Cryptographic Properties Research Articles

Enhancing Blockchain Security by Developing the SHA256 Algorithm

Security plays a vital role in various domains, including blockchain technology. The Blockchain serves as a secure data structure for storing transactional records. Hash functions are employed in cryptography to ensure integrity and authentication within the blockchain. The widely used SHA256 algorithm has faced recent attacks, prompting the development of stronger hash functions. This paper presents a novel modification approach to enhance the performance of SHA256 by introducing an extended mechanism for generating a 288-bit message digest and reducing the number of rounds to 44 instead of 64 while preserving the diffusion of data through its complex iterative process, which involves multiple rounds of bitwise and logical operations. The change makes sure that even small changes to the input data cause noticeable variations in the output hash, thereby maintaining cryptographic properties. The suggested hash function SHA288 achieves improved security, collision resistance, and preimage resistance, while maintaining a faster execution time compared to SHA256. The tables and tests conducted on the suggested algorithm have revealed its remarkable safety and robustness in countering attacks as well as demonstrated outstanding performance in random tests, which further enhances its security measures.

Design and implementation of dynamic s-boxes based on non-degenerate discrete chaotic systems

In recent years, digital chaotic systems have received considerable attention in the field of secure communications. However, during the digitalization of the system, the original properties of the chaotic system may change, resulting in the degradation of the dynamics. To address this problem, this paper designs a novel simplicial non-degenerate discrete chaotic system based on the inverse hyperbolic tangent function, and selects a three-dimensional discrete system as the object of analysis. Through the research conducted, it is found that the chaotic system exhibits high Lyapunov exponents under certain conditions. Furthermore, the excellent randomness of the system has been further validated by NIST SP800–22 tests. At the same time, this paper also proposes a dynamic S-box construction method based on the chaotic sequence generated from the three-dimensional chaotic mapping. By performing a series of basic operations and permutation treatment, many dynamic S-boxes can be generated. This paper comprehensively analyses the performance of S-boxes from two perspectives: single S-boxes and multiple S-boxes. The analysis covers issues such as bijective property, nonlinearity, strict avalanche criterion, differential approximation probability and bit independence criteria. The results of the performance analysis show that the dynamically generated S-boxes have excellent cryptographic properties, making them suitable for the design and application of cryptographic algorithms.

A comprehensive review of quantum image encryption methods: design characteristics, cryptographic properties, and AI integration

A comprehensive review of quantum image encryption methods: design characteristics, cryptographic properties, and AI integration

Reconstructing S-Boxes from Cryptographic Tables with Milp

Reconstructing an S-box from a cryptographic table such as difference distribution table (DDT), linear approximation table (LAT), differential-linear connectivity table (DLCT) or boomerang connectivity table (BCT) is one of the fundamental problems in symmetric-key cryptography. Till now, there are only very few known methods which can reconstruct an S-box from a given table: guess-and-determine algorithms of Boura et al. (DCC 2019) and Tian et al. (DCC 2020), sign determination algorithm of Dunkelman et al. (ToSC 2019) and STP based approach of Lu et al. (DCC 2022). In this paper we consider the reconstruction problem in an even more challenging setup where one needs to reconstruct S-boxes from a partial cryptographic table. We are able to reconstruct S-boxes when only a few number of rows of a cryptographic table is given. This problem has never been studied in the literature. We apply mixed integer linear programming (MILP) as the key tool for solving this problem. Needless to say that we can solve the reconstruction problem when the full table is given and this is the first ever application of MILP tool in solving such fundamental problems. As a further application of our method, we provide the generic MILP models which can search for S-boxes with a given cryptographic property such as differential uniformity, linearity, differential-linear uniformity or boomerang uniformity. Additionally, our method can recover a Boolean function from a given Walsh spectrum or a Boolean function with a given nonlinearity. We also introduce a new heuristic called Optimistic MILP objective that guides the model towards obtaining multiple S-boxes or Boolean functions with the same cryptographic property. We give detailed experimental results for up to 6-bit S-boxes showing the effectiveness of our technique.

Leveraging quantum uncertainty: Quantum randomness through the lens of classical communication networks

The generation of random numbers and the study of its properties have been an elusive field for a fair portion of the century. The application of random numbers is employed in many use cases such as cryptography, neural networks, numerical simulation, and gambling. The performance of each of these use cases is profoundly impacted by the employed random numbers; henceforth, the quality of randomness is of critical importance when it comes to their usage. A poorly generated random number can make a security system vulnerable, or any numerical or statistical evaluation misleading. Although various modes of classical random number generators exist and still function to provide strong cryptographic properties, emergence of quantum mechanical randomness has shed light on a novel path of generating certified randomness which supersedes the classical counterpart in terms of security. Harnessing quantum mechanical phenomena enables generation of true random numbers which can be certified and further implemented to elevate the net quality of the specific use cases. In this work, we generate and analyze random numbers from three different sources — 50: 50 beam splitter (BS), quantum key distribution (QKD) setup with classical post-processing scheme, and a commercially available quantum random number generator (QRNG) (ID Quantique (IDQ)). The quality of the generated random numbers from the various sources is checked in statistical tests and compared. Further on, we have developed a system which implements the QRNG-based random numbers to facilitate message authentication code (MAC) and one time password (OTP) protocols, demonstrating a communication network application. In this manner we discuss about a network which integrates quantum mechanics to the current classical networking approaches to enhance certain aspects of the networking protocol — in this case, the security.

An Innovative Algorithm Based on Chaotic Maps Amalgamated with Bit-Level Permutations for Robust S-Box Construction and Its Application in Medical Image Privacy

Data security and privacy have become essential due to the increasingly advanced interconnectivity in today’s world, hence the reliance on cryptography. This paper introduces a new algorithm that uses a novel hybrid Tent–May chaotic map to generate pseudo-random numbers, as well as block encryption. We design a robust S-box by combining the Tent and May Maps, which yields a chaotic system with improved cryptographic properties. This S-box is a critical cryptographic primitive that significantly improves encryption security and leverages the strengths of both maps. The encryption process involves two key steps: block-wise substitution and permutation. First, we divide the image into 16×16 blocks, then substitute each pixel with the 8−byte key and S-box. Next, we convert the encrypted image back into vector form, reorganize it using the permutation vector based on the subgroups of S16, and finally return it to its original form. This approach greatly improves block cipher security when used, especially to protect medical images by guaranteeing their confidentiality and noninterference. Performance measures like PSNR, UACI, MSE, NCC, AD, SC, MD, and NAE prove how immune our method is to various cryptographic and statistical attacks, making it more accurate and more secure than the existing techniques.

Security Analysis of the BB84 Protocol in IoT Networks

The Internet of Things (IoT) has experienced rapid growth, resulting in a proliferation of interconnected devices in domains such as smart homes, cities, and healthcare applications. Ensuring secure communication within IoT networks is crucial for protecting privacy and data. The BB84 protocol, a quantum key distribution (QKD) protocol, shows promise in enhancing IoT communication security. This paper presents a comprehensive security analysis of the BB84 protocol in IoT networks, examining key aspects including entropy, execution time, quantum bit error rate (QBER), and cryptographic key generation efficiency. The analysis aims to evaluate the protocol's resilience against potential attacks and its suitability for securing IoT communications. We delve into the evaluation of entropy levels in the key generation process to ensure strong cryptographic properties. We also examine the correlation between execution time and QBER to determine the protocol's efficiency and ability to counter timing-based attacks. Additionally, we analyze the efficiency of the keys generated by the BB84 protocol, taking into consideration factors such as key size, generation rate, and computational overhead. These evaluations allow us to assess the protocol's suitability for resource-constrained IoT devices

Cryptanalysis of Selected ARX-Based Block Ciphers

The security of digital communication and information systems is mostly dependent on block ciphers. ARX-based ciphers are widely used due to their simplicity and efficiency. This paper provides an exhaustive cryptanalysis of a subset of ARX-based block ciphers, with particular emphasis on SIMON, SPECK, and IDEA. These ciphers need to be exposed for their weaknesses in algebraic attack resistance and cryptographic properties such as key sensitivity. In addition, we assess the resource utilization and speed of these ciphers, both of which are critical for practical implementation. SMT (Satisfiability Modulo Theories) framework is utilized to tackle constraint fulfillment problems based on first-order logic. The following cryptographic steps use SMT solvers: differential cryptanalysis, collision attack, pre-image attack, modular root-finding, and cryptographic primitive verification. We show that SMT solvers can solve block cipher cryptanalysis constraints. In a cryptanalytic attack, we convert block cipher boolean equations to Z3py. The proposed cryptanalysis method evaluates ARX cipher performance. This method recovers the partial secret key using plaintext and ciphertext pairs, partial key bits, and a predetermined number of rounds. To determine whether SIMON, SPECK, or IDEA are appropriate for distinct security requirements, we conducted a comparative analysis of the results and presented them in tabulated form. This research builds a better understanding of ARX-based block ciphers and allows us to develop more robust and efficient cryptographic algorithms to protect sensitive data in modern communication systems.

GFLE: a low-energy lightweight block cipher based on a variant of generalized Feistel structure

Low-energy lightweight block ciphers are essential for applications with extremely resource-constrained to reduce energy and maintain security. The trade-off between diffusion property and area is a widely studied issue in the design of low-energy block ciphers. In this paper, a low-energy lightweight block cipher named as GFLE is presented. The core cipher of GFLE uses a variant of the Generalized Feistel Structure (GFS) with 4-branch, which combines the Type-II GFS with the simplified Lai-Massey. The DRmax of GFLE has a one-round improvement over the Type-II GFS optimized by Suzaki et al and the security margin is achieved in a shorter number of rounds. Moreover, an S-box with low-energy and good cryptographic properties is proposed by searching combinations based on gate-level circuits using a depth-first strategy. It exhibits better security properties and hardware performance compared to other S-boxes. The block cipher GFLE is implemented in ASIC with UMC 0.18 μm. It has been proved that the energy of GFLE is lower than Midori, WARP, SKINNY, CRAFT, etc in unified encryption and decryption (ED) circuits. GFLE reduces energy by 61.59% compared with SKINNY. The results show that GFLE in ED circuits consumes only 1596 Gate Equivalents (GEs) and 6.36 μ J/bit in area and energy, respectively.

Group-Action-Based S-box Generation Technique for Enhanced Block Cipher Security and Robust Image Encryption Scheme

Data security is one of the biggest concerns in the modern world due to advancements in technology, and cryptography ensures that the privacy, integrity, and authenticity of such information are safeguarded in today’s digitally connected world. In this article, we introduce a new technique for the construction of non-linear components in block ciphers. The proposed S-box generation process is a transformational procedure through which the elements of a finite field are mapped onto highly nonlinear permutations. This transformation is achieved through a series of algebraic and combinatorial operations. It involves group actions on some pairs of two Galois fields to create an initial S-box Pr Sbox, which induces a rich algebraic structure. The post S-box Po Sbox, which is derived from heuristic group-based optimization, leads to high nonlinearity and other important cryptographic parameters. The proposed S-box demonstrates resilience against various attacks, making the system resistant to statistical vulnerabilities. The investigation reveals remarkable attributes, including a nonlinearity score of 112, an average Strict Avalanche Criterion score of 0.504, and LAP (Linear Approximation Probability) score of 0.062, surpassing well-established S-boxes that exhibit desired cryptographic properties. This novel methodology suggests an encouraging approach for enhancing the security framework of block ciphers. In addition, we also proposed a three-step image encryption technique comprising of Row Permutation, Bitwise XOR, and block-wise substitution using Po Sbox. These operations contribute to adding more levels of randomness, which improves the dispersion across the cipher image and makes it equally intense. Therefore, we were able to establish that the approach works to mitigate against statistical and cryptanalytic attacks. The PSNR, UACI, MSE, NCC, AD, SC, MD, and NAE data comparisons with existing methods are also provided to prove the efficiency of the encryption algorithm.

Block Cipher Nonlinear Component Generation via Hybrid Pseudo-Random Binary Sequence for Image Encryption

To analyze the security of encryption, an effectual encryption scheme based on colored images utilizing the hybrid pseudo-random binary sequence (HPRBS) and substitution boxes, known as S-boxes, is proposed. The presented work aims to design S-boxes using pseudo-random binary numbers acquired by Linear Feedback Shift Registers (LFSRs) in combination with a modified quadratic chaotic map. Firstly, cryptographically robust S-boxes are constructed by using binary pseudo-random number sequences, and then the cryptographic properties of the presented S-boxes are tested. The suggested S-boxes showed good results. Secondly, an RGB image encryption algorithm utilizing sequences generated by modified quadratic chaotic maps and S-boxes is offered. The new color image encryption techniques comprise two steps, including a permutation and a substitution step. The key association with the content of the image is also addressed. This strategy can result in a “one-time pad” effect and make the algorithm resistant to chosen-plaintext attack (CPA). The proposed scheme has been confirmed to be more valuable than most of the existing schemes. S-boxes are analyzed by the nonlinearity test, bit independence criterion (BIC), linear and differential approximation probabilities (LPs; DPs), and Strict-Avalanche Criterion (SAC) tests. A comparison with different S-boxes presented in the literature is also carried out. The comparison shows encouraging results about the quality of the proposed box. From security and experimental outcomes, the effectiveness of the presented color image encryption technique is verified. The proposed scheme has evident efficiency benefits, which implies that the proposed colored encryption of the image scheme has better potential for application in encryption schemes in real-time.

COBLAH: A chaotic OBL initialized hybrid algebraic-heuristic algorithm for optimal S-box construction

The Substitution box (S-box) is the main nonlinear component responsible for the cryptographic strength of any Substitution-Permutation Network (SPN) based block cipher. Generating the S-box with optimal cryptographic properties is one of cryptography's most challenging combinatorial problems because of its enormous search space, lack of guidance, and conflicting performance criteria. This paper introduces a novel Chaotic Opposition-based Learning Initialized Hybrid Algebraic-Heuristic (COBLAH) algorithm, combining the favorable traits of Algebraic and heuristics methods based on Galois field inversion, affine mapping, and Genetic Algorithm (GA). The Galois field inversion and affine mapping are used to construct the S-box, while the GA guides the algebraic construction to find the best bit-matrix and additive vector based on any irreducible polynomial for GF(28). GA initializes with a random population generated using a newly constructed cosine-cubic map incorporated with binarization and Opposition-based Learning (OBL). Further, Multi-Objective Optimization Ratio Analysis (MOORA) is utilized to identify the best S-box from the final optimized population. The performance of the proposed algorithm is evaluated by comparing the generated COBLAH S-box with more than twenty state-of-the-art S-boxes, including Advanced Encryption Standard (AES), Skipjack, Gray, and Affine Power Affine (APA). The COBLAH S-box has nonlinearity 112, Strict Avalanche Criterion (SAC) offset 0.0202, Distance to SAC (DSAC) 332, Differential Approximation Probability (DP) 0.0625, Linear Approximation Probability (LP) 0.0156, Bit Independence Criterion-Strict Avalanche Criterion (BIC-SAC) 0.50006, and Bit Independence Criterion-Nonlinearity (BIC-NL) 112, which stands as the optimal observed thus far. The absence of fixed and opposite fixed points and the fact that it adheres to a single cycle aligns the COBLAH S-box with an ideal S-box. In addition, an image encryption mechanism is utilized to encrypt and decrypt the different images sourced from the standard USC-SIPI image dataset using COBLAH S-box and compared against different state-of-the-art S-boxes based on various image characteristics.

Further construction of even-variable balanced rotation symmetric Boolean functions with optimal algebraic immunity

The field of cryptography has given a lot of attention to rotation symmetric Boolean functions (RSBFs) because they possess special structures and include many functions with good cryptographic properties. It is difficult to construct even-variable balanced RSBFs with optimal algebraic immunity in the study on RSBFs. Recently, Mesnager et al. proposed the first and only construction of balanced RSBFs with optimal algebraic immunity for arbitrary even variables (Des. Codes Cryptogr. 89 (1) (2021) 1-17). The nonlinearity of their functions is not high. In this paper, we develop further research based on their construction and present a fresh design of n-variable balanced RSBFs with optimal algebraic immunity for arbitrary even n. Our functions have higher nonlinearity compared to their functions. Furthermore, the algebraic degree and fast algebraic immunity of our functions are not less than n−2 for n not bigger than 16.

On the Pentanomial Power Mapping Classification of 8-bit to 8-bit S-Boxes

Substitution boxes, or S-boxes, are one of the most important mathematical primitives in modern symmetric cryptographic algorithms. Given their importance, in the past decades, they have been thoroughly analyzed and evaluated by the academic world. Thus, a lot of desirable characteristics a given S-box should possess have been found. This includes, as much as possible, higher nonlinearity and algebraic degrees as well as, as much as possible, lower values of differential uniformity, autocorrelation and sum of squares indicator values. In this work, we use power mappings over GF(28) to generate, enumerate and evaluate all bijective S-boxes yielded by pentanomials of the form f(x)=xa+xb+xc+xd+xe given 0<a<b<c<d<e<256. We find a total of 152,320 different bijective S-boxes, which are further classified into 41,458 different groups in terms of the aforementioned characteristics as well as the number of their fixed points. Having this data, an S-box designer can easily generate a bijective substitution S-box with parameters of their choice. By using pentanomials, we show how we can easily construct S-boxes with cryptographic properties similar to those found in some popular S-boxes like the Kuznyechik S-box proposed by the Russian Federation’s standardization agency as well as the Skipjack S-box proposed by the National Security Agency of the USA.

Pseudo-random number generation based on spatial chaotic map of Logistic type and its cryptographic application

The pseudo-random number generator (PRNG) based on chaos has been widely used in the fields of digital communication, cryptography and computer simulation. In this paper, we study a new PRNG based on spatial surface chaotic system (SSCS), which is constructed based on coupling mapping lattice (CML) and one-dimensional Logistic chaotic map. To verify the performance of this generator, we study its nonlinear dynamic properties, including the Lyapunov exponent, ergodicity and bifurcation phenomena. Moreover, we analyze the cryptographic properties such as key space, key sensitivity, correlation, histogram and information entropy, while performing NIST SP800-22 test and TestU01 test for the randomness of this system. Theoretical analysis and numerical simulation results show that the PRNG proposed in this paper has good complexity, ergodicity, sensitivity and randomness. Moreover, the key space can increase dynamically with the increase of encrypted data, especially suitable for the protection of big data such as multimedia, which is more advantageous than most existing chaotic mapping. The results of this paper will provide a new idea for the study of chaotic cryptography, and motivate the study of new discrete chaotic models with desirable statistical properties.

SBGen: A high-performance library for rapid generation of cryptographic S-boxes

In the realm of cryptographic research, the generation of S-boxes with high nonlinearity and optimal cryptographic properties remains a critical challenge. This paper presents a novel approach to S-box generation, leveraging the strengths of heuristic optimization methods. Through a meticulous integration of Simulated Annealing (SA) and Hill Climbing (HC) algorithms with sophisticated cost functions, we introduce an innovative software tool that significantly advances the efficiency of generating highly nonlinear S-boxes. Our methodology is distinguished by its ability to consistently produce S-boxes that meet stringent security criteria, with a remarkable 100 % success rate and minimized computational overhead. A comparative analysis reveals that our approach outperforms existing methods in terms of the probability of generating target S-boxes and the average number of iterations required. The software's practical implications extend beyond theoretical advancements, offering a valuable resource for cryptographic system designers in their quest to fortify cipher systems against linear and differential cryptanalysis. By setting new benchmarks for nonlinearity and search efficiency, our work paves the way for future research in cryptographic S-box generation, highlighting the potential of combining heuristic techniques with domain-specific cost functions to achieve superior security outcomes.

Filter generators with variable transition functions over finite fields of characteristic 2

Filter generators are a traditional basis for creating synchronous stream ciphers. They are built with the help of linear shift registers (usually over a field of two elements) and nonlinear complexity functions, which are subject to a number of requirements in terms of the generators security against known attacks. Intensive researches of filter generators during the last decades show that meeting these requirements without degrading the performance of the generators is a very difficult task. Despite a large number of publications devoted to the construction of complexity functions with known “good cryptographic properties”, the usage of such functions in practice often becomes unacceptable due to the bulkiness of their constructions, which slows down the functioning of the corresponding generators, especially during software implementation. The way to overcome the noted difficulties by using an additional secret parameter that determines the appearance of the generator transitions’ function is proposed. Such a modification makes it possible to increase the security of generator (compared to traditional filter generators) against known attacks without increasing the length of its initial state. In particular, a specific version of a generator construction with a complexity function, which is determined with the help of substitutions used in the “Kalyna” encryption scheme, is considered. A lower estimate of the output sequences periods of the proposed generators was obtained. A research of their security to known attacks, in particular, Babbage-Golic balancing attack; an attack associated with a small number of terms in the polynomial representation of the complexity function (which negatively affects the value of the equivalent linear complexity of the output sequences of the generator); a natural correlation attack associated with the specifics of the proposed generator construction scheme; algebraic attacks of the Courtois-Mayer type were also conducted. At the end of the article, it is indicated how to choose the components of the proposed generators to ensure their security at a predetermined level.

ANCHOR: A New Proposed Stream Cipher for Smart Cards with Crypto Co-Processor

A smart card is a small pocket-sized computer with limited resources used for secure data processing and storage. The card consists of different software and hardware components, including a microprocessor, crypto co-processor, RAM, secure ROM, and operating system. Even though smart cards have a lot of limitations in terms of processing power and small-sized memory, smart cards are widely used in many applications that require a high degree of security such as e-passports, citizen cards, e- banking, etc... Basically, the security of a smart card depends mainly on the security strength of the cipher algorithm implemented inside. This paper presents a new lightweight, high-speed, and cryptographically strong stream cipher algorithm (ANCHOR) suitable for implementation on smart cards. The building blocks of the proposed cipher were carefully built/chosen with high-valued cryptographic properties. The randomness and linear complexity properties of the proposed cipher algorithm have been successfully tested with statistical tests of the NIST suite and the Berlekamp- Massey algorithm respectively. In order to test the performance of the proposed cipher algorithm, the algorithm was developed in C language and executed on a Linux machine.

Semi-Permanent Stuck-At Fault injection attacks on Elephant and GIFT lightweight ciphers

Fault attacks pose a potent threat to modern cryptographic implementations, particularly those used in physically approachable embedded devices in IoT environments. Information security in such resource-constrained devices is ensured using lightweight ciphers, where combinational circuit implementations of SBox are preferable over look-up tables (LUT) as they are more efficient regarding area, power, and memory requirements. Most existing fault analysis techniques focus on fault injection in memory cells and registers. Recently, a novel fault model and analysis technique, namely Semi-Permanent Stuck-At (SPSA) fault analysis, has been proposed to evaluate the security of ciphers with combinational circuit implementation of Substitution layer elements, SBox. In this work, we propose optimized techniques to recover the key in a minimum number of ciphertexts in such implementations of lightweight ciphers. Based on the proposed techniques, a key recovery attack on the NIST lightweight cryptography (NIST-LWC) standardization process finalist, Elephant AEAD, has been proposed. The proposed key recovery attack is validated on two versions of Elephant cipher. The proposed fault analysis approach recovered the secret key within 85 − 240 ciphertexts, calculated over 1000 attack instances. To the best of our knowledge, this is the first work on fault analysis attacks on the Elephant scheme. Furthermore, an optimized combinational circuit implementation of Spongent SBox (SBox used in Elephant cipher) is proposed, having a smaller gate count than the optimized implementation reported in the literature. The proposed fault analysis techniques are validated on primary and optimized versions of Spongent SBox through Verilog simulations. Further, we pinpoint SPSA hotspots in the lightweight GIFT cipher SBox architecture. We observe that GIFT SBox exhibits resilience towards the proposed SPSA fault analysis technique under the single fault adversarial model. However, eight SPSA fault patterns reduce the nonlinearity of the SBox to zero, rendering it vulnerable to linear cryptanalysis. Conclusively, SPSA faults may adversely affect the cryptographic properties of an SBox, thereby leading to trivial key recovery. The GIFT cipher is used as an example to focus on two aspects: i) its SBox construction is resilient to the proposed SPSA analysis and therefore characterizing such constructions for SPSA resilience and, ii) an SBox even though resilient to the proposed SPSA analysis, may exhibit vulnerabilities towards other classical analysis techniques when subjected to SPSA faults. Our work reports new vulnerabilities in fault analysis in the combinational circuit implementations of cryptographic protocols.

SPN based RGB image encryption over Gaussian integers

This research paper proposes a novel approach for constructing substitution boxes (S-boxes) over Gaussian integers, which are complex numbers with integer coefficients. The proposed method is based on the properties of the Gaussian integers and their arithmetic operations and ensures the S-boxes exhibit strong cryptographic properties. Furthermore, the paper demonstrates how these S-boxes can be utilized for image encryption through a substitution-permutation network (SPN) over Gaussian integers. The SPN involves iteratively applying the S-box and a permutation layer to the input image, which effectively scrambles the image data. Experimental results show that the proposed method achieves high security and robustness against various attacks while providing efficient encryption and decryption performance. This research thus provides a promising avenue for developing secure image encryption schemes based on Gaussian integers.