(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 determine all planar functions on F_q of the form c → ac^t,  where q is a power of 2, t is a positive integer such that t⁴ ≤ q,  and a is any nonzero element of F_q.  This settles and strengthens a conjecture of Schmidt and Zhou.

Michael Zieve:  home page   publication list