array $x\r
+ * @param Array $n\r
+ * @return Array\r
+ */\r
+ function _prepMontgomery($x, $n)\r
+ {\r
+ $lhs = new Math_BigInteger();\r
+ $lhs->value = array_merge($this->_array_repeat(0, count($n)), $x);\r
+ $rhs = new Math_BigInteger();\r
+ $rhs->value = $n;\r
+\r
+ list(, $temp) = $lhs->divide($rhs);\r
+ return $temp->value;\r
+ }\r
+\r
+ /**\r
+ * Modular Inverse of a number mod 2**26 (eg. 67108864)\r
+ *\r
+ * Based off of the bnpInvDigit function implemented and justified in the following URL:\r
+ *\r
+ * {@link http://www-cs-students.stanford.edu/~tjw/jsbn/jsbn.js}\r
+ *\r
+ * The following URL provides more info:\r
+ *\r
+ * {@link http://groups.google.com/group/sci.crypt/msg/7a137205c1be7d85}\r
+ *\r
+ * As for why we do all the bitmasking... strange things can happen when converting from floats to ints. For\r
+ * instance, on some computers, var_dump((int) -4294967297) yields int(-1) and on others, it yields \r
+ * int(-2147483648). To avoid problems stemming from this, we use bitmasks to guarantee that ints aren't\r
+ * auto-converted to floats. The outermost bitmask is present because without it, there's no guarantee that\r
+ * the "residue" returned would be the so-called "common residue". We use fmod, in the last step, because the\r
+ * maximum possible $x is 26 bits and the maximum $result is 16 bits. Thus, we have to be able to handle up to\r
+ * 40 bits, which only 64-bit floating points will support.\r
+ *\r
+ * Thanks to Pedro Gimeno Fortea for input!\r
+ *\r
+ * @see _montgomery()\r
+ * @access private\r
+ * @param Array $x\r
+ * @return Integer\r
+ */\r
+ function _modInverse67108864($x) // 2**26 == 67108864\r
+ {\r
+ $x = -$x[0];\r
+ $result = $x & 0x3; // x**-1 mod 2**2\r
+ $result = ($result * (2 - $x * $result)) & 0xF; // x**-1 mod 2**4\r
+ $result = ($result * (2 - ($x & 0xFF) * $result)) & 0xFF; // x**-1 mod 2**8\r
+ $result = ($result * ((2 - ($x & 0xFFFF) * $result) & 0xFFFF)) & 0xFFFF; // x**-1 mod 2**16\r
+ $result = fmod($result * (2 - fmod($x * $result, 0x4000000)), 0x4000000); // x**-1 mod 2**26\r
+ return $result & 0x3FFFFFF;\r
+ }\r
+\r
+ /**\r
+ * Calculates modular inverses.\r
+ *\r
+ * Say you have (30 mod 17 * x mod 17) mod 17 == 1. x can be found using modular inverses.\r
+ *\r
+ * Here's an example:\r
+ * <code>\r
+ * <?php\r
+ * include('Math/BigInteger.php');\r
+ *\r
+ * $a = new Math_BigInteger(30);\r
+ * $b = new Math_BigInteger(17);\r
+ *\r
+ * $c = $a->modInverse($b);\r
+ * echo $c->toString(); // outputs 4\r
+ *\r
+ * echo "\r\n";\r
+ *\r
+ * $d = $a->multiply($c);\r
+ * list(, $d) = $d->divide($b);\r
+ * echo $d; // outputs 1 (as per the definition of modular inverse)\r
+ * ?>\r
+ * </code>\r
+ *\r
+ * @param Math_BigInteger $n\r
+ * @return mixed false, if no modular inverse exists, Math_BigInteger, otherwise.\r
+ * @access public\r
+ * @internal See {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=21 HAC 14.64} for more information.\r
+ */\r
+ function modInverse($n)\r
+ {\r
+ switch ( MATH_BIGINTEGER_MODE ) {\r
+ case MATH_BIGINTEGER_MODE_GMP:\r
+ $temp = new Math_BigInteger();\r
+ $temp->value = gmp_invert($this->value, $n->value);\r
+\r
+ return ( $temp->value === false ) ? false : $this->_normalize($temp);\r
+ }\r
+\r
+ static $zero, $one;\r
+ if (!isset($zero)) {\r
+ $zero = new Math_BigInteger();\r
+ $one = new Math_BigInteger(1);\r
+ }\r
+\r
+ // $x mod $n == $x mod -$n.\r
+ $n = $n->abs();\r
+\r
+ if ($this->compare($zero) < 0) {\r
+ $temp = $this->abs();\r
+ $temp = $temp->modInverse($n);\r
+ return $negated === false ? false : $this->_normalize($n->subtract($temp));\r
+ }\r
+\r
+ extract($this->extendedGCD($n));\r
+\r
+ if (!$gcd->equals($one)) {\r
+ return false;\r
+ }\r
+\r
+ $x = $x->compare($zero) < 0 ? $x->add($n) : $x;\r
+\r
+ return $this->compare($zero) < 0 ? $this->_normalize($n->subtract($x)) : $this->_normalize($x);\r
+ }\r
+\r
+ /**\r
+ * Calculates the greatest common divisor and Bézout's identity.\r
+ *\r
+ * Say you have 693 and 609. The GCD is 21. Bézout's identity states that there exist integers x and y such that\r
+ * 693*x + 609*y == 21. In point of fact, there are actually an infinite number of x and y combinations and which\r
+ * combination is returned is dependant upon which mode is in use. See\r
+ * {@link http://en.wikipedia.org/wiki/B%C3%A9zout%27s_identity Bézout's identity - Wikipedia} for more information.\r
+ *\r
+ * Here's an example:\r
+ * <code>\r
+ * <?php\r
+ * include('Math/BigInteger.php');\r
+ *\r
+ * $a = new Math_BigInteger(693);\r
+ * $b = new Math_BigInteger(609);\r
+ *\r
+ * extract($a->extendedGCD($b));\r
+ *\r
+ * echo $gcd->toString() . "\r\n"; // outputs 21\r
+ * echo $a->toString() * $x->toString() + $b->toString() * $y->toString(); // outputs 21\r
+ * ?>\r
+ * </code>\r
+ *\r
+ * @param Math_BigInteger $n\r
+ * @return Math_BigInteger\r
+ * @access public\r
+ * @internal Calculates the GCD using the binary xGCD algorithim described in\r
+ * {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=19 HAC 14.61}. As the text above 14.61 notes,\r
+ * the more traditional algorithim requires "relatively costly multiple-precision divisions".\r
+ */\r
+ function extendedGCD($n)\r
+ {\r
+ switch ( MATH_BIGINTEGER_MODE ) {\r
+ case MATH_BIGINTEGER_MODE_GMP:\r
+ extract(gmp_gcdext($this->value, $n->value));\r
+\r
+ return array(\r
+ 'gcd' => $this->_normalize(new Math_BigInteger($g)),\r
+ 'x' => $this->_normalize(new Math_BigInteger($s)),\r
+ 'y' => $this->_normalize(new Math_BigInteger($t))\r
+ );\r
+ case MATH_BIGINTEGER_MODE_BCMATH:\r
+ // it might be faster to use the binary xGCD algorithim here, as well, but (1) that algorithim works\r
+ // best when the base is a power of 2 and (2) i don't think it'd make much difference, anyway. as is,\r
+ // the basic extended euclidean algorithim is what we're using.\r
+\r
+ $u = $this->value;\r
+ $v = $n->value;\r
+\r
+ $a = '1';\r
+ $b = '0';\r
+ $c = '0';\r
+ $d = '1';\r
+\r
+ while (bccomp($v, '0', 0) != 0) {\r
+ $q = bcdiv($u, $v, 0);\r
+\r
+ $temp = $u;\r
+ $u = $v;\r
+ $v = bcsub($temp, bcmul($v, $q, 0), 0);\r
+\r
+ $temp = $a;\r
+ $a = $c;\r
+ $c = bcsub($temp, bcmul($a, $q, 0), 0);\r
+\r
+ $temp = $b;\r
+ $b = $d;\r
+ $d = bcsub($temp, bcmul($b, $q, 0), 0);\r
+ }\r
+\r
+ return array(\r
+ 'gcd' => $this->_normalize(new Math_BigInteger($u)),\r
+ 'x' => $this->_normalize(new Math_BigInteger($a)),\r
+ 'y' => $this->_normalize(new Math_BigInteger($b))\r
+ );\r
+ }\r
+\r
+ $y = $n->copy();\r
+ $x = $this->copy();\r
+ $g = new Math_BigInteger();\r
+ $g->value = array(1);\r
+\r
+ while ( !(($x->value[0] & 1)|| ($y->value[0] & 1)) ) {\r
+ $x->_rshift(1);\r
+ $y->_rshift(1);\r
+ $g->_lshift(1);\r
+ }\r
+\r
+ $u = $x->copy();\r
+ $v = $y->copy();\r
+\r
+ $a = new Math_BigInteger();\r
+ $b = new Math_BigInteger();\r
+ $c = new Math_BigInteger();\r
+ $d = new Math_BigInteger();\r
+\r
+ $a->value = $d->value = $g->value = array(1);\r
+ $b->value = $c->value = array();\r
+\r
+ while ( !empty($u->value) ) {\r
+ while ( !($u->value[0] & 1) ) {\r
+ $u->_rshift(1);\r
+ if ( (!empty($a->value) && ($a->value[0] & 1)) || (!empty($b->value) && ($b->value[0] & 1)) ) {\r
+ $a = $a->add($y);\r
+ $b = $b->subtract($x);\r
+ }\r
+ $a->_rshift(1);\r
+ $b->_rshift(1);\r
+ }\r
+\r
+ while ( !($v->value[0] & 1) ) {\r
+ $v->_rshift(1);\r
+ if ( (!empty($d->value) && ($d->value[0] & 1)) || (!empty($c->value) && ($c->value[0] & 1)) ) {\r
+ $c = $c->add($y);\r
+ $d = $d->subtract($x);\r
+ }\r
+ $c->_rshift(1);\r
+ $d->_rshift(1);\r
+ }\r
+\r
+ if ($u->compare($v) >= 0) {\r
+ $u = $u->subtract($v);\r
+ $a = $a->subtract($c);\r
+ $b = $b->subtract($d);\r
+ } else {\r
+ $v = $v->subtract($u);\r
+ $c = $c->subtract($a);\r
+ $d = $d->subtract($b);\r
+ }\r
+ }\r
+\r
+ return array(\r
+ 'gcd' => $this->_normalize($g->multiply($v)),\r
+ 'x' => $this->_normalize($c),\r
+ 'y' => $this->_normalize($d)\r
+ );\r
+ }\r
+\r
+ /**\r
+ * Calculates the greatest common divisor\r
+ *\r
+ * Say you have 693 and 609. The GCD is 21.\r
+ *\r
+ * Here's an example:\r
+ * <code>\r
+ * <?php\r
+ * include('Math/BigInteger.php');\r
+ *\r
+ * $a = new Math_BigInteger(693);\r
+ * $b = new Math_BigInteger(609);\r
+ *\r
+ * $gcd = a->extendedGCD($b);\r
+ *\r
+ * echo $gcd->toString() . "\r\n"; // outputs 21\r
+ * ?>\r
+ * </code>\r
+ *\r
+ * @param Math_BigInteger $n\r
+ * @return Math_BigInteger\r
+ * @access public\r
+ */\r
+ function gcd($n)\r
+ {\r
+ extract($this->extendedGCD($n));\r
+ return $gcd;\r
+ }\r
+\r
+ /**\r
+ * Absolute value.\r
+ *\r
+ * @return Math_BigInteger\r
+ * @access public\r
+ */\r
+ function abs()\r
+ {\r
+ $temp = new Math_BigInteger();\r
+\r
+ switch ( MATH_BIGINTEGER_MODE ) {\r
+ case MATH_BIGINTEGER_MODE_GMP:\r
+ $temp->value = gmp_abs($this->value);\r
+ break;\r
+ case MATH_BIGINTEGER_MODE_BCMATH:\r
+ $temp->value = (bccomp($this->value, '0', 0) < 0) ? substr($this->value, 1) : $this->value;\r
+ break;\r
+ default:\r
+ $temp->value = $this->value;\r
+ }\r
+\r
+ return $temp;\r
+ }\r
+\r
+ /**\r
+ * Compares two numbers.\r
+ *\r
+ * Although one might think !$x->compare($y) means $x != $y, it, in fact, means the opposite. The reason for this is\r
+ * demonstrated thusly:\r
+ *\r
+ * $x > $y: $x->compare($y) > 0\r
+ * $x < $y: $x->compare($y) < 0\r
+ * $x == $y: $x->compare($y) == 0\r
+ *\r
+ * Note how the same comparison operator is used. If you want to test for equality, use $x->equals($y).\r
+ *\r
+ * @param Math_BigInteger $x\r
+ * @return Integer < 0 if $this is less than $x; > 0 if $this is greater than $x, and 0 if they are equal.\r
+ * @access public\r
+ * @see equals()\r
+ * @internal Could return $this->subtract($x), but that's not as fast as what we do do.\r
+ */\r
+ function compare($y)\r
+ {\r
+ switch ( MATH_BIGINTEGER_MODE ) {\r
+ case MATH_BIGINTEGER_MODE_GMP:\r
+ return gmp_cmp($this->value, $y->value);\r
+ case MATH_BIGINTEGER_MODE_BCMATH:\r
+ return bccomp($this->value, $y->value, 0);\r
+ }\r
+\r
+ return $this->_compare($this->value, $this->is_negative, $y->value, $y->is_negative);\r
+ }\r
+\r
+ /**\r
+ * Compares two numbers.\r
+ *\r
+ * @param Array $x_value\r
+ * @param Boolean $x_negative\r
+ * @param Array $y_value\r
+ * @param Boolean $y_negative\r
+ * @return Integer\r
+ * @see compare()\r
+ * @access private\r
+ */\r
+ function _compare($x_value, $x_negative, $y_value, $y_negative)\r
+ {\r
+ if ( $x_negative != $y_negative ) {\r
+ return ( !$x_negative && $y_negative ) ? 1 : -1;\r
+ }\r
+\r
+ $result = $x_negative ? -1 : 1;\r
+\r
+ if ( count($x_value) != count($y_value) ) {\r
+ return ( count($x_value) > count($y_value) ) ? $result : -$result;\r
+ }\r
+ $size = max(count($x_value), count($y_value));\r
+\r
+ $x_value = array_pad($x_value, $size, 0);\r
+ $y_value = array_pad($y_value, $size, 0);\r
+\r
+ for ($i = count($x_value) - 1; $i >= 0; --$i) {\r
+ if ($x_value[$i] != $y_value[$i]) {\r
+ return ( $x_value[$i] > $y_value[$i] ) ? $result : -$result;\r
+ }\r
+ }\r
+\r
+ return 0;\r
+ }\r
+\r
+ /**\r
+ * Tests the equality of two numbers.\r
+ *\r
+ * If you need to see if one number is greater than or less than another number, use Math_BigInteger::compare()\r
+ *\r
+ * @param Math_BigInteger $x\r
+ * @return Boolean\r
+ * @access public\r
+ * @see compare()\r
+ */\r
+ function equals($x)\r
+ {\r
+ switch ( MATH_BIGINTEGER_MODE ) {\r
+ case MATH_BIGINTEGER_MODE_GMP:\r
+ return gmp_cmp($this->value, $x->value) == 0;\r
+ default:\r
+ return $this->value === $x->value && $this->is_negative == $x->is_negative;\r
+ }\r
+ }\r
+\r
+ /**\r
+ * Set Precision\r
+ *\r
+ * Some bitwise operations give different results depending on the precision being used. Examples include left\r
+ * shift, not, and rotates.\r
+ *\r
+ * @param Math_BigInteger $x\r
+ * @access public\r
+ * @return Math_BigInteger\r
+ */\r
+ function setPrecision($bits)\r
+ {\r
+ $this->precision = $bits;\r
+ if ( MATH_BIGINTEGER_MODE != MATH_BIGINTEGER_MODE_BCMATH ) {\r
+ $this->bitmask = new Math_BigInteger(chr((1 << ($bits & 0x7)) - 1) . str_repeat(chr(0xFF), $bits >> 3), 256);\r
+ } else {\r
+ $this->bitmask = new Math_BigInteger(bcpow('2', $bits, 0));\r
+ }\r
+\r
+ $temp = $this->_normalize($this);\r
+ $this->value = $temp->value;\r
+ }\r
+\r
+ /**\r
+ * Logical And\r
+ *\r
+ * @param Math_BigInteger $x\r
+ * @access public\r
+ * @internal Implemented per a request by Lluis Pamies i Juarez <lluis _a_ pamies.cat>\r
+ * @return Math_BigInteger\r
+ */\r
+ function bitwise_and($x)\r
+ {\r
+ switch ( MATH_BIGINTEGER_MODE ) {\r
+ case MATH_BIGINTEGER_MODE_GMP:\r
+ $temp = new Math_BigInteger();\r
+ $temp->value = gmp_and($this->value, $x->value);\r
+\r
+ return $this->_normalize($temp);\r
+ case MATH_BIGINTEGER_MODE_BCMATH:\r
+ $left = $this->toBytes();\r
+ $right = $x->toBytes();\r
+\r
+ $length = max(strlen($left), strlen($right));\r
+\r
+ $left = str_pad($left, $length, chr(0), STR_PAD_LEFT);\r
+ $right = str_pad($right, $length, chr(0), STR_PAD_LEFT);\r
+\r
+ return $this->_normalize(new Math_BigInteger($left & $right, 256));\r
+ }\r
+\r
+ $result = $this->copy();\r
+\r
+ $length = min(count($x->value), count($this->value));\r
+\r
+ $result->value = array_slice($result->value, 0, $length);\r
+\r
+ for ($i = 0; $i < $length; ++$i) {\r
+ $result->value[$i] = $result->value[$i] & $x->value[$i];\r
+ }\r
+\r
+ return $this->_normalize($result);\r
+ }\r
+\r
+ /**\r
+ * Logical Or\r
+ *\r
+ * @param Math_BigInteger $x\r
+ * @access public\r
+ * @internal Implemented per a request by Lluis Pamies i Juarez <lluis _a_ pamies.cat>\r
+ * @return Math_BigInteger\r
+ */\r
+ function bitwise_or($x)\r
+ {\r
+ switch ( MATH_BIGINTEGER_MODE ) {\r
+ case MATH_BIGINTEGER_MODE_GMP:\r
+ $temp = new Math_BigInteger();\r
+ $temp->value = gmp_or($this->value, $x->value);\r
+\r
+ return $this->_normalize($temp);\r
+ case MATH_BIGINTEGER_MODE_BCMATH:\r
+ $left = $this->toBytes();\r
+ $right = $x->toBytes();\r
+\r
+ $length = max(strlen($left), strlen($right));\r
+\r
+ $left = str_pad($left, $length, chr(0), STR_PAD_LEFT);\r
+ $right = str_pad($right, $length, chr(0), STR_PAD_LEFT);\r
+\r
+ return $this->_normalize(new Math_BigInteger($left | $right, 256));\r
+ }\r
+\r
+ $length = max(count($this->value), count($x->value));\r
+ $result = $this->copy();\r
+ $result->value = array_pad($result->value, 0, $length);\r
+ $x->value = array_pad($x->value, 0, $length);\r
+\r
+ for ($i = 0; $i < $length; ++$i) {\r
+ $result->value[$i] = $this->value[$i] | $x->value[$i];\r
+ }\r
+\r
+ return $this->_normalize($result);\r
+ }\r
+\r
+ /**\r
+ * Logical Exclusive-Or\r
+ *\r
+ * @param Math_BigInteger $x\r
+ * @access public\r
+ * @internal Implemented per a request by Lluis Pamies i Juarez <lluis _a_ pamies.cat>\r
+ * @return Math_BigInteger\r
+ */\r
+ function bitwise_xor($x)\r
+ {\r
+ switch ( MATH_BIGINTEGER_MODE ) {\r
+ case MATH_BIGINTEGER_MODE_GMP:\r
+ $temp = new Math_BigInteger();\r
+ $temp->value = gmp_xor($this->value, $x->value);\r
+\r
+ return $this->_normalize($temp);\r
+ case MATH_BIGINTEGER_MODE_BCMATH:\r
+ $left = $this->toBytes();\r
+ $right = $x->toBytes();\r
+\r
+ $length = max(strlen($left), strlen($right));\r
+\r
+ $left = str_pad($left, $length, chr(0), STR_PAD_LEFT);\r
+ $right = str_pad($right, $length, chr(0), STR_PAD_LEFT);\r
+\r
+ return $this->_normalize(new Math_BigInteger($left ^ $right, 256));\r
+ }\r
+\r
+ $length = max(count($this->value), count($x->value));\r
+ $result = $this->copy();\r
+ $result->value = array_pad($result->value, 0, $length);\r
+ $x->value = array_pad($x->value, 0, $length);\r
+\r
+ for ($i = 0; $i < $length; ++$i) {\r
+ $result->value[$i] = $this->value[$i] ^ $x->value[$i];\r
+ }\r
+\r
+ return $this->_normalize($result);\r
+ }\r
+\r
+ /**\r
+ * Logical Not\r
+ *\r
+ * @access public\r
+ * @internal Implemented per a request by Lluis Pamies i Juarez <lluis _a_ pamies.cat>\r
+ * @return Math_BigInteger\r
+ */\r
+ function bitwise_not()\r
+ {\r
+ // calculuate "not" without regard to $this->precision\r
+ // (will always result in a smaller number. ie. ~1 isn't 1111 1110 - it's 0)\r
+ $temp = $this->toBytes();\r
+ $pre_msb = decbin(ord($temp[0]));\r
+ $temp = ~$temp;\r
+ $msb = decbin(ord($temp[0]));\r
+ if (strlen($msb) == 8) {\r
+ $msb = substr($msb, strpos($msb, '0'));\r
+ }\r
+ $temp[0] = chr(bindec($msb));\r
+\r
+ // see if we need to add extra leading 1's\r
+ $current_bits = strlen($pre_msb) + 8 * strlen($temp) - 8;\r
+ $new_bits = $this->precision - $current_bits;\r
+ if ($new_bits <= 0) {\r
+ return $this->_normalize(new Math_BigInteger($temp, 256));\r
+ }\r
+\r
+ // generate as many leading 1's as we need to.\r
+ $leading_ones = chr((1 << ($new_bits & 0x7)) - 1) . str_repeat(chr(0xFF), $new_bits >> 3);\r
+ $this->_base256_lshift($leading_ones, $current_bits);\r
+\r
+ $temp = str_pad($temp, ceil($this->bits / 8), chr(0), STR_PAD_LEFT);\r
+\r
+ return $this->_normalize(new Math_BigInteger($leading_ones | $temp, 256));\r
+ }\r
+\r
+ /**\r
+ * Logical Right Shift\r
+ *\r
+ * Shifts BigInteger's by $shift bits, effectively dividing by 2**$shift.\r
+ *\r
+ * @param Integer $shift\r
+ * @return Math_BigInteger\r
+ * @access public\r
+ * @internal The only version that yields any speed increases is the internal version.\r
+ */\r
+ function bitwise_rightShift($shift)\r
+ {\r
+ $temp = new Math_BigInteger();\r
+\r
+ switch ( MATH_BIGINTEGER_MODE ) {\r
+ case MATH_BIGINTEGER_MODE_GMP:\r
+ static $two;\r
+\r
+ if (!isset($two)) {\r
+ $two = gmp_init('2');\r
+ }\r
+\r
+ $temp->value = gmp_div_q($this->value, gmp_pow($two, $shift));\r
+\r
+ break;\r
+ case MATH_BIGINTEGER_MODE_BCMATH:\r
+ $temp->value = bcdiv($this->value, bcpow('2', $shift, 0), 0);\r
+\r
+ break;\r
+ default: // could just replace _lshift with this, but then all _lshift() calls would need to be rewritten\r
+ // and I don't want to do that...\r
+ $temp->value = $this->value;\r
+ $temp->_rshift($shift);\r
+ }\r
+\r
+ return $this->_normalize($temp);\r
+ }\r
+\r
+ /**\r
+ * Logical Left Shift\r
+ *\r
+ * Shifts BigInteger's by $shift bits, effectively multiplying by 2**$shift.\r
+ *\r
+ * @param Integer $shift\r
+ * @return Math_BigInteger\r
+ * @access public\r
+ * @internal The only version that yields any speed increases is the internal version.\r
+ */\r
+ function bitwise_leftShift($shift)\r
+ {\r
+ $temp = new Math_BigInteger();\r
+\r
+ switch ( MATH_BIGINTEGER_MODE ) {\r
+ case MATH_BIGINTEGER_MODE_GMP:\r
+ static $two;\r
+\r
+ if (!isset($two)) {\r
+ $two = gmp_init('2');\r
+ }\r
+\r
+ $temp->value = gmp_mul($this->value, gmp_pow($two, $shift));\r
+\r
+ break;\r
+ case MATH_BIGINTEGER_MODE_BCMATH:\r
+ $temp->value = bcmul($this->value, bcpow('2', $shift, 0), 0);\r
+\r
+ break;\r
+ default: // could just replace _rshift with this, but then all _lshift() calls would need to be rewritten\r
+ // and I don't want to do that...\r
+ $temp->value = $this->value;\r
+ $temp->_lshift($shift);\r
+ }\r
+\r
+ return $this->_normalize($temp);\r
+ }\r
+\r
+ /**\r
+ * Logical Left Rotate\r
+ *\r
+ * Instead of the top x bits being dropped they're appended to the shifted bit string.\r
+ *\r
+ * @param Integer $shift\r
+ * @return Math_BigInteger\r
+ * @access public\r
+ */\r
+ function bitwise_leftRotate($shift)\r
+ {\r
+ $bits = $this->toBytes();\r
+\r
+ if ($this->precision > 0) {\r
+ $precision = $this->precision;\r
+ if ( MATH_BIGINTEGER_MODE == MATH_BIGINTEGER_MODE_BCMATH ) {\r
+ $mask = $this->bitmask->subtract(new Math_BigInteger(1));\r
+ $mask = $mask->toBytes();\r
+ } else {\r
+ $mask = $this->bitmask->toBytes();\r
+ }\r
+ } else {\r
+ $temp = ord($bits[0]);\r
+ for ($i = 0; $temp >> $i; ++$i);\r
+ $precision = 8 * strlen($bits) - 8 + $i;\r
+ $mask = chr((1 << ($precision & 0x7)) - 1) . str_repeat(chr(0xFF), $precision >> 3);\r
+ }\r
+\r
+ if ($shift < 0) {\r
+ $shift+= $precision;\r
+ }\r
+ $shift%= $precision;\r
+\r
+ if (!$shift) {\r
+ return $this->copy();\r
+ }\r
+\r
+ $left = $this->bitwise_leftShift($shift);\r
+ $left = $left->bitwise_and(new Math_BigInteger($mask, 256));\r
+ $right = $this->bitwise_rightShift($precision - $shift);\r
+ $result = MATH_BIGINTEGER_MODE != MATH_BIGINTEGER_MODE_BCMATH ? $left->bitwise_or($right) : $left->add($right);\r
+ return $this->_normalize($result);\r
+ }\r
+\r
+ /**\r
+ * Logical Right Rotate\r
+ *\r
+ * Instead of the bottom x bits being dropped they're prepended to the shifted bit string.\r
+ *\r
+ * @param Integer $shift\r
+ * @return Math_BigInteger\r
+ * @access public\r
+ */\r
+ function bitwise_rightRotate($shift)\r
+ {\r
+ return $this->bitwise_leftRotate(-$shift);\r
+ }\r
+\r
+ /**\r
+ * Set random number generator function\r
+ *\r
+ * $generator should be the name of a random generating function whose first parameter is the minimum\r
+ * value and whose second parameter is the maximum value. If this function needs to be seeded, it should\r
+ * be seeded prior to calling Math_BigInteger::random() or Math_BigInteger::randomPrime()\r
+ *\r
+ * If the random generating function is not explicitly set, it'll be assumed to be mt_rand().\r
+ *\r
+ * @see random()\r
+ * @see randomPrime()\r
+ * @param optional String $generator\r
+ * @access public\r
+ */\r
+ function setRandomGenerator($generator)\r
+ {\r
+ $this->generator = $generator;\r
+ }\r
+\r
+ /**\r
+ * Generate a random number\r
+ *\r
+ * @param optional Integer $min\r
+ * @param optional Integer $max\r
+ * @return Math_BigInteger\r
+ * @access public\r
+ */\r
+ function random($min = false, $max = false)\r
+ {\r
+ if ($min === false) {\r
+ $min = new Math_BigInteger(0);\r
+ }\r
+\r
+ if ($max === false) {\r
+ $max = new Math_BigInteger(0x7FFFFFFF);\r
+ }\r
+\r
+ $compare = $max->compare($min);\r
+\r
+ if (!$compare) {\r
+ return $this->_normalize($min);\r
+ } else if ($compare < 0) {\r
+ // if $min is bigger then $max, swap $min and $max\r
+ $temp = $max;\r
+ $max = $min;\r
+ $min = $temp;\r
+ }\r
+\r
+ $generator = $this->generator;\r
+\r
+ $max = $max->subtract($min);\r
+ $max = ltrim($max->toBytes(), chr(0));\r
+ $size = strlen($max) - 1;\r
+ $random = '';\r
+\r
+ $bytes = $size & 1;\r
+ for ($i = 0; $i < $bytes; ++$i) {\r
+ $random.= chr($generator(0, 255));\r
+ }\r
+\r
+ $blocks = $size >> 1;\r
+ for ($i = 0; $i < $blocks; ++$i) {\r
+ // mt_rand(-2147483648, 0x7FFFFFFF) always produces -2147483648 on some systems\r
+ $random.= pack('n', $generator(0, 0xFFFF));\r
+ }\r
+\r
+ $temp = new Math_BigInteger($random, 256);\r
+ if ($temp->compare(new Math_BigInteger(substr($max, 1), 256)) > 0) {\r
+ $random = chr($generator(0, ord($max[0]) - 1)) . $random;\r
+ } else {\r
+ $random = chr($generator(0, ord($max[0]) )) . $random;\r
+ }\r
+\r
+ $random = new Math_BigInteger($random, 256);\r
+\r
+ return $this->_normalize($random->add($min));\r
+ }\r
+\r
+ /**\r
+ * Generate a random prime number.\r
+ *\r
+ * If there's not a prime within the given range, false will be returned. If more than $timeout seconds have elapsed,\r
+ * give up and return false.\r
+ *\r
+ * @param optional Integer $min\r
+ * @param optional Integer $max\r
+ * @param optional Integer $timeout\r
+ * @return Math_BigInteger\r
+ * @access public\r
+ * @internal See {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap4.pdf#page=15 HAC 4.44}.\r
+ */\r
+ function randomPrime($min = false, $max = false, $timeout = false)\r
+ {\r
+ $compare = $max->compare($min);\r
+\r
+ if (!$compare) {\r
+ return $min;\r
+ } else if ($compare < 0) {\r
+ // if $min is bigger then $max, swap $min and $max\r
+ $temp = $max;\r
+ $max = $min;\r
+ $min = $temp;\r
+ }\r
+\r
+ // gmp_nextprime() requires PHP 5 >= 5.2.0 per <http://php.net/gmp-nextprime>.\r
+ if ( MATH_BIGINTEGER_MODE == MATH_BIGINTEGER_MODE_GMP && function_exists('gmp_nextprime') ) {\r
+ // we don't rely on Math_BigInteger::random()'s min / max when gmp_nextprime() is being used since this function\r
+ // does its own checks on $max / $min when gmp_nextprime() is used. When gmp_nextprime() is not used, however,\r
+ // the same $max / $min checks are not performed.\r
+ if ($min === false) {\r
+ $min = new Math_BigInteger(0);\r
+ }\r
+\r
+ if ($max === false) {\r
+ $max = new Math_BigInteger(0x7FFFFFFF);\r
+ }\r
+\r
+ $x = $this->random($min, $max);\r
+\r
+ $x->value = gmp_nextprime($x->value);\r
+\r
+ if ($x->compare($max) <= 0) {\r
+ return $x;\r
+ }\r
+\r
+ $x->value = gmp_nextprime($min->value);\r
+\r
+ if ($x->compare($max) <= 0) {\r
+ return $x;\r
+ }\r
+\r
+ return false;\r
+ }\r
+\r
+ static $one, $two;\r
+ if (!isset($one)) {\r
+ $one = new Math_BigInteger(1);\r
+ $two = new Math_BigInteger(2);\r
+ }\r
+\r
+ $start = time();\r
+\r
+ $x = $this->random($min, $max);\r
+ if ($x->equals($two)) {\r
+ return $x;\r
+ }\r
+\r
+ $x->_make_odd();\r
+ if ($x->compare($max) > 0) {\r
+ // if $x > $max then $max is even and if $min == $max then no prime number exists between the specified range\r
+ if ($min->equals($max)) {\r
+ return false;\r
+ }\r
+ $x = $min->copy();\r
+ $x->_make_odd();\r
+ }\r
+\r
+ $initial_x = $x->copy();\r
+\r
+ while (true) {\r
+ if ($timeout !== false && time() - $start > $timeout) {\r
+ return false;\r
+ }\r
+\r
+ if ($x->isPrime()) {\r
+ return $x;\r
+ }\r
+\r
+ $x = $x->add($two);\r
+\r
+ if ($x->compare($max) > 0) {\r
+ $x = $min->copy();\r
+ if ($x->equals($two)) {\r
+ return $x;\r
+ }\r
+ $x->_make_odd();\r
+ }\r
+\r
+ if ($x->equals($initial_x)) {\r
+ return false;\r
+ }\r
+ }\r
+ }\r
+\r
+ /**\r
+ * Make the current number odd\r
+ *\r
+ * If the current number is odd it'll be unchanged. If it's even, one will be added to it.\r
+ *\r
+ * @see randomPrime()\r
+ * @access private\r
+ */\r
+ function _make_odd()\r
+ {\r
+ switch ( MATH_BIGINTEGER_MODE ) {\r
+ case MATH_BIGINTEGER_MODE_GMP:\r
+ gmp_setbit($this->value, 0);\r
+ break;\r
+ case MATH_BIGINTEGER_MODE_BCMATH:\r
+ if ($this->value[strlen($this->value) - 1] % 2 == 0) {\r
+ $this->value = bcadd($this->value, '1');\r
+ }\r
+ break;\r
+ default:\r
+ $this->value[0] |= 1;\r
+ }\r
+ }\r
+\r
+ /**\r
+ * Checks a numer to see if it's prime\r
+ *\r
+ * Assuming the $t parameter is not set, this function has an error rate of 2**-80. The main motivation for the\r
+ * $t parameter is distributability. Math_BigInteger::randomPrime() can be distributed accross multiple pageloads\r
+ * on a website instead of just one.\r
+ *\r
+ * @param optional Integer $t\r
+ * @return Boolean\r
+ * @access public\r
+ * @internal Uses the\r
+ * {@link http://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test Miller-Rabin primality test}. See \r
+ * {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap4.pdf#page=8 HAC 4.24}.\r
+ */\r
+ function isPrime($t = false)\r
+ {\r
+ $length = strlen($this->toBytes());\r
+\r
+ if (!$t) {\r
+ // see HAC 4.49 "Note (controlling the error probability)"\r
+ if ($length >= 163) { $t = 2; } // floor(1300 / 8)\r
+ else if ($length >= 106) { $t = 3; } // floor( 850 / 8)\r
+ else if ($length >= 81 ) { $t = 4; } // floor( 650 / 8)\r
+ else if ($length >= 68 ) { $t = 5; } // floor( 550 / 8)\r
+ else if ($length >= 56 ) { $t = 6; } // floor( 450 / 8)\r
+ else if ($length >= 50 ) { $t = 7; } // floor( 400 / 8)\r
+ else if ($length >= 43 ) { $t = 8; } // floor( 350 / 8)\r
+ else if ($length >= 37 ) { $t = 9; } // floor( 300 / 8)\r
+ else if ($length >= 31 ) { $t = 12; } // floor( 250 / 8)\r
+ else if ($length >= 25 ) { $t = 15; } // floor( 200 / 8)\r
+ else if ($length >= 18 ) { $t = 18; } // floor( 150 / 8)\r
+ else { $t = 27; }\r
+ }\r
+\r
+ // ie. gmp_testbit($this, 0)\r
+ // ie. isEven() or !isOdd()\r
+ switch ( MATH_BIGINTEGER_MODE ) {\r
+ case MATH_BIGINTEGER_MODE_GMP:\r
+ return gmp_prob_prime($this->value, $t) != 0;\r
+ case MATH_BIGINTEGER_MODE_BCMATH:\r
+ if ($this->value === '2') {\r
+ return true;\r
+ }\r
+ if ($this->value[strlen($this->value) - 1] % 2 == 0) {\r
+ return false;\r
+ }\r
+ break;\r
+ default:\r
+ if ($this->value == array(2)) {\r
+ return true;\r
+ }\r
+ if (~$this->value[0] & 1) {\r
+ return false;\r
+ }\r
+ }\r
+\r
+ static $primes, $zero, $one, $two;\r
+\r
+ if (!isset($primes)) {\r
+ $primes = array(\r
+ 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, \r
+ 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, \r
+ 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, \r
+ 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, \r
+ 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, \r
+ 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, \r
+ 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, \r
+ 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, \r
+ 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, \r
+ 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, \r
+ 953, 967, 971, 977, 983, 991, 997\r
+ );\r
+\r
+ if ( MATH_BIGINTEGER_MODE != MATH_BIGINTEGER_MODE_INTERNAL ) {\r
+ for ($i = 0; $i < count($primes); ++$i) {\r
+ $primes[$i] = new Math_BigInteger($primes[$i]);\r
+ }\r
+ }\r
+\r
+ $zero = new Math_BigInteger();\r
+ $one = new Math_BigInteger(1);\r
+ $two = new Math_BigInteger(2);\r
+ }\r
+\r
+ if ($this->equals($one)) {\r
+ return false;\r
+ }\r
+\r
+ // see HAC 4.4.1 "Random search for probable primes"\r
+ if ( MATH_BIGINTEGER_MODE != MATH_BIGINTEGER_MODE_INTERNAL ) {\r
+ foreach ($primes as $prime) {\r
+ list(, $r) = $this->divide($prime);\r
+ if ($r->equals($zero)) {\r
+ return $this->equals($prime);\r
+ }\r
+ }\r
+ } else {\r
+ $value = $this->value;\r
+ foreach ($primes as $prime) {\r
+ list(, $r) = $this->_divide_digit($value, $prime);\r
+ if (!$r) {\r
+ return count($value) == 1 && $value[0] == $prime;\r
+ }\r
+ }\r
+ }\r
+\r
+ $n = $this->copy();\r
+ $n_1 = $n->subtract($one);\r
+ $n_2 = $n->subtract($two);\r
+\r
+ $r = $n_1->copy();\r
+ $r_value = $r->value;\r
+ // ie. $s = gmp_scan1($n, 0) and $r = gmp_div_q($n, gmp_pow(gmp_init('2'), $s));\r
+ if ( MATH_BIGINTEGER_MODE == MATH_BIGINTEGER_MODE_BCMATH ) {\r
+ $s = 0;\r
+ // if $n was 1, $r would be 0 and this would be an infinite loop, hence our $this->equals($one) check earlier\r
+ while ($r->value[strlen($r->value) - 1] % 2 == 0) {\r
+ $r->value = bcdiv($r->value, '2', 0);\r
+ ++$s;\r
+ }\r
+ } else {\r
+ for ($i = 0, $r_length = count($r_value); $i < $r_length; ++$i) {\r
+ $temp = ~$r_value[$i] & 0xFFFFFF;\r
+ for ($j = 1; ($temp >> $j) & 1; ++$j);\r
+ if ($j != 25) {\r
+ break;\r
+ }\r
+ }\r
+ $s = 26 * $i + $j - 1;\r
+ $r->_rshift($s);\r
+ }\r
+\r
+ for ($i = 0; $i < $t; ++$i) {\r
+ $a = $this->random($two, $n_2);\r
+ $y = $a->modPow($r, $n);\r
+\r
+ if (!$y->equals($one) && !$y->equals($n_1)) {\r
+ for ($j = 1; $j < $s && !$y->equals($n_1); ++$j) {\r
+ $y = $y->modPow($two, $n);\r
+ if ($y->equals($one)) {\r
+ return false;\r
+ }\r
+ }\r
+\r
+ if (!$y->equals($n_1)) {\r
+ return false;\r
+ }\r
+ }\r
+ }\r
+ return true;\r
+ }\r
+\r
+ /**\r
+ * Logical Left Shift\r
+ *\r
+ * Shifts BigInteger's by $shift bits.\r
+ *\r
+ * @param Integer $shift\r
+ * @access private\r
+ */\r
+ function _lshift($shift)\r
+ {\r
+ if ( $shift == 0 ) {\r
+ return;\r
+ }\r
+\r
+ $num_digits = (int) ($shift / 26);\r
+ $shift %= 26;\r
+ $shift = 1 << $shift;\r
+\r
+ $carry = 0;\r
+\r
+ for ($i = 0; $i < count($this->value); ++$i) {\r
+ $temp = $this->value[$i] * $shift + $carry;\r
+ $carry = (int) ($temp / 0x4000000);\r
+ $this->value[$i] = (int) ($temp - $carry * 0x4000000);\r
+ }\r
+\r
+ if ( $carry ) {\r
+ $this->value[] = $carry;\r
+ }\r
+\r
+ while ($num_digits--) {\r
+ array_unshift($this->value, 0);\r
+ }\r
+ }\r
+\r
+ /**\r
+ * Logical Right Shift\r
+ *\r
+ * Shifts BigInteger's by $shift bits.\r
+ *\r
+ * @param Integer $shift\r
+ * @access private\r
+ */\r
+ function _rshift($shift)\r
+ {\r
+ if ($shift == 0) {\r
+ return;\r
+ }\r
+\r
+ $num_digits = (int) ($shift / 26);\r
+ $shift %= 26;\r
+ $carry_shift = 26 - $shift;\r
+ $carry_mask = (1 << $shift) - 1;\r
+\r
+ if ( $num_digits ) {\r
+ $this->value = array_slice($this->value, $num_digits);\r
+ }\r
+\r
+ $carry = 0;\r
+\r
+ for ($i = count($this->value) - 1; $i >= 0; --$i) {\r
+ $temp = $this->value[$i] >> $shift | $carry;\r
+ $carry = ($this->value[$i] & $carry_mask) << $carry_shift;\r
+ $this->value[$i] = $temp;\r
+ }\r
+\r
+ $this->value = $this->_trim($this->value);\r
+ }\r
+\r
+ /**\r
+ * Normalize\r
+ *\r
+ * Removes leading zeros and truncates (if necessary) to maintain the appropriate precision\r
+ *\r
+ * @param Math_BigInteger\r
+ * @return Math_BigInteger\r
+ * @see _trim()\r
+ * @access private\r
+ */\r
+ function _normalize($result)\r
+ {\r
+ $result->precision = $this->precision;\r
+ $result->bitmask = $this->bitmask;\r
+\r
+ switch ( MATH_BIGINTEGER_MODE ) {\r
+ case MATH_BIGINTEGER_MODE_GMP:\r
+ if (!empty($result->bitmask->value)) {\r
+ $result->value = gmp_and($result->value, $result->bitmask->value);\r
+ }\r
+\r
+ return $result;\r
+ case MATH_BIGINTEGER_MODE_BCMATH:\r
+ if (!empty($result->bitmask->value)) {\r
+ $result->value = bcmod($result->value, $result->bitmask->value);\r
+ }\r
+\r
+ return $result;\r
+ }\r
+\r
+ $value = &$result->value;\r
+\r
+ if ( !count($value) ) {\r
+ return $result;\r
+ }\r
+\r
+ $value = $this->_trim($value);\r
+\r
+ if (!empty($result->bitmask->value)) {\r
+ $length = min(count($value), count($this->bitmask->value));\r
+ $value = array_slice($value, 0, $length);\r
+\r
+ for ($i = 0; $i < $length; ++$i) {\r
+ $value[$i] = $value[$i] & $this->bitmask->value[$i];\r
+ }\r
+ }\r
+\r
+ return $result;\r
+ }\r
+\r
+ /**\r
+ * Trim\r
+ *\r
+ * Removes leading zeros\r
+ *\r
+ * @return Math_BigInteger\r
+ * @access private\r
+ */\r
+ function _trim($value)\r
+ {\r
+ for ($i = count($value) - 1; $i >= 0; --$i) {\r
+ if ( $value[$i] ) {\r
+ break;\r
+ }\r
+ unset($value[$i]);\r
+ }\r
+\r
+ return $value;\r
+ }\r
+\r
+ /**\r
+ * Array Repeat\r
+ *\r
+ * @param $input Array\r
+ * @param $multiplier mixed\r
+ * @return Array\r
+ * @access private\r
+ */\r
+ function _array_repeat($input, $multiplier)\r
+ {\r
+ return ($multiplier) ? array_fill(0, $multiplier, $input) : array();\r
+ }\r
+\r
+ /**\r
+ * Logical Left Shift\r
+ *\r
+ * Shifts binary strings $shift bits, essentially multiplying by 2**$shift.\r
+ *\r
+ * @param $x String\r
+ * @param $shift Integer\r
+ * @return String\r
+ * @access private\r
+ */\r
+ function _base256_lshift(&$x, $shift)\r
+ {\r
+ if ($shift == 0) {\r
+ return;\r
+ }\r
+\r
+ $num_bytes = $shift >> 3; // eg. floor($shift/8)\r
+ $shift &= 7; // eg. $shift % 8\r
+\r
+ $carry = 0;\r
+ for ($i = strlen($x) - 1; $i >= 0; --$i) {\r
+ $temp = ord($x[$i]) << $shift | $carry;\r
+ $x[$i] = chr($temp);\r
+ $carry = $temp >> 8;\r
+ }\r
+ $carry = ($carry != 0) ? chr($carry) : '';\r
+ $x = $carry . $x . str_repeat(chr(0), $num_bytes);\r
+ }\r
+\r
+ /**\r
+ * Logical Right Shift\r
+ *\r
+ * Shifts binary strings $shift bits, essentially dividing by 2**$shift and returning the remainder.\r
+ *\r
+ * @param $x String\r
+ * @param $shift Integer\r
+ * @return String\r
+ * @access private\r
+ */\r
+ function _base256_rshift(&$x, $shift)\r
+ {\r
+ if ($shift == 0) {\r
+ $x = ltrim($x, chr(0));\r
+ return '';\r
+ }\r
+\r
+ $num_bytes = $shift >> 3; // eg. floor($shift/8)\r
+ $shift &= 7; // eg. $shift % 8\r
+\r
+ $remainder = '';\r
+ if ($num_bytes) {\r
+ $start = $num_bytes > strlen($x) ? -strlen($x) : -$num_bytes;\r
+ $remainder = substr($x, $start);\r
+ $x = substr($x, 0, -$num_bytes);\r
+ }\r
+\r
+ $carry = 0;\r
+ $carry_shift = 8 - $shift;\r
+ for ($i = 0; $i < strlen($x); ++$i) {\r
+ $temp = (ord($x[$i]) >> $shift) | $carry;\r
+ $carry = (ord($x[$i]) << $carry_shift) & 0xFF;\r
+ $x[$i] = chr($temp);\r
+ }\r
+ $x = ltrim($x, chr(0));\r
+\r
+ $remainder = chr($carry >> $carry_shift) . $remainder;\r
+\r
+ return ltrim($remainder, chr(0));\r
+ }\r
+\r
+ // one quirk about how the following functions are implemented is that PHP defines N to be an unsigned long\r
+ // at 32-bits, while java's longs are 64-bits.\r
+\r
+ /**\r
+ * Converts 32-bit integers to bytes.\r
+ *\r
+ * @param Integer $x\r
+ * @return String\r
+ * @access private\r
+ */\r
+ function _int2bytes($x)\r
+ {\r
+ return ltrim(pack('N', $x), chr(0));\r
+ }\r
+\r
+ /**\r
+ * Converts bytes to 32-bit integers\r
+ *\r
+ * @param String $x\r
+ * @return Integer\r
+ * @access private\r
+ */\r
+ function _bytes2int($x)\r
+ {\r
+ $temp = unpack('Nint', str_pad($x, 4, chr(0), STR_PAD_LEFT));\r
+ return $temp['int'];\r
+ }\r
+}
\ No newline at end of file
projects / friendica.git / blobdiff