Minimal logarithmic signatures for the unitary group $$U_n(q)$$Un(q)

As a special type of factorization of finite groups, logarithmic signature (LS) is used as one of the main components of the private key cryptosystem $$PGM$$PGM and the public key cryptosystems $$MST_1$$MST1, $$MST_2$$MST2 and $$MST_3$$MST3. An LS with the shortest length is called a minimal logarithmic signature (MLS) and is even desirable for cryptographic constructions. The MLS conjecture states that every finite simple group has an MLS. Recently, Singhi et al. proved that the MLS conjecture is true for some families of simple groups. In this paper, we prove the existence of MLSs for the unitary group $$U_n(q)$$Un(q) and construct MLSs for a type of simple groups—the projective special unitary group $$PSU_n(q)$$PSUn(q).

[1]  William M. Kantor,et al.  Spreads, Translation Planes and Kerdock Sets. I , 1982 .

[2]  Jan De Beule,et al.  Partial ovoids and partial spreads in finite classical polar spaces , 2008 .

[3]  Tran van Trung,et al.  Pseudorandom number generators based on random covers for finite groups , 2012, Des. Codes Cryptogr..

[4]  Petra E. Holmes On Minimal Factorisations of Sporadic Groups , 2004, Exp. Math..

[5]  K. Brown,et al.  Graduate Texts in Mathematics , 1982 .

[6]  Nasir D. Memon,et al.  Algebraic properties of cryptosystem PGM , 1992, Journal of Cryptology.

[7]  J. Thas Ovoids and spreads of finite classical polar spaces , 1981 .

[8]  Martin Rötteler,et al.  On Minimal Length Factorizations of Finite Groups , 2003, Exp. Math..

[9]  Robert A. Wilson,et al.  The finite simple groups , 2009 .

[10]  Tran van Trung,et al.  On Minimal Logarithmic Signatures of Finite Groups , 2005, Exp. Math..

[11]  Spyros S. Magliveras,et al.  A Public Key Cryptosystem Based on Non-abelian Finite Groups , 2008, Journal of Cryptology.

[12]  Paul B. Garrett,et al.  Buildings and Classical Groups , 1997 .

[13]  Spyros S. Magliveras,et al.  Minimal logarithmic signatures for finite groups of Lie type , 2010, Des. Codes Cryptogr..

[14]  R. H. Dye Maximal Subgroups of Finite Orthogonal Groups Stabilizing Spreads of Lines , 1986 .

[15]  Peter W. Shor,et al.  Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer , 1995, SIAM Rev..

[16]  Nidhi Singhi,et al.  Minimal logarithmic signatures for classical groups , 2011, Des. Codes Cryptogr..

[17]  Tran van Trung,et al.  Public key cryptosystem MST3: cryptanalysis and realization , 2010, J. Math. Cryptol..

[18]  László Babai,et al.  On the Number of p -Regular Elements in Finite Simple Groups , 2009 .

[19]  Douglas R. Stinson,et al.  New Approaches to Designing Public Key Cryptosystems Using One-Way Functions and Trapdoors in Finite Groups , 2001, Journal of Cryptology.

[20]  Antonio Cossidente,et al.  Remarks on Singer Cyclic Groups and Their Normalizers , 2004, Des. Codes Cryptogr..