The Existence of Minimal Logarithmic Signatures for Some Finite Simple Groups

Abstract

A logarithmic signature for a finite group G is a sequence α = [A₁, …, A_s] of subsets of G such that every element g ∈ G can be uniquely written in the form g = g₁…g_s, where g_i ∈ A_i, 1 ⩽ i ⩽ s. The number ∑^s_{i = 1}|A_i| is called the length of α and denoted by l(α). A logarithmic signature α is said to be minimal (MLS) if l(α) = ∑ⁿ_{i = 1}m_ip_i, where is the prime factorization of |G|. The MLS conjecture states that every finite simple group has an MLS. The aim of this article is proving the existence of a minimal logarithmic signature for the untwisted groups G₂(3ⁿ), the orthogonal groups Ω₇(q) and PΩ⁺₈(q), q is an odd prime power, the orthogonal groups Ω₉(3), PΩ⁺₁₀(3), and PΩ⁻₈(3), the Tits simple group ²F₄(2)′, the Janko group J₃, the twisted group ³D₄(2), the Rudvalis group Ru, and the Fischer group Fi₂₂. As a consequence of our results, it is proved that all finite groups of order ⩽ 10¹² other than the Ree group Ree(27), the O’Nan group O′N, and the untwisted group G₂(7) have MLS.

Keywords:

• cryptosystem
• minimal logarithmic signature
• orthogonal group
• sporadic group
• untwisted group

2000 AMS Subject Classification:

• 20D08
• 94A60

Acknowledgments

The authors are indebted to the referee for his/her suggestions and helpful remarks that leaded us to rearrange this article.

Additional information

Funding

The research of the first and second authors is partially supported by the Iran National Science Foundation under grant number 93010006.