Article

# On the cycle structure of permutation polynomials.

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

Finite Fields and Their Applications (Impact Factor: 0.46). 01/2008; 14:593-614. DOI: 10.1016/j.ffa.2007.08.003 Source: DBLP

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**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:**L. Carlitz proved that any permutation polynomial ff over a finite field FqFq is a composition of linear polynomials and inversions. Accordingly, the minimum number of inversions needed to obtain ff is defined to be the Carlitz rank of ff by Aksoy et al. The relation of the Carlitz rank of ff to other invariants of the polynomial is of interest. Here we give a new lower bound for the Carlitz rank of ff in terms of the number of nonzero coefficients of ff which holds over any finite field. We also show that this complexity measure can be used to study classes of permutations with uniformly distributed orbits, which, for simplicity, we consider only over prime fields. This new approach enables us to analyze the properties of sequences generated by a large class of permutations of FpFp, with the advantage that our bounds for the discrepancy and linear complexity depend on the Carlitz rank, not on the degree. Hence, the problem of the degree growth under iterations, which is the main drawback in all previous approaches, can be avoided.Journal of Complexity 01/2013; · 1.19 Impact Factor - [Show abstract] [Hide abstract]

**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. · 1.08 Impact Factor

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