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+<?php
+
+/**
+ * PHP Barrett Modular Exponentiation Engine
+ *
+ * PHP version 5 and 7
+ *
+ * @author Jim Wigginton <[email protected]>
+ * @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 <[email protected]>
+ */
+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
+ ];
+ }
+}
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