An algebraic approach to elliptic and hyperelliptic curve cryptography (2019)
During the past decades, cryptographic methods have radically improved as well as the mathematical tools employed in them. In this work we introduce elliptic curves as building blocks of cryptosystems and review their properties from a theoretical viewpoint. Moreover, we analyze the Discrete Logarithm Problem on groups of points on elliptic curves, and algorithms for its solution are implemented and compared. Furthermore, hyperelliptic curves are included as a generalization of elliptic curves for use in cryptography. The Jacobian is then described as an analogue of the group of points on elliptic curves for the case of higher genus curves. Two algorithms are also implemented and compared for the solution of the Discrete Logarithm Problem on the Jacobian of a general hyperelliptic curve. Finally, we find some conditions for curves to be employed in real system applications.