A symmetric key cryptosystem based on logarithmic signatures for finite permutation groups was described by the first author in [6], and its algebraic properties were studied in [7]. In this paper we describe two possible approaches to the construction of new public key cryptosystems with message space a large finite group G, using logarithmic signatures and their generalizations. The first approach relies on the fact that permutations of the message space G induced by transversal logarithmic signatures almost always generate the full symmetric group S G on the message space. The second approach could potentially lead to new ElGamal-like systems based on trapdoor, oneway functions induced by logarithmic signature-like objects we call meshes, which are uniform covers for G.