Abstract

This study examines the dynamics of quadratic functions over finite fields Fq, where q = p^n, p is an odd prime, and n is a positive integer, within a theoretical, analytical, and non-experimental framework. The analysis integrates concepts from number theory, group theory, and graph theory to characterize the iterative behavior of quadratic functions of the form phi_c(x) = x^2 + c, where c is an element of Fq. Initially, the dynamics of the mapping phi(x) = x^2 over the finite field Fp are analyzed to establish fundamental structural properties. The study focuses on functional graphs in which vertices represent elements of the finite field and directed edges correspond to successive iterations of the function. Key concepts, including orbits, periodic and preperiodic points, cycle lengths, and tail lengths, are formally described and examined. The results indicate that the structure of these functional graphs is strongly influenced by the arithmetic properties of the multiplicative group of nonzero elements of Fp, particularly the factorization of p - 1, which determines the distribution and lengths of cycles and associated tails for the squaring map. The functional graphs exhibit cycles with attached inverse trees, reflecting the quadratic nature of the mappings and the algebraic structure of finite fields. Computational validation using GAP supports the theoretical findings, while an illustrative example over the finite field with nine elements (F9) demonstrates the applicability of the analytical framework. Overall, the study provides a systematic mathematical characterization of quadratic dynamical systems over finite fields and contributes to understanding the relationship between algebraic structures, iterative mappings, and functional graph configurations.