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Some new techniques and progress towards the resolution of the conjecture of exceptional APN functions and absolutely irreducibility of a class of polynomials

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Abstract

Almost Perfect Nonlinear (APN) functions are very useful in cryptography, when they are used as S-Boxes due to their good resistance to differential cryptanalysis. Also, some of the curves and surfaces defined by the corresponding nonlinear functions have many rational points and have applications to Algebraic-Geometric (AG) codes (Janwa and Wilson in Applied algebra, algebraic algorithms and error-correcting codes (San Juan, PR, 1993), vol 673, pp 180–194, Springer, Berlin, 1993, in IEEE Trans Inform Theory, accepted). An APN function \(f:\mathbb {F}_{2^n}\rightarrow \mathbb {F}_{2^n}\) is called an exceptional APN if it is APN on infinitely many extensions of \(\mathbb {F}_{2^n}\). Aubry et al. (Contemp Math 518:23–31, 2010) conjectured that the only exceptional APN functions are the Gold and the Kasami–Welch monomial functions. They established that a polynomial function of odd degree is not an exceptional APN provided the degree is not a Gold number \((2^k+1)\) or a Kasami–Welch number \((2^{2k}-2^k+1)\). When the degree of the polynomial function is a Gold number, several partial results are known. Here, we prove the relative primeness conjecture of the Gold degree polynomials. This result helps us substantially to make advances toward the resolution of the exceptional APN conjecture in the Gold degree case. We prove that Gold degree polynomials of the form \(x^{2^k+1}+h(x)\), where \(\deg (h)\) is an odd integer, cannot be exceptional APN (with a few natural exceptions). We also prove that the absolutely irreducible components of the Kasami–Welch degree curves intersect transversally at a particular point. Consequently, we prove that Kasami–Welch degree functions of type \(x^{2^{2k}-2^k+1}+h(x)\) produce absolutely irreducible surfaces in the case \(\deg (h) \equiv 3\pmod {4}\); and also \(\deg (h) \equiv 2^{m-1}+1 \pmod {2^m}\) under certain conditions. As a result, we show that the exceptional APN conjecture is true for these classes as well.

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Acknowledgements

We thank the referee for a very thorough reading of the manuscript and for helpful suggestions. Carlos Agrinsoni is supported by the NASA Training Grant No. NNX15AI11H, and 80NSSC20M0052. Opinions, findings, conclusions, or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of NASA. The authors would like to express their thanks to Eric Ferard for helpful observations.

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Authors and Affiliations

  1. Department of Mathematics, University of Puerto Rico, Cayey, 205 Calle Antonio R. Barceló, Cayey, 00736, Puerto Rico

    Moises Delgado

  2. Department of Mathematics, University of Puerto Rico, Rio Piedras, 17 University Ave. Ste 1701, San Juan, 00925-2537, Puerto Rico

    Heeralal Janwa & Carlos Agrinsoni

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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 G. McGuire.

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Delgado, M., Janwa, H. & Agrinsoni, C. Some new techniques and progress towards the resolution of the conjecture of exceptional APN functions and absolutely irreducibility of a class of polynomials. Des. Codes Cryptogr. 91, 2481–2495 (2023). https://doi.org/10.1007/s10623-023-01202-y

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