1: <?php
   2: 
   3: namespace Liberty;
   4: 
   5: /**
   6:  * Pure-PHP arbitrary precision integer arithmetic library.  
   7:  *
   8:  * Supports base-2, base-10, base-16, and base-256 numbers.  Uses the GMP or BCMath extensions, if available,
   9:  * and an internal implementation, otherwise.
  10:  *
  11:  * PHP versions 4 and 5
  12:  *
  13:  * {@internal (all DocBlock comments regarding implementation - such as the one that follows - refer to the
  14:  * {@link MATH_BIGINTEGER_MODE_INTERNAL MATH_BIGINTEGER_MODE_INTERNAL} mode)
  15:  *
  16:  * BigInteger uses base-2**26 to perform operations such as multiplication and division and
  17:  * base-2**52 (ie. two base 2**26 digits) to perform addition and subtraction.  Because the largest possible
  18:  * value when multiplying two base-2**26 numbers together is a base-2**52 number, double precision floating
  19:  * point numbers - numbers that should be supported on most hardware and whose significand is 53 bits - are
  20:  * used.  As a consequence, bitwise operators such as >> and << cannot be used, nor can the modulo operator %,
  21:  * which only supports integers.  Although this fact will slow this library down, the fact that such a high
  22:  * base is being used should more than compensate.
  23:  *
  24:  * When PHP version 6 is officially released, we'll be able to use 64-bit integers.  This should, once again,
  25:  * allow bitwise operators, and will increase the maximum possible base to 2**31 (or 2**62 for addition /
  26:  * subtraction).
  27:  *
  28:  * Numbers are stored in {@link http://en.wikipedia.org/wiki/Endianness little endian} format.  ie.
  29:  * (new BigInteger(pow(2, 26)))->value = array(0, 1)
  30:  *
  31:  * Useful resources are as follows:
  32:  *
  33:  *  - {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf Handbook of Applied Cryptography (HAC)}
  34:  *  - {@link http://math.libtomcrypt.com/files/tommath.pdf Multi-Precision Math (MPM)}
  35:  *  - Java's BigInteger classes.  See /j2se/src/share/classes/java/math in jdk-1_5_0-src-jrl.zip
  36:  *
  37:  * Here's an example of how to use this library:
  38:  * <code>
  39:  * <?php
  40:  *    include('Math/BigInteger.php');
  41:  *
  42:  *    $a = new BigInteger(2);
  43:  *    $b = new BigInteger(3);
  44:  *
  45:  *    $c = $a->add($b);
  46:  *
  47:  *    echo $c->toString(); // outputs 5
  48:  * ?>
  49:  * </code>
  50:  *
  51:  * LICENSE: Permission is hereby granted, free of charge, to any person obtaining a copy
  52:  * of this software and associated documentation files (the "Software"), to deal
  53:  * in the Software without restriction, including without limitation the rights
  54:  * to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
  55:  * copies of the Software, and to permit persons to whom the Software is
  56:  * furnished to do so, subject to the following conditions:
  57:  *
  58:  * The above copyright notice and this permission notice shall be included in
  59:  * all copies or substantial portions of the Software.
  60:  *
  61:  * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
  62:  * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
  63:  * FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
  64:  * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
  65:  * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
  66:  * OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
  67:  * THE SOFTWARE.
  68:  *
  69:  * @category  Math
  70:  * @package   BigInteger
  71:  * @author    Jim Wigginton <terrafrost@php.net>, Crypto Liberty Group <cryptolibertygroup@gmail.com>
  72:  * @copyright MMVI Jim Wigginton, Crypto Liberty Group
  73:  * @license   http://www.opensource.org/licenses/mit-license.html  MIT License
  74:  */
  75: 
  76: /**#@+
  77:  * Reduction constants
  78:  *
  79:  * @access private
  80:  * @see BigInteger::_reduce()
  81:  */
  82: /**
  83:  * @see BigInteger::_montgomery()
  84:  * @see BigInteger::_prepMontgomery()
  85:  */
  86: define('MATH_BIGINTEGER_MONTGOMERY', 0);
  87: /**
  88:  * @see BigInteger::_barrett()
  89:  */
  90: define('MATH_BIGINTEGER_BARRETT', 1);
  91: /**
  92:  * @see BigInteger::_mod2()
  93:  */
  94: define('MATH_BIGINTEGER_POWEROF2', 2);
  95: /**
  96:  * @see BigInteger::_remainder()
  97:  */
  98: define('MATH_BIGINTEGER_CLASSIC', 3);
  99: /**
 100:  * @see BigInteger::__clone()
 101:  */
 102: define('MATH_BIGINTEGER_NONE', 4);
 103: /**#@-*/
 104: 
 105: /**#@+
 106:  * Array constants
 107:  *
 108:  * Rather than create a thousands and thousands of new BigInteger objects in repeated function calls to add() and
 109:  * multiply() or whatever, we'll just work directly on arrays, taking them in as parameters and returning them.
 110:  *
 111:  * @access private
 112:  */
 113: /**
 114:  * $result[MATH_BIGINTEGER_VALUE] contains the value.
 115:  */
 116: define('MATH_BIGINTEGER_VALUE', 0);
 117: /**
 118:  * $result[MATH_BIGINTEGER_SIGN] contains the sign.
 119:  */
 120: define('MATH_BIGINTEGER_SIGN', 1);
 121: /**#@-*/
 122: 
 123: /**#@+
 124:  * @access private
 125:  * @see BigInteger::_montgomery()
 126:  * @see BigInteger::_barrett()
 127:  */
 128: /**
 129:  * Cache constants
 130:  *
 131:  * $cache[MATH_BIGINTEGER_VARIABLE] tells us whether or not the cached data is still valid.
 132:  */
 133: define('MATH_BIGINTEGER_VARIABLE', 0);
 134: /**
 135:  * $cache[MATH_BIGINTEGER_DATA] contains the cached data.
 136:  */
 137: define('MATH_BIGINTEGER_DATA', 1);
 138: /**#@-*/
 139: 
 140: /**#@+
 141:  * Mode constants.
 142:  *
 143:  * @access private
 144:  * @see BigInteger::BigInteger()
 145:  */
 146: /**
 147:  * To use the pure-PHP implementation
 148:  */
 149: define('MATH_BIGINTEGER_MODE_INTERNAL', 1);
 150: /**
 151:  * To use the BCMath library
 152:  *
 153:  * (if enabled; otherwise, the internal implementation will be used)
 154:  */
 155: define('MATH_BIGINTEGER_MODE_BCMATH', 2);
 156: /**
 157:  * To use the GMP library
 158:  *
 159:  * (if present; otherwise, either the BCMath or the internal implementation will be used)
 160:  */
 161: define('MATH_BIGINTEGER_MODE_GMP', 3);
 162: /**#@-*/
 163: 
 164: /**
 165:  * Karatsuba Cutoff
 166:  *
 167:  * At what point do we switch between Karatsuba multiplication and schoolbook long multiplication?
 168:  *
 169:  * @access private
 170:  */
 171: define('MATH_BIGINTEGER_KARATSUBA_CUTOFF', 25);
 172: 
 173: /**
 174:  * Pure-PHP arbitrary precision integer arithmetic library. Supports base-2, base-10, base-16, and base-256
 175:  * numbers.
 176:  *
 177:  * @package BigInteger
 178:  * @author  Jim Wigginton <terrafrost@php.net>
 179:  * @version 1.0.0RC4
 180:  * @access  public
 181:  */
 182: class BigInteger
 183: {
 184:     /**
 185:      * Holds the BigInteger's value.
 186:      *
 187:      * @var Array
 188:      * @access private
 189:      */
 190:     var $value;
 191: 
 192:     /**
 193:      * Holds the BigInteger's magnitude.
 194:      *
 195:      * @var Boolean
 196:      * @access private
 197:      */
 198:     var $is_negative = false;
 199: 
 200:     /**
 201:      * Random number generator function
 202:      *
 203:      * @see setRandomGenerator()
 204:      * @access private
 205:      */
 206:     var $generator = 'mt_rand';
 207: 
 208:     /**
 209:      * Precision
 210:      *
 211:      * @see setPrecision()
 212:      * @access private
 213:      */
 214:     var $precision = -1;
 215: 
 216:     /**
 217:      * Precision Bitmask
 218:      *
 219:      * @see setPrecision()
 220:      * @access private
 221:      */
 222:     var $bitmask = false;
 223: 
 224:     /**
 225:      * Mode independent value used for serialization.
 226:      *
 227:      * If the bcmath or gmp extensions are installed $this->value will be a non-serializable resource, hence the need for
 228:      * a variable that'll be serializable regardless of whether or not extensions are being used.  Unlike $this->value,
 229:      * however, $this->hex is only calculated when $this->__sleep() is called.
 230:      *
 231:      * @see __sleep()
 232:      * @see __wakeup()
 233:      * @var String
 234:      * @access private
 235:      */
 236:     var $hex;
 237: 
 238:     /**
 239:      * Converts base-2, base-10, base-16, and binary strings (base-256) to BigIntegers.
 240:      *
 241:      * If the second parameter - $base - is negative, then it will be assumed that the number's are encoded using
 242:      * two's compliment.  The sole exception to this is -10, which is treated the same as 10 is.
 243:      *
 244:      * Here's an example:
 245:      * <code>
 246:      * &lt;?php
 247:      *    include('Math/BigInteger.php');
 248:      *
 249:      *    $a = new BigInteger('0x32', 16); // 50 in base-16
 250:      *
 251:      *    echo $a->toString(); // outputs 50
 252:      * ?&gt;
 253:      * </code>
 254:      *
 255:      * @param optional $x base-10 number or base-$base number if $base set.
 256:      * @param optional integer $base
 257:      * @return BigInteger
 258:      * @access public
 259:      */
 260:     function __construct($x = 0, $base = 10)
 261:     {
 262:         if ( !defined('MATH_BIGINTEGER_MODE') ) {
 263:             switch (true) {
 264:                 case extension_loaded('gmp'):
 265:                     define('MATH_BIGINTEGER_MODE', MATH_BIGINTEGER_MODE_GMP);
 266:                     break;
 267:                 case extension_loaded('bcmath'):
 268:                     define('MATH_BIGINTEGER_MODE', MATH_BIGINTEGER_MODE_BCMATH);
 269:                     break;
 270:                 default:
 271:                     define('MATH_BIGINTEGER_MODE', MATH_BIGINTEGER_MODE_INTERNAL);
 272:             }
 273:         }
 274: 
 275:         if (function_exists('openssl_public_encrypt') && !defined('MATH_BIGINTEGER_OPENSSL_DISABLE') && !defined('MATH_BIGINTEGER_OPENSSL_ENABLED')) {
 276:             // some versions of XAMPP have mismatched versions of OpenSSL which causes it not to work
 277:             ob_start();
 278:             phpinfo();
 279:             $content = ob_get_contents();
 280:             ob_end_clean();
 281: 
 282:             preg_match_all('#OpenSSL (Header|Library) Version(.*)#im', $content, $matches);
 283: 
 284:             $versions = array();
 285:             if (!empty($matches[1])) {
 286:                 for ($i = 0; $i < count($matches[1]); $i++) {
 287:                     $versions[$matches[1][$i]] = trim(str_replace('=>', '', strip_tags($matches[2][$i])));
 288:                 }
 289:             }
 290: 
 291:             // it doesn't appear that OpenSSL versions were reported upon until PHP 5.3+
 292:             switch (true) {
 293:                 case !isset($versions['Header']):
 294:                 case !isset($versions['Library']):
 295:                 case $versions['Header'] == $versions['Library']:
 296:                     define('MATH_BIGINTEGER_OPENSSL_ENABLED', true);
 297:                     break;
 298:                 default:
 299:                     define('MATH_BIGINTEGER_OPENSSL_DISABLE', true);
 300:             }
 301:         }
 302: 
 303:         if (!defined('PHP_INT_SIZE')) {
 304:             define('PHP_INT_SIZE', 4);
 305:         }
 306: 
 307:         if (!defined('MATH_BIGINTEGER_BASE') && MATH_BIGINTEGER_MODE == MATH_BIGINTEGER_MODE_INTERNAL) {
 308:             switch (PHP_INT_SIZE) {
 309:                 case 8: // use 64-bit integers if int size is 8 bytes
 310:                     define('MATH_BIGINTEGER_BASE',       31);
 311:                     define('MATH_BIGINTEGER_BASE_FULL',  0x80000000);
 312:                     define('MATH_BIGINTEGER_MAX_DIGIT',  0x7FFFFFFF);
 313:                     define('MATH_BIGINTEGER_MSB',        0x40000000);
 314:                     // 10**9 is the closest we can get to 2**31 without passing it
 315:                     define('MATH_BIGINTEGER_MAX10',      1000000000);
 316:                     define('MATH_BIGINTEGER_MAX10_LEN',  9);
 317:                     // the largest digit that may be used in addition / subtraction
 318:                     define('MATH_BIGINTEGER_MAX_DIGIT2', pow(2, 62));
 319:                     break;
 320:                 //case 4: // use 64-bit floats if int size is 4 bytes
 321:                 default:
 322:                     define('MATH_BIGINTEGER_BASE',       26);
 323:                     define('MATH_BIGINTEGER_BASE_FULL',  0x4000000);
 324:                     define('MATH_BIGINTEGER_MAX_DIGIT',  0x3FFFFFF);
 325:                     define('MATH_BIGINTEGER_MSB',        0x2000000);
 326:                     // 10**7 is the closest to 2**26 without passing it
 327:                     define('MATH_BIGINTEGER_MAX10',      10000000);
 328:                     define('MATH_BIGINTEGER_MAX10_LEN',  7);
 329:                     // the largest digit that may be used in addition / subtraction
 330:                     // we do pow(2, 52) instead of using 4503599627370496 directly because some
 331:                     // PHP installations will truncate 4503599627370496.
 332:                     define('MATH_BIGINTEGER_MAX_DIGIT2', pow(2, 52));
 333:             }
 334:         }
 335: 
 336:         switch ( MATH_BIGINTEGER_MODE ) {
 337:             case MATH_BIGINTEGER_MODE_GMP:
 338:                 if (is_resource($x) && get_resource_type($x) == 'GMP integer') {
 339:                     $this->value = $x;
 340:                     return;
 341:                 }
 342:                 $this->value = gmp_init(0);
 343:                 break;
 344:             case MATH_BIGINTEGER_MODE_BCMATH:
 345:                 $this->value = '0';
 346:                 break;
 347:             default:
 348:                 $this->value = array();
 349:         }
 350: 
 351:         // '0' counts as empty() but when the base is 256 '0' is equal to ord('0') or 48
 352:         // '0' is the only value like this per http://php.net/empty
 353:         if (empty($x) && (abs($base) != 256 || $x !== '0')) {
 354:             return;
 355:         }
 356: 
 357:         switch ($base) {
 358:             case -256:
 359:                 if (ord($x[0]) & 0x80) {
 360:                     $x = ~$x;
 361:                     $this->is_negative = true;
 362:                 }
 363:             case  256:
 364:                 switch ( MATH_BIGINTEGER_MODE ) {
 365:                     case MATH_BIGINTEGER_MODE_GMP:
 366:                         $sign = $this->is_negative ? '-' : '';
 367:                         $this->value = gmp_init($sign . '0x' . bin2hex($x));
 368:                         break;
 369:                     case MATH_BIGINTEGER_MODE_BCMATH:
 370:                         // round $len to the nearest 4 (thanks, DavidMJ!)
 371:                         $len = (strlen($x) + 3) & 0xFFFFFFFC;
 372: 
 373:                         $x = str_pad($x, $len, chr(0), STR_PAD_LEFT);
 374: 
 375:                         for ($i = 0; $i < $len; $i+= 4) {
 376:                             $this->value = bcmul($this->value, '4294967296', 0); // 4294967296 == 2**32
 377:                             $this->value = bcadd($this->value, 0x1000000 * ord($x[$i]) + ((ord($x[$i + 1]) << 16) | (ord($x[$i + 2]) << 8) | ord($x[$i + 3])), 0);
 378:                         }
 379: 
 380:                         if ($this->is_negative) {
 381:                             $this->value = '-' . $this->value;
 382:                         }
 383: 
 384:                         break;
 385:                     // converts a base-2**8 (big endian / msb) number to base-2**26 (little endian / lsb)
 386:                     default:
 387:                         while (strlen($x)) {
 388:                             $this->value[] = $this->_bytes2int($this->_base256_rshift($x, MATH_BIGINTEGER_BASE));
 389:                         }
 390:                 }
 391: 
 392:                 if ($this->is_negative) {
 393:                     if (MATH_BIGINTEGER_MODE != MATH_BIGINTEGER_MODE_INTERNAL) {
 394:                         $this->is_negative = false;
 395:                     }
 396:                     $temp = $this->add(new BigInteger('-1'));
 397:                     $this->value = $temp->value;
 398:                 }
 399:                 break;
 400:             case  16:
 401:             case -16:
 402:                 if ($base > 0 && $x[0] == '-') {
 403:                     $this->is_negative = true;
 404:                     $x = substr($x, 1);
 405:                 }
 406: 
 407:                 $x = preg_replace('#^(?:0x)?([A-Fa-f0-9]*).*#', '$1', $x);
 408: 
 409:                 $is_negative = false;
 410:                 if ($base < 0 && hexdec($x[0]) >= 8) {
 411:                     $this->is_negative = $is_negative = true;
 412:                     $x = bin2hex(~pack('H*', $x));
 413:                 }
 414: 
 415:                 switch ( MATH_BIGINTEGER_MODE ) {
 416:                     case MATH_BIGINTEGER_MODE_GMP:
 417:                         $temp = $this->is_negative ? '-0x' . $x : '0x' . $x;
 418:                         $this->value = gmp_init($temp);
 419:                         $this->is_negative = false;
 420:                         break;
 421:                     case MATH_BIGINTEGER_MODE_BCMATH:
 422:                         $x = ( strlen($x) & 1 ) ? '0' . $x : $x;
 423:                         $temp = new BigInteger(pack('H*', $x), 256);
 424:                         $this->value = $this->is_negative ? '-' . $temp->value : $temp->value;
 425:                         $this->is_negative = false;
 426:                         break;
 427:                     default:
 428:                         $x = ( strlen($x) & 1 ) ? '0' . $x : $x;
 429:                         $temp = new BigInteger(pack('H*', $x), 256);
 430:                         $this->value = $temp->value;
 431:                 }
 432: 
 433:                 if ($is_negative) {
 434:                     $temp = $this->add(new BigInteger('-1'));
 435:                     $this->value = $temp->value;
 436:                 }
 437:                 break;
 438:             case  10:
 439:             case -10:
 440:                 // (?<!^)(?:-).*: find any -'s that aren't at the beginning and then any characters that follow that
 441:                 // (?<=^|-)0*: find any 0's that are preceded by the start of the string or by a - (ie. octals)
 442:                 // [^-0-9].*: find any non-numeric characters and then any characters that follow that
 443:                 $x = preg_replace('#(?<!^)(?:-).*|(?<=^|-)0*|[^-0-9].*#', '', $x);
 444: 
 445:                 switch ( MATH_BIGINTEGER_MODE ) {
 446:                     case MATH_BIGINTEGER_MODE_GMP:
 447:                         $this->value = gmp_init($x);
 448:                         break;
 449:                     case MATH_BIGINTEGER_MODE_BCMATH:
 450:                         // explicitly casting $x to a string is necessary, here, since doing $x[0] on -1 yields different
 451:                         // results then doing it on '-1' does (modInverse does $x[0])
 452:                         $this->value = $x === '-' ? '0' : (string) $x;
 453:                         break;
 454:                     default:
 455:                         $temp = new BigInteger();
 456: 
 457:                         $multiplier = new BigInteger();
 458:                         $multiplier->value = array(MATH_BIGINTEGER_MAX10);
 459: 
 460:                         if ($x[0] == '-') {
 461:                             $this->is_negative = true;
 462:                             $x = substr($x, 1);
 463:                         }
 464: 
 465:                         $x = str_pad($x, strlen($x) + ((MATH_BIGINTEGER_MAX10_LEN - 1) * strlen($x)) % MATH_BIGINTEGER_MAX10_LEN, 0, STR_PAD_LEFT);
 466:                         while (strlen($x)) {
 467:                             $temp = $temp->multiply($multiplier);
 468:                             $temp = $temp->add(new BigInteger($this->_int2bytes(substr($x, 0, MATH_BIGINTEGER_MAX10_LEN)), 256));
 469:                             $x = substr($x, MATH_BIGINTEGER_MAX10_LEN);
 470:                         }
 471: 
 472:                         $this->value = $temp->value;
 473:                 }
 474:                 break;
 475:             case  2: // base-2 support originally implemented by Lluis Pamies - thanks!
 476:             case -2:
 477:                 if ($base > 0 && $x[0] == '-') {
 478:                     $this->is_negative = true;
 479:                     $x = substr($x, 1);
 480:                 }
 481: 
 482:                 $x = preg_replace('#^([01]*).*#', '$1', $x);
 483:                 $x = str_pad($x, strlen($x) + (3 * strlen($x)) % 4, 0, STR_PAD_LEFT);
 484: 
 485:                 $str = '0x';
 486:                 while (strlen($x)) {
 487:                     $part = substr($x, 0, 4);
 488:                     $str.= dechex(bindec($part));
 489:                     $x = substr($x, 4);
 490:                 }
 491: 
 492:                 if ($this->is_negative) {
 493:                     $str = '-' . $str;
 494:                 }
 495: 
 496:                 $temp = new BigInteger($str, 8 * $base); // ie. either -16 or +16
 497:                 $this->value = $temp->value;
 498:                 $this->is_negative = $temp->is_negative;
 499: 
 500:                 break;
 501:             default:
 502:                 // base not supported, so we'll let $this == 0
 503:         }
 504:     }
 505: 
 506:     /**
 507:      * This function exists to maintain backwards compatibility with older code
 508:      *
 509:      * @param int $x    base-10 number or base-$base number if $base set.
 510:      * @param int $base Number base
 511:      */
 512:     public function BigInteger($x = 0, $base = 10)
 513:     {
 514:         self::__construct($x, $base);
 515:     }
 516: 
 517:     /**
 518:      * Converts a BigInteger to a byte string (eg. base-256).
 519:      *
 520:      * Negative numbers are saved as positive numbers, unless $twos_compliment is set to true, at which point, they're
 521:      * saved as two's compliment.
 522:      *
 523:      * Here's an example:
 524:      * <code>
 525:      * <?php
 526:      *    include('Math/BigInteger.php');
 527:      *
 528:      *    $a = new BigInteger('65');
 529:      *
 530:      *    echo $a->toBytes(); // outputs chr(65)
 531:      * ?>
 532:      * </code>
 533:      *
 534:      * @param Boolean $twos_compliment
 535:      * @return String
 536:      * @access public
 537:      * @internal Converts a base-2**26 number to base-2**8
 538:      */
 539:     function toBytes($twos_compliment = false)
 540:     {
 541:         if ($twos_compliment) {
 542:             $comparison = $this->compare(new BigInteger());
 543:             if ($comparison == 0) {
 544:                 return $this->precision > 0 ? str_repeat(chr(0), ($this->precision + 1) >> 3) : '';
 545:             }
 546: 
 547:             $temp = $comparison < 0 ? $this->add(new BigInteger(1)) : $this->copy();
 548:             $bytes = $temp->toBytes();
 549: 
 550:             if (empty($bytes)) { // eg. if the number we're trying to convert is -1
 551:                 $bytes = chr(0);
 552:             }
 553: 
 554:             if (ord($bytes[0]) & 0x80) {
 555:                 $bytes = chr(0) . $bytes;
 556:             }
 557: 
 558:             return $comparison < 0 ? ~$bytes : $bytes;
 559:         }
 560: 
 561:         switch ( MATH_BIGINTEGER_MODE ) {
 562:             case MATH_BIGINTEGER_MODE_GMP:
 563:                 if (gmp_cmp($this->value, gmp_init(0)) == 0) {
 564:                     return $this->precision > 0 ? str_repeat(chr(0), ($this->precision + 1) >> 3) : '';
 565:                 }
 566: 
 567:                 $temp = gmp_strval(gmp_abs($this->value), 16);
 568:                 $temp = ( strlen($temp) & 1 ) ? '0' . $temp : $temp;
 569:                 $temp = pack('H*', $temp);
 570: 
 571:                 return $this->precision > 0 ?
 572:                     substr(str_pad($temp, $this->precision >> 3, chr(0), STR_PAD_LEFT), -($this->precision >> 3)) :
 573:                     ltrim($temp, chr(0));
 574:             case MATH_BIGINTEGER_MODE_BCMATH:
 575:                 if ($this->value === '0') {
 576:                     return $this->precision > 0 ? str_repeat(chr(0), ($this->precision + 1) >> 3) : '';
 577:                 }
 578: 
 579:                 $value = '';
 580:                 $current = $this->value;
 581: 
 582:                 if ($current[0] == '-') {
 583:                     $current = substr($current, 1);
 584:                 }
 585: 
 586:                 while (bccomp($current, '0', 0) > 0) {
 587:                     $temp = bcmod($current, '16777216');
 588:                     $value = chr($temp >> 16) . chr($temp >> 8) . chr($temp) . $value;
 589:                     $current = bcdiv($current, '16777216', 0);
 590:                 }
 591: 
 592:                 return $this->precision > 0 ?
 593:                     substr(str_pad($value, $this->precision >> 3, chr(0), STR_PAD_LEFT), -($this->precision >> 3)) :
 594:                     ltrim($value, chr(0));
 595:         }
 596: 
 597:         if (!count($this->value)) {
 598:             return $this->precision > 0 ? str_repeat(chr(0), ($this->precision + 1) >> 3) : '';
 599:         }
 600:         $result = $this->_int2bytes($this->value[count($this->value) - 1]);
 601: 
 602:         $temp = $this->copy();
 603: 
 604:         for ($i = count($temp->value) - 2; $i >= 0; --$i) {
 605:             $temp->_base256_lshift($result, MATH_BIGINTEGER_BASE);
 606:             $result = $result | str_pad($temp->_int2bytes($temp->value[$i]), strlen($result), chr(0), STR_PAD_LEFT);
 607:         }
 608: 
 609:         return $this->precision > 0 ?
 610:             str_pad(substr($result, -(($this->precision + 7) >> 3)), ($this->precision + 7) >> 3, chr(0), STR_PAD_LEFT) :
 611:             $result;
 612:     }
 613: 
 614:     /**
 615:      * Converts a BigInteger to a hex string (eg. base-16)).
 616:      *
 617:      * Negative numbers are saved as positive numbers, unless $twos_compliment is set to true, at which point, they're
 618:      * saved as two's compliment.
 619:      *
 620:      * Here's an example:
 621:      * <code>
 622:      * <?php
 623:      *    include('Math/BigInteger.php');
 624:      *
 625:      *    $a = new BigInteger('65');
 626:      *
 627:      *    echo $a->toHex(); // outputs '41'
 628:      * ?>
 629:      * </code>
 630:      *
 631:      * @param Boolean $twos_compliment
 632:      * @return String
 633:      * @access public
 634:      * @internal Converts a base-2**26 number to base-2**8
 635:      */
 636:     function toHex($twos_compliment = false)
 637:     {
 638:         return bin2hex($this->toBytes($twos_compliment));
 639:     }
 640: 
 641:     /**
 642:      * Converts a BigInteger to a bit string (eg. base-2).
 643:      *
 644:      * Negative numbers are saved as positive numbers, unless $twos_compliment is set to true, at which point, they're
 645:      * saved as two's compliment.
 646:      *
 647:      * Here's an example:
 648:      * <code>
 649:      * <?php
 650:      *    include('Math/BigInteger.php');
 651:      *
 652:      *    $a = new BigInteger('65');
 653:      *
 654:      *    echo $a->toBits(); // outputs '1000001'
 655:      * ?>
 656:      * </code>
 657:      *
 658:      * @param Boolean $twos_compliment
 659:      * @return String
 660:      * @access public
 661:      * @internal Converts a base-2**26 number to base-2**2
 662:      */
 663:     function toBits($twos_compliment = false)
 664:     {
 665:         $hex = $this->toHex($twos_compliment);
 666:         $bits = '';
 667:         for ($i = strlen($hex) - 8, $start = strlen($hex) & 7; $i >= $start; $i-=8) {
 668:             $bits = str_pad(decbin(hexdec(substr($hex, $i, 8))), 32, '0', STR_PAD_LEFT) . $bits;
 669:         }
 670:         if ($start) { // hexdec('') == 0
 671:             $bits = str_pad(decbin(hexdec(substr($hex, 0, $start))), 8, '0', STR_PAD_LEFT) . $bits;
 672:         }
 673:         $result = $this->precision > 0 ? substr($bits, -$this->precision) : ltrim($bits, '0');
 674: 
 675:         if ($twos_compliment && $this->compare(new BigInteger()) > 0 && $this->precision <= 0) {
 676:             return '0' . $result;
 677:         }
 678: 
 679:         return $result;
 680:     }
 681: 
 682:     /**
 683:      * Converts a BigInteger to a base-10 number.
 684:      *
 685:      * Here's an example:
 686:      * <code>
 687:      * <?php
 688:      *    include('Math/BigInteger.php');
 689:      *
 690:      *    $a = new BigInteger('50');
 691:      *
 692:      *    echo $a->toString(); // outputs 50
 693:      * ?>
 694:      * </code>
 695:      *
 696:      * @return String
 697:      * @access public
 698:      * @internal Converts a base-2**26 number to base-10**7 (which is pretty much base-10)
 699:      */
 700:     function toString()
 701:     {
 702:         switch ( MATH_BIGINTEGER_MODE ) {
 703:             case MATH_BIGINTEGER_MODE_GMP:
 704:                 return gmp_strval($this->value);
 705:             case MATH_BIGINTEGER_MODE_BCMATH:
 706:                 if ($this->value === '0') {
 707:                     return '0';
 708:                 }
 709: 
 710:                 return ltrim($this->value, '0');
 711:         }
 712: 
 713:         if (!count($this->value)) {
 714:             return '0';
 715:         }
 716: 
 717:         $temp = $this->copy();
 718:         $temp->is_negative = false;
 719: 
 720:         $divisor = new BigInteger();
 721:         $divisor->value = array(MATH_BIGINTEGER_MAX10);
 722:         $result = '';
 723:         while (count($temp->value)) {
 724:             list($temp, $mod) = $temp->divide($divisor);
 725:             $result = str_pad(isset($mod->value[0]) ? $mod->value[0] : '', MATH_BIGINTEGER_MAX10_LEN, '0', STR_PAD_LEFT) . $result;
 726:         }
 727:         $result = ltrim($result, '0');
 728:         if (empty($result)) {
 729:             $result = '0';
 730:         }
 731: 
 732:         if ($this->is_negative) {
 733:             $result = '-' . $result;
 734:         }
 735: 
 736:         return $result;
 737:     }
 738: 
 739:     /**
 740:      * Copy an object
 741:      *
 742:      * PHP5 passes objects by reference while PHP4 passes by value.  As such, we need a function to guarantee
 743:      * that all objects are passed by value, when appropriate.  More information can be found here:
 744:      *
 745:      * {@link http://php.net/language.oop5.basic#51624}
 746:      *
 747:      * @access public
 748:      * @see __clone()
 749:      * @return BigInteger
 750:      */
 751:     function copy()
 752:     {
 753:         $temp = new BigInteger();
 754:         $temp->value = $this->value;
 755:         $temp->is_negative = $this->is_negative;
 756:         $temp->generator = $this->generator;
 757:         $temp->precision = $this->precision;
 758:         $temp->bitmask = $this->bitmask;
 759:         return $temp;
 760:     }
 761: 
 762:     /**
 763:      *  __toString() magic method
 764:      *
 765:      * Will be called, automatically, if you're supporting just PHP5.  If you're supporting PHP4, you'll need to call
 766:      * toString().
 767:      *
 768:      * @access public
 769:      * @internal Implemented per a suggestion by Techie-Michael - thanks!
 770:      */
 771:     function __toString()
 772:     {
 773:         return $this->toString();
 774:     }
 775: 
 776:     /**
 777:      * __clone() magic method
 778:      *
 779:      * Although you can call BigInteger::__toString() directly in PHP5, you cannot call BigInteger::__clone()
 780:      * 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
 781:      * only syntax of $y = clone $x.  As such, if you're trying to write an application that works on both PHP4 and PHP5,
 782:      * call BigInteger::copy(), instead.
 783:      *
 784:      * @access public
 785:      * @see copy()
 786:      * @return BigInteger
 787:      */
 788:     function __clone()
 789:     {
 790:         return $this->copy();
 791:     }
 792: 
 793:     /**
 794:      *  __sleep() magic method
 795:      *
 796:      * Will be called, automatically, when serialize() is called on a BigInteger object.
 797:      *
 798:      * @see __wakeup()
 799:      * @access public
 800:      */
 801:     function __sleep()
 802:     {
 803:         $this->hex = $this->toHex(true);
 804:         $vars = array('hex');
 805:         if ($this->generator != 'mt_rand') {
 806:             $vars[] = 'generator';
 807:         }
 808:         if ($this->precision > 0) {
 809:             $vars[] = 'precision';
 810:         }
 811:         return $vars;
 812: 
 813:     }
 814: 
 815:     /**
 816:      *  __wakeup() magic method
 817:      *
 818:      * Will be called, automatically, when unserialize() is called on a BigInteger object.
 819:      *
 820:      * @see __sleep()
 821:      * @access public
 822:      */
 823:     function __wakeup()
 824:     {
 825:         $temp = new BigInteger($this->hex, -16);
 826:         $this->value = $temp->value;
 827:         $this->is_negative = $temp->is_negative;
 828:         $this->setRandomGenerator($this->generator);
 829:         if ($this->precision > 0) {
 830:             // recalculate $this->bitmask
 831:             $this->setPrecision($this->precision);
 832:         }
 833:     }
 834: 
 835:     /**
 836:      * Adds two BigIntegers.
 837:      *
 838:      * Here's an example:
 839:      * <code>
 840:      * <?php
 841:      *    include('Math/BigInteger.php');
 842:      *
 843:      *    $a = new BigInteger('10');
 844:      *    $b = new BigInteger('20');
 845:      *
 846:      *    $c = $a->add($b);
 847:      *
 848:      *    echo $c->toString(); // outputs 30
 849:      * ?>
 850:      * </code>
 851:      *
 852:      * @param BigInteger $y
 853:      * @return BigInteger
 854:      * @access public
 855:      * @internal Performs base-2**52 addition
 856:      */
 857:     function add($y)
 858:     {
 859:         switch ( MATH_BIGINTEGER_MODE ) {
 860:             case MATH_BIGINTEGER_MODE_GMP:
 861:                 $temp = new BigInteger();
 862:                 $temp->value = gmp_add($this->value, $y->value);
 863: 
 864:                 return $this->_normalize($temp);
 865:             case MATH_BIGINTEGER_MODE_BCMATH:
 866:                 $temp = new BigInteger();
 867:                 $temp->value = bcadd($this->value, $y->value, 0);
 868: 
 869:                 return $this->_normalize($temp);
 870:         }
 871: 
 872:         $temp = $this->_add($this->value, $this->is_negative, $y->value, $y->is_negative);
 873: 
 874:         $result = new BigInteger();
 875:         $result->value = $temp[MATH_BIGINTEGER_VALUE];
 876:         $result->is_negative = $temp[MATH_BIGINTEGER_SIGN];
 877: 
 878:         return $this->_normalize($result);
 879:     }
 880: 
 881:     /**
 882:      * Performs addition.
 883:      *
 884:      * @param Array $x_value
 885:      * @param Boolean $x_negative
 886:      * @param Array $y_value
 887:      * @param Boolean $y_negative
 888:      * @return Array
 889:      * @access private
 890:      */
 891:     function _add($x_value, $x_negative, $y_value, $y_negative)
 892:     {
 893:         $x_size = count($x_value);
 894:         $y_size = count($y_value);
 895: 
 896:         if ($x_size == 0) {
 897:             return array(
 898:                 MATH_BIGINTEGER_VALUE => $y_value,
 899:                 MATH_BIGINTEGER_SIGN => $y_negative
 900:             );
 901:         } else if ($y_size == 0) {
 902:             return array(
 903:                 MATH_BIGINTEGER_VALUE => $x_value,
 904:                 MATH_BIGINTEGER_SIGN => $x_negative
 905:             );
 906:         }
 907: 
 908:         // subtract, if appropriate
 909:         if ( $x_negative != $y_negative ) {
 910:             if ( $x_value == $y_value ) {
 911:                 return array(
 912:                     MATH_BIGINTEGER_VALUE => array(),
 913:                     MATH_BIGINTEGER_SIGN => false
 914:                 );
 915:             }
 916: 
 917:             $temp = $this->_subtract($x_value, false, $y_value, false);
 918:             $temp[MATH_BIGINTEGER_SIGN] = $this->_compare($x_value, false, $y_value, false) > 0 ?
 919:                                           $x_negative : $y_negative;
 920: 
 921:             return $temp;
 922:         }
 923: 
 924:         if ($x_size < $y_size) {
 925:             $size = $x_size;
 926:             $value = $y_value;
 927:         } else {
 928:             $size = $y_size;
 929:             $value = $x_value;
 930:         }
 931: 
 932:         $value[] = 0; // just in case the carry adds an extra digit
 933: 
 934:         $carry = 0;
 935:         for ($i = 0, $j = 1; $j < $size; $i+=2, $j+=2) {
 936:             $sum = $x_value[$j] * MATH_BIGINTEGER_BASE_FULL + $x_value[$i] + $y_value[$j] * MATH_BIGINTEGER_BASE_FULL + $y_value[$i] + $carry;
 937:             $carry = $sum >= MATH_BIGINTEGER_MAX_DIGIT2; // eg. floor($sum / 2**52); only possible values (in any base) are 0 and 1
 938:             $sum = $carry ? $sum - MATH_BIGINTEGER_MAX_DIGIT2 : $sum;
 939: 
 940:             $temp = (int) ($sum / MATH_BIGINTEGER_BASE_FULL);
 941: 
 942:             $value[$i] = (int) ($sum - MATH_BIGINTEGER_BASE_FULL * $temp); // eg. a faster alternative to fmod($sum, 0x4000000)
 943:             $value[$j] = $temp;
 944:         }
 945: 
 946:         if ($j == $size) { // ie. if $y_size is odd
 947:             $sum = $x_value[$i] + $y_value[$i] + $carry;
 948:             $carry = $sum >= MATH_BIGINTEGER_BASE_FULL;
 949:             $value[$i] = $carry ? $sum - MATH_BIGINTEGER_BASE_FULL : $sum;
 950:             ++$i; // ie. let $i = $j since we've just done $value[$i]
 951:         }
 952: 
 953:         if ($carry) {
 954:             for (; $value[$i] == MATH_BIGINTEGER_MAX_DIGIT; ++$i) {
 955:                 $value[$i] = 0;
 956:             }
 957:             ++$value[$i];
 958:         }
 959: 
 960:         return array(
 961:             MATH_BIGINTEGER_VALUE => $this->_trim($value),
 962:             MATH_BIGINTEGER_SIGN => $x_negative
 963:         );
 964:     }
 965: 
 966:     /**
 967:      * Subtracts two BigIntegers.
 968:      *
 969:      * Here's an example:
 970:      * <code>
 971:      * <?php
 972:      *    include('Math/BigInteger.php');
 973:      *
 974:      *    $a = new BigInteger('10');
 975:      *    $b = new BigInteger('20');
 976:      *
 977:      *    $c = $a->subtract($b);
 978:      *
 979:      *    echo $c->toString(); // outputs -10
 980:      * ?>
 981:      * </code>
 982:      *
 983:      * @param BigInteger $y
 984:      * @return BigInteger
 985:      * @access public
 986:      * @internal Performs base-2**52 subtraction
 987:      */
 988:     function subtract($y)
 989:     {
 990:         switch ( MATH_BIGINTEGER_MODE ) {
 991:             case MATH_BIGINTEGER_MODE_GMP:
 992:                 $temp = new BigInteger();
 993:                 $temp->value = gmp_sub($this->value, $y->value);
 994: 
 995:                 return $this->_normalize($temp);
 996:             case MATH_BIGINTEGER_MODE_BCMATH:
 997:                 $temp = new BigInteger();
 998:                 $temp->value = bcsub($this->value, $y->value, 0);
 999: 
1000:                 return $this->_normalize($temp);
1001:         }
1002: 
1003:         $temp = $this->_subtract($this->value, $this->is_negative, $y->value, $y->is_negative);
1004: 
1005:         $result = new BigInteger();
1006:         $result->value = $temp[MATH_BIGINTEGER_VALUE];
1007:         $result->is_negative = $temp[MATH_BIGINTEGER_SIGN];
1008: 
1009:         return $this->_normalize($result);
1010:     }
1011: 
1012:     /**
1013:      * Performs subtraction.
1014:      *
1015:      * @param Array $x_value
1016:      * @param Boolean $x_negative
1017:      * @param Array $y_value
1018:      * @param Boolean $y_negative
1019:      * @return Array
1020:      * @access private
1021:      */
1022:     function _subtract($x_value, $x_negative, $y_value, $y_negative)
1023:     {
1024:         $x_size = count($x_value);
1025:         $y_size = count($y_value);
1026: 
1027:         if ($x_size == 0) {
1028:             return array(
1029:                 MATH_BIGINTEGER_VALUE => $y_value,
1030:                 MATH_BIGINTEGER_SIGN => !$y_negative
1031:             );
1032:         } else if ($y_size == 0) {
1033:             return array(
1034:                 MATH_BIGINTEGER_VALUE => $x_value,
1035:                 MATH_BIGINTEGER_SIGN => $x_negative
1036:             );
1037:         }
1038: 
1039:         // add, if appropriate (ie. -$x - +$y or +$x - -$y)
1040:         if ( $x_negative != $y_negative ) {
1041:             $temp = $this->_add($x_value, false, $y_value, false);
1042:             $temp[MATH_BIGINTEGER_SIGN] = $x_negative;
1043: 
1044:             return $temp;
1045:         }
1046: 
1047:         $diff = $this->_compare($x_value, $x_negative, $y_value, $y_negative);
1048: 
1049:         if ( !$diff ) {
1050:             return array(
1051:                 MATH_BIGINTEGER_VALUE => array(),
1052:                 MATH_BIGINTEGER_SIGN => false
1053:             );
1054:         }
1055: 
1056:         // switch $x and $y around, if appropriate.
1057:         if ( (!$x_negative && $diff < 0) || ($x_negative && $diff > 0) ) {
1058:             $temp = $x_value;
1059:             $x_value = $y_value;
1060:             $y_value = $temp;
1061: 
1062:             $x_negative = !$x_negative;
1063: 
1064:             $x_size = count($x_value);
1065:             $y_size = count($y_value);
1066:         }
1067: 
1068:         // at this point, $x_value should be at least as big as - if not bigger than - $y_value
1069: 
1070:         $carry = 0;
1071:         for ($i = 0, $j = 1; $j < $y_size; $i+=2, $j+=2) {
1072:             $sum = $x_value[$j] * MATH_BIGINTEGER_BASE_FULL + $x_value[$i] - $y_value[$j] * MATH_BIGINTEGER_BASE_FULL - $y_value[$i] - $carry;
1073:             $carry = $sum < 0; // eg. floor($sum / 2**52); only possible values (in any base) are 0 and 1
1074:             $sum = $carry ? $sum + MATH_BIGINTEGER_MAX_DIGIT2 : $sum;
1075: 
1076:             $temp = (int) ($sum / MATH_BIGINTEGER_BASE_FULL);
1077: 
1078:             $x_value[$i] = (int) ($sum - MATH_BIGINTEGER_BASE_FULL * $temp);
1079:             $x_value[$j] = $temp;
1080:         }
1081: 
1082:         if ($j == $y_size) { // ie. if $y_size is odd
1083:             $sum = $x_value[$i] - $y_value[$i] - $carry;
1084:             $carry = $sum < 0;
1085:             $x_value[$i] = $carry ? $sum + MATH_BIGINTEGER_BASE_FULL : $sum;
1086:             ++$i;
1087:         }
1088: 
1089:         if ($carry) {
1090:             for (; !$x_value[$i]; ++$i) {
1091:                 $x_value[$i] = MATH_BIGINTEGER_MAX_DIGIT;
1092:             }
1093:             --$x_value[$i];
1094:         }
1095: 
1096:         return array(
1097:             MATH_BIGINTEGER_VALUE => $this->_trim($x_value),
1098:             MATH_BIGINTEGER_SIGN => $x_negative
1099:         );
1100:     }
1101: 
1102:     /**
1103:      * Multiplies two BigIntegers
1104:      *
1105:      * Here's an example:
1106:      * <code>
1107:      * <?php
1108:      *    include('Math/BigInteger.php');
1109:      *
1110:      *    $a = new BigInteger('10');
1111:      *    $b = new BigInteger('20');
1112:      *
1113:      *    $c = $a->multiply($b);
1114:      *
1115:      *    echo $c->toString(); // outputs 200
1116:      * ?>
1117:      * </code>
1118:      *
1119:      * @param BigInteger $x
1120:      * @return BigInteger
1121:      * @access public
1122:      */
1123:     function multiply($x)
1124:     {
1125:         switch ( MATH_BIGINTEGER_MODE ) {
1126:             case MATH_BIGINTEGER_MODE_GMP:
1127:                 $temp = new BigInteger();
1128:                 $temp->value = gmp_mul($this->value, $x->value);
1129: 
1130:                 return $this->_normalize($temp);
1131:             case MATH_BIGINTEGER_MODE_BCMATH:
1132:                 $temp = new BigInteger();
1133:                 $temp->value = bcmul($this->value, $x->value, 0);
1134: 
1135:                 return $this->_normalize($temp);
1136:         }
1137: 
1138:         $temp = $this->_multiply($this->value, $this->is_negative, $x->value, $x->is_negative);
1139: 
1140:         $product = new BigInteger();
1141:         $product->value = $temp[MATH_BIGINTEGER_VALUE];
1142:         $product->is_negative = $temp[MATH_BIGINTEGER_SIGN];
1143: 
1144:         return $this->_normalize($product);
1145:     }
1146: 
1147:     /**
1148:      * Performs multiplication.
1149:      *
1150:      * @param Array $x_value
1151:      * @param Boolean $x_negative
1152:      * @param Array $y_value
1153:      * @param Boolean $y_negative
1154:      * @return Array
1155:      * @access private
1156:      */
1157:     function _multiply($x_value, $x_negative, $y_value, $y_negative)
1158:     {
1159:         //if ( $x_value == $y_value ) {
1160:         //    return array(
1161:         //        MATH_BIGINTEGER_VALUE => $this->_square($x_value),
1162:         //        MATH_BIGINTEGER_SIGN => $x_sign != $y_value
1163:         //    );
1164:         //}
1165: 
1166:         $x_length = count($x_value);
1167:         $y_length = count($y_value);
1168: 
1169:         if ( !$x_length || !$y_length ) { // a 0 is being multiplied
1170:             return array(
1171:                 MATH_BIGINTEGER_VALUE => array(),
1172:                 MATH_BIGINTEGER_SIGN => false
1173:             );
1174:         }
1175: 
1176:         return array(
1177:             MATH_BIGINTEGER_VALUE => min($x_length, $y_length) < 2 * MATH_BIGINTEGER_KARATSUBA_CUTOFF ?
1178:                 $this->_trim($this->_regularMultiply($x_value, $y_value)) :
1179:                 $this->_trim($this->_karatsuba($x_value, $y_value)),
1180:             MATH_BIGINTEGER_SIGN => $x_negative != $y_negative
1181:         );
1182:     }
1183: 
1184:     /**
1185:      * Performs long multiplication on two BigIntegers
1186:      *
1187:      * Modeled after 'multiply' in MutableBigInteger.java.
1188:      *
1189:      * @param Array $x_value
1190:      * @param Array $y_value
1191:      * @return Array
1192:      * @access private
1193:      */
1194:     function _regularMultiply($x_value, $y_value)
1195:     {
1196:         $x_length = count($x_value);
1197:         $y_length = count($y_value);
1198: 
1199:         if ( !$x_length || !$y_length ) { // a 0 is being multiplied
1200:             return array();
1201:         }
1202: 
1203:         if ( $x_length < $y_length ) {
1204:             $temp = $x_value;
1205:             $x_value = $y_value;
1206:             $y_value = $temp;
1207: 
1208:             $x_length = count($x_value);
1209:             $y_length = count($y_value);
1210:         }
1211: 
1212:         $product_value = $this->_array_repeat(0, $x_length + $y_length);
1213: 
1214:         // the following for loop could be removed if the for loop following it
1215:         // (the one with nested for loops) initially set $i to 0, but
1216:         // doing so would also make the result in one set of unnecessary adds,
1217:         // since on the outermost loops first pass, $product->value[$k] is going
1218:         // to always be 0
1219: 
1220:         $carry = 0;
1221: 
1222:         for ($j = 0; $j < $x_length; ++$j) { // ie. $i = 0
1223:             $temp = $x_value[$j] * $y_value[0] + $carry; // $product_value[$k] == 0
1224:             $carry = (int) ($temp / MATH_BIGINTEGER_BASE_FULL);
1225:             $product_value[$j] = (int) ($temp - MATH_BIGINTEGER_BASE_FULL * $carry);
1226:         }
1227: 
1228:         $product_value[$j] = $carry;
1229: 
1230:         // the above for loop is what the previous comment was talking about.  the
1231:         // following for loop is the "one with nested for loops"
1232:         for ($i = 1; $i < $y_length; ++$i) {
1233:             $carry = 0;
1234: 
1235:             for ($j = 0, $k = $i; $j < $x_length; ++$j, ++$k) {
1236:                 $temp = $product_value[$k] + $x_value[$j] * $y_value[$i] + $carry;
1237:                 $carry = (int) ($temp / MATH_BIGINTEGER_BASE_FULL);
1238:                 $product_value[$k] = (int) ($temp - MATH_BIGINTEGER_BASE_FULL * $carry);
1239:             }
1240: 
1241:             $product_value[$k] = $carry;
1242:         }
1243: 
1244:         return $product_value;
1245:     }
1246: 
1247:     /**
1248:      * Performs Karatsuba multiplication on two BigIntegers
1249:      *
1250:      * See {@link http://en.wikipedia.org/wiki/Karatsuba_algorithm Karatsuba algorithm} and
1251:      * {@link http://math.libtomcrypt.com/files/tommath.pdf#page=120 MPM 5.2.3}.
1252:      *
1253:      * @param Array $x_value
1254:      * @param Array $y_value
1255:      * @return Array
1256:      * @access private
1257:      */
1258:     function _karatsuba($x_value, $y_value)
1259:     {
1260:         $m = min(count($x_value) >> 1, count($y_value) >> 1);
1261: 
1262:         if ($m < MATH_BIGINTEGER_KARATSUBA_CUTOFF) {
1263:             return $this->_regularMultiply($x_value, $y_value);
1264:         }
1265: 
1266:         $x1 = array_slice($x_value, $m);
1267:         $x0 = array_slice($x_value, 0, $m);
1268:         $y1 = array_slice($y_value, $m);
1269:         $y0 = array_slice($y_value, 0, $m);
1270: 
1271:         $z2 = $this->_karatsuba($x1, $y1);
1272:         $z0 = $this->_karatsuba($x0, $y0);
1273: 
1274:         $z1 = $this->_add($x1, false, $x0, false);
1275:         $temp = $this->_add($y1, false, $y0, false);
1276:         $z1 = $this->_karatsuba($z1[MATH_BIGINTEGER_VALUE], $temp[MATH_BIGINTEGER_VALUE]);
1277:         $temp = $this->_add($z2, false, $z0, false);
1278:         $z1 = $this->_subtract($z1, false, $temp[MATH_BIGINTEGER_VALUE], false);
1279: 
1280:         $z2 = array_merge(array_fill(0, 2 * $m, 0), $z2);
1281:         $z1[MATH_BIGINTEGER_VALUE] = array_merge(array_fill(0, $m, 0), $z1[MATH_BIGINTEGER_VALUE]);
1282: 
1283:         $xy = $this->_add($z2, false, $z1[MATH_BIGINTEGER_VALUE], $z1[MATH_BIGINTEGER_SIGN]);
1284:         $xy = $this->_add($xy[MATH_BIGINTEGER_VALUE], $xy[MATH_BIGINTEGER_SIGN], $z0, false);
1285: 
1286:         return $xy[MATH_BIGINTEGER_VALUE];
1287:     }
1288: 
1289:     /**
1290:      * Performs squaring
1291:      *
1292:      * @param Array $x
1293:      * @return Array
1294:      * @access private
1295:      */
1296:     function _square($x = false)
1297:     {
1298:         return count($x) < 2 * MATH_BIGINTEGER_KARATSUBA_CUTOFF ?
1299:             $this->_trim($this->_baseSquare($x)) :
1300:             $this->_trim($this->_karatsubaSquare($x));
1301:     }
1302: 
1303:     /**
1304:      * Performs traditional squaring on two BigIntegers
1305:      *
1306:      * Squaring can be done faster than multiplying a number by itself can be.  See
1307:      * {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=7 HAC 14.2.4} /
1308:      * {@link http://math.libtomcrypt.com/files/tommath.pdf#page=141 MPM 5.3} for more information.
1309:      *
1310:      * @param Array $value
1311:      * @return Array
1312:      * @access private
1313:      */
1314:     function _baseSquare($value)
1315:     {
1316:         if ( empty($value) ) {
1317:             return array();
1318:         }
1319:         $square_value = $this->_array_repeat(0, 2 * count($value));
1320: 
1321:         for ($i = 0, $max_index = count($value) - 1; $i <= $max_index; ++$i) {
1322:             $i2 = $i << 1;
1323: 
1324:             $temp = $square_value[$i2] + $value[$i] * $value[$i];
1325:             $carry = (int) ($temp / MATH_BIGINTEGER_BASE_FULL);
1326:             $square_value[$i2] = (int) ($temp - MATH_BIGINTEGER_BASE_FULL * $carry);
1327: 
1328:             // note how we start from $i+1 instead of 0 as we do in multiplication.
1329:             for ($j = $i + 1, $k = $i2 + 1; $j <= $max_index; ++$j, ++$k) {
1330:                 $temp = $square_value[$k] + 2 * $value[$j] * $value[$i] + $carry;
1331:                 $carry = (int) ($temp / MATH_BIGINTEGER_BASE_FULL);
1332:                 $square_value[$k] = (int) ($temp - MATH_BIGINTEGER_BASE_FULL * $carry);
1333:             }
1334: 
1335:             // the following line can yield values larger 2**15.  at this point, PHP should switch
1336:             // over to floats.
1337:             $square_value[$i + $max_index + 1] = $carry;
1338:         }
1339: 
1340:         return $square_value;
1341:     }
1342: 
1343:     /**
1344:      * Performs Karatsuba "squaring" on two BigIntegers
1345:      *
1346:      * See {@link http://en.wikipedia.org/wiki/Karatsuba_algorithm Karatsuba algorithm} and
1347:      * {@link http://math.libtomcrypt.com/files/tommath.pdf#page=151 MPM 5.3.4}.
1348:      *
1349:      * @param Array $value
1350:      * @return Array
1351:      * @access private
1352:      */
1353:     function _karatsubaSquare($value)
1354:     {
1355:         $m = count($value) >> 1;
1356: 
1357:         if ($m < MATH_BIGINTEGER_KARATSUBA_CUTOFF) {
1358:             return $this->_baseSquare($value);
1359:         }
1360: 
1361:         $x1 = array_slice($value, $m);
1362:         $x0 = array_slice($value, 0, $m);
1363: 
1364:         $z2 = $this->_karatsubaSquare($x1);
1365:         $z0 = $this->_karatsubaSquare($x0);
1366: 
1367:         $z1 = $this->_add($x1, false, $x0, false);
1368:         $z1 = $this->_karatsubaSquare($z1[MATH_BIGINTEGER_VALUE]);
1369:         $temp = $this->_add($z2, false, $z0, false);
1370:         $z1 = $this->_subtract($z1, false, $temp[MATH_BIGINTEGER_VALUE], false);
1371: 
1372:         $z2 = array_merge(array_fill(0, 2 * $m, 0), $z2);
1373:         $z1[MATH_BIGINTEGER_VALUE] = array_merge(array_fill(0, $m, 0), $z1[MATH_BIGINTEGER_VALUE]);
1374: 
1375:         $xx = $this->_add($z2, false, $z1[MATH_BIGINTEGER_VALUE], $z1[MATH_BIGINTEGER_SIGN]);
1376:         $xx = $this->_add($xx[MATH_BIGINTEGER_VALUE], $xx[MATH_BIGINTEGER_SIGN], $z0, false);
1377: 
1378:         return $xx[MATH_BIGINTEGER_VALUE];
1379:     }
1380: 
1381:     /**
1382:      * Divides two BigIntegers.
1383:      *
1384:      * Returns an array whose first element contains the quotient and whose second element contains the
1385:      * "common residue".  If the remainder would be positive, the "common residue" and the remainder are the
1386:      * same.  If the remainder would be negative, the "common residue" is equal to the sum of the remainder
1387:      * and the divisor (basically, the "common residue" is the first positive modulo).
1388:      *
1389:      * Here's an example:
1390:      * <code>
1391:      * <?php
1392:      *    include('Math/BigInteger.php');
1393:      *
1394:      *    $a = new BigInteger('10');
1395:      *    $b = new BigInteger('20');
1396:      *
1397:      *    list($quotient, $remainder) = $a->divide($b);
1398:      *
1399:      *    echo $quotient->toString(); // outputs 0
1400:      *    echo "\r\n";
1401:      *    echo $remainder->toString(); // outputs 10
1402:      * ?>
1403:      * </code>
1404:      *
1405:      * @param BigInteger $y
1406:      * @return Array
1407:      * @access public
1408:      * @internal This function is based off of {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=9 HAC 14.20}.
1409:      */
1410:     function divide($y)
1411:     {
1412:         switch ( MATH_BIGINTEGER_MODE ) {
1413:             case MATH_BIGINTEGER_MODE_GMP:
1414:                 $quotient = new BigInteger();
1415:                 $remainder = new BigInteger();
1416: 
1417:                 list($quotient->value, $remainder->value) = gmp_div_qr($this->value, $y->value);
1418: 
1419:                 if (gmp_sign($remainder->value) < 0) {
1420:                     $remainder->value = gmp_add($remainder->value, gmp_abs($y->value));
1421:                 }
1422: 
1423:                 return array($this->_normalize($quotient), $this->_normalize($remainder));
1424:             case MATH_BIGINTEGER_MODE_BCMATH:
1425:                 $quotient = new BigInteger();
1426:                 $remainder = new BigInteger();
1427: 
1428:                 $quotient->value = bcdiv($this->value, $y->value, 0);
1429:                 $remainder->value = bcmod($this->value, $y->value);
1430: 
1431:                 if ($remainder->value[0] == '-') {
1432:                     $remainder->value = bcadd($remainder->value, $y->value[0] == '-' ? substr($y->value, 1) : $y->value, 0);
1433:                 }
1434: 
1435:                 return array($this->_normalize($quotient), $this->_normalize($remainder));
1436:         }
1437: 
1438:         if (count($y->value) == 1) {
1439:             list($q, $r) = $this->_divide_digit($this->value, $y->value[0]);
1440:             $quotient = new BigInteger();
1441:             $remainder = new BigInteger();
1442:             $quotient->value = $q;
1443:             $remainder->value = array($r);
1444:             $quotient->is_negative = $this->is_negative != $y->is_negative;
1445:             return array($this->_normalize($quotient), $this->_normalize($remainder));
1446:         }
1447: 
1448:         static $zero;
1449:         if ( !isset($zero) ) {
1450:             $zero = new BigInteger();
1451:         }
1452: 
1453:         $x = $this->copy();
1454:         $y = $y->copy();
1455: 
1456:         $x_sign = $x->is_negative;
1457:         $y_sign = $y->is_negative;
1458: 
1459:         $x->is_negative = $y->is_negative = false;
1460: 
1461:         $diff = $x->compare($y);
1462: 
1463:         if ( !$diff ) {
1464:             $temp = new BigInteger();
1465:             $temp->value = array(1);
1466:             $temp->is_negative = $x_sign != $y_sign;
1467:             return array($this->_normalize($temp), $this->_normalize(new BigInteger()));
1468:         }
1469: 
1470:         if ( $diff < 0 ) {
1471:             // if $x is negative, "add" $y.
1472:             if ( $x_sign ) {
1473:                 $x = $y->subtract($x);
1474:             }
1475:             return array($this->_normalize(new BigInteger()), $this->_normalize($x));
1476:         }
1477: 
1478:         // normalize $x and $y as described in HAC 14.23 / 14.24
1479:         $msb = $y->value[count($y->value) - 1];
1480:         for ($shift = 0; !($msb & MATH_BIGINTEGER_MSB); ++$shift) {
1481:             $msb <<= 1;
1482:         }
1483:         $x->_lshift($shift);
1484:         $y->_lshift($shift);
1485:         $y_value = &$y->value;
1486: 
1487:         $x_max = count($x->value) - 1;
1488:         $y_max = count($y->value) - 1;
1489: 
1490:         $quotient = new BigInteger();
1491:         $quotient_value = &$quotient->value;
1492:         $quotient_value = $this->_array_repeat(0, $x_max - $y_max + 1);
1493: 
1494:         static $temp, $lhs, $rhs;
1495:         if (!isset($temp)) {
1496:             $temp = new BigInteger();
1497:             $lhs =  new BigInteger();
1498:             $rhs =  new BigInteger();
1499:         }
1500:         $temp_value = &$temp->value;
1501:         $rhs_value =  &$rhs->value;
1502: 
1503:         // $temp = $y << ($x_max - $y_max-1) in base 2**26
1504:         $temp_value = array_merge($this->_array_repeat(0, $x_max - $y_max), $y_value);
1505: 
1506:         while ( $x->compare($temp) >= 0 ) {
1507:             // calculate the "common residue"
1508:             ++$quotient_value[$x_max - $y_max];
1509:             $x = $x->subtract($temp);
1510:             $x_max = count($x->value) - 1;
1511:         }
1512: 
1513:         for ($i = $x_max; $i >= $y_max + 1; --$i) {
1514:             $x_value = &$x->value;
1515:             $x_window = array(
1516:                 isset($x_value[$i]) ? $x_value[$i] : 0,
1517:                 isset($x_value[$i - 1]) ? $x_value[$i - 1] : 0,
1518:                 isset($x_value[$i - 2]) ? $x_value[$i - 2] : 0
1519:             );
1520:             $y_window = array(
1521:                 $y_value[$y_max],
1522:                 ( $y_max > 0 ) ? $y_value[$y_max - 1] : 0
1523:             );
1524: 
1525:             $q_index = $i - $y_max - 1;
1526:             if ($x_window[0] == $y_window[0]) {
1527:                 $quotient_value[$q_index] = MATH_BIGINTEGER_MAX_DIGIT;
1528:             } else {
1529:                 $quotient_value[$q_index] = (int) (
1530:                     ($x_window[0] * MATH_BIGINTEGER_BASE_FULL + $x_window[1])
1531:                     /
1532:                     $y_window[0]
1533:                 );
1534:             }
1535: 
1536:             $temp_value = array($y_window[1], $y_window[0]);
1537: 
1538:             $lhs->value = array($quotient_value[$q_index]);
1539:             $lhs = $lhs->multiply($temp);
1540: 
1541:             $rhs_value = array($x_window[2], $x_window[1], $x_window[0]);
1542: 
1543:             while ( $lhs->compare($rhs) > 0 ) {
1544:                 --$quotient_value[$q_index];
1545: 
1546:                 $lhs->value = array($quotient_value[$q_index]);
1547:                 $lhs = $lhs->multiply($temp);
1548:             }
1549: 
1550:             $adjust = $this->_array_repeat(0, $q_index);
1551:             $temp_value = array($quotient_value[$q_index]);
1552:             $temp = $temp->multiply($y);
1553:             $temp_value = &$temp->value;
1554:             $temp_value = array_merge($adjust, $temp_value);
1555: 
1556:             $x = $x->subtract($temp);
1557: 
1558:             if ($x->compare($zero) < 0) {
1559:                 $temp_value = array_merge($adjust, $y_value);
1560:                 $x = $x->add($temp);
1561: 
1562:                 --$quotient_value[$q_index];
1563:             }
1564: 
1565:             $x_max = count($x_value) - 1;
1566:         }
1567: 
1568:         // unnormalize the remainder
1569:         $x->_rshift($shift);
1570: 
1571:         $quotient->is_negative = $x_sign != $y_sign;
1572: 
1573:         // calculate the "common residue", if appropriate
1574:         if ( $x_sign ) {
1575:             $y->_rshift($shift);
1576:             $x = $y->subtract($x);
1577:         }
1578: 
1579:         return array($this->_normalize($quotient), $this->_normalize($x));
1580:     }
1581: 
1582:     /**
1583:      * Divides a BigInteger by a regular integer
1584:      *
1585:      * abc / x = a00 / x + b0 / x + c / x
1586:      *
1587:      * @param Array $dividend
1588:      * @param Array $divisor
1589:      * @return Array
1590:      * @access private
1591:      */
1592:     function _divide_digit($dividend, $divisor)
1593:     {
1594:         $carry = 0;
1595:         $result = array();
1596: 
1597:         for ($i = count($dividend) - 1; $i >= 0; --$i) {
1598:             $temp = MATH_BIGINTEGER_BASE_FULL * $carry + $dividend[$i];
1599:             $result[$i] = (int) ($temp / $divisor);
1600:             $carry = (int) ($temp - $divisor * $result[$i]);
1601:         }
1602: 
1603:         return array($result, $carry);
1604:     }
1605: 
1606: 
1607: 
1608:     /**
1609:      * Performs Legendre Symbol.
1610:      *
1611:      * Here's an example:
1612:      * <code>
1613:      * <?php
1614:      *    include('Math/BigInteger.php');
1615:      *
1616:      *    $a = new BigInteger('2');
1617:      *    $b = new BigInteger('3');
1618:      *
1619:      *    $c = $a->legendre($b);
1620:      *
1621:      *    echo $c->toString(); // outputs -1
1622:      * ?>
1623:      * </code>
1624:      *
1625:      * @param BigInteger $m The odd and positive BigNumber number. 
1626:      * @return BigInteger
1627:      * @access public
1628:      */
1629:     function legendre($p)
1630:     {
1631:         if ( MATH_BIGINTEGER_MODE == MATH_BIGINTEGER_MODE_GMP ) {
1632:             $temp = new BigInteger();
1633:             $temp->value = gmp_legendre($this->value, $p->value);
1634:             return $this->_normalize($temp);
1635:         }
1636: 
1637:         if ( MATH_BIGINTEGER_MODE == MATH_BIGINTEGER_MODE_BCMATH ) {
1638:             $temp = new BigInteger();
1639:             
1640:             if ($p->value >= 3 && $p->value % 2 == 1) {
1641:                 $this->value = bcmod($this->value, $p->value);
1642:                 if($this->value == 0) {
1643:                     return 0;
1644:                 }
1645:                 if ($this->value == 1) {
1646:                     return 1;
1647:                 }
1648:                 $a1 = $this->value;
1649:                 $e = 0;
1650:                 while (bcmod($a1, 2) == 0) {
1651:                     $a1 = bcdiv($a1, 2);
1652:                     $e = bcadd($e, 1);
1653:                 }
1654:                 if (bcmod($e, 2) == 0 || bcmod($p->value, 8) == 1 || bcmod($p->value, 8) == 7) {
1655:                     $s = 1;
1656:                 } else {
1657:                     $s = -1;
1658:                 }
1659:                 if ($a1 == 1) {
1660:                     $temp->value = $s;
1661:                     return $this->_normalize($temp);
1662:                 }
1663:                 if (bcmod($p->value, 4) == 3 && bcmod($a1, 4) == 3) {
1664:                     $s = -$s;
1665:                 }
1666: 
1667:                 $x = bcmod($p->value, $a1);
1668: 
1669:                 $objx = new BigInteger($x);
1670:                 $obja1 = new BigInteger($a1);
1671: 
1672:                 $temp->value = bcmul($s, $objx->legendre($obja1)->value);
1673:             }
1674: 
1675:             return $this->_normalize($temp);
1676:         }
1677: 
1678: 
1679:         // Standalone
1680:         $temp = new BigInteger();
1681:         $x = new BigInteger();
1682:         
1683:         if ($p->value >= 3 && $p->value % 2 == 1) {
1684:             $temp = $this->mod($p);
1685: 
1686:             if($temp->value == 0) {
1687:                 return 0;
1688:             }
1689:             if ($temp->value == 1) {
1690:                 return 1;
1691:             }
1692:             $acopy = $temp->copy();
1693:             $e = new BigInteger(0, 10);
1694:             $one = new BigInteger(1, 10);
1695:             $two = new BigInteger(2, 10);
1696:             $four = new BigInteger(4, 10);
1697:             $eight = new BigInteger(8, 10);
1698: 
1699:             while ($acopy->mod($two)->value == 0) {
1700:                 list($quo, $rem) = $acopy->divide($two);
1701:                 $acopy = $quo;
1702:                 $e = $e->add($one);
1703:             }
1704:             if ($e->mod($two)->value == 0 || $p->mod($eight)->value == 1 || $p->mod($eight)->value == 7) {
1705:                 $s = new BigInteger(1, 10);
1706:             } else {
1707:                 $s = new BigInteger(-1, 10);
1708:             }
1709:             if ($acopy->value == 1) {
1710:                 return $s;
1711:             }
1712:             if ($p->mod($four)->value == 3 && $acopy->mod($four)->value == 3) {
1713:                 $s->value = -$s->value;
1714:             }
1715: 
1716:             $x = $p->mod($acopy);
1717: 
1718:             $temp = $s->mul($x->legendre($acopy));
1719:         }
1720: 
1721:         return $temp;
1722:     }
1723: 
1724: 
1725: 
1726: 
1727: 
1728:     /**
1729:      * Performs modulo.
1730:      *
1731:      * Here's an example:
1732:      * <code>
1733:      * <?php
1734:      *    include('Math/BigInteger.php');
1735:      *
1736:      *    $a = new BigInteger('8');
1737:      *    $b = new BigInteger('3');
1738:      *
1739:      *    $c = $a->mod($b);
1740:      *
1741:      *    echo $c->toString(); // outputs 2
1742:      * ?>
1743:      * </code>
1744:      *
1745:      * @param BigInteger $m The modulo that is being evaluated. 
1746:      * @return BigInteger
1747:      * @access public
1748:      */
1749:     function mod($m)
1750:     {
1751:         if ( MATH_BIGINTEGER_MODE == MATH_BIGINTEGER_MODE_GMP ) {
1752:             $temp = new BigInteger();
1753:             $temp->value = gmp_mod($this->value, $m->value);
1754:             return $this->_normalize($temp);
1755:         }
1756: 
1757:         if ( MATH_BIGINTEGER_MODE == MATH_BIGINTEGER_MODE_BCMATH ) {
1758:             $temp = new BigInteger();
1759:             $temp->value = bcmod($this->value, $m->value);
1760:             return $this->_normalize($temp);
1761:         }
1762: 
1763:         list($quo, $rem) = $this->divide($m);
1764:         return $this->subtract($quo->multiply($m));
1765:     }
1766: 
1767: 
1768: 
1769: 
1770:     /**
1771:      * Performs modular exponentiation.
1772:      *
1773:      * Here's an example:
1774:      * <code>
1775:      * <?php
1776:      *    include('Math/BigInteger.php');
1777:      *
1778:      *    $a = new BigInteger('10');
1779:      *    $b = new BigInteger('20');
1780:      *    $c = new BigInteger('30');
1781:      *
1782:      *    $c = $a->modPow($b, $c);
1783:      *
1784:      *    echo $c->toString(); // outputs 10
1785:      * ?>
1786:      * </code>
1787:      *
1788:      * @param BigInteger $e
1789:      * @param BigInteger $n
1790:      * @return BigInteger
1791:      * @access public
1792:      * @internal The most naive approach to modular exponentiation has very unreasonable requirements, and
1793:      *    and although the approach involving repeated squaring does vastly better, it, too, is impractical
1794:      *    for our purposes.  The reason being that division - by far the most complicated and time-consuming
1795:      *    of the basic operations (eg. +,-,*,/) - occurs multiple times within it.
1796:      *
1797:      *    Modular reductions resolve this issue.  Although an individual modular reduction takes more time
1798:      *    then an individual division, when performed in succession (with the same modulo), they're a lot faster.
1799:      *
1800:      *    The two most commonly used modular reductions are Barrett and Montgomery reduction.  Montgomery reduction,
1801:      *    although faster, only works when the gcd of the modulo and of the base being used is 1.  In RSA, when the
1802:      *    base is a power of two, the modulo - a product of two primes - is always going to have a gcd of 1 (because
1803:      *    the product of two odd numbers is odd), but what about when RSA isn't used?
1804:      *
1805:      *    In contrast, Barrett reduction has no such constraint.  As such, some bigint implementations perform a
1806:      *    Barrett reduction after every operation in the modpow function.  Others perform Barrett reductions when the
1807:      *    modulo is even and Montgomery reductions when the modulo is odd.  BigInteger.java's modPow method, however,
1808:      *    uses a trick involving the Chinese Remainder Theorem to factor the even modulo into two numbers - one odd and
1809:      *    the other, a power of two - and recombine them, later.  This is the method that this modPow function uses.
1810:      *    {@link http://islab.oregonstate.edu/papers/j34monex.pdf Montgomery Reduction with Even Modulus} elaborates.
1811:      */
1812:     function modPow($e, $n)
1813:     {
1814:         $n = $this->bitmask !== false && $this->bitmask->compare($n) < 0 ? $this->bitmask : $n->abs();
1815: 
1816:         if ($e->compare(new BigInteger()) < 0) {
1817:             $e = $e->abs();
1818: 
1819:             $temp = $this->modInverse($n);
1820:             if ($temp === false) {
1821:                 return false;
1822:             }
1823: 
1824:             return $this->_normalize($temp->modPow($e, $n));
1825:         }
1826: 
1827:         if ( MATH_BIGINTEGER_MODE == MATH_BIGINTEGER_MODE_GMP ) {
1828:             $temp = new BigInteger();
1829:             $temp->value = gmp_powm($this->value, $e->value, $n->value);
1830: 
1831:             return $this->_normalize($temp);
1832:         }
1833: 
1834:         if ($this->compare(new BigInteger()) < 0 || $this->compare($n) > 0) {
1835:             list(, $temp) = $this->divide($n);
1836:             return $temp->modPow($e, $n);
1837:         }
1838: 
1839:         if (defined('MATH_BIGINTEGER_OPENSSL_ENABLED')) {
1840:             $components = array(
1841:                 'modulus' => $n->toBytes(true),
1842:                 'publicExponent' => $e->toBytes(true)
1843:             );
1844: 
1845:             $components = array(
1846:                 'modulus' => pack('Ca*a*', 2, $this->_encodeASN1Length(strlen($components['modulus'])), $components['modulus']),
1847:                 'publicExponent' => pack('Ca*a*', 2, $this->_encodeASN1Length(strlen($components['publicExponent'])), $components['publicExponent'])
1848:             );
1849: 
1850:             $RSAPublicKey = pack('Ca*a*a*',
1851:                 48, $this->_encodeASN1Length(strlen($components['modulus']) + strlen($components['publicExponent'])),
1852:                 $components['modulus'], $components['publicExponent']
1853:             );
1854: 
1855:             $rsaOID = pack('H*', '300d06092a864886f70d0101010500'); // hex version of MA0GCSqGSIb3DQEBAQUA
1856:             $RSAPublicKey = chr(0) . $RSAPublicKey;
1857:             $RSAPublicKey = chr(3) . $this->_encodeASN1Length(strlen($RSAPublicKey)) . $RSAPublicKey;
1858: 
1859:             $encapsulated = pack('Ca*a*',
1860:                 48, $this->_encodeASN1Length(strlen($rsaOID . $RSAPublicKey)), $rsaOID . $RSAPublicKey
1861:             );
1862: 
1863:             $RSAPublicKey = "-----BEGIN PUBLIC KEY-----\r\n" .
1864:                              chunk_split(base64_encode($encapsulated)) .
1865:                              '-----END PUBLIC KEY-----';
1866: 
1867:             $plaintext = str_pad($this->toBytes(), strlen($n->toBytes(true)) - 1, "\0", STR_PAD_LEFT);
1868: 
1869:             if (openssl_public_encrypt($plaintext, $result, $RSAPublicKey, OPENSSL_NO_PADDING)) {
1870:                 return new BigInteger($result, 256);
1871:             }
1872:         }
1873: 
1874:         if ( MATH_BIGINTEGER_MODE == MATH_BIGINTEGER_MODE_BCMATH ) {
1875:                 $temp = new BigInteger();
1876:                 $temp->value = bcpowmod($this->value, $e->value, $n->value, 0);
1877: 
1878:                 return $this->_normalize($temp);
1879:         }
1880: 
1881:         if ( empty($e->value) ) {
1882:             $temp = new BigInteger();
1883:             $temp->value = array(1);
1884:             return $this->_normalize($temp);
1885:         }
1886: 
1887:         if ( $e->value == array(1) ) {
1888:             list(, $temp) = $this->divide($n);
1889:             return $this->_normalize($temp);
1890:         }
1891: 
1892:         if ( $e->value == array(2) ) {
1893:             $temp = new BigInteger();
1894:             $temp->value = $this->_square($this->value);
1895:             list(, $temp) = $temp->divide($n);
1896:             return $this->_normalize($temp);
1897:         }
1898: 
1899:         return $this->_normalize($this->_slidingWindow($e, $n, MATH_BIGINTEGER_BARRETT));
1900: 
1901:         // is the modulo odd?
1902:         if ( $n->value[0] & 1 ) {
1903:             return $this->_normalize($this->_slidingWindow($e, $n, MATH_BIGINTEGER_MONTGOMERY));
1904:         }
1905:         // if it's not, it's even
1906: 
1907:         // find the lowest set bit (eg. the max pow of 2 that divides $n)
1908:         for ($i = 0; $i < count($n->value); ++$i) {
1909:             if ( $n->value[$i] ) {
1910:                 $temp = decbin($n->value[$i]);
1911:                 $j = strlen($temp) - strrpos($temp, '1') - 1;
1912:                 $j+= 26 * $i;
1913:                 break;
1914:             }
1915:         }
1916:         // at this point, 2^$j * $n/(2^$j) == $n
1917: 
1918:         $mod1 = $n->copy();
1919:         $mod1->_rshift($j);
1920:         $mod2 = new BigInteger();
1921:         $mod2->value = array(1);
1922:         $mod2->_lshift($j);
1923: 
1924:         $part1 = ( $mod1->value != array(1) ) ? $this->_slidingWindow($e, $mod1, MATH_BIGINTEGER_MONTGOMERY) : new BigInteger();
1925:         $part2 = $this->_slidingWindow($e, $mod2, MATH_BIGINTEGER_POWEROF2);
1926: 
1927:         $y1 = $mod2->modInverse($mod1);
1928:         $y2 = $mod1->modInverse($mod2);
1929: 
1930:         $result = $part1->multiply($mod2);
1931:         $result = $result->multiply($y1);
1932: 
1933:         $temp = $part2->multiply($mod1);
1934:         $temp = $temp->multiply($y2);
1935: 
1936:         $result = $result->add($temp);
1937:         list(, $result) = $result->divide($n);
1938: 
1939:         return $this->_normalize($result);
1940:     }
1941: 
1942:     /**
1943:      * Performs modular exponentiation.
1944:      *
1945:      * Alias for BigInteger::modPow()
1946:      *
1947:      * @param BigInteger $e
1948:      * @param BigInteger $n
1949:      * @return BigInteger
1950:      * @access public
1951:      */
1952:     function powMod($e, $n)
1953:     {
1954:         return $this->modPow($e, $n);
1955:     }
1956: 
1957:     /**
1958:      * Sliding Window k-ary Modular Exponentiation
1959:      *
1960:      * Based on {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=27 HAC 14.85} /
1961:      * {@link http://math.libtomcrypt.com/files/tommath.pdf#page=210 MPM 7.7}.  In a departure from those algorithims,
1962:      * however, this function performs a modular reduction after every multiplication and squaring operation.
1963:      * As such, this function has the same preconditions that the reductions being used do.
1964:      *
1965:      * @param BigInteger $e
1966:      * @param BigInteger $n
1967:      * @param Integer $mode
1968:      * @return BigInteger
1969:      * @access private
1970:      */
1971:     function _slidingWindow($e, $n, $mode)
1972:     {
1973:         static $window_ranges = array(7, 25, 81, 241, 673, 1793); // from BigInteger.java's oddModPow function
1974:         //static $window_ranges = array(0, 7, 36, 140, 450, 1303, 3529); // from MPM 7.3.1
1975: 
1976:         $e_value = $e->value;
1977:         $e_length = count($e_value) - 1;
1978:         $e_bits = decbin($e_value[$e_length]);
1979:         for ($i = $e_length - 1; $i >= 0; --$i) {
1980:             $e_bits.= str_pad(decbin($e_value[$i]), MATH_BIGINTEGER_BASE, '0', STR_PAD_LEFT);
1981:         }
1982: 
1983:         $e_length = strlen($e_bits);
1984: 
1985:         // calculate the appropriate window size.
1986:         // $window_size == 3 if $window_ranges is between 25 and 81, for example.
1987:         for ($i = 0, $window_size = 1; $e_length > $window_ranges[$i] && $i < count($window_ranges); ++$window_size, ++$i);
1988: 
1989:         $n_value = $n->value;
1990: 
1991:         // precompute $this^0 through $this^$window_size
1992:         $powers = array();
1993:         $powers[1] = $this->_prepareReduce($this->value, $n_value, $mode);
1994:         $powers[2] = $this->_squareReduce($powers[1], $n_value, $mode);
1995: 
1996:         // we do every other number since substr($e_bits, $i, $j+1) (see below) is supposed to end
1997:         // in a 1.  ie. it's supposed to be odd.
1998:         $temp = 1 << ($window_size - 1);
1999:         for ($i = 1; $i < $temp; ++$i) {
2000:             $i2 = $i << 1;
2001:             $powers[$i2 + 1] = $this->_multiplyReduce($powers[$i2 - 1], $powers[2], $n_value, $mode);
2002:         }
2003: 
2004:         $result = array(1);
2005:         $result = $this->_prepareReduce($result, $n_value, $mode);
2006: 
2007:         for ($i = 0; $i < $e_length; ) {
2008:             if ( !$e_bits[$i] ) {
2009:                 $result = $this->_squareReduce($result, $n_value, $mode);
2010:                 ++$i;
2011:             } else {
2012:                 for ($j = $window_size - 1; $j > 0; --$j) {
2013:                     if ( !empty($e_bits[$i + $j]) ) {
2014:                         break;
2015:                     }
2016:                 }
2017: 
2018:                 for ($k = 0; $k <= $j; ++$k) {// eg. the length of substr($e_bits, $i, $j+1)
2019:                     $result = $this->_squareReduce($result, $n_value, $mode);
2020:                 }
2021: 
2022:                 $result = $this->_multiplyReduce($result, $powers[bindec(substr($e_bits, $i, $j + 1))], $n_value, $mode);
2023: 
2024:                 $i+=$j + 1;
2025:             }
2026:         }
2027: 
2028:         $temp = new BigInteger();
2029:         $temp->value = $this->_reduce($result, $n_value, $mode);
2030: 
2031:         return $temp;
2032:     }
2033: 
2034:     /**
2035:      * Modular reduction
2036:      *
2037:      * For most $modes this will return the remainder.
2038:      *
2039:      * @see _slidingWindow()
2040:      * @access private
2041:      * @param Array $x
2042:      * @param Array $n
2043:      * @param Integer $mode
2044:      * @return Array
2045:      */
2046:     function _reduce($x, $n, $mode)
2047:     {
2048:         switch ($mode) {
2049:             case MATH_BIGINTEGER_MONTGOMERY:
2050:                 return $this->_montgomery($x, $n);
2051:             case MATH_BIGINTEGER_BARRETT:
2052:                 return $this->_barrett($x, $n);
2053:             case MATH_BIGINTEGER_POWEROF2:
2054:                 $lhs = new BigInteger();
2055:                 $lhs->value = $x;
2056:                 $rhs = new BigInteger();
2057:                 $rhs->value = $n;
2058:                 return $x->_mod2($n);
2059:             case MATH_BIGINTEGER_CLASSIC:
2060:                 $lhs = new BigInteger();
2061:                 $lhs->value = $x;
2062:                 $rhs = new BigInteger();
2063:                 $rhs->value = $n;
2064:                 list(, $temp) = $lhs->divide($rhs);
2065:                 return $temp->value;
2066:             case MATH_BIGINTEGER_NONE:
2067:                 return $x;
2068:             default:
2069:                 // an invalid $mode was provided
2070:         }
2071:     }
2072: 
2073:     /**
2074:      * Modular reduction preperation
2075:      *
2076:      * @see _slidingWindow()
2077:      * @access private
2078:      * @param Array $x
2079:      * @param Array $n
2080:      * @param Integer $mode
2081:      * @return Array
2082:      */
2083:     function _prepareReduce($x, $n, $mode)
2084:     {
2085:         if ($mode == MATH_BIGINTEGER_MONTGOMERY) {
2086:             return $this->_prepMontgomery($x, $n);
2087:         }
2088:         return $this->_reduce($x, $n, $mode);
2089:     }
2090: 
2091:     /**
2092:      * Modular multiply
2093:      *
2094:      * @see _slidingWindow()
2095:      * @access private
2096:      * @param Array $x
2097:      * @param Array $y
2098:      * @param Array $n
2099:      * @param Integer $mode
2100:      * @return Array
2101:      */
2102:     function _multiplyReduce($x, $y, $n, $mode)
2103:     {
2104:         if ($mode == MATH_BIGINTEGER_MONTGOMERY) {
2105:             return $this->_montgomeryMultiply($x, $y, $n);
2106:         }
2107:         $temp = $this->_multiply($x, false, $y, false);
2108:         return $this->_reduce($temp[MATH_BIGINTEGER_VALUE], $n, $mode);
2109:     }
2110: 
2111:     /**
2112:      * Modular square
2113:      *
2114:      * @see _slidingWindow()
2115:      * @access private
2116:      * @param Array $x
2117:      * @param Array $n
2118:      * @param Integer $mode
2119:      * @return Array
2120:      */
2121:     function _squareReduce($x, $n, $mode)
2122:     {
2123:         if ($mode == MATH_BIGINTEGER_MONTGOMERY) {
2124:             return $this->_montgomeryMultiply($x, $x, $n);
2125:         }
2126:         return $this->_reduce($this->_square($x), $n, $mode);
2127:     }
2128: 
2129:     /**
2130:      * Modulos for Powers of Two
2131:      *
2132:      * Calculates $x%$n, where $n = 2**$e, for some $e.  Since this is basically the same as doing $x & ($n-1),
2133:      * we'll just use this function as a wrapper for doing that.
2134:      *
2135:      * @see _slidingWindow()
2136:      * @access private
2137:      * @param BigInteger
2138:      * @return BigInteger
2139:      */
2140:     function _mod2($n)
2141:     {
2142:         $temp = new BigInteger();
2143:         $temp->value = array(1);
2144:         return $this->bitwise_and($n->subtract($temp));
2145:     }
2146: 
2147:     /**
2148:      * Barrett Modular Reduction
2149:      *
2150:      * See {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=14 HAC 14.3.3} /
2151:      * {@link http://math.libtomcrypt.com/files/tommath.pdf#page=165 MPM 6.2.5} for more information.  Modified slightly,
2152:      * so as not to require negative numbers (initially, this script didn't support negative numbers).
2153:      *
2154:      * Employs "folding", as described at
2155:      * {@link http://www.cosic.esat.kuleuven.be/publications/thesis-149.pdf#page=66 thesis-149.pdf#page=66}.  To quote from
2156:      * 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."
2157:      *
2158:      * Unfortunately, the "Barrett Reduction with Folding" algorithm described in thesis-149.pdf is not, as written, all that
2159:      * usable on account of (1) its not using reasonable radix points as discussed in
2160:      * {@link http://math.libtomcrypt.com/files/tommath.pdf#page=162 MPM 6.2.2} and (2) the fact that, even with reasonable
2161:      * radix points, it only works when there are an even number of digits in the denominator.  The reason for (2) is that
2162:      * (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
2163:      * comments for details.
2164:      *
2165:      * @see _slidingWindow()
2166:      * @access private
2167:      * @param Array $n
2168:      * @param Array $m
2169:      * @return Array
2170:      */
2171:     function _barrett($n, $m)
2172:     {
2173:         static $cache = array(
2174:             MATH_BIGINTEGER_VARIABLE => array(),
2175:             MATH_BIGINTEGER_DATA => array()
2176:         );
2177: 
2178:         $m_length = count($m);
2179: 
2180:         // if ($this->_compare($n, $this->_square($m)) >= 0) {
2181:         if (count($n) > 2 * $m_length) {
2182:             $lhs = new BigInteger();
2183:             $rhs = new BigInteger();
2184:             $lhs->value = $n;
2185:             $rhs->value = $m;
2186:             list(, $temp) = $lhs->divide($rhs);
2187:             return $temp->value;
2188:         }
2189: 
2190:         // if (m.length >> 1) + 2 <= m.length then m is too small and n can't be reduced
2191:         if ($m_length < 5) {
2192:             return $this->_regularBarrett($n, $m);
2193:         }
2194: 
2195:         // n = 2 * m.length
2196: 
2197:         if ( ($key = array_search($m, $cache[MATH_BIGINTEGER_VARIABLE])) === false ) {
2198:             $key = count($cache[MATH_BIGINTEGER_VARIABLE]);
2199:             $cache[MATH_BIGINTEGER_VARIABLE][] = $m;
2200: 
2201:             $lhs = new BigInteger();
2202:             $lhs_value = &$lhs->value;
2203:             $lhs_value = $this->_array_repeat(0, $m_length + ($m_length >> 1));
2204:             $lhs_value[] = 1;
2205:             $rhs = new BigInteger();
2206:             $rhs->value = $m;
2207: 
2208:             list($u, $m1) = $lhs->divide($rhs);
2209:             $u = $u->value;
2210:             $m1 = $m1->value;
2211: 
2212:             $cache[MATH_BIGINTEGER_DATA][] = array(
2213:                 'u' => $u, // m.length >> 1 (technically (m.length >> 1) + 1)
2214:                 'm1'=> $m1 // m.length
2215:             );
2216:         } else {
2217:             extract($cache[MATH_BIGINTEGER_DATA][$key]);
2218:         }
2219: 
2220:         $cutoff = $m_length + ($m_length >> 1);
2221:         $lsd = array_slice($n, 0, $cutoff); // m.length + (m.length >> 1)
2222:         $msd = array_slice($n, $cutoff);    // m.length >> 1
2223:         $lsd = $this->_trim($lsd);
2224:         $temp = $this->_multiply($msd, false, $m1, false);
2225:         $n = $this->_add($lsd, false, $temp[MATH_BIGINTEGER_VALUE], false); // m.length + (m.length >> 1) + 1
2226: 
2227:         if ($m_length & 1) {
2228:             return $this->_regularBarrett($n[MATH_BIGINTEGER_VALUE], $m);
2229:         }
2230: 
2231:         // (m.length + (m.length >> 1) + 1) - (m.length - 1) == (m.length >> 1) + 2
2232:         $temp = array_slice($n[MATH_BIGINTEGER_VALUE], $m_length - 1);
2233:         // if even: ((m.length >> 1) + 2) + (m.length >> 1) == m.length + 2
2234:         // if odd:  ((m.length >> 1) + 2) + (m.length >> 1) == (m.length - 1) + 2 == m.length + 1
2235:         $temp = $this->_multiply($temp, false, $u, false);
2236:         // if even: (m.length + 2) - ((m.length >> 1) + 1) = m.length - (m.length >> 1) + 1
2237:         // if odd:  (m.length + 1) - ((m.length >> 1) + 1) = m.length - (m.length >> 1)
2238:         $temp = array_slice($temp[MATH_BIGINTEGER_VALUE], ($m_length >> 1) + 1);
2239:         // if even: (m.length - (m.length >> 1) + 1) + m.length = 2 * m.length - (m.length >> 1) + 1
2240:         // if odd:  (m.length - (m.length >> 1)) + m.length     = 2 * m.length - (m.length >> 1)
2241:         $temp = $this->_multiply($temp, false, $m, false);
2242: 
2243:         // at this point, if m had an odd number of digits, we'd be subtracting a 2 * m.length - (m.length >> 1) digit
2244:         // number from a m.length + (m.length >> 1) + 1 digit number.  ie. there'd be an extra digit and the while loop
2245:         // following this comment would loop a lot (hence our calling _regularBarrett() in that situation).
2246: 
2247:         $result = $this->_subtract($n[MATH_BIGINTEGER_VALUE], false, $temp[MATH_BIGINTEGER_VALUE], false);
2248: 
2249:         while ($this->_compare($result[MATH_BIGINTEGER_VALUE], $result[MATH_BIGINTEGER_SIGN], $m, false) >= 0) {
2250:             $result = $this->_subtract($result[MATH_BIGINTEGER_VALUE], $result[MATH_BIGINTEGER_SIGN], $m, false);
2251:         }
2252: 
2253:         return $result[MATH_BIGINTEGER_VALUE];
2254:     }
2255: 
2256:     /**
2257:      * (Regular) Barrett Modular Reduction
2258:      *
2259:      * For numbers with more than four digits BigInteger::_barrett() is faster.  The difference between that and this
2260:      * is that this function does not fold the denominator into a smaller form.
2261:      *
2262:      * @see _slidingWindow()
2263:      * @access private
2264:      * @param Array $x
2265:      * @param Array $n
2266:      * @return Array
2267:      */
2268:     function _regularBarrett($x, $n)
2269:     {
2270:         static $cache = array(
2271:             MATH_BIGINTEGER_VARIABLE => array(),
2272:             MATH_BIGINTEGER_DATA => array()
2273:         );
2274: 
2275:         $n_length = count($n);
2276: 
2277:         if (count($x) > 2 * $n_length) {
2278:             $lhs = new BigInteger();
2279:             $rhs = new BigInteger();
2280:             $lhs->value = $x;
2281:             $rhs->value = $n;
2282:             list(, $temp) = $lhs->divide($rhs);
2283:             return $temp->value;
2284:         }
2285: 
2286:         if ( ($key = array_search($n, $cache[MATH_BIGINTEGER_VARIABLE])) === false ) {
2287:             $key = count($cache[MATH_BIGINTEGER_VARIABLE]);
2288:             $cache[MATH_BIGINTEGER_VARIABLE][] = $n;
2289:             $lhs = new BigInteger();
2290:             $lhs_value = &$lhs->value;
2291:             $lhs_value = $this->_array_repeat(0, 2 * $n_length);
2292:             $lhs_value[] = 1;
2293:             $rhs = new BigInteger();
2294:             $rhs->value = $n;
2295:             list($temp, ) = $lhs->divide($rhs); // m.length
2296:             $cache[MATH_BIGINTEGER_DATA][] = $temp->value;
2297:         }
2298: 
2299:         // 2 * m.length - (m.length - 1) = m.length + 1
2300:         $temp = array_slice($x, $n_length - 1);
2301:         // (m.length + 1) + m.length = 2 * m.length + 1
2302:         $temp = $this->_multiply($temp, false, $cache[MATH_BIGINTEGER_DATA][$key], false);
2303:         // (2 * m.length + 1) - (m.length - 1) = m.length + 2
2304:         $temp = array_slice($temp[MATH_BIGINTEGER_VALUE], $n_length + 1);
2305: 
2306:         // m.length + 1
2307:         $result = array_slice($x, 0, $n_length + 1);
2308:         // m.length + 1
2309:         $temp = $this->_multiplyLower($temp, false, $n, false, $n_length + 1);
2310:         // $temp == array_slice($temp->_multiply($temp, false, $n, false)->value, 0, $n_length + 1)
2311: 
2312:         if ($this->_compare($result, false, $temp[MATH_BIGINTEGER_VALUE], $temp[MATH_BIGINTEGER_SIGN]) < 0) {
2313:             $corrector_value = $this->_array_repeat(0, $n_length + 1);
2314:             $corrector_value[] = 1;
2315:             $result = $this->_add($result, false, $corrector_value, false);
2316:             $result = $result[MATH_BIGINTEGER_VALUE];
2317:         }
2318: 
2319:         // at this point, we're subtracting a number with m.length + 1 digits from another number with m.length + 1 digits
2320:         $result = $this->_subtract($result, false, $temp[MATH_BIGINTEGER_VALUE], $temp[MATH_BIGINTEGER_SIGN]);
2321:         while ($this->_compare($result[MATH_BIGINTEGER_VALUE], $result[MATH_BIGINTEGER_SIGN], $n, false) > 0) {
2322:             $result = $this->_subtract($result[MATH_BIGINTEGER_VALUE], $result[MATH_BIGINTEGER_SIGN], $n, false);
2323:         }
2324: 
2325:         return $result[MATH_BIGINTEGER_VALUE];
2326:     }
2327: 
2328:     /**
2329:      * Performs long multiplication up to $stop digits
2330:      *
2331:      * If you're going to be doing array_slice($product->value, 0, $stop), some cycles can be saved.
2332:      *
2333:      * @see _regularBarrett()
2334:      * @param Array $x_value
2335:      * @param Boolean $x_negative
2336:      * @param Array $y_value
2337:      * @param Boolean $y_negative
2338:      * @param Integer $stop
2339:      * @return Array
2340:      * @access private
2341:      */
2342:     function _multiplyLower($x_value, $x_negative, $y_value, $y_negative, $stop)
2343:     {
2344:         $x_length = count($x_value);
2345:         $y_length = count($y_value);
2346: 
2347:         if ( !$x_length || !$y_length ) { // a 0 is being multiplied
2348:             return array(
2349:                 MATH_BIGINTEGER_VALUE => array(),
2350:                 MATH_BIGINTEGER_SIGN => false
2351:             );
2352:         }
2353: 
2354:         if ( $x_length < $y_length ) {
2355:             $temp = $x_value;
2356:             $x_value = $y_value;
2357:             $y_value = $temp;
2358: 
2359:             $x_length = count($x_value);
2360:             $y_length = count($y_value);
2361:         }
2362: 
2363:         $product_value = $this->_array_repeat(0, $x_length + $y_length);
2364: 
2365:         // the following for loop could be removed if the for loop following it
2366:         // (the one with nested for loops) initially set $i to 0, but
2367:         // doing so would also make the result in one set of unnecessary adds,
2368:         // since on the outermost loops first pass, $product->value[$k] is going
2369:         // to always be 0
2370: 
2371:         $carry = 0;
2372: 
2373:         for ($j = 0; $j < $x_length; ++$j) { // ie. $i = 0, $k = $i
2374:             $temp = $x_value[$j] * $y_value[0] + $carry; // $product_value[$k] == 0
2375:             $carry = (int) ($temp / MATH_BIGINTEGER_BASE_FULL);
2376:             $product_value[$j] = (int) ($temp - MATH_BIGINTEGER_BASE_FULL * $carry);
2377:         }
2378: 
2379:         if ($j < $stop) {
2380:             $product_value[$j] = $carry;
2381:         }
2382: 
2383:         // the above for loop is what the previous comment was talking about.  the
2384:         // following for loop is the "one with nested for loops"
2385: 
2386:         for ($i = 1; $i < $y_length; ++$i) {
2387:             $carry = 0;
2388: 
2389:             for ($j = 0, $k = $i; $j < $x_length && $k < $stop; ++$j, ++$k) {
2390:                 $temp = $product_value[$k] + $x_value[$j] * $y_value[$i] + $carry;
2391:                 $carry = (int) ($temp / MATH_BIGINTEGER_BASE_FULL);
2392:                 $product_value[$k] = (int) ($temp - MATH_BIGINTEGER_BASE_FULL * $carry);
2393:             }
2394: 
2395:             if ($k < $stop) {
2396:                 $product_value[$k] = $carry;
2397:             }
2398:         }
2399: 
2400:         return array(
2401:             MATH_BIGINTEGER_VALUE => $this->_trim($product_value),
2402:             MATH_BIGINTEGER_SIGN => $x_negative != $y_negative
2403:         );
2404:     }
2405: 
2406:     /**
2407:      * Montgomery Modular Reduction
2408:      *
2409:      * ($x->_prepMontgomery($n))->_montgomery($n) yields $x % $n.
2410:      * {@link http://math.libtomcrypt.com/files/tommath.pdf#page=170 MPM 6.3} provides insights on how this can be
2411:      * improved upon (basically, by using the comba method).  gcd($n, 2) must be equal to one for this function
2412:      * to work correctly.
2413:      *
2414:      * @see _prepMontgomery()
2415:      * @see _slidingWindow()
2416:      * @access private
2417:      * @param Array $x
2418:      * @param Array $n
2419:      * @return Array
2420:      */
2421:     function _montgomery($x, $n)
2422:     {
2423:         static $cache = array(
2424:             MATH_BIGINTEGER_VARIABLE => array(),
2425:             MATH_BIGINTEGER_DATA => array()
2426:         );
2427: 
2428:         if ( ($key = array_search($n, $cache[MATH_BIGINTEGER_VARIABLE])) === false ) {
2429:             $key = count($cache[MATH_BIGINTEGER_VARIABLE]);
2430:             $cache[MATH_BIGINTEGER_VARIABLE][] = $x;
2431:             $cache[MATH_BIGINTEGER_DATA][] = $this->_modInverse67108864($n);
2432:         }
2433: 
2434:         $k = count($n);
2435: 
2436:         $result = array(MATH_BIGINTEGER_VALUE => $x);
2437: 
2438:         for ($i = 0; $i < $k; ++$i) {
2439:             $temp = $result[MATH_BIGINTEGER_VALUE][$i] * $cache[MATH_BIGINTEGER_DATA][$key];
2440:             $temp = (int) ($temp - MATH_BIGINTEGER_BASE_FULL * ((int) ($temp / MATH_BIGINTEGER_BASE_FULL)));
2441:             $temp = $this->_regularMultiply(array($temp), $n);
2442:             $temp = array_merge($this->_array_repeat(0, $i), $temp);
2443:             $result = $this->_add($result[MATH_BIGINTEGER_VALUE], false, $temp, false);
2444:         }
2445: 
2446:         $result[MATH_BIGINTEGER_VALUE] = array_slice($result[MATH_BIGINTEGER_VALUE], $k);
2447: 
2448:         if ($this->_compare($result, false, $n, false) >= 0) {
2449:             $result = $this->_subtract($result[MATH_BIGINTEGER_VALUE], false, $n, false);
2450:         }
2451: 
2452:         return $result[MATH_BIGINTEGER_VALUE];
2453:     }
2454: 
2455:     /**
2456:      * Montgomery Multiply
2457:      *
2458:      * Interleaves the montgomery reduction and long multiplication algorithms together as described in
2459:      * {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=13 HAC 14.36}
2460:      *
2461:      * @see _prepMontgomery()
2462:      * @see _montgomery()
2463:      * @access private
2464:      * @param Array $x
2465:      * @param Array $y
2466:      * @param Array $m
2467:      * @return Array
2468:      */
2469:     function _montgomeryMultiply($x, $y, $m)
2470:     {
2471:         $temp = $this->_multiply($x, false, $y, false);
2472:         return $this->_montgomery($temp[MATH_BIGINTEGER_VALUE], $m);
2473: 
2474:         static $cache = array(
2475:             MATH_BIGINTEGER_VARIABLE => array(),
2476:             MATH_BIGINTEGER_DATA => array()
2477:         );
2478: 
2479:         if ( ($key = array_search($m, $cache[MATH_BIGINTEGER_VARIABLE])) === false ) {
2480:             $key = count($cache[MATH_BIGINTEGER_VARIABLE]);
2481:             $cache[MATH_BIGINTEGER_VARIABLE][] = $m;
2482:             $cache[MATH_BIGINTEGER_DATA][] = $this->_modInverse67108864($m);
2483:         }
2484: 
2485:         $n = max(count($x), count($y), count($m));
2486:         $x = array_pad($x, $n, 0);
2487:         $y = array_pad($y, $n, 0);
2488:         $m = array_pad($m, $n, 0);
2489:         $a = array(MATH_BIGINTEGER_VALUE => $this->_array_repeat(0, $n + 1));
2490:         for ($i = 0; $i < $n; ++$i) {
2491:             $temp = $a[MATH_BIGINTEGER_VALUE][0] + $x[$i] * $y[0];
2492:             $temp = (int) ($temp - MATH_BIGINTEGER_BASE_FULL * ((int) ($temp / MATH_BIGINTEGER_BASE_FULL)));
2493:             $temp = $temp * $cache[MATH_BIGINTEGER_DATA][$key];
2494:             $temp = (int) ($temp - MATH_BIGINTEGER_BASE_FULL * ((int) ($temp / MATH_BIGINTEGER_BASE_FULL)));
2495:             $temp = $this->_add($this->_regularMultiply(array($x[$i]), $y), false, $this->_regularMultiply(array($temp), $m), false);
2496:             $a = $this->_add($a[MATH_BIGINTEGER_VALUE], false, $temp[MATH_BIGINTEGER_VALUE], false);
2497:             $a[MATH_BIGINTEGER_VALUE] = array_slice($a[MATH_BIGINTEGER_VALUE], 1);
2498:         }
2499:         if ($this->_compare($a[MATH_BIGINTEGER_VALUE], false, $m, false) >= 0) {
2500:             $a = $this->_subtract($a[MATH_BIGINTEGER_VALUE], false, $m, false);
2501:         }
2502:         return $a[MATH_BIGINTEGER_VALUE];
2503:     }
2504: 
2505:     /**
2506:      * Prepare a number for use in Montgomery Modular Reductions
2507:      *
2508:      * @see _montgomery()
2509:      * @see _slidingWindow()
2510:      * @access private
2511:      * @param Array $x
2512:      * @param Array $n
2513:      * @return Array
2514:      */
2515:     function _prepMontgomery($x, $n)
2516:     {
2517:         $lhs = new BigInteger();
2518:         $lhs->value = array_merge($this->_array_repeat(0, count($n)), $x);
2519:         $rhs = new BigInteger();
2520:         $rhs->value = $n;
2521: 
2522:         list(, $temp) = $lhs->divide($rhs);
2523:         return $temp->value;
2524:     }
2525: 
2526:     /**
2527:      * Modular Inverse of a number mod 2**26 (eg. 67108864)
2528:      *
2529:      * Based off of the bnpInvDigit function implemented and justified in the following URL:
2530:      *
2531:      * {@link http://www-cs-students.stanford.edu/~tjw/jsbn/jsbn.js}
2532:      *
2533:      * The following URL provides more info:
2534:      *
2535:      * {@link http://groups.google.com/group/sci.crypt/msg/7a137205c1be7d85}
2536:      *
2537:      * As for why we do all the bitmasking...  strange things can happen when converting from floats to ints. For
2538:      * instance, on some computers, var_dump((int) -4294967297) yields int(-1) and on others, it yields
2539:      * int(-2147483648).  To avoid problems stemming from this, we use bitmasks to guarantee that ints aren't
2540:      * auto-converted to floats.  The outermost bitmask is present because without it, there's no guarantee that
2541:      * the "residue" returned would be the so-called "common residue".  We use fmod, in the last step, because the
2542:      * maximum possible $x is 26 bits and the maximum $result is 16 bits.  Thus, we have to be able to handle up to
2543:      * 40 bits, which only 64-bit floating points will support.
2544:      *
2545:      * Thanks to Pedro Gimeno Fortea for input!
2546:      *
2547:      * @see _montgomery()
2548:      * @access private
2549:      * @param Array $x
2550:      * @return Integer
2551:      */
2552:     function _modInverse67108864($x) // 2**26 == 67,108,864
2553:     {
2554:         $x = -$x[0];
2555:         $result = $x & 0x3; // x**-1 mod 2**2
2556:         $result = ($result * (2 - $x * $result)) & 0xF; // x**-1 mod 2**4
2557:         $result = ($result * (2 - ($x & 0xFF) * $result))  & 0xFF; // x**-1 mod 2**8
2558:         $result = ($result * ((2 - ($x & 0xFFFF) * $result) & 0xFFFF)) & 0xFFFF; // x**-1 mod 2**16
2559:         $result = fmod($result * (2 - fmod($x * $result, MATH_BIGINTEGER_BASE_FULL)), MATH_BIGINTEGER_BASE_FULL); // x**-1 mod 2**26
2560:         return $result & MATH_BIGINTEGER_MAX_DIGIT;
2561:     }
2562: 
2563:     /**
2564:      * Calculates modular inverses.
2565:      *
2566:      * Say you have (30 mod 17 * x mod 17) mod 17 == 1.  x can be found using modular inverses.
2567:      *
2568:      * Here's an example:
2569:      * <code>
2570:      * <?php
2571:      *    include('Math/BigInteger.php');
2572:      *
2573:      *    $a = new BigInteger(30);
2574:      *    $b = new BigInteger(17);
2575:      *
2576:      *    $c = $a->modInverse($b);
2577:      *    echo $c->toString(); // outputs 4
2578:      *
2579:      *    echo "\r\n";
2580:      *
2581:      *    $d = $a->multiply($c);
2582:      *    list(, $d) = $d->divide($b);
2583:      *    echo $d; // outputs 1 (as per the definition of modular inverse)
2584:      * ?>
2585:      * </code>
2586:      *
2587:      * @param BigInteger $n
2588:      * @return mixed false, if no modular inverse exists, BigInteger, otherwise.
2589:      * @access public
2590:      * @internal See {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=21 HAC 14.64} for more information.
2591:      */
2592:     function modInverse($n)
2593:     {
2594:         switch ( MATH_BIGINTEGER_MODE ) {
2595:             case MATH_BIGINTEGER_MODE_GMP:
2596:                 $temp = new BigInteger();
2597:                 $temp->value = gmp_invert($this->value, $n->value);
2598: 
2599:                 return ( $temp->value === false ) ? false : $this->_normalize($temp);
2600:         }
2601: 
2602:         static $zero, $one;
2603:         if (!isset($zero)) {
2604:             $zero = new BigInteger();
2605:             $one = new BigInteger(1);
2606:         }
2607: 
2608:         // $x mod -$n == $x mod $n.
2609:         $n = $n->abs();
2610: 
2611:         if ($this->compare($zero) < 0) {
2612:             $temp = $this->abs();
2613:             $temp = $temp->modInverse($n);
2614:             return $this->_normalize($n->subtract($temp));
2615:         }
2616: 
2617:         extract($this->extendedGCD($n));
2618: 
2619:         if (!$gcd->equals($one)) {
2620:             return false;
2621:         }
2622: 
2623:         $x = $x->compare($zero) < 0 ? $x->add($n) : $x;
2624: 
2625:         return $this->compare($zero) < 0 ? $this->_normalize($n->subtract($x)) : $this->_normalize($x);
2626:     }
2627: 
2628:     /**
2629:      * Calculates the greatest common divisor and Bezout's identity.
2630:      *
2631:      * Say you have 693 and 609.  The GCD is 21.  Bezout's identity states that there exist integers x and y such that
2632:      * 693*x + 609*y == 21.  In point of fact, there are actually an infinite number of x and y combinations and which
2633:      * combination is returned is dependant upon which mode is in use.  See
2634:      * {@link http://en.wikipedia.org/wiki/B%C3%A9zout%27s_identity Bezout's identity - Wikipedia} for more information.
2635:      *
2636:      * Here's an example:
2637:      * <code>
2638:      * <?php
2639:      *    include('Math/BigInteger.php');
2640:      *
2641:      *    $a = new BigInteger(693);
2642:      *    $b = new BigInteger(609);
2643:      *
2644:      *    extract($a->extendedGCD($b));
2645:      *
2646:      *    echo $gcd->toString() . "\r\n"; // outputs 21
2647:      *    echo $a->toString() * $x->toString() + $b->toString() * $y->toString(); // outputs 21
2648:      * ?>
2649:      * </code>
2650:      *
2651:      * @param BigInteger $n
2652:      * @return BigInteger
2653:      * @access public
2654:      * @internal Calculates the GCD using the binary xGCD algorithim described in
2655:      *    {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=19 HAC 14.61}.  As the text above 14.61 notes,
2656:      *    the more traditional algorithim requires "relatively costly multiple-precision divisions".
2657:      */
2658:     function extendedGCD($n)
2659:     {
2660:         switch ( MATH_BIGINTEGER_MODE ) {
2661:             case MATH_BIGINTEGER_MODE_GMP:
2662:                 extract(gmp_gcdext($this->value, $n->value));
2663: 
2664:                 return array(
2665:                     'gcd' => $this->_normalize(new BigInteger($g)),
2666:                     'x'   => $this->_normalize(new BigInteger($s)),
2667:                     'y'   => $this->_normalize(new BigInteger($t))
2668:                 );
2669:             case MATH_BIGINTEGER_MODE_BCMATH:
2670:                 // it might be faster to use the binary xGCD algorithim here, as well, but (1) that algorithim works
2671:                 // best when the base is a power of 2 and (2) i don't think it'd make much difference, anyway.  as is,
2672:                 // the basic extended euclidean algorithim is what we're using.
2673: 
2674:                 $u = $this->value;
2675:                 $v = $n->value;
2676: 
2677:                 $a = '1';
2678:                 $b = '0';
2679:                 $c = '0';
2680:                 $d = '1';
2681: 
2682:                 while (bccomp($v, '0', 0) != 0) {
2683:                     $q = bcdiv($u, $v, 0);
2684: 
2685:                     $temp = $u;
2686:                     $u = $v;
2687:                     $v = bcsub($temp, bcmul($v, $q, 0), 0);
2688: 
2689:                     $temp = $a;
2690:                     $a = $c;
2691:                     $c = bcsub($temp, bcmul($a, $q, 0), 0);
2692: 
2693:                     $temp = $b;
2694:                     $b = $d;
2695:                     $d = bcsub($temp, bcmul($b, $q, 0), 0);
2696:                 }
2697: 
2698:                 return array(
2699:                     'gcd' => $this->_normalize(new BigInteger($u)),
2700:                     'x'   => $this->_normalize(new BigInteger($a)),
2701:                     'y'   => $this->_normalize(new BigInteger($b))
2702:                 );
2703:         }
2704: 
2705:         $y = $n->copy();
2706:         $x = $this->copy();
2707:         $g = new BigInteger();
2708:         $g->value = array(1);
2709: 
2710:         while ( !(($x->value[0] & 1)|| ($y->value[0] & 1)) ) {
2711:             $x->_rshift(1);
2712:             $y->_rshift(1);
2713:             $g->_lshift(1);
2714:         }
2715: 
2716:         $u = $x->copy();
2717:         $v = $y->copy();
2718: 
2719:         $a = new BigInteger();
2720:         $b = new BigInteger();
2721:         $c = new BigInteger();
2722:         $d = new BigInteger();
2723: 
2724:         $a->value = $d->value = $g->value = array(1);
2725:         $b->value = $c->value = array();
2726: 
2727:         while ( !empty($u->value) ) {
2728:             while ( !($u->value[0] & 1) ) {
2729:                 $u->_rshift(1);
2730:                 if ( (!empty($a->value) && ($a->value[0] & 1)) || (!empty($b->value) && ($b->value[0] & 1)) ) {
2731:                     $a = $a->add($y);
2732:                     $b = $b->subtract($x);
2733:                 }
2734:                 $a->_rshift(1);
2735:                 $b->_rshift(1);
2736:             }
2737: 
2738:             while ( !($v->value[0] & 1) ) {
2739:                 $v->_rshift(1);
2740:                 if ( (!empty($d->value) && ($d->value[0] & 1)) || (!empty($c->value) && ($c->value[0] & 1)) ) {
2741:                     $c = $c->add($y);
2742:                     $d = $d->subtract($x);
2743:                 }
2744:                 $c->_rshift(1);
2745:                 $d->_rshift(1);
2746:             }
2747: 
2748:             if ($u->compare($v) >= 0) {
2749:                 $u = $u->subtract($v);
2750:                 $a = $a->subtract($c);
2751:                 $b = $b->subtract($d);
2752:             } else {
2753:                 $v = $v->subtract($u);
2754:                 $c = $c->subtract($a);
2755:                 $d = $d->subtract($b);
2756:             }
2757:         }
2758: 
2759:         return array(
2760:             'gcd' => $this->_normalize($g->multiply($v)),
2761:             'x'   => $this->_normalize($c),
2762:             'y'   => $this->_normalize($d)
2763:         );
2764:     }
2765: 
2766:     /**
2767:      * Calculates the greatest common divisor
2768:      *
2769:      * Say you have 693 and 609.  The GCD is 21.
2770:      *
2771:      * Here's an example:
2772:      * <code>
2773:      * <?php
2774:      *    include('Math/BigInteger.php');
2775:      *
2776:      *    $a = new BigInteger(693);
2777:      *    $b = new BigInteger(609);
2778:      *
2779:      *    $gcd = a->extendedGCD($b);
2780:      *
2781:      *    echo $gcd->toString() . "\r\n"; // outputs 21
2782:      * ?>
2783:      * </code>
2784:      *
2785:      * @param BigInteger $n
2786:      * @return BigInteger
2787:      * @access public
2788:      */
2789:     function gcd($n)
2790:     {
2791:         extract($this->extendedGCD($n));
2792:         return $gcd;
2793:     }
2794: 
2795:     /**
2796:      * Absolute value.
2797:      *
2798:      * @return BigInteger
2799:      * @access public
2800:      */
2801:     function abs()
2802:     {
2803:         $temp = new BigInteger();
2804: 
2805:         switch ( MATH_BIGINTEGER_MODE ) {
2806:             case MATH_BIGINTEGER_MODE_GMP:
2807:                 $temp->value = gmp_abs($this->value);
2808:                 break;
2809:             case MATH_BIGINTEGER_MODE_BCMATH:
2810:                 $temp->value = (bccomp($this->value, '0', 0) < 0) ? substr($this->value, 1) : $this->value;
2811:                 break;
2812:             default:
2813:                 $temp->value = $this->value;
2814:         }
2815: 
2816:         return $temp;
2817:     }
2818: 
2819:     /**
2820:      * Compares two numbers.
2821:      *
2822:      * Although one might think !$x->compare($y) means $x != $y, it, in fact, means the opposite.  The reason for this is
2823:      * demonstrated thusly:
2824:      *
2825:      * $x  > $y: $x->compare($y)  > 0
2826:      * $x  < $y: $x->compare($y)  < 0
2827:      * $x == $y: $x->compare($y) == 0
2828:      *
2829:      * Note how the same comparison operator is used.  If you want to test for equality, use $x->equals($y).
2830:      *
2831:      * @param BigInteger $y
2832:      * @return Integer < 0 if $this is less than $y; > 0 if $this is greater than $y, and 0 if they are equal.
2833:      * @access public
2834:      * @see equals()
2835:      * @internal Could return $this->subtract($x), but that's not as fast as what we do do.
2836:      */
2837:     function compare($y)
2838:     {
2839:         switch ( MATH_BIGINTEGER_MODE ) {
2840:             case MATH_BIGINTEGER_MODE_GMP:
2841:                 return gmp_cmp($this->value, $y->value);
2842:             case MATH_BIGINTEGER_MODE_BCMATH:
2843:                 return bccomp($this->value, $y->value, 0);
2844:         }
2845: 
2846:         return $this->_compare($this->value, $this->is_negative, $y->value, $y->is_negative);
2847:     }
2848: 
2849:     /**
2850:      * Compares two numbers.
2851:      *
2852:      * @param Array $x_value
2853:      * @param Boolean $x_negative
2854:      * @param Array $y_value
2855:      * @param Boolean $y_negative
2856:      * @return Integer
2857:      * @see compare()
2858:      * @access private
2859:      */
2860:     function _compare($x_value, $x_negative, $y_value, $y_negative)
2861:     {
2862:         if ( $x_negative != $y_negative ) {
2863:             return ( !$x_negative && $y_negative ) ? 1 : -1;
2864:         }
2865: 
2866:         $result = $x_negative ? -1 : 1;
2867: 
2868:         if ( count($x_value) != count($y_value) ) {
2869:             return ( count($x_value) > count($y_value) ) ? $result : -$result;
2870:         }
2871:         $size = max(count($x_value), count($y_value));
2872: 
2873:         $x_value = array_pad($x_value, $size, 0);
2874:         $y_value = array_pad($y_value, $size, 0);
2875: 
2876:         for ($i = count($x_value) - 1; $i >= 0; --$i) {
2877:             if ($x_value[$i] != $y_value[$i]) {
2878:                 return ( $x_value[$i] > $y_value[$i] ) ? $result : -$result;
2879:             }
2880:         }
2881: 
2882:         return 0;
2883:     }
2884: 
2885:     /**
2886:      * Tests the equality of two numbers.
2887:      *
2888:      * If you need to see if one number is greater than or less than another number, use BigInteger::compare()
2889:      *
2890:      * @param BigInteger $x
2891:      * @return Boolean
2892:      * @access public
2893:      * @see compare()
2894:      */
2895:     function equals($x)
2896:     {
2897:         switch ( MATH_BIGINTEGER_MODE ) {
2898:             case MATH_BIGINTEGER_MODE_GMP:
2899:                 return gmp_cmp($this->value, $x->value) == 0;
2900:             default:
2901:                 return $this->value === $x->value && $this->is_negative == $x->is_negative;
2902:         }
2903:     }
2904: 
2905:     /**
2906:      * Set Precision
2907:      *
2908:      * Some bitwise operations give different results depending on the precision being used.  Examples include left
2909:      * shift, not, and rotates.
2910:      *
2911:      * @param Integer $bits
2912:      * @access public
2913:      */
2914:     function setPrecision($bits)
2915:     {
2916:         $this->precision = $bits;
2917:         if ( MATH_BIGINTEGER_MODE != MATH_BIGINTEGER_MODE_BCMATH ) {
2918:             $this->bitmask = new BigInteger(chr((1 << ($bits & 0x7)) - 1) . str_repeat(chr(0xFF), $bits >> 3), 256);
2919:         } else {
2920:             $this->bitmask = new BigInteger(bcpow('2', $bits, 0));
2921:         }
2922: 
2923:         $temp = $this->_normalize($this);
2924:         $this->value = $temp->value;
2925:     }
2926: 
2927:     /**
2928:      * Logical And
2929:      *
2930:      * @param BigInteger $x
2931:      * @access public
2932:      * @internal Implemented per a request by Lluis Pamies i Juarez <lluis _a_ pamies.cat>
2933:      * @return BigInteger
2934:      */
2935:     function bitwise_and($x)
2936:     {
2937:         switch ( MATH_BIGINTEGER_MODE ) {
2938:             case MATH_BIGINTEGER_MODE_GMP:
2939:                 $temp = new BigInteger();
2940:                 $temp->value = gmp_and($this->value, $x->value);
2941: 
2942:                 return $this->_normalize($temp);
2943:             case MATH_BIGINTEGER_MODE_BCMATH:
2944:                 $left = $this->toBytes();
2945:                 $right = $x->toBytes();
2946: 
2947:                 $length = max(strlen($left), strlen($right));
2948: 
2949:                 $left = str_pad($left, $length, chr(0), STR_PAD_LEFT);
2950:                 $right = str_pad($right, $length, chr(0), STR_PAD_LEFT);
2951: 
2952:                 return $this->_normalize(new BigInteger($left & $right, 256));
2953:         }
2954: 
2955:         $result = $this->copy();
2956: 
2957:         $length = min(count($x->value), count($this->value));
2958: 
2959:         $result->value = array_slice($result->value, 0, $length);
2960: 
2961:         for ($i = 0; $i < $length; ++$i) {
2962:             $result->value[$i]&= $x->value[$i];
2963:         }
2964: 
2965:         return $this->_normalize($result);
2966:     }
2967: 
2968:     /**
2969:      * Logical Or
2970:      *
2971:      * @param BigInteger $x
2972:      * @access public
2973:      * @internal Implemented per a request by Lluis Pamies i Juarez <lluis _a_ pamies.cat>
2974:      * @return BigInteger
2975:      */
2976:     function bitwise_or($x)
2977:     {
2978:         switch ( MATH_BIGINTEGER_MODE ) {
2979:             case MATH_BIGINTEGER_MODE_GMP:
2980:                 $temp = new BigInteger();
2981:                 $temp->value = gmp_or($this->value, $x->value);
2982: 
2983:                 return $this->_normalize($temp);
2984:             case MATH_BIGINTEGER_MODE_BCMATH:
2985:                 $left = $this->toBytes();
2986:                 $right = $x->toBytes();
2987: 
2988:                 $length = max(strlen($left), strlen($right));
2989: 
2990:                 $left = str_pad($left, $length, chr(0), STR_PAD_LEFT);
2991:                 $right = str_pad($right, $length, chr(0), STR_PAD_LEFT);
2992: 
2993:                 return $this->_normalize(new BigInteger($left | $right, 256));
2994:         }
2995: 
2996:         $length = max(count($this->value), count($x->value));
2997:         $result = $this->copy();
2998:         $result->value = array_pad($result->value, $length, 0);
2999:         $x->value = array_pad($x->value, $length, 0);
3000: 
3001:         for ($i = 0; $i < $length; ++$i) {
3002:             $result->value[$i]|= $x->value[$i];
3003:         }
3004: 
3005:         return $this->_normalize($result);
3006:     }
3007: 
3008:     /**
3009:      * Logical Exclusive-Or
3010:      *
3011:      * @param BigInteger $x
3012:      * @access public
3013:      * @internal Implemented per a request by Lluis Pamies i Juarez <lluis _a_ pamies.cat>
3014:      * @return BigInteger
3015:      */
3016:     function bitwise_xor($x)
3017:     {
3018:         switch ( MATH_BIGINTEGER_MODE ) {
3019:             case MATH_BIGINTEGER_MODE_GMP:
3020:                 $temp = new BigInteger();
3021:                 $temp->value = gmp_xor($this->value, $x->value);
3022: 
3023:                 return $this->_normalize($temp);
3024:             case MATH_BIGINTEGER_MODE_BCMATH:
3025:                 $left = $this->toBytes();
3026:                 $right = $x->toBytes();
3027: 
3028:                 $length = max(strlen($left), strlen($right));
3029: 
3030:                 $left = str_pad($left, $length, chr(0), STR_PAD_LEFT);
3031:                 $right = str_pad($right, $length, chr(0), STR_PAD_LEFT);
3032: 
3033:                 return $this->_normalize(new BigInteger($left ^ $right, 256));
3034:         }
3035: 
3036:         $length = max(count($this->value), count($x->value));
3037:         $result = $this->copy();
3038:         $result->value = array_pad($result->value, $length, 0);
3039:         $x->value = array_pad($x->value, $length, 0);
3040: 
3041:         for ($i = 0; $i < $length; ++$i) {
3042:             $result->value[$i]^= $x->value[$i];
3043:         }
3044: 
3045:         return $this->_normalize($result);
3046:     }
3047: 
3048:     /**
3049:      * Logical Not
3050:      *
3051:      * @access public
3052:      * @internal Implemented per a request by Lluis Pamies i Juarez <lluis _a_ pamies.cat>
3053:      * @return BigInteger
3054:      */
3055:     function bitwise_not()
3056:     {
3057:         // calculuate "not" without regard to $this->precision
3058:         // (will always result in a smaller number.  ie. ~1 isn't 1111 1110 - it's 0)
3059:         $temp = $this->toBytes();
3060:         $pre_msb = decbin(ord($temp[0]));
3061:         $temp = ~$temp;
3062:         $msb = decbin(ord($temp[0]));
3063:         if (strlen($msb) == 8) {
3064:             $msb = substr($msb, strpos($msb, '0'));
3065:         }
3066:         $temp[0] = chr(bindec($msb));
3067: 
3068:         // see if we need to add extra leading 1's
3069:         $current_bits = strlen($pre_msb) + 8 * strlen($temp) - 8;
3070:         $new_bits = $this->precision - $current_bits;
3071:         if ($new_bits <= 0) {
3072:             return $this->_normalize(new BigInteger($temp, 256));
3073:         }
3074: 
3075:         // generate as many leading 1's as we need to.
3076:         $leading_ones = chr((1 << ($new_bits & 0x7)) - 1) . str_repeat(chr(0xFF), $new_bits >> 3);
3077:         $this->_base256_lshift($leading_ones, $current_bits);
3078: 
3079:         $temp = str_pad($temp, ceil($this->bits / 8), chr(0), STR_PAD_LEFT);
3080: 
3081:         return $this->_normalize(new BigInteger($leading_ones | $temp, 256));
3082:     }
3083: 
3084:     /**
3085:      * Logical Right Shift
3086:      *
3087:      * Shifts BigInteger's by $shift bits, effectively dividing by 2**$shift.
3088:      *
3089:      * @param Integer $shift
3090:      * @return BigInteger
3091:      * @access public
3092:      * @internal The only version that yields any speed increases is the internal version.
3093:      */
3094:     function bitwise_rightShift($shift)
3095:     {
3096:         $temp = new BigInteger();
3097: 
3098:         switch ( MATH_BIGINTEGER_MODE ) {
3099:             case MATH_BIGINTEGER_MODE_GMP:
3100:                 static $two;
3101: 
3102:                 if (!isset($two)) {
3103:                     $two = gmp_init('2');
3104:                 }
3105: 
3106:                 $temp->value = gmp_div_q($this->value, gmp_pow($two, $shift));
3107: 
3108:                 break;
3109:             case MATH_BIGINTEGER_MODE_BCMATH:
3110:                 $temp->value = bcdiv($this->value, bcpow('2', $shift, 0), 0);
3111: 
3112:                 break;
3113:             default: // could just replace _lshift with this, but then all _lshift() calls would need to be rewritten
3114:                      // and I don't want to do that...
3115:                 $temp->value = $this->value;
3116:                 $temp->_rshift($shift);
3117:         }
3118: 
3119:         return $this->_normalize($temp);
3120:     }
3121: 
3122:     /**
3123:      * Logical Left Shift
3124:      *
3125:      * Shifts BigInteger's by $shift bits, effectively multiplying by 2**$shift.
3126:      *
3127:      * @param Integer $shift
3128:      * @return BigInteger
3129:      * @access public
3130:      * @internal The only version that yields any speed increases is the internal version.
3131:      */
3132:     function bitwise_leftShift($shift)
3133:     {
3134:         $temp = new BigInteger();
3135: 
3136:         switch ( MATH_BIGINTEGER_MODE ) {
3137:             case MATH_BIGINTEGER_MODE_GMP:
3138:                 static $two;
3139: 
3140:                 if (!isset($two)) {
3141:                     $two = gmp_init('2');
3142:                 }
3143: 
3144:                 $temp->value = gmp_mul($this->value, gmp_pow($two, $shift));
3145: 
3146:                 break;
3147:             case MATH_BIGINTEGER_MODE_BCMATH:
3148:                 $temp->value = bcmul($this->value, bcpow('2', $shift, 0), 0);
3149: 
3150:                 break;
3151:             default: // could just replace _rshift with this, but then all _lshift() calls would need to be rewritten
3152:                      // and I don't want to do that...
3153:                 $temp->value = $this->value;
3154:                 $temp->_lshift($shift);
3155:         }
3156: 
3157:         return $this->_normalize($temp);
3158:     }
3159: 
3160:     /**
3161:      * Logical Left Rotate
3162:      *
3163:      * Instead of the top x bits being dropped they're appended to the shifted bit string.
3164:      *
3165:      * @param Integer $shift
3166:      * @return BigInteger
3167:      * @access public
3168:      */
3169:     function bitwise_leftRotate($shift)
3170:     {
3171:         $bits = $this->toBytes();
3172: 
3173:         if ($this->precision > 0) {
3174:             $precision = $this->precision;
3175:             if ( MATH_BIGINTEGER_MODE == MATH_BIGINTEGER_MODE_BCMATH ) {
3176:                 $mask = $this->bitmask->subtract(new BigInteger(1));
3177:                 $mask = $mask->toBytes();
3178:             } else {
3179:                 $mask = $this->bitmask->toBytes();
3180:             }
3181:         } else {
3182:             $temp = ord($bits[0]);
3183:             for ($i = 0; $temp >> $i; ++$i);
3184:             $precision = 8 * strlen($bits) - 8 + $i;
3185:             $mask = chr((1 << ($precision & 0x7)) - 1) . str_repeat(chr(0xFF), $precision >> 3);
3186:         }
3187: 
3188:         if ($shift < 0) {
3189:             $shift+= $precision;
3190:         }
3191:         $shift%= $precision;
3192: 
3193:         if (!$shift) {
3194:             return $this->copy();
3195:         }
3196: 
3197:         $left = $this->bitwise_leftShift($shift);
3198:         $left = $left->bitwise_and(new BigInteger($mask, 256));
3199:         $right = $this->bitwise_rightShift($precision - $shift);
3200:         $result = MATH_BIGINTEGER_MODE != MATH_BIGINTEGER_MODE_BCMATH ? $left->bitwise_or($right) : $left->add($right);
3201:         return $this->_normalize($result);
3202:     }
3203: 
3204:     /**
3205:      * Logical Right Rotate
3206:      *
3207:      * Instead of the bottom x bits being dropped they're prepended to the shifted bit string.
3208:      *
3209:      * @param Integer $shift
3210:      * @return BigInteger
3211:      * @access public
3212:      */
3213:     function bitwise_rightRotate($shift)
3214:     {
3215:         return $this->bitwise_leftRotate(-$shift);
3216:     }
3217: 
3218:     /**
3219:      * Set random number generator function
3220:      *
3221:      * This function is deprecated.
3222:      *
3223:      * @param String $generator
3224:      * @access public
3225:      */
3226:     function setRandomGenerator($generator)
3227:     {
3228:     }
3229: 
3230:     /**
3231:      * Generates a random BigInteger
3232:      *
3233:      * Byte length is equal to $length. Uses crypt_random if it's loaded and mt_rand if it's not.
3234:      *
3235:      * @param Integer $length
3236:      * @return BigInteger
3237:      * @access private
3238:      */
3239:     function _random_number_helper($size)
3240:     {
3241:         $crypt_random = function_exists('crypt_random_string') || (!class_exists('Crypt_Random') && function_exists('crypt_random_string'));
3242:         if ($crypt_random) {
3243:             $random = crypt_random_string($size);
3244:         } else {
3245:             $random = '';
3246: 
3247:             if ($size & 1) {
3248:                 $random.= chr(mt_rand(0, 255));
3249:             }
3250: 
3251:             $blocks = $size >> 1;
3252:             for ($i = 0; $i < $blocks; ++$i) {
3253:                 // mt_rand(-2147483648, 0x7FFFFFFF) always produces -2147483648 on some systems
3254:                 $random.= pack('n', mt_rand(0, 0xFFFF));
3255:             }
3256:         }
3257: 
3258:         return new BigInteger($random, 256);
3259:     }
3260: 
3261:     /**
3262:      * Generate a random number
3263:      *
3264:      * @param optional Integer $min
3265:      * @param optional Integer $max
3266:      * @return BigInteger
3267:      * @access public
3268:      */
3269:     function random($min = false, $max = false)
3270:     {
3271:         if ($min === false) {
3272:             $min = new BigInteger(0);
3273:         }
3274: 
3275:         if ($max === false) {
3276:             $max = new BigInteger(0x7FFFFFFF);
3277:         }
3278: 
3279:         $compare = $max->compare($min);
3280: 
3281:         if (!$compare) {
3282:             return $this->_normalize($min);
3283:         } else if ($compare < 0) {
3284:             // if $min is bigger then $max, swap $min and $max
3285:             $temp = $max;
3286:             $max = $min;
3287:             $min = $temp;
3288:         }
3289: 
3290:         static $one;
3291:         if (!isset($one)) {
3292:             $one = new BigInteger(1);
3293:         }
3294: 
3295:         $max = $max->subtract($min->subtract($one));
3296:         $size = strlen(ltrim($max->toBytes(), chr(0)));
3297: 
3298:         /*
3299:             doing $random % $max doesn't work because some numbers will be more likely to occur than others.
3300:             eg. if $max is 140 and $random's max is 255 then that'd mean both $random = 5 and $random = 145
3301:             would produce 5 whereas the only value of random that could produce 139 would be 139. ie.
3302:             not all numbers would be equally likely. some would be more likely than others.
3303: 
3304:             creating a whole new random number until you find one that is within the range doesn't work
3305:             because, for sufficiently small ranges, the likelihood that you'd get a number within that range
3306:             would be pretty small. eg. with $random's max being 255 and if your $max being 1 the probability
3307:             would be pretty high that $random would be greater than $max.
3308: 
3309:             phpseclib works around this using the technique described here:
3310: 
3311:             http://crypto.stackexchange.com/questions/5708/creating-a-small-number-from-a-cryptographically-secure-random-string
3312:         */
3313:         $random_max = new BigInteger(chr(1) . str_repeat("\0", $size), 256);
3314:         $random = $this->_random_number_helper($size);
3315: 
3316:         list($max_multiple) = $random_max->divide($max);
3317:         $max_multiple = $max_multiple->multiply($max);
3318: 
3319:         while ($random->compare($max_multiple) >= 0) {
3320:             $random = $random->subtract($max_multiple);
3321:             $random_max = $random_max->subtract($max_multiple);
3322:             $random = $random->bitwise_leftShift(8);
3323:             $random = $random->add($this->_random_number_helper(1));
3324:             $random_max = $random_max->bitwise_leftShift(8);
3325:             list($max_multiple) = $random_max->divide($max);
3326:             $max_multiple = $max_multiple->multiply($max);
3327:         }
3328:         list(, $random) = $random->divide($max);
3329: 
3330:         return $this->_normalize($random->add($min));
3331:     }
3332: 
3333:     /**
3334:      * Generate a random prime number.
3335:      *
3336:      * If there's not a prime within the given range, false will be returned.  If more than $timeout seconds have elapsed,
3337:      * give up and return false.
3338:      *
3339:      * @param optional Integer $min
3340:      * @param optional Integer $max
3341:      * @param optional Integer $timeout
3342:      * @return BigInteger
3343:      * @access public
3344:      * @internal See {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap4.pdf#page=15 HAC 4.44}.
3345:      */
3346:     function randomPrime($min = false, $max = false, $timeout = false)
3347:     {
3348:         if ($min === false) {
3349:             $min = new BigInteger(0);
3350:         }
3351: 
3352:         if ($max === false) {
3353:             $max = new BigInteger(0x7FFFFFFF);
3354:         }
3355: 
3356:         $compare = $max->compare($min);
3357: 
3358:         if (!$compare) {
3359:             return $min->isPrime() ? $min : false;
3360:         } else if ($compare < 0) {
3361:             // if $min is bigger then $max, swap $min and $max
3362:             $temp = $max;
3363:             $max = $min;
3364:             $min = $temp;
3365:         }
3366: 
3367:         static $one, $two;
3368:         if (!isset($one)) {
3369:             $one = new BigInteger(1);
3370:             $two = new BigInteger(2);
3371:         }
3372: 
3373:         $start = time();
3374: 
3375:         $x = $this->random($min, $max);
3376: 
3377:         // gmp_nextprime() requires PHP 5 >= 5.2.0 per <http://php.net/gmp-nextprime>.
3378:         if ( MATH_BIGINTEGER_MODE == MATH_BIGINTEGER_MODE_GMP && function_exists('gmp_nextprime') ) {
3379:             $p = new BigInteger();
3380:             $p->value = gmp_nextprime($x->value);
3381: 
3382:             if ($p->compare($max) <= 0) {
3383:                 return $p;
3384:             }
3385: 
3386:             if (!$min->equals($x)) {
3387:                 $x = $x->subtract($one);
3388:             }
3389: 
3390:             return $x->randomPrime($min, $x);
3391:         }
3392: 
3393:         if ($x->equals($two)) {
3394:             return $x;
3395:         }
3396: 
3397:         $x->_make_odd();
3398:         if ($x->compare($max) > 0) {
3399:             // if $x > $max then $max is even and if $min == $max then no prime number exists between the specified range
3400:             if ($min->equals($max)) {
3401:                 return false;
3402:             }
3403:             $x = $min->copy();
3404:             $x->_make_odd();
3405:         }
3406: 
3407:         $initial_x = $x->copy();
3408: 
3409:         while (true) {
3410:             if ($timeout !== false && time() - $start > $timeout) {
3411:                 return false;
3412:             }
3413: 
3414:             if ($x->isPrime()) {
3415:                 return $x;
3416:             }
3417: 
3418:             $x = $x->add($two);
3419: 
3420:             if ($x->compare($max) > 0) {
3421:                 $x = $min->copy();
3422:                 if ($x->equals($two)) {
3423:                     return $x;
3424:                 }
3425:                 $x->_make_odd();
3426:             }
3427: 
3428:             if ($x->equals($initial_x)) {
3429:                 return false;
3430:             }
3431:         }
3432:     }
3433: 
3434:     /**
3435:      * Make the current number odd
3436:      *
3437:      * If the current number is odd it'll be unchanged.  If it's even, one will be added to it.
3438:      *
3439:      * @see randomPrime()
3440:      * @access private
3441:      */
3442:     function _make_odd()
3443:     {
3444:         switch ( MATH_BIGINTEGER_MODE ) {
3445:             case MATH_BIGINTEGER_MODE_GMP:
3446:                 gmp_setbit($this->value, 0);
3447:                 break;
3448:             case MATH_BIGINTEGER_MODE_BCMATH:
3449:                 if ($this->value[strlen($this->value) - 1] % 2 == 0) {
3450:                     $this->value = bcadd($this->value, '1');
3451:                 }
3452:                 break;
3453:             default:
3454:                 $this->value[0] |= 1;
3455:         }
3456:     }
3457: 
3458:     /**
3459:      * Checks a numer to see if it's prime
3460:      *
3461:      * Assuming the $t parameter is not set, this function has an error rate of 2**-80.  The main motivation for the
3462:      * $t parameter is distributability.  BigInteger::randomPrime() can be distributed accross multiple pageloads
3463:      * on a website instead of just one.
3464:      *
3465:      * @param optional Integer $t
3466:      * @return Boolean
3467:      * @access public
3468:      * @internal Uses the
3469:      *     {@link http://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test Miller-Rabin primality test}.  See
3470:      *     {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap4.pdf#page=8 HAC 4.24}.
3471:      */
3472:     function isPrime($t = false)
3473:     {
3474:         $length = strlen($this->toBytes());
3475: 
3476:         if (!$t) {
3477:             // see HAC 4.49 "Note (controlling the error probability)"
3478:             // @codingStandardsIgnoreStart
3479:                  if ($length >= 163) { $t =  2; } // floor(1300 / 8)
3480:             else if ($length >= 106) { $t =  3; } // floor( 850 / 8)
3481:             else if ($length >= 81 ) { $t =  4; } // floor( 650 / 8)
3482:             else if ($length >= 68 ) { $t =  5; } // floor( 550 / 8)
3483:             else if ($length >= 56 ) { $t =  6; } // floor( 450 / 8)
3484:             else if ($length >= 50 ) { $t =  7; } // floor( 400 / 8)
3485:             else if ($length >= 43 ) { $t =  8; } // floor( 350 / 8)
3486:             else if ($length >= 37 ) { $t =  9; } // floor( 300 / 8)
3487:             else if ($length >= 31 ) { $t = 12; } // floor( 250 / 8)
3488:             else if ($length >= 25 ) { $t = 15; } // floor( 200 / 8)
3489:             else if ($length >= 18 ) { $t = 18; } // floor( 150 / 8)
3490:             else                     { $t = 27; }
3491:             // @codingStandardsIgnoreEnd
3492:         }
3493: 
3494:         // ie. gmp_testbit($this, 0)
3495:         // ie. isEven() or !isOdd()
3496:         switch ( MATH_BIGINTEGER_MODE ) {
3497:             case MATH_BIGINTEGER_MODE_GMP:
3498:                 return gmp_prob_prime($this->value, $t) != 0;
3499:             case MATH_BIGINTEGER_MODE_BCMATH:
3500:                 if ($this->value === '2') {
3501:                     return true;
3502:                 }
3503:                 if ($this->value[strlen($this->value) - 1] % 2 == 0) {
3504:                     return false;
3505:                 }
3506:                 break;
3507:             default:
3508:                 if ($this->value == array(2)) {
3509:                     return true;
3510:                 }
3511:                 if (~$this->value[0] & 1) {
3512:                     return false;
3513:                 }
3514:         }
3515: 
3516:         static $primes, $zero, $one, $two;
3517: 
3518:         if (!isset($primes)) {
3519:             $primes = array(
3520:                 3,    5,    7,    11,   13,   17,   19,   23,   29,   31,   37,   41,   43,   47,   53,   59,
3521:                 61,   67,   71,   73,   79,   83,   89,   97,   101,  103,  107,  109,  113,  127,  131,  137,
3522:                 139,  149,  151,  157,  163,  167,  173,  179,  181,  191,  193,  197,  199,  211,  223,  227,
3523:                 229,  233,  239,  241,  251,  257,  263,  269,  271,  277,  281,  283,  293,  307,  311,  313,
3524:                 317,  331,  337,  347,  349,  353,  359,  367,  373,  379,  383,  389,  397,  401,  409,  419,
3525:                 421,  431,  433,  439,  443,  449,  457,  461,  463,  467,  479,  487,  491,  499,  503,  509,
3526:                 521,  523,  541,  547,  557,  563,  569,  571,  577,  587,  593,  599,  601,  607,  613,  617,
3527:                 619,  631,  641,  643,  647,  653,  659,  661,  673,  677,  683,  691,  701,  709,  719,  727,
3528:                 733,  739,  743,  751,  757,  761,  769,  773,  787,  797,  809,  811,  821,  823,  827,  829,
3529:                 839,  853,  857,  859,  863,  877,  881,  883,  887,  907,  911,  919,  929,  937,  941,  947,
3530:                 953,  967,  971,  977,  983,  991,  997
3531:             );
3532: 
3533:             if ( MATH_BIGINTEGER_MODE != MATH_BIGINTEGER_MODE_INTERNAL ) {
3534:                 for ($i = 0; $i < count($primes); ++$i) {
3535:                     $primes[$i] = new BigInteger($primes[$i]);
3536:                 }
3537:             }
3538: 
3539:             $zero = new BigInteger();
3540:             $one = new BigInteger(1);
3541:             $two = new BigInteger(2);
3542:         }
3543: 
3544:         if ($this->equals($one)) {
3545:             return false;
3546:         }
3547: 
3548:         // see HAC 4.4.1 "Random search for probable primes"
3549:         if ( MATH_BIGINTEGER_MODE != MATH_BIGINTEGER_MODE_INTERNAL ) {
3550:             foreach ($primes as $prime) {
3551:                 list(, $r) = $this->divide($prime);
3552:                 if ($r->equals($zero)) {
3553:                     return $this->equals($prime);
3554:                 }
3555:             }
3556:         } else {
3557:             $value = $this->value;
3558:             foreach ($primes as $prime) {
3559:                 list(, $r) = $this->_divide_digit($value, $prime);
3560:                 if (!$r) {
3561:                     return count($value) == 1 && $value[0] == $prime;
3562:                 }
3563:             }
3564:         }
3565: 
3566:         $n   = $this->copy();
3567:         $n_1 = $n->subtract($one);
3568:         $n_2 = $n->subtract($two);
3569: 
3570:         $r = $n_1->copy();
3571:         $r_value = $r->value;
3572:         // ie. $s = gmp_scan1($n, 0) and $r = gmp_div_q($n, gmp_pow(gmp_init('2'), $s));
3573:         if ( MATH_BIGINTEGER_MODE == MATH_BIGINTEGER_MODE_BCMATH ) {
3574:             $s = 0;
3575:             // if $n was 1, $r would be 0 and this would be an infinite loop, hence our $this->equals($one) check earlier
3576:             while ($r->value[strlen($r->value) - 1] % 2 == 0) {
3577:                 $r->value = bcdiv($r->value, '2', 0);
3578:                 ++$s;
3579:             }
3580:         } else {
3581:             for ($i = 0, $r_length = count($r_value); $i < $r_length; ++$i) {
3582:                 $temp = ~$r_value[$i] & 0xFFFFFF;
3583:                 for ($j = 1; ($temp >> $j) & 1; ++$j);
3584:                 if ($j != 25) {
3585:                     break;
3586:                 }
3587:             }
3588:             $s = 26 * $i + $j - 1;
3589:             $r->_rshift($s);
3590:         }
3591: 
3592:         for ($i = 0; $i < $t; ++$i) {
3593:             $a = $this->random($two, $n_2);
3594:             $y = $a->modPow($r, $n);
3595: 
3596:             if (!$y->equals($one) && !$y->equals($n_1)) {
3597:                 for ($j = 1; $j < $s && !$y->equals($n_1); ++$j) {
3598:                     $y = $y->modPow($two, $n);
3599:                     if ($y->equals($one)) {
3600:                         return false;
3601:                     }
3602:                 }
3603: 
3604:                 if (!$y->equals($n_1)) {
3605:                     return false;
3606:                 }
3607:             }
3608:         }
3609:         return true;
3610:     }
3611: 
3612: 
3613: 
3614: 
3615: 
3616: 
3617:     /**
3618:      * Logical Left Shift
3619:      *
3620:      * Shifts BigInteger's by $shift bits.
3621:      *
3622:      * @param Integer $shift
3623:      * @access private
3624:      */
3625:     function _lshift($shift)
3626:     {
3627:         if ( $shift == 0 ) {
3628:             return;
3629:         }
3630: 
3631:         $num_digits = (int) ($shift / MATH_BIGINTEGER_BASE);
3632:         $shift %= MATH_BIGINTEGER_BASE;
3633:         $shift = 1 << $shift;
3634: 
3635:         $carry = 0;
3636: 
3637:         for ($i = 0; $i < count($this->value); ++$i) {
3638:             $temp = $this->value[$i] * $shift + $carry;
3639:             $carry = (int) ($temp / MATH_BIGINTEGER_BASE_FULL);
3640:             $this->value[$i] = (int) ($temp - $carry * MATH_BIGINTEGER_BASE_FULL);
3641:         }
3642: 
3643:         if ( $carry ) {
3644:             $this->value[] = $carry;
3645:         }
3646: 
3647:         while ($num_digits--) {
3648:             array_unshift($this->value, 0);
3649:         }
3650:     }
3651: 
3652:     /**
3653:      * Logical Right Shift
3654:      *
3655:      * Shifts BigInteger's by $shift bits.
3656:      *
3657:      * @param Integer $shift
3658:      * @access private
3659:      */
3660:     function _rshift($shift)
3661:     {
3662:         if ($shift == 0) {
3663:             return;
3664:         }
3665: 
3666:         $num_digits = (int) ($shift / MATH_BIGINTEGER_BASE);
3667:         $shift %= MATH_BIGINTEGER_BASE;
3668:         $carry_shift = MATH_BIGINTEGER_BASE - $shift;
3669:         $carry_mask = (1 << $shift) - 1;
3670: 
3671:         if ( $num_digits ) {
3672:             $this->value = array_slice($this->value, $num_digits);
3673:         }
3674: 
3675:         $carry = 0;
3676: 
3677:         for ($i = count($this->value) - 1; $i >= 0; --$i) {
3678:             $temp = $this->value[$i] >> $shift | $carry;
3679:             $carry = ($this->value[$i] & $carry_mask) << $carry_shift;
3680:             $this->value[$i] = $temp;
3681:         }
3682: 
3683:         $this->value = $this->_trim($this->value);
3684:     }
3685: 
3686:     /**
3687:      * Normalize
3688:      *
3689:      * Removes leading zeros and truncates (if necessary) to maintain the appropriate precision
3690:      *
3691:      * @param BigInteger
3692:      * @return BigInteger
3693:      * @see _trim()
3694:      * @access private
3695:      */
3696:     function _normalize($result)
3697:     {
3698:         $result->precision = $this->precision;
3699:         $result->bitmask = $this->bitmask;
3700: 
3701:         switch ( MATH_BIGINTEGER_MODE ) {
3702:             case MATH_BIGINTEGER_MODE_GMP:
3703:                 if (!empty($result->bitmask->value)) {
3704:                     $result->value = gmp_and($result->value, $result->bitmask->value);
3705:                 }
3706: 
3707:                 return $result;
3708:             case MATH_BIGINTEGER_MODE_BCMATH:
3709:                 if (!empty($result->bitmask->value)) {
3710:                     $result->value = bcmod($result->value, $result->bitmask->value);
3711:                 }
3712: 
3713:                 return $result;
3714:         }
3715: 
3716:         $value = &$result->value;
3717: 
3718:         if ( !count($value) ) {
3719:             return $result;
3720:         }
3721: 
3722:         $value = $this->_trim($value);
3723: 
3724:         if (!empty($result->bitmask->value)) {
3725:             $length = min(count($value), count($this->bitmask->value));
3726:             $value = array_slice($value, 0, $length);
3727: 
3728:             for ($i = 0; $i < $length; ++$i) {
3729:                 $value[$i] = $value[$i] & $this->bitmask->value[$i];
3730:             }
3731:         }
3732: 
3733:         return $result;
3734:     }
3735: 
3736:     /**
3737:      * Trim
3738:      *
3739:      * Removes leading zeros
3740:      *
3741:      * @param Array $value
3742:      * @return BigInteger
3743:      * @access private
3744:      */
3745:     function _trim($value)
3746:     {
3747:         for ($i = count($value) - 1; $i >= 0; --$i) {
3748:             if ( $value[$i] ) {
3749:                 break;
3750:             }
3751:             unset($value[$i]);
3752:         }
3753: 
3754:         return $value;
3755:     }
3756: 
3757:     /**
3758:      * Array Repeat
3759:      *
3760:      * @param $input Array
3761:      * @param $multiplier mixed
3762:      * @return Array
3763:      * @access private
3764:      */
3765:     function _array_repeat($input, $multiplier)
3766:     {
3767:         return ($multiplier) ? array_fill(0, $multiplier, $input) : array();
3768:     }
3769: 
3770:     /**
3771:      * Logical Left Shift
3772:      *
3773:      * Shifts binary strings $shift bits, essentially multiplying by 2**$shift.
3774:      *
3775:      * @param $x String
3776:      * @param $shift Integer
3777:      * @return String
3778:      * @access private
3779:      */
3780:     function _base256_lshift(&$x, $shift)
3781:     {
3782:         if ($shift == 0) {
3783:             return;
3784:         }
3785: 
3786:         $num_bytes = $shift >> 3; // eg. floor($shift/8)
3787:         $shift &= 7; // eg. $shift % 8
3788: 
3789:         $carry = 0;
3790:         for ($i = strlen($x) - 1; $i >= 0; --$i) {
3791:             $temp = ord($x[$i]) << $shift | $carry;
3792:             $x[$i] = chr($temp);
3793:             $carry = $temp >> 8;
3794:         }
3795:         $carry = ($carry != 0) ? chr($carry) : '';
3796:         $x = $carry . $x . str_repeat(chr(0), $num_bytes);
3797:     }
3798: 
3799:     /**
3800:      * Logical Right Shift
3801:      *
3802:      * Shifts binary strings $shift bits, essentially dividing by 2**$shift and returning the remainder.
3803:      *
3804:      * @param $x String
3805:      * @param $shift Integer
3806:      * @return String
3807:      * @access private
3808:      */
3809:     function _base256_rshift(&$x, $shift)
3810:     {
3811:         if ($shift == 0) {
3812:             $x = ltrim($x, chr(0));
3813:             return '';
3814:         }
3815: 
3816:         $num_bytes = $shift >> 3; // eg. floor($shift/8)
3817:         $shift &= 7; // eg. $shift % 8
3818: 
3819:         $remainder = '';
3820:         if ($num_bytes) {
3821:             $start = $num_bytes > strlen($x) ? -strlen($x) : -$num_bytes;
3822:             $remainder = substr($x, $start);
3823:             $x = substr($x, 0, -$num_bytes);
3824:         }
3825: 
3826:         $carry = 0;
3827:         $carry_shift = 8 - $shift;
3828:         for ($i = 0; $i < strlen($x); ++$i) {
3829:             $temp = (ord($x[$i]) >> $shift) | $carry;
3830:             $carry = (ord($x[$i]) << $carry_shift) & 0xFF;
3831:             $x[$i] = chr($temp);
3832:         }
3833:         $x = ltrim($x, chr(0));
3834: 
3835:         $remainder = chr($carry >> $carry_shift) . $remainder;
3836: 
3837:         return ltrim($remainder, chr(0));
3838:     }
3839: 
3840:     // one quirk about how the following functions are implemented is that PHP defines N to be an unsigned long
3841:     // at 32-bits, while java's longs are 64-bits.
3842: 
3843:     /**
3844:      * Converts 32-bit integers to bytes.
3845:      *
3846:      * @param Integer $x
3847:      * @return String
3848:      * @access private
3849:      */
3850:     function _int2bytes($x)
3851:     {
3852:         return ltrim(pack('N', $x), chr(0));
3853:     }
3854: 
3855:     /**
3856:      * Converts bytes to 32-bit integers
3857:      *
3858:      * @param String $x
3859:      * @return Integer
3860:      * @access private
3861:      */
3862:     function _bytes2int($x)
3863:     {
3864:         $temp = unpack('Nint', str_pad($x, 4, chr(0), STR_PAD_LEFT));
3865:         return $temp['int'];
3866:     }
3867: 
3868:     /**
3869:      * DER-encode an integer
3870:      *
3871:      * The ability to DER-encode integers is needed to create RSA public keys for use with OpenSSL
3872:      *
3873:      * @see modPow()
3874:      * @access private
3875:      * @param Integer $length
3876:      * @return String
3877:      */
3878:     function _encodeASN1Length($length)
3879:     {
3880:         if ($length <= 0x7F) {
3881:             return chr($length);
3882:         }
3883: 
3884:         $temp = ltrim(pack('N', $length), chr(0));
3885:         return pack('Ca*', 0x80 | strlen($temp), $temp);
3886:     }
3887: }
3888: