A nut graph is a nontrivial simple graph whose adjacency matrix has a simple eigenvalue zero such that the corresponding eigenvector has no zero entries. It was recently shown that the order n and degree d of a vertex-transitive nut graph satisfy 4 | d, d ≥ 4, 2 | n and n ≥ d + 4; or d ≡ 2 (mod 4), d ≥ 6, 4 | n and n ≥ d + 6. Here, we prove that for each such n and d, there is a d-regular Cayley nut graph of order n. As a direct consequence, we find all the pairs (n, d) for which there exists a d-regular vertex-transitive (resp. Cayley) nut graph of order n.

Datum in ura / Date and time: 8.12.25

(15:00-16:00)

Predavalnica / Location: FAMNIT-MP1

Predavatelj / Lecturer: Ajla Šehović (University of Primorska)

Naslov / Title: Weightwise (almost) perfectly balanced functions: t-concatenation and the general Maiorana-McFarland class

Vsebina / Abstract:

Boolean functions with good cryptographic properties are fundamental in the design of secure symmetric primitives. In 2017, Carlet, Méaux and Rotella studied cryptographic properties of restricted-domain functions in the context of the FLIP stream cipher and introduced the concept of weightwise (almost) perfectly balanced (W(A)PB) functions—Boolean functions that are (almost) balanced on each set of vectors of the same Hamming weight. Several approaches have since been proposed to construct new families of such functions.

In this talk, I will present some new constructions of W(A)PB functions using two approaches. The first class is built via the t-concatenation of Boolean functions, whereas the second one draws certain functions from the so-called general Maiorana-McFarland class. I will present examples of functions in 16 variables that attain some of the best overall nonlinearities, and more importantly, the highest slice nonlinearities among all of the constructions presented in the literature.

This is a joint work with René Rodriguez- Aldama and Leyla Işık.

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