Cryptanalysis of a New Variant of the RSA Cryptosystem

Let $$N=pq$$ be an RSA modulus which is the product of two balanced prime factors. In 2018, Murru and Saettone presented a variant of the RSA cryptosystem based on a cubic Pell equation in which the public exponent e and the private exponent d satisfy $$ed\equiv 1\left( \text {mod}{\frac{\left( p^3-1\right) \left( q^3-1\right) }{(p-1)(q-1)}}\right) $$ . In 2022, Cotan and Teşeleanu extended the scheme of Murru and Saettone and proposed a variant of the RSA cryptosystem where e and d satisfy the key equation $$ed\equiv 1\left( \text {mod}{\frac{\left( p^n-1\right) \left( q^n-1\right) }{(p-1)(q-1)}}\right) $$ for any $$n\ge 2$$ . Moreover, they presented an attack based on the continued fraction algorithm that factor N when $$2\le n\le 4$$ , $$e=N^{n-1}$$ and $$d