Known-plaintext attack to two cryptosystems based on the BB equation

Recently, Rama Murthy and Swamy proposed a symmetric cryptosystem based on the Brahmagupta-Bhaskara (BB) equation. The BB equation is the quadratic Diophantine equation nx(2) + k = y(2), where k is an integer and n is a positive integer such that root n is irrational. For the particular case k = 1, the equation is called the Pell equation. The proposed cryptosystem was modified later by the same authors in order to avoid the crypt-analysis given by Youssef. Below, a known-plaintext attack to both cryptosystems is presented.