, Volume 9, Issue 3, pp 315-320

First online:

Planar functions over finite fields

  • L. RónyaiAffiliated withDepartment of Computer Science, The University of ChicagoComputer and Automation Institute, Hungarian Academy of Sciences
  • , T. SzŐnyiAffiliated withComputer and Automation Institute, Hungarian Academy of Sciences

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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.

AMS subject classification (1980)

05 B 25 11 T 21 51 E 15