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On linguistic dynamical systems, families of graphs of large girth, and cryptography

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Abstract

The paper is devoted to the study of a linguistic dynamical system of dimension n ≥ 2 over an arbitrary commutative ring K, i.e., a family F of nonlinear polynomial maps f α : K n → K n depending on “time” α ∈ {K − 0} such that f α −1 = f −αM, the relation f α1 (x) = f α2 (x) for some x ∈ Kn implies α1 = α2, and each map f α has no invariant points. The neighborhood {f α (υ)∣α ∈ K − {0}} of an element v determines the graph Γ(F) of the dynamical system on the vertex set Kn. We refer to F as a linguistic dynamical system of rank d ≥ 1 if for each string a = (α1, υ, α2), s ≤ d, where αi + αi+1 is a nonzero divisor for i = 1, υ, d − 1, the vertices υ a = f α1 × ⋯ × f αs (υ) in the graph are connected by a unique path. For each commutative ring K and each even integer n ≠= 0 mod 3, there is a family of linguistic dynamical systems Ln(K) of rank d ≥ 1/3n. Let L(n, K) be the graph of the dynamical system Ln(q). If K = Fq, the graphs L(n, Fq) form a new family of graphs of large girth. The projective limit L(K) of L(n, K), n → ∞, is well defined for each commutative ring K; in the case of an integral domain K, the graph L(K) is a forest. If K has zero divisors, then the girth of K drops to 4. We introduce some other families of graphs of large girth related to the dynamical systems Ln(q) in the case of even q. The dynamical systems and related graphs can be used for the development of symmetric or asymmetric cryptographic algorithms. These graphs allow us to establish the best known upper bounds on the minimal order of regular graphs without cycles of length 4n, with odd n ≥ 3. Bibliography: 42 titles.

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Authors and Affiliations

  1. University of Maria Curie-Sklodowska, Lublin, Poland

    V. A. Ustimenko

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Published in Zapiski Nauchnykh Seminarov POMI, Vol. 326, 2005, pp. 214–234.

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Ustimenko, V.A. On linguistic dynamical systems, families of graphs of large girth, and cryptography. J Math Sci 140, 461–471 (2007). https://doi.org/10.1007/s10958-007-0453-2

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  • DOI: https://doi.org/10.1007/s10958-007-0453-2

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