Permutation binomials and their groups

Abstract

This paper is devoted to studying the properties of permutation binomials over finite fields and the possibility to use permutation binomials as encryption functions. We present an algorithm for enumeration of permutation binomials. Using this algorithm, all permutation binomials for finite fields up to order 15000 were generated. Using this data, we investigate the groups generated by the permutation binomials and discover that over some finite fields $$ {{\mathbb F}_q} $$ , every bijective function on [1..q − 1] can be represented as a composition of binomials. We study the problem of generating permutation binomials over large prime fields. We also prove that a generalization of RSA using permutation binomials is not secure. Bibliography: 9 titles.