Authentication system, client, and server

An authentication system including a client, the authentication system comprising:

• a second server that determines whether or not matching information matches registration information based on a concealed evaluation value of an evaluation value indicating similarity between the registration information and the matching information, the concealed evaluation value being calculated by the client;
• a a first server that:
• transmits a message to the client;
• determines whether or not a signature is a correct signature by using a first verification key, the message, and the signature generated by the client;
• and
• determines that authentication succeeds when the matching information matches the registration information and it is determined that the signature is the correct signature,
• wherein the client
• stores concealed information obtained by concealing the registration information;
• calculates the concealed evaluation value based on the matching information and the concealed information;
• and
• generates the signature based on the message by using a signature key,
• wherein the second server further transmits the signature generated in the client to the first server under a condition that the matching information and the registration information are determined to match.