Abstract
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 g = a 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.
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References
Artin M.: Algebra. Prentice Hall Inc, Englewood Cliffs, NJ (1991)
Bereczky Á.: Maximal overgroups of singer elements in classical groups. J. Algebra 234, 187–206 (2000)
Biggs N., White A.T.: Permutation groups and combinatorial structures. Cambridge University Press, Cambridge-New York (1979)
Bohli J.M., Steinwandt R., González Vasco M.I., Martínez C.: Weak Keys in MST 1. Des. Codes Cryptogr. 37, 509–524 (2005)
Carter R.W.: Finite Groups of Lie Type: Conjugacy Classes and Complex Characters. John Wiley & Sons Ltd, New York (1985)
Conway J.H., Curtis R.T., Norton S.P., Parker R.A., Wilson R.A.: Atlas of Finite Groups. Clarendon Press, Oxford (1985)
Cossidente A., De Resmini M.J.: Remarks on singer cyclic groups and their normalizers. Des. Codes Cryptogr. 32, 97–102 (2004)
Garrett P.: Buildings and classical groups. Chapman & Hall, London (1997)
González Vasco M.I., Steinwandt R.: Obstacles in two public key cryptosystems based on group factorizations. Tatra Mt. Math. Publ. 25, 23–37 (2002)
González Vasco M.I., Rötteler M., Steinwandt R.: On minimal length factorizations of finite groups. Exp. Math. 12, 1–12 (2003)
Gorenstein D.: Finite Groups. Chelsea Publ. Co., New York (1980)
Hajós G.: Többméretű terek befedése kockaráccsal. Mat. Fiz. Lapok. 45, 171–190 (1938)
Hestenes M.D.: Singer groups. Canad. J. Math. 22, 492–513 (1970)
Hirschfeld J.: Projective geometries over finite fields. The Clarendon Press Oxford University Press, New York (1998)
Holmes P.E.: On minimal factorisations of sporadic groups. Exp. Math. 13, 435–440 (2004)
Lempken W., van Trung T.: On minimal logarithmic signatures of finite groups. Exp. Math. 14, 257–269 (2005)
Lempken W., Magliveras S.S., van Trung T., Wei W.: A public key cryptosystem based on non-abelian finite groups. J. Cryptol. 22, 62–74 (2009)
Magliveras S.S.: A Cryptosystem from logarithmic signatures of finite groups. In Proceedings of the 29th Midwest Symposium on Circuits and Systems, pp. 972–975. Elsevier Publishing Company, Amsterdam (1986).
Magliveras S.S.: Secret and public-key cryptosystems from group factorizations. Tatra Mt. Math. Publ. 25, 11–22 (2002)
Magliveras S.S., Memon N.D: Algebraic properties of cryptosystem PGM. J. Cryptol. 5, 167–183 (1992)
Magliveras S.S., Stinson D.R., van Trung T.: New approaches to designing public key cryptosystems using one-way functions and trapdoors in finite groups. J. Cryptol. 15, 285–297 (2002)
Magliveras S.S., Svaba P., van Trung T., Zajac P.: On the security of a realization of cryptosystem MST 3. Tatra Mt. Math. Publ. 41, 1–13 (2008)
Minkowski H.: Geometrie der Zahlen. Teubner, Leipzig (1896)
Minkowski H.: Diophantische Approximationem. Teubner, Leipzig (1907)
Singer J.: A theorem in finite projective geometry and some applications to number theory. Trans. Am. Math. Soc. 43, 377–385 (1938)
Singhi N., Singhi N.: Minimal logarithmic signatures for classical groups (preprint).
Szabó S.: Topics in Factorization of Abelian Groups. Birkhäuser Verlag, Basel (2004)
Wilson R.A.: The finite simple groups (in preparation). http://www.maths.qmul.ac.uk/~raw/fsgs.html.
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Communicated by Ron Mullin, Rainer Steinwandt.
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Singhi, N., Singhi, N. & Magliveras, S. Minimal logarithmic signatures for finite groups of Lie type. Des. Codes Cryptogr. 55, 243–260 (2010). https://doi.org/10.1007/s10623-010-9369-9
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DOI: https://doi.org/10.1007/s10623-010-9369-9