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Entropy Generation Optimization in Multidomain Systems: A Generalized Gouy-Stodola Theorem and Optimal Control
Intrinsic and Measured Information in Separable Quantum Processes
Two Types of Temporal Symmetry in the Laws of Nature
Quantum Electrodynamics from Quantum Cellular Automata, and the Tension Between Symmetry, Locality, and Positive Energy

Journal Description

Entropy

Entropy is an international and interdisciplinary peer-reviewed open access journal of entropy and information studies, published monthly online by MDPI. The International Society for the Study of Information (IS4SI) and Spanish Society of Biomedical Engineering (SEIB) are affiliated with Entropy and their members receive a discount on the article processing charge.

Open Access— free for readers, with article processing charges (APC) paid by authors or their institutions.
High Visibility: indexed within Scopus, SCIE (Web of Science), Inspec, PubMed, PMC, Astrophysics Data System, and other databases.
Journal Rank: JCR - Q2 (Physics, Multidisciplinary) / CiteScore - Q1 (Mathematical Physics)
Rapid Publication: manuscripts are peer-reviewed and a first decision is provided to authors approximately 21.8 days after submission; acceptance to publication is undertaken in 2.6 days (median values for papers published in this journal in the first half of 2025).
Recognition of Reviewers: reviewers who provide timely, thorough peer-review reports receive vouchers entitling them to a discount on the APC of their next publication in any MDPI journal, in appreciation of the work done.
Testimonials: See what our editors and authors say about Entropy.
Companion journals for Entropy include: Foundations, Thermo and Complexities.

Impact Factor: 2.0 (2024); 5-Year Impact Factor: 2.2 (2024)

Imprint Information Journal Flyer Open Access ISSN: 1099-4300

Latest Articles

16 pages, 386 KiB

Open AccessArticle

State Space Correspondence and Cross-Entropy Methods in the Assessment of Bidirectional Cardiorespiratory Coupling in Heart Failure

by Beatrice Cairo, Riccardo Pernice, Nikola N. Radovanović, Luca Faes, Alberto Porta and Mirjana M. Platiša

Entropy 2025, 27(7), 770; https://doi.org/10.3390/e27070770 (registering DOI) - 20 Jul 2025

The complex interplay between the cardiac and the respiratory systems, termed cardiorespiratory coupling (CRC), is a bidirectional phenomenon that can be affected by pathologies such as heart failure (HF). In the present work, the potential changes in strength of directional CRC were assessed [...] Read more.

The complex interplay between the cardiac and the respiratory systems, termed cardiorespiratory coupling (CRC), is a bidirectional phenomenon that can be affected by pathologies such as heart failure (HF). In the present work, the potential changes in strength of directional CRC were assessed in HF patients classified according to their cardiac rhythm via two measures of coupling based on k-nearest neighbor (KNN) estimation approaches, cross-entropy (CrossEn) and state space correspondence (SSC), applied on the heart period (HP) and respiratory (RESP) variability series, while also accounting for the complexity of the cardiac and respiratory rhythms. We tested the measures on 25 HF patients with sinus rhythm (SR, age: 58.9 ± 9.7 years; 23 males) and 41 HF patients with ventricular arrhythmia (VA, age 62.2 ± 11.0 years; 30 males). A predominant directionality of interaction from the cardiac to the respiratory rhythm was observed in both cohorts and using both methodologies, with similar statistical power, while a lower complexity for the RESP series compared to HP series was observed in the SR cohort. We conclude that CrossEn and SSC can be considered strictly related to each other when using a KNN technique for the estimation of the cross-predictability markers. Full article

(This article belongs to the Special Issue Entropy Methods for Cardiorespiratory Coupling Analysis)

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22 pages, 428 KiB

Open AccessArticle

ApproximateSecret Sharing in Field of Real Numbers

by Jiaqi Wan, Ziyue Wang, Yongqiang Yu and Xuehu Yan

Entropy 2025, 27(7), 769; https://doi.org/10.3390/e27070769 (registering DOI) - 20 Jul 2025

In the era of big data, the security of information encryption systems has garnered extensive attention, particularly in critical domains such as financial transactions and medical data management. While traditional Shamir’s Secret Sharing (SSS) ensures secure integer sharing through threshold cryptography, it exhibits [...] Read more.

In the era of big data, the security of information encryption systems has garnered extensive attention, particularly in critical domains such as financial transactions and medical data management. While traditional Shamir’s Secret Sharing (SSS) ensures secure integer sharing through threshold cryptography, it exhibits inherent limitations when applied to floating-point domains and high-precision numerical scenarios. To address these issues, this paper proposes an innovative algorithm to optimize SSS via type-specific coding for real numbers. By categorizing real numbers into four types—rational numbers, special irrationals, common irrationals, and general irrationals—our approach achieves lossless transmission for rational numbers, special irrationals, and common irrationals, while enabling low-loss recovery for general irrationals. The scheme leverages a type-coding system to embed data category identifiers in polynomial coefficients, combined with Bernoulli-distributed random bit injection to enhance security. The experimental results validate its effectiveness in balancing precision and security across various real-number types. Full article

(This article belongs to the Section Information Theory, Probability and Statistics)

12 pages, 493 KiB

Open AccessArticle

Exploring Non-Gaussianity Reduction in Quantum Channels

by Micael Andrade Dias and Francisco Marcos de Assis

Entropy 2025, 27(7), 768; https://doi.org/10.3390/e27070768 (registering DOI) - 20 Jul 2025

The quantum relative entropy between a quantum state and its Gaussian equivalent is a quantifying function of the system’s non-Gaussianity, a useful resource in several applications, such as quantum communication and computation. One of its most fundamental properties is to be monotonically decreasing [...] Read more.

The quantum relative entropy between a quantum state and its Gaussian equivalent is a quantifying function of the system’s non-Gaussianity, a useful resource in several applications, such as quantum communication and computation. One of its most fundamental properties is to be monotonically decreasing under Gaussian evolutions. In this paper, we develop the conditions for a non-Gaussian quantum channel to preserve the monotonically decreasing property. We propose a necessary condition to classify between Gaussian and non-Gaussian channels and use it to define a class of quantum channels that decrease the system’s non-Gaussianity. We also discuss how this property, combined with a restriction on the states at the channel’s input, can be applied to the security analysis of continuous-variable quantum key distribution protocols. Full article

(This article belongs to the Special Issue Continuous Variables for Quantum Key Distribution and Quantum Random Number Generators)

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16 pages, 1539 KiB

Open AccessArticle

A Quantum-like Approach to Semantic Text Classification

by Anastasia S. Gruzdeva, Rodion N. Iurev, Igor A. Bessmertny, Andrei Y. Khrennikov and Alexander P. Alodjants

Entropy 2025, 27(7), 767; https://doi.org/10.3390/e27070767 (registering DOI) - 19 Jul 2025

In this work, we conduct a sentiment analysis of English-language reviews using a quantum-like (wave-based) model of text representation. This model is explored as an alternative to machine learning (ML) techniques for text classification and analysis tasks. Special attention is given to the [...] Read more.

In this work, we conduct a sentiment analysis of English-language reviews using a quantum-like (wave-based) model of text representation. This model is explored as an alternative to machine learning (ML) techniques for text classification and analysis tasks. Special attention is given to the problem of segmenting text into semantic units, and we illustrate how the choice of segmentation algorithm is influenced by the structure of the language. We investigate the impact of quantum-like semantic interference on classification accuracy and compare the results with those obtained using classical probabilistic methods. Our findings show that accounting for interference effects improves accuracy by approximately

15 %

. We also explore methods for reducing the computational cost of algorithms based on the wave model of text representation. The results demonstrate that the quantum-like model can serve as a viable alternative or complement to traditional ML approaches. The model achieves classification precision and recall scores of around 0.8. Furthermore, the classification algorithm is readily amenable to optimization: the proposed procedure reduces the estimated computational complexity from

O (n^{2})

to

O (n)

. Full article

(This article belongs to the Section Multidisciplinary Applications)

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8 pages, 296 KiB

Open AccessCommunication

Equivalence of Informations Characterizes Bregman Divergences

by Philip S. Chodrow

Entropy 2025, 27(7), 766; https://doi.org/10.3390/e27070766 (registering DOI) - 19 Jul 2025

Bregman divergences form a class of distance-like comparison functions which plays fundamental roles in optimization, statistics, and information theory. One important property of Bregman divergences is that they generate agreement between two useful formulations of information content (in the sense of variability or [...] Read more.

Bregman divergences form a class of distance-like comparison functions which plays fundamental roles in optimization, statistics, and information theory. One important property of Bregman divergences is that they generate agreement between two useful formulations of information content (in the sense of variability or non-uniformity) in weighted collections of vectors. The first of these is the Jensen gap information, which measures the difference between the mean value of a strictly convex function evaluated on a weighted set of vectors and the value of that function evaluated at the centroid of that collection. The second of these is the divergence information, which measures the mean divergence of the vectors in the collection from their centroid. In this brief note, we prove that the agreement between Jensen gap and divergence informations in fact characterizes the class of Bregman divergences; they are the only divergences that generate this agreement for arbitrary weighted sets of data vectors. Full article

(This article belongs to the Section Information Theory, Probability and Statistics)

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21 pages, 459 KiB

Open AccessArticle

Post-Quantum Secure Multi-Factor Authentication Protocol for Multi-Server Architecture

by Yunhua Wen, Yandong Su and Wei Li

Entropy 2025, 27(7), 765; https://doi.org/10.3390/e27070765 (registering DOI) - 18 Jul 2025

The multi-factor authentication (MFA) protocol requires users to provide a combination of a password, a smart card and biometric data as verification factors to gain access to the services they need. In a single-server MFA system, users accessing multiple distinct servers must register [...] Read more.

The multi-factor authentication (MFA) protocol requires users to provide a combination of a password, a smart card and biometric data as verification factors to gain access to the services they need. In a single-server MFA system, users accessing multiple distinct servers must register separately for each server, manage multiple smart cards, and remember numerous passwords. In contrast, an MFA system designed for multi-server architecture allows users to register once at a registration center (RC) and then access all associated servers with a single smart card and one password. MFA with an offline RC addresses the computational bottleneck and single-point failure issues associated with the RC. In this paper, we propose a post-quantum secure MFA protocol for a multi-server architecture with an offline RC. Our MFA protocol utilizes the post-quantum secure Kyber key encapsulation mechanism and an information-theoretically secure fuzzy extractor as its building blocks. We formally prove the post-quantum semantic security of our MFA protocol under the real or random (ROR) model in the random oracle paradigm. Compared to related protocols, our protocol achieves higher efficiency and maintains reasonable communication overhead. Full article

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20 pages, 7353 KiB

Open AccessArticle

Comparative Analysis of Robust Entanglement Generation in Engineered XX Spin Chains

by Eduardo K. Soares, Gentil D. de Moraes Neto and Fabiano M. Andrade

Entropy 2025, 27(7), 764; https://doi.org/10.3390/e27070764 (registering DOI) - 18 Jul 2025

We present a numerical investigation comparing two entanglement generation protocols in finite XX spin chains with varying spin magnitudes (

s = 1 / 2, 1, 3 / 2

). Protocol 1 (P1) relies on staggered couplings to steer correlations toward [...] Read more.

We present a numerical investigation comparing two entanglement generation protocols in finite XX spin chains with varying spin magnitudes (

s = 1 / 2, 1, 3 / 2

). Protocol 1 (P1) relies on staggered couplings to steer correlations toward the ends of the chain. At the same time, Protocol 2 (P2) adopts a dual-port architecture that uses optimized boundary fields to mediate virtual excitations between terminal spins. Our results show that P2 consistently outperforms P1 in all spin values, generating higher-fidelity entanglement in shorter timescales when evaluated under the same system parameters. Furthermore, P2 exhibits superior robustness under realistic imperfections, including diagonal and off-diagonal disorder, as well as dephasing noise. To further assess the resilience of both protocols in experimentally relevant settings, we employ the pseudomode formalism to characterize the impact of non-Markovian noise on the entanglement dynamics. Our analysis reveals that the dual-port mechanism (P2) remains effective even when memory effects are present, as it reduces the excitation of bulk modes that would otherwise enhance environment-induced backflow. Together, the scalability, efficiency, and noise resilience of the dual-port approach position it as a promising framework for entanglement distribution in solid-state quantum information platforms. Full article

(This article belongs to the Special Issue Entanglement in Quantum Spin Systems)

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20 pages, 3787 KiB

Open AccessArticle

Enhancing Robustness of Variational Data Assimilation in Chaotic Systems: An α-4DVar Framework with Rényi Entropy and α-Generalized Gaussian Distributions

by Yuchen Luo, Xiaoqun Cao, Kecheng Peng, Mengge Zhou and Yanan Guo

Entropy 2025, 27(7), 763; https://doi.org/10.3390/e27070763 - 18 Jul 2025

Traditional 4-dimensional variational data assimilation methods have limitations due to the Gaussian distribution assumption of observation errors, and the gradient of the objective functional is vulnerable to observation noise and outliers. To address these issues, this paper proposes a non-Gaussian nonlinear data assimilation [...] Read more.

Traditional 4-dimensional variational data assimilation methods have limitations due to the Gaussian distribution assumption of observation errors, and the gradient of the objective functional is vulnerable to observation noise and outliers. To address these issues, this paper proposes a non-Gaussian nonlinear data assimilation method called α-4DVar, based on Rényi entropy and the α-generalized Gaussian distribution. By incorporating the heavy-tailed property of Rényi entropy, the objective function and its gradient suitable for non-Gaussian errors are derived, and numerical experiments are conducted using the Lorenz-63 model. Experiments are conducted with Gaussian and non-Gaussian errors as well as different initial guesses to compare the assimilation effects of traditional 4DVar and α-4DVar. The results show that α-4DVar performs as well as traditional method without observational errors. Its analysis field is closer to the truth, with RMSE rapidly dropping to a low level and remaining stable, particularly under non-Gaussian errors. Under different initial guesses, the RMSE of both the background and analysis fields decreases quickly and stabilizes. In conclusion, the α-4DVar method demonstrates significant advantages in handling non-Gaussian observational errors, robustness against noise, and adaptability to various observational conditions, thus offering a more reliable and effective solution for data assimilation. Full article

(This article belongs to the Section Complexity)

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18 pages, 821 KiB

Open AccessArticle

Joint Iterative Decoding Design of Cooperative Downlink SCMA Systems

by Hao Cheng, Min Zhang and Ruoyu Su

Entropy 2025, 27(7), 762; https://doi.org/10.3390/e27070762 - 18 Jul 2025

Sparse code multiple access (SCMA) has been a competitive multiple access candidate for future communication networks due to its superiority in spectrum efficiency and providing massive connectivity. However, cell edge users may suffer from great performance degradations due to signal attenuation. Therefore, a [...] Read more.

Sparse code multiple access (SCMA) has been a competitive multiple access candidate for future communication networks due to its superiority in spectrum efficiency and providing massive connectivity. However, cell edge users may suffer from great performance degradations due to signal attenuation. Therefore, a cooperative downlink SCMA system is proposed to improve transmission reliability. To the best of our knowledge, multiuser detection is still an open issue for this cooperative downlink SCMA system. To this end, we propose a joint iterative decoding design of the cooperative downlink SCMA system by using the joint factor graph stemming from direct and relay transmission. The closed form bit-error rate (BER) performance of the cooperative downlink SCMA system is also derived. Simulation results verify that the proposed cooperative downlink SCMA system performs better than the non-cooperative one. Full article

(This article belongs to the Special Issue Wireless Communications: Signal Processing Perspectives, 2nd Edition)

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14 pages, 3176 KiB

Open AccessArticle

Impact of Data Distribution and Bootstrap Setting on Anomaly Detection Using Isolation Forest in Process Quality Control

by Hyunyul Choi and Kihyo Jung

Entropy 2025, 27(7), 761; https://doi.org/10.3390/e27070761 - 18 Jul 2025

This study investigates the impact of data distribution and bootstrap resampling on the anomaly detection performance of the Isolation Forest (iForest) algorithm in statistical process control. Although iForest has received attention for its multivariate and ensemble-based nature, its performance under non-normal data distributions [...] Read more.

This study investigates the impact of data distribution and bootstrap resampling on the anomaly detection performance of the Isolation Forest (iForest) algorithm in statistical process control. Although iForest has received attention for its multivariate and ensemble-based nature, its performance under non-normal data distributions and varying bootstrap settings remains underexplored. To address this gap, a comprehensive simulation was performed across 18 scenarios involving log-normal, gamma, and t-distributions with different mean shift levels and bootstrap configurations. The results show that iForest substantially outperforms the conventional Hotelling’s T² control chart, especially in non-Gaussian settings and under small-to-medium process shifts. Enabling bootstrap resampling led to marginal improvements across classification metrics, including accuracy, precision, recall, F1-score, and average run length (ARL)₁. However, a key limitation of iForest was its reduced sensitivity to subtle process changes, such as a 1σ mean shift, highlighting an area for future enhancement. Full article

(This article belongs to the Section Multidisciplinary Applications)

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18 pages, 1438 KiB

Open AccessArticle

Maximum Entropy Estimates of Hubble Constant from Planck Measurements

by David P. Knobles and Mark F. Westling

Entropy 2025, 27(7), 760; https://doi.org/10.3390/e27070760 - 16 Jul 2025

A maximum entropy (ME) methodology was used to infer the Hubble constant from the temperature anisotropies in cosmic microwave background (CMB) measurements, as measured by the Planck satellite. A simple cosmological model provided physical insight and afforded robust statistical sampling of a parameter [...] Read more.

A maximum entropy (ME) methodology was used to infer the Hubble constant from the temperature anisotropies in cosmic microwave background (CMB) measurements, as measured by the Planck satellite. A simple cosmological model provided physical insight and afforded robust statistical sampling of a parameter space. The parameter space included the spectral tilt and amplitude of adiabatic density fluctuations of the early universe and the present-day ratios of dark energy, matter, and baryonic matter density. A statistical temperature was estimated by applying the equipartition theorem, which uniquely specifies a posterior probability distribution. The ME analysis inferred the mean value of the Hubble constant to be about 67 km/sec/Mpc with a conservative standard deviation of approximately 4.4 km/sec/Mpc. Unlike standard Bayesian analyses that incorporate specific noise models, the ME approach treats the model error generically, thereby producing broader, but less assumption-dependent, uncertainty bounds. The inferred ME value lies within 1σ of both early-universe estimates (Planck, Dark Energy Signal Instrument (DESI)) and late-universe measurements (e.g., the Chicago Carnegie Hubble Program (CCHP)) using redshift data collected from the James Webb Space Telescope (JWST). Thus, the ME analysis does not appear to support the existence of the Hubble tension. Full article

(This article belongs to the Special Issue Insight into Entropy)

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20 pages, 2382 KiB

Open AccessArticle

Heterogeneity-Aware Personalized Federated Neural Architecture Search

by An Yang and Ying Liu

Entropy 2025, 27(7), 759; https://doi.org/10.3390/e27070759 - 16 Jul 2025

Federated learning (FL), which enables collaborative learning across distributed nodes, confronts a significant heterogeneity challenge, primarily including resource heterogeneity induced by different hardware platforms, and statistical heterogeneity originating from non-IID private data distributions among clients. Neural architecture search (NAS), particularly one-shot NAS, holds [...] Read more.

Federated learning (FL), which enables collaborative learning across distributed nodes, confronts a significant heterogeneity challenge, primarily including resource heterogeneity induced by different hardware platforms, and statistical heterogeneity originating from non-IID private data distributions among clients. Neural architecture search (NAS), particularly one-shot NAS, holds great promise for automatically designing optimal personalized models tailored to such heterogeneous scenarios. However, the coexistence of both resource and statistical heterogeneity destabilizes the training of the one-shot supernet, impairs the evaluation of candidate architectures, and ultimately hinders the discovery of optimal personalized models. To address this problem, we propose a heterogeneity-aware personalized federated NAS (HAPFNAS) method. First, we leverage lightweight knowledge models to distill knowledge from clients to server-side supernet, thereby effectively mitigating the effects of heterogeneity and enhancing the training stability. Then, we build random-forest-based personalized performance predictors to enable the efficient evaluation of candidate architectures across clients. Furthermore, we develop a model-heterogeneous FL algorithm called heteroFedAvg to facilitate collaborative model training for the discovered personalized models. Comprehensive experiments on CIFAR-10/100 and Tiny-ImageNet classification datasets demonstrate the effectiveness of our HAPFNAS, compared to state-of-the-art federated NAS methods. Full article

(This article belongs to the Section Signal and Data Analysis)

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16 pages, 862 KiB

Open AccessArticle

Random Search Walks Inside Absorbing Annuli

by Anderson S. Bibiano-Filho, Jandson F. O. de Freitas, Marcos G. E. da Luz, Gandhimohan M. Viswanathan and Ernesto P. Raposo

Entropy 2025, 27(7), 758; https://doi.org/10.3390/e27070758 - 15 Jul 2025

We revisit the problem of random search walks in the two-dimensional (2D) space between concentric absorbing annuli, in which a searcher performs random steps until finding either the inner or the outer ring. By considering step lengths drawn from a power-law distribution, we [...] Read more.

We revisit the problem of random search walks in the two-dimensional (2D) space between concentric absorbing annuli, in which a searcher performs random steps until finding either the inner or the outer ring. By considering step lengths drawn from a power-law distribution, we obtain the exact analytical result for the search efficiency

η

in the ballistic limit, as well as an approximate expression for

η

in the regime of searches starting far away from both rings, and the scaling behavior of

η

for very small initial distances to the inner ring. Our numerical results show good overall agreement with the theoretical findings. We also analyze numerically the absorbing probabilities related to the encounter of the inner and outer rings and the associated Shannon entropy. The power-law exponent marking the crossing of such probabilities (equiprobability) and the maximum entropy condition grows logarithmically with the starting distance. Random search walks inside absorbing annuli are relevant, since they represent a mean-field approach to conventional random searches in 2D, which is still an open problem with important applications in various fields. Full article

(This article belongs to the Special Issue Transport in Complex Environments)

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33 pages, 1024 KiB

Open AccessArticle

Graph-Theoretic Limits of Distributed Computation: Entropy, Eigenvalues, and Chromatic Numbers

by Mohammad Reza Deylam Salehi and Derya Malak

Entropy 2025, 27(7), 757; https://doi.org/10.3390/e27070757 - 15 Jul 2025

We address the problem of the distributed computation of arbitrary functions of two correlated sources,

X_{1}

and

X_{2}

, residing in two distributed source nodes, respectively. We exploit the structure of a computation task by coding source characteristic graphs (and multiple [...] Read more.

We address the problem of the distributed computation of arbitrary functions of two correlated sources,

X_{1}

and

X_{2}

, residing in two distributed source nodes, respectively. We exploit the structure of a computation task by coding source characteristic graphs (and multiple instances using the n-fold OR product of this graph with itself). For regular graphs and general graphs, we establish bounds on the optimal rate—characterized by the chromatic entropy for the n-fold graph products—that allows a receiver for asymptotically lossless computation of arbitrary functions over finite fields. For the special class of cycle graphs (i.e., 2-regular graphs), we establish an exact characterization of chromatic numbers and derive bounds on the required rates. Next, focusing on the more general class of d-regular graphs, we establish connections between d-regular graphs and expansion rates for n-fold graph products using graph spectra. Finally, for general graphs, we leverage the Gershgorin Circle Theorem (GCT) to provide a characterization of the spectra, which allows us to derive new bounds on the optimal rate. Our codes leverage the spectra of the computation and provide a graph expansion-based characterization to succinctly capture the computation structure, providing new insights into the problem of distributed computation of arbitrary functions. Full article

(This article belongs to the Special Issue Information Theory and Data Compression)

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41 pages, 1006 KiB

Open AccessArticle

A Max-Flow Approach to Random Tensor Networks

by Khurshed Fitter, Faedi Loulidi and Ion Nechita

Entropy 2025, 27(7), 756; https://doi.org/10.3390/e27070756 - 15 Jul 2025

The entanglement entropy of a random tensor network (RTN) is studied using tools from free probability theory. Random tensor networks are simple toy models that help in understanding the entanglement behavior of a boundary region in the anti-de Sitter/conformal field theory (AdS/CFT) context. [...] Read more.

The entanglement entropy of a random tensor network (RTN) is studied using tools from free probability theory. Random tensor networks are simple toy models that help in understanding the entanglement behavior of a boundary region in the anti-de Sitter/conformal field theory (AdS/CFT) context. These can be regarded as specific probabilistic models for tensors with particular geometry dictated by a graph (or network) structure. First, we introduce a model of RTN obtained by contracting maximally entangled states (corresponding to the edges of the graph) on the tensor product of Gaussian tensors (corresponding to the vertices of the graph). The entanglement spectrum of the resulting random state is analyzed along a given bipartition of the local Hilbert spaces. The limiting eigenvalue distribution of the reduced density operator of the RTN state is provided in the limit of large local dimension. This limiting value is described through a maximum flow optimization problem in a new graph corresponding to the geometry of the RTN and the given bipartition. In the case of series-parallel graphs, an explicit formula for the limiting eigenvalue distribution is provided using classical and free multiplicative convolutions. The physical implications of these results are discussed, allowing the analysis to move beyond the semiclassical regime without any cut assumption, specifically in terms of finite corrections to the average entanglement entropy of the RTN. Full article

(This article belongs to the Section Quantum Information)

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31 pages, 2957 KiB

Open AccessArticle

Nash Equilibria in Four-Strategy Quantum Extensions of the Prisoner’s Dilemma Game

by Piotr Frąckiewicz, Anna Gorczyca-Goraj, Krzysztof Grzanka, Katarzyna Nowakowska and Marek Szopa

Entropy 2025, 27(7), 755; https://doi.org/10.3390/e27070755 - 15 Jul 2025

The concept of Nash equilibria in pure strategies for quantum extensions of the general form of the Prisoner’s Dilemma game is investigated. The process of quantization involves incorporating two additional unitary strategies, which effectively expand the classical game. We consider five classes of [...] Read more.

The concept of Nash equilibria in pure strategies for quantum extensions of the general form of the Prisoner’s Dilemma game is investigated. The process of quantization involves incorporating two additional unitary strategies, which effectively expand the classical game. We consider five classes of such quantum games, which remain invariant under isomorphic transformations of the classical game. The resulting Nash equilibria are found to be more closely aligned with Pareto-optimal solutions than those of the conventional Nash equilibrium outcome of the classical game. Our results demonstrate the complexity and diversity of strategic behavior in the quantum setting, providing new insights into the dynamics of classical decision-making dilemmas. In particular, we provide a detailed characterization of strategy profiles and their corresponding Nash equilibria, thereby extending the understanding of quantum strategies’ impact on traditional game-theoretical problems. Full article

(This article belongs to the Special Issue Decision Optimization in Information Theory and Game Theory, 2nd Edition)

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21 pages, 7084 KiB

Open AccessArticle

Chinese Paper-Cutting Style Transfer via Vision Transformer

by Chao Wu, Yao Ren, Yuying Zhou, Ming Lou and Qing Zhang

Entropy 2025, 27(7), 754; https://doi.org/10.3390/e27070754 - 15 Jul 2025

Style transfer technology has seen substantial attention in image synthesis, notably in applications like oil painting, digital printing, and Chinese landscape painting. However, it is often difficult to generate migrated images that retain the essence of paper-cutting art and have strong visual appeal [...] Read more.

Style transfer technology has seen substantial attention in image synthesis, notably in applications like oil painting, digital printing, and Chinese landscape painting. However, it is often difficult to generate migrated images that retain the essence of paper-cutting art and have strong visual appeal when trying to apply the unique style of Chinese paper-cutting art to style transfer. Therefore, this paper proposes a new method for Chinese paper-cutting style transformation based on the Transformer, aiming at realizing the efficient transformation of Chinese paper-cutting art styles. Specifically, the network consists of a frequency-domain mixture block and a multi-level feature contrastive learning module. The frequency-domain mixture block explores spatial and frequency-domain interaction information, integrates multiple attention windows along with frequency-domain features, preserves critical details, and enhances the effectiveness of style conversion. To further embody the symmetrical structures and hollowed hierarchical patterns intrinsic to Chinese paper-cutting, the multi-level feature contrastive learning module is designed based on a contrastive learning strategy. This module maximizes mutual information between multi-level transferred features and content features, improves the consistency of representations across different layers, and thus accentuates the unique symmetrical aesthetics and artistic expression of paper-cutting. Extensive experimental results demonstrate that the proposed method outperforms existing state-of-the-art approaches in both qualitative and quantitative evaluations. Additionally, we created a Chinese paper-cutting dataset that, although modest in size, represents an important initial step towards enriching existing resources. This dataset provides valuable training data and a reference benchmark for future research in this field. Full article

(This article belongs to the Section Multidisciplinary Applications)

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21 pages, 877 KiB

Open AccessArticle

Identity-Based Provable Data Possession with Designated Verifier from Lattices for Cloud Computing

by Mengdi Zhao and Huiyan Chen

Entropy 2025, 27(7), 753; https://doi.org/10.3390/e27070753 - 15 Jul 2025

Provable data possession (PDP) is a technique that enables the verification of data integrity in cloud storage without the need to download the data. PDP schemes are generally categorized into public and private verification. Public verification allows third parties to assess the integrity [...] Read more.

Provable data possession (PDP) is a technique that enables the verification of data integrity in cloud storage without the need to download the data. PDP schemes are generally categorized into public and private verification. Public verification allows third parties to assess the integrity of outsourced data, offering good openness and flexibility, but it may lead to privacy leakage and security risks. In contrast, private verification restricts the auditing capability to the data owner, providing better privacy protection but often resulting in higher verification costs and operational complexity due to limited local resources. Moreover, most existing PDP schemes are based on classical number-theoretic assumptions, making them vulnerable to quantum attacks. To address these challenges, this paper proposes an identity-based PDP with a designated verifier over lattices, utilizing a specially leveled identity-based fully homomorphic signature (IB-FHS) scheme. We provide a formal security proof of the proposed scheme under the small-integer solution (SIS) and learning with errors (LWE) within the random oracle model. Theoretical analysis confirms that the scheme achieves security guarantees while maintaining practical feasibility. Furthermore, simulation-based experiments show that for a 1 MB file and lattice dimension of n = 128, the computation times for core algorithms such as TagGen, GenProof, and CheckProof are approximately 20.76 s, 13.75 s, and 3.33 s, respectively. Compared to existing lattice-based PDP schemes, the proposed scheme introduces additional overhead due to the designated verifier mechanism; however, it achieves a well-balanced optimization among functionality, security, and efficiency. Full article

(This article belongs to the Section Information Theory, Probability and Statistics)

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17 pages, 583 KiB

Open AccessArticle

Cross-Domain Feature Enhancement-Based Password Guessing Method for Small Samples

by Cheng Liu, Junrong Li, Xiheng Liu, Bo Li, Mengsu Hou, Wei Yu, Yujun Li and Wenjun Liu

Entropy 2025, 27(7), 752; https://doi.org/10.3390/e27070752 - 15 Jul 2025

As a crucial component of account protection system evaluation and intrusion detection, the advancement of password guessing technology encounters challenges due to its reliance on password data. In password guessing research, there is a conflict between the traditional models’ need for large training [...] Read more.

As a crucial component of account protection system evaluation and intrusion detection, the advancement of password guessing technology encounters challenges due to its reliance on password data. In password guessing research, there is a conflict between the traditional models’ need for large training samples and the limitations on accessing password data imposed by privacy protection regulations. Consequently, security researchers often struggle with the issue of having a very limited password set from which to guess. This paper introduces a small-sample password guessing technique that enhances cross-domain features. It analyzes the password set using probabilistic context-free grammar (PCFG) to create a list of password structure probabilities and a dictionary of password fragment probabilities, which are then used to generate a password set structure vector. The method calculates the cosine similarity between the small-sample password set B from the target area and publicly leaked password sets

A_{i}

using the structure vector, identifying the set

A_{m a x}

with the highest similarity. This set is then utilized as a training set, where the features of the small-sample password set are enhanced by modifying the structure vectors of the training set. The enhanced training set is subsequently employed for PCFG password generation. The paper uses hit rate as the evaluation metric, and Experiment I reveals that the similarity between B and

A_{i}

can be reliably measured when the size of B exceeds 150. Experiment II confirms the hypothesis that a higher similarity between

A_{i}

and B leads to a greater hit rate of

A_{i}

on the test set of B, with potential improvements of up to 32% compared to training with B alone. Experiment III demonstrates that after enhancing the features of

A_{m a x}

, the hit rate for the small-sample password set can increase by as much as 10.52% compared to previous results. This method offers a viable solution for small-sample password guessing without requiring prior knowledge. Full article

(This article belongs to the Section Information Theory, Probability and Statistics)

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17 pages, 300 KiB

Open AccessArticle

Commitment Schemes from OWFs with Applications to Quantum Oblivious Transfer

by Thomas Lorünser, Sebastian Ramacher and Federico Valbusa

Entropy 2025, 27(7), 751; https://doi.org/10.3390/e27070751 - 15 Jul 2025

Commitment schemes (CSs) are essential to many cryptographic protocols and schemes with applications that include privacy-preserving computation on data, privacy-preserving authentication, and, in particular, oblivious transfer protocols. For quantum oblivious transfer (qOT) protocols, unconditionally binding commitment schemes that do not rely on hardness [...] Read more.

Commitment schemes (CSs) are essential to many cryptographic protocols and schemes with applications that include privacy-preserving computation on data, privacy-preserving authentication, and, in particular, oblivious transfer protocols. For quantum oblivious transfer (qOT) protocols, unconditionally binding commitment schemes that do not rely on hardness assumptions from structured mathematical problems are required. These additional constraints severely limit the choice of commitment schemes to random oracle-based constructions or Naor’s bit commitment scheme. As these protocols commit to individual bits, the use of such commitment schemes comes at a high bandwidth and computational cost. In this work, we investigate improvements to the efficiency of commitment schemes used in qOT protocols and propose an extension of Naor’s commitment scheme requiring the existence of one-way functions (OWFs) to reduce communication complexity for 2-bit strings. Additionally, we provide an interactive string commitment scheme with preprocessing to enable the fast and efficient computation of commitments. Full article

(This article belongs to the Special Issue Information-Theoretic Cryptography and Security)

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