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<?php
/**
* PHP Modular Exponentiation Engine
*
* PHP version 5 and 7
*
* @category Math
* @package BigInteger
* @author Jim Wigginton <terrafrost@php.net>
* @copyright 2017 Jim Wigginton
* @license http://www.opensource.org/licenses/mit-license.html MIT License
* @link http://pear.php.net/package/Math_BigInteger
*/
namespace phpseclib3\Math\BigInteger\Engines\PHP;
use phpseclib3\Math\BigInteger\Engines\PHP;
/**
* PHP Modular Exponentiation Engine
*
* @package PHP
* @author Jim Wigginton <terrafrost@php.net>
* @access public
*/
abstract class Base extends PHP
{
/**
* Cache constants
*
* $cache[self::VARIABLE] tells us whether or not the cached data is still valid.
*
* @access private
*/
const VARIABLE = 0;
/**
* $cache[self::DATA] contains the cached data.
*
* @access private
*/
const DATA = 1;
/**
* Test for engine validity
*
* @return bool
*/
public static function isValidEngine()
{
return static::class != __CLASS__;
}
/**
* Performs modular exponentiation.
*
* The most naive approach to modular exponentiation has very unreasonable requirements, and
* and although the approach involving repeated squaring does vastly better, it, too, is impractical
* for our purposes. The reason being that division - by far the most complicated and time-consuming
* of the basic operations (eg. +,-,*,/) - occurs multiple times within it.
*
* Modular reductions resolve this issue. Although an individual modular reduction takes more time
* then an individual division, when performed in succession (with the same modulo), they're a lot faster.
*
* The two most commonly used modular reductions are Barrett and Montgomery reduction. Montgomery reduction,
* although faster, only works when the gcd of the modulo and of the base being used is 1. In RSA, when the
* base is a power of two, the modulo - a product of two primes - is always going to have a gcd of 1 (because
* the product of two odd numbers is odd), but what about when RSA isn't used?
*
* In contrast, Barrett reduction has no such constraint. As such, some bigint implementations perform a
* Barrett reduction after every operation in the modpow function. Others perform Barrett reductions when the
* modulo is even and Montgomery reductions when the modulo is odd. BigInteger.java's modPow method, however,
* uses a trick involving the Chinese Remainder Theorem to factor the even modulo into two numbers - one odd and
* the other, a power of two - and recombine them, later. This is the method that this modPow function uses.
* {@link http://islab.oregonstate.edu/papers/j34monex.pdf Montgomery Reduction with Even Modulus} elaborates.
*
* @param \phpseclib3\Math\BigInteger\Engines\PHP $x
* @param \phpseclib3\Math\BigInteger\Engines\PHP $e
* @param \phpseclib3\Math\BigInteger\Engines\PHP $n
* @param string $class
* @return \phpseclib3\Math\BigInteger\Engines\PHP
*/
protected static function powModHelper(PHP $x, PHP $e, PHP $n, $class)
{
if (empty($e->value)) {
$temp = new $class();
$temp->value = [1];
return $x->normalize($temp);
}
if ($e->value == [1]) {
list(, $temp) = $x->divide($n);
return $x->normalize($temp);
}
if ($e->value == [2]) {
$temp = new $class;
$temp->value = $class::square($x->value);
list(, $temp) = $temp->divide($n);
return $x->normalize($temp);
}
return $x->normalize(static::slidingWindow($x, $e, $n, $class));
}
/**
* Modular reduction preparation
*
* @param array $x
* @param array $n
* @param string $class
* @see self::slidingWindow()
* @return array
*/
protected static function prepareReduce(array $x, array $n, $class)
{
return static::reduce($x, $n, $class);
}
/**
* Modular multiply
*
* @param array $x
* @param array $y
* @param array $n
* @param string $class
* @see self::slidingWindow()
* @return array
*/
protected static function multiplyReduce(array $x, array $y, array $n, $class)
{
$temp = $class::multiplyHelper($x, false, $y, false);
return static::reduce($temp[self::VALUE], $n, $class);
}
/**
* Modular square
*
* @param array $x
* @param array $n
* @param string $class
* @see self::slidingWindow()
* @return array
*/
protected static function squareReduce(array $x, array $n, $class)
{
return static::reduce($class::square($x), $n, $class);
}
}
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