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On the number of the rational zeros of linearized polynomials and the second-order nonlinearity of cubic Boolean functions

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Abstract

Determine the number of the rational zeros of any given linearized polynomial is one of the vital problems in finite field theory, with applications in modern symmetric cryptosystems. But, the known general theory for this task is much far from giving the exact number when applied to a specific linearized polynomial. The first contribution of this paper is a better general method to get a more precise upper bound on the number of rational zeros of any given linearized polynomial over arbitrary finite field. We anticipate this method would be applied as a useful tool in many research branches of finite field and cryptography. Really we apply this result to get tighter estimations of the lower bounds on the second-order nonlinearities of general cubic Boolean functions, which has been an active research problem during the past decade. Furthermore, this paper shows that by studying the distribution of radicals of derivatives of a given Boolean function one can get a better lower bound of the second-order nonlinearity, through an example of the monomial Boolean functions \(g_{\mu }=Tr(\mu x^{2^{2r}+2^{r}+1})\) defined over the finite field \({\mathbb F}_{2^{n}}\).

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References

  1. Berlekamp, E.R., Welch, L.R.: Weight distributions of the cosets of the (32; 6) Reed-Muller code. IEEE Trans. Inf. Theory 18(1), 203–207 (1972)

    Article  MathSciNet  Google Scholar 

  2. Bracken, C., Byrne, E., Markin, N., McGuire, G.: Determining the nonlinearity of a new family of APN functions. AAECC 2007. LNCS 4851, 72–79 (2007)

    MATH  Google Scholar 

  3. Bracken, C., Leander, G.: A highly nonlinear differentially 4 uniform power mapping that permutes fields of even degree. Finite Fields Applic. 16, 231–242 (2010)

    Article  MathSciNet  Google Scholar 

  4. Canteaut, A., Charpin, P., Kyureghyan, G.M.: A new class of monomial bent functions. Finite Fields Applic. 14, 221–241 (2008)

    Article  MathSciNet  Google Scholar 

  5. Carlet, C.: Boolean Functions for Cryptography and Error Correcting Codes. ChapterinBoolean Models and Methods in Mathematics, Computer Science, and Engineering. In: Crama, Y., Hammer, P.L. (eds.) , pp 257–397. Cambridge University Press (2010)

  6. Carlet, C.: On the higher order nonlinearities of algebraic immune functions. CRYPTO 2006. LNCS 4117, 584–601 (2006)

    MATH  Google Scholar 

  7. Carlet, C.: On the nonlinearity profile of the Dillon function. http://eprint.iacr.org/2009/577.pdf (2009)

  8. Carlet, C.: Recursive lower bounds on the nonlinearity profile of Boolean functions and their applications. IEEE Trans. Inf. Theory 54(3), 1262–1272 (2008)

    Article  MathSciNet  Google Scholar 

  9. Carlet, C.: On the nonlinearity of monotone Boolean functions. Cryptogr. Commun. 10(6), 1051–1061 (2018)

    Article  MathSciNet  Google Scholar 

  10. Carlet, C., Mesnager, S.: Improving the upper bounds on the covering radii of binary Reed-Muller codes. IEEE Trans. Inf. Theory 53(1), 162–173 (2007)

    Article  MathSciNet  Google Scholar 

  11. Carlet, C.: On the higher order nonlinearities of algebraic immune Boolean functions, CRYPTO 2006, ser. Lect. Notes Comput. Sci. 4117(2006), 584–601 (2006)

    Article  Google Scholar 

  12. Carlet, C., Dalai, D.K., Gupta, K.C., Maitra, S.: Algebraic immunity for cryptographically significant Boolean functions: Analysis and construction. IEEE Trans. Inf. Theory 52(7), 3105–3121 (2006)

    Article  MathSciNet  Google Scholar 

  13. Dobbertin, H.: One-to-one highly nonlinear power functions on GF(2n). Applic. Algebra Eng. Commun. Comput. 9(2), 139–152 (1998)

    Article  Google Scholar 

  14. Fu, S., Feng, X., Wu, B.: Differentially 4-uniform permutations with the best known nonlinearity from butterflies. http://eprint.iacr.org/2017/449.pdf (2017)

  15. Gangopadhyay, S., Garg, M.: The good lower bound of second-order nonlinearity of a class of Boolean function. http://eprint.iacr.org/2011/452.pdf (2011)

  16. Gangopadhyay, S., Sarkar, S., Telang, R.: On the lower bounds of the second order nonlinearity of some Boolean functions. Inform. Sci. 180(2), 266–273 (2010)

    Article  MathSciNet  Google Scholar 

  17. Garg, M., Gangopadhyay, S.: Good second-order nonlinearity of a bent function via Niho power function. http://eprint.iacr.org/2011/171.pdf (2011)

  18. Gode, R., Gangopadhyay, S.: On second-order nonlinearities of cubic monomial Boolean functions. http://eprint.iacr.org/2009/502.pdf (2009)

  19. Gow, R., Quinlan, R.: Galois extensions and subspaces of alternating bilinear forms with special rank properties. Linear Algebra Appl.s 430(8), 2212–2224 (2009)

    Article  MathSciNet  Google Scholar 

  20. Hou, X.: GL(m, 2) acting on R(r, m)/R(r − 1, m). Discret. Math. 149, 99–122 (1996)

    Article  MathSciNet  Google Scholar 

  21. Iwata, T., Kurosawa, K.: Probabilistic higher order differential attack and higher order bent functions. ASIACRYPT 1999, pp. 62–74. Springer. LNCS 1716 (1999)

  22. Kolokotronis, N., Limniotis, K.: Maiorana-McFarland functions with high second-order nonlinearity. http://eprint.iacr.org/2011/212.pdf (2011)

  23. Li, X., Hu, Y., Gao, J.: The lower bounds on the second-order nonlinearity of cubic Boolean functions. Lower bounds on the second order nonlinearity of Boolean functions. Int. J. Found. Comput. Sci. 22(6), 1331–1349 (2011). https://eprint.iacr.org/2010/009.pdf

    Article  Google Scholar 

  24. Lobanov, M.: Exact relation between nonlinearity and algebraic immunity. Discret. Math. Appl. 16(5), 453–460 (2006)

    Article  MathSciNet  Google Scholar 

  25. McEliece, R.J.: Finite Fields for Computer Scientists and Engineers. Kluwer Academic Publishers (1987)

  26. Mesnager, S.: Improving the lower bound on the higher order nonlinearity of Boolean functions with prescribed algebraic immunity. IEEE Trans. Inf. Theory 54(8), 3656–3662 (2008)

    Article  MathSciNet  Google Scholar 

  27. Pless, V.S., Huffman, W.C.: Handbook of Coding Theory. Elsevier, Amsterdam (1998)

    MATH  Google Scholar 

  28. Schatz, J.: The second-order Reed-Muller code of length 64 has covering radius 18. IEEE Trans. Inf. Theory 27, 529–530 (1981)

    Article  MathSciNet  Google Scholar 

  29. Singh, D.: Second-order nonlinearities of some classes of cubic Boolean functions based on secondary constructions. Int. J. Comput. Sci. Inf. Secur. 2(2), 786–791 (2011)

    MathSciNet  Google Scholar 

  30. Schmidt, K.U.: Nonlinearity measures of random Boolean functions. Cryptogr. Commun. 8(4), 637–645 (2016)

    Article  MathSciNet  Google Scholar 

  31. Sun, G., Wu, C.: The lower bounds on the second-order nonlinearity of three classes of Boolean functions with high nonlinearity. Inform. Sci. 179(3), 267–278 (2010)

    Article  MathSciNet  Google Scholar 

  32. Sun, G., Wu, C.: The lower bound on the second-order nonlinearity of a class of Boolean functions with high nonlinearity. Applic. Algebra Eng. Commun. Comput. 22, 37–45 (2011)

    Article  MathSciNet  Google Scholar 

  33. Wang, Q., Johansson, T.: A note on fast algebraic attacks and higher order nonlinearities, INSCRYPT 2010. Lect. Notes Comput. Sci 6584, 84–98 (2010)

    Article  Google Scholar 

Download references

Acknowledgements

The authors thank the Assoc. Edit. and the anonymous reviewers for their valuable comments which have highly improved the manuscript.

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Authors and Affiliations

  1. LAGA, Department of Mathematics, University of Paris VIII, 93526, Saint-Denis, France

    Sihem Mesnager

  2. University of Paris XIII, CNRS, LAGA UMR 7539, Sorbonne Paris Cite, 93430, Villetaneuse, France

    Sihem Mesnager

  3. Telecom ParisTech, 75013, Paris, France

    Sihem Mesnager

  4. Institute of Mathematics, State Academy of Sciences, Pyongyang, Democratic People’s Republic of Korea

    Kwang Ho Kim

  5. PGItech Corp., Pyongyang, Democratic People’s Republic of Korea

    Kwang Ho Kim

  6. KumSong School, Pyongyang, Democratic People’s Republic of Korea

    Myong Song Jo

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  1. Sihem Mesnager
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  2. Kwang Ho Kim
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  3. Myong Song Jo
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Correspondence to Sihem Mesnager.

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Mesnager, S., Kim, K.H. & Jo, M.S. On the number of the rational zeros of linearized polynomials and the second-order nonlinearity of cubic Boolean functions. Cryptogr. Commun. 12, 659–674 (2020). https://doi.org/10.1007/s12095-019-00410-5

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