  As a special type of factorization of finite groups, logarithmic signature
(LS) is used as the main component of cryptographic keys for secret key
cryptosystems such as PGM and public key cryptosystems like MST1, MST2 and
MST3. An LS with the shortest length, called a minimal logarithmic signature
(MLS), is even desirable for cryptographic applications. The MLS conjecture
states that every finite simple group has an MLS. Recently, the conjecture has
been shown to be true for general linear groups GLn(q), special linear groups
SLn(q), and symplectic groups Spn(q) with q a power of primes and for
orthogonal groups On(q) with q as a power of 2. In this paper, we present new
constructions of minimal logarithmic signatures for the orthogonal group On(q)
and SOn(q) with q as a power of odd primes. Furthermore, we give constructions
of MLSs for a type of classical groups projective commutator subgroup.
