Cryptoanalysis of RSA variants with special structure of RSA primes

In this paper, we present attacks on three types of RSA modulus when the least significant bits of the prime factors of RSA modulus satisfy some conditions. Let $p,$ and $q$ be primes of the form $p=a^{m_1}+r_p$ and $q=b^{m_2}+r_q$ respectively, where $a,b,m_{1},m_{2} \in \mathbb{Z^+}$ $r_p,$ and $r_q$ are known. The first attack is when the RSA modulus is $N=pq$ where $m_1$ or $m_2$ is an even number. If $\left(r_{p}r_{q}\right)^\frac{1}{2}$ is sufficiently small, then $N$ can be factored in polynomial time. The second attack is when $N=p^{s}q,$ where $q>p$ and $s$ divides $m_2.$ If $r_pr_q$ is sufficiently small, then $N$ can be factored in polynomial time. The third attack is when $N=p^{s+l}q^{s},$ where $p>q,$ $s,l \in \mathbb{Z^+},$ $l < \frac{s}{2}$ and $s$ divides $m_1l.$ If $a^{m_1}>qa^{\frac{m_1l}{s}},$ and $lr^3_p$ is sufficiently small, then $N$ can be factored in polynomial time.