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This Lambert Publication is a part of a continuing program of paperbound textbooks especially designed for students in developing countries at a reduced price. This book presents a novel public key cryptosystem (based on a Finite Non-Abelian Groups). There may be other Non-Abelian Groups can be used in our system. However we must be careful in applying a Non-Abelian Group to our cryptosystem in order that the cryptosystem is secure. The existence of Abelian normal subgroup reduces the security of the cryptosystems. So any Abelian normal subgroup must be of small order. In the proposed algorithm expressing an element of G is efficient and selection of parameter also easier comparatively. In future work, we have planned to do time complexity analysis and security analysis using cryptographic softwares.