EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6555 ISSN 1307-5543 – ejpam.com Published by New York Business Global 1 A Multi-Level Optimization Framework for Blockchain2 Security: Integrating Metaheuristics, Reinforcement3 Learning, and Game Theory4 Kassem Danach1,∗, Abbas Tarhini2, Wael Hosny Fouad Aly3,5 Hussin Jose Hejase16 1 Basic and Applied Sciences Research Center, Al Maaref University, Beirut, Lebanon7 2 Department of Computer Science, Lebanese American University, Beirut, Lebanon8 3 College of Engineering and Technology, American University of the Middle East, Kuwait9 10 Abstract. Blockchain technology relies on cryptographic mechanisms for transaction security and data integrity. However, the growing computational complexity, high transaction costs, and scalability issues pose significant challenges to blockchain adoption. Traditional cryptographic methods—such as hashing, key generation, encryption, and decryption—introduce excessive com- putational overhead, leading to energy inefficiencies and increased latency. This research proposes an optimization-driven crypto analysis framework that integrates metaheuristic algorithms, com- binatorial optimization, reinforcement learning, and game theory to enhance the efficiency and se- curity of blockchain cryptographic processes. The framework focuses on optimized cryptographic computation, gas fee reduction in smart contracts, security enhancement against cryptanalysis, and improved scalability of consensus mechanisms. Experimental evaluations demonstrate up to 39.4% reduction in cryptographic execution time, 29.4% savings in smart contract gas fees, and 33.3% improvement in decentralization of Proof-of-Stake validators. These results validate the effectiveness of the proposed framework in achieving secure, scalable, and cost-efficient blockchain operations. 2020 Mathematics Subject Classifications: 68M14, 68T20, 90C5911 Key Words and Phrases: cryptography, blockchain security, optimization, metaheuristics, re-12 inforcement learning, game theory13 14 1. Introduction15 Blockchain technology has revolutionized digital transactions by enabling secure, de-16 centralized, and tamper-resistant data management [1, 2]. Cryptographic mechanisms,17 including hash functions, asymmetric encryption, digital signatures, and zero-knowledge18 ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6555 Email addresses: kassem.danach@mu.edu.lb (K. Danach), wael.aly@aum.edu.kw (W. H. F. Aly) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) K. Danach et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6555 2 of 19 proofs (ZKPs), form the backbone of blockchain security [3, 4]. These methods ensure19 transaction integrity, identity authentication, and resistance against malicious attacks [5].20 Despite these advantages, blockchain suffers from notable limitations. Computational21 inefficiency, especially in cryptographic operations such as SHA-256 hashing in Bitcoin,22 hampers scalability and increases energy usage [6]. Smart contracts on Ethereum, for23 instance, incur high gas fees due to suboptimal cryptographic execution, limiting the24 practical scalability of decentralized applications (DApps) [7]. Additionally, the rise of25 quantum computing and (artficial intelligence) AI-driven cryptanalysis introduces new26 threats to traditional cryptographic schemes [8].27 Recent studies, [9, 10], have investigated metaheuristic optimization and AI-enhanced28 cryptographic strategies to address these inefficiencies. However, most of these solutions29 treat optimization as an isolated layer, failing to unify cryptographic efficiency, security30 robustness, and computational cost under a comprehensive framework.31 Reinforcement learning (RL), known for its effectiveness in complex decision-making,32 has shown promise in blockchain applications [11, 12]. Unlike static heuristics, RL enables33 systems to adapt dynamically through real-time feedback, allowing continuous optimiza-34 tion of cryptographic parameters, gas fees, and consensus strategies. Advanced RL models35 such as Deep Q-Networks (DQN) and Proximal Policy Optimization (PPO) have been36 employed to fine-tune elliptic curve cryptography (ECC) parameters and smart contract37 logic [13]. Moreover, multi-agent RL approaches have enhanced load balancing and energy38 efficiency in Proof of Work (PoW) and Proof of Stake (PoS) systems [14].39 Nevertheless, challenges remain. Many RL models lack explainability, struggle with40 convergence in cryptographic contexts, and are difficult to align with evolving threat mod-41 els. These gaps motivate the current research, which proposes a multi-level optimization42 framework that integrates metaheuristics, reinforcement learning, and game-theoretic rea-43 soning to holistically improve blockchain security and efficiency.44 This paper addresses key challenges in blockchain systems, including computational45 inefficiency in cryptographic operations, high smart contract gas fees, security threats from46 emerging technologies, and suboptimal consensus mechanism performance [2, 15]. The47 proposed framework introduces four core innovations: (1) metaheuristic optimization (e.g.,48 Genetic Algorithms, Simulated Annealing, Particle Swarm Optimization) for enhancing49 cryptographic performance; (2) multi-objective optimization for reducing smart contract50 gas fees in Ethereum and Hyperledger systems; (3) reinforcement learning-based detection51 of cryptographic vulnerabilities; and (4) game-theoretic models (e.g., Nash Equilibrium,52 Stackelberg games) for optimizing validator strategies in consensus mechanisms.53 The structure of the paper is as follows. Section 2 provides a comprehensive review of54 existing work in cryptographic optimization and blockchain security. Section 3 introduces55 the proposed multi-level optimization framework in detail. Section 4 describes the ex-56 perimental setup and presents the performance evaluation results. Section 5 analyzes the57 improvements in performance and efficiency achieved through the multi-level optimization58 approach. Finally, Section 6 summarizes the key findings and suggests avenues for future59 research.60 K. Danach et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6555 3 of 19 2. Related Work61 Blockchain security is fundamentally built on cryptographic primitives, yet the in-62 creasing complexity of blockchain networks demands efficient cryptographic optimization.63 Recent research has explored the application of metaheuristics, AI-driven cryptanalysis,64 and game-theoretic optimizations to enhance blockchain security and efficiency. This sec-65 tion reviews existing approaches in cryptographic optimization for blockchain security,66 smart contract efficiency, and consensus mechanisms.67 2.1. Cryptographic Optimization in Blockchain Security68 Cryptographic techniques such as hash functions, asymmetric encryption, and zero-69 knowledge proofs (ZKPs) are fundamental to ensuring the security and integrity of blockchain70 transactions [3, 4]. These mechanisms safeguard data authenticity, prevent double-spending,71 and enable privacy-preserving transactions in decentralized environments. However, tra-72 ditional cryptographic methods are computationally intensive, leading to high latency,73 resource consumption, and scalability issues, particularly in high-throughput blockchain74 networks such as Ethereum and Hyperledger [8]. Additionally, the rise of AI-driven crypt-75 analysis and quantum computing threats has raised concerns about the long-term viability76 of existing cryptographic protocols [16, 17].77 To address these challenges, researchers have explored metaheuristic optimization tech-78 niques to enhance cryptographic efficiency while maintaining security guarantees. For in-79 stance, Das et al. [9] applied Genetic Algorithms (GA) and Simulated Annealing (SA) to80 optimize hash function performance, resulting in a 30% reduction in computational over-81 head while preserving cryptographic strength. Similarly, Mukherjee et al. [10] proposed82 an adaptive cryptographic tuning mechanism using Deep Q-Networks (DQN), which dy-83 namically adjusts cryptographic parameters such as key size, iteration count, and entropy84 levels, improving security without excessive computational costs. Their findings demon-85 strate that reinforcement learning (RL) models can effectively balance security robustness86 and performance efficiency.87 Beyond hash function optimization, quantum-safe cryptographic protocols have also88 been explored. Dai et al. [18] investigated lattice-based encryption schemes optimized89 using Swarm Intelligence (SI) techniques, achieving significant improvements in encryp-90 tion speed and post-quantum security. Another study by Wang et al. [19] introduced91 a hybrid cryptographic framework combining Elliptic Curve Cryptography (ECC) with92 Reinforcement Learning, which enhances transaction security while reducing encryption93 overhead by up to 25%.94 Despite these advancements, most existing studies focus on isolated cryptographic95 functions rather than developing an integrated framework that simultaneously optimizes96 multiple security layers, including hashing, encryption, and privacy-preserving mecha-97 nisms. Additionally, the interaction between cryptographic optimization and blockchain98 scalability remains an underexplored area. Future research should investigate hybrid99 metaheuristic-AI optimization frameworks that integrate evolutionary algorithms, rein-100 K. Danach et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6555 4 of 19 forcement learning, and quantum-resistant cryptographic techniques to enhance blockchain101 security while ensuring computational feasibility.102 This research proposes a multi-layer cryptographic optimization framework that lever-103 ages metaheuristic search algorithms, reinforcement learning, and game-theoretic security104 models to dynamically adjust cryptographic parameters based on blockchain network con-105 ditions. The proposed approach seeks to reduce gas fees, improve transaction efficiency,106 and enhance resistance to AI-driven and quantum-enabled cryptographic attacks.107 2.2. Smart Contract Optimization108 Smart contracts enable automated transactions on blockchain platforms such as Ethereum109 and Hyperledger, facilitating trustless execution of decentralized applications (DApps).110 However, their execution is often computationally expensive due to inefficient crypto-111 graphic operations, redundant computation, and high gas fees, which hinder large-scale112 adoption [2, 20]. The cost of executing smart contracts is primarily influenced by the113 underlying virtual machine execution, storage operations, and cryptographic verifications,114 leading to performance bottlenecks and increased transaction fees [7].115 To mitigate these inefficiencies, researchers have explored multi-objective optimization116 techniques and resource allocation strategies. For instance, Androulaki et al. [7] proposed117 an optimized cryptographic execution model in Hyperledger Fabric, demonstrating a 25%118 reduction in computational overhead and gas fees by leveraging improved storage handling119 and optimized cryptographic signatures. Similarly, Dai et al. [21] investigated transaction120 batching strategies and adaptive gas limit adjustments to minimize blockchain transaction121 costs while maintaining security guarantees. Their study revealed that a dynamic gas122 pricing strategy based on predictive models could reduce execution costs by up to 30%123 compared to static gas pricing mechanisms.124 Recent advancements in AI-driven contract optimization have further improved smart125 contract efficiency. Reinforcement learning (RL)-based models have been employed to126 dynamically adjust gas fees and execution priorities, enhancing throughput and cost-127 effectiveness [8]. Specifically, Wang et al. [22] applied Deep Q-Learning to Ethereum128 smart contracts, achieving an 18% reduction in average execution costs by intelligently129 selecting optimal execution pathways. Additionally, actor-critic RL models have been130 integrated into smart contract execution engines to optimize transaction bundling, signif-131 icantly reducing network congestion [23].132 Despite these advances, most existing studies focus on isolated optimizations and do133 not holistically integrate metaheuristic optimization with cryptographic efficiency improve-134 ments. Hybrid approaches that combine genetic algorithms (GA), simulated annealing135 (SA), and swarm intelligence techniques with reinforcement learning could further enhance136 the adaptability and efficiency of smart contracts [24]. Furthermore, game-theoretic mod-137 els have been proposed to optimize miner incentives for cost-effectively executing smart138 contract, yet they remain underexplored in real-world implementations [25].139 This research aims to bridge these gaps by proposing a hybrid optimization framework140 that integrates metaheuristics, reinforcement learning, and cryptographic enhancements141 K. Danach et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6555 5 of 19 to optimize smart contract execution. By leveraging adaptive gas pricing, transaction pri-142 oritization, and cryptographic cost minimization, the proposed framework seeks to achieve143 efficient, scalable, and cost-effective smart contract execution in blockchain environments.144 2.3. Blockchain Consensus Mechanisms and Optimization145 Consensus mechanisms, including Proof-of-Work (PoW) and Proof-of-Stake (PoS), are146 foundational components in blockchain networks, responsible for maintaining security,147 achieving distributed agreement, and validating transactions [1, 15]. PoW mechanisms,148 as employed by Bitcoin, rely on computationally intensive puzzles that ensure security149 through substantial energy expenditure. While effective, this approach has been widely150 criticized for its unsustainable energy consumption and environmental impact, limiting151 scalability and practical deployment in resource-constrained settings.152 In contrast, PoS consensus protocols select validators based on the amount of cryp-153 tocurrency they stake, significantly reducing energy usage compared to PoW. However,154 PoS systems introduce their own challenges, primarily related to the fairness and efficiency155 of validator selection processes. One major concern is stake centralization, where large156 stakeholders can disproportionately influence consensus outcomes, potentially undermin-157 ing the decentralization and trustworthiness of the network [6].158 To mitigate these issues, researchers have increasingly turned to game-theoretic mod-159 eling and optimization techniques. Game theory provides a rigorous framework to analyze160 strategic interactions among validators, enabling the design of incentive mechanisms that161 promote fair stake distribution and robust network participation. For instance, Bano et162 al. [6] applied Nash equilibrium and Stackelberg game models to optimize validator selec-163 tion, demonstrating how these approaches can effectively reduce centralization risks while164 preserving consensus security. Their work highlights the importance of equilibrium-based165 strategies to balance validator rewards and operational costs.166 Further advancements have explored the integration of metaheuristic optimization al-167 gorithms to enhance PoS performance metrics. Hassanzadeh-Nazarabadi et al. [8] inves-168 tigated the use of metaheuristics for tuning mining reward schemes, resulting in a signif-169 icant 18% increase in transaction throughput. This approach exemplifies how adaptive170 optimization techniques can dynamically adjust system parameters to improve scalability171 and responsiveness without compromising security.172 These studies underscore the potential of combining game theory and metaheuristic173 optimization to address critical limitations in PoS consensus protocols, paving the way for174 more efficient, fair, and sustainable blockchain architectures.175 2.4. Research Gaps and Proposed Framework176 Although significant progress has been made in blockchain cryptographic optimization,177 several critical gaps persist in the literature. First, there is a lack of an integrated opti-178 mization framework that jointly considers cryptographic efficiency, consensus mechanism179 performance, and transaction cost reduction. Most existing studies [26–29] tend to focus180 on individual components in isolation—such as improving encryption speed or refining181 K. Danach et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6555 6 of 19 validator selection—without adopting a unified strategy that balances security, efficiency,182 and scalability.183 Second, the application of machine learning, particularly reinforcement learning, in184 real-time cryptanalysis and dynamic security adaptation remains underexplored. While AI185 techniques have been successfully applied to parameter tuning and heuristic selection, their186 full potential in adaptive threat detection and mitigation within blockchain environments187 is yet to be realized. Third, although game theory has been widely used for theoretical188 modeling of Proof-of-Stake (PoS) consensus, practical implementations of these models in189 real-world blockchain systems like Ethereum 2.0 or Tezos are still lacking [30]. This limits190 the effectiveness of validator incentive schemes and resource allocation strategies.191 To address these limitations, this research proposes an Optimization-Driven Crypto192 Analysis Framework that integrates metaheuristic algorithms, reinforcement learning, and193 game-theoretic modeling. The framework introduces three core innovations: (1) meta-194 heuristic and AI-based cryptographic optimization to reduce computational overhead and195 improve transaction throughput; (2) multi-objective optimization for smart contract ex-196 ecution, aimed at minimizing gas fees without compromising security; and (3) game-197 theoretic consensus optimization, improving validator selection, stake distribution, and198 overall network resilience. By bridging these complementary approaches, the proposed199 framework aims to deliver a secure, efficient, and scalable blockchain architecture capable200 of withstanding evolving cryptographic threats.201 Recent studies have explored a wide range of optimization and intelligence-driven tech-202 niques to improve network performance, cryptographic efficiency, and system resilience203 in emerging computing paradigms. Aly et al. [31, 32] examined SDN controller place-204 ment and architecture using dynamic feedback and machine learning models to reduce205 latency and improve adaptability, which aligns with our focus on cryptographic optimiza-206 tion within complex, distributed networks. Similarly, multi-agent and agent-based mod-207 eling approaches have been employed for risk analysis and system planning in uncertain208 environments [33, 34], offering valuable insights into designing decentralized and resilient209 frameworks akin to those used in blockchain. Studies on trust-aware task offloading and210 fairness in edge-fog-cloud systems [35] further reinforce the utility of multi-criteria decision-211 making (MCDM) frameworks, an idea echoed in our game-theoretic optimization layer.212 Other works have addressed secure and low-complexity communication schemes [36], UAV213 service location planning during crises [37], and fake news detection through embedding-214 based learning models [38], all of which underline the broader applicability of AI and215 optimization in enhancing the security, efficiency, and intelligence of distributed digital216 systems. In addition, investigations into specialized modeling techniques, such as Hidden217 Markov Models [39] and spectral element methods [40], demonstrate the growing empha-218 sis on adaptive computational frameworks for robust analysis—core to the reinforcement219 learning and optimization techniques employed in our proposed system.220 K. Danach et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6555 7 of 19 3. Methodology of the Optimization-Driven Crypto Analysis221 Framework for Blockchain Security and Efficiency222 This section presents the proposedOptimization-Driven Crypto Analysis Frame-223 work, designed to enhance blockchain security and efficiency through the integration of224 metaheuristic optimization, reinforcement learning-based cryptanalysis, and game-theoretic225 consensus modeling. The methodology is organized into six key components: problem for-226 mulation, proposed framework, optimization techniques, implementation strategy, math-227 ematical formulation, and expected outcomes.228 3.1. Problem Formulation229 Blockchain systems rely heavily on cryptographic computations, smart contract exe-230 cution, and consensus mechanisms to ensure security, trust, and efficiency. However, the231 increasing complexity of these components introduces significant computational overhead,232 energy consumption, and scalability limitations. The central objective of this research233 is to minimize computational cost while maximizing security and scalability by234 optimizing the following three aspects:235 • Cryptographic Computation: Reduce the execution time of encryption, hashing,236 and signature verification while preserving cryptographic strength.237 • Smart Contract Optimization: Minimize gas fees and latency in Ethereum-based238 smart contracts using multi-objective learning.239 • Consensus Mechanism Efficiency: Improve Proof-of-Stake (PoS) validator se-240 lection and reward fairness using game-theoretic equilibrium models.241 This problem is formalized as a constrained multi-objective optimization task:242 min x C(x) = α · T (x) + β · E(x) + γ · S(x) (1) 243 Subject to: T (x) ≤ Tmax, E(x) ≤ Emax, S(x) ≥ Smin (2) where244 • T (x): Execution time, E(x): Energy usage, S(x): Security level.245 • α, βandγ: Weights balancing the trade-offs.246 3.2. Proposed Framework247 The proposedOptimization-Driven Crypto Analysis Framework comprises three248 interlinked optimization layers, each responsible for improving a specific aspect of blockchain249 performance. These layers function synergistically, governed by a unified cost model.250 K. Danach et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6555 8 of 19 1. Cryptographic Optimization Layer: This layer employs Ant Colony Optimiza-251 tion (ACO) to tune cryptographic parameters xc such as key sizes, hash iterations, and252 encryption rounds:253 min xc Cc(xc) = αcTc(xc) + βcEc(xc)− δcSc(xc) (3) Subject to: Sc(xc) ≥ Sc,min254 2. Smart Contract Cost Optimization Layer: Deep Q-Networks (DQN) are trained255 to adjust smart contract execution parameters xs to reduce gas consumption:256 min xs Cs(xs) = γsG(xs) + βsTs(xs) (4) With G(xs) as gas cost and Ts(xs) as latency.257 3. Consensus Mechanism Optimization Layer: Using game-theoretic constructs,258 each validator i maximizes utility Ui while contributing to network decentralization:259 Ui(si, s−i) = Ri − Ci(si) + λDi(s−i) (5) A Nash equilibrium s∗ satisfies:260 Ui(s ∗ i , s ∗ −i) ≥ Ui(si, s ∗ −i) ∀i Global Objective: All three layers contribute to a composite cost:261 min x Ctotal(x) = w1Cc(xc) + w2Cs(xs)− w3 ∑ i Ui(si, s−i) (6) With constraints:262 Sc(xc) ≥ Smin, G(xs) ≤ Gmax, andD(s−i) ≥ Dmin 3.3. Optimization Techniques263 The proposed framework employs a combination of advanced optimization techniques,264 each selected for its suitability to address specific challenges within blockchain systems.265 Ant Colony Optimization (ACO): ACO is utilized for tuning cryptographic pa-266 rameters such as key sizes, hash iterations, and encryption rounds. Inspired by the foraging267 behavior of ants, ACO is well-suited for combinatorial optimization problems with discrete268 search spaces. In this context, it efficiently explores possible configurations to minimize269 execution time and energy consumption while maintaining required security levels. The270 pheromone updating mechanism guides the search towards optimal or near-optimal cryp-271 tographic settings, balancing performance and security constraints.272 Deep Q-Networks (DQN): To optimize smart contract execution, the framework273 leverages reinforcement learning via DQNs. This approach enables dynamic adjustment of274 K. Danach et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6555 9 of 19 contract parameters to minimize gas fees and latency during runtime. The DQN models275 the smart contract environment as a Markov Decision Process, where states represent276 contract execution contexts, actions correspond to parameter changes, and rewards reflect277 cost savings and performance improvements. Through iterative training, the DQN learns278 policies that adapt to varying contract workloads and blockchain conditions, enhancing279 economic efficiency without compromising functionality.280 Game-Theoretic Nash Equilibrium Modeling: For the Proof-of-Stake (PoS) con-281 sensus mechanism, game theory is applied to model validator interactions and incentives.282 Validators are represented as rational agents aiming to maximize their utilities, which283 balance rewards against operational costs and network decentralization metrics. By char-284 acterizing the system as a non-cooperative game, the framework identifies Nash equilibria285 where no validator can improve its utility unilaterally. This ensures fair stake distribution286 and robust consensus, mitigating centralization risks and promoting network trustworthi-287 ness.288 Each optimization technique is carefully tailored with domain-specific state spaces,289 objective functions, and constraints to capture the unique characteristics of the targeted290 blockchain components. By integrating these complementary methods, the framework291 achieves a holistic improvement in blockchain performance, security, and fairness.292 3.4. Implementation Strategy293 The implementation strategy follows a systematic approach to validate the proposed294 framework across multiple blockchain layers. Initially, real-world datasets from leading295 platforms such as Ethereum and Bitcoin are gathered to ensure the relevance and robust-296 ness of the optimization processes. Subsequently, metaheuristic algorithms, reinforcement297 learning models, and game-theoretic simulations are employed to target cryptographic298 efficiency, smart contract cost reduction, and consensus fairness, respectively. Finally,299 comprehensive evaluations measure improvements in computational performance, energy300 consumption, transaction fees, and network decentralization to demonstrate the practical301 benefits of the framework.302 (i) Collect blockchain datasets from Ethereum and Bitcoin.303 (ii) Apply ACO to optimize AES and SHA-based configurations.304 (iii) Train DQNs on Ethereum smart contracts.305 (iv) Simulate PoS game models for validator equilibrium.306 (v) Evaluate improvements in runtime, energy, fees, and decentralization.307 3.5. Expected Outcomes308 The expected outcomes highlight the multifaceted benefits of the proposed Optimization-309 Driven Crypto Analysis Framework. By leveraging advanced optimization techniques310 K. Danach et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6555 10 of 19 across cryptographic processes, smart contract execution, and consensus mechanisms, the311 framework aims to deliver significant improvements in performance, cost-efficiency, and312 decentralization. These enhancements collectively contribute to the robustness, scalability,313 and economic sustainability of blockchain networks, addressing critical challenges faced by314 current decentralized systems.315 • Up to 35% faster cryptographic operations.316 • 20–30% reduction in smart contract gas costs.317 • Fairer stake distribution and improved validator performance.318 • Scalable and adaptive security for future blockchain systems.319 4. Experimental Results320 This section presents a comprehensive evaluation of the proposed Optimization-321 Driven Crypto Analysis Framework, focusing on its performance across three pri-322 mary domains: cryptographic efficiency, smart contract cost reduction, and Proof-of-Stake323 (PoS) consensus optimization. The evaluation includes empirical measurements, algorith-324 mic comparisons, and statistical tests to assess both effectiveness and significance.325 4.1. Experimental Setup326 All experiments were executed on a high-performance computing platform equipped327 with an Intel Core i9-12900K CPU (16 cores, 3.2 GHz), NVIDIA RTX 3090 GPU (24GB328 VRAM), and 64GB DDR5 RAM, running Ubuntu 22.04 LTS. Optimization and ma-329 chine learning modules were implemented using Python 3.9, TensorFlow 2.9, and PyTorch330 1.12. Blockchain simulations employed the Ethereum (Ganache) and Hyperledger Fabric331 testbeds.332 Three datasets were utilized:333 • Ethereum Smart Contract Dataset: Gas fees and execution logs for over 10,000334 contracts.335 • Blockchain Transaction Dataset: Bitcoin and Ethereum logs used for through-336 put and cryptographic benchmarking.337 • Simulated PoS Network: Includes 1000 validators and 50,000 transactions to test338 stake distribution and performance.339 4.2. Evaluation Metrics340 To rigorously assess the effectiveness of the proposed Optimization-Driven Crypto341 Analysis Framework, we adopt a comprehensive set of evaluation metrics spanning cryp-342 tographic performance, smart contract execution efficiency, consensus mechanism effec-343 K. Danach et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6555 11 of 19 tiveness, system-level resource usage, and statistical validity. Each metric is defined as344 follows:345 1. Execution Time (ms): This metric captures the computational latency for core346 cryptographic operations including hashing, encryption, and digital signature verification.347 Let Tbaseline and Toptimized denote the average execution time before and after optimization,348 respectively. The improvement ratio is calculated as:349 ∆T = Tbaseline − Toptimized Tbaseline × 100% 2. Gas Fee (ETH): Gas consumption is a critical metric in Ethereum-based smart350 contract execution. We evaluate both the average and maximum gas usage across multiple351 smart contract types. If Gi denotes the gas used by contract instance i, the average gas352 fee is computed as:353 Gavg = 1 n n∑ i=1 Gi where n is the number of executed contracts.354 3. Throughput (TPS): Throughput is measured in transactions per second (TPS),355 indicating the scalability of the consensus mechanism. It is defined as:356 TPS = Ntx Tinterval where Ntx is the number of confirmed transactions within a fixed time window Tinterval.357 4. Stake Centralization Index (SCI): To assess the fairness and decentralization358 of Proof-of-Stake (PoS) consensus, we introduce the Stake Centralization Index (SCI),359 computed as the proportion of total stake held by the top 10% of validators:360 SCI = ∑k i=1 Si∑N j=1 Sj , k = ⌈0.1N⌉ where Si is the stake of validator i, N is the total number of validators, and k is the count361 of top-staked nodes. Lower SCI values indicate improved stake distribution.362 5. System Utilization (CPU/GPU): Resource consumption is measured in terms363 of average processor (CPU) and graphics processor (GPU) utilization percentages during364 cryptographic computations. Let UCPU and UGPU represent mean utilization:365 Uavg = 1 T ∫ T 0 u(t) dt where u(t) is instantaneous utilization and T is the duration of observation.366 6. Statistical Significance: To validate the improvements across metrics, we use the367 non-parametric Wilcoxon signed-rank test. For paired samples X = {x1, x2, ..., xn} and368 Y = {y1, y2, ..., yn} representing baseline and optimized results, the test assesses whether369 the median of the differences D = {xi − yi} is significantly different from zero:370 H0 : median(D) = 0 vs H1 : median(D) ̸= 0 A resulting p-value p < 0.05 indicates statistically significant improvements.371 K. Danach et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6555 12 of 19 4.3. Cryptographic Computation Optimization372 Metaheuristic tuning using Ant Colony Optimization (ACO) significantly reduced the373 processing time of fundamental cryptographic primitives, which are critical to blockchain374 security and performance. The optimizations targeted three core operations: AES key375 scheduling, SHA-256 hashing, and ECDSA signature verification.376 AES Key Scheduling involves generating a set of round keys from the original encryp-377 tion key. This process is essential for each block encryption in AES and directly impacts378 encryption speed. Optimization reduced the scheduling time from 15.3 ms to 9.1 ms,379 representing a 40.5% improvement.380 SHA-256 Hashing is a core cryptographic function used in transaction verification and381 block hashing. It converts input data into a fixed-size hash, ensuring data integrity and382 security. The optimized version cut execution time by 36.4%, from 22.8 ms to 14.5 ms.383 ECDSA Verification is used for authenticating digital signatures, ensuring that trans-384 actions are initiated by legitimate private key holders. It is computationally intensive,385 especially during consensus processes. The optimization yielded a 41.4% reduction in386 verification time, from 37.2 ms to 21.8 ms.387 Table 1: Cryptographic Operation Performance Operation Baseline (ms) Optimized (ms) Improvement (%) AES Key Scheduling 15.3 9.1 40.5 SHA-256 Hashing 22.8 14.5 36.4 ECDSA Verification 37.2 21.8 41.4 Overall average of optimization improvement equals 39.4%. These results demon-388 strate that even small-scale optimizations at the cryptographic layer can yield significant389 improvements in overall blockchain efficiency and responsiveness.390 4.4. Smart Contract Gas Optimization391 Using Deep Q-Networks (DQN), smart contract gas consumption was optimized across392 various categories (ERC-20, NFT, utility contracts).393 K. Danach et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6555 13 of 19 Figure 1: Average Gas Fee Reduction Across Contract Types The average gas fee was reduced by 28.7%, with peak reductions up to 45% in utility394 contracts.395 4.5. Consensus Mechanism Optimization396 A Stackelberg-Nash equilibrium model was used to improve PoS performance:397 Table 2: PoS Validator Performance Comparison Metric Baseline Optimized Change (%) Stake Centralization 47.2% 31.5% -33.3 Throughput (TPS) 175 232 +32.6 Validator Efficiency 82.1% 94.3% +14.9 4.6. Computational Resource Analysis398 Figure 2 shows the average resource usage:399 K. Danach et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6555 14 of 19 Figure 2: System Utilization During Cryptographic Tasks Optimized models achieved a 41.2% reduction in CPU time and a 29.6% decrease in400 GPU utilization.401 4.7. Statistical Significance Testing402 Wilcoxon signed-rank tests confirm the results:403 Table 3: Wilcoxon Signed-Rank Test Results Metric Category p-value Cryptographic Efficiency 0.0018 Smart Contract Gas Fee 0.0046 PoS Optimization 0.0023 All p-values <0.01, confirming high statistical significance. The experimental eval-404 uation confirms the effectiveness of the proposed framework across multiple blockchain405 dimensions. Cryptographic task execution times were reduced by approximately 40%,406 enhancing overall efficiency without compromising security. In the domain of smart con-407 tracts, the framework achieved a notable reduction in gas fees by around 29%, improving408 affordability and supporting wider adoption. Consensus mechanism performance also im-409 proved, with increased transaction throughput and a marked reduction in stake centraliza-410 tion, contributing to better scalability and decentralization. Additionally, the framework411 led to significant computational savings, reducing CPU usage by about 41% and GPU412 usage by 30%, thereby promoting energy-efficient operations. These performance gains413 were statistically validated with p-values less than 0.01, reinforcing the robustness and414 reliability of the proposed optimization-driven approach.415 K. Danach et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6555 15 of 19 5. Performance and Efficiency Gains through Multi-Level Optimization416 in Blockchain417 The results obtained in this study confirm the tangible benefits of the Optimization-418 Driven Crypto Analysis Framework across critical blockchain dimensions. By integrating419 metaheuristic algorithms, reinforcement learning, and game-theoretic modeling, the frame-420 work successfully enhances cryptographic performance, reduces smart contract execution421 costs, and improves the scalability and fairness of consensus mechanisms.422 In particular, the optimization of cryptographic functions through Ant Colony Opti-423 mization demonstrated a substantial reduction in execution time—up to 39.4%—for AES-424 256 key scheduling, SHA-256 hashing, and ECDSA verification, without compromising425 security standards. Smart contract gas fees were also significantly lowered by an aver-426 age of 29.4%, validating the potential of reinforcement learning to dynamically manage427 contract parameters in real-time deployment scenarios. The integration of game-theoretic428 models into the PoS consensus process not only improved validator efficiency but also re-429 duced stake centralization by over 30%, thus enhancing decentralization and trust within430 the network. Notably, the overall computational burden—measured via CPU and GPU431 utilization—was reduced by 40.6%, indicating the energy-efficient nature of the proposed432 approach.433 These improvements have broader implications for blockchain ecosystems. By lowering434 computational overhead, the framework contributes to higher transaction throughput and435 system responsiveness. The reduction in gas fees promotes the economic sustainability436 of decentralized applications, making them more accessible to users and developers. Fur-437 thermore, the balanced stake allocation introduced through strategic modeling mitigates438 centralization risks and promotes validator diversity, a vital factor for long-term network439 stability.440 Despite these encouraging outcomes, the study has some limitations that warrant fu-441 ture exploration. While the framework was validated using Ethereum and a simulated442 PoS environment, its applicability to alternative consensus models—such as those used443 in DAG-based or hybrid blockchain platforms—requires further investigation. Addition-444 ally, the reinforcement learning module, although effective in controlled settings, must445 be adapted for real-time cryptanalysis to respond dynamically to evolving threats. The446 integration of post-quantum cryptographic elements also remains an open challenge, es-447 sential for ensuring long-term resilience against quantum computing risks. Moreover, as448 blockchain networks and transaction datasets continue to grow, future research should449 explore hybrid solutions that combine metaheuristics with deep learning to maintain scal-450 ability.451 In summary, this work provides a strong foundation for enhancing the operational452 integrity and economic feasibility of blockchain systems. The framework’s modular design453 and multi-level optimization strategy make it well-suited for further development and454 integration into next-generation decentralized infrastructures.455 K. Danach et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 6555 16 of 19 6. Conclusions and Future Work456 This study introduced an Optimization-Driven Crypto Analysis Framework that inte-457 grates metaheuristic algorithms, machine learning-based cryptanalysis, and game-theoretic458 consensus modeling to improve the security, scalability, and efficiency of blockchain net-459 works. The evaluation focused on three core components: cryptographic computation,460 smart contract execution, and Proof-of-Stake validator optimization. The results demon-461 strated that the proposed methods significantly reduce execution times for cryptographic462 operations, lower transaction costs, and enhance the fairness and performance of consensus463 mechanisms.464 Through metaheuristic tuning, cryptographic functions such as AES-256 and SHA-465 256 experienced execution time reductions of nearly 40%, while Ethereum-based smart466 contracts saw an average gas fee decrease of 29.4%. The game-theoretic optimization of467 PoS consensus reduced stake centralization by 33.3% and increased validator efficiency,468 contributing to a more decentralized and resilient blockchain environment. Additionally,469 the overall computational footprint dropped by over 40%, indicating improved energy470 efficiency—a critical consideration for the sustainability of blockchain infrastructures.471 Beyond its technical contributions, the framework represents a broader step toward472 enabling scalable and economically viable decentralized systems. By optimizing crypto-473 graphic and consensus operations, it contributes to more responsive networks with lower474 barriers to participation for developers and users. The modular structure of the frame-475 work also allows for adaptation across various blockchain platforms, making it a versatile476 foundation for future innovation.477 Further research should aim to extend the framework to incorporate quantum-resistant478 cryptographic primitives, develop AI-driven intrusion detection systems for real-time threat479 analysis, and explore the fusion of metaheuristics with deep learning for even more adap-480 tive parameter optimization. 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