Planar functions over fields of characteristic two

2013/01/29 by Kai‐Uwe Schmidt, Yue Zhou, Schmidt, Kai-Uwe +1 · 2 citations
Computer Science · Engineering · #Coding theory and cryptography #graph theory and CDMA systems #Cryptographic Implementations and Security

paper · pdf · doi:10.48550/arxiv.1301.6999

Abstract

Classical planar functions are functions from a finite field to itself and give rise to finite projective planes. They exist however only for fields of odd characteristic. We study their natural counterparts in characteristic two, which we also call planar functions. They again give rise to finite projective planes, as recently shown by the second author. We give a characterisation of planar functions in characteristic two in terms of codes over ℤ₄. We then specialise to planar monomial functions f(x)=cx^t and present constructions and partial results towards their classification. In particular, we show that t=1 is the only odd exponent for which f(x)=cx^t is planar (for some nonzero c) over infinitely many fields. The proof techniques involve methods from algebraic geometry.

Cited by

• On homogeneous planar functions
• Exceptional planar polynomials

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• Low-degree planar polynomials over finite fields of characteristic two
• Exceptional planar polynomials
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• Trinomial Planar Functions on Cubic and Quartic Extensions of Finite Fields
• Further Study of Planar Functions in Characteristic Two