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+<?php
+
+/**
+ * BCMath 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\BCMath\Reductions;
+
+use phpseclib3\Math\BigInteger\Engines\BCMath\Base;
+
+/**
+ * PHP Barrett Modular Exponentiation Engine
+ *
+ * @author Jim Wigginton <[email protected]>
+ */
+abstract class Barrett extends Base
+{
+ /**
+ * Cache constants
+ *
+ * $cache[self::VARIABLE] tells us whether or not the cached data is still valid.
+ *
+ */
+ const VARIABLE = 0;
+ /**
+ * $cache[self::DATA] contains the cached data.
+ *
+ */
+ const DATA = 1;
+
+ /**
+ * 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 string $n
+ * @param string $m
+ * @return string
+ */
+ protected static function reduce($n, $m)
+ {
+ static $cache = [
+ self::VARIABLE => [],
+ self::DATA => []
+ ];
+
+ $m_length = strlen($m);
+
+ if (strlen($n) > 2 * $m_length) {
+ return bcmod($n, $m);
+ }
+
+ // if (m.length >> 1) + 2 <= m.length then m is too small and n can't be reduced
+ if ($m_length < 5) {
+ return self::regularBarrett($n, $m);
+ }
+ // n = 2 * m.length
+
+ if (($key = array_search($m, $cache[self::VARIABLE])) === false) {
+ $key = count($cache[self::VARIABLE]);
+ $cache[self::VARIABLE][] = $m;
+
+ $lhs = '1' . str_repeat('0', $m_length + ($m_length >> 1));
+ $u = bcdiv($lhs, $m, 0);
+ $m1 = bcsub($lhs, bcmul($u, $m));
+
+ $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);
+
+ $lsd = substr($n, -$cutoff);
+ $msd = substr($n, 0, -$cutoff);
+
+ $temp = bcmul($msd, $m1); // m.length + (m.length >> 1)
+ $n = bcadd($lsd, $temp); // m.length + (m.length >> 1) + 1 (so basically we're adding two same length numbers)
+ //if ($m_length & 1) {
+ // return self::regularBarrett($n, $m);
+ //}
+
+ // (m.length + (m.length >> 1) + 1) - (m.length - 1) == (m.length >> 1) + 2
+ $temp = substr($n, 0, -$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
+ $temp = bcmul($temp, $u);
+ // 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)
+ $temp = substr($temp, 0, -($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)
+ $temp = bcmul($temp, $m);
+
+ // 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).
+
+ $result = bcsub($n, $temp);
+
+ //if (bccomp($result, '0') < 0) {
+ if ($result[0] == '-') {
+ $temp = '1' . str_repeat('0', $m_length + 1);
+ $result = bcadd($result, $temp);
+ }
+
+ while (bccomp($result, $m) >= 0) {
+ $result = bcsub($result, $m);
+ }
+
+ return $result;
+ }
+
+ /**
+ * (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 string $x
+ * @param string $n
+ * @return string
+ */
+ private static function regularBarrett($x, $n)
+ {
+ static $cache = [
+ self::VARIABLE => [],
+ self::DATA => []
+ ];
+
+ $n_length = strlen($n);
+
+ if (strlen($x) > 2 * $n_length) {
+ return bcmod($x, $n);
+ }
+
+ if (($key = array_search($n, $cache[self::VARIABLE])) === false) {
+ $key = count($cache[self::VARIABLE]);
+ $cache[self::VARIABLE][] = $n;
+ $lhs = '1' . str_repeat('0', 2 * $n_length);
+ $cache[self::DATA][] = bcdiv($lhs, $n, 0);
+ }
+
+ $temp = substr($x, 0, -$n_length + 1);
+ $temp = bcmul($temp, $cache[self::DATA][$key]);
+ $temp = substr($temp, 0, -$n_length - 1);
+
+ $r1 = substr($x, -$n_length - 1);
+ $r2 = substr(bcmul($temp, $n), -$n_length - 1);
+ $result = bcsub($r1, $r2);
+
+ //if (bccomp($result, '0') < 0) {
+ if ($result[0] == '-') {
+ $q = '1' . str_repeat('0', $n_length + 1);
+ $result = bcadd($result, $q);
+ }
+
+ while (bccomp($result, $n) >= 0) {
+ $result = bcsub($result, $n);
+ }
+
+ return $result;
+ }
+}