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Low-degree planar polynomials over finite fields of characteristic two
- Publication Year :
- 2018
-
Abstract
- Planar functions are mappings from a finite field $\mathbb{F}_q$ to itself with an extremal differential property. Such functions give rise to finite projective planes and other combinatorial objects. There is a subtle difference between the definitions of these functions depending on the parity of $q$ and we consider the case that $q$ is even. We classify polynomials of degree at most $q^{1/4}$ that induce planar functions on $\mathbb{F}_q$, by showing that such polynomials are precisely those in which the degree of every monomial is a power of two. As a corollary we obtain a complete classification of exceptional planar polynomials, namely polynomials over $\mathbb{F}_q$ that induce planar functions on infinitely many extensions of~$\mathbb{F}_q$. The proof strategy is to study the number of $\mathbb{F}_q$-rational points of an algebraic curve attached to a putative planar function.~Our methods also give a simple proof of a new partial result for the classification of almost perfect nonlinear~functions.
- Subjects :
- Mathematics - Combinatorics
Mathematics - Algebraic Geometry
51E20, 11T71
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.1809.06271
- Document Type :
- Working Paper