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   1  <?php
   2  
   3  /**
   4   * PHP Montgomery Modular Exponentiation Engine
   5   *
   6   * PHP version 5 and 7
   7   *
   8   * @author    Jim Wigginton <terrafrost@php.net>
   9   * @copyright 2017 Jim Wigginton
  10   * @license   http://www.opensource.org/licenses/mit-license.html  MIT License
  11   * @link      http://pear.php.net/package/Math_BigInteger
  12   */
  13  
  14  namespace phpseclib3\Math\BigInteger\Engines\PHP\Reductions;
  15  
  16  use phpseclib3\Math\BigInteger\Engines\PHP\Montgomery as Progenitor;
  17  
  18  /**
  19   * PHP Montgomery Modular Exponentiation Engine
  20   *
  21   * @author  Jim Wigginton <terrafrost@php.net>
  22   */
  23  abstract class Montgomery extends Progenitor
  24  {
  25      /**
  26       * Prepare a number for use in Montgomery Modular Reductions
  27       *
  28       * @param array $x
  29       * @param array $n
  30       * @param string $class
  31       * @return array
  32       */
  33      protected static function prepareReduce(array $x, array $n, $class)
  34      {
  35          $lhs = new $class();
  36          $lhs->value = array_merge(self::array_repeat(0, count($n)), $x);
  37          $rhs = new $class();
  38          $rhs->value = $n;
  39  
  40          list(, $temp) = $lhs->divide($rhs);
  41          return $temp->value;
  42      }
  43  
  44      /**
  45       * Montgomery Multiply
  46       *
  47       * Interleaves the montgomery reduction and long multiplication algorithms together as described in
  48       * {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=13 HAC 14.36}
  49       *
  50       * @param array $x
  51       * @param array $n
  52       * @param string $class
  53       * @return array
  54       */
  55      protected static function reduce(array $x, array $n, $class)
  56      {
  57          static $cache = [
  58              self::VARIABLE => [],
  59              self::DATA => []
  60          ];
  61  
  62          if (($key = array_search($n, $cache[self::VARIABLE])) === false) {
  63              $key = count($cache[self::VARIABLE]);
  64              $cache[self::VARIABLE][] = $x;
  65              $cache[self::DATA][] = self::modInverse67108864($n, $class);
  66          }
  67  
  68          $k = count($n);
  69  
  70          $result = [self::VALUE => $x];
  71  
  72          for ($i = 0; $i < $k; ++$i) {
  73              $temp = $result[self::VALUE][$i] * $cache[self::DATA][$key];
  74              $temp = $temp - $class::BASE_FULL * ($class::BASE === 26 ? intval($temp / 0x4000000) : ($temp >> 31));
  75              $temp = $class::regularMultiply([$temp], $n);
  76              $temp = array_merge(self::array_repeat(0, $i), $temp);
  77              $result = $class::addHelper($result[self::VALUE], false, $temp, false);
  78          }
  79  
  80          $result[self::VALUE] = array_slice($result[self::VALUE], $k);
  81  
  82          if (self::compareHelper($result, false, $n, false) >= 0) {
  83              $result = $class::subtractHelper($result[self::VALUE], false, $n, false);
  84          }
  85  
  86          return $result[self::VALUE];
  87      }
  88  
  89      /**
  90       * Modular Inverse of a number mod 2**26 (eg. 67108864)
  91       *
  92       * Based off of the bnpInvDigit function implemented and justified in the following URL:
  93       *
  94       * {@link http://www-cs-students.stanford.edu/~tjw/jsbn/jsbn.js}
  95       *
  96       * The following URL provides more info:
  97       *
  98       * {@link http://groups.google.com/group/sci.crypt/msg/7a137205c1be7d85}
  99       *
 100       * As for why we do all the bitmasking...  strange things can happen when converting from floats to ints. For
 101       * instance, on some computers, var_dump((int) -4294967297) yields int(-1) and on others, it yields
 102       * int(-2147483648).  To avoid problems stemming from this, we use bitmasks to guarantee that ints aren't
 103       * auto-converted to floats.  The outermost bitmask is present because without it, there's no guarantee that
 104       * the "residue" returned would be the so-called "common residue".  We use fmod, in the last step, because the
 105       * maximum possible $x is 26 bits and the maximum $result is 16 bits.  Thus, we have to be able to handle up to
 106       * 40 bits, which only 64-bit floating points will support.
 107       *
 108       * Thanks to Pedro Gimeno Fortea for input!
 109       *
 110       * @param array $x
 111       * @param string $class
 112       * @return int
 113       */
 114      protected static function modInverse67108864(array $x, $class) // 2**26 == 67,108,864
 115      {
 116          $x = -$x[0];
 117          $result = $x & 0x3; // x**-1 mod 2**2
 118          $result = ($result * (2 - $x * $result)) & 0xF; // x**-1 mod 2**4
 119          $result = ($result * (2 - ($x & 0xFF) * $result))  & 0xFF; // x**-1 mod 2**8
 120          $result = ($result * ((2 - ($x & 0xFFFF) * $result) & 0xFFFF)) & 0xFFFF; // x**-1 mod 2**16
 121          $result = $class::BASE == 26 ?
 122              fmod($result * (2 - fmod($x * $result, $class::BASE_FULL)), $class::BASE_FULL) : // x**-1 mod 2**26
 123              ($result * (2 - ($x * $result) % $class::BASE_FULL)) % $class::BASE_FULL;
 124          return $result & $class::MAX_DIGIT;
 125      }
 126  }