(Both the published version and the arXiv version are available online.)

Planar functions are a much-studied class of functions from a finite field to itself, which have applications in combinatorics, coding theory, and cryptography. The classical definition of planar functions precluded the existence of any planar functions in characteristic 2. Recently Zhou introduced a new class of functions from even-order finite fields to themselves, which have the same types of applications as do classical planar functions. Following Zhou, if q is even then we say that a function  f : F_q →F_q  is planar if, for every nonzero b  in F_q, the function c → f(c+b) + f(c) +bc  is a bijection on F_q.  We prove a conjecture of Schmidt and Zhou which asserts that if q=Q^3 and t=Q+Q²  then certain functions of the form c → ac^t  are planar on F_q. Intriguingly, our proof relies on the following new result about Fermat curves: if  u  and v  are elements of  F_q  (with q even) such that u^Q-1 + v^Q-1 = 1,  then uv  is a cube in F_q. This result has subsequently been generalized to arbitrary q by Voloch and Zieve.

Michael Zieve:  home page   publication list