Discrete Logarithms in Finite Fields Some Algorithms for Computing New Public Key Cryptosystem

Abstract

Let p be a prime, Fp be a finite field, g be a primitive element of Fp and let h be a nonzero element of Fp. The discrete logarithm problem (DLP) is the problem of finding that an exponent k for which gk≡h (mod p). The well‐known problem of computing discrete logarithms has additional importance in recent years due to its applicability in cryptography. Several cryptographic systems would become insecure if an efficient discrete logarithm algorithm were discovered. In this paper are discused some known algorithms in this area.Most public key cryptosystems have been constructed based on abelian groups. Here we introduce how the discrete logarithm problem over a group can be seen as a special instance of an action by an abelian semigroup on finite set. The proposed new public key cryptosystem generalized the semigroup action problem due to Rosenlicht (see [8]) and shows how every semigroup action by an abelian semigroup gives rise to a Diffie‐Hellman key exchange.