As a special type of factorization of finite groups, logarithmic signature(LS) is used as the main component of cryptographic keys for secret keycryptosystems such as PGM and public key cryptosystems like MST1, MST2 andMST3. An LS with the shortest length, called a minimal logarithmic signature(MLS), is even desirable for cryptographic applications. The MLS conjecturestates that every finite simple group has an MLS. Recently, the conjecture hasbeen shown to be true for general linear groups GLn(q), special linear groupsSLn(q), and symplectic groups Spn(q) with q a power of primes and fororthogonal groups On(q) with q as a power of 2. In this paper, we present newconstructions of minimal logarithmic signatures for the orthogonal group On(q)and SOn(q) with q as a power of odd primes. Furthermore, we give constructionsof MLSs for a type of classical groups projective commutator subgroup.
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