  The study of finite projective planes involves planar functions, namely,
functions f : F_q --> F_q such that, for each nonzero a in F_q, the function c
--> f(c+a) - f(c) is a bijection on F_q. Planar functions are also used in the
construction of DES-like cryptosystems, where they are called perfect nonlinear
functions. We determine all planar functions on F_q of the form c --> c^t,
under the assumption that q >= (t-1)^4. This implies two conjectures of
Hernando, McGuire and Monserrat. Our arguments also yield a new proof of a
conjecture of Segre and Bartocci from 1971 about monomial hyperovals in finite
Desarguesian projective planes.
