Skip to main content

On the Algebraic Normal Form and Walsh Spectrum of Symmetric Functions over Finite Rings

  • Conference paper
  • 1000 Accesses

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 7369)

Abstract

A function over finite rings is a function from a ring \(E_{q}^{n}\) to a ring E r , where E k is ℤ /k ℤ. These functions are well used in cryptography: cipher design, hash function design and in theoretical computer science. In this paper, we are especially interested in symmetric functions. We give practical ways of computing their ANF and their Walsh Spectrum in \(\mathcal{O}\left({ n+q-1 \choose q-1 }^2\right)\) using linear algebra. Thus, we achieve a better complexity both in time and memory than the fast Fourier transform which is in \(\mathcal{O}\left( q^nn\log(q) \right)\).

Keywords

  • Boolean Function
  • Symmetric Function
  • Symmetry Class
  • Gray Code
  • Algebraic Degree

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (Canada)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   49.99
Price excludes VAT (Canada)
  • 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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Andrews, G.E.: The Theory of Partitions. Encyclopedia of Mathematics and its Applications, vol. 2. Addison-Wesley Publishing Co., Reading (1976); Reprinted by Cambridge University Press, Cambridge (1998)

    MATH  Google Scholar 

  2. Ars, G., Faugère, J.-C.: Algebraic immunities of functions over finite fields. Research Report RR-5532, INRIA (2005)

    Google Scholar 

  3. Camion, P., Canteaut, A.: Generalization of Siegenthaler Inequality and Schnorr-Vaudenay Multipermutations. In: Koblitz, N. (ed.) CRYPTO 1996. LNCS, vol. 1109, pp. 372–386. Springer, Heidelberg (1996), 10.1007/3-540-68697-5_28

    Google Scholar 

  4. Canteaut, A., Videau, M.: Symmetric boolean functions. IEEE Transactions on Information Theory 51(8), 2791–2811 (2005)

    CrossRef  MathSciNet  Google Scholar 

  5. Carlet, C.: The complexity of boolean functions from cryptographic viewpoint. In: Krause, M., Pudlák, P., Reischuk, R., van Melkebeek, D. (eds.) Complexity of Boolean Functions, Dagstuhl, Germany. Dagstuhl Seminar Proceedings, vol. 06111. Internationales Begegnungs- und Forschungszentrum für Informatik (IBFI), Schloss Dagstuhl, Germany (2006)

    Google Scholar 

  6. Cusick, T.W., Li, Y., Stanica, P.: Balanced symmetric functions over GF(p). IEEE Transactions on Information Theory 54(3), 1304–1307 (2008)

    CrossRef  MathSciNet  Google Scholar 

  7. Fu, S., Li, C., Sun, B.: Enumeration of Homogeneous Rotation Symmetric Functions over f p . In: Franklin, M.K., Hui, L.C.K., Wong, D.S. (eds.) CANS 2008. LNCS, vol. 5339, pp. 278–284. Springer, Heidelberg (2008)

    CrossRef  Google Scholar 

  8. Gopalakrishnan, K., Stinson, D.R.: Three characterizations of non-binary correlation-immune and resilient functions. Designs, Codes and Cryptography 5, 241–251 (1997)

    CrossRef  MathSciNet  Google Scholar 

  9. Hu, Y., Xiao, G.: Resilient functions over finite fields. IEEE Transactions on Information Theory 49(8), 2040–2046 (2003)

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. Krawtchouk, M.: Sur une généralisation des polynômes d’Hermite. C.R. Acad. Sci. Paris 189, 620–622 (1929)

    MATH  Google Scholar 

  11. Li, Y., Cusick, T.W.: Strict avalanche criterion over finite fields, submitted. Journal of Mathematical Cryptology 1, 65–78 (2005)

    CrossRef  MathSciNet  Google Scholar 

  12. Meier, W., Pasalic, E., Carlet, C.: Algebraic Attacks and Decomposition of Boolean Functions. In: Cachin, C., Camenisch, J.L. (eds.) EUROCRYPT 2004. LNCS, vol. 3027, pp. 474–491. Springer, Heidelberg (2004)

    CrossRef  Google Scholar 

  13. Mouffron, M.: Transitive q-Ary Functions over Finite Fields or Finite Sets: Counts, Properties and Applications. In: von zur Gathen, J., Imaña, J.L., Koç, Ç.K. (eds.) WAIFI 2008. LNCS, vol. 5130, pp. 19–35. Springer, Heidelberg (2008)

    CrossRef  Google Scholar 

  14. Rovetta, C., Mouffron, M.: De Bruijn sequences and complexity of symmetric functions. Cryptography and Communications, 1–19 (2011), 10.1007/s12095-011-0054-2

    Google Scholar 

  15. Sagan, B.E.: The symmetric group - representations, combinatorial algorithms, and symmetric functions. Wadsworth & Brooks/Cole mathematics series. Wadsworth (1991)

    Google Scholar 

  16. Sarkar, S., Maitra, S.: Efficient search for symmetric boolean functions under constraints on walsh spectra values. In: Michon, J.-F., Valarcher, P., Yunès, J.-B. (eds.) Proceedings of BFCA 2006 Conference, Rouen, France, March 13-15, pp. 29–50 (2006)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Batteux, B. (2012). On the Algebraic Normal Form and Walsh Spectrum of Symmetric Functions over Finite Rings. In: Özbudak, F., Rodríguez-Henríquez, F. (eds) Arithmetic of Finite Fields. WAIFI 2012. Lecture Notes in Computer Science, vol 7369. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31662-3_7

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-31662-3_7

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-31661-6

  • Online ISBN: 978-3-642-31662-3

  • eBook Packages: Computer ScienceComputer Science (R0)