Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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).
Beals R.M., Zieve M.E.: Decompositions of polynomials (2007) (Preprint).
Cohen S.D.: Review of [5] , Math. Rev. 2890555.
Coulter R.S.: The classification of planar monomials over fields of prime square order. Proc. Am. Math. Soc. 134, 3373–3378 (2006).
Coulter R.S., Lazebnik F.: On the classification of planar monomials over fields of square order. Finite Fields Appl. 18, 316–336 (2012).
Dembowski P., Ostrom T.G.: Planes of order n with collineation groups of order \(n^2\). Math. Z. 103, 239–258 (1968).
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).
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).
Guralnick R.M., Rosenberg J.E., Zieve M.E.: A new family of exceptional polynomials in characteristic two. Ann. Math. 172, 1367–1396 (2010).
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).
Guralnick R.M., Zieve M.E.: Polynomials with \(\operatorname{PSL}(2)\) monodromy. Ann. Math. 172, 1321–1365 (2010).
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).
Hernando F., McGuire G., Monserrat F.: On the classification of exceptional planar functions over \(F_p\) (2013) (arXiv:1301.4016v1).
Johnson N.L.: Projective planes of order \(p\) that admit collineation groups of order \(p^2\). J. Geom. 30, 49–68 (1987).
Klyachko A.A.: Monodromy groups of polynomial mappings. In: Studies in Number Theory, pp. 82–91. Saratov State University, Saratov (1975).
Leducq E.: Functions which are PN on infinitely many extensions of \(F_p\), \(p\) odd (2012) (arXiv:1006.2610v2).
Lidl R., Mullen G.L., Turnwald G.: Dickson Polynomials. Longman Scientific & Technical, Essex (1993).
Lidl R., Niederreiter H.: Finite Fields. Addison-Wesley, Reading (1983).
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).
Müller P.: A Weil-bound free proof of Schur’s conjecture. Finite Fields Appl. 3, 25–32 (1997).
Müller P.: Permutation groups of prime degree, a quick proof of Burnside’s theorem. Arch. Math. (Basel) 85, 15–17 (2005).
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).
Segre B., Bartocci U.: Ovali ed altre curve nei piani di Galois di caratteristica due. Acta Arith. 18, 423–449 (1971).
Turnwald G.: A new criterion for permutation polynomials. Finite Fields Appl. 1, 64–82 (1995).
Zieve M.E.: Exceptional polynomials. In: Mullen G.L., Panario D. (eds.) Handbook of Finite Fields, pp. 229–233. CRC Press, Boca Raton (2013).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by J. D. Key.
The author thanks Elodie Leducq and the referees for helpful remarks, and also thanks Gary McGuire and Peter Mueller for their interest. The author was partially supported by NSF Grant DMS-1162181.
Rights and permissions
About this article
Cite this article
Zieve, M.E. Planar functions and perfect nonlinear monomials over finite fields. Des. Codes Cryptogr. 75, 71–80 (2015). https://doi.org/10.1007/s10623-013-9890-8
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-013-9890-8