Abstract
This project focuses on the study of polynomial maps over finite fields and their connections and applications in Cryptography, Dynamics and Algebraic Curves. Our objects of study include: 1) linear maps over finite fields; 2) linearized permutation polynomials over finite fields; 3) algebraic curves over finite fields. In general, we are interested in the study of each of these objects in at least one of the following aspects: characterization, construction, and existence. 1. in the study of linear maps over finite fields, we are interested in the characterization of the functional graph associated with linear maps. The functional graph of a map over a finite field describes the dynamics of this map on the field. Two previous works (including one of the candidate) suggest that certain classes of linear maps produce functional graphs full of symmetries. Our aim is to extend, as general as possible, the results on these symmetries to a wider class of linear maps; 2. in the study of linearized permutation polynomials, we are interested in the construction and characterization of this kind of permutations, focusing on their applications to Cryptography which includes the construction of involutions in binary fields and permutations that are represented by polynomials with few nonzero coefficients (e.g. binomials, trinomials, and quadrinomials). Classical constructions and characterizations of linearized permutation polynomials explore several aspects in linear algebra, like invertible matrices and bases in finite vector spaces. Our study relies on a polynomial approach that was recently suggested by the candidate in one of his most recent publications: we introduce the class of nilpotent linearized polynomials and we show how to produce linearized permutations from these nilpotent polynomials; 3. in the study of curves over finite fields, we are interested in the characterization and construction of the automorphisms of algebraic curves C: y^m=f(x) over finite fields. We see that, in a general situation (i.e., C does not have any redundant or odd property), the automorphism group of C is obtained from the set of Mobius maps that permutes the roots of f(x). This correspondence suggests a polynomial approach in the study of the automorphism group of these curves: our study focuses on the characterization and construction of polynomials f(x) whose roots are permuted by a given set of Mobius maps. Two main questions will be discussed: a) the characterization of the Mobius map that permute the roots of a classical family of polynomials (e.g., Chebyshev or Cyclotomic polynomials); b) construction of polynomials whose roots are permuted by a given set of Mobius maps. In the sense of algebraic curves, question (a) concerns on the characterization of the automorphism group of a specific class of curves and question (b) concerns on the construction of algebraic curves with prescribed automorphism group. (AU)
