Applications of new criteria for absolute irreducibility of multivariate polynomials over finite fields to the conjecture of APN functions.
Discipline: Mathematics and Statistics
Subcategory: Mathematics and Statistics
Session: 1
Room: Chinatown
Yaniel Rivera Vega - University of Puerto Rico at Cayey
Co-Author(s): Moises Delgado, University of Puerto Rico at Cayey, Cayey, Puerto Rico
Absolute irreducibility of multivariate polynomials over a field F is defined as irreducibility over F and irreducibility over any extension of F. Criteria for absolute irreducibility is fundamental for applications in pure and applied mathematics as algebraic geometry, combinatorics, coding theory, cryptography, and finite geometry. On previous research, we proved new criteria for absolute irreducibility using the concept of “degree gap of a polynomial” and some recent results from Agrinsoni, Delgado and Janwa about this concept. The goal of this research was to apply these new criteria with some modifications towards a conjecture of APN functions. APN functions as defined by Nyberg, have the property of being high resistant against differential attacks when they are used as substitution components of block ciphers. One way to classify APN functions is by finding which of them have the property of being exceptional. Aubry, McGuire and Rodier conjectured that up to equivalence the Gold and Kasami-Welch functions are the only exceptional APN functions. Results from Rodier, Lang-Weil and Ghorpade Lachaud establish that if a surface associated to the function is absolutely irreducible, then the function is not exceptional APN. Our criteria helped prove the absolute irreducibility property in the surface associated to some open cases in the conjecture.
Funder Acknowledgement(s): Undergraduate Research Training Initiative for Student Enhancement (U-RISE)
Faculty Advisor: Moises Delgado, moises.delgado@upr.edu
Role: Worked on modifying previous results in absolute irreducibility using the degree gap to apply them to an associated surface of some Gold and Kasami-Welch functions in order to prove some open cases in a conjecture of APN functions. Also, I analyzed the new criteria that we developed before and experimented with some cases in order to obtain some new generalizations of it.