Expanding graphs of the Extremal Graph Theory and expanded platforms of Post Quantum Cryptography

Abstract

Explicit constructions in Extremal Graph Theory give appropriate lower bounds for Turan type problems. In the case of prohibited cycles, the explicit constructions can be used for various problems of Information Security. We observe recent applications of algebraic constructions of regular graphs of large girth and graphs with large cycle indicator to Coding Theory and Cryptography. In particular, we present a new multivariate platforms of postquantum Non-commutative Cryptography defined in graph theoretical terms. Index Terms-graphs of large girth, graphs of large cycle indicator, graph based stream ciphers, multivariate cryptography, non-commutative cryptography I. SOME DEFINITIONS OF EXTREMAL GRAPH THEORY T HE missing definitions of graph-theoretical concepts in the case of simple graphs which appears in this paper can be found in [1]. All graphs we consider are simple ones, i. e. undirected without loops and multiple edges. When it is convenient, we shall identify Γ with the corresponding antireflexive binary relation on V (Γ), i.e. E(Γ) is a subset of V (Γ)×V (Γ). The girth of a graph Γ, denoted by g = g(Γ), is the length of the shortest cycle in Γ. The diameter d = d(Γ) of the graph Γ is the maximal length of the shortest pass between its two vertices. Let g x = g x (Γ) be the length of the minimal cycle through the vertex x from the set V (Γ) of vertices in graph Γ. We refer to Cind(Γ) = max {g x , x ∈ V (Γ)} as cycle indicator of the graph Γ. The family Γ i of connected k-regular graphs of constant degree is a family of small world graphs, if d(Γ i) ≤ c log k (v i), for some constant c, c > 0. Recall that family of regular graphs Γ i of degree k and increasing order v i is a family of graphs of large girth, if g(Γ i) ≥ c log k (v i), for some independent constant c, c > 0. We refer to the family of regular simple graphs Γ i of degree k and order v i as a family of graphs of large cycle indicator, if Cind(Γ i) ≥ c log k (v i) for some independent constant c, c > 0. Notice that for vertex-transitive graph its girth and cycle indicator coincide. Defined above families plays an important role in Extremal Graph Theory, Theory of LDPC codes and Cryptography (see [2] and further references).

On the usage of postquantum protocols defined in terms of transformation semi- groups and their homomophisms

“…Defined above families plays an important role in Extremal Graph Theory, Theory of LDPC codes and Cryptography. (see [30], [33] and further references).…”

“…We hope that introduced above graphs ( , * ) and ( , * ) possess interesting extremal and spectral properties Groups ( , ) and ( , ) of cubical transformations of affine space associated with graphs ( , ) and ( , ) are interesting objects of algebraic transformation group theory because of composition of two maps of degree 3 for vast majority of pairs will have degree 9. Constructions and applications of these families of transformations groups are recently observed in [33] where some extensions of these groups are introduced.…”

On computations with double Schubert automaton and stable maps of multivariate cryptography

“…They contribute valuable applications of theory of Extremal graphs (see [20]) and Expanding graphs [21] to Theoretical Computer Science. We have been working on applications of these graphs to Cryptography (see [22], [23], [24], [25] and further references). This paper is dedicated to the problem of postquantum secure encryption of rather large files in terms of Multivariate Cryptography but with usage of ideas of Noncommutative Cryptography.…”