A counterexample to Tokareva's conjecture on sums of two bent functions

Source abstract

It is well known that the sum of two bent functions on $\mathbb{F}^n$ has algebraic degree at most $n/2$ for even $n\ge 4$. Tokareva conjectured that the converse holds: every Boolean function with degree at most $n/2$ can be written as a sum of two bent functions. The conjecture has been verified for even $n\leq 6$ and for several special classes of Boolean functions. We disprove this conjecture by showing that $x_1x_2(x_3x_4+x_5x_6+x_7x_8)$ is not a sum of two bent functions on $\mathbb{F}^8$. We extend this construction to obtain counterexamples in every even dimension $n\ge 8$. Using Shaporenko's reduction, we also obtain explicit counterexamples to Tokareva's derivative conjecture in every even dimension $n\ge 10$.