Known-Plaintext Attack to Two Cryptosystems Based on the BB Equation

Abstract

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 sqrt(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.