Explicit Criteria for Construction of Plateaued Functions

Plateaued functions are very important cryptographic functions due to their desirable cryptographic characteristics. We find explicit criteria for the construction of p-ary r-plateaued functions with an odd prime p. We point out that 0-plateaued functions are bent functions, and so plateaued functions generalize the notion of bent functions. We first derive an explicit form for the Walsh-Hadamard transform of a p-ary r-plateaued function. We then obtain an upper bound on the degree of p-ary r-plateaued functions, and we classify p-ary (n - 1)-plateaued functions in n variables. We also obtain explicit criteria for the existence of p-ary r-plateaued functions. Accordingly, these results lead to improved bounds on the existence of p-ary bent functions.