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

Let q = p^r  where  p  is prime and  r  is a positive integer. A planar function on F_q is a function  f : F_q →F_q  such that, for every nonzero element b  in F_q, the function c → f(c+b) - f(c)  is a bijection on F_q.  Planar functions can be used to construct finite projective planes. They also arise in cryptography, where they are called perfect nonlinear functions since these functions are optimally resistant to linear and differential cryptanalysis when used in DES-like cryptosystems.

We determine all planar functions on F_q which have the form c→c^t where t  is coprime to  p  and  q ≥ (t-1)⁴.  As a consequence, we prove two conjectures of Hernando, Mcguire and Monserrat. We also give a simple proof of a 1971 conjecture of Segre and Bartocci about monomial hyperovals in Desarguesian projective planes, which was first proved recently by Hernando and McGuire via a different argument.

Trivia: this paper established a personal record in an event which I hadn't previously conceived of. Namely, out of all my papers, this one involved the shortest length of time between the day I learned of the question (which occurred when I saw this paper) and the day on which my paper answering the question was cited in someone else's paper (namely, in this paper).

Michael Zieve:  home page   publication list