Constructing pairing-friendly elliptic curves with embedding degree 10

TL;DR

This paper introduces a new framework for constructing elliptic curves with specific embedding degrees, successfully creating curves with degree 10 and connecting to existing methods for degrees 3, 4, 6, and 12.

Contribution

The paper presents a general framework for constructing elliptic curves with prescribed embedding degrees, solving an open problem for degree 10 and analyzing limitations for higher degrees.

Findings

Constructed elliptic curves with embedding degree 10.

Framework unifies existing constructions for degrees 3, 4, 6, and 12.

Evidence suggests limited potential for infinite families with degree > 12.

Abstract

We present a general framework for constructing families of elliptic curves of prime order with prescribed embedding degree. We demonstrate this method by constructing curves with embedding degree k = 10, which solves an open problem posed by Boneh, Lynn, and Shacham. We show that our framework incorporates existing constructions for k = 3, 4, 6, and 12, and we give evidence that the method is unlikely to produce infinite families of curves with embedding degree k > 12.