Skip to main content
SpringerLink
Log in

FuLeeca: A Lee-Based Signature Scheme

  • Conference paper
  • First Online:
Code-Based Cryptography (CBCrypto 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14311))

Included in the following conference series:

Abstract

In this work, we introduce a new code-based signature scheme, called FuLeeca, based on the NP-hard problem of finding codewords of given Lee-weight. The scheme follows the Hash-and-Sign approach applied to quasi-cyclic codes. Similar approaches in the Hamming metric have suffered statistical attacks, which revealed the small support of the secret basis. Using the Lee metric, we are able to thwart such attacks. We use existing hardness results on the underlying problem and study adapted statistical attacks. We propose parameters for FuLeeca and compare them to an extensive list of proposed post-quantum secure signature schemes including the ones already standardized by NIST. This comparison reveals that FuLeeca is competitive. For example, for NIST category I, i.e., 160 bit of classical security, we obtain an average signature size of 1100 bytes and public key sizes of 1318 bytes. Comparing the total communication cost, i.e., the sum of the signature and public key size, we see that FuLeeca is only outperformed by Falcon while the other standardized schemes Dilithium and SPHINCS+ show higher communication costs than FuLeeca.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 1.

    See for example [31], where the CFS scheme using high rate Goppa codes has been attacked.

  2. 2.

    https://groups.google.com/a/list.nist.gov/g/pqc-forum/c/KvIege2EbuM.

  3. 3.

    https://git.math.uzh.ch/isd/lee-isd/lee-isd-algorithm-complexities/-/blob/master/Lee-ISD-restricted.nb.

  4. 4.

    https://github.com/setinski/Information-Set-Decoding-Analysis.

  5. 5.

    https://git.math.uzh.ch/isd/lee-isd/lee-isd-algorithm-complexities/-/blob/master/Lee-ISD-restricted.nb.

  6. 6.

    The implementation is publicly available at https://cpucycles.cr.yp.to/.

References

  1. Aguilar-Melchor, C., Gama, N., Howe, J., Hülsing, A., Joseph, D., Yue, D.: The return of the SDitH. Cryptology ePrint Archive (2022)

    Google Scholar 

  2. Aragon, N., Blazy, O., Gaborit, P., Hauteville, A., Zémor, G.: Durandal: a rank metric based signature scheme. In: Ishai, Y., Rijmen, V. (eds.) EUROCRYPT 2019. LNCS, vol. 11478, pp. 728–758. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-17659-4_25

    Chapter  Google Scholar 

  3. Aragon, N., Dyseryn, V., Gaborit, P.: Analysis of the security of the PSSI problem and cryptanalysis of the Durandal signature scheme. Cryptology ePrint Archive (2023)

    Google Scholar 

  4. Astola, J.: On the asymptotic behaviour of Lee-codes. Discret. Appl. Math. 8(1), 13–23 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  5. Aumasson, J.P., et al.: \(\text{SPHINCS}^+\), submission to the NIST post-quantum project, vol. 3 (2020). https://csrc.nist.gov/Projects/post-quantum-cryptography/selected-algorithms-2022

  6. Baldi, M., Bianchi, M., Chiaraluce, F., Rosenthal, J., Schipani, D.: Using LDGM codes and sparse syndromes to achieve digital signatures. In: Gaborit, P. (ed.) PQCrypto 2013. LNCS, vol. 7932, pp. 1–15. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-38616-9_1

    Chapter  Google Scholar 

  7. Baldi, M., Bitzer, S., Pavoni, A., Santini, P., Wachter-Zeh, A., Weger, V.: Zero knowledge protocols and signatures from the restricted syndrome decoding problem. Cryptology ePrint Archive (2023)

    Google Scholar 

  8. Barenghi, A., Biasse, J.-F., Persichetti, E., Santini, P.: LESS-FM: fine-tuning signatures from the code equivalence problem. In: Cheon, J.H., Tillich, J.-P. (eds.) PQCrypto 2021 2021. LNCS, vol. 12841, pp. 23–43. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-81293-5_2

    Chapter  MATH  Google Scholar 

  9. Barg, A.: Complexity issues in coding theory. Technical report TR97-046, Electronic Colloquium on Computational Complexity (ECCC) (1997). https://eccc.weizmann.ac.il/eccc-reports/1997/TR97-046/index.html. ISSN 1433-8092

  10. Barg, A.: Some new NP-complete coding problems. Problemy Peredachi Informatsii 30(3), 23–28 (1994). https://www.mathnet.ru/eng/ppi241

  11. Bariffi, J., Bartz, H., Liva, G., Rosenthal, J.: On the properties of error patterns in the constant Lee weight channel. In: International Zurich Seminar on Information and Communication (IZS 2022) Proceedings, pp. 44–48. ETH Zurich (2022)

    Google Scholar 

  12. Bariffi, J., Khathuria, K., Weger, V.: Information set decoding for Lee-metric codes using restricted balls. In: Deneuville, J.C. (ed.) CBCrypto 2022. LNCS, vol. 13839, pp. 110–136. Springer, Cham (2022). https://doi.org/10.1007/978-3-031-29689-5_7

    Chapter  Google Scholar 

  13. Becker, A., Joux, A., May, A., Meurer, A.: Decoding random binary linear codes in \(2^{n/20}\): how 1+1=0 improves information set decoding. In: Pointcheval, D., Johansson, T. (eds.) EUROCRYPT 2012. LNCS, vol. 7237, pp. 520–536. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-29011-4_31

    Chapter  MATH  Google Scholar 

  14. Bellare, M., Rogaway, P.: The exact security of digital signatures-how to sign with RSA and Rabin. In: Maurer, U. (ed.) EUROCRYPT 1996. LNCS, vol. 1070, pp. 399–416. Springer, Heidelberg (1996). https://doi.org/10.1007/3-540-68339-9_34

    Chapter  Google Scholar 

  15. Berlekamp, E.R., McEliece, R.J., van Tilborg, H.C.A.: On the inherent intractability of certain coding problems. IEEE Trans. Inf. Theory 24(3), 384–386 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  16. Bernstein, D.J.: Grover vs. McEliece. In: Sendrier, N. (ed.) PQCrypto 2010. LNCS, vol. 6061, pp. 73–80. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-12929-2_6

    Chapter  Google Scholar 

  17. Beullens, W.: Sigma protocols for MQ, PKP and SIS, and fishy signature schemes. In: Canteaut, A., Ishai, Y. (eds.) EUROCRYPT 2020. LNCS, vol. 12107, pp. 183–211. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-45727-3_7

    Chapter  Google Scholar 

  18. Bhattacharyya, M., Raina, A.: A quantum algorithm for syndrome decoding of classical error-correcting linear block codes. In: 2022 IEEE/ACM 7th Symposium on Edge Computing (SEC), pp. 456–461 (2022). https://doi.org/10.1109/SEC54971.2022.00069

  19. Bidoux, L., Gaborit, P.: Shorter signatures from proofs of knowledge for the SD, MQ, PKP and RSD problems. arXiv preprint arXiv:2204.02915 (2022)

  20. Blömer, J., Naewe, S.: Sampling methods for shortest vectors, closest vectors and successive minima. Theoret. Comput. Sci. 410(18), 1648–1665 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  21. Byrne, E., Horlemann, A.L., Khathuria, K., Weger, V.: Density of free modules over finite chain rings. Linear Algebra Appl. 651, 1–25 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  22. Chailloux, A., Debris-Alazard, T., Etinski, S.: Classical and Quantum algorithms for generic Syndrome Decoding problems and applications to the Lee metric (2021). https://eprint.iacr.org/2021/552. Report Number: 552

  23. Cho, J., No, J.S., Lee, Y., Koo, Z., Kim, Y.S.: Enhanced pqsigRM: code-based digital signature scheme with short signature and fast verification for post-quantum cryptography. Cryptology ePrint Archive (2022)

    Google Scholar 

  24. Courtois, N.T., Finiasz, M., Sendrier, N.: How to achieve a McEliece-based digital signature scheme. In: Boyd, C. (ed.) ASIACRYPT 2001. LNCS, vol. 2248, pp. 157–174. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-45682-1_10

    Chapter  Google Scholar 

  25. Dachman-Soled, D., Ducas, L., Gong, H., Rossi, M.: LWE with side information: attacks and concrete security estimation. In: Micciancio, D., Ristenpart, T. (eds.) CRYPTO 2020. LNCS, vol. 12171, pp. 329–358. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-56880-1_12

    Chapter  Google Scholar 

  26. Debris-Alazard, T., Sendrier, N., Tillich, J.-P.: Wave: a new family of trapdoor one-way preimage sampleable functions based on codes. In: Galbraith, S.D., Moriai, S. (eds.) ASIACRYPT 2019. LNCS, vol. 11921, pp. 21–51. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-34578-5_2

    Chapter  Google Scholar 

  27. Deneuville, J.-C., Gaborit, P.: Cryptanalysis of a code-based one-time signature. Des. Codes Crypt. 88(9), 1857–1866 (2020). https://doi.org/10.1007/s10623-020-00737-8

    Article  MathSciNet  MATH  Google Scholar 

  28. Ducas, L., et al.: Crystals-dilithium - algorithm specifications and supporting documentation (version 3.1) (2021). https://pq-crystals.org/dilithium/resources.shtml

  29. Dumer, I.I.: Two decoding algorithms for linear codes. Problemy Peredachi Informatsii 25(1), 24–32 (1989)

    MathSciNet  MATH  Google Scholar 

  30. Espitau, T., Tibouchi, M., Wallet, A., Yu, Y.: Shorter hash-and-sign lattice-based signatures. In: Dodis, Y., Shrimpton, T. (eds.) CRYPTO 2022. LNCS, vol. 13508, pp. 245–275. Springer, Cham (2022). https://doi.org/10.1007/978-3-031-15979-4_9

    Chapter  Google Scholar 

  31. Faugere, J.C., Gauthier-Umana, V., Otmani, A., Perret, L., Tillich, J.P.: A distinguisher for high-rate McEliece cryptosystems. IEEE Trans. Inf. Theory 59(10), 6830–6844 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  32. Feneuil, T.: Building MPCitH-based signatures from MQ, MinRank. Rank SD and PKP. Cryptology ePrint Archive (2022)

    Google Scholar 

  33. Feneuil, T., Joux, A., Rivain, M.: Shared permutation for syndrome decoding: new zero-knowledge protocol and code-based signature. Des. Codes Cryptogr. 91, 1–46 (2022)

    MathSciNet  MATH  Google Scholar 

  34. Feneuil, T., Joux, A., Rivain, M.: Syndrome decoding in the head: Shorter signatures from zero-knowledge proofs. Cryptology ePrint Archive (2022)

    Google Scholar 

  35. Fiat, A., Shamir, A.: How to prove yourself: practical solutions to identification and signature problems. In: Odlyzko, A.M. (ed.) CRYPTO 1986. LNCS, vol. 263, pp. 186–194. Springer, Heidelberg (1987). https://doi.org/10.1007/3-540-47721-7_12

    Chapter  Google Scholar 

  36. Fouque, P.A., et al.: FALCON: fast-fourier lattice-based compact signatures over NTRU, specification v1.2 (2020). https://csrc.nist.gov/Projects/post-quantum-cryptography/selected-algorithms-2022

  37. Gardy, D., Solé, P.: Saddle point techniques in asymptotic coding theory. In: Cohen, G., Lobstein, A., Zémor, G., Litsyn, S. (eds.) Algebraic Coding 1991. LNCS, vol. 573, pp. 75–81. Springer, Heidelberg (1992). https://doi.org/10.1007/BFb0034343

    Chapter  MATH  Google Scholar 

  38. Gentry, C.: Key recovery and message attacks on NTRU-composite. In: Pfitzmann, B. (ed.) EUROCRYPT 2001. LNCS, vol. 2045, pp. 182–194. Springer, Heidelberg (2001). https://doi.org/10.1007/3-540-44987-6_12

    Chapter  Google Scholar 

  39. Gentry, C., Peikert, C., Vaikuntanathan, V.: Trapdoors for hard lattices and new cryptographic constructions. In: Proceedings of the Fortieth Annual ACM Symposium on Theory of Computing, pp. 197–206 (2008)

    Google Scholar 

  40. Gligoroski, D., Samardjiska, S., Jacobsen, H., Bezzateev, S.: McEliece in the world of Escher. Cryptology ePrint Archive (2014)

    Google Scholar 

  41. Gueron, S., Persichetti, E., Santini, P.: Designing a practical code-based signature scheme from zero-knowledge proofs with trusted setup. Cryptography 6(1), 5 (2022)

    Article  Google Scholar 

  42. Horlemann-Trautmann, A.L., Weger, V.: Information set decoding in the Lee metric with applications to cryptography. Adv. Math. Commun. 15(4) (2021)

    Google Scholar 

  43. Jang, K., Baksi, A., Kim, H., Song, G., Seo, H., Chattopadhyay, A.: Quantum analysis of AES. Cryptology ePrint Archive (2022)

    Google Scholar 

  44. Löndahl, C., Johansson, T., Koochak Shooshtari, M., Ahmadian-Attari, M., Aref, M.R.: Squaring attacks on McEliece public-key cryptosystems using quasi-cyclic codes of even dimension. Des. Codes Crypt. 80, 359–377 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  45. May, A., Meurer, A., Thomae, E.: Decoding random linear codes in \(\tilde{\cal{O}}(2^{0.054n})\). In: Lee, D.H., Wang, X. (eds.) ASIACRYPT 2011. LNCS, vol. 7073, pp. 107–124. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-25385-0_6

    Chapter  MATH  Google Scholar 

  46. Moody, D., Perlner, R.: Vulnerabilities of “McEliece in the world of escher’’. In: Takagi, T. (ed.) PQCrypto 2016. LNCS, vol. 9606, pp. 104–117. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-29360-8_8

    Chapter  Google Scholar 

  47. Persichetti, E.: Efficient one-time signatures from quasi-cyclic codes: a full treatment. Cryptography 2(4), 30 (2018)

    Article  MathSciNet  Google Scholar 

  48. Phesso, A., Tillich, J.-P.: An efficient attack on a code-based signature scheme. In: Takagi, T. (ed.) PQCrypto 2016. LNCS, vol. 9606, pp. 86–103. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-29360-8_7

    Chapter  Google Scholar 

  49. Prange, E.: The use of information sets in decoding cyclic codes. IRE Trans. Inf. Theory 8(5), 5–9 (1962)

    Article  MathSciNet  Google Scholar 

  50. Regev, O., Rosen, R.: Lattice problems and norm embeddings. In: Proceedings of the Thirty-Eighth Annual ACM Symposium on Theory of Computing, pp. 447–456 (2006)

    Google Scholar 

  51. Santini, P., Baldi, M., Chiaraluce, F.: Cryptanalysis of a one-time code-based digital signature scheme. In: 2019 IEEE International Symposium on Information Theory (ISIT), pp. 2594–2598. IEEE (2019)

    Google Scholar 

  52. Schnorr, C.P., Euchner, M.: Lattice basis reduction: improved practical algorithms and solving subset sum problems. Math. Program. 66, 181–199 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  53. Sendrier, N.: Decoding one out of many. In: Yang, B.-Y. (ed.) PQCrypto 2011. LNCS, vol. 7071, pp. 51–67. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-25405-5_4

    Chapter  Google Scholar 

  54. National Institute of Standards and Technology: SHA-3 Standard: Permutation-Based Hash and Extendable-Output Functions. Technical report (2015). https://doi.org/10.6028/nist.fips.202

  55. Stern, J.: A method for finding codewords of small weight. Coding Theory Appl. 388, 106–113 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  56. Weger, V., Khathuria, K., Horlemann, A.L., Battaglioni, M., Santini, P., Persichetti, E.: On the hardness of the Lee syndrome decoding problem. Adv. Math. Commun. (2022). https://doi.org/10.3934/amc.2022029. https://www.aimsciences.org/en/article/doi/10.3934/amc.2022029

Download references

Acknowledgments

We would like to thank Sabine Pircher, Georg Sigl, Thomas Debris-Alazard and Wessel van Woerden for meaningful discussions.

Violetta Weger is supported by the European Union’s Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement no. 899987. Sebastian Bitzer, Georg Maringer, Stefan Ritterhoff and Antonia Wachter-Zeh were supported by the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG) under Grant No. WA3907/4-1, the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement no. 801434), and acknowledge the financial support by the Federal Ministry of Education and Research of Germany in the program of “Souverän. Digital. Vernetzt.”. Joint project 6G-life, project identification number: 16KISK002. Patrick Karl acknowledges the financial support by the Federal Ministry of Education and Research of Germany in the program of “Souverän. Digital. Vernetzt.”. Joint project 6G-life, project identification number: 16KISK002.

The authors would like to thank Wessel van Woerden and Felicitas Hörmann for pointing out the possible attack on FuLeeca.

Author information

Authors and Affiliations

  1. TUM School of Computation, Information and Technology, Technical University of Munich, Munich, Germany

    Stefan Ritterhoff, Georg Maringer, Sebastian Bitzer, Violetta Weger, Patrick Karl, Thomas Schamberger, Jonas Schupp & Antonia Wachter-Zeh

Authors
  1. Stefan Ritterhoff
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Georg Maringer
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sebastian Bitzer
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Violetta Weger
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Patrick Karl
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Thomas Schamberger
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. Jonas Schupp
    View author publications

    You can also search for this author in PubMed Google Scholar

  8. Antonia Wachter-Zeh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Georg Maringer .

Editor information

Editors and Affiliations

  1. Technology Innovation Institute, Abu Dhabi, United Arab Emirates

    Andre Esser

  2. Marche Polytechnic University, Ancona, Italy

    Paolo Santini

Rights and permissions

Reprints and permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Ritterhoff, S. et al. (2023). FuLeeca: A Lee-Based Signature Scheme. In: Esser, A., Santini, P. (eds) Code-Based Cryptography. CBCrypto 2023. Lecture Notes in Computer Science, vol 14311. Springer, Cham. https://doi.org/10.1007/978-3-031-46495-9_4

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-46495-9_4

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-46494-2

  • Online ISBN: 978-3-031-46495-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation