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Construction of resilient Boolean functions in odd variables with strictly almost optimal nonlinearity

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Abstract

Construction of resilient Boolean functions in odd variables having strictly almost optimal (SAO) nonlinearity is a challenging problem in coding theory and symmetric ciphers. In this paper, we propose a new method to obtain SAO resilient Boolean functions. By combining this method with High-Meets-Low construction technique, we can obtain resilient functions with better resiliency order and currently best known nonlinearity.

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References

  1. Dillon J.F.: Elementary Hadamard difference sets. PhD dissertation, Dept. Comput. Sci., Univ. Maryland, College Park, MD (1974).

  2. Ding C., Xiao G., Shan W.: The Stability Theory of Stream Ciphers, vol. 561. LNCSSpringer, New York (1991).

    Book  Google Scholar 

  3. Kavut S., Maitra S.: Patterson–Wiedemann type functions on 21 variables with nonlinearity greater than bent concatenation bound. IEEE Trans. Inf. Theory 62, 2277–2282 (2016).

    Article  MathSciNet  Google Scholar 

  4. Kavut S., Maitra S., Yücel M.D.: Search for Boolean functions with excellent profiles in the rotation symmetric class. IEEE Trans. Inf. Theory 53, 1743–1751 (2007).

    Article  MathSciNet  Google Scholar 

  5. Kavut S., Yücel M.D.: Generalized rotation symmetric and dihedral symmetric Boolean functions—9 variable Boolean functions with nonlinearity 242. In: Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, LNCS, vol. 4851, pp. 321–329. Springer, New York (2007).

  6. Meier W., Staffelbach O.: Nonlinearity criteria for cryptographic functions. In: Advances in Cryptology—EUROCRYPT’89. LNCS, vol. 434, pp. 549–562. Springer, New York (1990).

  7. Mykkelveit J.: The covering radius of the \((128,8)\) Reed–Muller code is \(56\). IEEE Trans. Inf. Theory 26, 359–362 (1980).

    Article  MathSciNet  Google Scholar 

  8. Pang S., Wang X., Wang J., Du J., Feng M.: Construction and count of 1-resilient rotation symmetric Boolean functions. Inf. Sci. 450, 36–342 (2018).

    Article  MathSciNet  Google Scholar 

  9. Patterson N.J., Wiedemann D.H.: The covering radius of the \((2^{15}, 16)\) Reed–Muller code is at least 16276. IEEE Trans. Inf. Theory 29, 354–356 (1983).

    Article  Google Scholar 

  10. Rothaus O.S.: On ‘bent’ functions. J. Comb. Theory Ser. A 20, 300–305 (1976).

    Article  Google Scholar 

  11. Siegenthaler T.: Correlation-immunity of nonlinear combining functions for cryptographic applications. IEEE Trans. Inf. Theory 30, 776–780 (1984).

    Article  MathSciNet  Google Scholar 

  12. Sarkar P., Maitra S.: Construction of nonlinear Boolean functions with important cryptographic properties. In: Advances in Cryptology—EUROCRYPT 2000. LNCS, vol. 1807, pp. 485–506. Springer, New York (2000).

    Chapter  Google Scholar 

  13. Sarkar S., Maitra S.: Idempotents in the neighbourhood of Patterson–Wiedemann functions having walsh spectra zeros. Des. Codes Cryptogr. 49, 95–103 (2008).

    Article  MathSciNet  Google Scholar 

  14. Wei Y., Pasalic E., Zhang F., Wu W., Wang C.: New constructions of resilient functions with strictly almost optimal nonlinearity via non-overlap spectra functions. Inf. Sci. 415, 377–396 (2017).

    Article  Google Scholar 

  15. Xiao G., Massey J.L.: A spectral characterization of correlation-immune combining functions. IEEE Trans. Inf. Theory 34, 569–571 (1988).

    Article  MathSciNet  Google Scholar 

  16. Zhang F., Carlet C., Hu Y., Cao T.: Secondary constructions of highly nonlinear Boolean functions and disjoint spectra plateaued functions. Inf. Sci. 283, 94–106 (2014).

    Article  MathSciNet  Google Scholar 

  17. Zhang F., Wei Y., Pasalic E., Xia S.: Large sets of disjoint spectra plateaued functions inequivalent to partially linear functions. IEEE Trans. Inf. Theory 64, 2987–2999 (2018).

    Article  MathSciNet  Google Scholar 

  18. Zhang W.-G.: High-Meets-Low: construction of strictly almost optimal resilient Boolean functions via fragmentary walsh spectra. IEEE Trans. Inf. Theory. https://doi.org/10.1109/TIT.2019.2899397

    Article  Google Scholar 

  19. Zhang W.-G., Pasalic E.: Improving the lower bound on the maximum nonlinearity of 1-resilient Boolean functions and designing functions satisfying all cryptographic criteria. Inf. Sci. 376, 21–30 (2017).

    Article  Google Scholar 

  20. Zhang W.-G., Pasalic E.: Generalized Maiorana–McFarland construction of resilient Boolean functions with high nonlinearity and good algebraic properties. IEEE Trans. Inf. Theory 60, 6681–6695 (2014).

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This study was funded by National Natural Science Foundation of China (Grant Nos. 61672414, U1604180) and National Cryptography Development Fund (Grant No. MMJJ20170113).

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Authors and Affiliations

  1. State Key Laboratory of Integrated Services Networks, Xidian University, Xi’an, 710071, China

    Yujuan Sun

  2. State Key Laboratory of Cryptology, P.O. Box 5159, Beijing, 100878, China

    Yujuan Sun

  3. School of Mathematics and Statistics, Henan University, Kaifeng, 475001, China

    Jiafang Zhang

  4. Indian Institute of Technology Roorkee, Roorkee, Uttarakhand, 247667, India

    Sugata Gangopadhyay

Authors
  1. Yujuan Sun
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  2. Jiafang Zhang
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  3. Sugata Gangopadhyay
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Correspondence to Yujuan Sun.

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Communicated by A. Pott.

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Sun, Y., Zhang, J. & Gangopadhyay, S. Construction of resilient Boolean functions in odd variables with strictly almost optimal nonlinearity. Des. Codes Cryptogr. 87, 3045–3062 (2019). https://doi.org/10.1007/s10623-019-00662-5

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  • DOI: https://doi.org/10.1007/s10623-019-00662-5

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