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Residue Number Systems (RNS) based on ChineseRemainder Theorem (CRT) permits the representation of large integers in terms of combinations of smaller ones. The set of all integers from 0 to M-1 with RNS representation and component wise modular addition and multiplication constitutes direct sum of smaller commutative rings. Encryption and decryption algorithm based on the properties of direct sum of smaller rings offers distinct advantages over decimal or fixed radix arithmetic. In this paper representation of integer using RNS, is successfully utilized in additive, multiplicative and affine stream cipher systems.The property of the cipher system based on RNS number system allow speeding up the encryption / decryption algorithm, reduce the time complexity and provides immunity to side channel, algebraic, and known plain text attacks. In this paper, the characteristics of additive, multiplicative and affine stream cipher systems, the key generation, and encryption and decryption based on RNS number system representation are discussed.

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Cryptography systems face new threats with the transformation of time and technology. Each innovation tries to contest challenges posed by the previous system by analyzing approaches that are able to provide impressive outcomes. The prime aim of this work is to urge ways in which the concept of Pell’s equation can be used in Public key Cryptography techniques.The main aim of this approach is secure and can be computed very fast. Using Cubic Pell’s equation defined in Quadratic Case, a secure public key technique for Key generation process is showcased. The paper highlights that a key generation time of proposed scheme using Pell’s Quadratic case equation is fast compared to existing methods.The strength and quality  of the proposed method is proved and analyzed by obtaining the results of entropy, differential analysis, correlation analysis and avalanche effect. The superiority of the proposed method over the conventional AES and DES is confirmed by a 50% increase in the execution speed and shows that Standard diviation and Entropy analysis of proposed scheme gives immunity to guess the encryption key and also it is hard to deduce the private key from public key using  Diffrential analysis.