analyze boolean functions

This work investigates the limitations of greedy strategies in constructing parity decision trees (PDTs), focusing on the existence of strongly folding directions within the Fourier spectrum support of Boolean functions. The authors present the first explicit class of Boolean functions devoid of ambient spectral components, for which the intersection of spectral supports under any two distinct translations is at most $O(|S|^{1/2})$, substantially tightening known upper bounds on spectral folding. By leveraging an affine subspace partitioning technique—APLPS—derived from full linear expansions, they achieve precise control over spectral support structure. Their results show that under the “lazy” maximal folding assumption, greedy methods cannot surpass the $O(|S|^{1/2})$ PDT depth upper bound, matching the best-known general bound and exposing the inadequacy of this assumption, though leaving open the potential advantages of adaptive greedy strategies.

This work addresses the challenge of simultaneously optimizing nonlinearity, resiliency order, and algebraic immunity in lightweight cryptographic Boolean functions. We propose an efficient, provably secure construction framework that achieves controllable trade-offs among these three critical cryptographic criteria. By integrating algebraic immunity analysis, linear bias control, and resilient structure design, we construct multiple families of n-variable Boolean functions where the number of variables scales linearly with the security parameters—specifically, n = O(r + d + t), with r, d, and t denoting the resiliency order, lower bound on nonlinearity, and algebraic immunity order, respectively. The resulting functions admit O(n)-size circuit implementations. To the best of our knowledge, this is the first unified construction enabling flexible, provable trade-offs among all three metrics, thereby significantly enhancing the synergy between security and efficiency. The approach is particularly suited for cryptographic algorithm design in resource-constrained environments.

Lightweight cryptographic hardware demands both high security and low computational overhead, yet existing Boolean function constructions struggle to simultaneously optimize nonlinearity, algebraic immunity, and implementation efficiency. Method: This paper proposes a novel construction framework that uniquely integrates integer arithmetic (addition and bit-shift operations) with polynomial operations over the binary field GF(2), enabling synergistic optimization of nonlinearity and algebraic immunity. Contribution/Results: For input sizes (n leq 20), the proposed functions achieve optimal trade-offs among implementation complexity, nonlinearity, and algebraic immunity. All constructed functions significantly outperform state-of-the-art efficient designs while requiring only basic arithmetic operations—addition, subtraction, multiplication, division, and bit-shifts—thus ensuring low hardware cost. Crucially, they exhibit strong resistance against fast algebraic attacks and high unpredictability, satisfying stringent security requirements. The approach is particularly suited for designing lightweight distinguishers and predicate functions in resource-constrained environments.

This work investigates the structural properties of Fourier-sparse Boolean functions over general finite abelian groups $mathbb{Z}_{p_1}^{n_1} imes cdots imes mathbb{Z}_{p_t}^{n_t}$ and their applications to property testing. Addressing odd prime-power-order groups, we first extend Granularity theory to this broader class. We refute the existence of a universal $O(1/s)$ lower bound on the smallest non-zero Fourier coefficient, constructing explicit counterexamples showing that in $mathbb{Z}_p^n$ ($p>2$), this coefficient can be as small as $1/omega(n)$. We establish a tight lower bound of $1/(m^2 s)^{lceil varphi(m)/2 ceil}$, where $m = mathrm{lcm}(p_1,dots,p_t)$. Leveraging this, we design an efficient property tester with query complexity $mathrm{poly}((ms)^{varphi(m)}, 1/varepsilon)$. Furthermore, we prove an $Omega(sqrt{s})$ lower bound on adaptive queries, demonstrating inherent limitations for testing Fourier sparsity in this setting.

This paper investigates translation-invariant (i.e., rotation-symmetric) vectorial Boolean functions on $n$ bits induced by $k$-bit Boolean functions, focusing on their cryptographic suitability in the non-bijective setting. Method: We introduce the notion of *almost lifting*—a weakly non-bijective construction whose induced mapping has collision number strictly bounded by $2^{k-1}$, independent of $n$. We prove this bound is tight, develop a systematic collision statistical model, and generalize Keccak’s $chi$ mapping into a family of cryptographic functions with controllable non-bijectivity. Contribution/Results: Leveraging Boolean function analysis and cryptographic evaluation (nonlinearity, differential uniformity, etc.), we construct a new family of efficiently computable, highly nonlinear, and collision-controlled functions. This work provides the first theoretically grounded and practically viable near-bijective framework for lightweight hash and permutation design.

This work investigates the tractability and computational hardness of Promise Constraint Satisfaction Problems (PCSPs) over Boolean domains. By introducing Fourier-analytic techniques into the PCSP framework—specifically leveraging influence measures of Boolean functions in conjunction with random 2-to-1 minors and sharp threshold theory—the study uncovers two universal mechanisms that govern whether a given problem is efficiently solvable or computationally intractable: the preservation of coordinate influences and the existence of sharp thresholds. This approach extends the prevailing paradigm for ordered PCSPs and, for the first time, establishes a clear dichotomy of tractability within broader classes of Boolean functions, including unate functions and polynomial threshold functions, thereby yielding new complexity-theoretic characterizations.

This study addresses the efficient construction of Boolean functions with flat nega-Hadamard spectra, particularly those simultaneously possessing bent and negabent properties. To overcome the limitations of conventional algebraic approaches, the work proposes a novel paradigm by systematically introducing evolutionary algorithms—specifically genetic programming—into this domain. A fitness function tailored to the nega-Hadamard transform is designed to guide the automated evolution of Boolean functions satisfying the desired spectral properties. Experimental results demonstrate the successful generation of both negabent and bent-negabent functions across multiple dimensions, confirming the effectiveness and generality of the proposed method. This approach establishes a new computational framework for constructing Boolean functions with prescribed cryptographic criteria.

This study investigates the circuit complexity bounds of fundamental Boolean operators in digital circuit design. By systematically analyzing known upper and lower complexity bounds for canonical Boolean functions—such as counters, adders, encoders, and multiplexers—and integrating both classical and modern Boolean function synthesis techniques, the work proposes efficient circuit synthesis strategies. Emphasizing the interplay between theoretical complexity and practical synthesis efficiency for basic operators, this research not only consolidates existing results but also provides a theoretical foundation and practical guidance for optimizing the design of complex digital circuits.