Designs, Codes and Cryptography

, Volume 75, Issue 1, pp 71–80 | Cite as

Planar functions and perfect nonlinear monomials over finite fields

  • Michael E. Zieve


The study of finite projective planes involves planar functions, namely, functions \(f:\mathbb {F}_q\rightarrow \mathbb {F}_q\) such that, for each \(a\in \mathbb {F}_q^*\), the function \(c\mapsto f(c+a)-f(c)\) is a bijection on \(\mathbb {F}_q\). Planar functions are also used in the construction of DES-like cryptosystems, where they are called perfect nonlinear functions. We determine all planar functions on \(\mathbb {F}_q\) of the form \(c\mapsto c^t\), under the assumption that \(q\ge (t-1)^4\). This resolves two conjectures of Hernando, McGuire and Monserrat. Our arguments also yield a new proof of a conjecture of Segre and Bartocci about monomial hyperovals in finite Desarguesian projective planes.


Planar functions Perfect nonlinear functions Monomial hyperovals 

Mathematics Subject Classification

51E20 11T06 11T71 05B05 


  1. 1.
    Abhyankar S.S., Cohen S.D., Zieve M.E.: Bivariate factorizations connecting Dickson polynomials and Galois theory. Trans. Am. Math. Soc. 352, 2871–2887 (2000).Google Scholar
  2. 2.
    Beals R.M., Zieve M.E.: Decompositions of polynomials (2007) (Preprint).Google Scholar
  3. 3.
    Cohen S.D.: Review of [5] , Math. Rev. 2890555.Google Scholar
  4. 4.
    Coulter R.S.: The classification of planar monomials over fields of prime square order. Proc. Am. Math. Soc. 134, 3373–3378 (2006).Google Scholar
  5. 5.
    Coulter R.S., Lazebnik F.: On the classification of planar monomials over fields of square order. Finite Fields Appl. 18, 316–336 (2012).Google Scholar
  6. 6.
    Dembowski P., Ostrom T.G.: Planes of order n with collineation groups of order \(n^2\). Math. Z. 103, 239–258 (1968).Google Scholar
  7. 7.
    Dickson L.E.: The analytic representation of substitutions on a power of a prime number of letters with a discussion of the linear group. Ann. Math. 11, 65–120, (1896–1897).Google Scholar
  8. 8.
    Granville A.: Arithmetic properties of binomial coefficients. I. Binomial coefficients modulo prime powers. In: Organic Mathematics, CMS Conference Proceedings, vol. 20, pp. 253–276. American Mathematical Society, Providence (1997).Google Scholar
  9. 9.
    Guralnick R.M., Rosenberg J.E., Zieve M.E.: A new family of exceptional polynomials in characteristic two. Ann. Math. 172, 1367–1396 (2010).Google Scholar
  10. 10.
    Guralnick R.M., Tucker T.J., Zieve M.E.: Exceptional covers and bijections on rational points. Int. Math. Res. Not. 2007, Article ID rnm004 (2007).Google Scholar
  11. 11.
    Guralnick R.M., Zieve M.E.: Polynomials with \(\operatorname{PSL}(2)\) monodromy. Ann. Math. 172, 1321–1365 (2010).Google Scholar
  12. 12.
    Hernando F., McGuire G.: Proof of a conjecture of Segre and Bartocci on monomial hyperovals in projective planes. Des. Codes Cryptogr. 65, 275–289 (2012).Google Scholar
  13. 13.
    Hernando F., McGuire G., Monserrat F.: On the classification of exceptional planar functions over \(F_p\) (2013) (arXiv:1301.4016v1).Google Scholar
  14. 14.
    Johnson N.L.: Projective planes of order \(p\) that admit collineation groups of order \(p^2\). J. Geom. 30, 49–68 (1987).Google Scholar
  15. 15.
    Klyachko A.A.: Monodromy groups of polynomial mappings. In: Studies in Number Theory, pp. 82–91. Saratov State University, Saratov (1975).Google Scholar
  16. 16.
    Leducq E.: Functions which are PN on infinitely many extensions of \(F_p\), \(p\) odd (2012) (arXiv:1006.2610v2).Google Scholar
  17. 17.
    Lidl R., Mullen G.L., Turnwald G.: Dickson Polynomials. Longman Scientific & Technical, Essex (1993).Google Scholar
  18. 18.
    Lidl R., Niederreiter H.: Finite Fields. Addison-Wesley, Reading (1983).Google Scholar
  19. 19.
    Lucas É.: Sur les congruences des nombres eulériens et des coefficients différentiels des fonctions trigonométriques, suivant un module premier, Bull. Soc. Math. France 6, 49–54 (1877–1878).Google Scholar
  20. 20.
    Müller P.: A Weil-bound free proof of Schur’s conjecture. Finite Fields Appl. 3, 25–32 (1997).Google Scholar
  21. 21.
    Müller P.: Permutation groups of prime degree, a quick proof of Burnside’s theorem. Arch. Math. (Basel) 85, 15–17 (2005).Google Scholar
  22. 22.
    Nyberg K., Knudsen L.R.: Provable security against differential cryptanalysis. In: Brickell E.F. (ed.) Advances in Cryptology (CRYPTO ’92), Lecture Notes in Computer Science, vol. 740, pp. 566–574. Springer-Verlag, Berlin (1992).Google Scholar
  23. 23.
    Segre B., Bartocci U.: Ovali ed altre curve nei piani di Galois di caratteristica due. Acta Arith. 18, 423–449 (1971).Google Scholar
  24. 24.
    Turnwald G.: A new criterion for permutation polynomials. Finite Fields Appl. 1, 64–82 (1995).Google Scholar
  25. 25.
    Zieve M.E.: Exceptional polynomials. In: Mullen G.L., Panario D. (eds.) Handbook of Finite Fields, pp. 229–233. CRC Press, Boca Raton (2013).Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MichiganAnn ArborUSA
  2. 2.Mathematical Sciences CenterTsinghua UniversityBeijingChina

Personalised recommendations