* @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 * * @package PHP * @author Jim Wigginton * @access public */ 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 $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 ]; } }