Bent functions from spreads
Sihem Mesnager · Contemporary mathematics - American Mathematical Society · 2015
Bent functions are optimal combinatorics objects. Since the introduction of these functions, substantial efforts have been directed towards their study in the last three decades. In this paper, we are interested firstly in bent functions on F 2 n \mathbb {F}_{2^n} whose restriction to n 2 \frac {n}2 -spreads are constant. The study of such bent functions motivates the clarification of connections between various subclasses of the class of partial bent functions and relations to the class of hyper-bent functions. We investigate their logic relations and state results giving more insight. We also draw a Venn diagram which explains the relations between these classes. Secondly, we present in a synthetic way the most important progresses obtained about the bent functions on F 2 n \mathbf {F}_{2^n} whose restrictions to n 2 \frac {n}2 -spreads are linear. Finally, we present our advances obtained about the bent functions on F 2 n \mathbb {F}_{2^n} whose restrictions to n 2 \frac {n}2 -spreads are affine.