Skip to main content

Construction of even-variable rotation symmetric Boolean functions with maximum algebraic immunity

Science China Information Sciences volume 56, pages 1–9 (2013)Cite this article

Abstract

Rotation symmetric Boolean functions (RSBFs) have been used as components of different cryptosystems. In this paper, we investigate n-variable (n even and n ⩾ 12) RSBFs to achieve maximum algebraic immunity (AI), and provide a construction of RSBFs with maximum AI and nonlinearity. These functions have higher nonlinearity than the previously known nonlinearity of RSBFs with maximum AI. We also prove that our construction provides high algebraic degree in some case.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

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

References

  1. Stanica P, Maitra S. Rotation symmetric Boolean functions—count and cryptographic properties. Electron Notes Discrete Math, 2003, 15: 139–145

    MathSciNet  Article  Google Scholar 

  2. Dalai D K, Maitra S, Sarkar S. Results on rotation symmetric bent functions. In: Proceedings of the 2nd International Workshop on Boolean Functions: Cryptography and Applications, Rouen, France, 2006. 137–156

  3. Maximov A, Hell M, Maitra S. Plateaued rotation symmetric Boolean functions on odd number of variables. In: Proceedings of the 1st Workshop on Boolean Functions: Cryptography and Applications, Rouen, France, 2005. 83–104

  4. Stanica P, Maitra S, Clark J. Results on rotation symmetric bent and correlation immune Boolean functions. In: Proceedings of Fast Software Encryption Workshop, Delhi, India, 2004. 161–177

  5. Pieprzyk J, Qu C X. Fast hashing and rotation-symmetric functions. J Univ Comput Sci, 1999, 5: 20–31

    MathSciNet  Google Scholar 

  6. Cusick T W, Stanica P. Fast evaluation, weights and nonlinearity of rotation-symmetric functions. Discrete Math, 2002, 258: 289–301

    MathSciNet  MATH  Article  Google Scholar 

  7. Courtois N, Pieprzyk J. Cryptanalysis of block ciphers with overdefined systems of equations. In: Advances in Cryptology-ASIACRYPT, Queenstown, New Zealand, 2002. 267–287

  8. Courtois N, Meier W. Algebraic attacks on stream ciphers with linear feedback. In: Advances in Cryptology-EUROCRYPT, Warsaw, Poland, 2003. 345–359

  9. Dalai D K, Gupta K C, Maitra S. Results on algebraic immunity for cryptographically significant Boolean functions. In: Proceedings of the 5th International Conference on Cryptology, Chennai, India, 2004. 92–106

  10. Meier W, Pasalic E, Carlet C. Algebraic attacks and decomposition of Boolean functions. In: Advances in Cryptology-EUROCRYPT, Santa Barbara, USA, 2004. 474–491

  11. Carlet C, Zeng X Y, Li C L, et al. Further properties of several classes of Boolean functions with optimum algebraic immunity. Des Codes Cryptogr, 2009, 52: 303–338

    MathSciNet  MATH  Article  Google Scholar 

  12. Carlet C, Dalai D K, Gupta K C, et al. Algebraic immunity for cryptographically significant Boolean functions: Analysis and construction. IEEE Trans Inf Theory, 2006, 52: 3105–3121

    MathSciNet  MATH  Article  Google Scholar 

  13. Dalai D K, Maitra S, Sarkar S. Basic theory in construction of Boolean functions with maximum possible annihilator immunity. Des Codes Cryptogr, 2006, 40: 41–58

    MathSciNet  MATH  Article  Google Scholar 

  14. Qu L J, Li C, Feng K Q. A note on symmetric Boolean functions with maximum algebraic immunity in odd number of variables. IEEE Trans Inf Theory, 2007, 53: 2908–2910

    MathSciNet  Article  Google Scholar 

  15. Qu L J, Feng K Q, Liu F, et al. Constructing symmetric Boolean functions with maximum algebraic immunity. IEEE Trans Inf Theory, 2009, 55: 2406–2412

    MathSciNet  Article  Google Scholar 

  16. Sarkar S, Maitra S. Construction of rotation symmetric Boolean functions with maximum algebraic immunity on odd number of variables. In: Proceedings of International Symposium on Applied Algebra, Algebraic Algorithms and Error-Correcting codes, Bangalore, India, 2007. 271–280

  17. Fu S J, Li C, Matsuura K, et al. Construction of rotation symmetric Boolean functions with maximum algebraic immunity. In: Proceedings of the 8th International Conference on Cryptology and Network Security, Kanazawa, Ishikawa, Japan, 2009. 402–412

  18. Sarkar S, Maitra S. Construction of rotation symmetric Boolean functions with optimal algebraic immunity. Comput Syst, 2009, 12: 267–284

    Google Scholar 

  19. Carlet C. A method of construction of balanced functions with optimum algebraic immunity. In: Proceedings of the International Workshop on Coding and Cryptography, The Wuyi Mountain, Fujiang, China, 2007

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics and System Science, National University of Defense Technology, Changsha, 410073, China

    ShaoJing Fu, Chao Li & LongJiang Qu

  2. Institute of Industrial Science, University of Tokyo, Tokyo, 153-8505, Japan

    ShaoJing Fu & Kanta Matsuura

  3. State Key Laboratory of Information Security, Graduate University of Chinese Academy of Sciences, Beijing, 100190, China

    Chao Li

Authors
  1. ShaoJing Fu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Chao Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Kanta Matsuura
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. LongJiang Qu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to ShaoJing Fu.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Fu, S., Li, C., Matsuura, K. et al. Construction of even-variable rotation symmetric Boolean functions with maximum algebraic immunity. Sci. China Inf. Sci. 56, 1–9 (2013). https://doi.org/10.1007/s11432-011-4350-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11432-011-4350-4

Keywords

  • Boolean function
  • rotation symmetry
  • algebraic immunity
  • nonlinearity