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

  • Sihem MesnagerEmail author
  • Kwang Ho Kim
  • Myong Song Jo


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 \({\mathbb F}_{2^{n}}\).


Boolean functions Nonlinearity Linearized polynomial Root number 

Mathematics Subject Classification (2010)

Primary: 06E30 11T71 94A60 Secondary: 11T24 11T22 



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. 1.LAGA, Department of MathematicsUniversity of Paris VIIISaint-DenisFrance
  2. 2.University of Paris XIIIVilletaneuseFrance
  3. 3.Telecom ParisTechParisFrance
  4. 4.Institute of MathematicsState Academy of SciencesPyongyangDemocratic People’s Republic of Korea
  5. 5.PGItech Corp.PyongyangDemocratic People’s Republic of Korea
  6. 6.KumSong SchoolPyongyangDemocratic People’s Republic of Korea

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