Abstract: Any Boolean function can be uniquely expressed as a univariate polynomial function on finite field. By using this representation and algebraic coding theory to discuss the criterion, by which a Boolean function achieves its Maximum Algebraic Immunity (MAI), the authors provided an equivalent criterion by which the algebraic immunity of a Boolean function with odd variables reaches MAI. According to the criterion, an equivalent condition to Boolean function of three variables that achieves MAI was reached. Thus, all MAI Boolean functions of three variables got constructed.

Key words: Boolean function, algebraic immunity, algebraic attack, Reed-Soloman codes, algebraic code

摘要： 任意的布尔函数可以唯一地表示成有限域上的单变元多项式函数，利用布尔函数的单变元多项式表示和代数编码理论，讨论了布尔函数的代数免疫达到最优的判别条件，得到了布尔函数的变元个数为奇数时，布尔函数具有最优代数免疫(MAI)的等价判别条件。利用该等价判别条件，给出3元布尔函数满足MAI的等价判别条件，进而构造出所有3元的MAI布尔函数。

关键词: 布尔函数, 代数免疫, 代数攻击, Reed-Soloman码, 代数编码