Asymptotic Bound for RSA Variant with Three Decryption Exponents

This paper presents a cryptanalysis attack on the RSA variant with modulus $N=p^rq$ for $r\geq 2$ with three public and private exponents $(e_1,d_1),$ $(e_2,d_2),$ $(e_3,d_3)$ sharing the same modulus $N$ where $p$ and $q$ are consider to prime having the same bit size. Our attack shows that we get the private exponent \sigma_1\sigma_2\sigma_3<\left(\frac{r-1}{r+1}\right)^4, which makes the modulus vulnerable to Coppersmith's attacks and can lead to the factorization of $N$ efficiently where d_1 The asymptotic bound of our attack is greater than the bounds for May \cite{May}, Zheng and Hu \cite{Z}, and Lu et al. \cite{Y} for 2\leq r \leq 10 and greater than Sarkar's \cite{Sarkar1} and \cite{Sarkar} bounds for 5 \leq r \leq10$