Joomla 3 External authentication script

jAuthentication.php

<?php
  /**
   * Joomla! External authentication script
   *
   * @author vdespa
   * Version 1.0
   *
   * Code adapted from /index.php
   *
   * @package    Joomla.Site
   *
   * @copyright  Copyright (C) 2005 - 2014 Open Source Matters, Inc. All rights reserved.
   * @license    GNU General Public License version 2 or later; see LICENSE.txt
   */

  if (version_compare(PHP_VERSION, '5.3.1', '<'))
  {
      die('Your host needs to use PHP 5.3.1 or higher to run this version of Joomla!');
  }

  /**
   * Constant that is checked in included files to prevent direct access.
   * define() is used in the installation folder rather than "const" to not error for PHP 5.2 and lower
   */
  define('_JEXEC', 1);

  if (file_exists(__DIR__ . '/defines.php'))
  {
      include_once __DIR__ . '/defines.php';
  }

  if (!defined('_JDEFINES'))
  {
      define('JPATH_BASE', __DIR__);
      require_once JPATH_BASE . '/includes/defines.php';
  }

  require_once JPATH_BASE . '/includes/framework.php';

  // Instantiate the application.
  $app = JFactory::getApplication('site');

  // JFactory
  require_once (JPATH_BASE .'/libraries/joomla/factory.php');


  // Hardcoded for now
  $credentials['username'] = 'adam';
  $credentials['password'] = 'adam';

  /**
   * Code adapted from plugins/authentication/joomla/joomla.php
   *
   * @package     Joomla.Plugin
   * @subpackage  Authentication.joomla
   *
   * @copyright   Copyright (C) 2005 - 2014 Open Source Matters, Inc. All rights reserved.
   * @license     GNU General Public License version 2 or later; see LICENSE.txt
   */

  // Get a database object
  $db    = JFactory::getDbo();
  $query = $db->getQuery(true)
      ->select('id, password')
      ->from('#__users')
      ->where('username=' . $db->quote($credentials['username']));

  $db->setQuery($query);
  $result = $db->loadObject();

  if ($result)
  {
      $match = JUserHelper::verifyPassword($credentials['password'], $result->password, $result->id);

      if ($match === true)
      {
          // Bring this in line with the rest of the system
          $user = JUser::getInstance($result->id);
          echo 'Joomla! Authentication was successful!';
      }
      else
      {
          // Invalid password
          // Prmitive error handling
          die('Invalid password');
      }
  } else {
      // Invalid user
      // Prmitive error handling
      die('Cound not find user in the database');
  }
?>

Instead of the database query and the verifyPassword() function, an easier way would be $loggedIn = $app->login($credentials);. This logs the user in and returns true if successful, so you can simply do if ($loggedIn) { ... }.