Paper 2026/2844

Antipodal necessity for rotation symmetric bent functions of degree at most three

Abstract

The conjecture that every homogeneous rotation symmetric bent Boolean function has degree two remains open in general. We prove a necessary condition that settles the degree-three case of this conjecture. Let $n>0$ be even. If a rotation symmetric Boolean function of degree at most three in $n$ variables is bent, then its algebraic normal form contains the antipodal quadratic orbit $\sum_{i<n/2} x_i x_{i+n/2}$. A homogeneous cubic has no quadratic terms, so no homogeneous rotation symmetric cubic Boolean function is bent, in any even number of variables. The condition also applies to nonhomogeneous functions with arbitrary lower-degree terms compatible with rotation symmetry; it neither classifies these functions nor addresses degree four and higher. The proof uses only finite Fourier analysis and linear algebra over the binary field. Restriction to the fixed space of a rotation of odd order reduces the problem to a power-of-two number of variables. There, Parseval's identity and a rigidity property of rotation symmetric quadratic forms impose incompatible conditions on a single fiber of the function. A second proof replaces the final fiber argument by descent through a single unipotent Jordan block. Examples illustrate the hypotheses. A Lean formalization of the original proof is provided as supplementary material; the Jordan proof has not been formalized.

Note: This PDF is identical to the author preprint archived on Figshare. Supplementary material and reproducibility files: https://doi.org/10.6084/m9.figshare.34165260.v4.