On the Planarity of Certain Dembowski-Ostrom Polynomials

Year 2022, Volume: 17 Issue: 2, 261 - 269, 25.11.2022

Zehra Aksoy , Barış Bülent Kırlar

https://doi.org/10.29233/sdufeffd.1053097

Abstract

Planar mappings, defined by Dembowski and Ostrom, are identified as a means to construct projective planes. Then, many important applications of planar mappings appear in different fields such as cryptography and coding theory. In this paper, we provide sufficient and necessary conditions for the planarity of certain Dembowski-Ostrom polynomials over the finite field extension of degree three with odd characteristic. In particular, we completely determine the coefficients of the given Dembowski-Ostrom polynomials to be planar.

Keywords

Planar mappings, Linearized polynomials, Dembowski-Ostrom polynomials

Supporting Institution

Scientific Research Fund of Suleyman Demirel University

Project Number

FYL-2020-7985

Bazı Dembowski-Ostrom Polinomlarının Planaritesi Üzerine

Abstract

Dembowski ve Ostrom tarafından tanımlanan planar dönüşümler projektif düzlemler oluşturmanın bir yolu olarak ortaya çıkmıştır. Sonrasında, planar dönüşümlerin kriptografi ve kodlama teorisi gibi farklı alanlarda birçok önemli uygulaması yapılmıştır. Bu çalışmada, tek karakteristiğe sahip üçüncü dereceden sonlu cisim genişlemeleri üzerinde tanımlanan belirli bir formdaki Dembowski-Ostrom polinomlarının planaritesi için gerek ve yeter koşullar elde edilmiştir. Özel olarak, verilen Dembowski-Ostrom polinomlarının planar olmasını sağlayan katsayılar tamamıyla belirlenmiştir.

Keywords

Lineerize polinomlar, Dembowski-Ostrom polinomları, Planar dönüşümler

Project Number

FYL-2020-7985

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