Niho Bent Functions and Subiaco/Adelaide Hyperovals

In this paper, the relation between binomial Niho bent functions discovered by Dobbertin et al. and o-polynomials that give rise to the Subiaco and Adelaide classes of hyperovals is found. This allows to expand the class of bent functions that corresponds to Subiaco hyperovals, in the case when $m\equiv 2 (\bmod 4)$.

💡 Research Summary

The paper establishes a precise algebraic link between a family of binomial Niho bent functions, originally discovered by Dobbertin and collaborators, and the o‑polynomials that generate the Subiaco and Adelaide classes of hyperovals. Bent functions are Boolean functions on the vector space 𝔽₂ⁿ whose Walsh spectrum has constant absolute value; they are central to cryptographic design because of their maximal non‑linearity and optimal differential properties. Niho bent functions are characterized by exponents of the form d = 2ⁱ(2ᵐ−1)+1, and Dobbertin et al. showed that a linear combination of two such monomials, \