Survey on SAP and its application in public-key cryptography

The concept of the semigroup action problem (SAP) was first introduced by Monico in 2002. Monico explained in his paper that the discrete logarithm problem (DLP) can be generalized to SAP. After defining the action problem in a semigroup, the concept was extended using different mathematical structures. In this paper, we discuss the concept of SAP and present a detailed survey of the work which has been done using it in public-key cryptography.

The multi-transitivity of free semigroup actions

In this paper, we introduce the conceptions of multi-transitivity, [Formula: see text]-transitivity and [Formula: see text]-mixing property for free semigroup actions and give some equivalent conditions for a free semigroup action to be multi-transitive, multi-transitive with respect to vectors and strongly multi-transitive, respectively. For instance, we prove that a free semigroup action is multi-transitive or multi-transitive with respect to a vector if and only if its corresponding skew product system is multi-transitive or multi-transitive with respect to the same vector.

A bridge theorem for the entropy of semigroup actions

The topological entropy of a semigroup action on a totally disconnected locally compact abelian group coincides with the algebraic entropy of the dual action. This relation holds both for the entropy relative to a net and for the receptive entropy of finitely generated monoid actions.

Topological and metric entropy for group and semigroup actions

It is well-known that the classical definition of topological entropy for group and semigroup actions is frequently zero in some rather interesting situations, e.g. smooth actions of ℤk+ (k >1) on manifolds. Different definitions have been considered by several authors. In the present article we describe the one proposed in 1995 by K.H.Hofmann and the author which produces topological entropy not trivially zero for such smooth actions. We discuss this particular approach, and also some of the main properties of the topological entropy defined in this way, its advantages and disadvantages compared with the classical definition. We also discuss some recent results, obtained jointly with Andrzej Biś, Dikran Dikranjan and Anna Giordano Bruno, of a similar definition of metric entropy, i.e. entropy with respect to an invariant measure for a group or a semigroup action, and some of its properties.

Some dynamical properties for free semigroup actions

In this paper, we introduce the notions of weakly mixing and totally transitivity for a free semigroup action. Let [Formula: see text] be a free semigroup acting on a compact metric space generated by continuous open self-maps. Assuming shadowing for [Formula: see text] we relate the average shadowing property of [Formula: see text] to totally transitivity and its variants. Also, we study some properties such as mixing, shadowing and average shadowing properties, transitivity, chain transitivity, chain mixing and specification property for a free semigroup action.

Expanding actions: minimality and ergodicity

We prove that every expanding minimal semigroup action of [Formula: see text] diffeomorphisms of a compact manifold (resp. [Formula: see text] conformal) is robustly minimal (resp. ergodic with respect to the Lebesgue emeasure). We also show how, locally, a blending region yields the robustness of the minimality and implies ergodicity.