, Volume 9, Issue 3, pp 315–320

Planar functions over finite fields

  • L. Rónyai
  • T. SzŐnyi

DOI: 10.1007/BF02125898

Cite this article as:
Rónyai, L. & SzŐnyi, T. Combinatorica (1989) 9: 315. doi:10.1007/BF02125898


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 orderp2, 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.

AMS subject classification (1980)

05 B 2511 T 2151 E 15

Copyright information

© Akadémiai Kiadó 1989

Authors and Affiliations

  • L. Rónyai
    • 1
    • 2
  • T. SzŐnyi
    • 2
  1. 1.Department of Computer ScienceThe University of ChicagoChicagoUSA
  2. 2.Computer and Automation InstituteHungarian Academy of SciencesBudapestHungary