Create Authentication for REST API

Authentication for REST API like done in amazon, facebook, dropbox they most probably conform to HMAC pattern. See here, here, here, and here.

alternatively
there is oauth which redirects end user to another site to verify their password

if you would like to have user name passwords, check this out on how to transmit. e.g. do oauth send token to use it later on. See here and here.

Incorporating Hmac See here, here, and here.