Skip to main content
Springer Nature Link
Log in

A Theory of Highly Nonlinear Functions

  • Conference paper
Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (AAECC 2006)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3857))

  • 1086 Accesses

Abstract

Highly nonlinear functions are important as sources of low-correlation sequences, high-distance codes and cryptographic primitives, as well as for applications in combinatorics and finite geometry.

We argue that the theory of such functions is best seen in terms of splitting factor pairs. This introduces an extra degree of freedom, through the pairing of a normalised function φ : G → N between groups with a homomorphism \(\varrho : {G} \rightarrow Aut{(N)}\).

From this perspective we introduce a new definition of equivalence for functions, relative to \(\varrho\), and show it preserves their difference distributions. When \(\varrho \equiv 1\) it includes CCZ and generalised linear equivalence, as well as planar and linear equivalence.

More generally, we use splitting factor pairs to relate several important measures of nonlinearity. We propose approaches to both linear approximation theory and bent functions, and to difference distribution theory and perfect nonlinear functions, which encompass the current approaches.

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.

Similar content being viewed by others

References

  1. Breveglieri, L., Cherubini, A., Macchetti, M.: On the generalized linear equivalence of functions over finite fields. In: Lee, P.J. (ed.) ASIACRYPT 2004. LNCS, vol. 3329, pp. 79–91. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  2. Budaghyan, L., Carlet, C., Pott, A.: New classes of almost bent and almost perfect nonlinear polynomials (2005) (preprint)

    Google Scholar 

  3. Carlet, C., Charpin, P., Zinoviev, V.: Codes, bent functions and permutations suitable for DES-like cryptosystems. Des., Codes Cryptogr. 15, 125–156 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  4. Carlet, C., Ding, C.: Highly nonlinear mappings. J. Complexity 20, 205–244 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  5. Coulter, R.S., Matthews, R.W.: Planar functions and planes of Lenz-Barlotti Class II. Des., Codes Cryptogr. 10, 167–184 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  6. de Launey, W.: Generalised Hadamard matrices which are developed modulo a group. Discrete Math. 104, 49–65 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  7. Galati, J.C.: A group extensions approach to relative difference sets. J. Combin. Designs 12, 279–298 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  8. Horadam, K.J.: Equivalence classes of central semiregular relative difference sets. J. Combin. Des. 8, 330–346 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  9. Horadam, K.J.: Hadamard Matrices, 9 Chapters, approx. 250 pp. Princeton University Press, Princeton (under review)

    Google Scholar 

  10. Hughes, G.: Characteristic functions of relative difference sets, correlated sequences and Hadamard matrices. In: Fossorier, M.P.C., Imai, H., Lin, S., Poli, A. (eds.) AAECC 1999. LNCS, vol. 1719, pp. 346–354. Springer, Heidelberg (1999)

    Chapter  Google Scholar 

  11. Hughes, G.: The equivalence of certain auto-correlated quaternary and binary arrays. Australas. J. Combin. 22, 37–40 (2000)

    MATH  MathSciNet  Google Scholar 

  12. Jungnickel, D.: On automorphism groups of divisible designs. Can. J. Math. 34, 257–297 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  13. Logachev, O.A., Salnikov, A.A., Yashchenko, V.V.: Bent functions on a finite abelian group. Discrete Math. Appl. 7, 547–564 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  14. Nyberg, K.: Perfect nonlinear S-boxes. In: Davies, D.W. (ed.) EUROCRYPT 1991. LNCS, vol. 547, pp. 378–386. Springer, Heidelberg (1991)

    Google Scholar 

  15. Perera, A.A.I., Horadam, K.J.: Cocyclic generalised Hadamard matrices and central relative difference sets. Des., Codes Cryptogr. 15, 187–200 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  16. Pott, A.: Nonlinear functions in abelian groups and relative difference sets. Discr. Appl. Math. 138, 177–193 (2004)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Royal Melbourne Institute of Technology, Melbourne, VIC, 3001, Australia

    K. J. Horadam

Authors
  1. K. J. Horadam
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Electrical Engineering, University of Hawaii, Honolulu, USA

    Marc P. C. Fossorier

  2. National Institute of Advanced Industrial Science and Technology (AIST), Research Center for Information Security (RCIS),  

    Hideki Imai

  3. Department of Electrical and Computer Engineering, University of California, CA 95616, Davis,  

    Shu Lin

  4. University Paul Sabatier, AAECC/IRIT, 118 route de Narbonne, 31300, Toulouse cedex, France

    Alain Poli

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Horadam, K.J. (2006). A Theory of Highly Nonlinear Functions. In: Fossorier, M.P.C., Imai, H., Lin, S., Poli, A. (eds) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC 2006. Lecture Notes in Computer Science, vol 3857. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11617983_8

Download citation

  • DOI: https://doi.org/10.1007/11617983_8

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-31423-3

  • Online ISBN: 978-3-540-31424-0

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics