Related papers: Lower Bounds on the Algebraic Immunity of Boolean …
A Boolean function on n variables is q-resilient if for any subset of at most q variables, the function is very likely to be determined by a uniformly random assignment to the remaining n-q variables; in other words, no coalition of at most…
We describe and analyze a numerical algorithm for computing the homology (Betti numbers and torsion coefficients) of semialgebraic sets given by Boolean formulas. The algorithm works in weak exponential time. This means that outside a…
Recently, Arjevani et al. [1] established a lower bound of iteration complexity for the first-order optimization under an $L$-smooth condition and a bounded noise variance assumption. However, a thorough review of existing literature on…
We study functions on the infinite-dimensional Hamming cube $\{-1,1\}^\infty$, in particular Boolean functions into $\{-1,1\}$, generalising results on analysis of Boolean functions on $\{-1,1\}^n$ for $n\in\mathbb{N}$. The notion of noise…
Reconstruction attacks and defenses are essential in understanding the data leakage problem in machine learning. However, prior work has centered around empirical observations of gradient inversion attacks, lacks theoretical grounding, and…
We prove a lower bound of $\Omega(n^{1/2 - c})$, for all $c>0$, on the query complexity of (two-sided error) non-adaptive algorithms for testing whether an $n$-variable Boolean function is monotone versus constant-far from monotone. This…
We give two approximation algorithms solving the Stochastic Boolean Function Evaluation (SBFE) problem for symmetric Boolean functions. The first is an $O(\log n)$-approximation algorithm, based on the submodular goal-value approach of…
Exhibiting an explicit Boolean function with a large high-order nonlinearity is an important problem in cryptography, coding theory, and computational complexity. We prove lower bounds on the second-order, third-order, and higher-order…
Bent Boolean functions are important objects in cryptography and coding theory, and there are several general approaches for constructing such functions. Metaheuristics proved to be a strong choice as they can provide many bent functions,…
This paper studies the hazard-free formula complexity of Boolean functions. Our first result shows that unate functions are the only Boolean functions for which the monotone formula complexity of the hazard-derivative equals the hazard-free…
This paper provides a general technique for lower bounding the Bayes risk of statistical estimation, applicable to arbitrary loss functions and arbitrary prior distributions. A lower bound on the Bayes risk not only serves as a lower bound…
A class of quantum protocols of bit commitment is constructed based on the nonorthogonal states coding and the correlation immunity of some Boolean functions. The binding condition of these protocols is guaranteed mainly by the law of…
We study the problem of modeling a binary operation that satisfies some algebraic requirements. We first construct a neural network architecture for Abelian group operations and derive a universal approximation property. Then, we extend it…
A subset $S$ of the Boolean hypercube $\mathbb{F}_2^n$ is a sumset if $S = \{a + b : a, b\in A\}$ for some $A \subseteq \mathbb{F}_2^n$. Sumsets are central objects of study in additive combinatorics, featuring in several influential…
A Boolean function is symmetric if it is invariant under all permutations of its arguments; it is quasi-symmetric if it is symmetric with respect to the arguments on which it actually depends. We present a test that accepts every…
A fuzzy Boolean function is a map $f:\cube^n\to [0,1]$, where $n\in\mathbb N$. We introduce and compare three ways of saying that such a function has bounded complexity. The first is a sampling property: the value $f(x)$ can be recovered,…
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…
Artificial Intelligence (AI) relies heavily on deep learning - a technology that is becoming increasingly popular in real-life applications of AI, even in the safety-critical and high-risk domains. However, it is recently discovered that…
Research in adversarial machine learning (AML) has shown that statistical models are vulnerable to maliciously altered data. However, despite advances in Bayesian machine learning models, most AML research remains concentrated on classical…
This research started with an algebra for reasoning about rely/guarantee concurrency for a shared memory model. The approach taken led to a more abstract algebra of atomic steps, in which atomic steps synchronise (rather than interleave)…