Explicit Criteria for Construction of Plateaued Functions

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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.