On the near prime-order MNT curves

TL;DR

This paper presents an explicit algorithm for generating near prime-order MNT elliptic curves with any cofactor, analyzes potential families, and discusses construction methods via Pell equations, enhancing curve generation techniques.

Contribution

It extends existing methods to produce all MNT curve families with any cofactor and analyzes their potential, providing a comprehensive approach to near prime-order curve construction.

Findings

Algorithm for generating all MNT curve families with any cofactor.

Analysis of potential curve families for given embedding degree and cofactor.

Discussion of Pell equations enabling specific curve constructions.

Abstract

In their seminar paper, Miyaji, Nakabayashi and Takano introduced the first method to construct families of prime-order elliptic curves with small embedding degrees, namely k = 3, 4, and 6. These curves, so-called MNT curves, were then extended by Scott and Barreto, and also Galbraith, McKee and Valenca to near prime-order curves with the same embedding degrees. In this paper, we extend the method of Scott and Barreto to introduce an explicit and simple algorithm that is able to generate all families of MNT curves with any given cofactor. Furthermore, we analyze the number of potential families of these curves that could be obtained for a given embedding degree $k$ and a cofactor h. We then discuss the generalized Pell equations that allow us to construct particular curves. Finally, we provide statistics of the near prime-order MNT curves.