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 \(g(x) \in \mathbb {F}[x_{1}, \ldots , x_{n}]\) 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 \(\mathbb {F}\).
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Notes
By nontrivial, we mean there exists some x ∈ {0, 1}n such that g(x) = 0.
f g = 0 should be understood semantically, i.e., for every x ∈ {0, 1}n, f(x)g(x) = 0; alternatively, f g = 0 could be understood as multiplication of polynomials over the quotient ring \(F[x_{1}, \ldots , x_{n}]/({x_{1}^{2}} = x_{1}, \ldots , {x_{n}^{2}} = x_{n})\).
The rows are indexed by subsets of [n] of size ≤ d, the columns are indexed by points x ∈ {0, 1}n such that f(x) = 1, and the entry (S, x) is exactly \({\prod }_{i \in S} x_{i}\).
The converse is not true, that is, there are 2k-variable symmetric boolean functions with maximum algebraic immunity k which are not k robust immune. However, they are “close” to some k robust immune functions.
The computation consists of some simple manipulations reducing to Vandermonde matrix.
Or equivalently, embed \(\mathbb {Z}_{\ge 0}\) into the ring of p-adic integers \(\mathbb {Z}_{p}\), which is a formal series \(x = {\sum }_{i \ge 0} x_{i} p^{i}\).
In the following inequality, we could have written x instead of x ≥ i + 1. We are denoting the variable by x ≥ i + 1 for bit alignment.
In abuse of notation, the term (1i, 0, ?e − 1 − i, x ≥2) means the sum over all 01 strings by replacing ? by 0 or 1.
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Li, Y. Characterization of robust immune symmetric boolean functions. Cryptogr. Commun. 7, 297–315 (2015). https://doi.org/10.1007/s12095-014-0120-7
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DOI: https://doi.org/10.1007/s12095-014-0120-7