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On the conjecture on APN functions and absolute irreducibility of polynomials

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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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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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Authors and Affiliations

  1. Department of Mathematics and Physics, University of Puerto Rico (UPR), Cayey, PR, 00736, USA

    Moises Delgado

  2. Department of Mathematics, University of Puerto Rico (UPR), Rio Piedras, Box 70377, San Juan, PR, 00936-8377, USA

    Heeralal Janwa

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  1. Moises Delgado
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  2. Heeralal Janwa
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Correspondence to Heeralal Janwa.

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Communicated by A. Pott.

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Delgado, M., Janwa, H. On the conjecture on APN functions and absolute irreducibility of polynomials. Des. Codes Cryptogr. 82, 617–627 (2017). https://doi.org/10.1007/s10623-015-0168-1

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