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)].


12K10 Semifields
51E15 Finite affine and projective planes (geometric aspects)
51A40 Translation planes and spreads in linear incidence geometry
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