On the implementation of cryptoalgorithms based on algebraic graphs over some commutative rings

Abstract

The paper is devoted to computer implementation of some graph based stream ciphers. We compare the time performance of this new algorithm with fast, but no very secure RC4, and with DES. It turns out that some of new algorithms are faster than RC4. They satisfy the Madryga requirements, which is unusual for stream ciphers (like RC4). The software package with new encryption algorithms is ready for the demonstration. We will study the properties of stream ciphers defined via finite automata corresponding to the family of algebraic graphs of high girth defined in (34). The sequence of the graphs gives a well defined projective limit (infinite graphs) which is useful for theoretical studies of Turing machine corresponding to the graph based stream cipher. Algebraic nature of graphs implies the polyno- miality of encryption scheme, but combinatorial properties make it possible to prove the absence of fixed points, and to establish that different keys produce distinct ciphertext. The transition functions of automaton related to graph form an arithmetical dynamic system in the sense of (33). We consider the results of desynchronization of such infinite dynamical system, via applications of graph automorphisms and graph deformation, its effect on the security level and chaotical structure of cipher strings. The time evaluation of these algorithms defined via directed asymmetrical graphs compares well with the performance of fast but not very secure RC4, DES, algorithms based on simple graphs (symmetric anti-reflexive binary relations) developed during the recent ten years. In section 2 we present the ideology of cryptography based on Extremal Graph Theory created by P. Erdos' and his school and some new results on extremal directed graphs, observe the results on cryptographical properties of graphs of high girth and implementations of algorithms based on automata related to such graphs. In section 3, the reader can find basic cryptographical terminology. Next section contains definitions of girth indicator and girth for directed graphs, concept of family of directed graphs of high girth, encryption automata related to members of such a family defined via special colouring of edges. Section 5 is devoted to explicit constructions of algebraic families of graphs of large girth. For each commutative ring K we define two related families of algebraic directed graphs RED(K) and RDF(K) of large girth. In section 5 and 6 we discuss the implementation of general stream cipher based of family of directed graphs of large girth RDEn(K), n = 2,3,... in case of rings K = Z2k, k 2 {8,16,32}. Graph RDFt(K), which is just a union of some connected components of RDEn(K), is useful in evaluating the girth indicator and the size of connected components of RDEn(K). We evaluate