Combinatorial Classification of Optimal Authentication Codes with Arbitration

Unconditionally secure authentication codes with arbitration (A²-codes) protect against deceptions from the transmitter and the receiver as well as that from the opponent. We first show that an optimal A²-code implies an orthogonal array and an affine alpha-resolvable design. Next we define a new design, an affine alpha-resolvable + BIBD, and prove that optimal A²-codes are equivalent to this new design. From this equivalence, we derive a condition on the parameters for the existence of optimal A²-codes. Further, we show tighter lower bounds on the size of keys than before for large sizes of source states which can be considered as an extension of the bounds on the related designs