Common Modulus Attack on the Elliptic Curve-Based RSA Algorithm Variant

This paper investigates the vulnerability of elliptic curve-based RSA to the common modulus attack, a well-known cryptanalytic method. RSA is a commonly utilized public-key encryption system that is based on the challenge of factoring large composite numbers, securing digital signatures, key exchanges, and data transmission. The security of the RSA algorithm has been thoroughly analyzed and continues to be robust against numerous types of attacks. However, in 2023, Boudabra and Nitaj introduced a variant of RSA employing elliptic curves within the ring <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbb{Z}/n\mathbb{Z}$</tex>, aiming to enhance both security and efficiency. Despite these advancements, this paper demonstrates that the common modulus attack can still compromise the system. By leveraging a shared modulus in multiple encryptions, the attack successfully recovers the plaintext without requiring the trapdoor information. We conduct simulations using specific parameters, with exponents <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$e_{1} = 233, e_{2} = 151$</tex>, and modulus <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n = 181603559630213323475279432919469869812801$</tex>, showing how the attack is feasible even for elliptic curve-based RSA. The results indicate that elliptic curve-enhanced RSA, while improving efficiency, remains susceptible to the common modulus attack. Our findings underscore the importance of careful implementation strategies and the need for greater attention to this vulnerability in practical cryptographic applications.