PHP Math_BigInteger::modPow方法代码示例

 /**
  * Raise an arbitrary precision number to another, reduced by a specified modulus
  *
  * @param string $base The left operand, as a string.
  * @param string $exp The right operand, as a string.
  * @param string $mod The modulus, as a string.
  * @access public
  * @return string|null Returns the result as a string, or <b>NULL</b> if modulus is 0.
  */
 public function powmod($base, $exp, $mod)
 {
     //FIXME
     $base = new Math_BigInteger($base);
     $exp = new Math_BigInteger($exp);
     $mod = new Math_BigInteger($mod);
     $mod = $base->modPow($exp, $mod);
     return $mod->toString();
 }

//.........这里部分代码省略.........
         return $this->_disconnect(NET_SSH2_DISCONNECT_KEY_EXCHANGE_FAILED);
     }
     switch ($kex_algorithms[$i]) {
         // see http://tools.ietf.org/html/rfc2409#section-6.2 and
         // http://tools.ietf.org/html/rfc2412, appendex E
         case 'diffie-hellman-group1-sha1':
             $prime = 'FFFFFFFFFFFFFFFFC90FDAA22168C234C4C6628B80DC1CD129024E088A67CC74' . '020BBEA63B139B22514A08798E3404DDEF9519B3CD3A431B302B0A6DF25F1437' . '4FE1356D6D51C245E485B576625E7EC6F44C42E9A637ED6B0BFF5CB6F406B7ED' . 'EE386BFB5A899FA5AE9F24117C4B1FE649286651ECE65381FFFFFFFFFFFFFFFF';
             break;
             // see http://tools.ietf.org/html/rfc3526#section-3
         // see http://tools.ietf.org/html/rfc3526#section-3
         case 'diffie-hellman-group14-sha1':
             $prime = 'FFFFFFFFFFFFFFFFC90FDAA22168C234C4C6628B80DC1CD129024E088A67CC74' . '020BBEA63B139B22514A08798E3404DDEF9519B3CD3A431B302B0A6DF25F1437' . '4FE1356D6D51C245E485B576625E7EC6F44C42E9A637ED6B0BFF5CB6F406B7ED' . 'EE386BFB5A899FA5AE9F24117C4B1FE649286651ECE45B3DC2007CB8A163BF05' . '98DA48361C55D39A69163FA8FD24CF5F83655D23DCA3AD961C62F356208552BB' . '9ED529077096966D670C354E4ABC9804F1746C08CA18217C32905E462E36CE3B' . 'E39E772C180E86039B2783A2EC07A28FB5C55DF06F4C52C9DE2BCBF695581718' . '3995497CEA956AE515D2261898FA051015728E5A8AACAA68FFFFFFFFFFFFFFFF';
             break;
     }
     // For both diffie-hellman-group1-sha1 and diffie-hellman-group14-sha1
     // the generator field element is 2 (decimal) and the hash function is sha1.
     $g = new Math_BigInteger(2);
     $prime = new Math_BigInteger($prime, 16);
     $kexHash = new Crypt_Hash('sha1');
     //$q = $p->bitwise_rightShift(1);
     /* To increase the speed of the key exchange, both client and server may
                reduce the size of their private exponents.  It should be at least
                twice as long as the key material that is generated from the shared
                secret.  For more details, see the paper by van Oorschot and Wiener
                [VAN-OORSCHOT].
     
                -- http://tools.ietf.org/html/rfc4419#section-6.2 */
     $one = new Math_BigInteger(1);
     $keyLength = min($keyLength, $kexHash->getLength());
     $max = $one->bitwise_leftShift(16 * $keyLength);
     // 2 * 8 * $keyLength
     $max = $max->subtract($one);
     $x = $one->random($one, $max);
     $e = $g->modPow($x, $prime);
     $eBytes = $e->toBytes(true);
     $data = pack('CNa*', NET_SSH2_MSG_KEXDH_INIT, strlen($eBytes), $eBytes);
     if (!$this->_send_binary_packet($data)) {
         user_error('Connection closed by server');
         return false;
     }
     $response = $this->_get_binary_packet();
     if ($response === false) {
         user_error('Connection closed by server');
         return false;
     }
     extract(unpack('Ctype', $this->_string_shift($response, 1)));
     if ($type != NET_SSH2_MSG_KEXDH_REPLY) {
         user_error('Expected SSH_MSG_KEXDH_REPLY');
         return false;
     }
     $temp = unpack('Nlength', $this->_string_shift($response, 4));
     $this->server_public_host_key = $server_public_host_key = $this->_string_shift($response, $temp['length']);
     $temp = unpack('Nlength', $this->_string_shift($server_public_host_key, 4));
     $public_key_format = $this->_string_shift($server_public_host_key, $temp['length']);
     $temp = unpack('Nlength', $this->_string_shift($response, 4));
     $fBytes = $this->_string_shift($response, $temp['length']);
     $f = new Math_BigInteger($fBytes, -256);
     $temp = unpack('Nlength', $this->_string_shift($response, 4));
     $this->signature = $this->_string_shift($response, $temp['length']);
     $temp = unpack('Nlength', $this->_string_shift($this->signature, 4));
     $this->signature_format = $this->_string_shift($this->signature, $temp['length']);
     $key = $f->modPow($x, $prime);
     $keyBytes = $key->toBytes(true);
     $this->exchange_hash = pack('Na*Na*Na*Na*Na*Na*Na*Na*', strlen($this->identifier), $this->identifier, strlen($this->server_identifier), $this->server_identifier, strlen($kexinit_payload_client), $kexinit_payload_client, strlen($kexinit_payload_server), $kexinit_payload_server, strlen($this->server_public_host_key), $this->server_public_host_key, strlen($eBytes), $eBytes, strlen($fBytes), $fBytes, strlen($keyBytes), $keyBytes);
     $this->exchange_hash = $kexHash->hash($this->exchange_hash);
     if ($this->session_id === false) {

 /**
  * Performs RSA Blinding
  *
  * Protects against timing attacks by employing RSA Blinding.
  * Returns $x->modPow($this->exponents[$i], $this->primes[$i])
  *
  * @access private
  * @param Math_BigInteger $x
  * @param Math_BigInteger $r
  * @param Integer $i
  * @return Math_BigInteger
  */
 function _blind($x, $r, $i)
 {
     $x = $x->multiply($r->modPow($this->publicExponent, $this->primes[$i]));
     $x = $x->modPow($this->exponents[$i], $this->primes[$i]);
     $r = $r->modInverse($this->primes[$i]);
     $x = $x->multiply($r);
     list(, $x) = $x->divide($this->primes[$i]);
     return $x;
 }

 /**
  * DSA verify.
  *
  * @param string $message     Message.
  * @param string $hash_alg    Hash algorithm.
  * @param Math_BigInteger $r  r.
  * @param Math_BigInteger $s  s.
  *
  * @return bool  True if verified.
  */
 public function verify($message, $hash_alg, $r, $s)
 {
     $hash = new Crypt_Hash($hash_alg);
     $hash_m = new Math_BigInteger($hash->hash($message), 256);
     $g = new Math_BigInteger($this->_key->key['g'], 256);
     $p = new Math_BigInteger($this->_key->key['p'], 256);
     $q = new Math_BigInteger($this->_key->key['q'], 256);
     $y = new Math_BigInteger($this->_key->key['y'], 256);
     $w = $s->modInverse($q);
     $hash_m_mul = $hash_m->multiply($w);
     $u1_base = $hash_m_mul->divide($q);
     $u1 = $u1_base[1];
     $r_mul = $r->multiply($w);
     $u2_base = $r_mul->divide($q);
     $u2 = $u2_base[1];
     $g_pow = $g->modPow($u1, $p);
     $y_pow = $y->modPow($u2, $p);
     $g_pow_mul = $g_pow->multiply($y_pow);
     $g_pow_mul_mod_base = $g_pow_mul->divide($p);
     $g_pow_mul_mod = $g_pow_mul_mod_base[1];
     $v_base = $g_pow_mul_mod->divide($q);
     $v = $v_base[1];
     return $v->compare($r) == 0;
 }

$result = bcsub($result, $y);
$_result = $_result->subtract($_y);
echo "\$result = \$result-\$y;\r\n";
echo "{$result}\r\n";
echo $_result->toString();
echo "\r\n\r\n";
$result = bcdiv($x, $y);
list($_result, ) = $_x->divide($_y);
echo "\$result = \$x/\$y;\r\n";
echo "{$result}\r\n";
echo $_result->toString();
echo "\r\n\r\n";
$result = bcmod($y, $z);
list(, $_result) = $_y->divide($_z);
echo "\$result = \$x%\$y;\r\n";
echo "{$result}\r\n";
echo $_result->toString();
echo "\r\n\r\n";
$result = bcmul($x, $z);
$_result = $_x->multiply($_z);
echo "\$result = \$x*\$z;\r\n";
echo "{$result}\r\n";
echo $_result->toString();
echo "\r\n\r\n";
$result = bcpowmod($x, $y, $result);
$_result = $_x->modPow($_y, $_result);
echo "\$result = (\$x**\$y)%\$result;\r\n";
echo "{$result}\r\n";
echo $_result->toString();
echo "\r\n\r\n";
// modInverse isn't demo'd because no equivalent to it exists in BCMath.

//.........这里部分代码省略.........
         return $this->_disconnect(NET_SSH2_DISCONNECT_KEY_EXCHANGE_FAILED);
     }
     switch ($kex_algorithms[$i]) {
         // see http://tools.ietf.org/html/rfc2409#section-6.2 and
         // http://tools.ietf.org/html/rfc2412, appendex E
         case 'diffie-hellman-group1-sha1':
             $p = pack('H256', 'FFFFFFFFFFFFFFFFC90FDAA22168C234C4C6628B80DC1CD129024E088A67CC74' . '020BBEA63B139B22514A08798E3404DDEF9519B3CD3A431B302B0A6DF25F1437' . '4FE1356D6D51C245E485B576625E7EC6F44C42E9A637ED6B0BFF5CB6F406B7ED' . 'EE386BFB5A899FA5AE9F24117C4B1FE649286651ECE65381FFFFFFFFFFFFFFFF');
             $keyLength = $keyLength < 160 ? $keyLength : 160;
             $hash = 'sha1';
             break;
             // see http://tools.ietf.org/html/rfc3526#section-3
         // see http://tools.ietf.org/html/rfc3526#section-3
         case 'diffie-hellman-group14-sha1':
             $p = pack('H512', 'FFFFFFFFFFFFFFFFC90FDAA22168C234C4C6628B80DC1CD129024E088A67CC74' . '020BBEA63B139B22514A08798E3404DDEF9519B3CD3A431B302B0A6DF25F1437' . '4FE1356D6D51C245E485B576625E7EC6F44C42E9A637ED6B0BFF5CB6F406B7ED' . 'EE386BFB5A899FA5AE9F24117C4B1FE649286651ECE45B3DC2007CB8A163BF05' . '98DA48361C55D39A69163FA8FD24CF5F83655D23DCA3AD961C62F356208552BB' . '9ED529077096966D670C354E4ABC9804F1746C08CA18217C32905E462E36CE3B' . 'E39E772C180E86039B2783A2EC07A28FB5C55DF06F4C52C9DE2BCBF695581718' . '3995497CEA956AE515D2261898FA051015728E5A8AACAA68FFFFFFFFFFFFFFFF');
             $keyLength = $keyLength < 160 ? $keyLength : 160;
             $hash = 'sha1';
     }
     $p = new Math_BigInteger($p, 256);
     //$q = $p->bitwise_rightShift(1);
     /* To increase the speed of the key exchange, both client and server may
                reduce the size of their private exponents.  It should be at least
                twice as long as the key material that is generated from the shared
                secret.  For more details, see the paper by van Oorschot and Wiener
                [VAN-OORSCHOT].
     
                -- http://tools.ietf.org/html/rfc4419#section-6.2 */
     $q = new Math_BigInteger(1);
     $q = $q->bitwise_leftShift(2 * $keyLength);
     $q = $q->subtract(new Math_BigInteger(1));
     $g = new Math_BigInteger(2);
     $x = new Math_BigInteger();
     $x->setRandomGenerator('crypt_random');
     $x = $x->random(new Math_BigInteger(1), $q);
     $e = $g->modPow($x, $p);
     $eBytes = $e->toBytes(true);
     $data = pack('CNa*', NET_SSH2_MSG_KEXDH_INIT, strlen($eBytes), $eBytes);
     if (!$this->_send_binary_packet($data)) {
         user_error('Connection closed by server', E_USER_NOTICE);
         return false;
     }
     $response = $this->_get_binary_packet();
     if ($response === false) {
         user_error('Connection closed by server', E_USER_NOTICE);
         return false;
     }
     extract(unpack('Ctype', $this->_string_shift($response, 1)));
     if ($type != NET_SSH2_MSG_KEXDH_REPLY) {
         user_error('Expected SSH_MSG_KEXDH_REPLY', E_USER_NOTICE);
         return false;
     }
     $temp = unpack('Nlength', $this->_string_shift($response, 4));
     $this->server_public_host_key = $server_public_host_key = $this->_string_shift($response, $temp['length']);
     $temp = unpack('Nlength', $this->_string_shift($server_public_host_key, 4));
     $public_key_format = $this->_string_shift($server_public_host_key, $temp['length']);
     $temp = unpack('Nlength', $this->_string_shift($response, 4));
     $fBytes = $this->_string_shift($response, $temp['length']);
     $f = new Math_BigInteger($fBytes, -256);
     $temp = unpack('Nlength', $this->_string_shift($response, 4));
     $signature = $this->_string_shift($response, $temp['length']);
     $temp = unpack('Nlength', $this->_string_shift($signature, 4));
     $signature_format = $this->_string_shift($signature, $temp['length']);
     $key = $f->modPow($x, $p);
     $keyBytes = $key->toBytes(true);
     $source = pack('Na*Na*Na*Na*Na*Na*Na*Na*', strlen($this->identifier), $this->identifier, strlen($this->server_identifier), $this->server_identifier, strlen($kexinit_payload_client), $kexinit_payload_client, strlen($kexinit_payload_server), $kexinit_payload_server, strlen($this->server_public_host_key), $this->server_public_host_key, strlen($eBytes), $eBytes, strlen($fBytes), $fBytes, strlen($keyBytes), $keyBytes);
     $source = pack('H*', $hash($source));
     if ($this->session_id === false) {

//.........这里部分代码省略.........
         user_error('No compatible key exchange algorithms found', E_USER_NOTICE);
         return $this->_disconnect(NET_SSH2_DISCONNECT_KEY_EXCHANGE_FAILED);
     }
     switch ($kex_algorithms[$i]) {
         // see http://tools.ietf.org/html/rfc2409#section-6.2 and
         // http://tools.ietf.org/html/rfc2412, appendex E
         case 'diffie-hellman-group1-sha1':
             $p = pack('N32', 4294967295.0, 4294967295.0, 3373259426.0, 0x2168c234, 3301335691.0, 2161908945.0, 0x29024e08, 2322058356.0, 0x20bbea6, 0x3b139b22, 0x514a0879, 0.0, 0.0, 3443147547.0, 0x302b0a6d, 4066317367.0, 0x4fe1356d, 0x6d51c245, 0.0, 0x625e7ec6, 0.0, 0.0, 0xbff5cb6, 0.0, 0.0, 0x5a899fa5, 0.0, 0x7c4b1fe6, 0x49286651, 0.0, 4294967295.0, 4294967295.0);
             $keyLength = $keyLength < 160 ? $keyLength : 160;
             $hash = 'sha1';
             break;
             // see http://tools.ietf.org/html/rfc3526#section-3
         // see http://tools.ietf.org/html/rfc3526#section-3
         case 'diffie-hellman-group14-sha1':
             $p = pack('N64', 4294967295.0, 4294967295.0, 3373259426.0, 0x2168c234, 3301335691.0, 2161908945.0, 0x29024e08, 2322058356.0, 0x20bbea6, 0x3b139b22, 0x514a0879, 0.0, 0.0, 3443147547.0, 0x302b0a6d, 4066317367.0, 0x4fe1356d, 0x6d51c245, 0.0, 0x625e7ec6, 0.0, 0.0, 0xbff5cb6, 0.0, 0.0, 0x5a899fa5, 0.0, 0x7c4b1fe6, 0x49286651, 0.0, 3254811832.0, 2707668741.0, 2564442166.0, 0x1c55d39a, 0x69163fa8, 4247048031.0, 2204458275.0, 3701714326.0, 0x1c62f356, 0x208552bb, 0.0, 0x7096966d, 0x670c354e, 0x4abc9804, 4050938888.0, 3390579068.0, 0x32905e46, 0x2e36ce3b, 0.0, 0x180e8603, 2603058082.0, 0.0, 3049610736.0, 0x6f4c52c9, 0.0, 2505578264.0, 0x3995497c, 0.0, 0x15d22618, 2566522128.0, 0x15728e5a, 2326571624.0, 4294967295.0, 4294967295.0);
             $keyLength = $keyLength < 160 ? $keyLength : 160;
             $hash = 'sha1';
     }
     $p = new Math_BigInteger($p, 256);
     //$q = $p->bitwise_rightShift(1);
     /* To increase the speed of the key exchange, both client and server may
                reduce the size of their private exponents.  It should be at least
                twice as long as the key material that is generated from the shared
                secret.  For more details, see the paper by van Oorschot and Wiener
                [VAN-OORSCHOT].
     
                -- http://tools.ietf.org/html/rfc4419#section-6.2 */
     $q = new Math_BigInteger(1);
     $q = $q->bitwise_leftShift(2 * $keyLength);
     $q = $q->subtract(new Math_BigInteger(1));
     $g = new Math_BigInteger(2);
     $x = new Math_BigInteger();
     $x = $x->random(new Math_BigInteger(1), $q, 'crypt_random');
     $e = $g->modPow($x, $p);
     $eBytes = $e->toBytes(true);
     $data = pack('CNa*', NET_SSH2_MSG_KEXDH_INIT, strlen($eBytes), $eBytes);
     if (!$this->_send_binary_packet($data)) {
         user_error('Connection closed by server', E_USER_NOTICE);
         return false;
     }
     $response = $this->_get_binary_packet();
     if ($response === false) {
         user_error('Connection closed by server', E_USER_NOTICE);
         return false;
     }
     list(, $type) = unpack('C', $this->_string_shift($response, 1));
     if ($type != NET_SSH2_MSG_KEXDH_REPLY) {
         user_error('Expected SSH_MSG_KEXDH_REPLY', E_USER_NOTICE);
         return false;
     }
     $temp = unpack('Nlength', $this->_string_shift($response, 4));
     $this->server_public_host_key = $server_public_host_key = $this->_string_shift($response, $temp['length']);
     $temp = unpack('Nlength', $this->_string_shift($server_public_host_key, 4));
     $public_key_format = $this->_string_shift($server_public_host_key, $temp['length']);
     $temp = unpack('Nlength', $this->_string_shift($response, 4));
     $fBytes = $this->_string_shift($response, $temp['length']);
     $f = new Math_BigInteger($fBytes, -256);
     $temp = unpack('Nlength', $this->_string_shift($response, 4));
     $signature = $this->_string_shift($response, $temp['length']);
     $temp = unpack('Nlength', $this->_string_shift($signature, 4));
     $signature_format = $this->_string_shift($signature, $temp['length']);
     $key = $f->modPow($x, $p);
     $keyBytes = $key->toBytes(true);
     $source = pack('Na*Na*Na*Na*Na*Na*Na*Na*', strlen($this->identifier), $this->identifier, strlen($this->server_identifier), $this->server_identifier, strlen($kexinit_payload_client), $kexinit_payload_client, strlen($kexinit_payload_server), $kexinit_payload_server, strlen($this->server_public_host_key), $this->server_public_host_key, strlen($eBytes), $eBytes, strlen($fBytes), $fBytes, strlen($keyBytes), $keyBytes);
     $source = pack('H*', $hash($source));
     if ($this->session_id === false) {

<?php

/*
On a 1.6GHz Pentium M, the following takes about 1.33 seconds with the pure-PHP implementation,
0.66 seconds with BCmath, and 0.001 seconds with GMP.
*/
include '../Math/BigInteger.php';
//define('MATH_BIGINTEGER_MODE', MATH_BIGINTEGER_MODE_INTERNAL);
//define('MATH_BIGINTEGER_MODE', MATH_BIGINTEGER_MODE_BCMATH);
$a = '0b078d385e9d05d9e029dc9732e75f94f59fdcfb989fe25e81edcb4f93c1dc53a9bb6ba09b5799bd' . 'aa9e35cd4e00a8200b720d9c6034da9819a5c84e3c7106fcdf5e64c975221bfd9b606bf924bc2971' . 'de66c470b88221b419ad32e0bff8fb234cbfa0f99e0909d46855a6751b7660b7178f0a661265ad23' . '8433331edb99e0ff';
$b = 'a6b9ac382a5f8d394ee83d9e6e21e993c8d240e1';
$c = 'aebbcd9a69b5116ce60400b4126c9e84173635abde4bfa56da904e75d752a51a47d3f088f13299a0' . '3b6bf66bf77a6accddeb16fc46a8a32164d7fad2ce4bb159e5caeddb40c25ae02c19e7426bd26398' . '14d36ead09509031fc423852c33ff0e6d402b2af825acc03ad6ad234eb5d269c17a026bd37c1b6e2' . '4c8c7248d09e12ef';
$a = new Math_BigInteger($a, 16);
$b = new Math_BigInteger($b, 16);
$c = new Math_BigInteger($c, 16);
$start = microtime(true);
$d = $a->modPow($b, $c);
$elapsed = microtime(true) - $start;
echo "took {$elapsed} seconds\r\n";
echo md5($d->toString());
// should equal aab326a2511ee857e16ce0cdd3243779

 /**
  * Decrypt data.
  *
  * @param string $text  PKCS1-v1_5 encoded text.
  *
  * @return string  Plaintext.
  */
 public function decrypt($text)
 {
     $out = '';
     $p_len = strlen($this->_key->key['p']);
     $text = str_split($text, $p_len);
     $text[count($text) - 1] = str_pad($text[count($text) - 1], $p_len, chr(0), STR_PAD_LEFT);
     $p = new Math_BigInteger($this->_key->key['p'], 256);
     $x = new Math_BigInteger($this->_key->key['x'], 256);
     for ($i = 0, $j = count($text); $i < $j; $i += 2) {
         $c1 = new Math_BigInteger($text[$i], 256);
         $c2 = new Math_BigInteger($text[$i + 1], 256);
         $s = $c1->modPow($x, $p);
         $m_prime = $s->modInverse($p)->multiply($c2)->divide($p);
         $em = str_pad($m_prime[1]->toBytes(), $p_len, chr(0), STR_PAD_LEFT);
         // EME-PKCS1-v1_5 decoding
         if (ord($em[0]) !== 0 || ord($em[1]) !== 2) {
             throw new RuntimeException();
         }
         $out .= substr($em, strpos($em, chr(0), 2) + 1);
     }
     return $out;
 }

 /**
  * Calculates pow($num, $pow) (mod $mod)
  *
  * @param string $num
  * @param string $pow
  * @param string $mod
  * @return string
  * @access public
  */
 function powmod($num, $pow, $mod)
 {
     $num = new Math_BigInteger($num, 10);
     $pow = new Math_BigInteger($pow, 10);
     $mod = new Math_BigInteger($mod, 10);
     $res = $num->modPow($pow, $mod);
     return $res->toString();
     /*        if (function_exists('bcpowmod')) {
                 // bcpowmod is only available under PHP5
                 return bcpowmod($num, $pow, $mod);
             }
     
             // emulate bcpowmod
             $result = '1';
             do {
                 if (!bccomp(bcmod($pow, '2'), '1')) {
                     $result = bcmod(bcmul($result, $num), $mod);
                 }
                 $num = bcmod(bcpow($num, '2'), $mod);
                 $pow = bcdiv($pow, '2');
             } while (bccomp($pow, '0'));
             return $result;*/
 }

 /**
  * Exponentiate with or without Chinese Remainder Theorem
  *
  * See {@link http://tools.ietf.org/html/rfc3447#section-5.1.1 RFC3447#section-5.1.2}.
  *
  * @access private
  * @param Math_BigInteger $x
  * @return Math_BigInteger
  */
 function _exponentiate($x)
 {
     if (empty($this->primes) || empty($this->coefficients) || empty($this->exponents)) {
         return $x->modPow($this->exponent, $this->modulus);
     }
     $num_primes = count($this->primes);
     $m_i = array(1 => $x->modPow($this->exponents[1], $this->primes[1]), 2 => $x->modPow($this->exponents[2], $this->primes[2]));
     $h = $m_i[1]->subtract($m_i[2]);
     $h = $h->multiply($this->coefficients[2]);
     list(, $h) = $h->divide($this->primes[1]);
     $m = $m_i[2]->add($h->multiply($this->primes[2]));
     $r = $this->primes[1];
     for ($i = 3; $i <= $num_primes; $i++) {
         $m_i = $x->modPow($this->exponents[$i], $this->primes[$i]);
         $r = $r->multiply($this->primes[$i - 1]);
         $h = $m_i->subtract($m);
         $h = $h->multiply($this->coefficients[$i]);
         list(, $h) = $h->divide($this->primes[$i]);
         $m = $m->add($r->multiply($h));
     }
     return $m;
 }