Factorizations in the elementary Abelian p-group and their cryptographic significance

Abstract

Let G be a finite group and let A i 1 ≤ i ≤ s, be subsets of G where ¦A i ¦ ? 2, 1 ≤ i ≤ s and s ? 2. We say that (A1, A2,..., A3) is a factorization of G if and only if for each g ? G there is exactly one way to express g = a 1 a 1 a 2··· a 3, where a j ? A i , 1 ≤ i ≤ s. The problem of finding factorizations of this type was first introduced by Hajos [3] in 1941. Since then a number of papers have appeared on the subject. More recently, Magliveras [6] has applied factorization of permutation groups to cryptography to obtain a private-key cryptosystem. Factorizations in the elementary abelian p-group were exploited (but not explicitly stated in these terms) by Webb [13] to produce a public-key cryptosystem conceptually similar to cryptosystems based on the knapsack problem. Using the result that certain types of factorizations in the elementary abelian p-group are necessarily transversal (a term introduced by Magliveras), this paper shows that the public-key system proposed by Webb is insecure.