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Notes on APN functions, semibiplanes and dimensional dual hyperovals

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Abstract

Two geometric objects, incidence graphs of semibiplanes and dimensional dual hyperovals, are respectively associated with APN and quadratic APN functions. From Proposition 2 (resp. Proposition 5), two APN (resp. quadratic APN) functions are CCZ (resp. extended affine) equivalent if and only if the associated graphs (resp. dimensional dual hyperovals) are isomorphic. The former graphs for almost bent functions are distance regular graphs by Proposition 4. The structures of automorphism groups of these geometric objects are investigated in Proposition 3 and Lemma 7. In particular, (Edel and Pott, Adv Math Commun 3:59–81 (2009), Question 2) is negatively answered.

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Authors and Affiliations

  1. Department of Mathematics, Tokyo Woman’s Christian University, Suginami-ku, 167-8585, Tokyo, Japan

    Satoshi Yoshiara

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  1. Satoshi Yoshiara
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Correspondence to Satoshi Yoshiara.

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Communicated by Leo Storme.

Dedicated to the memory of András Gács (1969-2009).

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Yoshiara, S. Notes on APN functions, semibiplanes and dimensional dual hyperovals. Des. Codes Cryptogr. 56, 197–218 (2010). https://doi.org/10.1007/s10623-010-9402-z

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  • DOI: https://doi.org/10.1007/s10623-010-9402-z

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