Asymmetric cryptography based on groups is mainly concerned with the role of noncommutative groups. The origin of group based cryptography goes back in the 1980s. Since then, numerous cryptographic proposals based on noncommutative groups have been evolved. In this thesis, we consider several noncommutative settings in di erent cryptographic context. This study adds to the cryptographic literature by demonstrating new asymmetric cryptographic schemes. On the one hand, two new public key exchange protocols and two asymmetric cryptosystems are constructed. On the other hand, di erent primitives like suggestion of plat- form, size of di erent parameters, security and e ciency aspects regarding these proposals, are also elaborated.