Let Fq be the finite field with q elements and n≥ 2 be a positive integer. We study the functional graph associated to linear maps over finite fields. In particular, we describe the functional graph Gf(Fqn) associated to the map induced by Lf on Fqn, where f is any irreducible divisor of xn- 1 over Fq and Lf is the q-associate of f. This description derives interesting information on the graph Gf(Fqn), such as the number of cycles and the average of the preperiod length. When gcd (f, xn- 1) = 1 , Lf is a permutation on Fqn and the cycle decomposition of Gf(Fqn) is well known. In this case, we present some applications of this result, such as the construction of linear involutions over odd characteristic and permutations with few fixed points.

Dynamical systems over finite fields, Involutions, Linear maps over finite fields, Permutation polynomials
Designs, Codes and Cryptography
School of Mathematics and Statistics

Panario, D, & Reis, L. (Lucas). (2018). The functional graph of linear maps over finite fields and applications. Designs, Codes and Cryptography. doi:10.1007/s10623-018-0547-5