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Designs, Codes and Cryptography

, Volume 82, Issue 3, pp 617–627 | Cite as

On the conjecture on APN functions and absolute irreducibility of polynomials

  • Moises Delgado
  • Heeralal Janwa
Article

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.

Keywords

Almost perfect nonlinear (APN) Cyclic codes Deligne estimate  Lang–Weil estimate Absolutely irreducible polynomial CCZ-equivalence  EA-equivalence Gold function Kasami function 

Mathematics Subject Classification

94A60 20C05 05B10 11T71 

Notes

Acknowledgments

The authors are thankful to R.M. Wilson, B. Mishra, H.F. Mattson, Jr., F. Castro, F. Piñero for helpful discussions, and the referees for helpful suggestions.

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of Mathematics and PhysicsUniversity of Puerto Rico (UPR)CayeyUSA
  2. 2.Department of MathematicsUniversity of Puerto Rico (UPR)San JuanUSA

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