Recent Results on Balanced Symmetric Boolean Functions
Recent Results on Balanced Symmetric Boolean Functions
复制标题
DOI：
10.1109/tit.2015.2455052
复制
发表时间：
2016-09
影响因子：
2.5
通讯作者：
Guang-pu Gao;Ying-ming Guo;Ya-qun Zhao
中科院分区：
文献类型：
--
作者：
Guang-pu Gao;Ying-ming Guo;Ya-qun Zhao
本文主要研究对称布尔函数的平衡性。证明了Canteaut和Videau提出的一个猜想，即当变量个数足够大时，代数次数固定的平衡对称布尔函数是平凡平衡的.记为σn，d，n元d次初等对称布尔函数。作为这一结果在初等对称布尔函数上的应用，我们证明了所有平凡平衡的初等对称布尔函数都具有σ 2 t +1 l-1，2 t的形式，其中t和l是任意正整数.这意味着Cusick et al.的猜想，即σ 2 t +1 l-1，2 t是唯一的非线性平衡初等对称布尔函数，等价于所有平衡初等对称布尔函数都是平凡平衡的猜想。
This paper focuses on the balancedness of symmetric Boolean functions. We prove a conjecture presented by Canteaut and Videau, which states that the balanced symmetric Boolean functions of fixed algebraic degree are trivially balanced when the number of variables is large enough. Denoted by σn,d, the n-variable elementary symmetric Boolean function of degree d. As an application of this result to elementary symmetric Boolean functions, we show that all the trivially balanced elementary symmetric Boolean functions are of the form σ2t+1l-1,2t, where t and l are any positive integers. It implies that Cusick et al.'s conjecture, which claims that σ2t+1l-1,2t is the only nonlinear balanced elementary symmetric Boolean functions, is equivalent to the conjecture that all the balanced elementary symmetric Boolean functions are trivially balanced.