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Cryptography is one of the major concerned areas of computer and data security and a very promising direction in cryptography research is known as DNA Cryptography. DNA computational logic can be used in cryptography for encrypting, storing and transmitting the information, as well as for computation. Although in its primitive stage, DNA cryptography is shown to be very effective. In this paper, a proposal is given where the concept of DNA is being used in the encryption and decryption process. The theoretical analysis and implementations shows this method to be efficient in computation, storage and transmission; and it is very powerful against certain attacks. This paper also proposes a unique cipher text generation procedure as well as a new key generation procedure. Finally, to demonstrate the performance of the proposed method, its implementation is explained and the results are analyzed.