Notes on APN functions, semibiplanes and dimensional dual hyperovals

Authors:
Satoshi Yoshiara
Affiliations:
Department of Mathematics, Tokyo Woman's Christian University, Tokyo, Japan 167-8585
Venue:
Designs, Codes and Cryptography
Year:
2010

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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.