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SQIsign
Post-quantum digital signature scheme

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 of an elliptic curve endomorphism 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 and quaternions.

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Security

SQIsign's security relies on the hardness of the endomorphism ring problem, which is currently considered hard.56

The authors also provide a rationale for the chosen parameters in the last chapter of the specification.7

While SQIsign makes use of a similar construction, the weaknesses of SIDH do not translate to it.8

There is a security proof for SQIsign.9

Implementations

There is a reference implementation hosted on GitHub.

SQIsign 2.0

The team behind SQIsign improved the original design in their round 2 submission and incorporated improvements from the SQIsign2D-West variant.10

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%.11: 6 

Variants

There are a couple of variants based on the original SQIsign:12

  • SQIsignHD: New dimensions in cryptography13
  • SQIsign2D-West: The fast, the small, and the safer14
  • SQIsign2D‑East: A new signature scheme using 2-dimensional isogenies15
  • SQIPrime: A dimension 2 variant of SQISignHD with non-smooth challenge isogenies16

References

  1. specifically supersingular elliptic curves /wiki/Supersingular_elliptic_curve ↩

  2. Westerbaan, Bas; Larisch, James; Ahmad, Suleman; Fayed, Marwan; Westerbaan, Bas; Valenta, Luke; Krivit, Alex (2021-11-08). "Sizing Up Post-Quantum Signatures". The Cloudflare Blog. Retrieved 2024-11-15. https://blog.cloudflare.com/sizing-up-post-quantum-signatures/ ↩

  3. Feo, Luca De; Kohel, David; Leroux, Antonin; Petit, Christophe; Wesolowski, Benjamin (2020). "SQISign: compact post-quantum signatures from quaternions and isogenies". Cryptology ePrint Archive. Retrieved 2024-11-18. https://eprint.iacr.org/2020/1240 ↩

  4. Nakagawa, Kohei; Onuki, Hiroshi (2024). "SQIsign2D-East: A New Signature Scheme Using 2-dimensional Isogenies". Cryptology ePrint Archive. Retrieved 2024-11-15. https://eprint.iacr.org/2024/771 ↩

  5. Page, Aurel; Wesolowski, Benjamin (2023). "The supersingular Endomorphism Ring and One Endomorphism problems are equivalent". Cryptology ePrint Archive. arXiv:2309.10432. Retrieved 2024-11-15. https://eprint.iacr.org/2023/1399 ↩

  6. "THE SUPERSINGULAR ENDOMORPHISM RING PROBLEM GIVEN ONE ENDOMORPHISM" (PDF). Retrieved 2024-11-15. https://eprint.iacr.org/2023/1448.pdf ↩

  7. "SQIsign - Algorithm specifications and supporting documentation - Version 1.0" (PDF). Retrieved 2024-11-15. https://sqisign.org/spec/sqisign-20230601.pdf ↩

  8. "SQIsign - Algorithm specifications and supporting documentation - Version 1.0" (PDF). Retrieved 2024-11-15. https://sqisign.org/spec/sqisign-20230601.pdf ↩

  9. Aardal, Marius A.; Basso, Andrea; Feo, Luca De; Patranabis, Sikhar; Wesolowski, Benjamin (2025). "A Complete Security Proof of SQIsign". Cryptology ePrint Archive. Retrieved May 16, 2025. https://eprint.iacr.org/2025/379 ↩

  10. "SQIsign - Algorithm specifications and supporting documentation - Version 2.0" (PDF). Retrieved May 16, 2025. https://sqisign.org/spec/sqisign-20250205.pdf ↩

  11. "SQIsign - Algorithm specifications and supporting documentation - Version 2.0" (PDF). Retrieved May 16, 2025. https://sqisign.org/spec/sqisign-20250205.pdf ↩

  12. "SQIsign". SQIsign. 2023-06-01. Retrieved 2024-11-17. https://sqisign.org/ ↩

  13. Dartois, Pierrick; Leroux, Antonin; Robert, Damien; Wesolowski, Benjamin (2023). "SQISignHD: New Dimensions in Cryptography". Cryptology ePrint Archive. Retrieved 2024-11-17. https://eprint.iacr.org/2023/436 ↩

  14. Basso, Andrea; Feo, Luca De; Dartois, Pierrick; Leroux, Antonin; Maino, Luciano; Pope, Giacomo; Robert, Damien; Wesolowski, Benjamin (2024). "SQIsign2D-West: The Fast, the Small, and the Safer". Cryptology ePrint Archive. Retrieved 2024-11-17. https://eprint.iacr.org/2024/760 ↩

  15. Nakagawa, Kohei; Onuki, Hiroshi (2024). "SQIsign2D-East: A New Signature Scheme Using 2-dimensional Isogenies". Cryptology ePrint Archive. Retrieved 2024-11-17. https://eprint.iacr.org/2024/771 ↩

  16. Duparc, Max; Fouotsa, Tako Boris (2024). "SQIPrime: A dimension 2 variant of SQISignHD with non-smooth challenge isogenies". Cryptology ePrint Archive. Retrieved 2024-11-17. https://eprint.iacr.org/2024/773 ↩