Strong external difference families in abelian and non-abelian groups

Abstract

Strong external difference families (SEDFs) have applications to cryptography and are rich combinatorial structures in their own right; until now, all SEDFs have been in abelian groups. In this paper, we consider SEDFs in both abelian and non-abelian groups. We characterize the order of groups possessing admissible parameters for non-trivial SEDFs, develop non-existence and existence results, several of which extend known results, and present the first family of non-abelian SEDFs. We introduce the concept of equivalence for EDFs and SEDFs, and begin the task of enumerating SEDFs. Complete results are presented for all groups up to order 24, underpinned by a computational approach.

Nonexistence of Strong External Difference Families in Abelian Groups of Order Being Product of At Most Three Primes

“…For example, we are able to eliminate a number of plausible parameter sets of (v, m, k, λ)-SEDF in abelian groups listed in the table [12,Remark 5.17]. At this point, we conjecture that the (243,11,22,20)-SEDF in C 5 3 constructed in [12,22] is the only SEDF with v ≤ 10 4 and m > 2.…”

“…In contrast, when m > 2, there is only one known example of such SEDF. More precisely, there is a (243,11,22,20)-SEDF in the group C 5 3 , which was discovered by two groups of researchers independently and simultaneously [12,Theorem 3.1], [22,Theorem 3.6]. In the following, we summarize some nonexistence results of SEDF with m > 2.…”

“…As an emerging combinatorial configuration proposed in [18], strong external difference family (SEDF) is under intensive study recently [1,9,11,12,16,18,22]. Roughly speaking, an SEDF is a collection of disjoint subsets of the same size in a group, so that the differences generated from these subsets cover each non-identity elements of the group the same number of times.…”

“…So far, the research in this area mostly focuses on SEDFs in abelian groups. On the other hand, a recent paper initiates the study of SEDFs in nonabelian groups [11]. It is worthy to note that various extensions of SEDFs have been intensively investigated in [10,15,18,21,22].…”