Abstract
Planar functions over finite fields give rise to finite projective planes and other combinatorial objects. They exist only in odd characteristic, but recently Zhou introduced an even characteristic analogue which has similar applications. In this paper we determine all planar functions on \({\mathbb {F}}_q\) of the form \(c\mapsto ac^t\), where \(q\) is a power of \(2\), \(t\) is an integer with \(0< t\le q^{1/4}\), and \(a\in {\mathbb {F}}_q^*\). This settles and sharpens a conjecture of Schmidt and Zhou.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aubry, Y., Perret, M.: A Weil theorem for singular curves. In: Arithmetic, Geometry and Coding Theory, pp. 1–7. de Gruyte (1996)
Capelli, A.: Sulla riduttibilità delle equazioni algebriche. Ren. Accad. Sc. Fis. Mat. Soc. Napoli 4, 84–90 (1898)
Carlet, C., Ding, C., Yuan, J.: Linear codes from perfect nonlinear mappings and their secret sharing schemes. IEEE Trans. Inform. Theory 51, 2089–2102 (2005)
Dembowski, P., Ostrom, T.G.: Planes of order \(n\) with collineation groups of order \(n^2\). Math. Z. 103, 239–258 (1968)
Fried, M., Jarden, M.: Field Arithmetic. Springer, Berlin (1986)
Ganley, M.J., Spence, E.: Relative difference sets and quasiregular collineation groups. J. Combin. Theory Ser. A 19, 134–153 (1975)
Lang, S.: Algebra, revised third edition. Springer, New York (2002)
Leep, D.B., Yeomans, C.C.: The number of points on a singular curve over a finite field. Arch. Math. (Basel) 63, 420–426 (1994)
Lidl, R., Niederreiter, H.: Finite Fields. Addison-Wesley, Reading (1983)
Lorenzini, D.: Reducibility of polynomials in two variables. J. Algebra 156, 65–75 (1993)
Mit’kin, D.A.: Polynomials with minimal set of values and the equation \(f(x)=f(y)\) in a finite prime field. Math. Notes 38, 513–520 (1985)
Nyberg, K., Knudsen, L.R.: Provable security against differential cryptanalysis. In: Advances in Cryptology (CRYPTO ’92), Lecture Notes in Computer Science 740, pp. 566–574. Springer (1992)
Scherr, Z., Zieve, M.E.: Planar monomials in characteristic \(2\). Ann. Comb. 18, 723–729 (2014)
Schmidt, K.-U., Zhou, Y.: Planar functions over fields of characteristic two. J. Algebr. Comb. 40, 503–526 (2014)
Weil, A.: Sur les Courbes Algébriques et les Variétés qui s’en Déduisent, Paris (1948)
Zhou, Y.: \((2^n,2^n,2^n,1)\)-relative difference sets and their representations. J. Combin. Des. 21, 563–584 (2013)
Zieve, M.E.: Exceptional polynomials. In: Handbook of Finite Fields, 229–233. CRC Press (2013)
Zieve, M.E.: Planar functions and perfect nonlinear monomials over finite fields. Des. Codes Cryptogr. 75, 71–80 (2015)
Author information
Authors and Affiliations
Corresponding author
Additional information
The authors thank the referees for their useful comments. The second author thanks the NSF for support under Grant DMS-1162181.
Rights and permissions
About this article
Cite this article
Müller, P., Zieve, M.E. Low-degree planar monomials in characteristic two. J Algebr Comb 42, 695–699 (2015). https://doi.org/10.1007/s10801-015-0597-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10801-015-0597-y