How to log users into different sites?

I’m using Multisite with each site a subdirectory off of the root. I’d like the root ( to be a login screen and, depending upon what username and password a user enters, they are logged in and redirected to the appropriate, protected site (, or

Do I need to make the entire network private, or is there a plugin or two that will handle the login and the redirection?

Thanks for the help!