A system for playing a gambling game over a network. A plurality of clients are connected to a server by the network. The client personally authenticates a person using the client by analyzing a fingerprint collected by a fingerprint sensor. The client then generates elemental information in accordance with a predetermined condition. Authentication-information is generated irreversibly and univocally on the basis of the elemental information generated by the client. The server receives a plurality of pieces of the authentication information from the clients which it transmits to the clients. The client then transmits the elemental information to the server after reception of all the pieces of the authentication information.