Some new absolutely irreducibility testing criteria and their applications to the proof of a conjecture on exceptional almost perfect nonlinear (E-APN) function

Abstract

Our research work is on the construction of new absolute irreducible testing criteria and the creation of criteria that guarantee the existence of an absolute irreducible factor defined over $\ff_q$ and its applications towards the exceptional almost perfect nonlinear (APN) conjecture. Our results have direct implications and applications to algebraic geometry, algebraic number theory, coding theory, cryptography, sequence design, exceptional polynomials, finite geometry and combinatorics, where absolute irreducibility is critical. We use these new criteria and previous well establish results to solve many pending cases of the exceptional APN conjecture. We resolved the conjecture completely when the polynomial degree is Gold, and the second term is an odd degree term. We do this by generalizing a previous result by Delgado and Janwa. When the degree is Gold, and the second term is an even degree term, we use a method designed by Delgado and Janwa to prove the conjecture of all the possible cases with three exceptions. In these three cases, we gave a series of conditions the polynomials left need to fulfill. For the Kasami-Welch degree case, first, we extend the criteria for factorization into absolutely irreducible factors for the monomial case. When the degree of the polynomial is a Kasami-Welch exponent, and the degree of the second term is $1 \pmod{4}$, we generalize a result by Delgado and Janwa in two different ways. First, we give a bound on the degree of the second term that allows us to cover more cases than the one of degree $5 \pmod{8}$. Second, we gave a condition on the Kasami exponent, which allows us to guarantee the existence of an absolutely irreducible factor defined over $\ff_q$. Using a technique similar to the Gold, we manage to provide an upper bound on the multiplicity of the point $\{(1,1,1)\}$ for the second term. If this bound is not met, then we can guarantee the existence of an absolutely irreducible factor defined over $\ff_q$. Using this bound, we can partially prove the conjecture when the second term has an even degree. For the even degree case, we provide a characterization of the factorization for an infinite family of cases. We also give a conditional proof for the general case. Using this characterization and the results of Caullery and Rodier, for the case, when the degree of the polynomial is $4e$, when $e$ is a Gold or Kasami-Welch exponent, we prove that under a certain condition in the second term, the polynomial is not exceptional APN.