New approaches to designing public key cryptosystems using one-way functions and trapdoors in finite groups

A symmetric key cryptosystem based on logarithmic signature s 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 signature s and their generalizations. The first approach relies on the fact that permutations of the message space G induced by transversal logarithmic signature s almost always generate the full symmetric group SG on the message space. The second approach could potentially lead to new ElGamal-like systems based on trapdoor, one-way functions induced by logarithmic signature -like objects we call meshes , which are uniform covers for G .