Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 5, 436-443 2025 Publisher: Learning Gate DOI: 10.55214/25768484.v9i5.6886 © 2025 by the authors; licensee Learning Gate © 2025 by the authors; licensee Learning Gate History: Received: 19 February 2025; Revised: 26 April 2025; Accepted: 30 April 2025; Published: 5 May 2025 * Correspondence: s.naserelden@gmail.com Enhancing block cipher diffusion: A mathematical approach to key- dependent mix column transformations Shiraz Naserelden1*, Norma Alias1,2, Abdelrahamn Altigani3, Maher Waheeb3, Azmi Alazzam3, Muawia A. Elsadig4, Hasan Abu Hilal5 1Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia (UTM), Johor Bahru, 81310, Malaysia; s.naserelden@gmail.com (S.N.). 2Center of Engineering Education, Universiti Teknologi Malaysia (UTM), Johor Bahru, 81310, Malaysia; normaalias@utm.my (N.A.). 3Computer Information Science, Higher Colleges of Technology, 25026, Al Ain, Abu Dhabi, UAE; aabdelgader@hct.ac.ae (A.A.) mwaheeb@hct.ac.ae (M.W.) aalazzam@hct.ac.ae (A.Z.) 4Imam Abdulrahman Bin Faisal University, Dammam 31441, Saudi Arabia; makhlafalla@iau.edu.sa (M.A.E.). 5Engineering Technology & Science, Higher Colleges of Technology, Abu Dhabi, UAE; habuhilal@hct.ac.ae (H.A.H.). Abstract: In this paper, we propose a novel key-dependent MixColumn transformation to enhance the diffusion properties of block ciphers, specifically within the AES framework. The proposed method introduces dynamic variation into the MixColumn step by extracting a pseudo-random value from the round key, which is used to perform cyclic permutations on the MixColumn coefficients. This leads to the construction of a round-dependent transformation matrix, mathematically defined and proven to maintain invertibility over GF(2⁸). We provide a formal mathematical representation of the proposed model and analyze its diffusion characteristics, linear independence, and resistance to cryptanalytic attacks. Our approach eliminates static transformation patterns and introduces key-driven unpredictability, offering a promising direction for adaptive cryptographic primitives without increasing algorithmic complexity or requiring hardware modifications. Keywords: Block ciphers, Advance encryption standard, Finite fields, MixColumn transformation, Encryption, Maximum distance separable. 1. Introduction Block ciphers are a cornerstone of contemporary cryptographic infrastructure, underpinning data security in countless digital systems. Their effectiveness relies heavily on two fundamental properties defined by Claude Shannon: confusion, which conceals the correlation between the encryption key and ciphertext, and diffusion, which ensures that changes in the plaintext are spread across the ciphertext in a complex and non-linear manner [1]. One of the most prominent examples of these principles in action is the Advanced Encryption Standard (AES) [2] which integrates multiple transformations including SubBytes, ShiftRows, MixColumns and AddRoundKey to achieve strong security. In particular, the MixColumns operation introduces diffusion through a fixed linear transformation over a finite field. However, as cryptanalysis continues to evolve, static diffusion techniques such as these are increasingly scrutinized for their potential vulnerability to attacks [3, 4] especially those that exploit weaknesses related to key structure or predictability, such as related-key and weak-key attacks [5]. To counter these risks, researchers have turned their attention to key-dependent diffusion techniques [6], which adjust the transformation dynamically based on the secret key. These approaches aim to increase cryptographic strength by introducing an additional layer of unpredictability [7]. 437 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 5: 436-443, 2025 DOI: 10.55214/25768484.v9i5.6886 © 2025 by the authors; licensee Learning Gate Nevertheless, achieving a balance between adaptability, security, and implementation efficiency is far from trivial, and poorly constructed solutions may inadvertently introduce new weaknesses. This study introduces a mathematically rigorous method for constructing key-dependent MixColumn transformations that not only enhance diffusion but also preserve critical cryptographic qualities such as reversibility, optimal branch number, and robustness against both linear and differential cryptanalysis. The methodology is built on well-established mathematical structures, including circulant matrices, Maximum Distance Separable (MDS) codes, and algebraic frameworks like polynomial rings, enabling the development of dynamic yet computationally efficient diffusion layers. Our key contributions have threefold: First, we present a theoretical framework for key-dependent diffusion by integrating adaptive diffusion into broader cipher designs. 2. Literature Review A number of symmetric encryption algorithms and mechanisms have been introduced in the last few decades [8-12]. However, there have been special interest on enhancing the efficiency, adaptability, and security of the Advanced Encryption Standard (AES), particularly in the context of constrained environments such as embedded systems and IoT devices. To address the limitations of resource-constrained environments, multiple works have explored architectural and hardware-based improvements. One such study introduces an optimized AES implementation using a Low-Transition Linear Feedback Shift Register (LFSR), hardware design techniques, and low-power strategies including subthreshold voltage operation and back-biasing. These modifications collectively reduce power consumption and enhance latency performance, making the algorithm more suitable for IoT devices [13]. developing a formal model for MixColumn operations whose structure is dynamically influenced by the encryption key, ensuring that essential diffusion and security properties are preserved. Second, we propose efficient matrix constructions using lightweight and structured designs that support key- dependent behavior while maintaining compatibility with high-performance cryptographic implementations. Finally, we conduct a comprehensive security assessment, evaluating the proposed approach against a range of classical and modern cryptanalytic techniques, including differential, linear, and algebraic attacks, demonstrating improved resilience compared to traditional fixed diffusion layers. Key-dependent MixColumns transformations represent a promising advancement in cryptographic security by enhancing resistance to various attacks, yet they also introduce added complexity and computational demands, highlighting the ongoing need for research into more efficient algorithms, hardware implementations, and practical design trade-offs to ensure their widespread applicability [14, 15]. By introducing key-driven variability into the diffusion process, our approach paves the way for more resilient cryptographic primitives, capable of withstanding increasingly sophisticated adversarial techniques while retaining the efficiency demanded by real-world applications. The rest of this paper is organized as follows: Section two is the literature review, which discusses a number of similar research work in this field. Section three details our methodology and presents the mathematical foundation for dynamic MixColumn transformations. Section four provides a security analysis. Section five concludes the work, highlighting future research directions and the potential for Similarly, another study eliminates lookup tables in the MixColumns operation and parallelizes major AES transformations within the encryption and decryption processes. Implemented on FPGA devices, the design achieves reduced resource usage and high processing speed, demonstrating its practicality for secure embedded applications with limited hardware capacity [15]. Further extending these improvements, research utilizing Virtex-6 and Spartan-6 FPGA platforms proposes a modified SubBytes stage to minimize power, area, and delay. The resulting AES structure improves throughput and is tested using real-world medical imaging data, highlighting its suitability for high-speed, power-sensitive applications [16]. 438 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 5: 436-443, 2025 DOI: 10.55214/25768484.v9i5.6886 © 2025 by the authors; licensee Learning Gate Others research focus on enhancing MixColumns and diffusion layers. A key area of innovation lies in the transformation stages, particularly MixColumns. One work strengthens AES’s resistance to cryptanalytic attacks by replacing the standard MixColumn matrix with Maximum Distance Separable (MDS) matrices of varying sizes. These changes improve the cipher's branch number and diffusion characteristics, enhancing protection against linear and differential cryptanalysis while maintaining efficiency [17]. In a related approach, researchers propose using 8×8 circulant matrices in the MixColumns step, offering a faster and more secure alternative to traditional 4×4 matrices. This method reduces arithmetic operations and boosts processing speeds by up to 79%, especially beneficial for applications involving key exchange protocols such as Elliptic Curve Diffie-Hellman (ECDH) [18]. Complementing these matrix-based improvements, another work replaces MixColumns with a transformation derived from a magic cube over GF(28). Designed specifically for voice message encryption on social media platforms, the approach uses cube-generated keys to enhance execution speed and encryption complexity, demonstrating promising results in time reduction and security metrics [19]. Addressing power efficiency from a different angle, one study explores the use of Quantum Dot Cellular Automata (QCA) for implementing an Xtime multiplier in Galois Field arithmetic. QCA’s high frequency and low energy requirements make it a compelling alternative to CMOS technology. The proposed QCA-based multiplier significantly reduces energy dissipation and quantum cost, showcasing its potential for future lightweight AES implementations [20]. In the context of lightweight cryptography, another study examines the PRINCE block cipher, focusing on vulnerabilities to side-channel attacks in unrolled architectures. By employing a chosen- input attack and analyzing Threshold Implementations (TI), the researchers highlight the critical trade- off between performance and security in lightweight designs and emphasize the importance of protecting multiple cipher rounds against key leakage [21]. To further complicate adversarial analysis, recent work introduces a polymorphic AES variant (P- AES), where core transformations—SubBytes, ShiftRows, and MixColumns—are made key-dependent. Each encryption instance varies based on the key, thereby significantly reducing predictability and increasing resistance to cryptanalytic attacks. The approach maintains AES’s performance advantages while embedding an additional layer of dynamic security [22]. Collectively, these studies offer diverse and innovative strategies for evolving the AES encryption standard. From energy-efficient hardware designs and high-speed transformations to enhanced security through polymorphism and diffusion, the modifications address critical challenges in modern cryptography. These advancements not only improve AES’s adaptability for constrained environments but also strengthen its resilience against emerging security threats, affirming its continued relevance in secure digital communications. 3. Methodology 3.1. Proposed Method: Key-Dependent Cyclic Mixcolumn Transformation Let the AES MixColumn transformation be defined over the finite field 𝔽28. The standard AES MixColumn matrix 𝑀 is: 𝑀 = [ 2 3 1 1 1 2 3 1 1 1 2 3 3 1 1 2 ] Let 𝐾𝑟 ∈ 𝔽28 16 be the round key at round 𝑟. From 𝐾𝑟, extract a permutation index 𝜌𝑟 ∈ {0,1,2,3} as follows: 439 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 5: 436-443, 2025 DOI: 10.55214/25768484.v9i5.6886 © 2025 by the authors; licensee Learning Gate 𝜌𝑟 = (∑ 𝑘𝑟,𝑖 15 𝑖=0 ) 𝑚𝑜𝑑 4 Where 𝑘𝑟,𝑖 is the 𝑖𝑡ℎ byte of 𝐾𝑟. Define 𝜋𝜌𝑟 (𝑀) to be the matrix formed by applying a cyclic row permutation to 𝑀 by 𝜌𝑟 positions. For instance: • If 𝜌𝑟 = 1 , then row 0 moves to row 1, row 1 → row 2, ..., row 3 → row 0. • If 𝜌𝑟 = 2, rows are rotated by 2 positions, and so on. Let this dynamically generated matrix be 𝑀𝑟 = 𝜋𝜌𝑟 (𝑀). Then the state transformation becomes: 𝑆′ = 𝑀𝑟. 𝑆 𝑆𝜖𝔽28 4×1 is a column of the state matrix, and 𝑆′ is the output column after the MixColumn operation 3.2. Invertibility of the Dynamic MixColumn Transformation A core requirement of any linear transformation in a block cipher is invertibility. The original AES MixColumn matrix 𝑀 is invertible over the finite field 𝔽28, with a known inverse matrix 𝑀−1 used during decryption. In our proposed scheme, we define a set of transformation matrices {𝑀𝑟} generated by cyclically permuting the rows of 𝑀 by a round-specific index 𝜌𝑟 ∈ {0,1,2,3}. Each 𝑀𝑟 is defined as: 𝑀𝑟 = 𝜋𝜌𝑟 (𝑀) Where 𝜋𝜌𝑟 denotes a cyclic permutation of the rows of 𝑀 by 𝜌𝑟 positions. Proposition 1. If 𝑀𝜖𝔽28 4×4 is invertible, then 𝑀𝑟 = 𝜋𝜌𝑟 (𝑀) is also invertible for any 𝜌𝑟 ∈ {0,1,2,3}. Proof. Cyclic row permutation is equivalent to left-multiplying 𝑀 by a permutation matrix 𝑃 𝜌𝑟 , which is invertible and orthogonal (i.e., 𝑃𝜌𝑟 −1 = 𝑃𝜌𝑟 𝑇 ): 𝑀𝑟 = 𝜋𝜌𝑟 (𝑀) Since both 𝑃 𝜌𝑟 and 𝑀 are invertible, their product 𝑀𝑟 is also invertible: 𝑀𝑟 −1 = 𝑀−1𝑃𝜌𝑟 −1 Therefore, each 𝑀𝑟 used in encryption has a well-defined inverse transformation for decryption: 𝑆 = 𝑀𝑟 −1. 𝑆′ = 𝑀−1𝑃𝜌𝑟 −1. 𝑆′ Hence, the decryption process can be symmetrically defined using the inverse permutation and the inverse MixColumn matrix. 3.3. Diffusion Properties One of the primary goals of the MixColumn operation is to ensure that a single-byte change in the input affects multiple bytes in the output, a property known as diffusion. In standard AES, each column 𝑆𝜖𝔽28 4×1 is transformed by a fixed MDS (Maximum Distance Separable) matrix M. The diffusion effect is measured by the branch number 𝐵, defined as: 𝐵(𝑀) = min 𝑥≠0 (𝑤𝑡(𝑥) + 𝑤𝑡(𝑀. 𝑥)) Where 𝑤𝑡(𝑥) is the Hamming weight (number of non-zero components) of the vector 𝑥. Since our transformation is based on permutations of the rows of 𝑀, the resulting matrix 𝑀𝑟 maintains the same set of rows, only reordered. This implies that the branch number is preserved under permutation. Proposition 2. For all 𝜌𝑟 ∈ {0,1,2,3}, the branch number satisfies 𝐵(𝑀𝑟) = 𝐵(𝑀). Proof. Row permutation does not alter the linear dependencies between input and output components, nor does 440 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 5: 436-443, 2025 DOI: 10.55214/25768484.v9i5.6886 © 2025 by the authors; licensee Learning Gate it affect the weight distribution of the transformation matrix. Therefore, the minimum combined weight remains unchanged across all 𝑀𝑟, preserving diffusion strength. 3.4. Key-Dependent Unpredictability By varying 𝑀𝑟 per round based on a value 𝜌𝑟 extracted from the round key, we introduce round- dependent structural variation. Since the round key 𝐾𝑟 is a function of the secret master key, the permutation index 𝜌𝑟 becomes unpredictable to an adversary without key knowledge. Furthermore, this variability introduces a layer of non-linearity in the key schedule interaction, increasing resistance to: • Linear cryptanalysis: due to variability in the diffusion layer. • Algebraic attacks: because the transformation matrix is no longer constant and cannot be eliminated or algebraically modeled as fixed coefficients. 4. Security Discussion The security of block ciphers relies heavily on confusion and diffusion principles as articulated by Claude Shannon. While AES exhibits strong resistance to most classical attacks, the static nature of its MixColumn operation may provide structural predictability exploitable in certain advanced attack scenarios. In this section, we discuss how the proposed key-dependent cyclic MixColumn transformation enhances security by introducing dynamic variability, maintaining diffusion strength, and complicating adversarial modeling. 4.1. Resistance to linear and differential cryptanalysis In classical differential cryptanalysis, an attacker studies how input differences propagate through the cipher to infer key-related information. Similarly, linear cryptanalysis seeks affine approximations between plaintext, ciphertext, and key bits. Because the proposed transformation matrix 𝑀𝑟 = 𝜋𝜌𝑟 (𝑀) varies each round based on a value 𝜌𝑟 ∈ {0,1,2,3} extracted from the round key, the adversary no longer operates against a static linear transformation. The diffusion characteristics remain preserved due to the invariant branch number (as proven in Section 3), but the attack surface becomes more complex. Let ∆𝑆 represent a difference in input state columns. The output difference after the MixColumn becomes: ∆𝑆′ = 𝑀𝑟 . ∆𝑆 Since 𝑀𝑟 varies with 𝐾𝑟, each round uses a distinct transformation. Therefore, an attacker attempting to build a differential characteristic must account for a sequence of different matrices {𝑀1, 𝑀2, … , 𝑀3}, each dependent on round keys, which are not known in advance. This disrupts the construction of stable differential trails or linear approximations across rounds. Thus, the cipher exhibits increased round unpredictability, a critical advantage in defending against statistical attacks. 4.2. Resistance to Algebraic Attacks Algebraic attacks exploit systems of equations describing the cipher's operations. In AES, fixed S- box and linear layer components allow attackers to build compact algebraic representations for analysis. Our method disrupts this model: since 𝑀𝑟 changes every round, the system of equations modeling the cipher becomes non-uniform and parameterized by secret key-derived values. Specifically, the cyclic row permutation index 𝜌𝑟 changes the coefficients of the transformation matrix in a key-dependent manner, increasing the algebraic complexity. 441 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 5: 436-443, 2025 DOI: 10.55214/25768484.v9i5.6886 © 2025 by the authors; licensee Learning Gate If an attacker models the encryption as a polynomial system 𝒫 over 𝔽28, then each 𝑀𝑟 introduces fresh variables into 𝒫𝑟 . Hence, the overall system cannot be statically precomputed, reducing the feasibility of Gröbner basis or SAT-based algebraic attacks. 4.3. Related-Key and Structural Attacks Related-key attacks rely on observing the behavior of the cipher under known relations between keys. In traditional AES, the linearity of MixColumn and its independence from the key may allow certain relations to persist. By tying the MixColumn matrix directly to the round key through 𝜌𝑟, such relations are disrupted. Even related round keys 𝐾𝑟 and 𝐾𝑟 ′ may yield different 𝜌𝑟 values, thus different transformation matrices 𝑀𝑟, resulting in unrelated diffusion paths. Moreover, structural attacks that exploit repeated internal patterns are mitigated, since our transformation ensures that no two rounds necessarily share the same MixColumn transformation, assuming the round keys are non-repetitive (which is generally true in AES key schedules). 4.4. Implementation Considerations and Overhead While this work is theoretical in nature and does not require implementation, we note that the proposed method introduces no increase in matrix size or field complexity. The operations are performed over 𝔽28 as in standard AES. The only added operation is the generation of the permutation index 𝜌𝑟 from the round key a simple byte-level modular operation. This suggests that even if implemented, the method would incur negligible overhead while offering enhanced variability and improved resistance to structure-dependent attacks. 5. Conclusion and Future Work In this paper, we introduced a mathematically grounded enhancement to the AES MixColumn transformation by proposing a key-dependent cyclic permutation mechanism. The core idea is to extract a round-specific permutation index from the round key and apply it to the rows of the standard MixColumn matrix, resulting in a dynamic, round-varying linear transformation. We formally proved that the proposed transformation preserves invertibility and diffusion strength, specifically maintaining the branch number of the AES MixColumn matrix. Moreover, the variation introduced at each round is driven entirely by the key schedule, rendering the cipher more resilient to differential, linear, and algebraic attacks, while incurring no increase in matrix complexity or underlying field operations. The theoretical contribution lies in showing that simple structural variation, when tied to key material, can enhance cryptographic robustness without sacrificing formal properties such as linear independence or efficient invertibility. The following are a few directions for potential areas for future research: • Theoretical Bound on Branch Number Stability: While the branch number is preserved under row permutations, further generalization to other types of key-dependent transformations (e.g., column permutations or coefficient rotations) warrants formal analysis. • Provable Security under Composition: A formal security proof under existing cryptographic models (e.g., PRP or IND-CPA frameworks) would offer a deeper understanding of this transformation’s impact in composition with the rest of AES operations. • Matrix Class Characterization: A mathematical characterization of the class of all invertible matrices obtainable via cyclic permutations of AES's MixColumn could help in assessing the total variability and potential design space. 442 Edelweiss Applied Science and Technology ISSN: 2576-8484 Vol. 9, No. 5: 436-443, 2025 DOI: 10.55214/25768484.v9i5.6886 © 2025 by the authors; licensee Learning Gate • Generalization to Other Block Ciphers: The approach may be extended to other SPN-based ciphers that utilize MDS matrices in their linear layer, allowing for broader applicability of key- dependent transformations. 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Elshoush, "A polymorphic advanced encryption standard – A novel approach," IEEE Access, vol. 9, pp. 20191-20207, 2021. 10.1109/access.2021.3051556 Appendix A. Example of Key-Dependent MixColumn Permutations. Let the original AES MixColumn matrix 𝑀𝜖𝔽28 4×4 be: 𝑀 = [ 2 3 1 1 1 2 3 1 1 1 2 3 3 1 1 2 ] Let the cyclic row permutation index 𝜌𝑟 for each round 𝑟 be extracted from a round key byte 𝑘𝑟 using: 𝜌𝑟 = 𝑘𝑟 𝑚𝑜𝑑 4 Example: Round-wise Matrix Transformations Round r Key Byte Kr 𝝆𝒓 = 𝒌𝒓 𝒎𝒐𝒅 𝟒 Permutation Description Transformed Mr 1 0x73 3 Rotate rows down by 3 𝑀1 = 𝜋3(𝑀) 2 0xB1 1 Rotate rows down by 1 𝑀2 = 𝜋1(𝑀) 3 0x2A 2 Rotate rows down by 2 𝑀3 = 𝜋2(𝑀) 4 0x0C 0 No rotation (standard AES) 𝑀4 = 𝑀 The actual row-permuted matrices are: 1. 𝑀1 = 𝜋3(𝑀) → [ 3 1 1 2 2 3 1 1 1 2 3 1 1 1 2 3 ] 2. 𝑀2 = 𝜋1(𝑀) → [ 1 2 3 1 3 1 1 2 2 3 1 1 1 2 3 1 ] 3. 𝑀3 = 𝜋2(𝑀) → 𝑀 = [ 1 1 2 3 1 2 3 1 3 1 1 2 2 3 1 1 ] 4. 𝑀4 = 𝑀 → 𝑀 = [ 2 3 1 1 1 2 3 1 1 1 2 3 3 1 1 2 ] https://doi.org/10.32985/ijeces.16.3.6 https://doi.org/10.1007/s10825-024-02248-4 https://doi.org/10.3390/genes1030413