Coding theory seminar
Monday, 11 December, 2 pm
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Venue: Ramanujan Hall
Host: Sudhir Ghorpade
Title: The Exceptional Almost Perfect Nonlinear Function Conjecture
Speaker: Heeralal Janwa, University of Puerto Rico, the Main Campus at Rio Piedras
Abstract: Let F be a finite field of characteristic 2. A function f from F to F is called an almost perfect nonlinear (APN) function if the equation f(x+a) -f(x) = b has at most 2 solutions for every a,b in F, with a nonzero. APN functions are important in coding theory, cryptography, and combinatorics. We call a function an exceptional APN if it is an APN on F and on infinitely many extensions of F. We can transform the problem of finding the APN function into the problem of finding rational points in the variety X given by the corresponding multivariate polynomial G(x,y z) that lies outside the affine surface given by (x+y)(y+z)(x+z)=0. Using Lang-Weil, Deligne, and Ghoparde-Lachaud bounds one can estimate the number of rational points of the variety X when it is absolutely irreducible. These estimates allow us to transform the problem of whether a function is an exceptional APN to the problem of finding an absolute irreducible factor of the polynomial G(x,y z) different from (x+y), (y+z), (x+z).
The monomial exceptional APN functions had been classified up to CCZ equivalence in 2011, proving the conjecture of Janwa and Wilson (1993). The main tools used were the computation and classification of the singularities of X and a new algorithm for the absolute irreducibility testing using Bezout's Theorem. Aubry, McGuire, and Rodier (2010) conjectured that the only exceptional APN functions up to CCZ equivalence are the Gold exponent (2^k + 1) and the Kasami-Welch exponent (2^{2k} - 2^k + 1) monomial functions.
We will present the resolution of this conjecture in the Gold degree exponent case. We have also made substantial progress in resolving the exceptional APN conjecture for the Kasami-Welch case. In addition, we will present some results for the even degree cases for both the Gold and the Kasami-Welch degree exponents.
As a consequence of these results, we prove part of a conjecture on exceptional crooked functions. One of the main tools in our proofs is our new absolute irreducibility criteria. We will remind the audience that absolute irreducibility property is fundamental to number theory and arithmetic algebraic geometry, for example, as a necessary condition for applying the bounds of Deligne, Bombieri, Lang-Weil, and Ghorpade-Lachaud on rational points and exponential sums. We will present some open problems on singular zeta functions of curves over finite fields. (Joint work with Carlos Agrinsoni and Moisés Delgado)