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

*Longjiang Qu and Shaojing Fu and Qingping Dai and Chao Li*

**Abstract: **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. Firstly, many functions, such as
quadratic Boolean functions, Maiorana-MacFarland 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 $n\leq 6$. 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.

**Category / Keywords: **foundations / Bent functions, Sum of bent functions, Maiorana-MacFarland bent function, Partial spread function.

**Date: **received 20 Jan 2014

**Contact author: **ljqu_happy at hotmail com

**Available format(s): **PDF | BibTeX Citation

**Version: **20140121:135341 (All versions of this report)

**Discussion forum: **Show discussion | Start new discussion

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