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Low-degree planar monomials in characteristic two. (English) Zbl 1343.51005
Planar functions over finite fields give rise to finite projective planes and other combinatorial objects. They exist only in odd characteristic, but recently Y. Zhou [J. Combin. Des. 21, No. 12, 563–584 (2013; Zbl 1290.05038)] introduced an even characteristic analogue which has similar applications. In this paper, the authors focus on monomial functions on \(\mathbb F_q\) of the form \(f(x) = cx^t\). In particular, they show that, if \(q\) is a power of \(2\), \(t\) is an integer with \(0 < t \leq q^{1/4}\) and \(c \in \mathbb F_q^*\), then a monomial function is planar on \(\mathbb F_q\) if and only if \(t\) is a power of \(2\). This extends a result in [K.-U. Schmidt and Y. Zhou, J. Algebr. Comb. 40, No. 2, 503–526 (2014; Zbl 1319.51008)], where the authors proved that \(t=1\) is the only odd value such that \(f(x)\) is planar for some \(c \neq 0\) over infinitely many fields.

51E20 Combinatorial structures in finite projective spaces
11T06 Polynomials over finite fields
11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
05B05 Combinatorial aspects of block designs
Full Text: DOI arXiv
[1] Aubry, Y., Perret, M.: A Weil theorem for singular curves. In: Arithmetic, Geometry and Coding Theory, pp. 1-7. de Gruyte (1996) · Zbl 0873.11037
[2] Capelli, A, Sulla riduttibilità delle equazioni algebriche, Ren. Accad. Sc. Fis. Mat. Soc. Napoli, 4, 84-90, (1898) · JFM 29.0071.03
[3] 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) · Zbl 1192.94114
[4] Dembowski, P; Ostrom, TG, Planes of order \(n\) with collineation groups of order \(n^2\), Math. Z., 103, 239-258, (1968) · Zbl 0163.42402
[5] Fried, M., Jarden, M.: Field Arithmetic. Springer, Berlin (1986) · Zbl 0625.12001
[6] Ganley, MJ; Spence, E, Relative difference sets and quasiregular collineation groups, J. Combin. Theory Ser. A, 19, 134-153, (1975) · Zbl 0313.50014
[7] Lang, S.: Algebra, revised third edition. Springer, New York (2002)
[8] Leep, DB; Yeomans, CC, The number of points on a singular curve over a finite field, Arch. Math. (Basel), 63, 420-426, (1994) · Zbl 0819.11023
[9] Lidl, R., Niederreiter, H.: Finite Fields. Addison-Wesley, Reading (1983) · Zbl 0554.12010
[10] Lorenzini, D, Reducibility of polynomials in two variables, J. Algebra, 156, 65-75, (1993) · Zbl 0791.12001
[11] Mit’kin, DA, Polynomials with minimal set of values and the equation \(f(x)=f(y)\) in a finite prime field, Math. Notes, 38, 513-520, (1985) · Zbl 0591.12023
[12] 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) · Zbl 0824.68037
[13] Scherr, Z; Zieve, ME, Planar monomials in characteristic \(2\), Ann. Comb., 18, 723-729, (2014) · Zbl 1310.51006
[14] Schmidt, K-U; Zhou, Y, Planar functions over fields of characteristic two, J. Algebr. Comb., 40, 503-526, (2014) · Zbl 1319.51008
[15] Weil, A.: Sur les Courbes Algébriques et les Variétés qui s’en Déduisent, Paris (1948) · Zbl 0791.12001
[16] Zhou, Y, \((2^n,2^n,2^n,1)\)-relative difference sets and their representations, J. Combin. Des., 21, 563-584, (2013) · Zbl 1290.05038
[17] Zieve, M.E.: Exceptional polynomials. In: Handbook of Finite Fields, 229-233. CRC Press (2013) · Zbl 1314.51003
[18] Zieve, ME, Planar functions and perfect nonlinear monomials over finite fields, Des. Codes Cryptogr., 75, 71-80, (2015) · Zbl 1314.51003
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