Skip to main content
SpringerLink
Log in

On the Properties of Vectorial Functions with Plateaued Components and Their Consequences on APN Functions

  • Conference paper
Book cover Codes, Cryptology, and Information Security (C2SI 2015)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 9084))

Abstract

[This is an extended abstract of paper [15], which has been submitted to a journal] Boolean plateaued functions and vectorial functions with plateaued components, that we simply call plateaued, play a significant role in cryptography, but little is known on them. We give here, without proofs, new characterizations of plateaued Boolean and vectorial functions, by means of the value distributions of derivatives and of power moments of the Walsh transform. This allows us to derive several characterizations of APN functions in this framework, showing that all the main results known for quadratic APN functions extend to plateaued functions. Moreover, we prove that the APN-ness of those plateaued vectorial functions whose component functions are unbalanced depends only on their value distribution. This proves that any plateaued (n,n)-function, n even, having same value distribution as APN power functions, is APN and has same extended Walsh spectrum as the APN Gold functions.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • 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

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Berger, T., Canteaut, A., Charpin, P., Laigle-Chapuy, Y.: On almost perfect nonlinear functions. IEEE Trans. Inform. Theory 52(9), 4160–4170 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  2. Biham, E., Shamir, A.: Differential Cryptanalysis of DES-like Cryptosystems. Journal of Cryptology 4(1), 3–72 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  3. Budaghyan, L.: Construction and Analysis of Cryptographic Functions. Springer, (200 pages) (to appear)

    Google Scholar 

  4. Budaghyan, L., Carlet, C.: Classes of Quadratic APN Trinomials and Hexanomials and Related Structures. IEEE Trans. Inform. Theory 54(5), 2354–2357 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  5. Budaghyan, L., Carlet, C.: CCZ-equivalence of Bent Vectorial Functions and Related Constructions. Designs, Codes and Cryptography 59(1-3), 69–87 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  6. Budaghyan, L., Carlet, C., Felke, P., Leander, G.: An infinite class of quadratic APN functions which are not equivalent to power functions. In: Proceedings of IEEE International Symposium on Information Theory (ISIT) (2006)

    Google Scholar 

  7. Budaghyan, L., Carlet, C., Leander, G.: Two classes of quadratic APN binomials inequivalent to power functions. IEEE Trans. Inform. Theory 54(9), 4218–4229 (2008), This paper is a completed and merged version of [6] and [8]

    Google Scholar 

  8. Budaghyan, L., Carlet, C., Leander, G.: Another class of quadratic APN binomials over \(\mathbb{F}_{2^n}\): the case n divisible by 4. In: Proceedings of the Workshop on Coding and Cryptography, WCC 2007, pp. 49–58 (2007)

    Google Scholar 

  9. Budaghyan, L., Carlet, C., Leander, G.: Constructing new APN functions from known ones. Finite Fields and Applications 15(2), 150–159 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  10. Budaghyan, L., Carlet, C., Pott, A.: New Classes of Almost Bent and Almost Perfect Nonlinear Polynomials. In: Proceedings of the Workshop on Coding and Cryptography 2005, Bergen, pp. 306–315 (2005)

    Google Scholar 

  11. Budaghyan, L., Carlet, C., Pott, A.: New Classes of Almost Bent and Almost Perfect Nonlinear Functions. IEEE Trans. Inform. Theory 52(3), 1141–1152 (2006), This is a completed version of [10]

    Google Scholar 

  12. Carlet, C.: A transformation on Boolean functions, its consequences on some problems related to Reed-Muller codes. In: Charpin, P., Cohen, G. (eds.) EUROCODE 1990. LNCS, vol. 514, pp. 42–50. Springer, Heidelberg (1991)

    Chapter  Google Scholar 

  13. Carlet, C.: Boolean Functions for Cryptography and Error Correcting Codes. Chapter of the monography. In: Crama, Y., Hammer, P. (eds.) Boolean Models and Methods in Mathematics, Computer Science, and Engineering, pp. 257–397. Cambridge University Press (2010), Preliminary version available at http://www.math.univ-paris13.fr/~carlet/pubs.html

  14. Carlet, C.: Vectorial Boolean Functions for Cryptography. Chapter of the monography. In: Crama, Y., Hammer, P. (eds.) Boolean Methods and Models. Cambridge University Press, Preliminary version available at http://www.math.univ-paris13.fr/~carlet/pubs.html

  15. Carlet, C.: Boolean and vectorial plateaued functions and APN functions. Preprint

    Google Scholar 

  16. Carlet, C., Charpin, P., Zinoviev, V.: Codes, bent functions and permutations suitable for DES-like cryptosystems. Designs, Codes and Cryptography 15(2), 125–156 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  17. Carlet, C., Prouff, E.: On plateaued functions and their constructions. In: Johansson, T. (ed.) FSE 2003. LNCS, vol. 2887, pp. 54–73. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  18. Carlet, C., Gong, G., Tan, Y.: Quadratic Zero-Difference Balanced Functions, APN functions and Strongly Regular Graphs. To appear in Designs, Codes and Cryptography

    Google Scholar 

  19. Chabaud, F., Vaudenay, S.: Links between Differential and Linear Cryptanalysis. In: De Santis, A. (ed.) EUROCRYPT 1994. LNCS, vol. 950, pp. 356–365. Springer, Heidelberg (1995)

    Chapter  Google Scholar 

  20. Ding, C., Tan, Y.: Zero-Difference Balanced Functions With Applications. Journal of Statistical Theory and Practice 6(1), 3–19 (2012)

    Article  MathSciNet  Google Scholar 

  21. Knudsen, L.: Truncated and higher order differentials. In: Preneel, B. (ed.) FSE 1994. LNCS, vol. 1008, pp. 196–211. Springer, Heidelberg (1995)

    Chapter  Google Scholar 

  22. Lai, X.: Higher order derivatives and differential cryptanalysis. In: Proceedings of the Symposium on Communication, Coding and Cryptography (1994), in Honor of J. L. massey on the occasion of his 60’th birthday

    Google Scholar 

  23. Matsui, M.: Linear cryptanalysis method for DES cipher. In: Helleseth, T. (ed.) EUROCRYPT 1993. LNCS, vol. 765, pp. 386–397. Springer, Heidelberg (1994)

    Chapter  Google Scholar 

  24. Mesnager, S.: Characterizations of plateaued and bent functions in characteristic p. In: Schmidt, K.-U., Winterhof, A. (eds.) SETA 2014. LNCS, vol. 8865, pp. 71–81. Springer, Heidelberg (2014)

    Google Scholar 

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

    Chapter  Google Scholar 

  26. Nyberg, K.: On the construction of highly nonlinear permutations. In: Rueppel, R.A. (ed.) EUROCRYPT 1992. LNCS, vol. 658, pp. 92–98. Springer, Heidelberg (1993)

    Chapter  Google Scholar 

  27. Yu, Y., Wang, M., Li, Y.: A matrix approach for constructing quadratic APN functions. In: Proceedings of International Workshop on Coding and Cryptography, pp. 39–47 (2013)

    Google Scholar 

  28. Weng, G., Tan, Y., Gong, G.: On almost perfect nonlinear functions and their related algebraic objects. In: Proceedings of International Workshop on Coding and Cryptography, pp. 48–57 (2013)

    Google Scholar 

  29. Zheng, Y., Zhang, X.-M.: Plateaued functions. In: Varadharajan, V., Mu, Y. (eds.) ICICS 1999. LNCS, vol. 1726, pp. 284–300. Springer, Heidelberg (1999)

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. LAGA, UMR 7539, CNRS, Universities of Paris 8 and Paris 13, Department of Mathematics, University of Paris 8, 2 rue de laliberté, 93526, Saint-Denis cedex 02, France

    Claude Carlet

Authors
  1. Claude Carlet
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Claude Carlet .

Editor information

Editors and Affiliations

  1. University of Mohammed V, Rabat, Morocco

    Said El Hajji

  2. University of Caen, Caen, France

    Abderrahmane Nitaj

  3. LAGA,Universities of Paris 8 and Paris 13, Saint-Denis Cedex 02, France

    Claude Carlet

  4. University of Mohammed V, Rabat, Morocco

    El Mamoun Souidi

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Carlet, C. (2015). On the Properties of Vectorial Functions with Plateaued Components and Their Consequences on APN Functions. In: El Hajji, S., Nitaj, A., Carlet, C., Souidi, E. (eds) Codes, Cryptology, and Information Security. C2SI 2015. Lecture Notes in Computer Science(), vol 9084. Springer, Cham. https://doi.org/10.1007/978-3-319-18681-8_5

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-18681-8_5

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-18680-1

  • Online ISBN: 978-3-319-18681-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation