Discrete logarithm problem on elliptic curves over finite fields

Official URL: https://risc01.sabanciuniv.edu/record=b2771027_(Table of contents)

The main focus of this thesis is the so-called elliptic curve discrete logarithm problem. The statement of the problem is that given a point P and a k-multiple of P on an elliptic curve defined over a finite field, can we recover k? There has been no general algorithm that solves this problem in subexponential time. For this reason, the problem has been conjectured to be hard, and it is used to provide the security of many cryptosystems for classical computers. In this thesis, we study several algorithms, and the theory behind them, that are used to solve the problem under certain conditions. We also provide a relatively new algorithm that can be implemented to solve the discrete logarithm problem for specific elliptic curves. Additionally, we discuss the fundamental theory of elliptic curves defined over a commutative ring with unity, as they provide a useful tool for the solution of the discrete logarithm problem for a certain family of elliptic curves over finite fields.