EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6470 ISSN 1307-5543 – ejpam.com Published by New York Business Global Multiple RGB Images Security based on Substitution–Permutation Network over the Residue Classes of Eisenstein Integer Z[Ω]π Maha Alammari1, Muhammad Sajjad2,∗, Mushtaq K. Abdalrahem3, Robinson-Julian Serna4 1 Department of Mathematics, College of Science, King Saud University, P.O. Box 22452 Riyadh 11495, Saudi Arabia 2 NUTECH School of Applied Science and Humanities, National University of Technology, Islamabad, 44000, Pakistan 3 College of Pharmacy, University of Al-Ameed 4 Escuela de Matemáticas y Estad́ıstica, Universidad Pedagógica y Tecnológica de Colombia, Tunja, Colombia Abstract. This paper presents a multiple RGB image encryption scheme that utilizes a pair of 8×8 S-boxes constructed over the residue classes of Eisenstein integers Z[Ω]π, implemented within a three-stage Substitution–Permutation Network (SPN) architecture. The S-boxes are generated using Eisenstein integer algebra through affine transformations and their corresponding inverse functions, ensuring strong nonlinearity. The first S-box serves as a substitution function, while the second contributes to both permutation and diffusion. Enhanced cryptographic strength is achieved through modular arithmetic in Z[Ω]π, which supports essential encryption properties such as confusion and diffusion. Further complexity is introduced by combining the two S-boxes via an XOR operation to construct a third S-box, promoting greater inter-channel diffusion among the RGB components. The proposed SPN framework is designed to resist differential and linear cryptanalysis through its layered substitution, permutation, and XOR-based mixing operations. Separate yet interlinked processing pathways for each image channel ensure secure and efficient encryption. Experimental evaluations validate the proposed method, demonstrating high entropy, low inter-channel correlation, and robust resistance to various attacks, making it a strong candidate for secure multimedia communication applications. 2020 Mathematics Subject Classifications: 11T71, 14G50, 94A60, 81P94, 16S38, 97G70 Key Words and Phrases: Eisenstein integers, Multiple Image Encryption, Substitution Permu- tation Network, Security Analysis ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6470 Email addresses: malammari@ksu.edu.sa (M. Alammari), muhammad.sajjad@nutech.edu.pk (M. Sajjad), mushtaq.k@alameed.edu.iq (M. K. Abdalrahem), robinson.serna@uptc.edu.co (R-J. Serna) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) M. Alammari et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6470 2 of 30 1. Introduction Cryptography is the practice of transforming information into a format that is un- readable to unauthorized users, ensuring confidentiality and security. Historically, crypto- graphic techniques have evolved from simple substitution ciphers to complex mechanical and digital systems. With the rise of digital communication, modern cryptography has be- come essential for protecting privacy, ensuring data integrity, and enabling secure authen- tication. Contemporary encryption systems incorporate advanced mathematical founda- tions and sophisticated algorithms that far surpass traditional approaches. Cryptographic schemes are broadly categorized into symmetric and asymmetric systems. Symmetric cryptography uses a single key for both encryption and decryption, while asymmetric cryptography employs a public-private key pair. Both types play a crucial role in securing data through hash functions and digital signatures, which verify authenticity and prevent tampering. As digital environments continue to expand, cryptography evolves to include quantum-resistant methods and stronger encryption strategies [1, 2]. Substitution–permutation networks (SPNs) are a fundamental cryptographic frame- work employed in modern block ciphers. Within SPNs, substitution boxes (S-boxes) serve as nonlinear components that significantly enhance security by introducing con- fusion—scrambling input so that even small changes produce vastly different outputs. Strong S-boxes are designed to ensure balanced output distribution, high nonlinearity, and avalanche effects. Algebraic structures, random mappings, and optimization tech- niques are used to construct secure S-boxes, and current research explores novel designs over alternative mathematical domains to improve both robustness and efficiency [3]. RGB image encryption is a critical subfield of cryptography concerned with securing digital image content against unauthorized access or tampering. An RGB image comprises three color channels—red, green, and blue—each with its own pixel intensity matrix. Un- like text encryption, image encryption must handle large data volumes, spatial redundancy, and high inter-pixel correlation. To address these challenges, encryption schemes employ pixel-level substitution, permutation, and diffusion techniques. When encrypting multiple RGB images, added complexity is required to prevent inter-image correlation. XOR-based transformations, key-dependent operations, and multi-channel mixing enhance security by neutralizing differential attacks and preserving uniqueness across encrypted datasets. This demand is especially high in domains such as cloud storage, secure image sharing, and medical imaging [4, 5]. Beyond conventional number systems, Eisenstein integers—complex numbers of the form a + bω, where ω is a primitive cube root of unity—offer a rich algebraic struc- ture with notable symmetry and well-defined modular arithmetic. These properties make Eisenstein integers suitable for secure cryptographic system design, particularly for non- linear transformations within SPNs. Their inherent mathematical features provide natural resistance against common cryptographic attacks, and they continue to attract attention for applications in coding theory, encryption, and signal processing [6–10]. With the growing demands for secure storage and transmission of visual data, substi- tution–permutation structures remain central to modern cryptographic design. The role M. Alammari et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6470 3 of 30 of chaotic maps, conservative hyperchaotic systems, Knuth–Durstenfeld algorithms, and Hopfield neural networks in image encryption has also been widely studied [11–16]. Re- cent works have integrated one-time keys, spatial permutations, DNA-based operations, and Boolean networks for enhancing encryption resilience. Optical asymmetric key cryp- tosystems, pixel exchange schemes, and spectral transformations are also employed to address efficiency and attack resistance [17–20]. The Eisenstein integer domain has in- creasingly been explored for cryptographic component design, extending from its roots in error-correcting code construction to advanced image encryption systems [6, 7, 9]. Furthermore, recent contributions have shown the feasibility of RGB image encryption using Gaussian integers within SPN frameworks [21–24]. The use of quaternion integers and BCH-code constructions over finite fields has strengthened digital communication se- curity by reducing vulnerability to analytical attacks [25]. Integration of chaotic maps with elliptic curve cryptography, quantum-walk-based pseudo-random number generators, and multimedia steganography has enabled encrypted image concealment within diverse content [26–28]. Parallel computing techniques have also accelerated encryption algo- rithms, enabling real-time applications [29]. Other breakthroughs include compressive ghost imaging and 3D permutation models for multi-image protection [30, 31]. The increasing reliance on multiple-image encryption in military communication, med- ical diagnostics, and secure cloud systems reveals the limitations of early cryptographic methods. Despite the robustness of classical approaches such as RSA and protocols in the Handbook of Applied Cryptography, they can fall short when processing high-volume multimedia data [1, 2]. While chaotic maps [4], DNA coding [15, 32], and neural net- works [13] have addressed image security to some extent, they often suffer from limited key sensitivity or high computational costs. Recent developments involving quaternion and Gaussian integers offer promising alternatives [22–25], and Eisenstein integers present a fresh avenue for developing secure block ciphers [7, 9]. In this paper, we propose a novel encryption framework based on Eisenstein integers, implemented within a three-stage SPN architecture for the encryption of multiple RGB images. Unlike conventional designs, our scheme employs a pair of 8× 8 S-boxes over the residue classes of Eisenstein integers Z[Ω]π, ensuring high nonlinearity and strong confu- sion properties. Affine transformations and their inverses enhance resistance to cryptana- lytic attacks. Additionally, an XOR-derived third S-box is used to increase inter-channel diffusion across RGB components. The system is designed for high entropy, minimal correlation, and effective defence against differential and linear cryptanalysis. The remainder of this paper is structured as follows. Section 2 introduces the math- ematical background, including Eisenstein integers and relevant residue class theorems. Section 3 describes the construction of cryptographically secure S-boxes over Eisenstein primes. Section 4 details the encryption process for multiple RGB images using the pro- posed SPN model. Section 5 presents a comprehensive security and performance evalua- tion. Finally, Section 6 concludes the paper and outlines future research directions. Our results demonstrate that modular arithmetic over Eisenstein residue classes can offer ro- bust multimedia encryption with strong empirical validation and high resistance to known attacks. M. Alammari et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6470 4 of 30 2. Eisenstein Integers and Their Properties Following the discussions in [6–10], Eisenstein integers (EIs) form a distinguished subset of the complex numbers, denoted by Z[Ω] = {a+ bΩ | a, b ∈ Z}, where Ω = (−1 + √ 3i)/2 is a primitive cube root of unity. This set forms a commutative ring with identity under addition and multiplication. Let z = a+ bΩ be an Eisenstein integer. Its complex conjugate is defined as: z̄ = a+ bΩ̄ = a+ bΩ2, where Ω2 is the complex conjugate of Ω. The norm of z is then given by: N(z) = zz̄ = (a+ bΩ)(a+ bΩ2) = a2 − ab+ b2. Theorem 1 ([7, 8, 10]). Let p be a rational prime. Then there exists an Eisenstein prime c ∈ Z[Ω] such that N(c) = cc̄ = p if and only if p is not a prime in Z[Ω]. Theorem 2 ([7, 8, 10]). If the norm N(c) of an Eisenstein integer c is a rational prime, then c is a prime in Z[Ω]. 2.1. Residue Class of Eisenstein Integers [7, 8] Given c = a+ bΩ ∈ Z[Ω], the residue class modulo c is denoted as Z[Ω]c. The modulo operation f : Z[Ω] → Z[Ω]c is defined by: f(x) = y mod c = x− ⌊xc̄ cc̄ ⌋ c, where the floor operation is applied separately to the real and imaginary parts to ensure that y ∈ Z[Ω]c remains an Eisenstein integer. 2.2. Eisenstein Mannheim Weight and Distance [7, 8] Let β, γ ∈ Z[Ω]c and define α = γ−β = c+dΩ as an Eisenstein integer. The Eisenstein Mannheim weight WEM(α) is defined by: WEM(α) = |c|+ |d|. The Eisenstein Mannheim distance between β and γ is: dEM(β, γ) = WEM(γ − β). Proposition 1 ([7, 8]). Let δk = ck + dkΩ be distinct primes in Z[Ω]c such that pk = c2k − ckdk + d2k are distinct rational primes. If Z[Ω]∗ (δk) is generated by α, then: αϕ(pk)/2 ≡ −1 (mod δk). M. Alammari et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6470 5 of 30 Proposition 2 ([7, 8]). Let δ1 = c1 + d1Ω and δ2 = c2 + d2Ω be two prime Eisenstein integers with corresponding rational primes p = c21 − c1d1 + d21 and q = c22 − c2d2 + d22. Then, there exist elements e, f ∈ Z[Ω]∗(δ1δ2) such that: fϕ(p) ≡ 1 (mod δ1δ2), eϕ(q) ≡ 1 (mod δ1δ2). Proposition 3 ([7]). Let δk = ck+dkΩ be distinct Eisenstein primes and pk = c2k−ckdk+d2k distinct rational primes. Then there exists an element τk ∈ Z[Ω]∗(δ1δ2···δk) such that: τ ϕ(pk) k ≡ 1 (mod δ1δ2 · · · δk), for k = 1, 2, . . . ,m. Theorem 3 ([7]). Let c be an Eisenstein prime with norm N(c), and let α ̸= 0 be a nonzero Eisenstein integer. Then: αN(c) ≡ α (mod c). Theorem 4 ([7, 8]). If ⟨α⟩ = ⟨a+bΩ⟩ is a principal ideal in Z[Ω] with gcd(a, b) = 1, then: Z[Ω]/⟨a+ bΩ⟩ ∼= Za2−ab+b2 . 3. Redesigning of n× n S-boxes over the Eisenstein Integers Ensuring strong security remains a fundamental requirement in modern cryptographic systems. Among the core techniques that enhance cryptographic resilience, the introduc- tion of confusion plays a central role. Substitution boxes (S-boxes) are widely employed nonlinear components in symmetric ciphers and contribute significantly to the overall se- curity by disrupting statistical patterns and increasing unpredictability. The strength of an S-box improves significantly when constructed using robust algebraic structures, such as the Eisenstein integers (EIs). The rich mathematical properties of EIs support the development of cryptographically strong S-boxes with enhanced nonlinearity, optimal differential uniformity, and high avalanche effect. This section presents a step-by-step algorithm for constructing n× n S-boxes over the Eisenstein integer residue class ring. 3.1. Step-by-Step Construction Methodology (i) Group Formation: Use the definitions and theorems outlined in Section 2 to form a cyclic group G ⊂ Z[Ω]∗ of order p− 1, where p = N(E) is a rational prime derived from the norm of an Eisenstein prime E. (ii) Mapping Function: Define a nonlinear mapping over the cyclic groupG = {x1, x2, . . . , xp−1} by: g(xi) = 1 ax−1 i + b , where x−1 i is the multiplicative inverse of xi ∈ G, and a, b ∈ Z[Ω]\{0} are parameters chosen to ensure bijectivity and cryptographic complexity. M. Alammari et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6470 6 of 30 (iii) Component Separation: Decompose each g(xi) into its non-Ω (real) and Ω-part (imaginary) components: g(xi) = ui + viΩ. (iv) Modulo Operation: Apply modular reduction over 2n to both components inde- pendently to generate two sets: g1 = {ui mod 2n}, g2 = {vi mod 2n}. (v) Affine Transformation: Define affine transformations over the sets g1 and g2 as: h(xi) = (cxi + d) mod 2n, where c, d ∈ Z are carefully selected constants that maximize nonlinearity and cryp- tographic resistance [24]. (vi) Final S-box Pair: Construct the final S-boxes S1 and S2 by applying the affine transformation to the elements of g1 and g2, respectively: S1(i) = h(ui), S2(i) = h(vi), for i = 0, 1, . . . , 2n − 1. 3.2. Significance of Eisenstein Integer-Based S-boxes The proposed method yields S-boxes with strong cryptographic characteristics, includ- ing: • Enhanced arithmetic capabilities through Eisenstein modular operations, • Improved resistance to differential and linear attacks via optimized algebraic struc- ture, • Increased statistical randomness and nonlinearity due to the combined mapping and transformation stages. These properties make Eisenstein integer-based S-boxes highly suitable for use in sub- stitution–permutation networks (SPNs), block ciphers, and multimedia security systems. 3.3. Construction of 8× 8 S-boxes over Eisenstein Integers To demonstrate the practicality of the proposed algorithm, we construct a pair of 8×8 S-boxes over the Eisenstein residue class ring using an Eisenstein prime: E = 73 + 31Ω. The norm of this Eisenstein integer is: N(E) = 732 − 73 · 31 + 312 = 4027, which is a prime number in Z. Therefore, the residue class ring Z[Ω]E forms a finite field, and the generator δ ∈ Z[Ω]∗E is selected accordingly. By applying the aforementioned construction steps and selecting appropriate transfor- mation constants a, b, c, d, we generate the final S-box pair S1 and S2. These are presented in Tables 1 and 2, respectively. M. Alammari et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6470 7 of 30 Table 1: S1 over the non-Omega Part of EI 250 59 179 169 149 22 214 248 6 137 237 138 14 16 68 129 2 163 37 231 75 12 81 4 102 100 108 86 26 216 91 173 255 193 51 15 139 57 160 210 21 143 95 199 241 132 63 64 178 98 65 176 166 85 42 183 17 191 221 218 242 54 168 38 41 69 130 141 88 202 181 245 56 148 180 253 31 233 171 89 87 49 48 189 55 110 119 27 201 46 34 114 156 225 7 219 44 164 23 243 162 125 10 128 36 251 8 246 92 213 52 157 40 107 235 220 53 120 134 232 249 198 99 190 9 61 206 142 71 159 165 155 228 106 150 203 236 5 24 28 1 111 226 83 101 230 97 121 3 96 186 29 153 123 172 118 144 167 184 254 131 196 234 135 66 174 47 182 94 208 170 79 13 77 90 187 43 145 112 0 122 18 33 154 39 50 158 67 74 115 244 152 212 45 229 70 136 25 192 205 204 73 105 252 124 133 211 209 72 146 127 93 185 151 224 217 19 84 140 60 103 82 177 58 175 117 104 247 239 116 30 207 80 197 194 188 147 35 76 200 20 238 11 113 240 32 126 78 109 215 222 227 195 223 161 62 3.4. Nonlinearity Analysis of Eisenstein Integer-Based S-boxes S-boxes derived from Eisenstein integers demonstrate strong nonlinearity properties, which critically enhance their resistance against linear and differential cryptanalysis. Due to their inherent hexagonal lattice structure, Eisenstein integers provide a unique mathe- matical foundation for S-box construction that supports advanced security features. Non- linearity in an S-box is quantitatively defined as the smallest Hamming distance between the output Boolean functions of the S-box and the set of all affine functions. Higher non- linearity values directly correlate with stronger resistance against linear approximation attacks. The symmetric and multiplicative properties of Eisenstein integers, when applied to S-box construction, yield nonlinearity metrics that match or surpass those observed in classical S-boxes defined over finite fields of characteristic two. Through Eisenstein integer-based transformations, the input differentials are dispersed more uniformly, en- abling the S-box to achieve both high nonlinearity and low differential uniformity. These characteristics are essential for thwarting differential attacks by reducing the probability of predictable output differences. Simulation-based experiments and theoretical evaluations confirm that symmetric-key cryptographic schemes employing Eisenstein-integer-based S- boxes offer elevated levels of security. The comparative analysis with standard S-box constructions is presented in Tables 3 and 4, highlighting the robustness of the proposed S-box architecture against known cryptanalytic techniques [24, 27, 31]. 3.5. Bit Independence Criterion (BIC) The Bit Independence Criterion (BIC) is a vital statistical measure for evaluating the resistance of S-boxes against linear and differential cryptanalysis. It assesses how inde- M. Alammari et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6470 8 of 30 Table 2: S2 over the Omega Part of EI 122 187 51 41 21 150 86 120 134 9 109 10 142 144 196 1 130 35 165 103 203 140 209 132 230 228 236 214 154 88 219 45 127 65 179 143 11 185 32 82 149 15 223 71 113 4 191 192 50 226 193 48 38 213 170 55 145 63 93 90 114 182 40 166 169 197 2 13 216 74 53 117 184 20 52 125 159 105 43 217 215 177 176 61 183 238 247 155 73 174 162 242 28 97 135 91 172 36 151 115 34 253 138 0 164 123 136 118 220 85 180 29 168 235 107 92 181 248 6 104 121 70 227 62 137 189 78 14 199 31 37 27 100 234 22 75 108 133 152 156 129 239 98 211 229 102 225 249 131 224 58 157 25 251 44 246 16 39 56 126 3 68 106 7 194 46 175 54 222 80 42 207 141 205 218 59 171 17 240 128 250 146 161 26 167 178 30 195 202 243 116 24 84 173 101 198 8 153 64 77 76 201 233 124 252 5 83 81 200 18 255 221 57 23 96 89 147 212 12 188 231 210 49 186 47 245 232 119 111 244 158 79 208 69 66 60 19 163 204 72 148 110 139 241 112 160 254 206 237 87 94 99 67 95 33 190 Table 3: Nonlinearity of Proposed S-boxes Functions S-boxes g1 g2 g3 g4 g5 g6 g7 g8 S1 106.00 106.00 108.00 108.00 106.00 108.00 106.00 108.00 S2 106.00 106.00 108.00 108.00 106.00 108.00 106.00 108.00 pendently output bits respond when a single input bit is flipped, revealing the presence or absence of linear correlations or predictable dependencies among output bits. Eisenstein- integer-based S-boxes utilize their rich algebraic and symmetric structures to disrupt these correlations effectively. Their hexagonal lattice foundation and modular arithmetic con- tribute to a highly nonlinear and unpredictable mapping between inputs and outputs. These mathematical features lead to enhanced propagation of bit differences across the output, minimizing potential vulnerabilities exploitable by attackers. The BIC evaluation for the constructed S-boxes S1 and S2 demonstrates their strong statistical independence among output bits. Computational simulations and empirical analyses validate that both S-boxes meet high BIC standards, indicating robust protection against both linear and Table 4: Average Nonlinearity Comparisons S-boxes Schemes Nonlinearity S1 Proposed (EI) 107.00 S2 Proposed (EI) 107.00 [27] Elliptic Curve 104.00 [24] Gaussian Integers 106.50 [31] Chaotic Map 104.70 M. Alammari et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6470 9 of 30 differential attack strategies [25, 27]. The BIC matrices for S1 and S2 are presented in Tables 5 and 6, respectively, followed by a comparative summary with existing literature in Table 7. Table 5: BIC Analysis Matrix for S-box S1 0.000 0.506 0.516 0.514 0.516 0.486 0.512 0.490 0.506 0.000 0.490 0.486 0.510 0.504 0.498 0.473 0.516 0.490 0.000 0.502 0.529 0.479 0.516 0.471 0.514 0.486 0.502 0.000 0.510 0.500 0.512 0.484 0.516 0.510 0.529 0.510 0.000 0.527 0.518 0.518 0.486 0.504 0.479 0.500 0.527 0.000 0.502 0.506 0.512 0.498 0.516 0.512 0.518 0.502 0.000 0.492 0.490 0.473 0.471 0.484 0.518 0.506 0.492 0.000 Table 6: BIC Analysis Matrix for S-box S2 0.000 0.506 0.516 0.514 0.516 0.486 0.512 0.490 0.506 0.000 0.490 0.486 0.510 0.504 0.498 0.473 0.516 0.490 0.000 0.502 0.529 0.479 0.516 0.471 0.514 0.486 0.502 0.000 0.510 0.500 0.512 0.484 0.516 0.510 0.529 0.510 0.000 0.527 0.518 0.518 0.486 0.504 0.479 0.500 0.527 0.000 0.502 0.506 0.512 0.498 0.516 0.512 0.518 0.502 0.000 0.492 0.490 0.473 0.471 0.484 0.518 0.506 0.492 0.000 Table 7: Comparison of BIC Scores with Existing Literature S-boxes Maximum Minimum Average S1 (Proposed) 0.594 0.391 0.502 S2 (Proposed) 0.594 0.391 0.502 [27] 0.543 0.473 0.503 [25] 0.609 0.375 0.505 3.6. Strict Avalanche Criterion (SAC) A secure cryptographic evaluation of S-box remaining strength depends on the SAC, which guarantees input bit alterations result in output bit changes with a 50% probabil- ity for each bit. The algebraic features of Eisenstein integers create a special foundation for building S-boxes which demonstrate exceptional diffusion properties. Complex trans- formations in this structure distribute non-linearly the input variations throughout the output bits, thus improving avalanche results. A SAC-compliant S-box requires that each output bit position shows complete independence to input changes and maintains equal M. Alammari et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6470 10 of 30 probability of change, which makes it impossible for adversaries to predict bit transitions. Arithmetic operations inside this Eisenstein lattice domain establish complex bit interre- lationships which make the system secure. Results from computational assessments show that Eisenstein-integer-based S-boxes achieve an almost perfect SAC because their output bits display uniform change distribution after any input bit modification. SAC resistance against differential cryptanalysis depends on this property, which makes symmetric cryp- tographic systems incorporating the mentioned S-box design methods more secure [25, 27]. SAC reports a comparison of proposed results against existing literature through Tables 8, 9, and 10. Table 8: SAC Analysis of S1 0.484 0.500 0.516 0.531 0.531 0.484 0.500 0.547 0.531 0.500 0.438 0.563 0.484 0.500 0.547 0.500 0.578 0.422 0.563 0.500 0.500 0.563 0.531 0.422 0.516 0.500 0.500 0.500 0.531 0.422 0.516 0.500 0.406 0.500 0.563 0.547 0.500 0.500 0.438 0.500 0.516 0.500 0.547 0.516 0.578 0.484 0.531 0.500 0.469 0.469 0.516 0.516 0.531 0.563 0.531 0.375 0.531 0.453 0.531 0.547 0.516 0.484 0.594 0.531 Table 9: SAC Analysis of S2 0.484 0.500 0.516 0.531 0.531 0.484 0.500 0.547 0.531 0.500 0.438 0.563 0.484 0.500 0.547 0.500 0.578 0.422 0.563 0.500 0.500 0.563 0.531 0.422 0.516 0.500 0.500 0.500 0.531 0.422 0.516 0.500 0.406 0.500 0.563 0.547 0.500 0.500 0.438 0.500 0.516 0.500 0.547 0.516 0.578 0.484 0.531 0.500 0.469 0.469 0.516 0.516 0.531 0.563 0.531 0.375 0.531 0.453 0.531 0.547 0.516 0.484 0.594 0.531 Table 10: Comparison of SAC with Existing Literature S-boxes Maximum Minimum Average S1 0.594 0.375 0.508 S2 0.594 0.375 0.508 [27] 0.610 0.422 0.516 [25] 0.594 0.406 0.504 M. Alammari et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6470 11 of 30 3.7. Linear Approximation Probability The Linear Approximation Probability (LAP) serves as a key cryptographic property of S-boxes made from Eisenstein integers because it measures the highest possible corre- lation differences between linear input-output expressions, thereby determining the resis- tance against linear cryptanalysis. Eisenstein integers provide complex algebraic properties which increase the mapping complexities of S-boxes to protect them from linear approx- imation attacks. A Low Probability of Approximation (LAP) in an S-box functions to protect security because it prevents the determination of output-input relationships with probabilities greater than 0.5 through linear equations. S-boxes built from Eisenstein field arithmetic show highly non-linear behavior because they spread input variations through nonlinear modular arithmetic operations. The selection and optimization process for Eisenstein integer mappings enables these S-boxes to demonstrate minimal bias in lin- ear approximation, which results in low maximal LAP metrics. Eisenstein-integer-based S-boxes demonstrate strong resistance against linear attacks through empirical testing and theoretical evaluation because they function well in secure encryption algorithms [25, 27]. A comparison between the proposed work and existing literature regarding the LAP can be seen in Table 11. Table 11: Comparison of LAP with Existing Literature S-boxes LAP S1 0.125 S2 0.125 [27] 0.148 [25] 0.133 3.8. Differential Approximation Probability The Differential Approximation Probability (DAP) represents a fundamental criterion for evaluating the resistance of S-boxes generated by Eisenstein integers because it de- termines the maximum probability of detecting output differentials from particular input differentials. Eisenstein integers establish an original mathematical framework which leads to better distribution of data while minimizing the ability to predict differential charac- teristics. A low DAP value stops attackers from detecting any specific differential output relation based on input variations since it generates unknown output differences based on every input combination. The nonlinear outcome of complex multiplication with Eisen- stein integers within the domain uses modular arithmetic to distribute input differences randomly throughout the output dimensions. The precise construction of Eisenstein inte- ger mapping functions leads these S-boxes to possess minimal differential uniformity which ensures that any differential pair remains unlikely to happen. The strong resistance to differential cryptanalysis demonstrated by Eisenstein-integer-based S-boxes qualifies them as ideal components for secure encryption algorithms that need powerful nonlinearity and M. Alammari et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6470 12 of 30 diffusion characteristics [25, 27]. The proposed results undergo DAP with literature com- parison through Tables 12, 13, and 14. Table 12: DAP Analysis of S1 0.02344 0.02344 0.03125 0.03906 0.02344 0.02344 0.02344 0.03125 0.02344 0.03125 0.02344 0.03125 0.02344 0.02344 0.02344 0.02344 0.02344 0.03906 0.02344 0.02344 0.02344 0.03125 0.02344 0.03125 0.03125 0.02344 0.02344 0.02344 0.02344 0.02344 0.03125 0.02344 0.02344 0.03125 0.02344 0.03125 0.03125 0.02344 0.02344 0.02344 0.02344 0.03125 0.02344 0.02344 0.02344 0.03125 0.02344 0.03125 0.02344 0.02344 0.03125 0.03125 0.02344 0.03125 0.03125 0.02344 0.03906 0.02344 0.03906 0.03125 0.02344 0.02344 0.03125 0.03125 0.03125 0.02344 0.02344 0.02344 0.02344 0.02344 0.03125 0.02344 0.02344 0.02344 0.02344 0.02344 0.02344 0.02344 0.02344 0.02344 0.02344 0.02344 0.03125 0.03125 0.02344 0.02344 0.02344 0.02344 0.02344 0.02344 0.02344 0.02344 0.02344 0.02344 0.02344 0.02344 0.02344 0.03125 0.03125 0.02344 0.03125 0.03125 0.03125 0.02344 0.02344 0.03906 0.02344 0.02344 0.03125 0.02344 0.02344 0.03125 0.02344 0.03125 0.02344 0.03125 0.03125 0.02344 0.02344 0.02344 0.03125 0.03125 0.02344 0.02344 0.03125 0.02344 0.03125 0.03125 0.02344 0.02344 0.03125 0.03125 0.03906 0.02344 0.03125 0.03125 0.03125 0.02344 0.03906 0.02344 0.02344 0.02344 0.02344 0.02344 0.02344 0.02344 0.02344 0.02344 0.03125 0.02344 0.03125 0.02344 0.02344 0.02344 0.02344 0.03125 0.02344 0.03125 0.02344 0.02344 0.02344 0.02344 0.03125 0.02344 0.02344 0.01562 0.03125 0.02344 0.03906 0.03125 0.03125 0.03125 0.02344 0.02344 0.02344 0.02344 0.02344 0.03125 0.03125 0.02344 0.03125 0.02344 0.02344 0.02344 0.02344 0.03125 0.02344 0.03125 0.03906 0.02344 0.03125 0.02344 0.02344 0.03125 0.03125 0.03125 0.03125 0.02344 0.03125 0.02344 0.02344 0.03125 0.02344 0.02344 0.03125 0.02344 0.03125 0.03125 0.02344 0.02344 0.03125 0.03125 0.03125 0.03125 0.02344 0.02344 0.03906 0.03125 0.03125 0.03125 0.02344 0.02344 0.02344 0.02344 0.02344 0.02344 0.02344 0.02344 0.03125 0.02344 0.02344 0.02344 0.02344 0.02344 0.03125 0.02344 0.02344 0.02344 0.02344 0.02344 0.03906 0.02344 0.02344 0.03125 0.02344 0.02344 0.02344 0.03125 0.02344 0.03125 0.03125 0.03125 0.03906 0.03125 0.03125 0 Table 13: DAP Analysis of S2 0.02344 0.03906 0.02344 0.03125 0.02344 0.02344 0.03125 0.02344 0.03125 0.02344 0.02344 0.02344 0.02344 0.02344 0.03906 0.02344 0.02344 0.02344 0.02344 0.02344 0.02344 0.03125 0.03125 0.02344 0.02344 0.03125 0.02344 0.02344 0.03906 0.02344 0.03906 0.02344 0.02344 0.03125 0.03125 0.02344 0.02344 0.03125 0.02344 0.02344 0.02344 0.02344 0.02344 0.03125 0.03125 0.02344 0.03125 0.03125 0.03125 0.02344 0.03125 0.02344 0.03125 0.03125 0.02344 0.02344 0.03125 0.02344 0.02344 0.02344 0.02344 0.02344 0.02344 0.03906 0.02344 0.02344 0.02344 0.03125 0.02344 0.03125 0.03125 0.03125 0.02344 0.02344 0.03125 0.02344 0.03125 0.02344 0.01562 0.02344 0.02344 0.02344 0.02344 0.02344 0.03125 0.02344 0.03125 0.02344 0.02344 0.03125 0.02344 0.03125 0.03125 0.02344 0.03906 0.02344 0.02344 0.02344 0.02344 0.02344 0.02344 0.03125 0.03125 0.03125 0.03125 0.02344 0.02344 0.02344 0.02344 0.02344 0.02344 0.02344 0.02344 0.02344 0.03125 0.03125 0.03125 0.02344 0.03125 0.02344 0.02344 0.02344 0.02344 0.03125 0.02344 0.02344 0.02344 0.02344 0.02344 0.02344 0.02344 0.02344 0.03125 0.03125 0.03125 0.03125 0.03125 0.02344 0.02344 0.02344 0.02344 0.02344 0.02344 0.02344 0.02344 0.02344 0.03125 0.03125 0.03125 0.03125 0.03125 0.03125 0.02344 0.03125 0.02344 0.03125 0.02344 0.02344 0.02344 0.03125 0.04688 0.03125 0.02344 0.02344 0.02344 0.02344 0.02344 0.02344 0.01562 0.03125 0.03125 0.02344 0.02344 0.03125 0.02344 0.02344 0.02344 0.02344 0.02344 0.03125 0.02344 0.02344 0.02344 0.02344 0.02344 0.02344 0.03125 0.03125 0.02344 0.03125 0.03125 0.03125 0.01562 0.03125 0.02344 0.02344 0.02344 0.03125 0.03906 0.03125 0.03125 0.02344 0.03125 0.02344 0.03125 0.02344 0.02344 0.02344 0.02344 0.02344 0.03125 0.02344 0.02344 0.02344 0.02344 0.03906 0.02344 0.02344 0.02344 0.02344 0.02344 0.03906 0.02344 0.02344 0.03125 0.03125 0.02344 0.02344 0.03906 0.03125 0.03125 0.02344 0.02344 0.03125 0.03125 0.02344 0.03125 0.02344 0.02344 0.02344 0.02344 0.03125 0.02344 0.03125 0.03125 0.03125 0.03125 0.02344 0.02344 0.02344 0.02344 0.02344 0.02344 0.03125 0.03125 0.00000 Table 14: Comparison of DAP with Existing Literature S-boxes DAP S1 0.039 S2 0.047 [27] 0.047 [25] 0.039 3.9. Fixed Point, DBN, LBN, and Linear Structure A fixed point in an S-box is defined as a value S(x) = x. The presence of fixed points can weaken the cryptographic strength of an S-box, as they provide predictable mappings that may be exploited by adversaries. Secure S-box designs aim to avoid or minimize fixed points. However, in certain cryptographic systems, the inclusion of fixed points is intentional to satisfy specific design objectives [22, 33]. Table 15 presents a comparative analysis of fixed points. M. Alammari et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6470 13 of 30 The Differential Branch Number (DBN) is a key cryptographic metric used to evaluate an S-box’s resistance to differential cryptanalysis. It is defined as the minimum sum of active input and output bits for all nonzero input differences. An S-box with a high DBN ensures strong diffusion, making it more difficult for an attacker to exploit differential patterns since a minor change in the input causes widespread changes in the output. Thus, optimizing DBN improves the diffusion characteristics of S-boxes and enhances security against differential attacks [22, 33]. The linear approximation is used to evaluate resistance against linear cryptanalysis through the Linear Branch Number (LBN). It is defined as the minimum sum of active input and output bits over all nonzero linear masks. A high LBN indicates strong resistance to linear approximations, meaning that linear relationships between input and output bits are minimized. This significantly reduces the probability of successful linear attacks, thereby enhancing the security of symmetric key cryptosystems [22, 33]. Table 15 compares the DBN and LBN values of different S-boxes. An S-box is said to exhibit a Linear Structure (LS) if there exists an input difference a and an output difference b such that for all x, S(x ⊕ a) ⊕ S(x) = b. The existence of such structures indicates predictability in transformations, making the S-box vulnerable to linear cryptanalysis. Hence, a secure S-box should have no or very few linear structures to ensure high nonlinearity and resistance to such attacks [22, 33]. A comparison of the LS property is also presented in Table 15. Table 15: Comparison of FP, DBN, LBN, and LS with Existing Literature S-boxes DBN LBN FP LS S1 1 1 1 0 S2 3 1 1 0 [33] 1 2 2 0 [22] 2 2 2 0 4. Multiple RGB Image Algorithm over Eisenstein Integer The encryption process of multiple RGB images using Eisenstein integers Z[ω] consists of the following sequential steps: (i) Input Preparation The encryption process begins with importing the RGB images requiring protection. Each image is composed of Red, Green, and Blue components. The pixel values are transformed by mapping the color channels to elements of the Eisenstein integer ring Z[ω], preparing the data for secure encoding. (ii) S-box Construction As detailed in Section 3, two 8 × 8 S-boxes, denoted as S1 and S2, are constructed over the residue classes of Eisenstein integers Z[ω]. These structures are integral to increasing the cryptographic strength of the encryption scheme. M. Alammari et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6470 14 of 30 (iii) SPN Framework The proposed Substitution-Permutation Network (SPN) framework applies the con- structed S-boxes to enhance the security of multiple RGB images using three primary stages: (a) Substitution Phase The substitution phase begins with applying S-box S1 to introduce non-linearity (confusion). All pixel values from the Red, Green, and Blue channels are substi- tuted such that the output values exhibit a non-linear dependency on the input. The usage of Z[ω] increases security due to its rich algebraic structure, making cryptanalysis (especially linear and differential attacks) significantly harder. (b) Permutation Phase The next stage applies S-box S2 to perform permutations that enhance dif- fusion. Pixel values from each channel are spatially rearranged so that pat- terns are removed, and substitution effects are spread across the entire image. A distinct permutation is applied to each color channel to avoid predictable transformations. (c) Final XOR Operation A third S-box S3 is dynamically generated by XORing S1 and S2. This S- box is then used to XOR the pixel values of the image. This introduces further randomness and enhances complexity. The XOR operation ensures that without the proper key (which generates S3), the transformation cannot be reversed. (iv) Final Transformation The SPN process thoroughly transforms each channel, optimizing both confusion and diffusion, and thereby safeguarding the image from a broad range of cryptanalytic attacks. (v) Output the Encrypted Image After applying the transformations, the modified Red, Green, and Blue channels are combined to form the final encrypted image. This encryption process is applied across all images in the dataset. (vi) Final Output The resulting encrypted images exhibit significantly increased entropy, lower correla- tion between adjacent pixels, and enhanced resistance against differential and linear attacks, ensuring the robustness of multimedia data transmission. Significance of the SPN Framework: The proposed three-stage SPN achieves an optimal balance between confusion and dif- fusion, critical for effective cryptographic systems. The dynamic switching and shuffling operations eliminate sequential predictability and ensure that visual transformations are distributed uniformly throughout the image. The incorporation of the XOR operation via S3 introduces high unpredictability, minimizing susceptibility to cryptanalytic techniques. M. Alammari et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6470 15 of 30 Leveraging Eisenstein integer residue classes Z[ω] constructs encryption layers that are far more complex than traditional RSA, thus substantially enhancing security. Sections 3 and 4 S-box construction joins the production process with multiple RGB image encryption methods as described in Section 5 according to Figure 1. Figure 1: Flowchart of the Proposed Study 5. Multiple RGB Image Encryption Implementation and Analysis over Eisenstein Integers This innovative form of encryption protects digital images using the mathematical structure of Eisenstein integers for multiple RGB image files. The encryption method processes the color channels—Red, Green, and Blue—through a higher-dimensional al- gebraic space, providing powerful diffusion and confusion security capabilities. The pro- cess employs several transformations, including substitution-permutation networks (SPNs) and chaotic mappings, and performs modular arithmetic over Eisenstein integers to build highly sensitive systems resistant to known cryptographic attacks such as differential and statistical threats. By leveraging the non-trivial number-theoretic properties of Eisenstein integers, the encryption scheme introduces complexity that makes keyless decryption of protected images highly impractical for adversaries. This technique offers superior secu- rity features compared to traditional cryptographic methods due to increased randomness and entropy. Furthermore, the method demonstrates optimal performance in concurrent image processing, enabling the encryption of multiple images in parallel by exploiting the inherent algebraic structure for efficient key management. This contributes significantly to M. Alammari et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6470 16 of 30 secure image transmission, multimedia security, and cloud storage applications by ensur- ing both confidentiality and integrity of visual data. Figure 2 displays two original RGB images labeled A and B, alongside their corresponding encrypted versions, illustrating the effectiveness of the proposed encryption scheme. (a) Original Image A (b) Original Image B (c) Encrypted Image A (d) Encrypted Image B Figure 2: Original RGB images (a) and (b), and their encrypted versions (c) and (d) using the proposed Eisenstein-integer-based encryption. 5.1. Histogram Analysis Histogram analysis serves as a key statistical tool in evaluating the security of multiple RGB image encryption schemes based on Eisenstein integers. Specifically, it assesses the M. Alammari et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6470 17 of 30 distribution of pixel intensities across the red, green, and blue channels in the encrypted images. An effective encryption method produces histograms with uniform distributions for all channels, indicating the elimination of original image patterns and correlations. In the proposed method, complex arithmetic operations over Eisenstein integers ensure high randomness in the distribution of pixel values. This randomness disrupts statistical regularities, thereby thwarting adversaries from exploiting histogram-based cryptanalytic techniques. The visual comparison of the histograms of original and encrypted images offers a direct measure of encryption strength. A robust encryption approach is validated when the histogram of the encrypted image diverges significantly from the original, re- sembling a noise-like uniform distribution. Such transformation demonstrates the success of the diffusion and confusion mechanisms embedded in the encryption algorithm. The histogram analysis, therefore, confirms the encryption system’s resilience against statisti- cal attacks and its ability to preserve data confidentiality during storage or transmission [24, 25]. The histogram results for Image B are illustrated in Figure 3. Figure 3: Histogram comparison of original and encrypted Image B across RGB channels 5.2. NPCR and UACI The Number of Pixel Change Rate (NPCR) is a fundamental metric for evaluating the security of multiple RGB image encryption schemes, especially those based on Eisenstein integers. It quantifies the degree of pixel modification resulting from slight changes in the input image. A robust encryption system should yield NPCR values close to 99%, indicating that even minimal changes in the plaintext image cause widespread alterations in the encrypted image. This characteristic enhances resistance against differential attacks, since it becomes nearly impossible for an attacker to infer how small changes in the input propagate through the encryption process. The arithmetic operations over Eisenstein integers contribute to this unpredictability by spreading changes across all three RGB channels. The Unified Average Changing Intensity (UACI) quantifies the average intensity vari- ation between original and encrypted images. A high UACI value (ideally around 33% for M. Alammari et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6470 18 of 30 8-bit images) signifies that the encryption process successfully disrupts pixel intensities, preventing any correlation or leakage of statistical patterns from the original image. The use of Eisenstein integer arithmetic further reinforces encryption strength by applying modular transformations that obscure relationships between the input and output data. Table 16 provides a comparative analysis of NPCR and UACI values for two encrypted images (Image A and Image B), alongside results from previously published methods [5, 22, 24, 26, 34]. These results confirm the effectiveness of the proposed encryption method in achieving high security through high NPCR and UACI values. Table 16: NPCR and UACI analysis NPCR UACI Images Red Green Blue Red Green Blue Image A 0.9961 0.9962 0.9961 0.2990 0.3151 0.3178 Image B 0.9963 0.9961 0.9962 0.2964 0.3091 0.3115 [26] 0.9960 0.9961 0.9961 0.3347 0.3347 0.3346 [5] 0.9969 0.9969 0.9966 0.3367 0.3332 0.3367 [22] 0.9961 0.9961 0.9961 0.3544 0.3177 0.3419 [24] 0.9959 0.9964 0.9962 0.3269 0.3037 0.2762 [34] 0.9960 0.9961 0.9963 0.2956 0.3094 0.3112 5.3. Maximum Deviation and Irregular Deviation Maximum deviation serves as a fundamental metric to assess multiple RGB image encryption over Eisenstein integers because it determines the biggest possible difference between pixel distributions of plaintext and ciphertext images. An encryption process achieves effective pixel value randomization when maximum deviation numbers are ele- vated, which reduces the correlation between plaintext and ciphertext visuals. The encryp- tion process needs this randomness to stop attacks based on statistical analysis because adversaries could spot patterns between pixels to discover keys or rebuild parts of the original content. Eisenstein integer-based encryption employs algebraic structure for pixel changes that enhance pixel intensity differences between color channels of images. Strong diffusion properties of an encryption algorithm across multiple images are verified through continuing maximum deviation assessments of the encryption process. The security as- sessment of the encryption scheme benefits from this metric, and it pairs effectively with NPCR and UACI to show the system’s effectiveness in producing secure image encryption results [22, 24, 34]. The deviations from MD testing are provided within Table 17. The assessment of encryption effectiveness relies on irregular deviation, as it measures the irregular pixel intensity variations between original and encrypted images. The en- cryption algorithm’s ability to interrupt adjacent pixel value connections becomes visible through Eisenstein integer-based multiple RGB image encryption since irregular deviation measures pixel variation discrepancies. A high irregular deviation results from encryption schemes that create complicated color channel transformations to disrupt all substantial or M. Alammari et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6470 19 of 30 linear relationships between neighbouring image pixels. Eisenstein integers augment the encryption security through their non-commutative, non-associative mathematical prop- erties, which fragment pixel intensity distribution and impede attacks on encrypted data. The security of the encryption process strengthens as multiple image encryptions pro- duce high irregular deviation values throughout all channels, which creates randomized ciphertext appearance similar to noise patterns [22, 24, 34]. Table 17 presents the results regarding ID. Table 17: Maximum Deviation and Irregular Deviation Comparative Analysis MD ID Images Red Green Blue Red Green Blue Encrypted Image A 58211 49079 54201 27373 27770 27072 Encrypted Image B 60779 58417 57547 38072 38030 38125 [22] 53397 48329 53529 29231 25127 28374 [24] 60210 47069 62218 26266 19443 27027 [34] 52021 61841 61742 38097 37989 37924 5.4. Correlation Analysis A critical security evaluation of multiple RGB image encryption over Eisenstein in- tegers depends on correlation analysis which analyzes horizontal, vertical, and diagonal directions. The research analyzes the connection patterns between image pixel values across multiple image orientations. A reliable encryption solution should produce no sub- stantial relationships between pixels of an encrypted image because this implies pixels should not affect neighboring pixels. The encrypted pixel relationships in Eisenstein inte- ger systems become harder to break because the system’s algebraic functionality combines complex mathematics with modular arithmetic. The encrypted image shows complete pixel independence in all directions, as its horizontal, vertical, and diagonal correlation values tend toward zero. The absence of pattern definitions between any pair of horizon- tal, vertical, or diagonal pixels enhances security by resisting multiple statistical attacks. Without these image correlations, attackers struggle to predict pixel values based on po- sitions and surrounding pixels. This robustness improves further when the encryption method simultaneously processes multiple RGB images, as the lack of correlation affects every color channel and direction. After encryption, the images transform into random noise patterns, ensuring higher security and protection against standard attacks. This mechanism safeguards image integrity in practical scenarios like secure transmission and encrypted storage [5, 22, 24, 26, 34]. The analysis is supported by the correlation values and images presented in Figure 5 and Table 18. M. Alammari et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6470 20 of 30 (a) Horizontal Correlation (b) Diagonal Correlation (c) Vertical Correlation Figure 4: Correlation analysis of Image A in horizontal, diagonal, and vertical directions 5.5. Information Entropy The assessment of encrypted image unpredictability and randomness depends upon in- formation entropy, which shows how successfully encryption methods remove discernible patterns from original images. The multiple RGB image encryption over Eisenstein in- tegers uses entropy to determine the uncertainty and disorder of encrypted image pixel intensity distributions. A perfect entropy rating near the maximum value indicates a uni- form pixel intensity distribution, which represents maximal disorder and randomness in the image. The protection of encrypted images depends heavily on high entropy because in- creased randomness makes statistical analysis insufficient to reveal important information about original images. By implementing Eisenstein integers in the encryption process, the scheme achieves randomness through complex arithmetic operations that destroy image structure and produce encrypted content with distributed pixel values. The encryption method preserves security when applied to RGB image collections because it generates equally high entropy values across all red, green, and blue channels, demonstrating com- plete masking of predictable data. Encryption methods with this feature provide strong protection by making images resistant to cryptanalysis and unauthorized statistical anal- M. Alammari et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6470 21 of 30 (a) Horizontal Correlation (b) Diagonal Correlation (c) Vertical Correlation Figure 5: Correlation analysis of Image B in horizontal, vertical, and diagonal directions ysis [5, 24, 26, 34]. The entropy evaluation of the proposed work is in agreement with existing research, as shown in Table 19. 5.6. MSE and PSNR The evaluation of encryption effectiveness depends heavily on the Mean Squared Error (MSE), as this metric quantifies the average pixel intensity deviations between the origi- nal and encrypted images through squared differences. The proposed encryption method demonstrates significant modification to pixel intensities; a high MSE value indicates that the encrypted image exhibits considerable differences from its original form. In the case of multiple RGB image encryption over Eisenstein integers, such alterations enhance se- curity by ensuring that all discernible patterns and structural similarities vanish. The complex non-linear transformations using Eisenstein integers induce maximum diffusion, transforming the original image into an encrypted form with substantial variations. The high MSE values across red, green, and blue channels confirm the robustness of the en- cryption against statistical attacks [22, 34]. Furthermore, consistently high MSE values across different images signify the reliability of the method in preserving security across M. Alammari et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6470 22 of 30 Table 18: Comparative Analysis of Horizontal, Diagonal, and Vertical Correlation Horizontal Diagonal Vertical Images Red Green Blue Red Green Blue Red Green Blue Image A Original Red 0.5620 0.9271 0.9426 0.5351 0.9108 0.9153 0.5420 0.9474 0.9431 Green 0.5420 0.9474 0.9431 0.5494 0.9068 0.9054 0.5506 0.9410 0.9388 Blue 0.5089 0.7093 0.9524 0.5234 0.7250 0.8961 0.5423 0.7284 0.9462 Image A Encrypted Red -0.0098 0.0084 0.0187 -0.0572 0.0584 -0.0036 -0.0238 0.0209 -0.0200 Green 0.0365 0.0077 0.0653 -0.0037 0.0120 0.0005 -0.0037 -0.0565 -0.0145 Blue -0.0269 0.0285 0.0286 -0.0044 -0.0220 -0.0877 -0.0194 0.0168 0.0103 Image B Original Red 0.9169 0.5977 0.3950 0.8780 0.6129 0.3067 0.9302 0.5772 0.4273 Green 0.3930 0.9145 0.9449 0.3178 0.8745 0.8974 0.3839 0.9018 0.9323 Blue 0.4005 0.7157 0.9434 0.3418 0.6757 0.9139 0.3793 0.7213 0.9276 Image B Encrypted Red -0.0117 -0.0382 0.0538 0.0122 0.0282 0.0625 0.0297 -0.0072 -0.0500 Green 0.0503 0.0148 -0.0193 -0.0327 0.0185 0.0353 -0.0236 0.0494 -0.0306 Blue -0.0539 0.0508 -0.0197 0.0015 0.0488 0.0018 0.0063 0.0147 0.0331 Table 19: Information Entropy comparison Information Entropy Images Red Green Blue Average Image A 7.9993 7.9993 7.9993 7.9993 Image B 7.9993 7.9993 7.9994 7.9994 [26] 7.9913 7.9914 7.9916 7.9916 [5] 7.9995 7.9995 7.9994 7.9995 [24] 7.9976 7.9967 7.9976 7.9987 [34] 7.9992 7.9994 7.9993 7.9993 varied inputs. The MSE evaluation of the proposed work is in agreement with existing research, as shown in Table 20. Peak Signal-to-Noise Ratio (PSNR) serves as a complementary metric to MSE, mea- suring the ratio of maximum possible pixel intensity to the MSE on a logarithmic scale. In secure image encryption, low PSNR values are preferred, as they signify large deviations between original and encrypted images. The use of Eisenstein integers ensures non-linear and unpredictable pixel alterations, resulting in minimal resemblance between the two. The inverse relationship between MSE and PSNR validates the effectiveness of the en- cryption; higher MSE corresponds to lower PSNR, reflecting greater data obfuscation. All RGB channels show uniformly low PSNR values, indicating strong diffusion and confusion properties. The encryption process becomes more secure when such consistency is ob- served across multiple images, ensuring that no visual or statistical reconstruction of the original image is feasible. This confirms the capability of the proposed encryption scheme in safeguarding RGB image confidentiality [22, 34]. The PSNR evaluation of the proposed work is in agreement with existing research, as shown in Table 20. 5.7. Contrast and Energy The assessment of multiple RGB image encryption over Eisenstein integers relies sig- nificantly on contrast measurements, as this metric reflects the pixel intensity variations M. Alammari et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6470 23 of 30 Table 20: MSE and PSNR Comparative Analysis MSE PSNR Images Red Green Blue Red Green Blue Original Image A 44.1849 44.4965 44.1297 2.50 2.33 2.53 Encrypted Image A 44.2659 44.3553 44.2291 2.45 2.40 2.47 Original Image B 44.1320 43.9747 44.2144 2.53 2.62 2.48 Encrypted Image B 44.3440 44.3418 44.2445 2.41 2.41 2.47 Original [22] 44.6746 44.2313 44.9505 2.23 2.47 2.10 Encrypted [22] 44.3020 44.3185 44.4057 2.43 2.42 2.38 Original [34] 44.1849 44.4965 44.1297 2.50 2.33 2.53 Encrypted [34] 44.1080 44.2180 44.3952 2.54 2.48 2.38 between adjacent pixels in an image. An effective encryption process should drastically al- ter the contrast levels of the original image, aiming to minimize any perceptible patterns in the ciphered output. The proposed encryption method, which employs Eisenstein integers, applies complex mathematical transformations that introduce widespread changes in the image, effectively removing all original intensity gradients—even those arising from subtle pixel value differences. This encryption technique produces images where pixel intensi- ties are uniformly randomized, leading to noise-like patterns that obscure any identifiable structures or features. Consequently, contrast analysis of several RGB images processed by this method shows no meaningful intensity correlation, validating the cryptographic strength of the proposed approach. By suppressing contrast-based patterns, the encryp- tion ensures protection against attacks that exploit visual characteristics or perceptual cues, thereby safeguarding image privacy during storage and transmission [5, 22, 34]. The contrast evaluation results are presented in Table 21. In addition, the encryption of multiple RGB images over Eisenstein integers is evalu- ated using energy as a complementary statistical measure. Energy quantifies the unifor- mity and intensity distribution within an image by computing the sum of squared pixel values. After encryption, the resulting image exhibits lower energy due to the uniform distribution of pixel values, induced by the complex algebraic operations involved in the Eisenstein integer framework. This reduction in energy confirms the randomness of the ciphered image, eliminating recognizable intensity concentrations. The encryption pro- cess consistently maintains this low-energy distribution across all RGB components, thus providing a uniform ciphertext with minimal structural cues. Such randomness resists cryptanalysis attempts based on statistical inference, ensuring high data confidentiality in practical applications such as secure transmission and storage of digital images [5, 22, 34]. The energy evaluation results are presented in Table 21. 5.8. Homogeneity and Standard Deviation The security evaluation of RGB image encryption based on Eisenstein integers de- pends on analyzing image pixel uniformity through homogeneity. Homogeneity measures the closeness of similar intensity values in an image; encrypted images should ideally ex- M. Alammari et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6470 24 of 30 Table 21: Contrast and Energy Analysis of Different Images Contrast Energy Images Red Green Blue Red Green Blue Original Image A 0.5693 0.6411 0.6196 0.0752 0.0735 0.0713 Encrypted Image A 10.5229 10.5093 10.5358 0.0156 0.0156 0.0156 Original Image B 0.6062 0.7930 0.6063 0.0828 0.0748 0.0906 Encrypted Image B 10.5357 10.4913 10.4710 0.0156 0.0156 0.0156 Original [5] 0.5439 0.5000 0.4726 0.0779 0.0821 0.1708 Encrypted [5] 10.5114 10.4770 10.4894 0.1051 0.1048 0.1049 Original [22] 0.4717 0.4879 0.4261 0.0838 0.0834 0.1242 Encrypted [22] 10.4878 10.4861 10.5034 0.0156 0.0156 0.0156 Original [34] 0.5693 0.6411 0.6196 0.0752 0.0735 0.0713 Encrypted [34] 10.5357 10.4913 10.4710 0.0156 0.0156 0.0156 hibit low homogeneity to demonstrate that uniform patterns are thoroughly disrupted. A robust encryption algorithm using Eisenstein integers randomizes pixel intensity values across the full range, causing neighboring pixels to become decorrelated and visually noisy. This outcome confirms effective confusion and diffusion in all three RGB channels. Consis- tently low homogeneity values across images indicate that the encryption technique masks structured image content and defends against unauthorized statistical analysis [22, 34]. The homogeneity evaluation results are presented in Table 22. In parallel, standard deviation (SD) serves as a statistical measure to evaluate the dispersion of pixel intensities. A high SD in encrypted images implies that pixel values are spread over the entire possible range, supporting the presence of randomness and infor- mation diffusion. Eisenstein-based transformations induce such statistical unpredictability via their nonlinear arithmetic properties. A secure image encryption process should exhibit high SD across all RGB channels, aligning the encrypted image statistics with white noise and minimizing the risk of pattern leakage [22, 34]. The standard deviation evaluation results are presented in Table 22. Table 22: Homogeneity and Standard Deviation Analysis of Different Images Homogeneity Standard Deviation Images Red Green Blue Red Green Blue Original A 0.8335 0.8324 0.8293 57.4332 64.5652 65.8041 Encrypted A 0.3890 0.3891 0.3880 73.9658 73.9845 73.9391 Original B 0.8384 0.8198 0.8499 54.7733 63.0304 63.5231 Encrypted B 0.3891 0.3892 0.3899 73.9073 73.9022 73.9116 Original [22] 0.8855 0.8726 0.8855 9.5315 ×103 8.5820 ×103 1.0732 ×104 Encrypted [22] 0.3892 0.3897 0.3889 72.3821 67.0023 61.9652 Original [34] 0.8835 0.8324 0.8293 57.4332 64.5652 65.8041 Encrypted [34] 0.3891 0.3892 0.3899 73.8951 73.9193 73.9622 M. Alammari et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6470 25 of 30 5.9. NIST Test The NIST test suite serves as a comprehensive benchmark to evaluate the statistical strength of encryption algorithms, especially for RGB image encryption using Eisenstein integers. This suite encompasses various randomness tests to ensure that encrypted im- ages exhibit properties similar to truly random sequences, thereby making them resilient to cryptographic attacks. The frequency and block frequency tests assess whether pixel values are evenly distributed. The rank test evaluates linear independence of pixel groups, while the runs test (with M = 10, 000) checks the occurrence of consecutive identical bits. The long runs of ones test identifies long uninterrupted sequences, ensuring that no patterns survive encryption. Template-based tests like overlapping and non-overlapping templates detect fixed patterns within encrypted data. The Discrete Fourier Transform (DFT) test checks for periodic features in pixel distributions. The approximate entropy test measures data complexity, while the universal test and serial test examine compres- sion potential and sequence structure, respectively. Both forward and reverse cumulative sum tests identify systematic biases. The random excursions and random excursions vari- ants assess the behavior of cumulative sums across specific states. Successful performance across these tests indicates effective diffusion and confusion. Eisenstein integer-based encryption demonstrates uniformly high security performance across RGB channels, as shown in Table 23. The algebraic complexity and chaotic properties of Eisenstein opera- tions significantly improve encryption robustness, leading to highly unpredictable output that resists statistical and differential cryptanalysis [5, 22, 24, 26, 34]. M. Alammari et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6470 26 of 30 Table 23: NIST Analysis of RGB Image A Tests P-values Remarks Red Green Blue Frequency 0.41097 0.19479 0.80028 ✓ Block frequency 0.96476 0.74492 0.02102 ✓ Rank 0.29191 0.29191 0.29191 ✓ Runs (M=10,000) 0.74369 0.70048 0.61359 ✓ Long runs of ones 0.71270 0.71270 0.71270 ✓ Overlapping templates 0.85988 0.85988 0.85988 ✓ No overlapping templates 0.92285 0.99561 0.96777 ✓ Spectral DFT 0.46816 0.11048 0.24574 ✓ Approximate entropy 0.00610 0.68249 0.50842 ✓ Universal 0.98125 0.98914 0.98654 ✓ Serial p values 1 0 0.26606 0.22531 ✓ Serial p values 2 0 0.47179 0.06546 ✓ Cumulative sums forward 0.23783 0.36684 0.18702 ✓ Cumulative sums reverse 1.06720 1.62490 0.88700 ✓ Random excursions X = -4 0.87792 0 0.91256 ✓ X = -3 0.36782 0.22540 0.83339 ✓ X = -2 0.62224 0.79201 0.35516 ✓ X = -1 0.48909 0.87691 0.85346 ✓ X = 1 0.76990 0.87691 0.71270 ✓ X = 2 0.47291 0.82820 0.86295 ✓ X = 3 0.73730 0.98202 0.86069 ✓ X = 4 0.85692 0.98894 0.59778 ✓ Random excursions variants X = -5 0.91388 1.00000 0.59588 ✓ X = -4 0.80626 0.89369 0.59298 ✓ X = -3 0.77167 0.52709 0.69263 ✓ X = -2 0.92538 0.10247 0.75946 ✓ X = -1 0.62650 0.07710 0.85968 ✓ X = 1 0.33039 0.47950 0.72367 ✓ X = 2 0.22339 0.54029 1.00000 ✓ X = 3 0.16808 0.52709 0.75183 ✓ X = 4 0.24403 0.59298 0.89369 ✓ X = 5 0.30423 0.63735 0.76828 ✓ 6. Conclusion and Future Directions The proposed encryption method based on a Substitution-Permutation Network (SPN) operating over Eisenstein integers Z[ω]π establishes a robust framework for securing multi- ple RGB images. The cryptographic strength of the system is enhanced by its novel S-box architecture, which leverages the algebraic properties of Eisenstein integers to achieve high M. Alammari et al. / Eur. J. Pure Appl. Math, 18 (3) (2025), 6470 27 of 30 nonlinearity and effective modular arithmetic. By integrating substitution operations with permutations and XOR mechanisms in an SPN structure, the scheme achieves substan- tial diffusion and confusion properties, improving its resistance to linear and differential cryptanalysis. The encryption system ensures data integrity and processing efficiency by independently securing each color channel, making it a viable solution for encrypted multimedia transmission. The method demonstrates practical potential for image secu- rity through high entropy, low correlation, and strong resistance against statistical and structural attacks. Although promising, the algorithm could benefit from further development, partic- ularly for real-time implementation and hardware acceleration to enhance performance in large-scale applications. Additionally, future research may explore the application of Eisenstein integers in cryptographic primitives such as key exchange protocols and digital signature schemes. A comprehensive security analysis against contemporary and emerg- ing attack techniques would further establish its applicability in secure communication systems. Acknowledgements This research is supported by Universidad Pedagógica y Tecnológica de Colombia (SGI 3725) and Minciencias (Conv. 934). Data availability The images used in this study were obtained from ’The USC-SIPI Image Database’ (https://sipi.usc.edu/database). 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