Almost perfect and planar functions
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.
scientific article; zbMATH DE number 6538695
scientific article