The RSA is one of the current default cryptosystems that provides security with applications such as encryptions and digital signatures. It is important to further study the weak characteristics of the RSA to ensure correct utilisation in order not to be susceptible to adversaries. In this paper, we give detailed analysis on security of the Murru-Saettone variant of the RSA cryptosystem that utilised a cubic Pell $ed-k\left ({p^{2}+p+1}\right )\left ({q^{2}+q+1}\right )=1$ as key equation and $N=pq$ as RSA modulus. We propose some attacks on this variant when the prime difference $|p-q|$ is small. Our first approach is to utilise the continued fractions algorithm to determine the parameter $d$ which enables us to determine the secret $p$ and $q$ . Our second approach considers the Coppersmith's method and lattice basis reduction to factor the modulus $N$ . Our attacks improve recent cryptanalyses on the cubic Pell equation variant of RSA. Furthermore, our attacks prove that under small prime difference scenario, the number of susceptible private exponents for the cubic Pell equation variant of RSA is much larger than the standard RSA.

We propose a rank metric codes based encryption based on the hard problem of rank syndrome decoding problem. We propose a new encryption with a public key matrix by considering the adding of a random distortion matrix over ${\mathbb{F}}_{q^m}$ of full column rank n. We show that IND-CPA security is achievable for our encryption under assumption of the Decisional Rank Syndrome Decoding problem. Furthermore, we also prove some bounds for the number of matrices of a fixed rank with entries over a finite field. Our proposal allows the choice of the error terms with rank up to $\frac{r}{2}$ , where r is the error-correcting capability of a code. Our encryption based on Gabidulin codes has public key size of $13.68$ KB, which is 82 times smaller than the public key size of McEliece Cryptosystem based on Goppa codes. For similar post-quantum security level of $2^{140}$ bits, our encryption scheme has a smaller public key size than the key size suggested by LOI17 Encryption.