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

Abstract

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