Characterizations of Partially Bent and Plateaued Functions over Finite Fields

Partially bent and plateaued functions over finite fields have significant applications in cryptography, sequence theory, coding theory, design theory and combinatorics. They have been extensively studied due to their various desirable cryptographic properties. In this paper, we study on characterizations of partially bent and plateaued functions over finite fields, with the aim of clarifying their structure. We first redefine the notion of partially bent functions over any finite field Fq , with q a prime power, and then provide a few characterizations of these functions in terms of their derivatives, Walsh power moments and autocorrelation functions. We next characterize partially bent (vectorial) functions over Fp , with p a prime, by means of their derivatives and Walsh power moments. We finally characterize plateaued functions over Fp in terms of their Walsh power moments, derivatives and autocorrelation functions