On the conjecture on APN functions and absolute irreducibility of polynomials |
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Authors: | Moises Delgado Heeralal Janwa |
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Affiliation: | 1.Department of Mathematics and Physics,University of Puerto Rico (UPR),Cayey,USA;2.Department of Mathematics,University of Puerto Rico (UPR),San Juan,USA |
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Abstract: | An almost perfect nonlinear (APN) function (necessarily a polynomial function) on a finite field (mathbb {F}) is called exceptional APN, if it is also APN on infinitely many extensions of (mathbb {F}). In this article we consider the most studied case of (mathbb {F}=mathbb {F}_{2^n}). A conjecture of Janwa–Wilson and McGuire–Janwa–Wilson (1993/1996), settled in 2011, was that the only monomial exceptional APN functions are the monomials (x^n), where (n=2^k+1) or (n={2^{2k}-2^k+1} ) (the Gold or the Kasami exponents, respectively). A subsequent conjecture states that any exceptional APN function is one of the monomials just described. One of our results is that all functions of the form (f(x)=x^{2^k+1}+h(x)) (for any odd degree h(x), with a mild condition in few cases), are not exceptional APN, extending substantially several recent results towards the resolution of the stated conjecture. We also show absolute irreducibility of a class of multivariate polynomials over finite fields (by repeated hyperplane sections, linear transformations, and reductions) and discuss their applications. |
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