Evaluation of Computational Approaches of Short Weierstrass Elliptic Curves for Cryptography

TL;DR

This survey reviews the evolution, security considerations, and standards of Short Weierstrass elliptic curves in cryptography, highlighting recent approaches and analyzing their usability for secure cryptographic applications.

Contribution

It provides a comprehensive overview of the development, attack countermeasures, and selection criteria of elliptic curves, including evaluation of deterministic and random selection methods.

Findings

Analysis of attack evolution on elliptic curve discrete logarithm problem

Evaluation of deterministic and random curve selection approaches

Insight into the usability of standard and non-standard elliptic curves

Abstract

The survey presents the evolution of Short Weierstrass elliptic curves after their introduction in cryptography. Subsequently, this evolution resulted in the establishment of present elliptic curve computational standards. We discuss the chronology of attacks on Elliptic Curve Discrete Logarithm Problem and investigate their countermeasures to highlight the evolved selection criteria of cryptographically safe elliptic curves. Further, two popular deterministic and random approaches for selection of Short Weierstrass elliptic curve for cryptography are evaluated from computational, security and trust perspectives and a trend in existent computational standards is demonstrated. Finally, standard and non-standard elliptic curves are analysed to add a new insight into their usability. There is no such survey conducted in past to the best of our knowledge.