On minimal logarithmic signatures of finite groups

Abstract

Logarithmic signatures (LS) are a kind of factorization of finite groups which are used as a main component of cryptographic keys for secret key cryptosystems such as PGM and public key cryptosystems like MST1. As such, logarithmic signatures of short length are of special interest. In the present paper we deal with the fundamental question of the existence of logarithmic signatures of shortest length, called minimal logarithmic signatures (MLS), for finite groups. Studies of the problem can be found in several articles, especially in “On Minimal Length Factorizations of Finite Groups,” where González Vasco, Rö}tteler, and Steinwandt show that minimal logarithmic signatures exist for all groups of order less than 175,560 by direct computation using the method of factorization of a group into "disjoint'' subgroups. We introduce new approaches to deal with the question. The first method uses the double coset decomposition to construct minimal logarithmic signatures. This method allows one to prove, for instance, that if {\small $\gcd(n,q-1)\in \{1,4, p \ | \ p \ \mbox{prime} \}$}, then the projective special linear groups {\small $L_n(q)$} have an MLS. Another main goal of this paper is to construct MLS for all finite groups of order {\small $\leq 10^{10}$}. Surprisingly, the method of double coset decomposition turns out to be very effective, as we can construct MLS for all groups in the range except eight groups. We are also able to prove that if an MLS for any of these eight groups exists, then it cannot be constructed by the method of double coset decomposition. We further discuss a method of construction of MLS for groups of the form {\small $G=A.B$} with subgroups {\small $A$}, {\small $B$} and {\small $A\cap B \not= 1$}, by building suitable MLS for {\small $A$} and {\small $B$} and "gluing'' them together.