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ChessCrypt: enhancing wireless communication security in smart cities through dynamically generated S-Box with chess-based nonlinearity | Scientific Reports
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ChessCrypt: enhancing wireless communication security in smart cities through dynamically generated S-Box with chess-based nonlinearity | Scientific Reports

Feb 22, 2025

Scientific Reports volume 14, Article number: 28205 (2024) Cite this article

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In the contemporary landscape of smart cities, ensuring the security and confidentiality of transmitted data has become paramount. The proliferation of Internet of Things (IoT) devices, mobile communication platforms, and wireless sensor networks has magnified the need for robust cryptographic solutions to safeguard sensitive information from malicious adversaries. In response to these challenges, we introduce ChessCrypt—a novel approach to enhancing wireless communication security through the development of a new S-Box algorithm inspired by the nonlinear movement patterns of chess pieces. ChessCrypt leverages the dynamic and unpredictable nature of chess piece movements, namely knights, kings, and bishops, to introduce a high degree of nonlinearity and confusion into the encryption process. By incorporating elements of randomness and unpredictability, our proposed S-Box algorithm offers robust protection against a wide range of cyber threats, including eavesdropping, data interception, and cryptographic attacks. We demonstrate the effectiveness and resilience of ChessCrypt through rigorous testing and analysis, showcasing its superiority over traditional cryptographic methods in enhancing the security of wireless communication networks. Our results underscore the significance of ChessCrypt as a promising solution for fortifying wireless communication security in an increasingly interconnected world of ours.

Wireless communication security in smart cities has transformed the global landscape, impacting everything from IoT devices and robotics to laptops and smartphones. The pervasive nature of this connectivity presents numerous security challenges, including unauthorized access, malicious attacks, eavesdropping, and data interception. As wireless communication serves as a cornerstone of modern communication systems within smart cities, it is imperative to take robust measures to ensure the security and integrity of these networks1.

In the past, many efforts have been made to impart wireless communication security in the smart cities2. The work3, for example, discusses the emerging wireless technology called long range (LoRa). Moreover, the reported work demonstrates that the IoT technology based on the LoRa has the potential to boost the smart cities’ security. This work rendered a new backoff algorithm called as REBEB. Besides, the fairness of the reported algorithm is above the 0.4 in varied competing windows and nodes. Moreover, the work4 carried out a comprehensive survey regarding the state of the art technologies for the burgeoning era of smart cities. These technologies span from deep learning (DL), machine learning (ML), internet of things (IoT) to mobile computing, big data, blockchain, sixth-generation (6G) networks, WiFi-7, etc.

Confusion and diffusion are the two principal operations for ensuring the security of the communication systems. The application of these two operations completely morphs the precious data. In this way, the potential hacker may not understand it upon the unauthorized access. Central to these two operations lies the core constructs of substitution boxes (S-Boxes)5,6,7. S-Boxes are a great resources to introduce nonlinearity for the varied encryption processes which, in turn, boost security. Strong S-Box is a vital element against varied dynamic cyberattacks.

In the past, plethora of S-Boxes have been written based on the varied theories, constructs and other mathematical notions. They include, for instance, cyclic group8, Rabinovich-Fabrikant system9, cellular automaton10, complete latin square11, chess piece castle6 to name a few. The work8 introduced four S-Boxes by exploiting the cyclic group of residue class of noncommutative quaternion integers. Apart from that, the S-Boxes were analyzed through an array of benchmarks like bit independence criterion, differential approximation probability, nonlinearity, linear approximation probability, and strict avalanche criterion which are the avalanche effect tests. Moreover, the study9 employed Rabinovich-Fabrikant (RF) system of coupled ordinary differential equations which used third order nonlinearity generating rich chaotic and complex dynamics. The particular modus operandi of the RF system worked like this. First of all, random integers were spawned. After that, these random integers were subjected to permutation for getting highly nonlinear chaotic Substitution box (S-box). The work10 used the constructs of algebraic group structure, cellular automata theory and discrete chaotic maps. Besides, the metrics claimed by this study are 110 (minimum non-linearity), satisfaction of strict avalanche and bits independence criterions, differential uniformity as low as 6, linear approximation probability as low as 0.0703, and auto-correlation function (absolute indicator) of 40. The mathematical construct of complete latin square has also been employed by the study11 to come up with novel S-Box. The reported work generated a complete latin square through the usage of sequences of random numbers spawned by the chaotic system. Using this complete latin square, the required S-Box was created. Performance analyses rendered very promising results. Recently, the work6 developed a novel algorithm for the S-Box using the chess piece castle for the generation of nonlinearity in the S-Box. Apart from that, the five wing hyperchaotic map was employed to spawn the streams of random numbers. The simulation and security analysis proved that the S-Box is furnished with sufficient security. Here are a few scenarios in which S-Boxes’ built-in capabilities may be used to help counter cyber security threats.

Unauthorized individuals may intercept wireless communications to have an unauthorized access to the precious data12—a phenomenon known as Eavesdropping. Apart from that, adversaries and other cyber criminals have a lot of tactics at their disposal to intercept data (called as Data Interception) packets travelling over the wireless networks. Later on, they may exploit the information so obtained to materialize their nefarious designs13. Sometimes, intermediaries may position themselves on the various points of communication and intercept it (Man-in-the-Middle Attacks). Further, they may fabricate the illegally retrieved data14. On the other hand, cyber-criminals may capture data packets and replay them to gain unauthorized access or obstruct communication altogether15. This phenomenon is termed as Replay Attacks. In a yet another act called Cryptanalysis, cyber assailants may utilize advanced cryptographic methods to decrypt encoded information and jeopardize communication security16 all together. Additionally, in a cyber attack called as Key Management, encryption keys are compromised due to poor key management procedures. In this way, encrypted data becomes susceptible to decryption17. There exists an other set of attacks called as Denial of Service (DoS) Attacks. These attacks are a tactic used by hackers to interfere with wireless networks and prevent authorized users from accessing them18. Sometimes, unscrupulous insiders who have legitimate access to wireless communication systems could misuse their permissions to undermine data security19, an act dubbed as Insider Threats. Very often, attackers may intercept or obstruct data transmission by taking advantage of flaws in the physical layer of wireless communication systems20 (Physical Layer Attacks). Besides, hackers need to obtain sensitive data from wireless devices, they may take advantage of side-channel information, such as power consumption or electromagnetic radiation21—technical term characterizing this act is called as Side-Channel Attacks.

Information-Theoretic Security refers to a level of security in cryptographic systems where the encryption scheme is mathematically proven to be secure, even if an adversary has infinite computational power. It guarantees that no amount of computing resources or advanced algorithms can break the encryption or reveal any meaningful information from the ciphertext, except by brute force guessing of the secret key (if it’s very short). The security is based solely on the information content, not on the computational difficulty of certain problems, which is typical in most modern cryptographic systems. Many efforts have been made to achieve this security,22,23,24,25, for instance.

Inspired by the above discussion, in particular the work6, this study ventures to craft a yet another S-Box by the clever engineering of the constructs of chess pieces and 5D hyperchaotic map. The chess pieces being employed are knight, king and bishop.

This work introduces a novel S-Box algorithm inspired by the dynamic movement patterns of chess pieces to enhance wireless communication security in smart cities.

Nonlinear movement of the knight in chess, enhances non-linearity and resistance against diverse cyber attacks. The utilization of the knight’s unique movement pattern represents a significant advancement in S-Box construction, contributing to unprecedented security measures in the field.

Three related theories will be employed in this study. As the name suggests, these theories would yield the necessary support for the construction of new S-Box algorithm. These three theories include the chess game, 5D multi-wing hyperchaotic system and wireless communication security in smart cities.

Chessboard and movements of its pieces king, knight and bishop.

Chess is a classical game whom normally two persons play with each other using a board26. Figure 1a depicts the board along with the pieces. An integer range \(\{1, 2, 3, 4, 5, 6, 7, 8\}\) acts as a label for both the columns and rows of the game. Each player contains 16 pieces totalling to 32 pieces. Pawn, knight, bishop, castle (aka rook), queen and king are the names of these pieces. Each piece/player contains its unique movement in the game. Queen and pawn are thought of as the most potent and the weakest pieces respectively. Besides, the colors of pieces are different for each player. Figure 1a shows a particular instance of the game with white colored players lying on the rows 1 and 2. In the same way, black colored players are lying on seventh and eighth rows. Each piece occupies a distinct address on the chessboard. The initial address of white king is (5, 1), for example. In this study, three pieces of king, knight and bishop have been used to heighten the element of non-linearity in suggested S-Box. Figure 1b–d show all the possible moves of the chess pieces king, knight and bishop respectively. Besides, all the three chosen pieces have been depicted in the single Fig. 1e.

Theory of chaos contends that faintest alteration in system parameters and initial values results in a phenomenal and a marked change in ensuing output27. In compliance with this notion, researchers have come up with too many chaotic systems and maps. The mentioned maps and systems are furnished with nice features of mixing, ergodicity, randomness, aperiodicity, and unpredictability28. These systems and maps are normally categorized as 2-dimensional, 1-dimensional, and higher dimensional. To spawn the streams of random data, these maps have been ubiquitously employed by the cryptographers. Once these random numbers are generated, they are used to conduct the two principal operations of substitution and permutation.

The current research endeavor has chosen this map29. This map is characterized through the following set of equations.

where the sets of values \(a_1, a_2, a_3, a_4, a_5, a_6, a_7, a_8\) and p, q, r, s, t correspond to system’s parameters and initial values of map in a respective way. Moreover, qr, pq and \(p^{2}q\) are nonlinear terms of chaotic system. Apart from that, the study30 describes the features like periodic orbit etc of the chaotic map. Step of time for system’s solution has been kept as 0.001 for drawing attractors of 5D multi-wing hyperchaotic map. Additionally, Fig. 2 exhibits system (1)’s behavior of chaoticity. Moreover, Lyapunov exponents of the chaotic map under consideration are \(\{L_{1}, L_{2}, L_{3}, L_{4}, L_{5} = 9.979, 1.96, 0.005362, -19.13, -27.82 \}\) (Fig. 3).

Attractors (projection) of the system (1) in different planes and spaces: (a) Plane of pq ; (b) Plane of pr; (c) Plane of pt; (d) Plane of qt; (e) Plane of rs; (f) Plane of rt; (g) Space of 3D prt; (h) Space of 3D pqr.

System 1’s Lyapunov exponents.

In the complex ecosystem of a smart city, where the inter-connectivity of devices and services is pivotal, the security of wireless communications emerges as a cornerstone for ensuring the integrity and confidentiality of transmitted data. The novel S-Box, particularly generated with unique properties like chess-based non-linearity, presents a promising solution to the challenges posed by potential cyber threats. This innovative S-Box enhances cryptographic measures by leveraging the unpredictable and complex movement patterns of chess pieces to introduce high levels of non-linearity and confusion in the encryption process. This approach not only mitigates risks such as eavesdropping and data manipulation but also fortifies the encryption against more sophisticated cryptographic attacks, ensuring a robust defense mechanism for the secure exchange of information across a smart city’s wireless networks.

Smart cities leverage advanced technologies to improve urban infrastructure, services, and quality of life. Key components include smart schools, hospitals, markets, libraries, police stations, and emergency services, all of which rely heavily on wireless communication for data transmission and connectivity. However, this dependence on wireless networks introduces significant security challenges, such as eavesdropping, data breaches, and unauthorized access.

Data Integrity and Privacy: Ensuring that data transmitted over wireless networks remains unaltered and confidential31.

Authentication and Access Control: Verifying the identity of users and devices to prevent unauthorized access32.

Resistance to Attacks: Protecting against various attacks, including eavesdropping, man-in-the-middle attacks, and denial-of-service attacks33.

Wired Equivalent Privacy (WEP) is known due to varied security weaknesses34. Besides, it uses RC4 stream cipher to encrypt the data. Additionally, this encryption is compromised due to intrinsic loopholes in the algorithm and its implementation. Apart from that, major issue which plagues the WEP is the usage of static encryption key distributed among its users. This exposes the network to the key recovery attack.

Introduced as an improvement over WEP, Wi-Fi Protected Access (WPA) incorporates the Temporal Key Integrity Protocol (TKIP) to rectify some of the vulnerabilities present in the WEP35. Inspite of this, WPA shows some lacunas in its design principle. Moreover, TKIP is not impervious to varied attacks like the Michael attack which exploits the weaknesses in its integrity check. WPA’s design for backward compatibility with WEP meant that some security flaws from the earlier protocol were retained. Additionally, early implementations of WPA had their own security issues, which could be exploited to compromise network integrity.

By introducing the Advanced Encryption Standard (AES) for encryption, Wi-Fi Protected Access II (WPA2) further improved the security36. In this way, WPA2 is significantly stronger than TKIP. Unluckily, WPA2 has its own issues. An attack called as Key Reinstallation Attack (KRACK) in 2017 showed the varied loopholes in the WPA2 handshake process. In this way, it allows hackers to intercept and decrypt traffic. Additionally, WPA2’s security depends on the strength of the password used. The weak passwords can be vulnerable to brute-force attacks.

Wi-Fi Protected Access III (WPA3) introduces newer advancements in the Wi-Fi security37. Moreover, WPA3 is furnished with enhanced protection and stronger encryption against brute-force attacks. In contrast to that, initial deployments faced some issues like vulnerabilities in early firmware versions. WPA3 includes a transition mode to support devices only compatible with WPA2, which can potentially introduce vulnerabilities if not properly managed. The requirement for updated hardware and software to fully support WPA3 features has also posed challenges, as partial support can result in weaker security.

S-Boxes (Substitution Boxes) are critical components in many encryption algorithms, including the Advanced Encryption Standard (AES). In the context of smart cities, S-Boxes must meet specific requirements to address the unique challenges posed by wireless communication. For example, they should provide High Non-Linearity38 and confusion, which are essential for creating secure cryptographic systems. Besides, they must render Low Differential Uniformity to minimize the probability of differential attacks39. Additionally, they are supposed to resist algebraic attacks40—a phenomenon called as Strong Algebraic Complexity.

The standard AES S-Box is vulnerable to certain cryptographic attacks, such as differential and linear cryptanalysis, due to its algebraic structure, which can be exploited to find weaknesses in the encryption process41.

Our proposed S-Box addresses the limitations of the standard AES S-Box by introducing a dynamic and adaptable design:

Dynamic Non-Linearity: Using the random movement of chess pieces combined with chaotic maps generates non-linear transformations that are difficult to predict or reverse-engineer. Besides, just changing the initial values and the system parameters, we can get more S-Boxes.

Adaptive Structure: The S-Box can be dynamically adjusted based on real-time security requirements, providing enhanced flexibility and resilience against evolving threats.

Efficient Implementation: Designed to be lightweight, ensuring minimal performance overhead and suitability for resource-constrained devices.

Our novel S-box is designed to enhance the security of wireless communication systems, which are integral to the infrastructure of smart cities. Figure 4 illustrates various smart city components such as smart schools, helicopters, Wi-Fi networks, libraries, and police stations etc., all of which rely heavily on secure wireless communication.

Wireless communication security in a typical smart city.

Smart Schools and Libraries: These institutions require secure wireless networks for data transmission, including sensitive information about students and educational resources42. Our S-box enhances the encryption algorithms used in these networks, ensuring data integrity and privacy.

Helicopters and Emergency Services: Secure communication is crucial for coordinating emergency responses43. Integrating the S-Box into encryption protocols for communication systems used by police stations and medical services protects against eavesdropping and unauthorized access.

Public Wi-Fi Networks: Common in smart cities and susceptible to various attacks44, public Wi-Fi networks benefit from the improved encryption robustness provided by the S-Box, which safeguards user data and maintains service reliability.

Markets and Smart Buildings: These environments involve numerous IoT devices and sensors communicating wirelessly to manage resources efficiently45. The S-Box ensures secure data transmission between devices, preventing unauthorized access and data breaches.

Fire Stations: Secure and reliable communication is essential for coordinating emergency responses46. The S-Box enhances the security of communication channels, ensuring that emergency response information remains confidential and tamper-proof.

Stadiums: Large public venues like stadiums rely on secure wireless communication for operations and public safety47. The S-Box provides the necessary encryption to protect communications from potential threats.

Hospitals: Medical facilities depend on secure wireless networks to transmit patient data, communicate with medical devices, and coordinate care48. The S-Box strengthens the encryption of these communications, protecting sensitive medical information from unauthorized access.

In this section, algorithms of key stream generation and construction of S-Box will be discussed in detail.

In this subsection, we explain the way, random numbers’ key streams have been generated for the construction of S-Box. Ignite the Chaotic Map (1) along with values: \(p = 1.0, q = 1.0, r = 1.0, s = 1.0, t = 1.0, a_1 = 10.0,\)\(a_2 = 60.0, a_3 = 20.0, a_4 = 15.0, a_5 = 40.0, a_6 = 1.0, a_7 = 50.0, a_8 = 10.0\). The streams rendered by the System (1) are very raw. They must be customized to make them useful. For this purpose, Algorithm 1 (Customizing Key Streams) is being invoked for the construction of new key streams with tuple \(\{p, q, r, s, t, \Psi \}\) of parameters, where \(\Psi\) corresponds to the number of elements in the proposed S-Box. for loop (Line 1) is iterating for \(\Psi ^2\) times. Five new streams \(\{walk_t\}_{t=1}^{\Psi }\), \(\{maneuver_t\}_{t=1}^{\Psi }\), \(\{move_t\}_{t=1}^{\Psi }\), \(\{direct_t\}_{t=1}^{\Psi }\) and \(\{selector_t\}_{t=1}^{\Psi }\) have been introduced at lines numbering 2, 3, 4, 5 and 6 respectively. These five streams contain integers in ranges of \([1, 2,\ldots , 4]\), \([1, 2, \ldots , 8]\),\([-(\sqrt{\Psi }-1), \ldots , (\sqrt{\Psi }-1)]\),[1, 2] and [1, 2, 3] respectively. Symbol \(\lfloor .\rfloor\) refers to floor function, where as operator mod(a, b) returns the remainder when a is divided by b.

Customizing Key Streams

This research study has unlocked the dormant potential of random walk of three chess pieces for the construction of an S-Box as already described. Figure 5 shows the flowchart for the suggested algorithm of the S-Box.

Here, we explore the way, knight moves on the chessboard and how this move can be imported in the construction of the proposed box.

Proposed S-Box scheme.

Only knight moves non-linearly among all the pieces of chess. This non-linearity bears great implications for the enterprise of cryptography. As far as the move of knight is concerned, it moves two squares horizontally or vertically followed by one move either to the left or to the right as depicted in the Fig. 6a. Colloquially speaking, sometimes, its move is dubbed as an L shaped move. Figure 6a depicts all the possible eight moves of this chess piece. We have named them as maneuver1, maneuver2,maneuver3,maneuver4,maneuver5,maneuver6, maneuver7, and maneuver8. Shortly, these moves will be formalized through the eight definitions. In addition, because a knight has less options for moves when it gets closer to the board’s edge, we have designed the knight to appear on the other edge of the board in the event that a particular move option fails. This deviation from the chess rules would yield greater security to the potential S-Box. To demonstrate the positioning of a knight on opposite edges of a chessboard, Fig. 6 presents six examples of \(8 \times 8\) chessboards, each illustrating the eight possible knight moves. For example, in Fig. 6b, the knight attempting the move maneuver1, finds itself at a position two places below the position (2, 1), therefore it has nowhere to land. This is how we have responded to the situation. There is a “reflection” at position (2, 7), two positions down from position (2, 1). Knight has so been forced to take this role. Similarly, when the knight endeavors to execute the maneuver2 move, as depicted in Fig. 6b, it encounters a lack of landing space. This is because it arrives at a position one square to the left and two squares below its starting point (1, 1). Consequently, after two consecutive reflections, it will settle in the position (8, 7). With first reflection, it reaches two positions below the position (8, 1) and with second reflection it lands at (8, 7).

Knight will land on row 8 if it travels one position below row 1. Knight will land at row 1 if it travels one position above row 8. In a similar vein, the knight will arrive on row 7 if it travels two places below row 1. Knight will land on row 2 if it travels two positions above row 8. Columns have been handled using the same method.

Similarly, we have addressed the other knight moves. Specifically, we have focused on the cyclic moves of the knight near a single corner, namely (1, 1). Analogously, one can comprehend the knight’s behavior near the other corners.

We now provide formal definitions for each of the eight knight moves. These definitions serve to delineate all potential knight moves. Each move is defined with respect to the initial position of the knight, denoted as (p, q), on a chessboard of size \(\sqrt{\Psi } \times \sqrt{\Psi }\).

Illustration of the eight normal and cyclic moves of a knight on an \(8 \times 8\) chessboard from different starting positions: (a) all eight natural moves when the knight is initially positioned at (4, 3); (b) two normal and six cyclic moves when the knight starts at position (1, 1); (c) three normal and five cyclic moves when the knight starts at position (2, 1); (d) four normal and four cyclic moves when the knight starts at position (3, 1); (e) four normal and four cyclic moves when the knight starts at position (2, 2); (f) six normal and two cyclic moves when the knight starts at position (3, 2).

According to the Fig. 6a, knight can move to any of the eight addresses. So, following definitions characterize all the eight moves of knight. Every definition is taking the three parameters p, q and \(\Psi\). p, q corresponds to the current address of the knight and \(\Psi\) to the square of the side of the proposed S-Box.

The move of knight is declared as \(maneuver_1(p,q,\Psi )\) if it lands on the address (\(1, q-2\)) if (\(p \leftarrow \sqrt{\Psi }\) && \(q > 2\)) || (\(1, \sqrt{\Psi }\)) if (\(p \leftarrow \sqrt{\Psi }\) && \(q \leftarrow 2\)) || (\(1, \sqrt{\Psi }-1\)) if (\(p \leftarrow \sqrt{\Psi }\) && \(q \leftarrow 1\)) || (\(\sqrt{\Psi }, \sqrt{\Psi }\)) if (\(p \leftarrow \sqrt{\Psi }-1\) && \(q \leftarrow 2\)) || (\(\sqrt{\Psi }, \sqrt{\Psi }-1\)) if (\(p \leftarrow \sqrt{\Psi }-1\) && \(q \leftarrow 1\)) || (\(p+1, \sqrt{\Psi }-1\)) if (\(p \le \sqrt{\Psi }-2\) && \(q \leftarrow 1\)) || (\(p+1, \sqrt{\Psi }\)) if (\(p \le \sqrt{\Psi }-2\) & \(q \leftarrow 2\)) else (\(p+1, q-2\)).

The move of knight is declared as \(maneuver_2(p,q,\Psi )\) if it lands on the address (\(\sqrt{\Psi }, q-2\)) if (\(p \leftarrow 1\) && \(q > 2\)) || (\(\sqrt{\Psi }, \sqrt{\Psi }\)) if (\(p \leftarrow 1\) && \(q \leftarrow 2\)) || (\(\sqrt{\Psi }, \sqrt{\Psi }-1\)) if (\(p \leftarrow 1\) && \(q \leftarrow 1\)) || (\(1, \sqrt{\Psi }\)) if (\(p \leftarrow 2\) && \(q \leftarrow 2\)) || (\(1, \sqrt{\Psi }-1\)) if (\(p \leftarrow 2\) && \(q \leftarrow 1\)) || (\(p-1, \sqrt{\Psi }-1\)) if (\(p > 2\) && \(q \leftarrow 1\)) || (\(p-1, \sqrt{\Psi }\)) if (\(p > 2\) && \(q \leftarrow 2\)) else (\(p-1, q-2\)).

The move of knight is declared as \(maneuver_3(p,q,\Psi )\) if it lands on the address (\(\sqrt{\Psi }-1, q-1\)) if (\(p \leftarrow 1\) && \(q > 1\)) || (\(\sqrt{\Psi }-1, \sqrt{\Psi }\)) if (\(p \leftarrow 1\) && \(q \leftarrow 1\)) || (\(\sqrt{\Psi }, q-1\)) if (\(p \leftarrow 2\) && \(q > 1\)) || (\(\sqrt{\Psi }, \sqrt{\Psi }\)) if (\(p \leftarrow 2\) && \(q \leftarrow 1\)) || (\(p-2, \sqrt{\Psi }\)) if (\(p > 2\) && \(q \leftarrow 1\)) else (\(p-2, q-1\)).

The move of knight is declared as \(maneuver_4(p,q,\Psi )\) if it lands on the address (\(\sqrt{\Psi }-1, q+1\)) if (\(p \leftarrow 1\) && \(q \le \sqrt{\Psi }-1\)) || (\(\sqrt{\Psi }-1, 1\)) if (\(p \leftarrow 1\) && \(q \leftarrow \sqrt{\Psi }\)) || (\(\sqrt{\Psi }, q+1\)) if (\(p \leftarrow 2\) && \(q \le \sqrt{\Psi }-1\)) || (\(\sqrt{\Psi }, 1\)) if (\(p \leftarrow 2\) && \(q \leftarrow \sqrt{\Psi }\)) || (\(p-2, 1\)) if (\(p>2\) && \(q \leftarrow \sqrt{\Psi }\)) else (\(p-2, q+1\)).

The move of knight is declared as \(maneuver_5(p,q,\Psi )\) if it lands on the address (\(\sqrt{\Psi }, q+2\)) if (\(p \leftarrow 1\) && \(q \le \sqrt{\Psi }-2\)) || (\(\sqrt{\Psi }, 1\)) if (\(p \leftarrow 1\) && \(q \leftarrow \sqrt{\Psi }-1\)) || (\(\sqrt{\Psi }, 2\)) if (\(p \leftarrow 1\) && \(q \leftarrow \sqrt{\Psi }\)) || (1, 1) if (\(p \leftarrow 2\) && \(q \leftarrow \sqrt{\Psi }-1\)) || (1, 2) if (\(p \leftarrow 2\) && \(q \leftarrow \sqrt{\Psi }\)) || (\(p-1, 2\)) if (\(p > 2\) && \(q \leftarrow \sqrt{\Psi }\)) || (\(p-1, 1\)) if (\(p > 2\) && \(q \leftarrow \sqrt{\Psi }-1\)) else (\(p-1, q+2\)).

The move of knight is declared as \(maneuver_6(p,q,\Psi )\) if it lands on the address (\(1, q+2\)) if (\(p \leftarrow \sqrt{\Psi }\) && \(q \le \sqrt{\Psi }-2\)) || (1, 1) if (\(p \leftarrow \sqrt{\Psi }\) && \(q \leftarrow \sqrt{\Psi }-1\)) || (1, 2) if (\(p \leftarrow \sqrt{\Psi }\) && \(q \leftarrow \sqrt{\Psi }\)) || (\(\sqrt{\Psi }, 1\)) if (\(p \leftarrow \sqrt{\Psi }-1\) && \(q \leftarrow \sqrt{\Psi }-1\)) || (\(\sqrt{\Psi }, 2\)) if (\(p \leftarrow \sqrt{\Psi }-1\) && \(q \leftarrow \sqrt{\Psi }\)) || (\(p+1, 2\)) if (\(p \le \sqrt{\Psi }-2\) && \(q \leftarrow \sqrt{\Psi }\)) || (\(p+1, 1\)) if (\(p \le \sqrt{\Psi }-2\) && \(q \leftarrow \sqrt{\Psi }-1\)) else (\(p+1, q+2\)).

The move of knight is declared as \(maneuver_7(p,q,\Psi )\) if it lands on the address (\(2, q+1\)) if (\(p \leftarrow \sqrt{\Psi }\) && \(q \le \sqrt{\Psi }-1\)) || (2, 1) if (\(p \leftarrow \sqrt{\Psi }\) && \(q \leftarrow \sqrt{\Psi }\)) || (\(1, q+1\)) if (\(p \leftarrow \sqrt{\Psi }-1\) && \(q \le \sqrt{\Psi }-1\)) || (1, 1) if (\(p \leftarrow \sqrt{\Psi }-1\) && \(q \leftarrow \sqrt{\Psi }\)) || (\(p+2, 1\)) if (\(p \le \sqrt{\Psi }-2\) && \(q \leftarrow \sqrt{\Psi }\)) else (\(p+2, q+1\)).

The move of knight is declared as \(maneuver_8(p,q,\Psi )\) if it lands on the address (\(2, q-1\)) if (\(p \leftarrow \sqrt{\Psi }\) && \(q > 1\)) || (\(2, \sqrt{\Psi }\)) if (\(p \leftarrow \sqrt{\Psi }\) && \(q \leftarrow 1\)) || (\(1, q-1\)) if (\(p \leftarrow \sqrt{\Psi }-1\) && \(q > 1\)) || (\(1, \sqrt{\Psi }\)) if (\(p \leftarrow \sqrt{\Psi }-1\) && \(q \leftarrow 1\)) || (\(p+2, \sqrt{\Psi }\)) if (\(p \le \sqrt{\Psi }-2\) && \(q \leftarrow 1\)) else (\(p+2, q-1\)).

The operators && and || denotes the logical and and logical or operators respectively.

It is to be noted that, if knight happens to be near to any of the four edges of the board, the number of its moves has been made intact by letting it to appear from the oppose side of the board. In such cases, no address is found for a particular move. This act would boost the non-linearity of the S-Box.

Call the Algorithm 2 with the following list of parameters.

The first three pairs denote the initial addresses of the king, knight and bishop on the chessboard. Besides, the key streams walk, maneuver, move refer to the random numbers which would govern the different moves of the chess pieces on the board. Additionally, the variable direct is Boolean and it controls the way, the Bishop moves diagonally or anti-diagonally on the board. Apart from that, the variable selector selects any one pairing out of the total three pairs of the chess pieces, i.e., (king, knight), (knight, bishop) and (bishop, king). Lastly, the variable \(\Psi\) denotes the total number of elements in the S-Box as described earlier.

Non-Linearity Based S-Box Maker

The output of the Algorithm 2 is a \(\sqrt{\Psi } \times \sqrt{\Psi }\) S-Box containing the integers \(0, 1, 2, \ldots , \Psi - 1\) in a scrambled way. As the algorithm is sparked, an S-Box variable say sbox is initialized with the integers \(0, 1, 2, \ldots , \Psi -1\). Moreover, any two arbitrary chess pieces out of the three chess pieces king, knight and bishop are selected randomly. Further, these two chess pieces walk on the hypothetical chessboard according to the chess rules and the Fig. 1e. As they occupy the new addresses, the corresponding integers in the S-Box sbox are swapped with each other (Lines 9, 15 and 21, for example). This process has been iterated \(\Psi ^2\) times to embed the non-linearity in the suggested S-Box. Here we explain the Algorithm 2.

Line 1 initializes the variable sbox with the values \(0, 1, 2, \ldots , \Psi -1\). Moreover, the line 2 reshapes the sbox to the size of \(\sqrt{\Psi } \times \sqrt{\Psi }\). Step 2: The for loop at line 3 iterates for \(\Psi ^2\) times. In each \(k^{th}\) iteration, the value of random number from the array selector has been given to the index of the swtich statement.

The corresponding case statements have been fired at the lines 6, 12 and 18. The statements against these cases are self-explanatory. For example, if case 1 gets fired, two algorithms of Address of \(king(walk_k, \{king_x, king_y\},\) \(\Psi\)) and Address of \(knight(maneuver_k, \{knight_x, knight_y\},\) \(\Psi\)) have been invoked. Both of these algorithms return the new addresses of the pieces king and knight in the pairs \(\{p_1,q_1\}\) and \(\{p_2,q_2\}\) respectively.

The statement at line 9 swaps the integers of the sbox at the addresses \((p_1, q_1)\) and \((p_2, q_2)\). Analogously, other cases deal with the moves of the other pairs of pieces. Lastly, the statement at line 25 returns the S-Box sbox.

Besides, the Algorithm 3 uses the chess piece king. Its parameters are walk, \(\{king_x, king_y\}\), \(\Psi\). The pair \(\{king_x, king_y\}\) refers to the initial address of the king. Here we explain its working.

Line 1 defines the three static variables p, q, flag. The reason for taking the static variables is that we need to maintain the previous addresses of the pieces for all the invocations of these algorithms.

Lines 2 - 7 will be executed only one time. Lines 3 and 4 are assigning the given address \(\{king_x, king_y\}\) to the pair of variables p, q. Line 6 returns the current address to the calling program.

Lines 8 - 19 update the address of king depending upon the value of walk. Ternary operator has been employed. Only four moves of the king have been entertained. In case, king happens to be at one of the edges of the board, it has been made to appear from the opposite side of the edge. The peculiar anatomy of these ternary conditions vividly indicates this logic. The value of \(\sqrt{\Psi }\) indicates the side of the S-Box. Line 20 returns the new address of the king.

Address of King

Algorithm 4 finds the new address of the piece bishop. Its parameters are move, \(\{bishop_x, bishop_y\}\), direction, \(\Psi\). The pair \(\{bishop_x, bishop_y\}\) contains the initial address of this piece. If the value of variable direction is 1, the Algorithm 5 has been called, otherwise the Algorithm 6 has been invoked.

Address of Bishop

Address of Bishop in Diagonal Direction

Address of Bishop in Anti-Diagonal Direction

Figure 7 explains the way, these two algorithms for the move of bishop have been designed. Moreover, the variables length and actualdist denote the length of the diagonal formed and the actual distance, the bishop has to travel. Lines 64 and 62 return the new addresses to the calling algorithms.

Demo of starting and ending positions of the bishop in Diagonal and Anti-Diagonal directions for \(\Psi = 64\): (a) (2, 4), (5,7), \(move = 3\), Diagonal direction; (b) (2, 4), (1,3), \(move = 5\), Diagonal direction; (c) (5, 5), (2,8), \(move = 3\), Anti-Diagonal direction; (d) (5, 5), (7,3), \(move = 5\), Anti-Diagonal direction.

Address of Knight

Lastly, the Algorithm 7 returns the address of the knight to the calling algorithm depending upon the subscript t of the definition \(maneuver_t\).

In a smart hospital environment, the secure transmission of sensitive medical data is crucial. Consider a medical image p generated in a smart laboratory equipped with advanced imaging technologies. This image must be transmitted securely to the concerned doctor in the hospital without any risk of interception or tampering. Medical images, such as MRI scans or X-rays, contain highly sensitive patient information. Unauthorized access or tampering can lead to severe privacy breaches and misdiagnosis, making robust encryption essential. Traditional encryption methods might not be sufficient to protect against sophisticated attacks, especially in a wireless communication setup. To address these challenges, our novel S-Box can be integrated into the encryption algorithm used for securing the transmission of medical images. The proposed S-Box enhances the security by introducing dynamic non-linearity and adaptive structure, which are crucial for resisting various cryptanalytic attacks. Invoke Algorithm 8 to encrypt the medical image p using our novel S-Box. This algorithm ensures that the image is securely encrypted before transmission. The encrypted image is then transmitted over the hospital’s secure network to the doctor. Figure 8 demonstrates the workability of the encryption process, showing how the medical image is effectively encrypted and secured.

Algorithm 9 is about the decryption algorithm. Since the private key cryptography has been employed, so just reversal of the steps of the encryption algorithm renders its decryption counterpart. Moreover, Fig. 9 elucidates the encryption and decryption processes by involving the concerned stakeholders.

Securing a Medical Image through Encryption

Decryption algorithm

This sophisticated approach to encrypting medical images demonstrates the practical applicability of our novel S-Box in real-world scenarios, ensuring the secure transmission of sensitive data in a smart hospital environment.

A medical image, its scrambling and unscrambling through the novel S-Box: (a) medical plaintext image; (b) scrambled image by using the novel S-Box; (c) cipher medical image; (d) unencrypted/original image.

Proposed security apparatus of the medical image.

Cryptanalysis savvy have plethora of methods for cracking the ciphers. From these methods, chosen plaintext, known plaintext and ciphertext only attacks are routinely employed by them49. Here, their approach would be discussed. In the scenarios where ciphertext only attack is chosen by the hackers, they arrange few ciphertexts. In the scenarios of known plaintext attacks, these hackers arrange pairs of plaintexts and corresponding ciphertexts. Additionally, in the scenarios of chosen plaintext attack, they have encryption paraphernalia on their disposal. In this method, they can have as many ciphertexts as they desire. In case, if it is shown that proposed image cipher defies the chosen plaintext attack, then defiance of known plaintext only and ciphertext only attacks are assured in a straightforward manner. Its reason is that ciphertext only and known plaintext attacks comprise a sort of “subset” of chosen plaintext attack.

For initiating the chosen plaintext attack, hackers and other adversaries may choose a special image, like Black image (Fig. 10a) for encryption and later on tracing secret key which was used. The moment, Black image is encrypted (Fig. 10b), random numbers employed in encryption scheme are retrieved. Later on, hackers may choose other plaintext image say the medical image and encrypt it (Fig. 10c) by launching known plaintext attack, along with key employed in encryption operation of Black image. The hacker in question, would be disappointed. Its reason is that cipher image of medical image could not be decrypted using the key (Fig. 10d). In the same way, steps were taken upon some other special image White (Fig. 10e–h). This experiment indicates the inherent defiance of the proposed image cipher against chosen plaintext, known plaintext and ciphertext only attacks.

Using black and white images, demo of defiance of chosen plaintext attack: (a) black image; (b) cipher black image; (c) Cipher medical image; (d) decrypted medical image with possible secret key from the black image; (e) white image; (f) cipher white image; (g) cipher medical image; (h) decrypted medical image with possible secret key from white image.

In this section, the proposed S-Box would be simulated and its security analysis will be carried out. We have taken the value of \(\Psi = 256\) for simulation. The resulting S-Box can be seen in the Fig. 11 with hexadecimal integers. One can observe that the numbers 0 to FF have been scrambled sufficiently in the required \(16 \times 16\) S-Box.

S-box generated using the proposed algorithm—ChessCrypt.

Merely creating cryptographic goods is insufficient; instead, they need to meet the most recent standards, benchmarks, and evaluation standards decided upon by analysts, security professionals, and cryptographers. Using these benchmarks, we will show in this section the defiance and robustness of suggested S-Box. These benchmarks consist of differential probability (DP), bijectivity, non-linearity (NL), Bit Independence Criterion (BIC), Strict Avalanche Criterion (SAC), and linear probability (LP)6.

Occasionally, potential hackers target security goods with linear cryptanalysis assaults50. The created S-Boxes need to have a significant amount of non-linearity encoded in order to counter this hazard. Should a linear mapping exist between the plaintext and the ciphertext, then adversaries and other opponents may be able to attack the S-Box that is being used.

Mathematical equation (2) is normally used to assess the intrinsic non-linearity of any Boolean function b(k) (n-bit)51.

In the equation provided, \(WS_b(H)\) denotes Walsh spectrum of some function b. Additionally, subsequent mathematical equation aids in determining non-linearity of Boolean function b(k).

In the above equation \(H \in \{0,1\}^n\). Additionally, the dot product of H and X is denoted as H.X. This dot product can be found as

Non-linearity values for suggested S-Box are: 104, 105, 103, 109, 110, 104, 106, and 105. Among these values, the minimum, maximum, and average values are 103, 110, and 105.75, respectively. Furthermore, Table 1 displays non-linearity values for all 8 constituent Boolean functions.

In general, an S-Box of size \(n \times n\) is considered bijective if it encompasses \(2^{n-1}\) unique integers52. Moreover, these integers are confined within the range of \([0, 2^{n-1}]\). Figure 11 demonstrably satisfies this bijectivity criterion.

This is one of the frequently employed criteria to test the robustness of some S-Box. It works like this. Upon changing one input bit n, there is 50% chance that ensuing output bit m will also alter in order to meet the rigorous avalanche requirement51. Stated otherwise, S-Box in question is equipped with an adequate degree of chaoticity and randomness if its SAC value approaches 0.5. Additionally, the calculated values of recommended S-Box are displayed in Table 2. Another name for this matrix is dependence matrix. Additionally, suggested S-Box’s average SAC value of 0.5082 satisfies the set standard.

The cryptographers utilize this additional standard to assess the stability of their product. According to this benchmark, S-Box is deemed effective in segregating the output bits from one another if a modification in an input bit, such as q, results in distinct modification of s and r—the output bits51.

To meet this requirement, the constituent Boolean functions of the S-Box must adhere to the conditionality of non-linearity. The expression \((T_a[p] \oplus T_b[q]) - (T_a[p] \oplus T_b[p])\) is computed across the entire range of input values for p, ranging from 0 to 255, where T represents the S-Box. This calculation serves to evaluate the BIC-SAC performance of an S-Box.

It is important to note that q and p differ by exactly one bit in this evaluation. Moreover, the average BIC-SAC values computed across all input values define the effectiveness of an S-Box; values close to 0.5 suggest ideal performance. The criteria used to assess the non-linearity and SAC for the component Boolean functions inside the suggested S-Box are shown in Tables 3 and 4.

The BIC property is considered to be fulfilled by an S-box that satisfies the non-linearity and SAC criteria, as per the study findings of Carlisle and Stafford53. The figures of 103.125 and 0.5058 in the case of the suggested S-box indicate a noticeably poor linear relationship between the output bits. These results clearly confirm that the proposed S-box fulfills the BIC property.

The concept of linear probability is utilized to determine the intrinsic correlation between the input and output of an S-Box5. A lower LP value is desirable for a robust S-Box. In the case of the proposed S-Box, Eq. (5) yields a maximum LP value of 0.1320. This suggests that the S-Box is strong enough to withstand linear cryptanalysis attempts.

Within this equation, T, \(a_z\), and \(b_z\) symbolize the S-Box, input mask, and output mask, respectively. Moreover, the variable N denotes a collection of integers ranging from 0 to 255.

By examining the variations between the pairs of ciphertexts and the matching plaintexts, the original plaintext is attempted to be recovered from the provided ciphertext in this cryptanalysis54. Potential hackers can access the secret key by examining these discrepancies. This statistic should have a comparatively lower value for the robust S-Box. Equation (6) is used to find the differential probability.

The data presented in Table 5 reveals that the proposed S-box exhibits a differential probability of \(12/256 = 0.0469\). This outcome indicates a strong resistance against various forms of differential cryptanalysis attacks. Equation (6) below specifies computation of differential probability (DP):

The input and output differentials are denoted by \(\triangle z\) and \(\triangle y\) in this equation, respectively.

In this study, we introduced ChessCrypt, a novel approach to enhancing wireless communication security in smart cities through the development of a new S-Box algorithm. By leveraging the dynamic and unpredictable nature of chess piece movements, namely knights, kings, and bishops, ChessCrypt introduces a high degree of non-linearity and confusion into the encryption process. Our results demonstrate the effectiveness and resilience of ChessCrypt in mitigating common vulnerabilities of cybersecurity, including eavesdropping, data interception, and cryptographic attacks.

One of the key contributions of ChessCrypt lies in its emphasis on non-linearity as a fundamental aspect of cryptographic strength. Traditional cryptographic methods often rely on linear transformations and algorithms, which may be susceptible to attacks that exploit linear relationships in the data. ChessCrypt, on the other hand, takes advantage of the chess piece’s intrinsic non-linearity (particularly of knight) to add complexity and unpredictable movements to the encryption process. By using this method, the cryptographic system becomes more resilient to different types of attacks, such as algebraic, differential, and brute force attacks.

We evaluate ChessCrypt experimentally using bit independence criterion, strict avalanche criterion, bijectivity, non-linearity, linear probability, and differential probability as validation metrics. These metrics show how reliable and strong ChessCrypt is at improving wireless communication security, and their results are very encouraging. ChessCrypt exhibits significant non-linearity, which highlights its potential as a viable solution for strengthening wireless communication networks in smart cities against constantly evolving cyber threats.

In summary, the introduction of ChessCrypt marks a substantial progression in wireless communication security, harnessing nonlinear transformations inspired by the erratic movements of chess piece knight. Future research avenues could delve into additional refinements of ChessCrypt and its utilization across diverse cybersecurity domains, thereby perpetuating the advancement and refinement of cryptographic methodologies in our ever-expanding interconnected landscape. One can see all the results through a panoramic view in the Table 6. Besides, results of the published works have also been written in this table. One can see that the results of the proposed work are comparable to those of published ones.

Our research represents a significant contribution to the field of wireless communication security, offering a novel and effective solution for protecting sensitive data transmitted over wireless channels in smart cities. Through the development and evaluation of our proposed S-Box algorithm, inspired by the nonlinear movement patterns of chess pieces, we have demonstrated its efficacy in enhancing the resilience of wireless communication networks against evolving cyber threats. In order to reduce cyber risks, our research first identified the fundamental weaknesses of wireless communication networks and the urgent need for strong security measures. After realizing how important S-Boxes are to cryptographic systems, we set out to create a brand-new S-Box algorithm that is especially suited to the difficulties presented by contemporary wireless networks. Conclusively, our study constitutes a noteworthy progression in the domain of wireless communication security, providing an innovative and efficacious resolution for safeguarding confidential information transported via wireless channels.

Despite the significant contributions of our research to enhancing wireless communication security through the novel S-Box algorithm, several challenges and limitations must be acknowledged. The primary challenge we faced, was the complexity in designing and implementing the S-Box algorithm in the smart hospital setting. The integration of our proposed algorithm into real-world applications necessitates thorough validation and adjustment to address potential interoperability issues. Another challenge and limitation is related to the computational overhead associated with our S-Box algorithm. While the algorithm has been designed to offer robust security, its performance in high-throughput environments needs further evaluation. The additional computational requirements may impact the efficiency and speed of data transmission, particularly in resource-constrained devices or networks with high traffic volumes.

Future work will build upon the foundation established by our research to address the identified challenges and expand the applicability of our S-Box algorithm. One key area for further exploration is the optimization of the algorithm to reduce computational overhead while maintaining its security effectiveness. This includes investigating potential improvements in algorithm efficiency and performance, such as parallel processing techniques or hardware acceleration. Moreover, future research will focus on the integration of the S-Box algorithm with advanced cryptographic frameworks and security protocols. This includes exploring its compatibility with emerging technologies and standards in wireless communication, such as 5G and beyond, to ensure its relevance and effectiveness in future network architectures. Another promising avenue for future work is the investigation of the S-Box algorithm’s application to other areas of cybersecurity, such as securing Internet of Things (IoT) devices, cloud computing environments, and data storage systems. The adaptability of the algorithm to different security domains could offer additional benefits and contribute to broader advancements in the field.

Data is provided within the manuscript or supplementary information files.

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Ala Saleh Alluhaidan would like to express sincere gratitude to Princess Nourah bint Abdulrahman University, Saudi Arabia, for supporting this research through project number (PNURSP2024R234). Nisreen Innab would like to express sincere gratitude to AlMaarefa University, Riyadh, Saudi Arabia, for supporting this research.

College of Computing and Informatics (CCI), Saudi Electronic University, Riyadh, 11673, Saudi Arabia

Abdulbasid Banga

Department of Computer Science and IT, The University of Lahore, Lahore, 54000, Pakistan

Nadeem Iqbal & Atif Ikram

Faculty of Ocean Engineering Technology and Informatics, Universiti Malaysia Terengganu, Kuala Terengganu, 21030, Malaysia

Atif Ikram

Department of Computer Science and Information Systems, College of Applied Sciences, AlMaarefa University, Riyadh, 13713, Saudi Arabia

Nisreen Innab

Department of Information Systems, College of Computer and Information Sciences, Princess Nourah bint Abdulrahman University, Riyadh, 11671, Saudi Arabia

Ala Saleh Alluhaidan

Department of Computer Science, Faculty of Computer Science and Information Technology, Jerash University, Jerash, 26150, Jordan

Bassam Mohammad ElZaghmouri

Math and Computer Science Department, Faculty of Science, Menoufia University, Menoufia, 32511, Egypt

Hossam Diab

Computer Science Department, Applied College, Taibah University, Madinah, 41477, Saudi Arabia

Hossam Diab

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A.B. and N.I. conceived the idea of this project, developed algorithms and supervised the project; A.I., N.I. and A.S.A developed few algorithms, validated the results and contributed writting the original draft; B.M.E. and H.D. contributed in writing the original draft, developed algorithms and proofread the manuscript.

Correspondence to Ala Saleh Alluhaidan.

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Banga, A., Iqbal, N., Ikram, A. et al. ChessCrypt: enhancing wireless communication security in smart cities through dynamically generated S-Box with chess-based nonlinearity. Sci Rep 14, 28205 (2024). https://doi.org/10.1038/s41598-024-77927-0

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Received: 29 June 2024

Accepted: 28 October 2024

Published: 15 November 2024

DOI: https://doi.org/10.1038/s41598-024-77927-0

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