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. We start from the 2012 paper of Bileschi, Cusick and Padgett, which gave an algorithm for finding a recursion for the truth table of any n-variable cubic rotation symmetric Boolean function generated by a monomial, as well as a homogeneous recursion for its (Hamming) weight as n increases. This greatly reduced the computational complexity of computing the weights of such functions for large n, but it was still necessary to calculate the truth tables of the functions for the values of n needed to give the initial conditions for the recursion. This computation could be infeasible if the recursion order is large, since the truth tables have 2n entries. The present paper shows how to use the roots of the characteristic polynomial of the recursion to find the initial conditions without looking at any truth tables, given the mild and plausible assumption that these roots are distinct. This results in a huge decrease in the computational complexity (including the time needed to find the roots) to something linear in n, apart from logarithmic factors.
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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)
Brown, A., Cusick, T.W.: Recursive weights for some Boolean functions. J. Math. Cryptol. (2012). doi:10.1515/jmc-2011-0020
Carlet, C.: Boolean functions for cryptography and error-correcting codes. In: Crama, Y., Hammer, P.L. (eds.) Boolean Models and Methods in Mathematics, Computer Science, and Engineering, pp. 257–397. Cambridge University Press, Cambridge (2010)
Cusick, T.W.: Affine equivalence of cubic homogeneous rotation symmetric Boolean functions. Inf. Sci. 181, 5067–5083 (2011)
Cusick, T.W., Brown, A.: Affine equivalence for rotation symmetric Boolean functions with p k variables. Finite Fields Appl. 18, 547–562 (2012)
Cusick, T.W., Cheon, Y.: Affine equivalence for for rotation symmetric Boolean functions with 2k variables. Des., Codes Cryptogr. 63, 273–294 (2012)
Cusick, T.W., Stănică, P.: Fast evaluation, weights and nonlinearity of rotation symmetric functions. Discrete Math. 258, 289–301 (2002)
Cusick, T.W., Stănică, P.: Cryptographic Boolean Functions and Applications. Academic Press, San Diego (2009)
Dumas, J.-G., et al.: LinBox founding scope allocation, parallel building blocks, and separate compilation. In: ICMS. LNCS, vol. 6237, pp. 77–83. Springer, Berlin (2010)
Dumas, J.-G., Pernet, C., Wan, Z.: Efficient computation of the characteristic polynomial. In: International Symposium on Symbolic and Algebraic Computation – ISSAC, 2005, pp. 140–147. Assoc. Computing Machinery Digital Library (2005)
Kavut, S., Maitra, S., Yücel, M.D.: Enumeration of 9-variable Rotation Symmetric Boolean Functions having Nonlinearity > 240. In: Advances in Cryptology – Indocrypt, 2006. LNCS, vol. 4329, pp. 266–279. Springer, Berlin (2006)
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)
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)
Maximov, A.: Classes of Plateaued Rotation Symmetric Boolean functions under Transformation of Walsh Spectra. In: Workshop on Coding and Cryptography WCC 2005. LNCS, vol. 3969, pp. 325–334. Springer, Berlin (2006)
Pan, V.Y.: Univariate polynomials: nearly optimal algorithms for numerical factorization and root-finding. J. Symb. Comput. 33, 701–733 (2002)
Pieprzyk, J., Qu, C.X.: Fast hashing and rotation-symmetric functions. J. Univers. Comput. Sci. 5(1), 20–31 (1999)
Stănică, P., Maitra, S.: Rotation symmetric Boolean functions - count and cryptographic properties. Discrete Appl. Math. 156, 1567–1580 (2008)
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Cusick, T.W. Finding Hamming weights without looking at truth tables. Cryptogr. Commun. 5, 7–18 (2013). https://doi.org/10.1007/s12095-012-0072-8
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DOI: https://doi.org/10.1007/s12095-012-0072-8