Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
Springer Nature Link
Log in

On the fast algebraic immunity of threshold functions

  • Published:
Cryptography and Communications Aims and scope Submit manuscript

Abstract

Motivated by the impact of fast algebraic attacks on stream ciphers, and recent constructions using a threshold function as main part of the filtering function, we study the fast algebraic immunity of threshold functions. As a first result, we determine exactly the fast algebraic immunity of all majority functions in more than 8 variables. Then, For all n ≥ 8 and all threshold value between 1 and n we exhibit the fast algebraic immunity for most of the thresholds, and we determine a small range for the value related to the few remaining cases. Finally, provided m ≥ 2, we determine exactly the fast algebraic immunity of all threshold functions in 3 ⋅ 2m or 3 ⋅ 2m + 1 variables.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

Data Availability

Not applicable.

Code Availability

Not applicable.

References

  1. Courtois, N.: Fast algebraic attacks on stream ciphers with linear feedback. In: Boneh, D. (ed.) CRYPTO 2003, volume 2729 of LNCS, pp 176–194. Springer, Heidelberg (2003)

  2. Courtois, N., Meier, W.: Algebraic attacks on stream ciphers with linear feedback. In: Biham, E. (ed.) EUROCRYPT 2003, volume 2656 of LNCS. Springer, Heidelberg (2003)

  3. Applebaum, B., Lovett, S.: Algebraic attacks against random local functions and their countermeasures. In: Wichs, Daniel, Mansour, Y (eds.) 48th ACM STOC. ACM Press, June (2016)

  4. Armknecht, F.: Improving fast algebraic attacks. In: Roy, B.K., Meier, Willi (eds.) FSE 2004, volume 3017 of LNCS, pp 65–82. Springer, Heidelberg (2004)

  5. Hawkes, P., Rose, G.G.: Rewriting variables: The complexity of fast algebraic attacks on stream ciphers. In: Franklin, M. (ed.) CRYPTO 2004, volume 3152 of LNCS, pp 390–406. Springer, Heidelberg (2004)

  6. Armknecht, F., Carlet, C., Gaborit, P., Künzli, S., Meier, W., Ruatta, O.: Efficient computation of algebraic immunity for algebraic and fast algebraic attacks. In: Vaudenay, S. (ed.) EUROCRYPT 2006, volume 4004 of LNCS. Springer, Heidelberg (2006)

  7. Méaux, P., Journault, A., Standaert, F.-X., Carlet, C.: Towards stream ciphers for efficient FHE with low-noise ciphertexts. In: Fischlin, M., Coron, J.-S. (eds.) EUROCRYPT 2016, Part I, volume 9665 of LNCS, pp 311–343. Springer, Heidelberg (2016)

  8. Méaux, P., Carlet, C., Journault, A., Standaert, F.-X.: Improved filter permutators for efficient FHE: better instances and implementations. In: Hao, F., Ruj, S., Gupta, S.S. (eds.) Progress in cryptology - INDOCRYPT, volume 11898 of LNCS, pp. 68–91. Springer (2019)

  9. Goldreich, O.: Candidate one-way functions based on expander graphs. Electron. Colloq. Comput. Complex. (ECCC) 7(90) (2000)

  10. Méaux, P., Carlet, C., Journault, A., Standaert, F.-X.: Improved filter permutators: Combining symmetric encryption design, boolean functions, low complexity cryptography, and homomorphic encryption, for private delegation of computations. Cryptol. ePrint Arch., Report 2019/483 (2019)

  11. Hoffmann, C., Méaux, P., Ricosset, T.: Transciphering, using filip and TFHE for an efficient delegation of computation. In: Progress in cryptology - INDOCRYPT 2020 - 21st International conference on cryptology in India, Bangalore, India, December 13-16, 2020, Proceedings, pp. 39–61 (2020)

  12. Applebaum, B., Lovett, S.: Algebraic attacks against random local functions and their countermeasures. SIAM J. Comput., 52–79 (2018)

  13. Maitra, S., Sarkar, P.: Maximum nonlinearity of symmetric boolean functions on odd number of variables. IEEE Trans. Inf. Theory 48, 2626–2630 (2002). https://doi.org/10.1109/TIT.2002.801482

    Article  MathSciNet  Google Scholar 

  14. Carlet, C.: On the degree, nonlinearity, algebraic thickness, and nonnormality of boolean functions, with developments on symmetric functions. IEEE Trans. Inf. Theory, 2178–2185 (2004)

  15. Canteaut, A., Videau, M.: Symmetric boolean functions. IEEE Trans. Inf. Theory, 2791–2811 (2005)

  16. An, B., Preneel, B.: On the algebraic immunity of symmetric boolean functions. In: Progress in cryptology - INDOCRYPT 2005, 6th International conference on cryptology in India, Bangalore, India, December 10-12, 2005, Proceedings, pp. 35–48 (2005)

  17. Qu, L., Feng, K., Liu, F., Wang, L.: Constructing symmetric boolean functions with maximum algebraic immunity. IEEE Trans. Inf. Theory, 2406–2412 (2009)

  18. Chen, Y., Lu, P.: Two classes of symmetric boolean functions with optimum algebraic immunity: Construction and analysis. IEEE Trans. Inf. Theory 57(4), 2522–2538 (2011). ISSN 1557-9654. https://doi.org/10.1109/TIT.2011.2111810

    Article  MathSciNet  Google Scholar 

  19. Gao, G., Guo, Y., Zhao, Y.: Recent results on balanced symmetric boolean functions. IEEE Trans. Inf. Theory 62(9), 5199–5203 (2016). ISSN 1557-9654. https://doi.org/10.1109/TIT.2015.2455052

    Article  MathSciNet  Google Scholar 

  20. Carlet, C., Méaux, P.: Boolean functions for homomorphic-friendly stream ciphers. In: Proceedings of the Conference on Algebra, Codes and Cryptology (A2C), pp 166–182. Springer, Cham (2019)

  21. Carlet, C., Méaux, P.: A complete study of two classes of boolean functions for homomorphic-friendly stream ciphers. IACR Cryptol. ePrint Arch. 2020, 1562 (2020)

    Google Scholar 

  22. Tang, D., Luo, R., Du, X.: The exact fast algebraic immunity of two subclasses of the majority function. IEICE Trans., 2084–2088 (2016)

  23. Chen, Y., Guo, F., Zhang, L., Fast algebraic immunity of 2m + 2 and 2m + 3 variables majority function. Cryptol. ePrint Arch. Report 2019/286 (2019)

  24. Méaux, P.: On the fast algebraic immunity of majority functions. In: Schwabe, P., Thériault, N. (eds.) Progress in cryptology - LATINCRYPT, volume 11774 of LNCS, pp. 86–105. Springer (2019)

  25. Carlet, C.: Boolean Functions for Cryptography and Coding Theory. Cambridge University Press, Cambridge (2020)

    Book  Google Scholar 

  26. Dalai, D.K., Maitra, S., Sarkar, S.: Basic theory in construction of boolean functions with maximum possible annihilator immunity. Des. Codes Cryptogr. (2006)

  27. Qu, L., Li, C., Feng, K.: A note on symmetric boolean functions with maximum algebraic immunity in odd number of variables. IEEE Trans. Inf. Theory 53 (2007)

  28. Sarkar, P., Maitra, S.: Balancedness and correlation immunity of symmetric boolean functions. Discrete Math. :2351–2358. ISSN 0012-365X (2007)

  29. Linial, N., Rothschild, B.: Incidence matrices of subsets—a rank formula. SIAM J. Algebraic. Discrete Methods 2, 09 (1981)

    Article  MathSciNet  Google Scholar 

Download references

Funding

The author is a beneficiary of a FSR (“Fonds spécial de recherche”, Belgium) incoming post-doctoral fellowship.

Author information

Authors and Affiliations

  1. ICTEAM/ELEN/Crypto Group, Université catholique de Louvain, Louvain-la-Neuve, Belgium

    Pierrick Méaux

Authors
  1. Pierrick Méaux
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pierrick Méaux.

Ethics declarations

Conflict of Interests

Not applicable.

Additional information

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Méaux, P. On the fast algebraic immunity of threshold functions. Cryptogr. Commun. 13, 741–762 (2021). https://doi.org/10.1007/s12095-021-00505-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12095-021-00505-y

Keywords

Mathematics Subject Classification (2010)

Search

Navigation