On the number of distinct values of a class of functions over a finite field

Abstract

Several authors have recently shown that a planar function over a finite field of order q must have at least ( q + 1 ) / 2 distinct values. In this note this result is extended by weakening the hypothesis significantly and strengthening the conclusion. We also give an algorithm for determining whether a given bivariate polynomial ϕ ( X , Y ) can be written as f ( X + Y ) − f ( X ) − f ( Y ) for some polynomial f. Using the ideas of the algorithm, we then show a Dembowski–Ostrom polynomial is planar over a finite field of order q if and only if it yields exactly ( q + 1 ) / 2 distinct values under evaluation; that is, it meets the lower bound of the image size of a planar function.