Planar functions over finite fields


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.

This is a preview of subscription content, access via your institution.


  1. [1]

    Z. I. Borevich andI. R. Shafarevich, Number theory,Academic Press, New York, 1966.

    Google Scholar 

  2. [2]

    P. Dembowski,Finite Geometries, Springer-Verlag, Berlin, 1968.

    Google Scholar 

  3. [3]

    P. Dembowski andT. G. Ostrom, Planes of ordern with collineation groups of ordern 2,Math. Zeit.,103 (1968), 239–258.

    Article  Google Scholar 

  4. [4]

    M. Ganley, Polarities of translation planes,Geom. Ded.,1 (1972), 103–116.

    Google Scholar 

  5. [5]

    A. Goncalves andC. Y. Ho, On collineation groups of a projective plane of prime order,Geom. Ded.,20 (1986), 357–366.

    Google Scholar 

  6. [6]

    J. W. P. Hirschfeld,Projective Geometries over Finite Fields, Clarendon Press, Oxford, 1979.

    Google Scholar 

  7. [7]

    K. Ireland andM. Rosen,A classical introduction to modern number theory, GTM 84, Springer-Verlag, Berlin 1982.

    Google Scholar 

  8. [8]

    N. L. Johnson, Projective planes of orderp that admit collineation groups of orderp 2,J. of Geom.,30 (1987), 49–68.

    Article  Google Scholar 

  9. [9]

    L. Rédei,Lacunary polynomials over finite fields, Akadémiai Kiadó, Budapest, 1973.

    Google Scholar 

  10. [10]

    H. Salzmann, Topological planes,Advances in Math.,2 (1967), 1–60.

    Article  Google Scholar 

  11. [11]

    B.Segre, Ovals in a finite projective plane,Can. J. Math. (1955), 414–416.

  12. [12]

    B.Segre,Lectures on Modern Geometry, Cremonese Roma, 1961.

  13. [13]

    K. Yamamoto, On Jacobi Sums and Difference Sets,J. Comb. Th. 3 (1967), 146–181.

    Google Scholar 

Download references

Author information



Rights and permissions

Reprints and Permissions

About this article

Cite this article

Rónyai, L., SzŐnyi, T. Planar functions over finite fields. Combinatorica 9, 315–320 (1989).

Download citation

AMS subject classification (1980)

  • 05 B 25
  • 11 T 21
  • 51 E 15