Realization of sporadic finite simple groups via families of exponential sums

Abstract: In a series of papers, joint with Nicholas M. Katz (Princeton U.) and Pham H. Tiep (Rutgers U.) we realize some of the sporadic simple groups as monodromy groups of parametric families of exponential sums: that is, these sums follow the same distributions as the traces of the elements of these sporadic groups on their usual representations. Finding local systems whose monodromy group is a finite group is quite unusual, and it is even more infrequent that this finite group is one of the sporadic ones. 

Some of these families have been applied to the study of almost perfect non-linear functions, a particular class of boolean function with great importance in cryptography and coding theory.