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 Rédei function over a nonbinary finite field $\mathbb{F}_q$. In particular, we obtain two descriptions of the tree attached to the cyclic nodes in these graphs and provide period and preperiod estimates for Rédei functions. We also extend characterizations of Rédei permutations by describing their decomposition into disjoint cycles. Finally, we obtain some results on the length of the cycles related to Rédei permutations and we give an algorithm to construct Rédei permutations with prescribed length cycles in a geometric progression.

Additional Metadata
Keywords Dynamical systems over finite fields, Multiplication map in cyclic groups, Redei functions over finite fields
Persistent URL dx.doi.org/10.1137/140993338
Journal SIAM Journal on Discrete Mathematics
Qureshi, C. (Claudio), & Panario, D. (2015). Rédei actions on finite fields and multiplication map in cyclic group. SIAM Journal on Discrete Mathematics, 29(3), 1486–1503. doi:10.1137/140993338