Abstract
It is a difficult challenge to find Boolean functions used in stream ciphers achieving all of the necessary criteria and the research of such functions has taken a significant delay with respect to cryptanalyses. Very recently, an infinite class of Boolean functions has been proposed by Tu and Deng having many good cryptographic properties under the assumption that the following combinatorial conjecture about binary strings is true:
Conjecture 0.1. Let S t,k be the following set:
Then:
The main contribution of the present paper is the reformulation of the problem in terms of carries which gives more insight on it than simple counting arguments. Successful applications of our tools include explicit formulas of \(\left|{S_{t,k}}\right|\) for numbers whose binary expansion is made of one block, a proof that the conjecture is asymptotically true and a proof that a family of numbers (whose binary expansion has a high number of 1s and isolated 0s) reaches the bound of the conjecture. We also conjecture that the numbers in that family are the only ones reaching the bound.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Courtois, N., Meier, W.: Algebraic attacks on stream ciphers with linear feedback. In: Biham, E. (ed.) EUROCRYPT 2003. LNCS, vol. 2656, pp. 345–359. Springer, Heidelberg (2003)
Courtois, N.: Fast algebraic attacks on stream ciphers with linear feedback. In: Boneh, D. (ed.) CRYPTO 2003. LNCS, vol. 2729, pp. 176–194. Springer, Heidelberg (2003)
Carlet, C., Feng, K.: An infinite class of balanced functions with optimal algebraic immunity, good immunity to fast algebraic attacks and good nonlinearity. In: Pieprzyk, J. (ed.) ASIACRYPT 2008. LNCS, vol. 5350, pp. 425–440. Springer, Heidelberg (2008)
Tu, Z., Deng, Y.: A conjecture on binary string and its applications on constructing boolean functions of optimal algebraic immunity. Cryptology ePrint Archive, Report 2009/272 (2009), http://eprint.iacr.org/
Dillon, J.: Elementary Hadamard Difference Sets. PhD thesis, University of Maryland (1974)
Carlet, C.: On a weakness of the Tu-Deng function and its repair. Cryptology ePrint Archive, Report 2009/606 (2009), http://eprint.iacr.org/
Flori, J.P., Randriambololona, H., Cohen, G., Mesnager, S.: On a conjecture about binary strings distribution. Cryptology ePrint Archive, Report 2010/170 (2010), http://eprint.iacr.org/
Monagan, M.B., Geddes, K.O., Heal, K.M., Labahn, G., Vorkoetter, S.M., McCarron, J., DeMarco, P.: Maple 10 Programming Guide. Maplesoft, Waterloo ON, Canada (2005)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Flori, JP., Randriam, H., Cohen, G., Mesnager, S. (2010). On a Conjecture about Binary Strings Distribution. In: Carlet, C., Pott, A. (eds) Sequences and Their Applications – SETA 2010. SETA 2010. Lecture Notes in Computer Science, vol 6338. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15874-2_30
Download citation
DOI: https://doi.org/10.1007/978-3-642-15874-2_30
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-15873-5
Online ISBN: 978-3-642-15874-2
eBook Packages: Computer ScienceComputer Science (R0)