Suzana Stojkovic

Contribution to the Study of Multiple-Valued Bent Functions

This paper is meant to be a contribution to the study of multiple-valued bent functions. Special care has been given to the introduction of a sound formalism to allow giving formal proof of some properties and to characterize bent functions. A class of bent functions, called strict bent is introduced and its characterization is given. It is shown that there are 486 two place ternary strict bent functions.

Hyper-bent Multiple-Valued Functions

Hyper-bent functions constitute a subset of bent functions and are harder to approximate than bent functions, making them particularly attractive for cryptographic applications. In the multiple-valued world, up to now, characterization and generation of hyper-bent functions represent an interesting challenging mathematical problem. We show that multiple-valued hyper-bent functions constitute a reduced subset of the multiple-valued bent functions and give a simple characterization lemma. Finally we introduce a new concept, that of strict hyper-bent functions, and study some of the properties of these functions. The only mathematical requirements of the paper are college algebra and a basic knowledge of Galois fields.

Application of Multi-valued Decision Diagrams in Computing the Direct Partial Logic Derivatives

Multi-State Systems (MSSs) are mathematical models often used in reliability engineering since allow a rather detailed evaluation of system reliability. These models represent the system reliability/availability behavior from the perfect functioning to the fault, but allow to distinguish several intermediate states. The calculation of related reliability measures can be performed in terms of Direct Partial Logic Derivatives (DPLDs). A drawback of this approach is high dimensionality of MSSs. To overcome this problem, we propose to use Multi-valued Decision Diagrams (MDDs) as a data structure to represent multi-state systems and perform computations of reliability measures in terms of DPLDs.

The Maiorana Method to Generate Multiple-Valued Bent Functions Revisited

A matrix-based formal representation of the Maiorana Method to generate MV-bent functions is introduced and a proof of correctness is given. The generation of bent, and strict bent multiple-valued functions is analyzed. It is shown that all bent functions generated with the Maiorana method are strict bent. Finally it is shown that the number of ternary bent functions of 4 arguments is at least 708,588. A general lower bound on the number of multiple-valued bent functions on an even and odd number of arguments is provided.

Representation of Incompletely Specified Binary and Multiple-Valued Logic Functions by Compact Decision Diagrams

Incompletely specified logic functions are often met in computer science and engineering and their compact representations are a subject of a wide research interest. In this paper, we present a method for determining an assignment of unspecified function values that will produce compact decision diagrams. The method is based on the analysis of Walsh and Vilenkin-Chrestenson spectral coefficients for binary and multiple-valued functions, respectively, and the operation of spectral translation of logic functions. Spectral transforms are used to identify the linear function with strongest correlation with the initial function and assign the unspecified values as well as to specify the spectral translation that will result in a compact decision diagram.

Linearization of Ternary Decision Diagrams by Using the Polynomial Chrestenson Spectrum

In the paper a procedure for linearization of ternary decision diagrams is proposed. Linearization is based on the properties of the polynomial Chrestenson spectral coefficients of the considered function. In the cases of some functions, by using the proposed procedure, linearized TDDs are reduced when compared with the original TDDs.

Mapping Decision Diagrams for Multiple-Valued Logic Functions into Threshold Logic Networks

The paper presents a method for threshold logic realization of multiple valued functions through decision diagrams. New threshold logic modules convenient for mapping to decision diagrams are introduced and it is shown that these modules allow to reduce the complexity of the realization by using heterogeneous decision diagrams.

An Improved Algorithm for the Construction of Decision Diagrams by Rearranging and Partitioning the Input Cube Set

Decision diagrams (DDs) are a data structure that allows compact representation of discrete functions. The efficient construction of DDs in terms of space and time is often considered problem. A particular problem is that during the construction of a DD, a large number of temporary nodes are created. We address this problem in the case when the functions are specified in the PLA format. A common practice is to construct a DD by recursively processing all the cubes in PLA specification. The DD representing a subfunction defined by a single cube is merged with the DD for the subfunction defined by all the previously processed cubes. We proposed a method of reordering and partitioning the set of cubes in PLA specification that results in the reduction of both space and time complexities of the construction of DDs. First, we arrange cubes by their suffices. Then we partition the set of cubes, construct DDs for the subfunctions representing each partition separately, and merge them into a final DD. The reordering and partitioning ensures that these intermediary decision diagrams never exceed a certain size which is controlled by the size of the partitions. In this way, the number of operations on the nodes during the merging decision diagrams is reduced. This reduction results in a decrease both in the number of temporary nodes and construction time. The proposed method is used for the construction of DDs for the set of standard benchmark functions. The experiments show that the total number of created nodes is reduced on average by 34.65 percent, while the construction time is decreased by 48.6 percent.

Spectral Analysis of Special Properties of Ternary Functions

This paper shows that particular classes of linear combinations of the coefficients of the circular Chrestenson-Vilenkin spectrum of ternary functions characterize whether cofactors of the function are constant or balanced. These results may be applied to take decisions related to hardware implementations or to simplify ternary decision diagrams.

On Fixed Points of the Reed-Muller-Fourier Transform

The Reed-Muller-Fourier transform combines relevant aspects of the RM transform and the DFT. It constitutes a bijection in the set of p-valued functions. Some properties of the transform matrix are formally analyzed and its eigenvectors with eigenvalue lambda = 1, which are its fixed points, are studied. Some methods to generate fixed points from known fixed points are presented and the number of fixed points for some values of p and n are given.