On the number of bent functions from iterative constructions: lower bounds and hypotheses

Abstract

In the paper we study lower bounds on the number of bent functionsthat can be obtained by iterative constructions, namely by theconstruction proposed by A. Canteaut and P. Charpin in 2003. Thenumber of bent iterative functions is expressed in terms of sizes offinite sets and it is shown that evaluation of this number isclosely connected to the problem of decomposing Boolean functioninto sum of two bent functions. A new lower bound for the number ofbent iterative functions that is supposed to be asymptotically tightis given. Applying Monte-Carlo methods the number of bent iterativefunctions in $8$ variables is counted. Based on the performedcalculations several hypotheses on the asymptotic value of thenumber of all bent functions are formulated.