Distortion maps for elliptic curves over finite fields

TL;DR

This paper investigates the existence of distortion maps for elliptic curves over finite fields, revisiting existing results and offering new insights to enhance cryptographic applications involving pairings.

Contribution

It provides a comprehensive analysis of distortion maps, including detailed proofs and new perspectives, advancing understanding in elliptic curve cryptography.

Findings

Revisited key results on distortion maps

Provided detailed proofs for existence conditions

Proposed new perspectives on the topic

Abstract

The Weil pairing on elliptic curves has deep links with discrete logarithm problems. In practice, to better suit the functionalities of cryptosystems, one often needs to modify the original Weil pairing via what is called a distortion map. We propose a study on the question of the existence of distortion maps for elliptic curves over finite fields. We revisit results from the literature and provide detailed proofs. We also propose new perspectives at times.