Homogeneous Bent Functions Research Articles

Relationship between homogeneous bent functions and Nagy graphs

Relationship between homogeneous bent functions and Nagy graphs

Relationship Between Homogeneous Bent Functions and Nagy Graphs

We study the relationship between homogeneous bent functions and some intersection graphs of a special type that are called Nagy graphs and denoted by Γ(n,k). The graph Γ(n,k) is the graph whose vertices correspond to (nk) unordered subsets of size k of the set 1,..., n. Two vertices of Γ(n,k) are joined by an edge whenever the corresponding k-sets have exactly one common element. Those n and k for which the cliques of size k + 1 are maximal in Γ(n,k) are identified. We obtain a formula for the number of cliques of size k + 1 in Γ(n,k) for n = (k + 1)k/2. We prove that homogeneous Boolean functions of 10 and 28 variables obtained by taking the complement to the cliques of maximal size in Γ(10,4) and Γ(28,7) respectively are not bent functions.

Связь однородных бент-функций и графов пересечений

Связь однородных бент-функций и графов пересечений

The eight variable homogeneous degree three bent functions

We determine the affine equivalence classes of the eight variable degree three homogeneous bent functions using a new algorithm. Our algorithm applies to general bent functions and can systematically determine the automorphism groups. We provide a partial verification of the enumeration of eight variable degree three homogeneous bent functions obtained by Meng et al. We determine the affine equivalence classes of these functions.

A proof for the nonexistence of some homogeneous bent functions

By the relationship between the first linear spectra of a function at partial points and the Hamming weights of the sub-functions, and by the Hamming weight of homogenous Boolean function, it is proved that there exist no homogeneous bent functions of degreem inn=2m variables form>3.

Homogeneous bent functions of degree n in 2 n variables do not exist for n>3

We prove that homogeneous bent functions f : GF(2) 2n→GF(2) of degree n do not exist for n>3. Consequently homogeneous bent functions must have degree < n for n>3.

Homogeneous Bent Functions, Invariants, and Designs

We establish a new connection between invariant theory and the theory of bent functions. This enables us to construct Boolean functions with a prescribed symmetry group action. Besides the quadratic bent functions the only other previously known homogeneous bent functions are the six variable degree three functions constructed in l16r. We show that these bent functions arise as invariants under an action of the symmetric group on four letters and determine the stabilizer which turns out to be a matrix group of order 10752. We apply the machinery of invariant theory in order to construct homogeneous bent functions of degree three in 8, 10, and 12 variables. This approach gives a great computational advantage over the unstructured search problem and yields Boolean functions which have a concise description in terms of certain designs and graphs. We consider the question of linear equivalence of the constructed bent functions and study the properties of the associated elementary abelian difference sets.