Breaking Modulus of the form \(N = p^rq^s\) with Improved Polynomial Attacks

Abstract

Let \(N=p^r q^s\) be prime power moduli where \(p\) and \(q\) are unbalance prime numbers for \(2 \leq s<r\). If \(q<p<\lambda q\) and \(q^s<p^r<\lambda q^s\), and\[\phi(N) \approx \lambda^{\frac{r-s}{2 r}}\left(N^{\frac{r+s}{2 r}+N^{\frac{r+s-2}{2 r}}}\right)-N^{\frac{r+s-1}{2 r}}\left(\lambda^{\frac{r-s+1}{2 r}}+\lambda^{\frac{r-s-1}{2 r}}\right)\]then\[x<\sqrt{\frac{\lambda^{\frac{r-s}{2 r}}\left(N^{\frac{r+s}{2 r}}+N^{\frac{r+s-2}{2 r}}\right)-N^{\frac{r+s-1}{2 r}}\left(\lambda^{\frac{r-s+1}{2 r}}+\lambda^{\frac{r-s-1}{2 r}}\right)}{2 N^{\frac{1+2 \alpha r}{2 r}}}}\] which leads to the factorization of the moduli \(N=p^r q^s\) in polynomial time. The second assaults on s multi prime power moduli are described \(N_i=p^r_i q^s_i\) for \(i=1,2,..., \omega.\) We use lattice basis reduction techniques to obtain the parameters (x; yi) or (y; xi) after transforming the system of equations into a simultaneous Diophantine approximation problem, and it resulted in simultaneous factorization of s moduli \(N_i\) in polynomial time.