  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.
