Security Issues of Novel RSA Variant

Abstract

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.

Keywords

• Continued fractions
• coppersmith’s method
• cubic Pell equation
• factorization
• RSA