A Search for Differentially-6 Uniform (n, n-2) Functions

Abstract

Finding cryptographic primitives satisfying certain properties is a difficult problem. In this domain, besides the algebraic constructions, researchers often use heuristics. There exists a set of interesting problems related to the notion of differential uniformity for a function F: F2n→ F2m. When n=m, then the best obtainable differential uniformity equals 2, since it is necessarily positive and even, and since examples of differentially 2-uniform functions are known. Heuristics are able to reach such functions; there is then some intuition that heuristics can be used for other open problems related to differential uniformity. When n > m > n/2, differential uniformity is bounded by 2n-m+2 from below (when m=n-2, by 6). Unfortunately, we know such functions only for dimensions equal to n=4,5. In this paper, we explore several evolutionary algorithms and problem sizes in order to find functions having differential uniformity equal to 6. Our results show that several solution encodings are able to find such functions but only in dimensions (4, 2) and (5, 3). Since differentially 6-uniform functions were known for those sizes before, our results can be used as a source of new functions in those dimensions and as an indicator that for (6, 4) such functions either do not exist or that it is extremely difficult to find them.