Skip to main content
SpringerLink
Log in

The autocorrelation properties of single cycle polynomial T-functions

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

Abstract

T-functions have been widely used in the design of symmetric ciphers, hash functions, and fast cryptographic primitives. Single cycle polynomial T-functions are a special category. If they are used as state transition functions of stream ciphers, the security of the generated sequences is crucial. In 2008, Kolokotronis proposed a conjecture regarding the autocorrelation function’s values of coordinate sequences generated by single cycle polynomial T-functions. In this paper, we show that the conjecture does not hold in general and prove the conditions under which it holds.

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.

Similar content being viewed by others

References

  1. Anashin V., Bogdanov A., Kizhvatov I., Kumar S.: ABC: A new fast flexible stream cipher. ECRYPT Stream Cipher Project Report. https://www.cosic.esat.kuleuven.be/ecrypt/stream/ciphers/abc/abc.pdf. Accessed 30 Aug 2015.

  2. Anashin V., Khrennikov A.: Applied Algebraic Dynamics. Walter de Gruyter, Berlin (2009).

    Book  MATH  Google Scholar 

  3. Anashin V., Khrennikov A., Yurova E.: T-functions revisited: new criteria for bijectivity/transitivity. Des. Codes Cryptogr. 71, 383–407 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  4. Anashin V.: Non-archimedean ergodic theory and pseudorandom generators. Comput. J. 53, 370–392 (2010).

    Article  Google Scholar 

  5. Anashin V.: The non-archimedean theory of discrete systems. Math. Comput. Sci. 6, 375–393 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  6. Anashin V.: Uniformly distributed sequences of p-adic integers. Math. Notes 55, 109–133 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  7. Anashin V.: Uniformly distributed sequences in computer algebra or how to construct program generators of random numbers. J. Math. Sci. 89, 1355–1390 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  8. Anashin V.: Uniformly distributed sequences of p-adic integers II. Discret. Math. Appl. 12, 527–590 (2002).

    MathSciNet  MATH  Google Scholar 

  9. Klimov A.: Applications of T-functions in cryptography. Ph.D. Thesis, Department of Applied Mathe-matics and Computer Science, Weizmann Institute of Science (2005).

  10. Klimov A., Shamir A.: A new class of invertible mappings. In: Kaliski Jr. B.S., (ed.) Proceedings of Workshop on Cryptographic Hardware and Embedded Systems 2002. Lecture Notes in Computer Science, vol. 2523, pp. 470–483. Springer, Berlin (2003).

  11. Klimov A., Shamir A.: Cryptographic applications of T-functions. In: Matsui M., Zuccherato R. (eds.) Proceedings of Workshop on Selected Areas in Cryptography 2003. Lecture Notes in Computer Science, vol. 3006, pp. 248–261. Springer, Berlin (2004).

  12. Klimov A., Shamir A.: New applications of T-functions in block ciphers and hash functions. In: Gilbert H., Handschuh H. (eds.) Proceedings of Fast Software Encryption 2005. Lecture Notes in Computer Science, vol. 3557, pp. 18–31. Springer, Berlin (2005).

  13. Kolokotronis N.: Cryptographic properties of nonlinear pseudorandom number generators. Des. Codes Cryptogr. 46, 353–363 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  14. Larin M.V.: Transitive polynomial transformations of residue class rings. Discret. Math. Appl. 12, 127–140 (2002).

    Article  MATH  Google Scholar 

  15. Lausch H., Nöbauer W.: Algebra of Polynomials. North-Holland, Amsterdam (1973).

    MATH  Google Scholar 

  16. Lerner E.E.: Uniform distribution of sequences generated by iterated polynomials. Doklady Math. 3, 704–706 (2015).

    Article  MATH  Google Scholar 

  17. Mayhew G.L.: Auto-correlation properties of modified de Bruijn sequences. In: Proceedings of 2000 IEEE Position Location and Navigation Symposium, pp. 349–354. IEEE Press (2000).

  18. Molland H., Helleseth T.: A linear weakness in the Klimov-Shamir T-function. In: Proceedings of 2005 IEEE International Symposium on Information Theory, pp. 1106–1110 (2005).

  19. Roy D., Chaturvedi A., Mukhopadhyay S.: New Constructions of T-function. Information Security Practice and Experience, pp. 395–405. Springer, Heidelberg (2015).

    Book  Google Scholar 

  20. Shi T., Anashin V., Lin D.D.: Linear Weaknesses in T-functions. In: Helleseth T., Jedwab J. (eds.) Proceedings of Sequences and Their Applications 2012. Lecture Notes in Computer Science, vol. 7280, pp. 279–290. Springer, Berlin (2012).

  21. Wang J.S., Qi W.F.: Linear equation on polynomial single cycle T-functions. In: Pei D.Y., et al. (eds.) Proceedings of Information Security and Cryptology 2007. Lecture Notes in Computer Science, vol. 4990, pp. 256–270. Springer, Berlin (2008).

  22. Zhang, H.N., Wang, X.Y.: Differential cryptanalysis of T-function based stream cipher TSC-4. In: Nam K.H., Rhee G. (eds.), Proceedings of Information Security and Cryptology 2007. Lecture Notes in Computer Science, vol. 4817, pp. 227–238. Springer, Berlin (2007).

  23. Zhang W., Wu C.K.: The algebraic normal form, linear complexity and k-error linear complexity of single-cycle T-function. In: Gong G. et al. (eds.) SETA 2006. Lecture Notes on Computer Science, vol. 4086, pp. 391–401. Springer, Berlin (2006).

Download references

Acknowledgements

The authors would like to thank the anonymous reviewers for their detailed comments and suggestions. This work was supported by the National Natural Science Foundation of China [Grant Nos. 61272041, 61502532].

Author information

Authors and Affiliations

  1. Zhengzhou Information Science and Technology Institute, Zhengzhou, China

    SenPeng Wang, Bin Hu & Yan Liu

Authors
  1. SenPeng Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Bin Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yan Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to SenPeng Wang.

Additional information

Communicated by C. Carlet.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, S., Hu, B. & Liu, Y. The autocorrelation properties of single cycle polynomial T-functions. Des. Codes Cryptogr. 86, 1527–1540 (2018). https://doi.org/10.1007/s10623-017-0410-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-017-0410-0

Keywords

Mathematics Subject Classification

Search

Navigation