Proof of Correspondence between Keys and Encoding Maps in an Authentication Code

Systematic authentication codes without secrecy were defined in [1]. In [2] two new systematic authentication codes based on the Gray map on a Galois ring are introduced with the purpose of optimally reducing the impersonation and substitution probabilities. The first code is another example of a previously constructed code using the Gray map on Galois rings and modules over these rings [3,4]. The second code generalises the construction in [3], on the assumption of the existence of an appropriate class of bent functions. For this code, the existence of the bijection between the key space and the set of encoding maps is proved in this paper in a rather long but exhaustive way.