On the Search for Supersingular Elliptic Curves and Their Applications

Elliptic curves with the special quality known as supersingularity have gained much popularity in the rapidly developing field of cryptography. The conventional method of employing random search is quite ineffective in finding these curves. This paper analyzes the search of supersingular elliptic curves in the space of curves over F p 2 . We show that naive random search is unsuitable to easily find any supersingular elliptic curves when the space size is greater than 10 13 . We improve the random search using a necessary condition for supersingularity. As our main result, we define for the first time an objective function to measure the supersingularity in ordinary curves, and we apply local search and a genetic algorithm using that function. The study not only finds these supersingular elliptic curves but also investigates possible uses for them. These curves were used to create cycles inside the isogeny graph in one particular application. The research shows how the design of S-boxes may strategically use these supersingular elliptic curves. The key components of replacement, which is a fundamental step in the encryption process that shuffles and encrypts the data inside images, are S-boxes. This work represents a major advancement in effectively identifying these useful elliptic curves, eventually leading to their wider application and influence in the rapidly expanding field of cryptography.