On some Representations of Quadratic APN Functions and Dimensional Dual Hyperovals
Yves Edel · Ghent University Academic Bibliography (Ghent University) · 2010
APN functions (see Definition 1) have become a quite popular area of research and especially quadratic APN functions have links to various other mathematical areas. One link is to geometry, namely dimensional dual hyperovals (see Definition 2). Quadratic APN functions are equivalent to certain subspaces of the alternating bilinear forms, and most examples of dual hyperoval are, or can be, constructed from certain subspaces of the bilinear forms. As consequence different related publications use substantially different notation. The present manuscript is based on the authors efforts to provide an explicit translation of the notation found in several papers on APN functions and dual hyperovals by Nakagawa, Taniguchi and Yoshiara, into his own notation and back. The focus of this manuscript are the representations. An other important point, the question of equivalence is not touched. We refer to [14] for the characterization of the dual hyperovals that arise from quadratic APN functions via the construction as described in Proposition 6, for the correspondence of their mutual equivalence classes as well as for an explicit construction of the APN function, given an ”APN-dual hyperoval”. Furthermore we restrict ourselves to the binary case, although some results hold also in more generality. Firstly as most results on APN functions are only for this case, secondly as the notation often gets substantially less complicated. For more extensive references, overviews over actual or more general results, open problems and motivations we refer to some recent overview articles [8, 9, 16,22].