As a special factorization category of finite groups, logarithmic signature(LS) is used as the main component of cryptographic keys that operate within secret key cryptosystems such as PGM and public key cryptosystems like M ST1, M ST2 and M ST3. An LS with the shortest length is called a minimal logarithmic signature(MLS) that constitutes of the smallest sized blocks and offers the lowest complexity, and is therefore desirable for cryptographic constructions. However, the existence of MLSs for finite groups should be firstly taken into an account. The MLS conjecture states that every finite simple group has an MLS. If it holds, then by the consequence of Jordan-H¨older Theorem, every finite group would have an MLS. In fact, many cryptographers and mathematicians are keen for solving this problem. Some effective work has already been done in search of MLSs for finite groups. Recently, we have made some progress towards searching a minimal length key for MST cryptosystems and presented a theoretical proof of MLS conjecture. As a special factorization category of finite groups, logarithmic signature (LS) is used as the main component of cryptographic keys that operate within secret key cryptosystems such as PGM and public key cryptosystems like M ST1, M ST2 and M ST3. An LS with the shortest length is called minimally logarithmic signature (MLS) that constitutes the smallest sized blocks and offers the lowest complexity, and is desirable for cryptographic constructions. However, the existence of MLSs for bespoke groups should be called taken into an account. MLS conjecture states that every finite simple group has an MLS. If it holds, then by the consequence of Jordan-H¨older Theorem, every finite group would have an MLS. In fact, many cryptographers and mathematicians are keen for solving this problem. Some effective work has already been done in search of MLSs for finite groups. Recently, we have made some progress towards searching a minimal length key for MST cryptosystems and presented a theoretical proof of MLS conjecture.