Characterization of robust immune symmetric boolean functions

Abstract

Fix a field �� $\mathbb {F}$ . The algebraic immunity over �� $\mathbb {F}$ of boolean function f : {0, 1} n � {0, 1} is defined as the minimal degree of a nontrivial (multilinear) polynomial g ( x ) � �� [ x 1 , � , x n ] $g(x) \in \mathbb {F}[x_{1}, \ldots , x_{n}]$ such that f(x) is a constant (either 0 or 1) for all x � {0, 1} n satisfying g(x) = 0. Function f is called k r o b u s t i m m u n e if the algebraic immunity of f is always not less than k no matter how one changes the value of f(x) for k ≤ |x| ≤ n � k. For any field �� $\mathbb {F}$ , any integers n, k � 0, we characterize all k robust immune symmetric boolean functions in n variables. The proof is based on a known symmetrization technique and constructing a partition of nonnegative integers satisfying certain (in)equalities about p-adic distance, where p is the characteristic of the field �� $\mathbb {F}$ .