On q-ary plateaued functions over Fq and their explicit characterizations

Plateaued and bent functions play a significant role in cryptography, sequence theory, coding theory and combinatorics. In 1997, Coulter and Matthews redefined bent functions over any finite field Fq where q is a prime power, and established their properties. The objective of this work is to redefine the notion of plateaued functions over Fq, and to present several explicit characterizations of those functions. We first give, over Fq, the notion of q-ary plateaued functions, which relies on the concept of the Walsh–Hadamard transform in terms of canonical additive character of Fq. We then give a concrete example of q-ary plateaued function, that is not vectorial p-ary plateaued function. This suggests that the study of plateaued-ness is also significant for q-ary functions over Fq. We finally characterize q-ary plateaued functions in terms of derivatives, Walsh power moments and autocorrelation functions.