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Counting permutation equivalent degree six binary polynomials invariant under the cyclic group

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Abstract

In this paper we find an exact formula for the number of affine equivalence classes under permutations for binary polynomials degree \(d=6\) invariant under the cyclic group (also, called monomial rotation symmetric), for a prime number of variables; this extends previous work for \(2\le d\le 5\).

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References

  1. Babai, L.: Isomorphism problem for a class of point-symmetric structures. Acta Math. Acad. Sci. Hung. 29, 329–336 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  2. Canright, D., Chung, J.H., Stănică, P.: Circulant matrices and affine equivalence of monomial rotation symmetric Boolean functions. Discrete Math. J. 338(12), 2197–2211 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  3. Carlitz, L.: Note on a quartic congruence. Am. Math. Mon. 63, 569–571 (1956)

    Article  MATH  MathSciNet  Google Scholar 

  4. Cusick, T.W.: Affine equivalence of cubic homogeneous rotation symmetric functions. Inform. Sci. 181(22), 5067–5083 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  5. Cusick, T.W., Brown, A.: Affine equivalence for rotation symmetric Boolean functions with \(p^k\) variables. Finite Fields Appl. 18(3), 547–562 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  6. Cusick, T.W., Cheon, Y.: Affine equivalence for rotation symmetric Boolean functions with \(2^{k}\) variables. Des. Codes Crypt. 63, 273–294 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  7. Cusick, T.W., Cheon, Y.: Affine equivalence of quartic homogeneous rotation symmetric Boolean functions. Inform. Sci. 259, 192–211 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  8. Cusick, T.W., Stănică, P.: Cryptographic Boolean Functions and Applications. Elsevier, Amsterdam (2009)

    MATH  Google Scholar 

  9. Cusick, T.W., Stănică, P.: Counting equivalence classes for monomial rotation symmetric boolean functions with prime dimension. Cryptogr. Commun. (Discrete Struct. Boolean Funct. Seq.) 1, 67–81 (2016)

    MATH  MathSciNet  Google Scholar 

  10. Driver, E., Leonard, P.A., Williams, K.S.: Irreducible quartic polynomials with factorizations modulo \(p\). Am. Math. Mon. 112(10), 876–890 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  11. Stănică, P.: Affine equivalence of quartic monomial rotation symmetric Boolean functions in prime power dimension. Inf. Sci. 314, 212–224 (2015)

    Article  MathSciNet  Google Scholar 

  12. Stănică, P., Maitra, S.: Rotation symmetric Boolean functions—count and cryptographic properties. Discrete Appl. Math. 156, 1567–1580 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  13. Wiedemann, D., Zieve, M.E.: Equivalence of sparse circulants: the bipartite Ádám problem. Manuscript. arXiv:0706.1567v1. www.math.lsa.umich.edu/~zieve/papers/circulants.html

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Acknowledgments

We thank the anonymous referees for their very detailed and helpful comments. Work on this paper was partially done during an enjoyable visit of the second named author to the School of Mathematics of the Wits University in September 2015. We thank the School for Hospitality and excellent working conditions.

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Authors and Affiliations

  1. School of Mathematics, University of the Witwatersrand, Private Bag X3, Wits, 2050, South Africa

    Florian Luca

  2. Applied Mathematics Department, Naval Postgraduate School, Monterey, CA, 93943-5216, USA

    Pantelimon Stănică

  3. Institute of Mathematics “Simion Stoilow” of the Romanian Academy, Bucharest, Romania

    Pantelimon Stănică

  4. Centro de Ciencias Matematicas UNAM, Morelia, Mexico

    Florian Luca

Authors
  1. Florian Luca
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  2. Pantelimon Stănică
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Correspondence to Pantelimon Stănică.

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Luca, F., Stănică, P. Counting permutation equivalent degree six binary polynomials invariant under the cyclic group. AAECC 28, 1–10 (2017). https://doi.org/10.1007/s00200-016-0294-7

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  • DOI: https://doi.org/10.1007/s00200-016-0294-7

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