In this thesis we present methods of fi nding discrete logarithms in fi elds GF(p) and GF(p^2). Based on the discrete logarithm, an identity-based (ID-based) cryptosystem is constructed. We shall illustrate the discrete logarithm method of decryption and ID-based method of encryption. The notion of the ID-based system can be applied to all of the discrete logarithm based cryptosystems without changing their original design. This is advantageous to public-key cryptosystems since the public-key authentication is easy and this reduces key management.