When a Boolean Function can be Expressed as the Sum of two Bent Functions

In this paper we study the problem that when a Boolean function can be represented as the sum of two bent functions. This problem was recently presented by N. Tokareva in studying the number of bent functions [20]. Firstly, many functions, such as quadratic Boolean functions, MaioranaMacFarland bent functions, partial spread functions etc, are proved to be able to be represented as the sum of two bent functions. Methods to construct such functions from low dimension ones are also introduced. N. Tokareva’s main hypothesis is proved for n ≤ 6. Moreover, two hypotheses which are equivalent to N. Tokareva’s main hypothesis are presented. These hypotheses may lead to new ideas or methods to solve this problem. At last, necessary and sufficient conditions on the problem when the sum of several bent functions is again a bent function are given.