FourQ

In cryptography, FourQ is an elliptic curve developed by Microsoft Research. It is designed for key agreements schemes (elliptic-curve Diffie–Hellman) and digital signatures (Schnorr), and offers about 128 bits of security. It is equipped with a reference implementation made by the authors of the original paper. The open source implementation is called FourQlib and runs on Windows and Linux and is available for x86, x64, and ARM. It is licensed under the MIT License and the source code is available on GitHub.

Its name is derived from the four dimensional Gallant–Lambert–Vanstone scalar multiplication, which allows high performance calculations. The curve is defined over a two dimensional extension of the prime field defined by the Mersenne prime ${\displaystyle 2^{127}-1}$.

The curve was published in 2015 by Craig Costello and Patrick Longa from Microsoft Research on ePrint.

The paper was presented in Asiacrypt in 2015 in Auckland, New Zealand, and consequently a reference implementation was published on Microsoft's website.

There were some efforts to standardize usage of the curve under IETF; these efforts were withdrawn in late 2017.

The curve is defined by a twisted Edwards equation

${\displaystyle d}$ is a non-square in ${\displaystyle \mathbb {F} _{p^{2}}}$, where ${\displaystyle p}$ is the Mersenne prime ${\displaystyle 2^{127}-1}$.

In order to avoid small subgroup attacks, all points are verified to lie in an N-torsion subgroup of the elliptic curve, where N is specified as a 246-bit prime dividing the order of the group.