Almost perfect and planar functions
type: Item
instance of: scholarly article
title: Almost perfect and planar functions
zbMATH Open document ID: 1351.51004
author: Alexander Pott
published in: Designs, Codes and Cryptography
publication date: 2016-02-02T00:00:00Z
review text: A function \(f\) from the finite field \(F_{p^n}\) to \(F_{p^n}\) is called planar if the equation \(f(x+a)-f(x)=b\) has exactly one solution for all \(b\) and all \(a\neq 0\), or almost perfect nonlinear (APN) if it has 0 or 2 solutions for all \(b\) and all \(a\neq 0\). Planar functions require \(p\) to be odd, and APN functions require \(p\) to be 2. Also, a function from \(F_{p^n}\) to \(F_{p^n}\) is called modified planar if the equation \(f(x+a)-f(x)+ax=b\) has one solution for all \(b\) and all \(a\neq 0\). This paper surveys several known results on these three function types. It starts by looking at ways that planar and modified planar functions can be used to obtain certain relative difference sets and projective planes. For APN functions, there is some discussion of how they are related to certain codes (in particular codes with small minimum weight) and methods of checking whether two APN functions are equivalent (Walsh spectra and codes sometimes help here). There are also tables of known APN functions (including those that are monomial, sporadic, or in infinite classes) and a discussion of general methods for constructing such functions. There are also other noteworthy results related to these functions; these include connections with other types of functions (for instance, crooked and almost bent functions possess the APN property), how planar functions and certain projective planes can be obtained from semifields, and a discussion of when functions can be planar, APN or modified planar over infinitely many fields.
Mathematics Subject Classification ID: 12E20
Mathematics Subject Classification ID: 11T71
Mathematics Subject Classification ID: 51A25
Mathematics Subject Classification ID: 05B05
Mathematics Subject Classification ID: 05B10
Mathematics Subject Classification ID: 05B25
Mathematics Subject Classification ID: 51A35
Mathematics Subject Classification ID: 51A40
Mathematics Subject Classification ID: 94A60
Mathematics Subject Classification ID: 94B05
zbMATH DE Number: 6538695
zbMATH Keywords: planar function
zbMATH Keywords: almost perfect nonlinear function
zbMATH Keywords: modified planar function
zbMATH Keywords: code
zbMATH Keywords: design
zbMATH Keywords: relative difference set
zbMATH Keywords: Walsh transform
zbMATH Keywords: finite field
zbMATH Keywords: semifield
reviewed by: R. Julian R. Abel
describes a project that uses: Magma
MaRDI profile type: Publication
full work available at URL: s10623-015-0151-x
OpenAlex ID: W2285925216
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Almost perfect and planar functions (DOI)
DOI: 10.1007/S10623-015-0151-X