Two-dimensional ferromagnetic materials hold great promise for advancing low-power, high-integrated spintronic devices due to their atomic flat surfaces and versatile interfacial modulation. However, achieving a combination of room-temperature, field-free spin-orbit torque switching with tunable polarity in wafer-scale vdW heterostructures remains a significant challenge. Here, we demonstrate polarity-tunable, field-free spin-orbit torque switching in an all-vdW Bi2Te3/Fe4GeTe2 heterostructure, grown by molecular beam epitaxy. Interfacial coupling induces perpendicular magnetic anisotropy in Bi2Te3/Fe4GeTe2 interface, while the rest in-plane magnetic anisotropy component of Fe4GeTe2 breaks the inversion symmetry, enabling field-free switching. By modulating the direction of in-plane component, magnetic switching with different polarity could be achieved at low current density (~1.55×106 A/cm2). This allows for 16 reconfigurable Boolean logic functions in a single device, paving a pathway for energy-efficient 2D spintronic memory and logic systems. Our findings highlight the potential of all-vdW spin-orbit torque devices to revolutionize spintronics with scalable, room-temperature electronic control.

A High Throughput and Saliency Preserving Lightweight Medical Image Encryption Scheme Using Hyper-Chaotic Maps and DNA Encoding Generated Nonlinear Boolean Function

We characterize semantically quantified subjects, type (et,t), in English and show that the Boolean closure of the generalized existential and universal quantifiers is exactly the conservative ones. We prove that all subjects are expressible as Boolean functions of Montagovian individuals and that all mathematically extend to objects, type (eet,et). But quantified objects also include many functions that are not subject extensions, contrary to usual textbook assumptions. This is because two-place predicates (P2s) have more structure than one-place ones (P1s), so quantified objects have more to vary with/depend on. For example, we illustrate how lexical P2s in English can force their models to be infinite; P1s provably cannot.

The online manipulation-resilient testing model, proposed by Kalemaj, Raskhodnikova, and Varma (Theory of Computing 2023), studies property testing in situations where access to the input degrades continuously and adversarially. Our main contributions are as follows: — An extension of the model, introducing batch queries where multiple queries are made and answered between each round of manipulation, and fractional manipulation rate , where the adversary makes less than one manipulation per round. — New optimal testers for linearity of Boolean functions in the original online and offline models. — A new lower bound in the original model for testing whether a Boolean function has low degree and an algorithm using batch queries that overcomes it. — Efficient testers for local properties of sequences when the manipulation rate is fractional. Specifically, for sortedness, we show a sharp transition from optimal query complexity to the impossibility of testability, depending on the manipulation rate.

The set of binary decision diagrams , an efficient data structure representing Boolean functions, is extensively used in many distinct contexts like model verification, machine learning, cryptography or also resolution of combinatorial problems. The most famous variant, called reduced ordered binary decision diagram ( robdd for short), can be viewed as the result of a specific compaction of a complete decision tree. A great property is that, once an order over the Boolean variables is fixed, each Boolean function is represented by exactly one robdd . In this paper we aim at computing the exact distribution of the Boolean functions in \(k\) variables according to the robdd size . Recall the number of Boolean functions with \(k\) variables is equal to \(2^{2^{k}}\) , which is of double exponential growth with respect to the number of variables. The maximal size of an robdd with \(k\) variables is \(M_{k}\approx 2^{k}/k\) . In this paper, we develop the first polynomial algorithm to derive the distribution of Boolean functions over \(k\) variables with respect to robdd size denoted by \(n\) . It performs \(O(k\;n^{3}\log n)\) arithmetic operations on integers and necessitates to store \(O(n^{2})\) integers in memory storage; note that the maximal size of integers involved in the computations is \(O(k\;2^{k})\) bits. Our new approach relies on a decomposition of robdd s layer by layer and on an enumerative inclusion-exclusion argument.

Using spectral techniques, H. Huang proved that every subgraph of \(H_n\) , the hypercube of dimension n , induced on more than half the vertices has maximum degree at least \(\sqrt {n}\) . Combined with earlier work, this completed a proof of the sensitivity conjecture. In this work we show how to derive Huang’s result using linear dependency and independence of vectors associated with the vertices of the hypercube. Our approach leads to several improvements of Huang’s result. In particular, we prove that in any induced subgraph of \(H_n\) with more than half the number of vertices, there are two vertices, one of odd parity and the other of even parity, each with at least n vertices at distance at most 2. As an application, we show that for any Boolean function f , the polynomial degree of f is bounded above by \(\text{s}_0(f) \text{s}_1(f)\) , a statement which implies the sensitivity conjecture (but not immediately implied by the sensitivity conjecture). Using these linear dependencies, we show structural relations about the neighborhoods on the induced subgraphs at distance at most three. A key implement in Huang’s proof is to assign signs ( \(+,-\) ) to the edges of \(H_n\) such that the product of the signs on each 4-cycle is \(-\) . With the set of negative edges being called a signature, one may observe that there are a total of \(2^{2^n-1}\) such signatures on \(H_n\) satisfying this condition and that the symmetric difference of any two such signatures is an edge cut. A question of high interest then is to find the smallest size among all these signatures. This is known as the frustration index in the study of signed graphs. Here we provide lower and upper bounds for this parameter, observing that the two bounds match when n is a power of 4. We then establish a strong connection with other studies: On one hand with a question of Erdős on the number of edges of a largest 4-cycle free subgraph of the hypercube. On the other hand with Ambainis functions which are used to show a separation between degree and adversary lower bounds on query complexity.

The integration of both memory and computation within memristive devices has emerged as a promising approach in the domain of in-memory computing (IMC). Among various architectures, memristor-based crossbars offer high packing density, scalability, and compatibility with CMOS technology, enabling not only dense storage but also efficient implementation of Boolean logic functions. Majority logic, in particular, has shown superior efficiency over conventional logic primitives across several nanotechnologies. This work presents an adder design leveraging the majority logic function (MJF), including the realization of a full adder and a ripple carry adder directly within the IMC framework. The design assumes the availability of both nominal and complementary input data within the crossbar, allowing computation to be performed entirely in memristors. To mitigate the sneak path problem and associated read disturbances, we employ a one-transistor-one-memristor (1T1R) crossbar structure using 45 nm nMOS transistors in conjunction with the VTEAM memristor model. Furthermore, we propose a resource-constrained mapping technique for implementing arbitrary logic functions using MJF within the 1T1R crossbar. Simulation results demonstrate substantial performance gains, achieving up to 90% reduction in computation steps and 70% improvement in memristor utilization compared to existing IMC approaches.

2-rotation symmetric Boolean functions have important applications in cryptography. This paper focuses on constructing 2-rotation symmetric Boolean functions having nice cryptographic properties. Firstly, we present some new constructions of 2-rotation symmetric bent functions with optimal algebraic degree. Secondly, we propose a class of 2-rotation symmetric resilient functions having high nonlinearity and optimal algebraic degree.

This paper is devoted to the study of complexity of finding a special covering for a set, as well as to obtaining some important applications of special decomposition. We formulate the problem of existence of a special covering as a decision problem. To determine the complexity class in which this problem is located, we study the relationship between this problem and the Boolean satisfiability problem, treating them as formal languages. We prove that these problems are polynomially equivalent, which means that the problem of existence of a special covering for a set is an -complete problem. In this article we also introduce a new concept ‘Replaceable Subsets’. The properties of such subsets are used to fill in the missing elements needed to obtain a special set covering. It is proved that when searching for missing elements to fill a special covering, the order in which these elements are considered does not matter. This result is of great importance in the search for satisfiability of Boolean functions.

ABSTRACT The scale‐up of computation performances required by the rapidly increasing demand for the analysis and management of large databases poses serious doubts about the sustainability of von Neumann hardware architectures. Unconventional computing, taking inspiration from biological models and relying on self‐assembled systems based on nanoparticles and nanowires, may offer interesting alternatives. Here, we report the experimental characterization of the mechanisms that regulate the bistable electrical behavior and the resistive switching of self‐assembled gold nanostructured thin films. We show that the adaptive reconfiguration properties of the nanostructured network under specific input stimuli drive the reprogrammability of the device. We demonstrate how this system can be employed for the implementation of polymorphic devices, which can be used both as unconventional multiplexers (MUX) and as reconfigurable threshold logic gates (TLG), able to generate a complete set of Boolean functions.

Discovering Boolean functions that satisfy properties such as balancedness and nonlinearity is a complex optimization problem, which is crucial to important cryptographic constructions like block and stream ciphers. The difficulty of this problem lies in the search space growing super-exponentially in the number of variables. Evolutionary approaches, including Genetic Algorithms (GAs) and Genetic Programming (GP), have been successfully applied to overcome this difficulty. The major drawback of these methods is that they evolve functions through encodings that are either exponential in the input size or hard to interpret. We address this problem as follows. (i) We propose a new encoding for Boolean functions as reaction systems, a bio-inspired computational model which can be directly translated into the compact and easily interpretable Disjunctive Normal Form (DNF). (ii) We design EvoBRS, an evolutionary optimization framework that exploits this new representation to discover Boolean functions with maximum nonlinearity (bent functions), possibly under the balancedness constraint. (iii) We back up our novel paradigm with a refined theoretical analysis of independent interest. (iv) We conduct a rigorous experimental study, demonstrating that EvoBRS consistently discovers diverse, highly nonlinear Boolean functions with and without the balancedness constraint. EvoBRS proves particularly effective on balanced functions, successfully identifying balanced maximally nonlinear instances and outperforming both GP and state-of-the-art GAs. All the discovered functions are returned in a compact and easily interpretable DNF. A preliminary version of this work appeared in Ascone et al., GECCO 2025.

Early in 1992, Deutsch-Jozsa algorithm computed a symmetric partial Boolean function with a single quantum query, and thus achieved the best separation between classical deterministic and exact quantum query complexity. Recently, it was clarified that all symmetric partial Boolean functions with a single quantum query can be computed exactly by Deutsch-Jozsa algorithm. For the general partial Boolean functions with a single quantum query, the latest characterizations is complex and not very satisfactory. Based on this, this paper proves and discovers three new results: (1) Under a new equivalence, each partial Boolean function with a single quantum query can be transformed to a simple partial Boolean function whose polynomial degree is just one; (2) For partial Boolean functions up to four bits, there are only 10 non-trivial partial Boolean functions with a single quantum query; (3) For each quantum 1-query algorithm with undefined measurement, there exists a constructive method for finding out all partial Boolean functions that can be computed exactly by the algorithm.

Extremal boolean functions with long prime implicants

In recent years the framework of learning from label proportions (LLP) has been gaining importance in machine learning. In this setting, the training examples are aggregated into subsets or bags and only the average label per bag is available for learning an example-level predictor. This generalizes traditional PAC learning which is the special case of unit-sized bags. The computational learning aspects of LLP were studied in recent works [22, 23] which showed algorithms and hardness for learning halfspaces in the LLP setting. In this work we focus on the intractability of LLP learning Boolean functions. Our first result shows that given a collection of bags of size at most 2 which are consistent with an OR function, it is NP-hard to find a CNF of constantly many clauses which satisfies any constant-fraction of the bags. This is in contrast with the work of [22] which gave a (2/5)-approximation for learning ORs using a halfspace. Thus, our result provides a separation between constant clause CNFs and halfspaces as hypotheses for LLP learning ORs. Next, we prove the hardness of satisfying more than 1/2 + o (1) fraction of such bags using a t -DNF (i.e. DNF where each term has ≤ t literals) for any constant t . In usual PAC learning such a hardness was known [16] only for learning noisy ORs. We also study the learnability of parities and show that it is NP-hard to satisfy more than ( q /2 q − 1 + o (1))-fraction of q -sized bags which are consistent with a parity using a parity, while a random parity based algorithm achieves a (1/2 q − 2 )-approximation.

Masking is a crucial countermeasure against side-channel attacks, yet it is challenging to implement in hardware. At CHES 2024, Kumar S.V. et al. introduced the concept of Time Sharing Masking (TSM) for constructing low-latency first-order PINI gadgets. While TSM has demonstrated its effectiveness, we propose a new approach called mirrored-circuits to further enhance hardware performance. This approach constructs two MIRROR circuits that share randomness between them. For Boolean functions represented in the Algebraic Normal Form (ANF), the mirroredcircuits approach outperforms previous works in terms of the number of randomness bits and registers. By reconstructing the ANF form, the above two metrics can be further optimized using an SMT approach. We provide experimental results from several case studies to evaluate the efficiency of our approach. For 4-bit S-boxes, we achieve substantial reductions in randomness, delay, and area. Specifically, the 1-cycle SKINNY S-box achieves reductions of 50%, 37%, and 29% in randomness, delay, and area, respectively. For the AES S-box, the 1-cycle design minimizes randomness by 49%, and the 2-cycle design reduces randomness by 35% and area by 47%. To ensure the security of our schemes, we conduct comprehensive verification through three dimensions: theoretical proof, formal verification using SILVER, and FPGA-based experiments.

For a function f \colon \{0,1\}^{n} \to \mathbb{R} with Fourier expansion f=\sum_{S \subset \{1,\ldots,n\}}\hat f(S)\chi_{S} , the hypercontractive inequality for the noise operator allows bounding norms of T_{\rho} f = \sum_{S} \rho^{|S|}\hat f(S)\chi_{S} in terms of norms of f . If f is Boolean-valued, the level- d inequality allows bounding the norm of f^{=d}=\sum_{|S|=d}\hat f(S)\chi_{S} in terms of \mathbb{E}[f] . These two inequalities play a central role in analysis of Boolean functions and its applications. While both inequalities hold in a sharp form when the hypercube \{0,1\}^{n} is endowed with the uniform measure, it is easy to show that they do not hold for more general discrete product spaces, and finding a ‘natural’ generalization was a long-standing open problem. Keevash, Lifshitz, Long, and Minzer [J. Amer. Math. Soc. 37, 245–279 (2024)] obtained a hypercontractive inequality for general discrete product spaces, that holds for functions which are ‘global’ – namely, are not significantly affected by a restriction of a small set of coordinates. This hypercontractive inequality is not sharp, which precludes applications to the symmetric group S_{n} and to other settings where sharpness of the bound is crucial. Also, no sharp level- d inequality for global functions over general discrete product spaces is known. We obtain sharp versions of the hypercontractive inequality and of the level- d inequality for global functions over discrete product spaces. Our inequalities open the way for diverse applications to extremal set theory, group theory, theoretical computer science, and number theory. We demonstrate this by proving quantitative bounds on the size of intersecting families of sets and vectors under weak symmetry conditions and by describing numerous applications that were obtained using our results. Those contain applications to the study of functions over the symmetric group S_{n} – including hypercontractivity and level- d inequalities, character bounds, variants of Roth’s theorem and of Bogolyubov’s lemma, and diameter bounds, as well as an application to the Furstenberg–Sárközy problem on the maximal size of a subset of \{1,\ldots,n\} which does not contain two elements that differ by a perfect square.

ABSTRACT The Erdős–Ko–Rado (EKR) theorem and its generalizations can be viewed as classifications of maximum independent sets in appropriately defined families of graphs, such as the Kneser graph . In this paper, we investigate the independence number of random spanning subgraphs of two other families of graphs whose maximum independent sets satisfy an EKR‐type characterization: the derangement graph on the set of permutations in and the derangement graph on the set of perfect matchings in the complete graph . In both cases, we show there is a sharp threshold probability for the event that the independence number of a random spanning subgraph is equal to that of the original graph. As a useful tool to aid our computations, we obtain a Friedgut–Kalai–Naor (FKN) type theorem on sparse boolean functions whose domain is the vertex set of . In particular, we show that boolean functions whose Fourier transforms are highly concentrated on the first two irreducible modules in the module , is close to being the characteristic function of a union of maximum independent sets in the derangement graph on perfect matchings.

Multiple threshold potentials in nanofluidic pores with negative differential resistance.

The problem of finding a minimal circuit to implement a given function is one of the oldest in electronics. It is known to be NP-hard. Still, many tools exist to find sub-optimal circuits to implement a function. In electronics, such tools are known as synthesisers. However, these synthesisers aim to implement very large functions (a whole electronic chip). In cryptography, the focus is on small functions, hence the necessity for new dedicated tools for small functions. Several tools exist to implement small functions. They differ by their algorithmic approach (some are based on Depth-First-Search as introduced by Ullrich in 2011, some are based on SAT-solvers like the tool desgined by Stoffelen in 2016, some non-generic tools use subfield decomposition) and by their optimisation criteria (some optimise for circuit size, others for circuit depth, and some for side-channel-protected implementations). However, these tools are limited to functions operating on less than 5 bits, sometimes 6 bits for quadratic functions, or to very simple functions. The limitation lies in a high computing time. We propose a new tool (The tool is provided alongside the IEEE article with CodeOcean and at <uri xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">https://github.com/seduval/implem-quad-sbox</uri>) to implement quadratic functions up to 9 bits within AND-depth 1, minimising the number of AND gates. This tool is more time-efficient than previous ones, allowing to explore larger implementations than others on 6 bits or less and allows to reach larger sizes, up to 9 bits.

Lower bounds on the third-order nonlinearities of biquadratic Boolean functions