Planar functions over finite fields

Abstract

Letp>2 be a prime. A functionf: GF(p)→GF(p) is planar if for everya∃GF(p) *, the functionf(x+a−f(x) is a permutation ofGF(p). Our main result is that every planar function is a quadratic polynomial. As a consequence we derive the following characterization of desarguesian planes of prime order. IfP is a protective plane of prime orderp admitting a collineation group of orderp 2, thenP is the Galois planePG(2,p). The study of such collineation groups and planar functions was initiated by Dembowski and Ostrom [3] and our results are generalizations of some results of Johnson [8].

We have recently learned that results equivalent to ours have simultaneously been obtained by Y. Hiramine and D. Gluck.

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Rónyai, L., SzŐnyi, T. Planar functions over finite fields. Combinatorica 9, 315–320 (1989). https://doi.org/10.1007/BF02125898

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AMS subject classification (1980)

  • 05 B 25
  • 11 T 21
  • 51 E 15