We consider a generalization of balancing sequences and investigate some properties of the generalised balancing sequences in this paper. For a positive integer p we solve for the Diophantine equations, x 2 ± px y + y 2 ± x = 0 and express its solutions in terms of generalised balancing sequences.

In this paper, we gave an attack on RSA (Rivest–Shamir–Adleman) Cryptosystem when φ(N) has small multiplicative inverse modulo e and the prime sum p + q is of the form p + q = 2nk0 + k1, where n is a given positive integer and k0 and k1 are two suitably small unknown integers using sublattice reduction techniques and Coppersmith’s methods for finding small roots of modular polynomial equations. When we compare this method with an approach using lattice based techniques, this procedure slightly improves the bound and reduces the lattice dimension. Employing the previous tools, we provide a new attack bound for the deciphering exponent when the prime sum p + q = 2nk0 + k1 and performed an analysis with Boneh and Durfee’s deciphering exponent bound for appropriately small k0 and k1.

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In this paper we propose a key stream for encryptions obtained by concatenation of solutions of Diophantineequations which are generated by using some properties of Generalised Balancing Sequences.