Skip to main content
SpringerLink
Log in

Idempotents in the neighbourhood of Patterson-Wiedemann functions having Walsh spectra zeros

  • Published:
Designs, Codes and Cryptography Aims and scope Submit manuscript

Abstract

In this paper we study the neighbourhood of 15-variable Patterson-Wiedemann (PW) functions, i.e., the functions that differ by a small Hamming distance from the PW functions in terms of truth table representation. We exploit the idempotent structure of the PW functions and interpret them as Rotation Symmetric Boolean Functions (RSBFs). We present techniques to modify these RSBFs to introduce zeros in the Walsh spectra of the modified functions with minimum reduction in nonlinearity. Our technique demonstrates 15-variable balanced and 1-resilient functions with currently best known nonlinearities 16272 and 16264 respectively. In the process, we find functions for which the autocorrelation spectra and algebraic immunity parameters are best known till date.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Berlekamp E.R., Welch L.R. (1972). Weight distributions of the cosets of the (32, 6) Reed-Muller code. IEEE Trans. Inform. Theory 18(1): 203–207

    Article  MATH  MathSciNet  Google Scholar 

  2. Carlet C., Dalai D.K., Gupta K.C., Maitra S. (2006). Algebraic immunity for cryptographically significant Boolean functions: analysis and construction. IEEE Trans. Inform. Theory 527: 3105–3121

    Article  MathSciNet  Google Scholar 

  3. Dillon J.F.: Elementary Hadamard difference sets. Ph.D. thesis, University of Maryland (1974).

  4. Dobbertin H.: Construction of bent functions and balanced Boolean functions with high nonlinearity. In: Preneel B. (ed.) Fast Software Encryption, Lecture Notes in Computer Science, vol. 1008, pp. 61–74. Springer (1994).

  5. Filiol E., Fontaine C.: Highly nonlinear balanced Boolean functions with a good correlation-immunity. In: Nyberg K. (ed.) EUROCRYPT, Lecture Notes in Computer Science, vol. 1403, pp. 475–488. Springer (1998).

  6. Fontaine C.: The nonlinearity of a class of Boolean functions with short representation. In: Přibyl J. (ed.) PRAGOCRYPT’96, pp. 129–144. CTU Publishing House (1996).

  7. Fontaine C. (1999). On some cosets of the first-order Reed-Muller code with high minimum weight. IEEE Trans. Inform. Theory 45(4): 1237–1243

    Article  MATH  MathSciNet  Google Scholar 

  8. Kavut S., Maitra S., Sarkar S., Yücel M.D.: Enumeration of 9-variable rotation symmetric Boolean functions having nonlinearity > 240. In: Barua R., Lange T. (eds.) INDOCRYPT, Lecture Notes in Computer Science, vol. 4329, pp. 266–279. Springer (2006).

  9. Kavut S., Maitra S., Yücel M.D. (2007). Search for Boolean functions with excellent profiles in the rotation symmetric class. IEEE Trans. Inform. Theory 53(5): 1743–1751

    Article  MathSciNet  Google Scholar 

  10. Kavut S., Yücel M.D.: Balanced Boolean functions with nonlinearity \( > 2^{n-1} - 2^{\frac{n-1}{2}}\) . Cryptology ePrint Archive, http://eprint.iacr.org/, Report 2007/321, 15 August (2007).

  11. Langevin P., Zanotti J.-P. (2005). Nonlinearity of some invariant Boolean functions. Des. Codes Cryptogr. 36(2): 131–146

    Article  MATH  MathSciNet  Google Scholar 

  12. Maitra S., Sarkar P. (2002). Modifications of Patterson-Wiedemann functions for cryptographic applications. IEEE Trans. Inform. Theory 48(1): 278–284

    Article  MATH  MathSciNet  Google Scholar 

  13. Mykkeltveit J.J. (1980). The covering radius of the (128, 8) Reed-Muller code is 56. IEEE Trans. Inform. Theory 26(3): 359–362

    Article  MATH  MathSciNet  Google Scholar 

  14. Pasalic E., Johansson T.: Further results on the relation between nonlinearity and resiliency for Boolean functions. In: Walker M. (ed.) IMA International Conference, Lecture Notes in Computer Science, vol. 1746, pp. 35–44. Springer (1999).

  15. Patterson N.J., Wiedemann D.H.: The covering radius of the (215, 16) Reed-Muller code is at least 16276. IEEE Trans. Inform. Theory 29(3), 354–356 (1983). See also the correction in 36(2): 443 (1990).

    Google Scholar 

  16. Rothaus O.S. (1976). On “bent” functions. J. Combin. Theory Ser. A 20(3): 300–305

    Article  MATH  MathSciNet  Google Scholar 

  17. Sarkar P., Maitra S.: Construction of nonlinear Boolean functions with important cryptographic properties. In: Advances in Cryptology-EUROCRYPT, Lecture Notes in Computer Science, vol. 1807, pp. 485–506 (2000).

  18. Sarkar P., Maitra S.: Nonlinearity bounds and constructions of resilient Boolean functions. In: Bellare M. (ed.) CRYPTO, Lecture Notes in Computer Science, vol. 1880, pp. 515–532. Springer (2000).

  19. Sarkar P., Maitra S. (2004). Construction of nonlinear resilient Boolean functions using “small” affine functions. IEEE Trans. Inform. Theory 50(9): 2185–2193

    Article  MathSciNet  Google Scholar 

  20. Sarkar S., Maitra S.: Idempotents in the neighbourhood of Patterson-Wiedemann functions having Walsh spectra zeros. In: WCC 2007, International Workshop on Coding and Cryptography,pp. 351–360, April 16-20, 2007, Versailles (France). A detailed version is available at Cryptology ePrint Archive, http://eprint.iacr.org/, Report 2007/427, 19 November, 2007.

  21. Seberry J., Zhang X., Zheng Y.: Nonlinearly balanced Boolean functions and their propagation characteristics (extended abstract). In: Stinson D.R. (ed.) CRYPTO, Lecture Notes in Computer Science, vol. 773, pp. 49–60. Springer (1993).

  22. Stănică P., Maitra S., Clark J.: Results on Rotation Symmetric Bent and Correlation Immune Boolean Functions. Fast Software Encryption Workshop (FSE 2004). Lecture Notes in Computer Science, vol. 3017, pp. 161–177. Springer Verlag (2004).

Download references

Author information

Authors and Affiliations

  1. Applied Statistics Unit, Indian Statistical Institute, 203 B T Road, Kolkata, 700 108, India

    Sumanta Sarkar & Subhamoy Maitra

Authors
  1. Sumanta Sarkar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Subhamoy Maitra
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Subhamoy Maitra.

Additional information

This is a revised version (with new results) of the paper that has been presented in WCC 2007, International Workshop on Coding and Cryptography, April 16-20, 2007, Versailles (France).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sarkar, S., Maitra, S. Idempotents in the neighbourhood of Patterson-Wiedemann functions having Walsh spectra zeros. Des. Codes Cryptogr. 49, 95–103 (2008). https://doi.org/10.1007/s10623-008-9181-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-008-9181-y

Keywords

AMS Classification

Search

Navigation