Abstract
The minimal logarithmic signature conjecture states that in any finite simple group there are subsets A i , 1 ≤ i ≤ k such that the size |A i | of each A i is a prime or 4 and each element of the group has a unique expression as a product \({\prod_{i=1}^k x_i}\) of elements \({x_i \in A_i}\). The conjecture is known to be true for several families of simple groups. In this paper the conjecture is shown to be true for the groups \({\Omega^-_{2m}(q), \Omega^+_{2m}(q)}\), when q is even, by studying the action on suitable spreads in the corresponding projective spaces. It is also shown that the method can be used for the finite symplectic groups. The construction in fact gives cyclic minimal logarithmic signatures in which each A i is of the form \({\{y_i^j \ |\ 0 \leq j < |A_i|\}}\) for some element y i of order ≥ |A i |.
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References
André J.: Über nicht-Desarguessche Ebenen mit transitiver Translationsgruppe. Math. Z. 60, 156–186 (1954)
Babai L., Pálfy P.P., Saxl J.: On the number of p regular elements in finite simple groups. LMS J. Comput. Math. 12, 82–119 (2009)
Bereczky Á.: Maximal overgroups of singer elements in classical groups. J. Algebra 234, 187–206 (2000)
De Beule J., Klein A., Metsch K., Storme L.: Partial ovoids and partial spreads of classical finite polar spaces. Serdica Math. J. 34, 689–714 (2008)
Bohli J.M., Steinwandt R., González Vasco M.I., Martínez C.: Weak keys in MST 1. Des. Codes Cryptogr. 37, 509–524 (2005)
Bruck R.H., Bose R.C.: The construction of translation planes from projective spaces. J. Algebra 1, 85–102 (1964)
Dye R.H.: Partitions and their stabilizers for line complexes and quadrics. Ann. Math. Pura Appl. 114(4), 173–194 (1977)
Dye R.H.: Maximal subgroups of finite orthogonal groups stabilizing spreads of lines. J. Lond. Math. Soc. 33(2), 279–293 (1986)
Dye R.H.: Spreads and classes of maximal subgroups of GL n (q), SL n (q), PGL n (q) and PSL n (q). Ann. Math. Pura Appl. 158(4), 33–50 (1991)
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)
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)
Kantor W.M.: Spreads, translation planes and Kerdock sets. I. SIAM J Algebraic Discret. Methods 3, 151–165 (1982)
Kleidman P., Liebeck M.: The subgroup structure of the finite classical groups. London Mathematical Society Lecture Note Series, vol. 129. Cambridge University Press, Cambridge (1990).
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., 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)
Singhi N., Singhi N., Magliveras S.S.: Minimal logarithmic signatures for finite groups of lie type. Des. Codes Cryptogr. 55, 243–260 (2010)
Thas J.A.: Ovoids and spreads of finite classical polar spaces. Geom. Dedicata 10, 135–143 (1981)
Wilson R.A.: The finite simple groups. Graduate Texts in Mathematics, vol 251. Springer-Verlag, London (2009)
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Communicated by J. D. Key.
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Singhi, N., Singhi, N. Minimal logarithmic signatures for classical groups. Des. Codes Cryptogr. 60, 183–195 (2011). https://doi.org/10.1007/s10623-010-9427-3
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DOI: https://doi.org/10.1007/s10623-010-9427-3