Explicit criteria for construction of plateaued functions

Plateaued functions are very important cryptographic functions due to theirdesirable cryptographic characteristics. We find explicit criteria forthe 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 forthe Walsh-Hadamard transform of a p-ary r-plateaued function. We then obtain an upperbound 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 forthe existence of p-ary r-plateaued functions. Accordingly, these results lead to improved bounds on the existence of p-ary bent functions.