This paper presents new short decryption exponent attacks on RSA, which successfully leads to the factorization of RSA modulus $N=pq$ in polynomial time. The paper has two parts. In the first part, we report the usage of the small prime difference method of the form $|b^{2}p-a^{2}q|<N^{\gamma }$ where the ratio of $\frac{q}{p}$ is close to $\frac{b^2}{a^2}$ , which yields a bound $d<\frac{\sqrt{3}}{\sqrt{2}}{N}^{\frac{3}{4}-\gamma }$ from the convergents of the continued fraction expansion of $\frac{e}{N-\lceil \frac{a^2+b^2}{ab}\sqrt{N}\rceil +1}$ . The second part of the paper reports four cryptanalytic attacks on t instances of RSA moduli $N_s=p_s{q}_s$ for $s=1,2,\dots ,t$ where we use $N-\lceil \frac{a^2+b^2}{ab}\sqrt{N}\rceil +1$ as an approximation of $\varphi (N)$ satisfying generalized key equations of the shape $e_{s}d-k_s\varphi (N_s)=1$ , $e_s{d}_s-k\varphi (N_s)=1$ , $e_{s}d-k_s\varphi (N_s)=z_s$ , and $e_s{d}_s-k\varphi (N_s)=z_s$ for unknown positive integers $d,k_s,d_s,k_s$ , and $z_s$ , where we establish that t RSA moduli can be simultaneously factored in polynomial time using combinations of simultaneous Diophantine approximations and lattice basis reduction methods. In all the reported attacks, we have found an improved short secret exponent bound, which is considered to be better than some bounds as reported in the literature.

The importance of keeping information secret cannot be overemphasized especially in today,s digital world where eavesdroppers are rampant in our chanels of communication. This made the use of strong encryption schemes inevitable in order to safeguard the security of our system. RSA cryptosystem and its variants have been designed to provide confidentiality and integrity of data in our medium of communication. This paper reports new short decryption exponent attack on prime power with modulus $N=p^rq$ for $r\geq 2$ using continued fraction method which makes it vulnerable to Diophantine attack and breaks the security of the cryptosystem by factoring the modulus into its prime factors since the hardness relies on the integer factorization problem. The paper also shows that if the short decryption exponent $d <\frac{1}{\sqrt{2}}\sqrt{N- 2^{\frac{2r+1}{r+1}} N^{\frac{r}{r+1}}}$, then one of the convergents $\frac{k}{d}$ can be found from the continued fraction expansion of $\frac{e}{N-\left\lceil 2^{\frac{2r+1}{r+1}} N^{\frac{r}{r+1}}\right\rceil}$ which leads to the successful factorization of prime power modulus $N=p^rq$ in polynomial time. The second part of the paper presents new findings on simultaneous factorization of $t$ prime power with moduli $N_s=p^r_sq_s$ for $s=1,\ldots, t$ using simultaneous Diophantine approximations and lattice basis reduction methods which produces the prime factors of the form $(p_s,q_s)$ for $s=1,\ldots, t$ in polynomial time where solutions of four system of equations of the form $e_sd-k_s\phi(N_s)=1$, $e_sd_s-k\phi(N_s)=1$, $e_sd-k_s\phi(N_s)=z_s$ and $e_sd_s-k\phi(N_s)=z_s$ are provided. Our results increases the short decryption exponent bounds of some reported works

The security of RSA cryptosystem and its variants rely on the intractability of integer factorization problem. Series of attacks have been reported to exploit the cryptosystem leading to polynomial time factorization of the composite integer into its prime factors p and q. This paper presents two cryptanalysis attacks on the prime power modulus as one of the RSA variants with moduli Ni = pi2qi2 where generalized key equations of the form eid - kiφ(Ni)=1 and eidi - kφ(Ni) = 1 can be exploited and factored simultaneously using simultaneous Diophantine approximation method and lattice basis reduction technique. The paper also gives numerical examples to demonstrate how the attacks work.