Constructing Functions with Low Differential Uniformity

The lower the differential uniformity of a function, the more resilient it is to differential cryptanalysis if used in a substitution box. APN functions and planar functions are specifically those functions which have optimal differential uniformity in even and odd characteristic, respectively. In this article, we provide two methods for constructing functions with low, but not necessarily optimal, differential uniformity. Our first method involves altering the coordinate functions of any known planar function and relies upon the relation between planar functions and orthogonal systems identified by Coulter and Matthews in 1997. As planar functions exist only over fields of odd order, the method works for odd characteristic only. The approach also leads us to a generalization of Dillon's Switching Technique for constructing APN functions. Our second construction method is motivated by a result of Coulter and Henderson, who showed in 2008 how commutative presemifields of odd order were in one-to-one correspondence with planar Dembowski-Ostrom polynomials via the multiplication of the presemifield. Using this connection as a starting point, we examine the functions arising from the multiplication of other well-structured algebraic objects such as non-commutative presemifields and planar nearfields. In particular, we construct a number of infinite classes of functions which have low, though not optimal, differential uniformity. This class of functions originally stems from the presemifields of Kantor and Williams of characteristic 2. Thus, regardless of the characteristic, between our two methods we are able to construct infinitely many functions which have low, though not optimal, differential uniformity over fields of arbitrarily large order.