Use HMAC. The shared secret key can be used to sign messages and they can be verified. Basically, every Gamma server has a id. This id is converted into a messa

Use HMAC. The shared secret key can be used to sign messages and they can be verified. Basically, every Gamma server has a id. This id is converted into a message (which includes expiry time). This message is then signed by the Alpha server. Gamma server takes this response+signature back to Beta server. Beta server verifies this and grants access. Depending on your design, you could have this signature sent on every request, or accept the message as confirmed and add it to beta server's knowledge (so next time gamma can authenticate much more easily via methods such as sessions or cookies or tokens). ie, Beta issues a token to Gamma once Gamma presents proof. The token is internally linked in Beta's knowledge to a certain expiry date and Gamma's ID. For every next request, Gamma simply presents this token.