A Note on a Combinatorial Conjecture
Author(s)
Guixin Deng
Affiliation(s)
School of Mathematical Science, Guangxi Teachers Education University, Nanning, China.
School of Mathematical Science, Guangxi Teachers Education University, Nanning, China.
ABSTRACT
It is difficult to find Boolean functions achieving many good cryptographic properties. Recently, Tu and Deng obtained two classes of Boolean functions with good properties based on a combinatorial conjecture about binary strings. In this paper, using different approaches, we prove this conjecture is true in some cases. This conjecture has resisted different attempts of proof since it is hard to find a recursive method. In this paper we give a recursive formula in a special case.
It is difficult to find Boolean functions achieving many good cryptographic properties. Recently, Tu and Deng obtained two classes of Boolean functions with good properties based on a combinatorial conjecture about binary strings. In this paper, using different approaches, we prove this conjecture is true in some cases. This conjecture has resisted different attempts of proof since it is hard to find a recursive method. In this paper we give a recursive formula in a special case.
Cite this paper
G. Deng, "A Note on a Combinatorial Conjecture," Open Journal of Discrete Mathematics, Vol. 3 No. 1, 2013, pp. 49-52. doi: 10.4236/ojdm.2013.31010.
G. Deng, "A Note on a Combinatorial Conjecture," Open Journal of Discrete Mathematics, Vol. 3 No. 1, 2013, pp. 49-52. doi: 10.4236/ojdm.2013.31010.
References
[1] Z. Tu and Y. Deng, “A Conjecture on Binary String and Its Application on Constructing Boolean Functions of Optimal Algebraic Immunity,” Designs, Codes and Cryptography, Vol. 60, No. 1, 2010, pp. 1-14. doi:10.1007/s10623-010-9413-9 [2] G. Cohen and J.-P. Flori, “On a Generalized Combinatorial Conjecture Involving Addition Mod ,” IACR Cryptology ePrint Archive, Vol. 400, 2011, in press. [3] T. W. Cusick, Y. Li and P. Stanica, “On a Combinatorial Conjecture,” Integers, Vol. 11, No. 2, 2011, pp. 185-203. doi:10.1515/integ.2011.017 [4] G. Deng and P. Yuan, “On a Combinatorial Conjecture of Tu and Deng,” Integers, Vol. 12, No. A48, 2012. [5] J.-P. Flori and H. Randriam, “On the Number of Carries Occuring in an Addition Mod ,” Integers, Vol. 12, No. A10, 2012.
[1] Z. Tu and Y. Deng, “A Conjecture on Binary String and Its Application on Constructing Boolean Functions of Optimal Algebraic Immunity,” Designs, Codes and Cryptography, Vol. 60, No. 1, 2010, pp. 1-14. doi:10.1007/s10623-010-9413-9 [2] G. Cohen and J.-P. Flori, “On a Generalized Combinatorial Conjecture Involving Addition Mod ,” IACR Cryptology ePrint Archive, Vol. 400, 2011, in press. [3] T. W. Cusick, Y. Li and P. Stanica, “On a Combinatorial Conjecture,” Integers, Vol. 11, No. 2, 2011, pp. 185-203. doi:10.1515/integ.2011.017 [4] G. Deng and P. Yuan, “On a Combinatorial Conjecture of Tu and Deng,” Integers, Vol. 12, No. A48, 2012. [5] J.-P. Flori and H. Randriam, “On the Number of Carries Occuring in an Addition Mod ,” Integers, Vol. 12, No. A10, 2012.