An algorithm for fast multiplication of elements in 2-groups based on the Zhegalkin polynomials

Abstract

Network design for a multiprocessor computing system or data center is an important problem where the search for graph models that have attractive topological properties and allow the use of efficient routing algorithms is carried out. Cayley graphs have the indicated properties, in particular such as high symmetry, hierarchical structure, recursive design, high connectivity and fault tolerance. The definition of the Cayley graph implies that the vertices of the graph are elements of some algebraic group. Selecting a group and its generating elements allows us to obtain a graph that meets the necessary requirements for diameter, degree of vertices, number of nodes, etc. A large number of scientific articles and monographs are devoted to solving this problem. The goal of this work is to create an algorithm for fast multiplication of elements in finite 2-groups whose exponent is 2n. The first section of the article provides a theoretical justification for the algorithm for fast multiplication in finite 2-groups. It is shown that elements of these groups can be represented in the form of bit strings, and their multiplication is carried out based on the Zhegalkin polynomials. The second section presents the pseudocode of the algorithm on the basis of which the Zhegalkin polynomials are calculated. The third section demonstrates an example of obtaining the Zhegalkin polynomials for a two-generated group of exponent 4. In conclusion, the prospects for using the algorithm on the real hardware are discussed.