$jwt = new JOSE_JWT(array(
    'foo' => 'bar'
));
$jwt->toString();

$jwt_string = 'eyJ...';
$jwt = JOSE_JWT::decode($jwt_string);

$private_key = "-----BEGIN RSA PRIVATE KEY-----\n....";
$jwt = new JOSE_JWT(array(
    'foo' => 'bar'
));
$jws = $jwt->sign($private_key, 'RS256');

$public_key = "-----BEGIN RSA PUBLIC KEY-----\n....";
$jwt_string = 'eyJ...';
$jwt = JOSE_JWT::decode($jwt_string);
$jws = new JOSE_JWS($jwt);
$jws->verify($public_key, 'RS256');

$jwe = new JOSE_JWE($plain_text);
$jwe->encrypt(file_get_contents('/path/to/public_key.pem'));
$jwe->toString();

$jwt_string = 'eyJ...';
$jwe = JOSE_JWT::decode($jwt_string);
$jwe->decrypt($private_key);

$public_key = new phpseclib\Crypt\RSA();
$public_key->loadKey('-----BEGIN RSA PUBLIC KEY-----\n...');
JOSE_JWK::encode($public_key); # => JOSE_JWK instance

$private_key = new phpseclib\Crypt\RSA();
$private_key->setPassword($pass_phrase); # skip if not encrypted
$private_key->loadKey('-----BEGIN RSA PRIVATE KEY-----\n...');
JOSE_JWK::encode($private_key); # => JOSE_JWK instance

# public key
$components = array(
    'kty' => 'RSA',
    'e' => 'AQAB',
    'n' => 'x9vNhcvSrxjsegZAAo4OEuo...'
);
JOSE_JWK::decode($components); # => phpseclib\Crypt\RSA instance