Constructions Of A~3-Codes From Projective Spaces And Symplectic Space

General authentication codes with arbitration (A2-codes) require the transmitter and receiver must fully trust the arbitrator. But in fact, the arbitrator may not be entirely credible in many cases and even produce attack to the certification system. A2-codes is powerless when the arbitrator is involved with cheat. A~3-code, in which the receiver, transmitter and arbitrator are dishonest, is an extension of the general authentication code with arbitration and the arbitrator may also attack the certification system. Therefore, the receiver, transmitter and arbitrator in A~3-model have their own key information to avoid attacks from the outside. The following works are presented based on the constructions of A2-codes and the relative definitions of A~3-codes:1. The construction from projective space.Let 0<t<r<m.Let F be a fixed m-flat in PG(n,Fq) and let R be a fixed t-flat contained in F. Define the source state s to be the r-flat contained in F and containing R. Define the transmitter's key et to be the s-flat intersecting F at R. Define the receiver's and the arbiter's key to be the different (s-1)-flats intersecting F at R, respectively. Define the message m to be the (r+s-t)-flat which contains R. Transmitter's encoding rules, receiver's deconding rules and arbitrator's arbitrating rules are defined. The rationality of construction schemes is proved, meanwhile, the parameters of A~3-code and the probabilities of success in different attacks are also computed.2. The construction from singular pseudo-symplectic space.Using the relative relationship in singular pseudo-symplectic geometry space, we construct a A~3-code. The rationality of the A~3-code is proved, meanwhile, the parameters of the code and the probabilities of success in different attacks are also computed, assuming that the probability distributions of source states and participants keys are uniform.