Redei actions on finite fields and multiplication map in cyclic group

Abstract

We describe the functional graph of the multiplication-by-n map in a cycle group and use this to obtain the structure of the functional graph associated with a Redei function over a nonbinary finite field F-q. In particular, we obtain two descriptions of the tree attached to the cyclic nodes in these graphs and provide period and preperiod estimates for Redei functions. We also extend characterizations of Redei permutations by describing their decomposition into disjoint cycles. Finally, we obtain some results on the length of the cycles related to Redei permutations and we give an algorithm to construct Redei permutations with prescribed length cycles in a geometric progression