Two Constructions of \(A^{2}\)-codes with Secrecy from Polynomials over Finite Fields

The authentication codes with arbitration are said to be $A^2$-codes. Two constructions of $A^2$-codes with secrecy from polynomials over finite fields are constructed to prevent communication systems from attacks which come from the opponent, the transmitter and the receiver. Parameters of the codes and probabilities of successful attacks are also computed. At last, two constructions are compared with a known one. It is important that a source state can’t be recovered from the message without the knowledge of the transmitter’s encoding rule or the receiver’s decoding rule. It must be decoded before verification.