Characterization of robust immune symmetric boolean functions
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 \(g(x) \in \mathbb {F}[x_{1}, \ldots , x_{n}]\) 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 \(\mathbb {F}\).
characterization symmetric boolean functions algebraic immunity 06e30 94a60 94c10