Abstract: By analyzing character of Boolean function, the connection between [n]-variables self-dual Boolean function and [n-1]-variables Boolean function is set up, and according to this connection the algebraic immunity and nonlinearity of [n]-variables self-dual Boolean function are discussed. It is also obtained that terms of monomials with algebraic degree nonzero are odd. The number, algebraic norm formation and cryptographical property of [n]-variables self-dual function with algebraic degree [n-1] are presented. A conjecture about terms of monomials with algebraic degree [t] is proposed, and two special cases are proved out.

Key words: self-dual Boolean function, linear structure, algebraic immunity, algebraic degree, terms of monomials

摘要： 通过分析布尔函数的特征，建立了[n]元自对偶布尔函数和[n-1]元布尔函数之间的关系，根据此关系讨论了[n]元自对偶布尔函数的代数免疫度及其非线性度，得出自对偶布尔函数的非零次单项式个数为奇数，给出了[n]元[n-1]次自对偶布尔函数的个数和代数正规型表示的特征及其密码学性质，对其代数次数为[t]的单项式个数提出了猜想，对其中两种特殊情况进行了证明。

关键词: 自对偶布尔函数, 线性结构, 代数免疫度, 代数次数, 单项式个数