Cryptographic Properties of Parastrophic Quasigroup Transformation V. Dimitrova 1 , V. Bakeva 1 , A. Popovska-Mitrovikj 1 , and A. Krapeˇ z 2 1 Faculty of Computer Science and Engineering, Ss. Cyril and Methodius University, Skopje, Macedonia 2 Serbian Academy of Sciences and Arts, Beograd, Serbia vesna.dimitrova@finki.ukim.mk,verica.bakeva@finki.ukim.mk, aleksandra.popovska.mitrovikj@finki.ukim.mk,sasa@mi.sanu.ac.rs Abstract. In this paper, we consider cryptographic properties of para- strophic quasigroup transformation defined elsewhere. With that trans- formation we classify the quasigroups of order 4 in three classes: 1) parastrophic fractal; 2) fractal and parastrophic non-fractal; and 3) non- fractal. We investigate the algebraic properties of the previous classes and present a connection between fractal properties and algebraic properties of quasigroups of order 4. Also, we find a number of different parastro- phes of each quasigroup of order 4 and divide the set of all quasigroups of order 4 in 4 classes. Using these classifications the number of quasigroups of order 4 which are suitable for designing of cryptographic primitives is increased. Keywords: quasigroup, parastrophic quasigroup transformations, cryp- tographic properties 1 Introduction Quasigroups and quasigroup transformations are very useful for construction of cryptographic primitives, error detecting and error correcting codes. The reasons for that are the structure of quasigroups, their large number, the properties of quasigroup transformations and so on. The quasigroup string transformations E and their properties were considered in several papers. A quasigroup (Q, ∗) is a groupoid (i.e. algebra with one binary operation ∗ on the finite set Q) satisfying the law: (∀u, v ∈ Q)(∃!x, y ∈ Q) (x ∗ u = v & u ∗ y = v) (1) In fact, (1) says that a groupoid (Q, ∗) is a quasigroup if and only if the equations x ∗ u = v and u ∗ y = v have unique solutions x and y for each given u, v ∈ Q. It has been noted that every quasigroup (Q, ∗) has a set of five quasi- groups, called parastrophes denoted with /, \, ·, //, \\ which are defined in Table 1.