Constructions of Almost Optimal Resilient Boolean Functions on Large Even Number of Variables

In this paper, a technique on constructing nonlinear resilient Boolean functions is described. By using several sets of disjoint spectra functions on a small number of variables, an almost optimal resilient function on a large even number of variables can be constructed. It is shown that given any $m$, one can construct infinitely many $n$-variable ($n$ even), $m$-resilient functions with nonlinearity $>2^{n-1}-2^{n/2}$. A large class of highly nonlinear resilient functions which were not known are obtained. Then one method to optimize the degree of the constructed functions is proposed. Last, an improved version of the main construction is given.

💡 Research Summary

The paper addresses the long‑standing problem of constructing Boolean functions that are simultaneously highly resilient, highly nonlinear, and of large algebraic degree—properties essential for cryptographic primitives such as stream ciphers and S‑boxes. After reviewing the state of the art (Maiorana‑McFarland, Partial‑Spread, and recent propagation‑resilient constructions), the authors introduce a novel block‑wise design based on “disjoint spectra functions”.

The core idea is to start with a small number $k$ of variables and build a set of $2^{k}$ Boolean functions whose Walsh spectra are pairwise disjoint. This can be achieved by selecting functions from known high‑nonlinearity classes and applying carefully chosen linear transformations that separate their spectra. The authors then replicate this set across $t=n/2$ blocks (assuming $n$ is even), each block receiving a distinct copy of the disjoint‑spectra family. The global function $F$ on $n$ variables is defined as the XOR of the block functions together with a global linear term $L(x)$.

The main theorem proves that for any fixed resilience order $m$, and for all sufficiently large even $n$, the constructed $F$ is $m$‑resilient and its nonlinearity satisfies
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