Externally Funded Project

Bent functions and generalizations, APN functions

# Project

Title: Bent functions and generalizations, APN functions
Number: FWF Project P 30966-NBL
Runtime: 01.09.2018-31.03.2021

# Abstract

Bent functions, vectorial bent functions, APN functions and related functions have applications in cryptography and in coding theory, and are also widely studied for connections to objects in combinatorics and finite geometry, like difference sets, designs or strongly regular graphs.

In the framework of this project it is investigated which bent functions are components of vectorial bent functions. There are many constructions of bent functions, a few of vectorial bent functions. It is though an open problem if there are bent functions that are lonely, i.e., not a component of a vectorial bent function. We present the first construction of vectorial bent functions with non-weakly regular bent component functions (but dual-bent) and the first one, of which the components are non-dual-bent. Coding-theoretic criteria were obtained for a Boolean bent function (not) to be lonely. Initiating a detailed study of vanishing flats, a quadruple system attached to vectorial Boolean functions, a further design-theoretic criterion was obtained. As another application of vanishing flats, totally skew covers of $n$-dimensional vector spaces were constructed.

As a further highlight, an upper bound was shown for the nonlinearity of a plateaued vectorial Boolean functions in dimension $n=2m$ with the maximal possible number $2^n-2^m$ of bent components, and a class of functions of which some attain this bound was analysed. As another result, further connections between vectorial bent functions and strongly regular graphs have been established.

Generalized bent functions, a superclass of bent functions which map into the cyclic group with $2^k$ elements, can be seen as a bent function $a$ with a corresponding partition of $\mathbb{F}_2^n$ with certain properties. In the framework of the project it is shown that Maiorana-McFarland (MMF) bent functions allow the largest possible such partitions.

The existence of bent functions into the cyclic group with $2^k$, $k\ge 3$, elements, which do not come from spreads, has been an open problem. We could give a positive answer by constructing some classes which provable do not come from the spread construction.

As a further achievement, a new method was introduced for determining the nonlinearity of some classes of quadratic functions. The nonlinearity of a new APN function presented by Taniguchi in 2019 is determined, and some shorter proofs for the nonlinearity of some other functions were found. Further, in the framework of the project, nonlinearity and differential uniformity of $(n,n)$ functions that contain vectorial MMF bent functions, were analysed in detail.

Within this project, in total 12 articles have been submitted to recognized scientific journals or peer-reviewed conference proceedings (8 appeared, 4 are still in the reviewing process). Several of the results were presented at international conferences and at seminar talks.

# Publications

1. A. Çeşmelioğlu, W. Meidl and A. Pott, A survey on bent functions and their duals. In: Combinatorics and Finite Fields, Radon Series on Computational and Applied Mathematics, de Gruyter 2019, pp. 39–56.
https://doi.org/10.1515/9783110642094-003
https://www.researchgate.net/publication/334305559
2. N. Anbar, T. Kalaycı, W. Meidl, Determining the Walsh spectra of Taniguchi’s and related APN-functions. Finite Fields Appl. 60 (2019), 101577, 20 pp.
https://doi.org/10.1016/j.ffa.2019.101577
arXiv:1904.12776
3. W. Meidl, A. Pott, Generalized bent functions into $\mathbb{Z}_{p^k}$ from the partial spread and the Maiorana-McFarland class. Cryptogr. Commun. 11 (2019), 1233–1245.
https://doi.org/10.1007/s12095-019-00370-w
https://www.researchgate.net/publication/333251935
4. I. Pirsic, Magische Quadrat-Quadrate und Divisionsalgebren. Math. Semesterber. 67 (2020), 169–183.
https://doi.org/10.1007/s00591-019-00268-x
https://www.researchgate.net/publication/337607193
5. A. Çeşmelioğlu, W. Meidl, A. Pott, Vectorial bent functions in odd characteristic and their components. Cryptogr. Commun. 12 (2020), 899–912.
https://doi.org/10.1007/s12095-020-00444-0
https://www.researchgate.net/publication/342947536
6. N. Anbar, C. Kaşıkcı, W. Meidl, A. Topuzoğlu, Shifted plateaued functions and their differential properties. Cryptogr. Commun. 12 (2020), 1091–1105.
https://doi.org/10.1007/s12095-020-00426-2
https://www.researchgate.net/publication/325441767
7. S. Li, W. Meidl, A. Polujan, A. Pott, C. Riera, P. Stănică, Vanishing flats: a combinatorial viewpoint on the planarity of functions and their application. IEEE Trans. Inform. Theory 66 (2020), 7101–7112.
DOI: 10.1109/TIT.2020.3002993
arXiv:2006.01941
8. W. Meidl, I. Pirsic, Bent and $\mathbb{Z}_{2^k}$-Bent functions from spread-like partitions. Designs, Codes, Cryptogr., 89 (2021), 75–89.
https://doi.org/10.1007/s10623-020-00805-z
arXiv:2009.11019

# Planned publications

1. N. Anbar, T. Kalaycı, W. Meidl, Analysis of $(n,n)$-functions obtained from the Maiorana-McFarland class. Peer Review. Submitted.
https://www.researchgate.net/publication/340063618
2. N. Anbar, T. Kalaycı, W. Meidl, L. Merai, On functions with the maximal number of bent components. Peer Review. Submitted.
arXiv:2010.03801
3. W. Meidl, A. Polujan, A. Pott, Linear codes and incidence structures of bent functions and their generalizations. Peer Review. Submitted.
arXiv:2012.06866
4. A. Çeşmelioğlu, W. Meidl, I. Pirsic, Vectorial bent functions and partial difference sets. Peer Review. Submitted.