Creating New Elliptic Curves for Uses in Cryptography

Abstract

Public key cryptography, is the basis of m odem cryptography, allows us to send and receive messages over public channels secretly, without requiring a meeting beforehand. Most public key cryptosystems, such as the Diffie- Hellman Key Exchange, rely on the difficulty of solving the Discrete Logarithm Problem (DLP). We can translate public key cryptosystems that rely on the DLP to Elliptic Curve cryptosystems as the Elliptic Curve Discrete Logarithm Problem (ECDLP) is believed to be more difficult and therefore harder to break. There are certain precautions we need to take when using Elliptic Curve Cryptography to safeguard against particular attacks on the cryptosystem. Therefore, picking a curve that is secure enough is crucial to a good cryptosystem. Unfortunately, there are only a handful of secure elliptic curves that are publicly known and used. The goal of this thesis is to generate more elliptic curves that are useful for our security systems.