Using the RSfiles! search plugin will enable you to search for files that have been added to download folder using the default Joomla! search module.
To install the plugin just download it from the View my downloads area (valid RSFiles! licenses), go to Extensions > Install/Uninstall, browse the file and click "Upload and Install".
This plugin only offers one configuration parameter: ItemId. Setting this up will enable SEF links on the RSFiles! search results (can be a fictional menu item).