Protected pages with user registered users passwords…

I have searched all over this site for this feature. I have looked at more than a few plugins, and they either do not work with the latest version of WordPress, do not offer what I am looking for, or add unnecessary bloat.

I would like to modify the built in page/post password feature. Instead of using an assigned password, I would like only registered members to have access to the pages with their passwords. People that are not members, will be unable to access the pages, while members will be able to with their password.

Any help with this will be appreciated.