In this paper, we first propose a rigorous polynomial time algorithm based on the univariate Coppersmith theorem that factorizes an integer \(N=PQ^r\) with unknown primes P, Q and \(r \ge 1\) , for an arbitrary given prime \(e>N^{\frac{1}{4r}}\) satisfying \(e\mid \phi (N)\) . Furthermore, the bound \(e>N^{\frac{1}{4r}}\) is improved to \(e>N^{\beta -r\beta ^2}\) if the value of \(\beta \) such that \(Q\ge N^\beta \) is known. Moreover, these two bounds can also be obtained respectively when e is no longer a prime number. Specifically, we address the scenario where e is a square-free composite number with known factorization, as well as the case where e is a composite number with known factorization, and the corresponding prime factors P and Q of N satisfy \(\gcd (P-1, Q-1)=2\) . In both cases, in order to make the corresponding algorithms polynomial time, we limit the number of prime factors of e to \(O (\log \log N)\) and r to be any integer constant. When applied to the \(\phi \) -hiding assumption, our bounds are significantly better than previous results. When applied to standard RSA moduli, we extend the case where e, which was involved in the previous best bound \(e>N^{\frac{1}{4}}\) , is no longer a prime number. This can highlight the security vulnerability introduced by the function \(\textrm{RSA}_{N,e}\) , as discussed by Kakvi et al. in [23], being at least \(e_i\) -to-1 lossy for some prime factors \(e_i\) of e. When applied to decompose the semi-smooth RSA subgroup moduli, we provide a rigorous proof for the second bound presented by Naccache and Stern for the first time. We validate the effectiveness of algorithms through experiments.