For authentication, the secret can be a user ID. A client with a logged-in user can store a token holding both a user ID and secret share. A server can hold another secret share for each user ID.
When requesting something that requires authentication, the client can send the user ID along with its' secret share. The server can use the client share and its' own share for the user to recreate the secret user ID. If the recreated user ID matches the ID that the client claims to be, then the server can proceed to fulfill the request. If not, then the request can be rejected.
Is this authentication scheme secure? If an adversary were to intercept a secret share for a user, would anything other than the user be compromised? Does this method have any practical use?