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 radic(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 cryptanalysis given by Youssef. Below, a known-plaintext attack to both cryptosystems is presented.