Skip to main content
SpringerLink
Log in

New lower bounds on nonlinearity and a class of highly nonlinear functions

  • Cryptographic Functions And Ciphers
  • Conference paper
  • First Online:
Information Security and Privacy (ACISP 1997)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1270))

Included in the following conference series:

  • 124 Accesses

Abstract

Highly nonlinear Boolean functions occupy an important position in the design of secure block as well as stream ciphers. This paper proves two new lower bounds on the nonlinearity of Boolean functions. Based on the study of these new lower bounds, we introduce a class of highly nonlinear Boolean functions on odd dimensional spaces and show examples of such functions.

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

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. C. M. Adams and S. E. Tavares. Generating and counting binary bent sequences. IEEE Transactions on Information Theory, IT-36 No. 5:1170–1173, 1990.

    Article  Google Scholar 

  2. K. G. Beauchamp. Applications of Walsh and Related Functions with an Introduction to Sequency Functions. Microelectronics and Signal Processing. Academic Press, London, New York, Tokyo, 1984.

    Google Scholar 

  3. Claude Carlet. Partially-bent functions. Designs, Codes and Cryptography, 3:135–145, 1993.

    Google Scholar 

  4. G. D. Cohen, M. G. Karpovsky, Jr. H. F. Mattson, and J. R. Schatz. Covering radius — survey and recent results. IEEE Transactions on Information Theory, IT-31(3):328–343, 1985.

    Article  Google Scholar 

  5. J. F. Dillon. A survey of bent functions. The NSA Technical Journal, pages 191–215, 1972. (unclassified).

    Google Scholar 

  6. C. Ding, G. Xiao, and W. Shan. The Stability Theory of Stream Ciphers, volume 561 of Lecture Notes in Computer Science. Springer-Verlag, Berlin, Heidelberg, New York, 1991.

    Google Scholar 

  7. F. J. MacWilliams and N. J. A. Sloane. The Theory of Error-Correcting Codes. North-Holland, Amsterdam, New York, Oxford, 1978.

    Google Scholar 

  8. M. Matsui. Linear cryptanalysis method for DES cipher. In Advances in Cryptology — EUROCRYPT'93, volume 765, Lecture Notes in Computer Science, pages 386–397. Springer-Verlag, Berlin, Heidelberg, New York, 1994.

    Google Scholar 

  9. B. Preneel, W. V. Leekwijck, L. V. Linden, R. Govaerts, and J. Vandewalle. Propagation characteristics of boolean functions. In Advances in Cryptology — EUROCRYPT'90, volume 437, Lecture Notes in Computer Science, pages 155–165. Springer-Verlag, Berlin, Heidelberg, New York, 1991.

    Google Scholar 

  10. O. S. Rothaus. On “bent” functions. Journal of Combinatorial Theory, Ser. A, 20:300–305, 1976.

    Google Scholar 

  11. J. Seberry, X. M. Zhang, and Y. Zheng. Nonlinearity and propagation characteristics of balanced boolean functions. Information and Computation, 119(1):1–13, 1995.

    Article  Google Scholar 

  12. T. Siegenthaler. Correlation-immunity of nonlinear combining functions for cryptographic applications. IEEE Transactions on Information Theory, IT-30 No. 5:776–779, 1984.

    Article  Google Scholar 

  13. H. Tanaka and T. Kaneko. A linear attack to the random generator by non linear combiner. In Proceedings of 1996 IEEE International Symposium on Information Theory and Its Applications, volume 1, pages 331–334, Victoria, B.C., Canada, September 1996.

    Google Scholar 

  14. R. Yarlagadda and J. E. Hershey. Analysis and synthesis of bent sequences. IEE Proceedings (Part E), 136:112–123, 1989.

    Google Scholar 

  15. X. M. Zhang and Y. Zheng. GAC — the criterion for global avalanche characteristics of cryptographic functions. Journal of Universal Computer Science, 1(5):316–333, 1995. (available at http://hgiicm.tu-graz.ac.at/).

    Google Scholar 

  16. X. M. Zhang and Y. Zheng. Auto-correlations and new bounds on the nonlinearity of boolean functions. In Advances in Cryptology — EUROCRYPT'96, volume 1070, Lecture Notes in Computer Science, pages 294–306. Springer-Verlag, Berlin, Heidelberg, New York, 1996.

    Google Scholar 

  17. X. M. Zhang and Y. Zheng. Characterizing the structures of cryptographic functions satisfying the propagation criterion for almost all vectors. Design, Codes and Cryptography, 7(1/2):111–134, 1996. special issue dedicated to Gus Simmons.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. The University of Wollongong, 2522, Wollongong, NSW, Australia

    Xian-Mo Zhang

  2. Monash University, VIC 3199, Frankston, Melbourne, Australia

    Yuliang Zheng

Authors
  1. Xian-Mo Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yuliang Zheng
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Vijay Varadharajan Josef Pieprzyk Yi Mu

Rights and permissions

Reprints and permissions

Copyright information

© 1997 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Zhang, XM., Zheng, Y. (1997). New lower bounds on nonlinearity and a class of highly nonlinear functions. In: Varadharajan, V., Pieprzyk, J., Mu, Y. (eds) Information Security and Privacy. ACISP 1997. Lecture Notes in Computer Science, vol 1270. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0027922

Download citation

  • DOI: https://doi.org/10.1007/BFb0027922

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-63232-0

  • Online ISBN: 978-3-540-69237-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation