On q-ary plateaued functions over F_q and their explicit characterizations
Résumé
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 F_q where q s a prime power, and established their properties. The objective of this work is to redefine the notion of plateaued functions over F_q, and to present several explicit characterizations of those functions. We first give, over F_q, the notion of q-ary plateaued functions, which relies on the concept of the Walsh–Hadamard transform in terms of canonical additive character of F_q.
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 F_q. We finally characterize q-ary
plateaued functions in terms of derivatives, Walsh power moments and autocorrelation functions.