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<?php
/**
* PHP Barrett Modular Exponentiation Engine
*
* PHP version 5 and 7
*
* @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\Reductions;
use phpseclib3\Math\BigInteger\Engines\PHP;
use phpseclib3\Math\BigInteger\Engines\PHP\Base;
/**
* PHP Barrett Modular Exponentiation Engine
*
* @author Jim Wigginton <terrafrost@php.net>
*/
abstract class Barrett extends Base
{
/**
* Barrett Modular Reduction
*
* See {@link http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf#page=14 HAC 14.3.3} /
* {@link http://math.libtomcrypt.com/files/tommath.pdf#page=165 MPM 6.2.5} for more information. Modified slightly,
* so as not to require negative numbers (initially, this script didn't support negative numbers).
*
* Employs "folding", as described at
* {@link http://www.cosic.esat.kuleuven.be/publications/thesis-149.pdf#page=66 thesis-149.pdf#page=66}. To quote from
* it, "the idea [behind folding] is to find a value x' such that x (mod m) = x' (mod m), with x' being smaller than x."
*
* Unfortunately, the "Barrett Reduction with Folding" algorithm described in thesis-149.pdf is not, as written, all that
* usable on account of (1) its not using reasonable radix points as discussed in
* {@link http://math.libtomcrypt.com/files/tommath.pdf#page=162 MPM 6.2.2} and (2) the fact that, even with reasonable
* radix points, it only works when there are an even number of digits in the denominator. The reason for (2) is that
* (x >> 1) + (x >> 1) != x / 2 + x / 2. If x is even, they're the same, but if x is odd, they're not. See the in-line
* comments for details.
*
* @param array $n
* @param array $m
* @param class-string<PHP> $class
* @return array
*/
protected static function reduce(array $n, array $m, $class)
{
static $cache = [
self::VARIABLE => [],
self::DATA => []
];
$m_length = count($m);
// if (self::compareHelper($n, $static::square($m)) >= 0) {
if (count($n) > 2 * $m_length) {
$lhs = new $class();
$rhs = new $class();
$lhs->value = $n;
$rhs->value = $m;
list(, $temp) = $lhs->divide($rhs);
return $temp->value;
}
// if (m.length >> 1) + 2 <= m.length then m is too small and n can't be reduced
if ($m_length < 5) {
return self::regularBarrett($n, $m, $class);
}
// n = 2 * m.length
if (($key = array_search($m, $cache[self::VARIABLE])) === false) {
$key = count($cache[self::VARIABLE]);
$cache[self::VARIABLE][] = $m;
$lhs = new $class();
$lhs_value = &$lhs->value;
$lhs_value = self::array_repeat(0, $m_length + ($m_length >> 1));
$lhs_value[] = 1;
$rhs = new $class();
$rhs->value = $m;
list($u, $m1) = $lhs->divide($rhs);
$u = $u->value;
$m1 = $m1->value;
$cache[self::DATA][] = [
'u' => $u, // m.length >> 1 (technically (m.length >> 1) + 1)
'm1' => $m1 // m.length
];
} else {
extract($cache[self::DATA][$key]);
}
$cutoff = $m_length + ($m_length >> 1);
$lsd = array_slice($n, 0, $cutoff); // m.length + (m.length >> 1)
$msd = array_slice($n, $cutoff); // m.length >> 1
$lsd = self::trim($lsd);
$temp = $class::multiplyHelper($msd, false, $m1, false); // m.length + (m.length >> 1)
$n = $class::addHelper($lsd, false, $temp[self::VALUE], false); // m.length + (m.length >> 1) + 1 (so basically we're adding two same length numbers)
//if ($m_length & 1) {
// return self::regularBarrett($n[self::VALUE], $m, $class);
//}
// (m.length + (m.length >> 1) + 1) - (m.length - 1) == (m.length >> 1) + 2
$temp = array_slice($n[self::VALUE], $m_length - 1);
// if even: ((m.length >> 1) + 2) + (m.length >> 1) == m.length + 2
// if odd: ((m.length >> 1) + 2) + (m.length >> 1) == (m.length - 1) + 2 == m.length + 1
$temp = $class::multiplyHelper($temp, false, $u, false);
// if even: (m.length + 2) - ((m.length >> 1) + 1) = m.length - (m.length >> 1) + 1
// if odd: (m.length + 1) - ((m.length >> 1) + 1) = m.length - (m.length >> 1)
$temp = array_slice($temp[self::VALUE], ($m_length >> 1) + 1);
// if even: (m.length - (m.length >> 1) + 1) + m.length = 2 * m.length - (m.length >> 1) + 1
// if odd: (m.length - (m.length >> 1)) + m.length = 2 * m.length - (m.length >> 1)
$temp = $class::multiplyHelper($temp, false, $m, false);
// at this point, if m had an odd number of digits, we'd be subtracting a 2 * m.length - (m.length >> 1) digit
// number from a m.length + (m.length >> 1) + 1 digit number. ie. there'd be an extra digit and the while loop
// following this comment would loop a lot (hence our calling _regularBarrett() in that situation).
$result = $class::subtractHelper($n[self::VALUE], false, $temp[self::VALUE], false);
while (self::compareHelper($result[self::VALUE], $result[self::SIGN], $m, false) >= 0) {
$result = $class::subtractHelper($result[self::VALUE], $result[self::SIGN], $m, false);
}
return $result[self::VALUE];
}
/**
* (Regular) Barrett Modular Reduction
*
* For numbers with more than four digits BigInteger::_barrett() is faster. The difference between that and this
* is that this function does not fold the denominator into a smaller form.
*
* @param array $x
* @param array $n
* @param string $class
* @return array
*/
private static function regularBarrett(array $x, array $n, $class)
{
static $cache = [
self::VARIABLE => [],
self::DATA => []
];
$n_length = count($n);
if (count($x) > 2 * $n_length) {
$lhs = new $class();
$rhs = new $class();
$lhs->value = $x;
$rhs->value = $n;
list(, $temp) = $lhs->divide($rhs);
return $temp->value;
}
if (($key = array_search($n, $cache[self::VARIABLE])) === false) {
$key = count($cache[self::VARIABLE]);
$cache[self::VARIABLE][] = $n;
$lhs = new $class();
$lhs_value = &$lhs->value;
$lhs_value = self::array_repeat(0, 2 * $n_length);
$lhs_value[] = 1;
$rhs = new $class();
$rhs->value = $n;
list($temp, ) = $lhs->divide($rhs); // m.length
$cache[self::DATA][] = $temp->value;
}
// 2 * m.length - (m.length - 1) = m.length + 1
$temp = array_slice($x, $n_length - 1);
// (m.length + 1) + m.length = 2 * m.length + 1
$temp = $class::multiplyHelper($temp, false, $cache[self::DATA][$key], false);
// (2 * m.length + 1) - (m.length - 1) = m.length + 2
$temp = array_slice($temp[self::VALUE], $n_length + 1);
// m.length + 1
$result = array_slice($x, 0, $n_length + 1);
// m.length + 1
$temp = self::multiplyLower($temp, false, $n, false, $n_length + 1, $class);
// $temp == array_slice($class::regularMultiply($temp, false, $n, false)->value, 0, $n_length + 1)
if (self::compareHelper($result, false, $temp[self::VALUE], $temp[self::SIGN]) < 0) {
$corrector_value = self::array_repeat(0, $n_length + 1);
$corrector_value[count($corrector_value)] = 1;
$result = $class::addHelper($result, false, $corrector_value, false);
$result = $result[self::VALUE];
}
// at this point, we're subtracting a number with m.length + 1 digits from another number with m.length + 1 digits
$result = $class::subtractHelper($result, false, $temp[self::VALUE], $temp[self::SIGN]);
while (self::compareHelper($result[self::VALUE], $result[self::SIGN], $n, false) > 0) {
$result = $class::subtractHelper($result[self::VALUE], $result[self::SIGN], $n, false);
}
return $result[self::VALUE];
}
/**
* Performs long multiplication up to $stop digits
*
* If you're going to be doing array_slice($product->value, 0, $stop), some cycles can be saved.
*
* @see self::regularBarrett()
* @param array $x_value
* @param bool $x_negative
* @param array $y_value
* @param bool $y_negative
* @param int $stop
* @param string $class
* @return array
*/
private static function multiplyLower(array $x_value, $x_negative, array $y_value, $y_negative, $stop, $class)
{
$x_length = count($x_value);
$y_length = count($y_value);
if (!$x_length || !$y_length) { // a 0 is being multiplied
return [
self::VALUE => [],
self::SIGN => false
];
}
if ($x_length < $y_length) {
$temp = $x_value;
$x_value = $y_value;
$y_value = $temp;
$x_length = count($x_value);
$y_length = count($y_value);
}
$product_value = self::array_repeat(0, $x_length + $y_length);
// the following for loop could be removed if the for loop following it
// (the one with nested for loops) initially set $i to 0, but
// doing so would also make the result in one set of unnecessary adds,
// since on the outermost loops first pass, $product->value[$k] is going
// to always be 0
$carry = 0;
for ($j = 0; $j < $x_length; ++$j) { // ie. $i = 0, $k = $i
$temp = $x_value[$j] * $y_value[0] + $carry; // $product_value[$k] == 0
$carry = $class::BASE === 26 ? intval($temp / 0x4000000) : ($temp >> 31);
$product_value[$j] = (int) ($temp - $class::BASE_FULL * $carry);
}
if ($j < $stop) {
$product_value[$j] = $carry;
}
// the above for loop is what the previous comment was talking about. the
// following for loop is the "one with nested for loops"
for ($i = 1; $i < $y_length; ++$i) {
$carry = 0;
for ($j = 0, $k = $i; $j < $x_length && $k < $stop; ++$j, ++$k) {
$temp = $product_value[$k] + $x_value[$j] * $y_value[$i] + $carry;
$carry = $class::BASE === 26 ? intval($temp / 0x4000000) : ($temp >> 31);
$product_value[$k] = (int) ($temp - $class::BASE_FULL * $carry);
}
if ($k < $stop) {
$product_value[$k] = $carry;
}
}
return [
self::VALUE => self::trim($product_value),
self::SIGN => $x_negative != $y_negative
];
}
}