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.