Abstract
In the current paper we consider the following properties of filters: perfect balancedness of a filter function (i.e. preserving pure randomness of the input sequence) and linearity of a filter function in the first or the last essential variable. Previous results on this subject are discussed, including misleading statements in Gouget and Sibert (LNCS, vol. 4876, 2007) about the connection between perfect balancedness and resistance to Anderson conditional correlation attack; the incorrectness of two known results, the sufficient condition of perfect balancedness in Golić (LNCS, vol. 1039, 1996) and the necessary condition of perfect balancedness in Dichtl (LNCS, vol. 1267, 1997), is demonstrated by providing counterexamples.
We present a novel method of constructing large classes of perfectly balanced functions that are nonlinear in the first and the last essential variable and obtain a new lower bound of the number of such functions.
Golić conjecture (LNCS, vol. 1039, 1996) states that the necessary and sufficient condition for a function to be perfectly balanced for any choice of a tapping sequence is linearity of a function in the first or the last essential variable. In the second part of the current paper we prove the Golić conjecture.
Article PDF
Similar content being viewed by others
References
R.J. Anderson, Searching for the optimum correlation attack, in Fast Software Encryption, ed. by B. Preneel. LNCS, vol. 1008 (Springer, Heidelberg, 1995), pp. 137–143
M. Dichtl, On nonlinear filter generators, in FSE 1997, ed. by E. Biham. LNCS, vol. 1267 (Springer, Heidelberg, 1997), pp. 103–106
J.Dj. Golić, On the security of nonlinear filter generators, in Proceedings of Fast Software Encryption 1996, ed. by D. Gollmann. LNCS, vol. 1039 (Springer, Heidelberg, 1996), pp. 173–188
J.Dj. Golić, Conditional correlation attack on combiners with memory. Electron. Lett. 32(24), 2193–2195 (1996)
J.Dj. Golić, A. Clark, E. Dawson, Generalized inversion attack on nonlinear filter generators. IEEE Trans. Comput. C-49, 1100–1109 (2000)
A. Gouget, H. Sibert, Revisiting correlation-immunity in filter generators, in Proceedings of SAC 2007, ed. by C. Adams, A. Miri, M. Wiener. LNCS, vol. 4876 (Springer, Heidelberg, 2007), pp. 378–395
O.A. Logachev, On perfectly balanced Boolean functions. Cryptology ePrint Archive, Report 2007/022. http://eprint.iacr.org/
O.A. Logachev, A.A. Salnikov, S.V. Smyshlyaev, V.V. Yashchenko, Perfectly balanced functions in symbolic dynamics, in Proceedings of the NATO Advanced Research Workshop on Enhancing Cryptographic Primitives with Techniques from Error Correcting Codes, Veliko Tarnovo, Bulgaria, 6–9 October 2008 (IOS Press, Amsterdam, 2009), pp. 222–233
O.A. Logachev, S.V. Smyshlyaev, V.V. Yashchenko, New methods of investigation of perfectly balanced Boolean functions. Discrete Math. Appl. 19(3), 237–262 (2009)
S.V. Smyshlyaev, Barriers of perfectly balanced Boolean functions. Discrete Math. Appl. 20(3), 321–336 (2010)
S.N. Sumarokov, Functions of defect zero and invertibility of one class of finite-memory encoders. Obozr. Prom. Prikl. Mat. 1(1), 33–55 (1994) (in Russian)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Willi Meier
Rights and permissions
About this article
Cite this article
Smyshlyaev, S.V. Perfectly Balanced Boolean Functions and Golić Conjecture. J Cryptol 25, 464–483 (2012). https://doi.org/10.1007/s00145-011-9100-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00145-011-9100-7