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In this book,research work in symmetric and asymmetric cryptography at elementary as well as advanced level has been propagated. Some problems discussed here are beyond DES (a symmetric-key cryptosystem), RSA (a public-key cryptosystem) and are on ongoing research areas in different phases of cryptography. Lattice-based systems and NTRU offer, in principle,very good speed. They are also expected to survive the development of fairly sized quantum computers, because they are based on problems for which no quantum algorithm is known for the general case. The RSA scheme relies on the difficult of factoring large numbers, while for the ElGamal and Elliptic Curve schemes the difficulty of solving discrete logarithms provide the basis. Given that the security of these public key schemes rely on such a small number of problems that are currently considered hard, research on new schemes that are based on other classes of problems is worthwhile. The idea behind voter-verifiability in very abstract term is to provide the voter, at the time of vote casting, with a unique receipt with their vote in encrypted form.