SQIsign is a post-quantum signature scheme submitted to first round of the post-quantum standardisation process. It is based around a

proof of knowledge In cryptography, a proof of knowledge is an interactive proof in which the prover succeeds in 'convincing' a verifier that the prover knows something. What it means for a machine to 'know something' is defined in terms of computation. A machine 'k ...

of an

elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If the ...

endomorphism In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a g ...

that can be transformed to a signature scheme using the Fiat–Shamir transform. It promises small key sizes between 64 and 128 bytes and small signature sizes between 177 and 335 bytes, which outperforms other post-quantum signature schemes that have a trade-off between signature and key sizes. SQIsign, however, has higher signing and verification times. The original paper concluded that their C implementation takes 0.6 s for key generation, 2.5 s for a sign operation and 0.05 s or 50 ms for a verification operation. These times have been improved with new variations like SQIsign-east. The name stands for "Short Quaternion and Isogeny Signature" as it makes use of

isogenies In mathematics, localization of a category consists of adding to a category inverse morphisms for some collection of morphisms, constraining them to become isomorphisms. This is formally similar to the process of localization of a ring; it in genera ...

and

quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. The algebra of quater ...

Security

SQIsign's security relies on the hardness of the endomorphism ring problem, which is currently considered hard. The authors also provide a rationale for the chosen parameters in the last chapter of the specification. While SQIsign makes use of a similar construction, the weaknesses of SIDH do not translate to it. There is a security proof for SQIsign.

Implementations

There is
reference implementation
hosted on

GitHub GitHub () is a Proprietary software, proprietary developer platform that allows developers to create, store, manage, and share their code. It uses Git to provide distributed version control and GitHub itself provides access control, bug trackin ...

SQIsign 2.0

The team behind SQIsign improved the original design in their round 2 submission and incorporated improvements from the SQIsign2D-West variant. This has improved the signing time by a factor of 20 and the verification time by a factor of 6 while increasing the security level and reducing the signature size by 14%.

Variants

There are a couple of variants based on the original SQIsign: * SQIsignHD: New dimensions in cryptography * SQIsign2D-West: The fast, the small, and the safer * SQIsign2D‑East: A new signature scheme using 2-dimensional isogenies * SQIPrime: A dimension 2 variant of SQISignHD with non-smooth challenge isogenies

References

Asymmetric-key algorithms Digital signature schemes {{crypto-stub