Advances in Mathematical Cryptography and Information Security Toward Industry 5.0

1. Introduction

Industry 5.0 builds on the success of Industry 4.0, envisioning human-centricity, sustainability, and resilience in manufacturing [1,2,3]. Modern industrial systems comprise complex networks and systems that integrate information technology, e.g., massive Internet of Things (IoT) devices, and interconnected industrial processes, which require robust security frameworks to ensure the security and privacy of sensitive user data [4]. As industrial systems architecture grows, the security landscape becomes more complex and the need to protect security-critical systems and infrastructure from unauthorized access becomes imperative. However, due to the proliferation of dense IoT devices operating on open wireless channels, designing a secure user authentication scheme in such a complex environment to achieve the desired forward secrecy poses several challenges that must be proactively addressed. Existing security authentication schemes do not adequately address the prevailing security issues in complex industrial systems, particularly considering the vast number of resource-limited sensor nodes that play active roles in such dynamic systems [5]. Additionally, traditional security architectures have not provided the desired forward secrecy, partly due to the enormous overhead costs and significant computational complexity.

Sensitive user data of all parties in the information security ecosystem need to be adequately secured from a multitude of sophisticated adversaries that proliferate cyberspace and industrial settings [6]. Consequently, it becomes expedient to authenticate and secure sensitive user information that is transmitted via these sensor nodes from potential intruders [7,8]. To achieve this objective, suitable authentication protocols and session keys are necessary to grant access only to authorized users of the system. In recent times, mathematical cryptography has been exploited to provide robust security authentication for complex systems, including industrial and manufacturing systems in Industry 5.0. Effectively, cryptography relies on mathematics and logic to design strong and robust security schemes. Modern cryptography and information security emphasize the mathematics behind the theory of these cryptosystems; their evolving integration in practical designs to provide adequate and provable security to authorized users of these systems is gaining widespread popularity in the research community.

This Special Issue presents original contributions focusing on the design and development of advanced mathematical cryptographic schemes for information security in relation to Industry 5.0. The following topics are covered: variable-step semi-implicit solvers with adjustable symmetry and their application in chaos-based communication, a blockchain-assisted secure and lightweight authentication scheme for multi-server internet of drones environments, an improved network intrusion detection technique using the feature engineering approach with boosting classifiers, a secure authentication scheme with local differential privacy in edge intelligence-enabled vehicular ad hoc networks, covert communication for dual images with two-tier bits flipping, design of a secure and privacy-preserving data sharing scheme based on key aggregation and private set intersection in a medical information system, mathematical model of the process of data transmission over the radio channel of cyber-physical systems, a reliable and privacy-preserving vehicular energy trading scheme using decentralized identifiers, embedding secret data in a vector quantization codebook using a novel thresholding scheme, and efficient parallel secure outsourcing of modular exponentiation to the cloud for IoT applications. This Special Issue features important techniques that can drive interesting results along with defined cutting-edge research, leading to the development of advanced mathematical cryptography and information security systems in support of the envisioned Industry 5.0.

2. Contributions

This Special Issue presents ten research papers from leading authors on this trending research topic. The contributions are outlined as follows.

First, Rybin et al. proposed a variable-step semi-implicit solver with adjustable symmetry and its application for chaos-based communication (Contribution 1). The work focuses on a variable-step, modified composite diagonal version of the existing solver, featuring a stable and simple Hairer step size controller. In particular, the authors demonstrated that the proposed method alters the dynamics of the solution by adjusting the symmetry coefficient, thereby fine-tuning the dynamics of the obtained discrete model with no influence from the appropriate step size. Furthermore, the authors constructed a direct chaotic communication system based on the Sprott Case S chaotic oscillator, demonstrating high values in the most prominent Lyapunov exponent. For the modulation parameter, the tolerance parameter of the step size controller is used to insert a message into the chaotic time series. In general, the main results show that the proposed modulation scheme exhibits competitive robustness to noise and return map attacks compared to modulation methods based on fixed-step solvers.

Secondly, Ju et al. (Contribution 2) studied a blockchain-assisted, secure, and lightweight authentication scheme for multi-server Internet of drones environments. The protocol provides robust security measures against threats and achieves a low computational and communication overhead. For validation processes, formal methods such as the Real-Or-Random (RoR) model and Burrow–Abadi–Needham (BAN) logic were employed. In comparison with existing security schemes, the proposed protocol demonstrates high levels of efficiency and robustness, making it suitable for resource-limited IoT applications.

The work by Rai et al. (Contribution 3) presented a network-assisted intrusion detection system that employs feature engineering and machine learning models. A transmission control protocol/internet protocol (TCP/IP)-based dataset from various protocols was used in the study. Several preprocessing techniques were employed to improve the quality of the dataset. Based on the preprocessed dataset, models were developed and their prediction accuracy was assessed using feature engineering. Overall, the trained model demonstrates robust effectiveness and accuracy in detecting network intrusions, positioning it well for real-time use.

Kwon et al. (Contribution 4) introduced an authentication scheme for privacy preservation in edge intelligence-enabled vehicular ad hoc networks. Specifically, the scheme utilizes fuzzy extractors, elliptic curve cryptography, and physical unclonable function technology to create a secure communication channel. Additionally, local differential privacy and symmetric key encryption were used to ensure the confidentiality of the data upload process. Security verification and validation processes were carried out and the computation and communication efficiency of the proposed scheme were evaluated relative to the existing schemes, leveraging the multi-precision integer and rational arithmetic cryptographic library software development kit.

Contribution 5 focuses on covert communication for dual images with two-tier bit flipping. Specifically, Xu et al. propose an innovative one-to-two data hiding scheme that employs a two-tier bit-flipping strategy to embed sensitive data in dual images. This produces two stego images, which are transmitted to two distinct recipients and can only be extracted when both recipients maintain mutual trust, thereby ensuring the security of the stego images. The results show that the proposed scheme produces a high visual quality and achieves an excellent data payload compared to the preliminary schemes.

The work of Oh et al. considered a secure and privacy-preserving data sharing scheme, guaranteeing data confidentiality and privacy (Contribution 6). Key aggregate encryption and private set intersection techniques were employed, enabling secure data sharing and offering the protection of sensitive user data. Additionally, informal and formal security analyses, leveraging the Burrow–Abadi–Needham logic and Scyther, were conducted to show the scheme’s resilience against potential adversarial attacks. Furthermore, the execution time for cryptographic operations was implemented using a multi-precision integer and a rational arithmetic cryptographic library, followed by a comparative analysis with related schemes.

Makhmudov et al. presented a modified mathematical model for evaluating the quality of mobile radio channels in the context of cyber-physical systems (Contribution 7). By conceptualizing the radio channel as a stochastic network, the authors were able to derive useful metrics, including the equivalent function, mathematical expectation, variance, and the time distribution function of the implemented processes. This is achieved by employing the Gamma distribution for the initial distribution functions of random variables, thereby achieving analytical precision.

Kim et al. recommended a reliable and privacy-preserving vehicular energy trading scheme that uses a decentralized identifier technology (Contribution 8). In the design, only users who complete successful energy trading are authorized to manage complete transaction information. The work employs both informal and formal security analyses to verify the proposed scheme and demonstrate its performance compared to its counterparts.

In Contribution 9, Lin et al. examined a data hiding scheme for vector quantization (VQ) codebooks, enabling a sender to perform an Exclusive OR (XOR) operation on most of the pixel values in a codebook and then apply a threshold to control data embedding. During this process, the generated auxiliary information is embedded alongside secret data in the index reordering phase. In this case, the recipient can extract the data and reconstruct the VQ-compressed image using the reverse process once the stego codebook and the reordered index table are received. The proposed scheme shows a superior embedding capacity when compared to state-of-the-art schemes.

Lastly, Rath et al. (Contribution 10) presented an efficient parallel secure outsourcing of modular exponentiation to the cloud for IoT applications. The work proposes a robust solution that strengthens the Rivest–Shamir–Adleman (RSA) algorithm and mitigates the flaws identified in existing schemes. The results indicate that the scheme is resilient against existing lattice-based attacks and provides a more secure and efficient method for IoT devices to utilize the capabilities of third-party servers while ensuring data integrity and confidentiality.

3. Conclusions and Future Scope

This Special Issue presents recent advances in the design and development of advanced mathematical cryptographic schemes for information security in relation to Industry 5.0. Specifically, the Special Issue features original contributions from academics and industry experts tailored to addressing critical security issues envisioned in Industry 5.0. These contributions will facilitate innovative research in mathematical cryptography and information security, aiming to enhance the security of Industry 5.0 systems and applications for a smart society.