Direct sums of balanced functions, perfect nonlinear functions, and orthogonal cocycles

Determining if a direct sum of functions inherits nonlinearity properties from its direct summands is a subtle problem. Here, we correct a statement by Nyberg on inheritance of balance and we use a connection between balanced derivatives and orthogonal cocycles to generalize Nyberg's result to orthogonal cocycles. We obtain a new search criterion for PN functions and orthogonal cocycles mapping to non-cyclic abelian groups and use it to find all the orthogonal cocycles over Z2t, 2 t 4. We conjecture that any orthogonal cocycle over Z2t, t 2, must be multiplicative

A polynomial approach to cocycles over elementary abelian groups

We derive bivariate polynomial formulae for cocycles and coboundaries in Z2(xs2124pn,xs2124pn), and a basis for the (pn-1-n)-dimensional GF(pn)-space of coboundaries. When p=2 we determine a basis for the $(2^n + {n\choose 2} -1)$-dimensional GF(2n)-space of cocycles and show that each cocycle has a unique decomposition as a direct sum of a coboundary and a multiplicative cocycle of restricted form

Finding maximal bicliques in bipartite networks using node similarity

In real world complex networks, communities are usually both overlapping and hierarchical. A very important class of complex networks is the bipartite networks. Maximal bicliques are the strongest possible structural communities within them. Here we consider overlapping communities in bipartite networks and propose a method that detects an order-limited number of overlapping maximal bicliques covering the network. We formalise a measure of relative community strength by which communities can be categorised, compared and ranked. There are very few real bipartite datasets for which any external ground truth about overlapping communities is known. Here we test three such datasets. We categorise and rank the maximal biclique communities found by our algorithm according to our measure of strength. Deeper analysis of these bicliques shows they accord with ground truth and give useful additional insight. Based on this we suggest our algorithm can find true communities at the first level of a hierarchy. We add a heuristic merging stage to the maximal biclique algorithm to produce a second level hierarchy with fewer communities and obtain positive results when compared with other overlapping community detection algorithms for bipartite networks

Relative difference sets, graphs and inequivalence of functions between groups

For cryptographic purposes, we want to find functions with both low differential uniformity and dissimilarity to all linear functions and to know when such functions are essentially different. For vectorial Boolean functions, extended affine equivalence and the coarser Carlet¿Charpin¿Zinoviev (CCZ) equivalence are both used to distinguish between nonlinear functions. It remains hard to tell when CCZ equivalent functions are EA-inequivalent. This paper presents a framework for solving this problem in full generality, for functions between arbitrary finite groups. This common framework is based on relative difference sets (RDSs). The CCZ and EA equivalence classes of perfect nonlinear (PN) functions are each derived, by quite different processes, from equivalence classes of splitting semiregular RDSs. By generalizing these processes, we obtain a much strengthened formula for all the graph equivalences which define the EA equivalence class of a given function, amongst those which define its CCZ equivalence class

A theory of highly nonlinear functions

Highly nonlinear functions are important as sources of low-correlation sequences, high-distance codes and cryptographic primitives, as well as for applications in combinatorics and finite geometry. We argue that the theory of such functions is best seen in terms of splitting factor pairs. This introduces an extra degree of freedom, through the pairing of a normalised function φ : G --> N between groups with a homomorphism e : G --> Aut(N). From this perspective we introduce a new definition of equivalence for functions, relative to e, and show it preserves their difference distributions. When e≡1 it includes CCZ and generalised linear equivalence, as well as planar and linear equivalence. More generally, we use splitting factor pairs to relate several important measures of nonlinearity. We propose approaches to both linear approximation theory and bent functions, and to difference distribution theory and perfect nonlinear functions, which encompass the current approaches