Multiple-valued bent functions are functions with highest nonlinearity which makes them interesting for multiple-valued cryptography. Since the general structure of bent functions is still unknown, methods for construction of bent functions are often based on some deterministic criteria. For practical applications, it is often necessary to be able to construct a bent function that does not belong to any specific class of functions. Thus, the criteria for constructions are combined with exhaustive search over all possible functions which can be very CPU time consuming. A solution is to restrict the search space by some conditions that should be satisfied by the produced bent functions. In this paper, we proposed the construction method based on spectral subsets of multiple-valued bent functions satisfying certain appropriately formulated restrictions in Galois field (GF) and Reed-Muller-Fourier (RMF) domains. Experimental results show that the proposed method efficiently constructs ternary and quaternary bent functions by using these restrictions.

Binary bent functions have a strictly specified number of non-zero values. In the same way, ternary bent functions satisfy certain requirements on the elements of their value vectors. These requirements can be used to specify six classes of ternary bent functions. Classes are mutually related by encoding of function values. Given a basic ternary bent function, other functions in the same class can be constructed by permutation matrices having a block structure similar to that of the factor matrices appearing in the Good-Thomas decomposition of Cooley-Tukey Fast Fourier transform and related algorithms.

This paper presents a group theoretic approach to the design of Decision diagrams (DDs) with increased functionality of nodes. Basic characteristics of DDs determine their applications, and thus, the optimization of DDs with respect to different characteristics is an important task. Increased functionality of nodes provides for optimization of DDs. In this paper, the methods for optimization of binary DDs by pairing of variables are interpreted as the optimization of DDs by changing the domain group for the represented functions. Then, it is pointed out that, for Abelian groups, the increased functionality of nodes by using larger subgroups may improve some of the characteristics of DDs at the price of other characteristics. With this motivation, we proposed the use of non-Abelian groups for the domain of represented functions by taking advantages from basic features of their group representations. At the same time, the present methods for optimization of DDs, do not offer any criterion or efficient algorithm to choose among a variety of possible different DDs for an assumed domain group. Therefore, we propose Fourier DDs on non-Abelian groups to exploit the reduced cardinality of the Fourier spectrum on these groups.

Decision diagrams (DDs) are data structures commonly used for representation of discrete functions with large number of variables. Binary DDs (BDDs) are used for representation and manipulation with Boolean functions. Complexity of a BDD is usually measured by its size, that is defined as the number of non-terminal nodes in the BDD. Minimization of the sizes of DDs is a problem greatly considered in literature and many related algorithms (exact and heuristic) have been proposed. However, there are many functions for which BDDs when minimized are still large and can have even an exponential size in the number of variables. An approach to derive compact decision diagram representations for such functions is transformation of BDDs into Multi-valued DDs (MDDs) and Heterogeneous MDDs (HMDDs). Complexity of MDDs and HMDDs is measured by the cost which is a generalization of the notion of the size by taking into account complexity of nodes in MDDs and HMDDs. This paper presents a method for transformation of BDD into HMDD with minimal cost. The proposed method reduces the time for determination of the type of nodes in HMDDs by introducing a matrix expressing dependency (interconnections) among nodes at different levels. Comparing to other methods for conversion of BDDs into HMDDs, the method reduces the number of traverses of a BDD necessary for collecting enough information to construct an equivalent HMDD. For an experimental verification of its efficiency, the method is applied to construction of HMDDs for some benchmark functions and their arithmetic and Walsh spectra.

This paper describes a method for the efficient computation of the total autocorrelation for large multiple-output Boolean functions over a Shared Binary Decision Diagram (SBDD). The existing methods for computing the total autocorrelation over decision diagrams are restricted to single output functions and in the case of multiple-output functions require repeating the procedure k times where k is the number of outputs. The proposed method permits to perform the computation in a single traversal of SBDD. In that order, compared to standard BDD packages, we modified the way of traversing sub-diagrams in SBDD and introduced an additional memory function kept in the hash table for storing results of the computation of the autocorrelation between two subdiagrams in the SBDD. Due to that, the total amount of computations is reduced which makes the method feasible in practical applications. Experimental results over standard benchmarks confirm the efficiency of the method.