Vulnerabilities of \(ex^2\) - \(y^2\)\(\phi\) (N)= z Using Modulus of the Form N= \(p^r\)\(q^s\)

The technical details of RSA works on the idea that it is easy to generate modulus by multiplying two sufficiently large prime numbers together, but factorizing number back into original extremely difficult. Suppose \(N=p^r q^s\) are modulus, where \(p\) and \(q\) product unknown unbalance primes for \(2 \leq s<r\). paper proves using an approximation \(\phi(N) \approx\) \(N-N^{\frac{r+8-1}{2 r}}\left(\lambda^{\frac{1-8}{2 r}}+\lambda^{\frac{-8}{2 r}}\right)+N^{\frac{r+8-2}{2 T}} \lambda^{\frac{1-8}{2 r}}\), private keys \(\frac{x^2}{y^2}\) can be found from convergents continued fractions expansion of\[\left|\frac{e}{N-N^{\frac{r+8-1}{2 r}} r}}}-\frac{y^2}{x^2}\right|<\frac{1}{2 x^4}\] which leads factorization moduli factors p q in polynomial time. second part this reseach report further, how generalized system equations form \(e_ux^2\) - \(y^2_u\phi(N_u)\) = \(z_u\) \(e_ux^2_u\) \(y^2\phi(N_u)\) simultaneous Diophantine method LLL algorithm values integers \(x,y_u\),\(\phi(N_u)\) \(x_u\),y,\(\phi(N_u)\) respectively, yeild successful k \(N_u=p^r_uq^s_u\) u 1,2, ...