Creating a restful Login API

I have read a few articles on the subject, but I am just looking for some more advice really.

I have a WordPress site with a login functionality and user database on that site.

What I need to do is allow a 3rd party application to authenticate details which it receives against my WordPress database.

I was thinking the best way to do this would be by sending the data to a REST API, but security wise I cannot think how I can return a successful login without this being easily duplicated by a user.

I have read about oAuth and HMAC, but am not sure which route I should be going down.

Any help greatly appreciated, thanks.

Ian