### Dynamical systems as the main instrument for the constructions of new quadratic families and their usage in cryptography

#### Abstract

Let K be a finite commutative ring and f = f(n) a bijective polynomial map f(n) of the Cartesian power K^n onto itself of a small degree c and of a large order. Let f^y be a multiple composition of f with itself in the group of all polynomial automorphisms, of free module K^n. The discrete logarithm problem with the pseudorandom base f(n) (solvef^y = b for y) is a hard task if n is sufficiently large. We will use families of algebraic graphs defined over K and corresponding dynamical systems for the explicit constructions of such maps f(n) of a large order with c = 2 such that all nonidentical powers f^y are quadratic polynomial maps. The above mentioned result is used in the cryptographical algorithms based on the maps f(n) – in the symbolic key exchange protocols and public keys algorithms.

#### Full Text:

PDFDOI: http://dx.doi.org/10.2478/v10065-012-0030-2

Data publikacji: 2012-01-01 00:00:00

Data złożenia artykułu: 2016-04-28 09:08:09

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