Elliptic Curve Cryptography: Pre and Post Quantum

Public-key cryptography has been at the center of online communication and information transfer for decades. With computing power growing at an exponential rate, some of the most widely used encryption schemes are starting to show their limits. The RSA algorithm, which is still widely used around the world, now requires very large keys to ensure security. Since these systems may appear on low computing power devices such as mobile phones, or chips, it has become essential to create protocols for which we can reach the same level of security without spending considerable computing power setting up the system in the first place. Elliptic curve cryptography (ECC) provides an exciting alternative to RSA, and has shown to be a lot more efficient in terms of key size. In this paper, we provide a description of how elliptic curves are used in modern cryptography, as well as their current limitations and future prospects. Because quantum computers pose a serious threat to the currently in use public-key systems, we also describe the recent progress on super singular elliptic curves isogenies, which may offer a quantum resistant cryptosystem and a viable alternative for the future of elliptic curve based cryptography.