Modern elliptic curve cryptography

Elliptic curve cryptography has raised attention as it allows for having shorter keys and ciphertexts. For example, to obtain similar security levels with 2048 bit RSA key, it is necessary to use only 256 bit keys using over elliptic curve cryptography. Additionally, developments in the index calculus method for solving a discrete logarithm problem increases the sizes of the keys to keep the security requirements. However, these methods do not apply to points on the elliptic curves, allowing better estimations for security levels over longer period of time. Up to now, only a small family of elliptic curves have been widely used for cryptographic purposes. However, as the construction of these curves have been lead by large intelligence agencies, skepticism has been rising against using these families for different purposes as the may contain unknown backdoors. These concerns can be understood as getting arithmetic right on these families is difficult and developer can add security bugs inadvertently. In this report we study different elliptic curve formulas, concentrating on Edward curve. Furthermore, we study the feasibility of implementing such curves on embedded hardware devices and discuss optimization methods for obtaining reasonable performance rates.