Analysis on the Efficiency and Security of the RSA and ECDH Key Exchange Algorithms

The Rivest-Shamir-Adleman (RSA) and Elliptic Curve Diffie-Hellman (ECDH) Key Exchange algorithms are examples of trapdoor algorithms, which use mathematical complexity to provide real life internet security. In this paper we will look at the efficiency and security considerations of the RSA and ECDH Key Exchange algorithms. These algorithms were coded in Java on Replit and ran with different message sizes and key sizes to compare their efficiency based on encryption time. Furthermore, the main attacks on these cryptosystems, the Quadratic Sieve algorithm for the RSA and the Baby Step Giant Step algorithm for the ECDH, were coded, and the run times of the attack algorithms were compared to determine the relative security. Regression analysis was performed to determine the curves of best fit to approximate the graphs of message sizes and key sizes compared to the runtime. According to industry standard, the ECDH has the advantage over the RSA for the same level of security given a smaller key size required. At the same 112-bit strength, the ECDH key size is 224 bits, and the RSA key size is 2028 bits. Based on the extrapolated data to this industry standard, my results showed that encryption for the ECDH was faster than the RSA but did not confirm the comparable security as hacking the RSA was slower. The Quadratic Sieve codes could be improved for faster attack. In the future, the development of quantum computers will necessitate quantum-resistant algorithms as the Shor's Algorithm can solve current problems faster.