Planar functions over finite fields

Combinatorica volume 9, pages315–320(1989)Cite this article

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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Affiliations

  1. Department of Computer Science, The University of Chicago, Ryerson Hall, 1100 E. 58th Street, 60637, Chicago, IL, USA

    L. Rónyai

  2. Computer and Automation Institute, Hungarian Academy of Sciences, P. O. B. 63, H-1502, Budapest, Hungary

    L. Rónyai & T. SzŐnyi

Authors
  1. L. Rónyai
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  2. T. SzŐnyi
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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