Designs, Codes and Cryptography

, Volume 55, Issue 2–3, pp 243–260 | Cite as

Minimal logarithmic signatures for finite groups of Lie type



A logarithmic signature (LS) for a finite group G is an ordered tuple α =  [A 1, A 2, . . . , A n ] of subsets A i of G, such that every element \({g \in G}\) can be expressed uniquely as a product ga 1 a 2 . . . a n , where \({a_i \in A_i}\). The length of an LS α is defined to be \({l(\alpha)= \sum^{n}_{i=1}|A_i|}\). It can be easily seen that for a group G of order \({\prod^k_{j=1}{p_j}^{m_j}}\), the length of any LS α for G, satisfies, \({l(\alpha) \geq \sum^k_{j=1}m_jp_j}\). An LS for which this lower bound is achieved is called a minimal logarithmic signature (MLS) (González Vasco et al., Tatra Mt. Math. Publ. 25:2337, 2002). The MLS conjecture states that every finite simple group has an MLS. This paper addresses the MLS conjecture for classical groups of Lie type and is shown to be true for the families PSL n (q) and PSp 2n (q). Our methods use Singer subgroups and the Levi decomposition of parabolic subgroups for these groups.


Logarithmic signatures Finite groups of Lie type Simple groups Singer groups 

Mathematics Subject Classification (2000)

94A60 11T71 14G50 20G40 05E20 


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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • Nidhi Singhi
    • 1
  • Nikhil Singhi
    • 1
  • Spyros Magliveras
    • 1
  1. 1.Department of Mathematical SciencesCenter for Cryptology and Information SecurityBoca RatonUSA

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