Externally Funded Project

Functions with a special Walsh transform

FWF Project M 1767-N26
Runtime: 01.02.2015-30.09.2017

###### Project Abstract

Boolean and $p$-ary bent functions have applications in cryptography and rich connections to other objects in mathematics, for instance, bent functions correspond to (relative) difference sets in elementary abelian groups. Many constructions of bent functions are known, all of which, but one, yield so-called (weakly) regular bent functions.
In the framework of this project, two new constructions of not weakly regular bent functions have been presented, the second one is the very first construction which yields bent functions for which the so-called dual function is in general not bent. Until then only sporadic examples of such bent functions, which were found via computer search, have been known.
A Boolean or $p$-ary function is called $k$-normal (weakly $k$-normal), if it is constant (affine) on a $k$-dimensional subspace. Most known bent functions exhibit a seemingly “typical” behaviour with respect to normality, for instance, most known Boolean bent functions in a dimension $n$ are $\frac{n}{2}$-normal. In this project, we developed an algorithm for testing normality for $p$-ary functions. With the help of this algorithm, we showed that some of our constructed bent functions exhibit a seemingly “non-typical” behaviour also with respect to normality.

In the framework of the project, a concept of duality for vectorial bent functions has been developed and analysed. As we could show, having a vectorial dual bent function is an exceptional property for a vectorial bent function $F$, but remarkably, exactly that property needed for obtaining strongly regular graphs from $F$.

As a further highlight, we completely solved the problem of counting so-called idempotent quadratic functions with prescribed co-dimension. Many results in the literature on this problem are now consequences of our more general results.

Generalized bent functions, which map into the cyclic group with $2^k$ elements, and have applications in CDMA-systems, have been investigated intensively in the last few years by various research groups in various countries. Meanwhile these functions are well-understood, and one also knows some constructions. A significant contribution was made within this project. Several results together with various co-authors were published in recognized journals, and were presented at international conferences.

Shifted bent functions and components of modified planar functions have been investigated in detail, the results have been published in recognized journals, and were presented at international conferences.

###### Publications
1. C. Kaşıkcı, W. Meidl, A. Topuzoğlu, Spectra of a Class of Quadratic Functions: Average Behaviour and Counting Functions. Cryptogr. Commun. 8 (2016), no.2, 191–214.
2. W. Meidl and H. Niederreiter, Multisequences with high joint non- linear complexity. Designs, Codes and Cryptography 81 (2016), no. 2, 337–346.
3. W. Meidl, Generalized Rothaus construction and non-weakly regular bent functions. J. of Combinat. Theory Series A, 141 (2016), 78–89.
4. A. Çeşmelioğlu, W. Meidl and A. Pott, There are infinitely many bent functions for which the dual is not bent. IEEE Trans. Inform. Theory 62 (2016), no. 9, 5204–5208.
5. T. Martinsen, W. Meidl and P. Stănică, Generalized bent functions and their Gray images, in (S. Duquesne, S. Petkova-Nikova, eds.) Arithmetic of Finite Fields, Proceedings of WAIFI 2016, Lecture Notes in Computer Science 10064, Springer-Verlag, Berlin Heidelberg, 2016.
6. N. Anbar, W. Meidl, A. Topuzoğlu, Idempotent and p-potent quadratic functions: Distribution of nonlinearity and co-dimension. Designs, Codes and Cryptography 82 (2017), no. 1-2, 265–291.
7. T. Martinsen, W. Meidl and P. Stănică, Partial spread and vecto- rial generalized bent functions. Designs, Codes and Cryptography 85 (2017), no. 1, 1–13.
8. N. Anbar and W. Meidl, Bent and Bent4 spectra of Boolean functions over finite fields, Finite Fields Appl. 46 (2017), 163–178.
9. T. Martinsen, W. Meidl, S. Mesnager and P. Stănică, Decompos- ing generalized bent and hyperbent functions. IEEE Trans. Inform. Theory 63 (2017), no. 12, 7804–7812.
10. W. Meidl, A secondary construction of bent functions, octal gbent functions and their duals. Math. Comput. Simulation 143 (2018), 57–64.
11. N. Anbar and W. Meidl, Modified planar functions and their com- ponents, Cryptogr. Commun., to appear.
###### Articles
• C. Kaşıkcı , W. Meidl , A. Topuzoğlu
Spectra of a Class of Quadratic Functions: Average Behaviour and Counting Functions (pdf)
• W. Meidl
Generalized Rothaus construction and non-weakly regular bent functions (pdf)
• W. Meidl
A secondary construction of bent functions, octal gbent functions and their duals (pdf)