Heuristics on pairing-friendly elliptic curves

TL;DR

This paper develops a heuristic asymptotic formula to estimate the number of pairing-friendly elliptic curves with specific properties, aiding in cryptographic curve selection.

Contribution

It introduces a heuristic asymptotic formula for counting certain classes of pairing-friendly elliptic curves with fixed parameters.

Findings

Provides an asymptotic estimate for the number of such elliptic curves

Focuses on curves with fixed embedding degree and discriminant

Considers curves with prime subgroup order up to x

Abstract

We present a heuristic asymptotic formula as $x\to \infty$ for the number of isogeny classes of pairing-friendly elliptic curves with fixed embedding degree $k\geq 3$, with fixed discriminant, with rho-value bounded by a fixed $\rho_0$ such that $1<\rho_0<2$, and with prime subgroup order at most $x$.