Algebraic properties of cryptosystem PGM

In the late 1970s Magliveras invented a private-key cryptographic system called Permutation Group Mappings (PGM). PGM is based on the prolific existence of certain kinds of factorization sets, called logarithmic signatures, for finite permutation groups. PGM is an endomorphic system with message space ℤ_|G| for a given finite permutation group G. In this paper we prove several algebraic properties of PGM. We show that the set of PGM transformations ℐ _G is not closed under functional composition and hence not a group. This set is 2-transitive on ℤ_|G| if the underlying group G is not hamiltonian and not abelian. Moreover, if the order of G is not a power of 2, then the set of transformations contains an odd permutation. An important consequence of these results is that the group generated by the set of transformations is nearly always the symmetric group ℒ_|G|. Thus, allowing multiple encryption, any permutation of the message space is attainable. This property is one of the strongest security conditions that can be offered by a private-key encryption system.