Article

# On the cycle structure of permutation polynomials.

Sabancı University, MDBF, Orhanlı, 34956 Tuzla, İstanbul, Turkey

Finite Fields and Their Applications 01/2008; 14:593-614. DOI: 10.1016/j.ffa.2007.08.003 Source: DBLP

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**ABSTRACT:**Classes of permutations of finite fields with various specific properties are often needed for applications. We use a recent classification of permutation polynomials using their Carlitz rank with advantage, to produce examples of classes of permutations of F"p, for odd p, which for instance are ''random'', have low differential uniformity, prescribed cycle structure, high polynomial degree, large weight and large dispersion. They are also easy to implement. We indicate applications in coding and cryptography.Journal of Computational and Applied Mathematics 03/2014; 259:536-545. · 0.99 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**L. Carlitz proved that any permutation polynomial f of a finite field FqFq is a composition of linear polynomials and the monomials xq−2xq−2. This result motivated the study of Carlitz rank of f, which is defined in 2009 to be the minimum number of inversions xq−2xq−2, needed to obtain f, by E. Aksoy et al. We give a survey of results obtained so far on natural questions related to this concept and indicate a variety of applications, which emerged recently.Journal of Symbolic Computation 01/2013; · 0.39 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**In this work we establish some new interleavers based on permutation functions. The inverses of these interleavers are known over a finite field $\mathbb{F}_q$. For the first time M\"{o}bius and R\'edei functions are used to give new deterministic interleavers. Furthermore we employ Skolem sequences in order to find new interleavers with known cycle structure. In the case of R\'edei functions an exact formula for the inverse function is derived. The cycle structure of R\'edei functions is also investigated. The self-inverse and non-self-inverse versions of these permutation functions can be used to construct new interleavers.11/2010;

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