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Weights of Boolean cubic monomial rotation symmetric functions

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Abstract

This paper studies degree 3 Boolean functions in n variables x 1, ..., x n which are rotation symmetric, that is, invariant under any cyclic shift of the indices of the variables. These rotation symmetric functions have been extensively studied in the last dozen years or so because of their importance in cryptography. Some of the cryptographic applications are described in a 2002 paper of Cusick and Stănică, which gave a recursion for the truth table and a nonhomogeneous recursion for the (Hamming) weight of the homogeneous cubic rotation symmetric function generated by the monomial x 1 x 2 x 3. Until now, this was the only investigation of the recursive structure of such functions. Here we provide an algorithm for finding a recursion for the truth table of any cubic rotation symmetric Boolean function generated by a monomial, as well as a homogeneous recursion for its weight as n increases; in doing so we greatly reduce the computational complexity of a problem that appeared to be exponential in the number of variables, as well as provide a new way of studying the structure of the functions. The method makes some computations practically accessible that were previously entirely unfeasible. Once the weights have been computed for the initial small values of n, the further weights can be computed from the recursion, without looking at the truth table at all.

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References

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

    Article  MathSciNet  Google Scholar 

  2. Cusick, T.W., Stănică, P.: Fast evaluation, weights and nonlinearity of rotation symmetric functions. Discrete Math. 258, 289–301 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Hell, M., Maximov, A., Maitra, S.: On efficient implementation of search strategy for rotation symmetric Boolean functions. In: 9th International Workshop on Algebraic and Combinatorial Coding Theory (ACCT), Black Sea Coast, Bulgaria (2004)

  4. Kavut, S., Maitra, S., Yücel, M.D.: Enumeration of 9-variable rotation symmetric Boolean functions having nonlinearity >240. In: Adv. in Cryptology – Indocrypt 2006. LNCS 4329, pp. 266–279. Springer, Berlin (2006)

    Chapter  Google Scholar 

  5. 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  Google Scholar 

  6. Kavut, S., Yücel, M.D.: 9-variable Boolean functions with nonlinearity 242 in the generalized rotation symmetric class. Inf. Comput. 208, 341–350 (2010)

    Article  MATH  Google Scholar 

  7. Kim, H., Park, S.-M., Hahn, S.G.: On the weight and nonlinearity of homogeneous rotation symmetric Boolean functions of degree 2. Discrete Appl. Math. 157, 428–432 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Maitra, S., Sarkar, S., Dalai, D.K.: On Dihedral group invariant Boolean functions. In: Boolean Functions and Cryptographic Applications (BFCA 2007), pp. 63–76. University of Rouen, France (2007)

    Google Scholar 

  9. Maximov, A.: Classes of plateaued rotation symmetric Boolean functions under transformation of Walsh spectra. In: Workshop on Coding and Cryptography WCC 2005. LNCS 3969, pp. 325–334. Springer, Berlin (2006)

    Google Scholar 

  10. Maximov, A., Hell, M., Maitra, S.: Plateaued rotation symmetric Boolean functions on odd number of variables. In: Boolean Functions and Cryptographic Applications (BFCA 2005), pp. 83–104. University of Rouen, France (2005)

    Google Scholar 

  11. Pieprzyk, J., Qu, C.X.: Fast hashing and rotation-symmetric functions. J. Univers. Comput. Sci. 5(1), 20–31 (1999)

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. Zhang, X., Guo, H., Feng, R., Li, Y.: Proof of a conjecture about rotation symmetric functions. Discrete Math. 311, 1281–1289 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. University at Buffalo, Buffalo, NY, USA

    Maxwell L. Bileschi, Thomas W. Cusick & Daniel Padgett

Authors
  1. Maxwell L. Bileschi
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  2. Thomas W. Cusick
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  3. Daniel Padgett
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Corresponding author

Correspondence to Thomas W. Cusick.

Additional information

Preparation of this paper was supported by NSF CSUMS grant 0802994 to University at Buffalo, where corresponding author Cusick was the faculty mentor for undergrad student authors Bileschi and Padgett.

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Bileschi, M.L., Cusick, T.W. & Padgett, D. Weights of Boolean cubic monomial rotation symmetric functions. Cryptogr. Commun. 4, 105–130 (2012). https://doi.org/10.1007/s12095-011-0060-4

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  • DOI: https://doi.org/10.1007/s12095-011-0060-4

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