# Curves

ECTester contains a collection of elliptic curve/point parameters, these parameters either come from standards or were generated manually or using ecgen.

These parameters can be found in the cz.crcs.ectester.data package.

## Standard

### SECG

SEC 2: Recommended Elliptic Curve Domain Parameters version 2.0 January 27, 2010

### NIST

RECOMMENDED ELLIPTIC CURVES FOR FEDERAL GOVERNMENT USE July 1999

### x962

ANSI X9.62 example curves.

### Brainpool

ECC Brainpool Standard Curves and Curve Generation v. 1.0 19.10.2005

### anssi

Agence nationale de la sécurité des systèmes d’information: Publication d’un paramétrage de courbe elliptique visant des applications de passeport électronique et de l’administration électronique française. 21 November 2011

### GOST

GOST R 34.10-2001: RFC5832 curves.

### Barreto-Naehrig

Barreto-Naehrig curves from: A Family of Implementation-Friendly BN Elliptic Curves

### Other

An assortment of some other curves. Montgomery curves transformed into short Weierstrass form from https://eprint.iacr.org/2013/647.pdf

Curve25519 transformed into short Weierstrass form.

## Generated

### anomalous

These prime field curves have the same order as the field order, and are susceptible to attacks reducing ECDLP over a multiplicative group of the curve, to DLP over an additive group of the underlying field, which is easy (linear time).

Some of these are from Atsuko Miyaji’s paper, others were generated using ecgen.

### invalid

This category contains pre-generated invalid curves for a large subset of NIST, SECG and Brainpool curves. Invalid curves for a given curve, are short Weierstrass curves with all parameters equal to the given curve except the `b`

parameter. These curves can be used to attack some implementations.

Generated using ecgen.

### composite

Contains curves of composite order, with small order points.

Generated using ecgen.

### wrong

Contains parameters that are not elliptic curves(over Fp and F2m), such as `p`

parameter that is not prime or an irreducible polynomial that is not irreducible.

Generated manually.

### twist

Contains pre-generated points on twists of known named curves from NIST, SECG. These points can be used to attack some implementations.

Generated using ecgen.

### degenerate

Contains pre-generated points on the line `Y: x = 0`

. These points are constructed from elements of prime
order in the multiplicative group F_p given a curve over it.

Generate manually using PARI/GP.

### cofactor

Contains curves that are composite order, with points not on the subgroup generated by the generator.

Generated using ecgen.

### supersingular

Contains supersingular curves, over F_p with order equal to p + 1. These have embedding degree equal to 2.

Generated using ecgen.

## Other

### Wycheproof

Contains some test vectors from the google/Wycheproof project.