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Commutative semifields from projection mappings. (English) Zbl 1241.12004

Let \(q\) be a power of the odd prime \(p\), \(m\) an odd natural number, and let \(F\) be the field with \(q^{2m}\) elements. Sitting inside \(F\) are the fields \(F_q\), \(K\), \(L\), with \(q\), \(q^2\), \(q^m\) elements, respectively.
For any map \(f: F\to F\) define the polarization \(x\ast y:=(f(x+y)-f(x)-f(y))/2\). Then \(f\) is called planar if the map \(x\mapsto x\ast a\) is bijective for each \(a\in F\setminus\{0\}\). Furthermore, \(f\) is called quadratic if a polynomial representation \(f(x)=\sum a_ix^i\) has the property \(a_i\neq 0\Longrightarrow i\) is a sum of two powers of \(p\). It is an easy exercise to show that the polarization of a quadratic map is symmetric bilinear over the prime field of \(F\). Thus the polarization of a quadratic planar map is a commutative presemifield multiplication on \(F\).
The author constructs a quadratic planar map \(f\) based on the trace \(F\to L\) in such a way that \(F_q\) is contained in the left nucleus and \(K\) in the middle nucleus of the semifield \((F,+,\ast)\). Moreover, we have \(u\ast x=ux\) for all \(u\in K\).
This way the author generalizes a construction of G. Lunardon et al. [“Symplectic spreads and quadric veroneseans” (manuscript)].
Modifying his own approach the author also gives a new way to generalize and simplify the construction of a family of commutative semifields due to L. Budaghyan and T. Helleseth [“New commutative semifields defined by new PN multinomials”, Cryptogr. Commun. 3, 1–16 (2011; Zbl 1291.12006); “New perfect nonlinear multinomials over \(\mathbb F_{p^{2k}}\) for any odd prime \(p\)”, Lect. Notes Comput. Sci. 5203, 403–414 (2008; Zbl 1177.94137)].

MSC:

12K10 Semifields
51E15 Finite affine and projective planes (geometric aspects)
51A40 Translation planes and spreads in linear incidence geometry
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[1] Albert A.A.: On nonassociative division algebras. Trans. Am. Math. Soc. 72, 296–309 (1952) · Zbl 0046.03601 · doi:10.1090/S0002-9947-1952-0047027-4
[2] Ball S., Brown M.R.: The six semifield planes associated with a semifield flock. Adv. Math. 189, 68–87 (2004) · Zbl 1142.12305 · doi:10.1016/j.aim.2003.11.006
[3] Bierbrauer J.: New semifields, PN and APN functions. Des. Codes Cryptogr. 54, 189–200 (2010) · Zbl 1269.12006 · doi:10.1007/s10623-009-9318-7
[4] Bierbrauer J.: New commutative semifields and their nuclei. In: Bras-Amorós, M., Høholdt, T. (eds) Proceedings of AAECC-18, Lecture Notes in Computer Science, vol. 5527, pp. 179–185. Tarragona, Spain (2009) · Zbl 1273.12006
[5] Budaghyan L., Helleseth T.: New commutative semifields defined by new PN multinomials, Cryptography and Communications (to appear). · Zbl 1291.12006
[6] Budaghyan L., Helleseth T.: New perfect nonlinear multinomials over $${\(\backslash\)mathbb{F}_{p\^{2k}}}$$ for any odd prime p. In: Sequences and their applications-SETA 2008. LNCS, vol. 5203, pp. 403–414. Springer, Heidelberg (2008). · Zbl 1177.94137
[7] Coulter R.S., Henderson M.: Commutative presemifields and semifields. Adv. Math. 217, 282–304 (2008) · Zbl 1194.12007 · doi:10.1016/j.aim.2007.07.007
[8] Coulter R.S., Henderson M., Kosick P.: Planar polynomials for commutative semifields with specified nuclei. Des. Codes Cryptogr. 44, 275–286 (2007) · Zbl 1215.12012 · doi:10.1007/s10623-007-9097-y
[9] Dembowski P., Ostrom T.G.: Planes of order n with collineation groups of order n 2. Math. Z. 103, 239–258 (1968) · Zbl 0163.42402 · doi:10.1007/BF01111042
[10] Dickson L.E.: On commutative linear algebras in which division is always uniquely possible. Trans. Am. Mat. Soc 7, 514–522 (1906) · doi:10.1090/S0002-9947-1906-1500764-6
[11] Lunardon G., Marino G., Polverino O., Trombetti R.: Symplectic spreads and quadric veroneseans (manuscript) · Zbl 1228.51003
[12] Zha Z., Kyureghyan G.M., Wang X.: Perfect nonlinear binomials and their semifields. Finite Fields Appl. 15, 125–133 (2009) · Zbl 1194.12003 · doi:10.1016/j.ffa.2008.09.002
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