A Comparison of Security and its Performance for Key Agreements in Post-Quantum Cryptography

Nowadays, we are surrounded by devices collecting and transmitting private information. Currently, the two main mathematical problems that guarantee security on the Internet are the Integer Factorization Problem and the Discrete Logarithm Problem. However, Shor's quantum algorithm can easily solve both problems. Therefore, research into cryptographic algorithms that run in classical computers and are resistant to quantum computers is extremely necessary. This area is known as post-quantum cryptography and usually studies asymmetric cryptography. By means of asymptotic analysis, the purpose of this paper is to provide an evaluation of security and its performance for the types of cryptographic systems considered safe against quantum attacks in the second-round NIST Post-Quantum Standardization Process, namely isogeny cryptosystems based on supersingular elliptic curves, error correction code-based encryption system, and lattice-based ring learning with errors. We performed a security comparison of Key Agreements protocols based on these three post-quantum cryptographic primitives and compared them with Discrete Logarithm Problem and Integer Factorization Problem. The comparison of security and its performance is presented by security level, the former by complexity analyses to achieve theoretical minimum key sizes, and the latter by simulation to assess a practical performance comparison. In the complexity analysis, as we increase the security level and then the size of the cryptographic keys increases, techniques based on isogeny outperform all other post-quantum algorithms in relation to key sizes at practical security level. In the performance comparison, the results show that the code-based protocol presents the best results among the others.