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MNT Elliptic Curves with Non-Prime Order

Maciej Grześkowiak

TL;DR

This work extends the Miyaji–Nakabayashi–Takano (MNT) framework to generalized MNT elliptic curves with non-prime orders by allowing a small cofactor $q$ so that the group order is $qn$, while maintaining a small embedding degree $k\in\{3,4,6\}$. It constructs explicit parametric polynomial families $(n_k(x), p_k(x), t_k(x))$ satisfying $qn_k(x)=p_k(x)+1-t_k(x)$ and $qn_k(x) \mid \Phi_k(p_k(x))$, with $t_k(x)^2-4p_k(x)\le 0$ and irreducibility, enabling CM-based curve generation via Pell-type equations. For each $k$, the paper provides modular-root–driven prescriptions to produce $p_k(x)$, $n_k(x)$, and $t_k(x)$, along with associated generalized Pell equations (e.g., $X^2+3\Delta Y^2=-8$ or $=24$) that underpin explicit curve construction over finite fields. The results yield efficient, scalable Families in $\mathcal{F}_k(h)$ for small $q$ and enable generation of pairing-friendly curves with non-prime orders, expanding practical options for cryptographic protocols requiring embedding-degree $k$ with controlled arithmetic in extension fields.

Abstract

Miyaji, Nakabayashi, and Takano proposed the algorithm for the construction of prime order pairing-friendly elliptic curves with embedding degrees $k=3,4,6$. We present a method for generating generalized MNT curves. The order of such pairing-friendly curves is the product of two prime numbers.

MNT Elliptic Curves with Non-Prime Order

TL;DR

This work extends the Miyaji–Nakabayashi–Takano (MNT) framework to generalized MNT elliptic curves with non-prime orders by allowing a small cofactor so that the group order is , while maintaining a small embedding degree . It constructs explicit parametric polynomial families satisfying and , with and irreducibility, enabling CM-based curve generation via Pell-type equations. For each , the paper provides modular-root–driven prescriptions to produce , , and , along with associated generalized Pell equations (e.g., or ) that underpin explicit curve construction over finite fields. The results yield efficient, scalable Families in for small and enable generation of pairing-friendly curves with non-prime orders, expanding practical options for cryptographic protocols requiring embedding-degree with controlled arithmetic in extension fields.

Abstract

Miyaji, Nakabayashi, and Takano proposed the algorithm for the construction of prime order pairing-friendly elliptic curves with embedding degrees . We present a method for generating generalized MNT curves. The order of such pairing-friendly curves is the product of two prime numbers.
Paper Structure (9 sections, 6 theorems, 28 equations, 2 tables)

This paper contains 9 sections, 6 theorems, 28 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Main theorems
  3. The case $k=6$
  4. The case $k=4$
  5. The case $k=3$
  6. Proof of Theorems
  7. The case $k=6$
  8. The case $k=4$
  9. The case $k=3$

Key Result

Theorem 2.1

Fix $j\in \{3,6\}$, a prime $q \equiv 1 \pmod 6$ or $q=3$. Let $s <q$ be a root of $\Phi_j(x)\pmod q$. If $p_6(x) = \Phi_4(qx+s)$, and then polynomials $(n_6(x),p_6(x), t_6(x))$ parameterizes a family of generalized MNT elliptic curves with embedding degree 6. Moreover, the family has the corresponding generalized Pell equations

Theorems & Definitions (14)

  • Definition 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.4
  • Remark 2.5
  • Remark 2.6
  • Theorem 2.7
  • ...and 4 more