Some Remarks on the Logarithmic Signatures of Finite Abelian Groups

In the paper about the cryptosystem MST3, Svaba and Trung proposed a way to build a cryptosystem based on the concept of logarithmic signatures, and they choose Suzuki’s group, which is not abelian for implementing. Recently, to reason why these methods cannot be applied to abelian groups; Svaba, Trung and Wolf developed some algorithms to factorize the fused transversal logarithmic signatures (FTLS). Their attacks can be avoided by some modifications, which is the aim of this paper, where we will use the weakness of the discrete logarithm problem (DLP) to propose two cryptosystems. The first one is based on the new concept about quasi-logarithmic signature of finite solvable groups, which is the generalization of logarithmic signatures. The second is built on the logarithmic signatures of finite cyclic 2-groups, which include two interesting examples on Pell’s curves and elliptic curves over finite fields.