A Generalised Balancing Sequence and Solutions of Diophantine Equations \(x^2\pm pxy + y^2\pm x = 0\)

Abstract

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.

Encryption with Generalised Balancing Sequences and Generalised Lucas Balancing Sequences

“…Balancing Numbers Bn are defined by Behera and Panda [7] as the natural numbers that satisfy the recurrence relations Bn+1 = 6Bn − Bn−1 with B0 = 0 and B1 = 1. In [2], Generalised Balancing Sequences are introduced as the numbers satisfying the recurrent relation Bn+1 = pBn − Bn−1 with B0 = 0, B1 = 1 where p is a positive integer. It also discusses the following properties of Generalised Balancing Sequences and Diophantine Equation x 2 − pxy + y 2 − x = 0 whose solutions are in terms of Generalised Balancing Sequences.…”

“…A Key Stream using a solution of a particular class of Diophantine Equation is generated as follows. Both sender and receiver agree upon positive integer α and compute all the Generalised Balancing sequences from B2α−1 using fast computation algorithm.Considers a class of solutions (B2n(p) 2 , B2n(p)B2n−1(p)) of the Diophantine Equation x 2 − pxy + y 2 − x = 0 as in [2] for α ≤ n . Then concatenate solutions (B2n(p) 2 , B2n(p)B2n−1(p)) of Diophantine equation for all α ≤ n and obtain key stream as…”

“…Diophantine Equations with infinitely many soluitons play a vital role in cryptography. In [1,2,3] we described the solutions of Diophantine Equations x 2 ± pxy + y 2 ± x = 0 and x 2 ± pxy + y 2 ± ( p 2 −4 4 )x = 0 that are generated from Generalised Balancing Sequences and Generalised Lucas Balancing Sequences respectively. In [4] P.Anuradha Kameswari and B.Ravi Theja came forward with algorithms for fast computation of Lucas sequences [5,6].…”