奇数变元plateaued函数代数免疫性质研究

Plateaued函数是包含Bent函数和部分Bent函数的更大函数类,具有许多优良的密码学性质。基于布尔函数非线性度与代数免疫阶之间的关系,利用Walsh谱等工具,讨论奇数变元的plateaued函数的代数免疫性质,得到其存在低次零化子的一个充分条件,并进一步刻画变元个数n与plateaued函数的阶r之间的具体关系,利用此关系可确定函数代数免疫阶的上界。

On algebraic immunity of Plateaued functions in odd variables

The plateaued function is a larger class of Boolean functions which include all partially bent functions and bent functions as a proper subset. The plateaued function has many cryptographically desirable properties. The algebraic immunity（a new criteria for evaluating the property of the Boolean function）of the plateaued function in odd variables is studied in this paper. Based on the relationship between the nonlinearity and algebraic immunity of the Boolean functions and by means of Walsh spectrum and other tools, a sufficient condition is given on which the plateaued function can not posses the optimized algebraic immunity, that is, there exists some annihilators of low degree. Moreover, an inequality is given to describe the relationship between n（the number of the variables）and the function’s degree r. According to this inequality, the upper bound of the function’s algebraic immunity can be determined.