Random Graph Construction for Cryptographic Applications

Abstract

The author in a number of his previous papers proposed methods to construct symmetric cryptographic algorithms based on the generalized cellular automata. To possess good cryptographic properties, graphs of such automata must meet a number of requirements. To construct such graphs both the deterministic methods and the randomized ones can be used. The paper deals with the randomized method of their construction. Briefly describing the generalized cellular automata it gives the requirements for graphs of such automata, including the following: the graph must have properties close to those of a randomly chosen graph; the graph diameter should be as small as possible; the graph must be regular; the graph should not be bipartite; the graph should have as minimum number of loops and multiple edges as possible; the graph degree should be (to reduce the communication complexity of the cellular automata) four, at least. In the family of graphs there must be a sufficiently large number of graphs with the number of vertices from several tens to several thousand. The expanding graphs, in particular, the so-called Ramanujan graphs meet these requirements. To build small Ramanujan graphs two methods are known, namely constructing a random regular graph with subsequent verification of spectral characteristics and constructing a graph by using a certain known deterministic algorithm. The paper considers the first method and gives an algorithm for generating such graphs. For practical purposes, it is important to know what characteristics random regular graphs have. In order to find this, a computer experiment was carried out. For each N of the set {256,512,1024,2048,4096} and each d of the set {3,4,5,6,7,8,9,10,11,12,13,14,15,16} was generated a large number of random d-regular graphs on N vertices. A total of 448 thousand graphs were generated. The paper presents graphs of the density distribution of λ parameters and diameters of these graphs. It can be seen from the graphs that among the randomly generated regular graphs, the Ramanujan graphs appear with a high probability, and the diameter of the graphs obtained is quite small. In general, such graphs are well suitable for the purposes considered. The research activity was supported by the Grant of the Russian Foundation for Basic Research within the framework of the scientific project No. 16-07-00542 a.