Families Of Bent Functions Research Articles

Bent Functions and Strongly Regular Graphs

The family of bent functions is a known class of Boolean functions, which have a great importance in cryptography. The Cayley graph defined on \(\mathbb{Z}_{2}^{n}\) by the support of a bent function is a strongly regular graph \(srg(v,k,\lambda,\mu)\), with \(\lambda=\mu\). In this paper we list the parameters of such Cayley graphs. Moreover, a condition is given on \((n,m)\)-bent functions \(F=(f_1,\ldots,f_m)\), involving the support of their components \(f_i\), and their \(n\)-ary symmetric differences.

Criteria for maximal nonlinearity of a function over a finite field

Abstract An n-place function over a field with q elements is called maximally nonlinear if it has the greatest nonlinearity among all such functions. Criteria and necessary conditions for maximal nonlinearity are obtained, which imply that, for even n, the maximally nonlinear functions are bent functions, but, for q > 2, the known families of bent functions are not maximally nonlinear. For an arbitrary finite field, a relationship between the Hamming distances from a function to all affine mappings and the Fourier spectra of the nontrivial characters of the function are found.

Special bent and near-bent functions

Starting from special near-bent functions in dimension $2t-1$ we construct bent functions in dimension $2t$ having a specific derivative. We deduce new families of bent functions.

Strongly regular graphs associated with ternary bent functions

We prove a new characterization of weakly regular ternary bent functions via partial difference sets. Partial difference sets are combinatorial objects corresponding to strongly regular graphs. Using known families of bent functions, we obtain in this way new families of strongly regular graphs, some of which were previously unknown. One of the families includes an example in [N. Hamada, T. Helleseth, A characterization of some { 3 v 2 + v 3 , 3 v 1 + v 2 , 3 , 3 } -minihypers and some [ 15 , 4 , 9 ; 3 ] -codes with B 2 = 0 , J. Statist. Plann. Inference 56 (1996) 129–146], which was considered to be sporadic; using our results, this strongly regular graph is now a member of an infinite family. Moreover, this paper contains a new proof that the Coulter–Matthews and ternary quadratic bent functions are weakly regular.

A New Class of Bent Functions

In this letter, using techniques from linear algebra and coding theory, we characterize the quadratic Boolean functions represented by trace. We show that a linear combination of trace-terms over finite field can be determined to be bent by a polynomial GCD computation. Then we derive some new families of bent functions.