khaotik/ellj0-ecdlp-stdcurve

Dataset of elliptic curves with J-invariant 0 over finite field

Introduction

Elliptic curve with J-invariant 0 is an important class of elliptic curve, both theoretically and in applications of cryptography.

An elliptic curve with Weierstrass equation $y^2 = x^3 + a x + b$ has J-invariant 0 when $a=0$ and $b \neq 0$.

Well known curves such as secp256k1 ($y^2=x^3+7$ over 256 bit prime field) and bls2-381 ($y^2=x^3+4$ over 381 bit prime field) are widely used in cryptocurrency projects.

During development of ECDLP solver for these curves, it often valuable to test the proposed algorithm on a smaller curve to gain insights.

Thus, it is beneficial to standarize curves used for testing purposes:

1. To gain insights on asymptotic performance of proposed algorithm as field size grows.
2. Simplify engineering hurdle during prototyping phase. Eg. 256 bit modular arithmetic may require specialized libraries on a GPU, but 64 and 32 bit integer support is out-of-box most times.
3. The standardized curves allows more precise comparison of different ECDLP solvers.

Notations

Notation	Meaning
$E/\mathbb{F}_p$	elliptic curve reduced modulo prime number $p$
$E/\mathbb{F}_q$	elliptic curve of finite field of size $q$
$E_\unicode{x2b23}$	elliptic curve of J-invariant 0 ^$\dagger 1$
$l(E/\mathbb{F}_q)$	point count of elliptic curve over $\mathbb{F}_q$
$G(E)$	generator of elliptic curve $E$
$a_q(E/\mathbb{F}_q)$	frobenius trace of elliptic curve $E$ modulo $q$
$[n]P$	scalar multiplication by $n$ of point $P$ on an elliptic curve

List of Datasets

ellws-a0b7-v1.csv

This is directly related to secp256k1 curve with Weierstrass equation $E:y^2=x^3+7$ . The dataset consists of curve $y^2=x^3+7$ reduced modulo various primes. The primes are from small bit length to large ones, up to 256 bits. The curve secp256k1 itself is also in the dataset.

The dataset is constructed via search algorithm, each curve $E_\unicode{x2b23}/\mathbb{F}_p$ in the dataset satisfies:

• Is neither a supersingular curve nor a anomalous curve.
• Embedding degree for MOV attack is high.
• $p-1$ and $l(E_\unicode{x2b23})-1$ are not smooth numbers.
• $p \equiv 7 \mod{24}$ and $l(E_\unicode{x2b23}) \equiv 1 \mod{24}$
• There are 6 isomorphism classes of all non-singular $E_\unicode{x2b23}/\mathbb{F}_p$
• When $x=1$, $y = \sqrt{1^3+7} = \sqrt{8} \in \mathbb{F}_p$, this allows point with x-coordinate 1 always a curve generator.
• Let the curve's bit length $L = \lceil log_2(p) \rceil$, then $2^{L-1} \leq l(E_\unicode{x2b23}) \lt 2^L$, this ensures no integer overflow happens with chosen bit length.

Most of above resembles cryptographic properties of secp256k1 itself.

The row schema for this dataset is:

COLUMN	bits	field	point_count	generator	embedding_degree
TYPE	integer	2-tuple of integer	integer	2-tuple of integer	integer
EXAMPLE	16	36607^1	36457	1;3610	9114
MEANING	bit length	field characteristic & extension degree	group order	group generator representative $^\dagger 2$	MOV attack embedding degree

More datasets

More dataset may be constructed as the need arise.

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$\dagger$ Footnotes

1. Hexagon symbol is used because elliptic curve with J-invariant 0 has Eisenstein integers as endomorphism ring. Eisenstein integers form a hexagonal lattice.

2. If group order $l(E)$ is for prime, any valid generator can be used without impact on curve security. Assuming we are finding $n$ for $[n]P = Q$ where $P,Q \in E$, we can find $u,v \in \mathbb{F}_{l(E)}$ such that $[u]G = P, [v]G = Q$, then $n = v/u$. Nevertheless, we put a canonical generator in the dataset.

License

MIT License