SRAM-based Gaussian Noise Generation for Post- Quantum Cryptography: A Lightweight Hardware Approach

SRAM-based Gaussian Noise Generation for Post- Quantum Cryptography: A Lightweight Hardware Approach | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Article SRAM-based Gaussian Noise Generation for Post- Quantum Cryptography: A Lightweight Hardware Approach Moon-Seok Kim, Seung-Bae Jeon, Sungho Kim This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7249561/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 11 Dec, 2025 Read the published version in Scientific Reports → Version 1 posted 11 You are reading this latest preprint version Abstract As quantum computing progresses, conventional public-key cryptographic schemes such as RSA and ECC face increasing vulnerability to quantum attacks. Post-quantum cryptography (PQC), especially schemes based on the learning with errors (LWE) problem, depends on Gaussian-distributed noise for security. However, traditional Gaussian noise generation methods—such as Box–Muller, rejection sampling, and Ziggurat—incur high computational and memory costs, making them unsuitable for lightweight or embedded systems. This paper proposes a hardware-based Gaussian noise generator that uses the inherent randomness of static random access memory (SRAM) power-on states. The method aggregates SRAM start-up bits and computes their Hamming weight to efficiently generate Gaussian-distributed integers without analog components, large lookup tables, or external random number generators. Experimental results show that the output closely matches a Gaussian distribution under various group sizes and environmental conditions. Statistical tests, including Shapiro–Wilk and Kolmogorov–Smirnov, achieve over 95% pass rates, while Kullback–Leibler divergence remains below 0.01. The generator also maintains Gaussian properties across a wide thermal range (−20 °C to 100 °C). These results demonstrate that the proposed SRAM-based generator offers a practical, lightweight, and thermally robust solution for PQC, particularly in lattice- and code-based cryptographic schemes. Physical sciences/Engineering Physical sciences/Mathematics and computing Physical sciences/Physics Gaussian error sampler post-quantum cryptography (PQC) learning with error (LWE) SRAM (Static Random Access Memory) Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 INTRODUCTION The emergence of quantum computing technology poses a substantial threat to conventional cryptographic systems, such as Rivest–Shamir–Adleman (RSA) and elliptic curve cryptography (ECC), which rely on the computational hardness of mathematical problems for security [1]–[3]. These classical cryptographic protocols, foundational to modern digital security, are vulnerable to quantum algorithms that can efficiently factor large integers and solve discrete logarithm problems [4]–[5]. Consequently, post-quantum cryptography (PQC) has gained attention as a promising alternative, ensuring resilience against quantum adversaries [6]–[7]. A fundamental principle of PQC is the learning with errors (LWE) problem, which introduces intentional noise in encryption and public-key generation operations to enhance security [7]–[9]. The LWE problem renders cryptanalysis infeasible for adversaries by introducing a Gaussian-distributed error term [7]–[9]. Thus, the ability to generate Gaussian-distributed signals is crucial for implementing PQC, particularly in lattice-based and code-based cryptographic schemes [10]–[11]. However, existing Gaussian noise generation techniques, such as Box–Muller transformation, rejection sampling, and Ziggurat sampling, impose significant computational and memory overheads, requiring hardware-intensive arithmetic operations or large precomputed lookup tables [12]–[15]. To address these limitations, we propose a hardware-based Gaussian noise generator that leverages the intrinsic power-on initialization characteristics of static random-access memory (SRAM). Unlike traditional Gaussian samplers that require dedicated analog-to-digital converters (ADCs), precomputed cumulative distribution tables (CDTs), or random number generators (RNGs), the proposed approach extracts Gaussian noise directly from SRAM initialization data, eliminating additional computational burdens. To our knowledge, this is the first demonstration of a fully digital Gaussian noise generator for PQC that requires no additional entropy source, CDTs, RNGs, or analog circuitry. The key idea behind this technique is the inherent randomness of SRAM power-on states, which result from process variations and intrinsic device-level fluctuations [16]–[17]. When an SRAM module is powered on, each memory cell unpredictably stabilizes in either logic “0” or logic “1” [16]–[17]. By aggregating the start-up states of multiple SRAM cells and computing their Hamming weight, the proposed module efficiently generates integer values that follow a Gaussian normal distribution. The contributions of this study are as follows: 1. We propose a novel SRAM-based Gaussian noise generator for PQC applications, leveraging power-on start-up variations in SRAM cells. 2. We demonstrate that computing the Hamming weight of SRAM power-on bits results in a distribution that follows a Gaussian shape, similar to a large-scale fair coin toss experiment governed by the binomial distribution. 3. We validate the statistical normality of the generated noise under different environmental conditions and operating temperatures, ensuring robustness for practical cryptographic implementations. MATERIALS AND METHODS Experimental setup The experimental setup is designed to generate and analyze SRAM-based Gaussian noise for PQC applications. SRAM initialization data serves as a physical entropy source, leveraging power-up state initialization variations to produce randomness. The SRAM chip model was selected based on the following criteria: 1. Models that allow the extraction of SRAM data before executing SRAM writing operations. 2. Industrial models that guarantee reliable operation within a tolerable temperature range of −20 °C to 80 °C. Based on these criteria, IS62WV51216 (Integrated Silicon Solution Inc.: ISSI) models were selected [18]. The Open103Z development board, built around an STM32 microcontroller, serves as the primary hardware interface, facilitating real-time communication with the SRAM module and executing Hamming weight extraction algorithms [19]. A host personal computer (PC) communicates with the Open103Z board to control power cycling and data extraction. The data acquisition process consists of the following steps: 1. Power cycling: The SRAM chips are powered on and off under controlled conditions. 2. Data extraction: After each power-up, the host PC extracts the SRAM initialization data. 3. Gaussian conversion: The extracted SRAM initialization data is processed to compute the Hamming weight, which is subsequently utilized in PQC applications as a Gaussian-distributed integer. To evaluate the temperature dependence of the SRAM-based Gaussian noise generator, a CTHC-65N temperature chamber was utilized. The chamber supports an operational temperature range of −30 °C to 150 °C. The SRAM chip and test boards were placed inside the chamber, while the host PC remained external for data extraction and system control. The target test temperatures were set at −20 °C, 0 °C, 20 °C, 40 °C, 60 °C, and 80 °C to ensure a comprehensive analysis of thermal effects. Simulation setup To analyze SRAM initialization dynamics and logic state transitions, simulations were conducted using LTSPICE (SPICE: Simulation Program with Integrated Circuit Emphasis), an analog circuit simulator from Analog Devices [20]. The Si7540DP_N and Si7540DP_P transistor models were used to represent NMOS and PMOS devices, respectively. To simulate ideal Gaussian noise and ideal fair coin toss behavior, MATLAB software was employed. Specifically: 1. The randn() function generated an ideal Gaussian-distributed dataset, serving as a statistically valid reference for comparison [21]. 2. The randi() function simulated an unbiased Bernoulli process, where each bit follows a fair coin toss distribution with equal probabilities of “0” and “1” [21]. These simulations provide a theoretical baseline for quantitative comparisons between SRAM-generated data, ideal Gaussian data, and unbiased binary data models. RESULTS AND DISCUSSION Figure 1 illustrates the overall architecture of a post-quantum cryptography (PQC) system designed to leverage an SRAM-based Gaussian noise generator. Figure 1(a) presents the schematic of a PQC system integrating an SRAM module as a Gaussian noise source, detailing the functional connections between the system bus, PQC engine, and the SRAM-based Gaussian noise generator. The system bus transmits the Gaussian noise signal from the SRAM module to the PQC engine. The SRAM module is a key component in ensuring cryptographic security by generating Gaussian noise, which mitigates adversarial decryption attempts, even in the presence of quantum computing capabilities. Figure 1(b) highlights the internal structure of the SRAM-based Gaussian noise generator. Raw SRAM power-on data is retrieved and processed through a Hamming weight extractor, which transforms the raw bits into a statistically Gaussian-distributed noise signal. The resulting noise, visualized in the schematic as a bell-shaped histogram, is critical for LWE or related lattice-based and code-based encryption schemes. This hardware-driven noise sampling approach significantly reduces the computational overhead associated with traditional software-based Gaussian samplers while leveraging the intrinsic stochastic behavior of SRAM devices to enhance security. Figure 1(c) illustrates the PQC encryption process using Gaussian noise errors. A plain image (left) is input into the PQC engine, which employs the SRAM-generated Gaussian noise to produce a ciphered image (right). Without the aid of SRAM-generated noise, quantum computing algorithms can perfectly decrypt the ciphered output back to the original image [4]–[5]. In contrast, incorporating SRAM-generated noise prevents quantum adversaries from achieving decryption. Thus, the SRAM-based Gaussian noise generator plays a vital role in PQC implementations, providing resilience against quantum computing threats [10]–[11]. Figure 2 provides a detailed analysis of SRAM initialization and logic state transitions. Figure 2(a) shows simulated voltage transient curves during SRAM initialization, illustrating the time-dependent variations of V S , V A , and V B . Figure 2(b) depicts the circuit structure of an SRAM unit cell, comprising PMOS transistors (M1 and M2) and NMOS transistors (M3 and M4). V S acts as the primary voltage supply, while V A and V B serve as internal node voltages between PMOS transistors M1 and M2, and NMOS transistors M3 and M4, respectively. The voltage levels at V A and V B dynamically adjust based on the threshold voltage ( V TH ) conditions of the transistors in the SRAM unit cell (Threshold voltages: V TH1 for M1, V TH2 for M2, V TH3 for M3, and V TH4 for M4). The simulated transient curve in Figure 2(a) represents time-dependent voltage variations when the threshold voltages are intentionally set to | V TH1 | < | V TH2 |, where absolute values are considered due to the negative V TH characteristics of typical PMOS transistors. Figures 2(c) and 2(d) describe the sequential transition process for initializing the SRAM cell into logic “1” and logic “0” states, respectively. The logic “1” state follows a four-step process: (1) power-off state, (2) activation of PMOS M1, which causes V A to track V S , (3) activation of NMOS M4, pulling V B to V GND , and (4) stabilization with V A = V DD and V B = V GND [22]–[23]. Conversely, the logic “0” transition sequence is as follows: (1) power-off, (2) activation of PMOS M2, causing V B to track V S , (3) activation of NMOS M3, pulling V A to V GND , and (4) final stabilization with V B = V DD and V A = V GND . Process variations influence whether the SRAM unit cell stabilizes in the logic “0” or “1” state, resulting in a statistical distribution that exhibits inherent unpredictability and unbiased characteristics, akin to a fair coin toss. Figure 3 provides an in-depth view of the implementation of the SRAM-based Gaussian noise generator, focusing on the extraction of Gaussian-distributed data. Figure 3(a) illustrates the operational workflow of the SRAM-based Gaussian noise generator, where raw SRAM initialization data is collected and processed to extract Gaussian-distributed integers. The Hamming weight extractor converts raw binary data into Gaussian-distributed integers by counting the number of logic “1” values within a defined bit group. These Gaussian-distributed integers are critical for PQC applications, particularly in the implementation of the LWE. Figure 3(b) presents the experimental setup for the SRAM-based Gaussian noise generator. An Open103Z development board interfaces with the SRAM chip, facilitating communication and executing the Hamming weight extraction process. Figure 3(c) compares two implementations of the Hamming weight extractor: one using the C programming language for processor-based applications and the other utilizing hardware description language (HDL) for FPGA and SoC-based cryptographic accelerators [24]–[25]. Both implementations demonstrate low overhead, making them well-suited for practical cryptographic systems. Figure 4 presents a comprehensive statistical analysis to validate the Gaussian characteristics of the SRAM-based noise generator. To ensure the reliability of the generated noise, three datasets are compared: (A group) experimentally obtained SRAM data, (B group) an ideal Gaussian distribution generated by simulation, and (C group) a fair coin toss simulation. The results confirm that the SRAM-generated noise closely follows a Gaussian distribution, reinforcing its applicability in PQC. Figure 4(a) illustrates the methodology for generating the experimental and simulated datasets. The (A group) dataset represents the raw Hamming weight data extracted from the SRAM module, following the methodology described in Figure 3. First, 256 SRAM power-on bits are collected, and the Hamming weight is computed, yielding a Gaussian-distributed integer ranging from 0 to 256. (B group) serves as an ideal Gaussian reference with the same mean ( μ = 129.8) and standard deviation ( σ = 9.2) as the SRAM dataset. This reference dataset is generated using MATLAB’s randn() function, which produces normally distributed random numbers [21]. (C group) is derived from an independent and identically distributed (IID) Bernoulli process, simulating a fair coin toss. Figure 4(b) compares the empirical distributions of the three datasets using histograms overlaid with theoretical Gaussian curves (red lines). Visually, all three datasets exhibit near-Gaussian distributions; however, slight deviations in the mean ( μ ) and standard deviation ( σ ) are observed between the SRAM-generated data (A group) and the IID binary sequence (C group). These deviations suggest that the SRAM experimental data (A group) has some intrinsic correlations among SRAM cells [26]–[27]. To rigorously validate the normality of these datasets, several statistical metrics are employed, including the quantile–quantile (Q–Q) plot, Shapiro–Wilk (SW) test, Kolmogorov–Smirnov (KS) test, and Kullback–Leibler (KL) divergence. Figure 4(c) presents Q–Q plots for each dataset, comparing the empirical quantiles to theoretical Gaussian quantiles. If the data follows a Gaussian distribution, the points should align closely with the y = x reference line [28]–[29]. All Q–Q plots in Figure 4(c) show straight-line alignment, confirming that the SRAM-generated noise (A group) adheres to a Gaussian distribution, making it suitable for PQC applications. Figure 4(d) presents the results of the statistical tests. The SW test assesses whether a dataset follows a normal distribution, while the KS test determines whether two datasets originate from the same probability distribution [30]–[31]. The pass rates for the SW and KS tests exceed 95%, demonstrating that the SRAM experimental data exhibits strong normality. Additionally, KL divergence quantifies the difference between a probability distribution and a reference distribution. A KL divergence value of 0 indicates a perfect match, while values below 0.01 suggest near-identical distributions [32]-[33]. In this study, KL divergence values remain well below 0.01, confirming that the SRAM-based Gaussian noise generator produces statistically valid Gaussian noise. Overall, Figure 4 provides strong empirical evidence that the SRAM-based noise generator effectively produces Gaussian-distributed integers. While the SRAM-based integer data shows slight deviations from ideal IID Bernoulli distributions, the observed statistical properties remain sufficiently close to a true Gaussian distribution to support its application in PQC. By employing multiple statistical validation methods, including Q–Q plots, SW tests, KS tests, and KL divergence, these results confirm the suitability of the SRAM-based noise generator for PQC. Figure 5 presents a detailed analysis of the Hamming weight distribution as a function of cell group size. This analysis evaluates whether the Gaussian characteristics of SRAM-generated noise are maintained across different grouping configurations. Figure 5(a) illustrates the process of computing the Hamming weight for different cell group sizes, where the cell group size represents the number of SRAM bits used in each calculation. For example, with an 8-bit cell group, the Hamming weight ranges from 0 to 8. The extracted SRAM experimental data exhibits a near-Gaussian distribution, closely following the theoretical Gaussian curve ( μ = 4.1 and σ = 1.7), as shown by the red lines. Figures 5(b), 5(c), and 5(d) provide a comparative statistical evaluation between the SRAM-based experimental data (shaded bars) and the fair coin toss simulation (solid red lines) across various cell group sizes. Figure 5(b) presents the average Hamming weight for each group size, plotted on a logarithmic scale. Both the SRAM data and the coin toss simulation show a linear increase in the mean Hamming weight with increasing group size, consistent with theoretical expectations. Figure 5(c) shows the standard deviation of the Hamming weight distribution as a function of group size, while Figure 5(d) presents the normalized standard deviation (NSD), defined as the standard deviation divided by the square root of the group size. Theoretically, for an ideal Bernoulli process, the NSD converges to a constant value of 0.5 [34]–[35]. The NSD values obtained from the SRAM experimental data in Figure 5(d) closely follow this theoretical trend, aligning consistently with the values derived from the fair coin toss simulation. This indicates that the statistical behavior of SRAM power-on states approximates that of an ideal fair Bernoulli process across different group sizes. Minor deviations from the theoretical model are observed, possibly due to intrinsic correlations among SRAM cells during the power-on process [26]–[27]. However, Figure 5 as a whole demonstrates that the SRAM-based noise follows a statistical trend remarkably similar to that of the fair coin toss model. Furthermore, the NSD results suggest suitability for application as a Gaussian error sampler in lattice-based and code-based PQC algorithms [36]–[37]. Overall, the results provide strong evidence that the SRAM-based data exhibits stable and reliable statistical properties, supporting its applicability for PQC, particularly in Gaussian error sampling for Kyber and NTRU algorithms [36]–[37]. Figure 6 presents a statistical assessment of the normality of SRAM-based Gaussian noise across different cell group sizes. The analysis includes three key metrics: (a) KL divergence, (b) SW test pass rate, and (c) KS test pass rate. The evaluation is conducted for three datasets: (A group) SRAM experimental data, (B group) an ideal Gaussian distribution, and (C group) a fair coin toss simulation. The objective is to validate the Gaussian characteristics of SRAM-generated noise and its applicability to PQC. Figure 6(a) presents the KL divergence values comparing the SRAM experimental data (A group) to the ideal Gaussian distribution (B group) and the fair coin toss dataset (C group) to the same Gaussian reference. The results indicate that KL divergence values remain below 10⁻² for all datasets, confirming that the SRAM-based data successfully produces Gaussian-distributed noise with a high degree of statistical fidelity [32]–[33]. Notably, the KL divergence reaches its lowest value for the SRAM data when the group size is 64 bits, suggesting optimal Gaussian conformity at this configuration. Figure 6(b) shows the SW test pass rates for each dataset. For smaller cell group sizes ( e.g. , 8 and 16 bits), the pass rates are lower due to the inherent variability associated with small sample sizes. However, as the group size increases to 32 bits and beyond, the pass rates for all three datasets exceed 95%, indicating strong normality. This suggests that SRAM-generated noise demonstrates Gaussian characteristics when the cell group size is sufficiently large, e.g. , 32 bits or more. Figure 6(c) displays the KS test pass rates across varying group sizes for all datasets. As the group size increases to 16 bits and above, the pass rates for the SRAM experimental data (A group) consistently exceed 95%, further confirming normality. These results align with those observed in the SW test, as shown in Figure 6(b). In summary, Figure 6 provides robust statistical validation that SRAM-based noise exhibits Gaussian-distributed properties, particularly for cell group sizes of 32 bits and above. The low KL divergence values (Figure 6a), high pass rates in the SW (Figure 6b), and KS (Figure 6c) tests confirm the suitability of SRAM-generated noise for PQC applications. Figure 7 presents a comprehensive evaluation of the impact of temperature variations on the statistical properties of SRAM-based Gaussian noise. The analysis spans a temperature range from –20 °C to 100 °C and examines its effects on the Hamming weight distribution, mean, standard deviation, and normality metrics, including Q–Q plots, KL divergence, SW test, and KS test. This investigation aims to validate the robustness and environmental stability of SRAM-generated noise, ensuring its reliability for PQC applications. Figures 7(a) and 7(b) illustrate the Hamming weight distributions obtained from SRAM data at –20 °C and 100 °C, respectively. Both distributions exhibit near-Gaussian shapes, with the annotated mean and standard deviation values demonstrating temperature dependence. Notably, the mean remains relatively stable ( μ = 131.6 at –20 °C and μ = 129.8 at 100 °C), while the standard deviation decreases from σ = 12.1 to σ = 8.9 as the temperature increases. Figure 7(c) provides a detailed analysis of the temperature-dependent trends in the mean and standard deviation of the Hamming weight distribution. The mean (gray bars) remains consistently stable across the full temperature range, confirming the temperature-invariant nature of the mean for SRAM-based Gaussian noise. In contrast, the standard deviation (red line) shows an increasing trend at lower temperatures, indicating broader distribution spreads in colder conditions. This behavior is attributed to temperature-induced changes in SRAM cell characteristics, such as data remanence and static noise margin [23], [38], [39]. Figure 7(d) presents Q–Q plots at three representative temperatures (–20 °C, 20 °C, and 100 °C), visually assessing the normality of the SRAM-generated noise. Across all temperature conditions, the data points align closely with the y = x reference line, indicating that Gaussian characteristics are preserved despite environmental variations. Figure 7(e) shows KL divergence values at each temperature point, quantifying the similarity between the SRAM data and an ideal Gaussian distribution. In all cases, KL divergence values remain well below 10⁻², indicating a high degree of statistical similarity between the measured and reference distributions [32]–[33]. Figure 7(f) displays the pass rates of the SW and KS tests across the tested temperature range. The pass rates exceed 95% in all cases, confirming strong normality and reinforcing the temperature resilience of the SRAM-based noise generator. In summary, Figure 7 provides compelling empirical evidence that the statistical properties of SRAM-generated Gaussian noise remain stable across a wide temperature range (–20 °C to 100 °C). Although slight increases in standard deviation occur at lower temperatures, overall Gaussian behavior is preserved, as confirmed by Q–Q plots (Figure 7d), KL divergence (Figure 7e), and SW/KS test results (Figure 7f). These findings validate the environmental robustness of the SRAM-based Gaussian noise generator, ensuring its suitability for secure and reliable PQC implementations across diverse operating conditions. CONCLUSION This study presents a hardware-efficient Gaussian noise generator based on static random access memory (SRAM) power-on behavior for secure post-quantum cryptography (PQC) implementations. By leveraging the inherent unpredictability in SRAM start-up states, the proposed system extracts Gaussian-distributed integers using a lightweight Hamming weight extraction method. Compared to conventional software-based Gaussian samplers, this approach significantly reduces computational overhead and is well-suited for resource-constrained PQC hardware platforms. Experimental validation confirms that the data generated from SRAM start-up sequences closely follows a Gaussian distribution across varying cell group sizes and environmental conditions. Statistical tests further support that the SRAM-based Gaussian integer generator is highly suitable for PQC applications, particularly lattice-based and code-based PQC algorithms. Kullback–Leibler (KL) divergence values consistently remain below 0.01, and both the Shapiro–Wilk (SW) and Kolmogorov–Smirnov (KS) tests report pass rates exceeding 95% across all cell group sizes. Robustness under environmental variation was also demonstrated. Across a temperature range from –20 °C to 100 °C, the mean Hamming weight remains nearly constant, while the standard deviation shifts modestly from σ = 12.1 to σ = 8.9. Despite these slight deviations, all statistical normality metrics remain stable, indicating strong environmental resilience. In conclusion, the proposed SRAM-based Gaussian noise generator offers a lightweight, statistically validated, and thermally robust solution for PQC systems, paving the way for practical and secure cryptographic implementations in system on chip (SoC) or microcontrollers in the post-quantum era. Declarations Data availability Statement The data that support the findings of this study are available from the corresponding author upon reasonable request. Declaration of Competing Interests The authors declare no competing financial interests. Acknowledgments This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (RS-2024-00449412). Declaration of competing interests The authors declare no competing financial interest. Author contribution declaration M. S. Kim performed conceptualization, investigation, writing-original draft, software, and visualization. S. B. Jeon performed data curation, and methodology. S. Kim performed Writing-Review & Editing, supervision, project administration, and funding acquisition. Funding declaration This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (RS-2024-00449412). References W. Buchanan and A. 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K. Wang, W. H. Hou, H. Y. Zhang, J. Q. Wang, M. Goh, Z. P. Tian, and K. W. Shen, "KDE-OCSVM model using Kullback-Leibler divergence to detect anomalies in medical claims," Expert Syst. Appl. , vol. 200, p. 117056, 2022. R. Arya, and B. Antonisamy, " Sample size estimation in prevalence studies," The Indian Journal of Pediatrics , vol. 79, no. 11, pp. 1482–1488, May 2012. A. Azzalini, “The skew‐normal distribution and related multivariate families,” Scandinavian journal of statistics , vol. 32, no. 2, pp. 159-188, 2005. M. Hamburg, J. Hermelink, R. Primas, S. Samardjiska, T. Schamberger, S. Streit, E. Strieder, and C. Vredendaal, "Chosen ciphertext k-trace attacks on masked CCA2 Secure KYBER," IACR Transactions on Cryptographic Hardware and Embedded Systems , pp. 88–113, 2021. Z. Li, X. Lu, Z. Wang, R. Wang, Y. Liu, Y. Zheng, L. Zhao, K. Wang, and R. Hou, "Faster NTRU-based Bootstrapping in less than 4 ms," in IACR Transactions on Cryptographic Hardware and Embedded Systems , 2024, pp. 418–451. B. H. Calhoun and A. P. Chandrakasan, "Static Noise Margin Variation for Sub-threshold SRAM in 65-nm CMOS," IEEE J. Solid-State Circuits , vol. 41, no. 7, pp. 1673–1679, 2006. Y. Kagiyama, S. Okumura, K. Yanagida, S. Yoshimoto, Y. Nakata, and S. Izumi, "Bit Error Rate Estimation in SRAM Considering Temperature Fluctuation," in Proc. Int. Symp. Qual. Electron. Des. (ISQED) , 2012, pp. 516–519. Additional Declarations No competing interests reported. Supplementary Files Supportinginformationver00.docx Cite Share Download PDF Status: Published Journal Publication published 11 Dec, 2025 Read the published version in Scientific Reports → Version 1 posted Editorial decision: Revision requested 01 Oct, 2025 Reviews received at journal 30 Sep, 2025 Reviews received at journal 23 Sep, 2025 Reviewers agreed at journal 08 Sep, 2025 Reviewers agreed at journal 02 Sep, 2025 Reviewers agreed at journal 02 Sep, 2025 Reviewers agreed at journal 23 Aug, 2025 Reviewers invited by journal 17 Aug, 2025 Editor assigned by journal 17 Aug, 2025 Submission checks completed at journal 11 Aug, 2025 First submitted to journal 11 Aug, 2025 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-7249561","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":504466383,"identity":"8fa1af51-a4ab-4579-aad3-6cafef8d1f35","order_by":0,"name":"Moon-Seok Kim","email":"","orcid":"","institution":"Hanbat National University","correspondingAuthor":false,"prefix":"","firstName":"Moon-Seok","middleName":"","lastName":"Kim","suffix":""},{"id":504466387,"identity":"786fcbc4-47cc-4033-bdb7-f69db3d2df04","order_by":1,"name":"Seung-Bae Jeon","email":"","orcid":"","institution":"Hanbat National University","correspondingAuthor":false,"prefix":"","firstName":"Seung-Bae","middleName":"","lastName":"Jeon","suffix":""},{"id":504466389,"identity":"451b41f5-facb-4224-80b8-3b030b0839cd","order_by":2,"name":"Sungho Kim","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA0klEQVRIiWNgGAWjYJACZgaGBDkGBh4IT4JYLcaka0lsIFqL/Izcw68LKtLS+yVyD35gqLFjkJx9AL8Wgxt5adYzzuTkzpyRlyzBcCyZQZovgYAWiRwzY962itwNt3MMJBjYDjDI8RB0GERLuv3tHOMfDP+I0MJwI8f4MW9bToKBdI6ZBGPbAQZpQloMzrwxY+Y5k2Y44/4bM4vEvmQeyR5CDmvPMf7MU5Esz99zxvjGh292chJnCDmMgYENERMJ8AjFD5g/EKNqFIyCUTAKRjAAAA4iOqDkk6UnAAAAAElFTkSuQmCC","orcid":"","institution":"Ewha Womans University","correspondingAuthor":true,"prefix":"","firstName":"Sungho","middleName":"","lastName":"Kim","suffix":""}],"badges":[],"createdAt":"2025-07-30 07:08:29","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-7249561/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7249561/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1038/s41598-025-27929-3","type":"published","date":"2025-12-11T15:58:09+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":89838714,"identity":"752df9cf-5f55-4fe2-b9d9-d141b579ec91","added_by":"auto","created_at":"2025-08-25 15:06:05","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":373225,"visible":true,"origin":"","legend":"\u003cp\u003eOverall architecture of post-quantum cryptography (PQC) systemleveraging SRAM-based Gaussian noise generator. (a) System-level schematic of PQC system, where SRAM module serves as source of Gaussian noise, interfacing with PQC engine through the system bus. (b) Workflow of SRAM-based Gaussian noise generator, where raw SRAM initialization data is processed \u003cem\u003evia\u003c/em\u003e Hamming weight extractor, yielding Gaussian-distributed noise signal. (c) Application of Gaussian noise in encryption, where plain image is encrypted by PQC engine using SRAM-generated noise, resulting in ciphered image resistant to cryptanalysis by quantum computing.\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-7249561/v1/433b31ad976d6ee0a46df92b.png"},{"id":89837616,"identity":"80bed8e2-2c88-4917-8768-a6f2b2550392","added_by":"auto","created_at":"2025-08-25 14:58:05","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":208670,"visible":true,"origin":"","legend":"\u003cp\u003eAnalysis of SRAM initialization and logic state transitions. (a) Simulated voltage transient curves during SRAM initialization, illustrating time-dependent voltage variations in voltages \u003cem\u003eV\u003c/em\u003e\u003csub\u003eS\u003c/sub\u003e, \u003cem\u003eV\u003c/em\u003e\u003csub\u003eA\u003c/sub\u003e, and \u003cem\u003eV\u003c/em\u003e\u003csub\u003eB\u003c/sub\u003e. (b) Schematic of SRAM unit cell, consisting of PMOS transistors (M1, M2) and NMOS transistors (M3, M4), where \u003cem\u003eV\u003c/em\u003e\u003csub\u003eA\u003c/sub\u003e and \u003cem\u003eV\u003c/em\u003e\u003csub\u003eB\u003c/sub\u003e serve as internal node voltages. (c) Sequential transition of SRAM cell into the logic “1” state and (d) sequential transition into the logic “0” state during SRAM startup, depicting transistor switching behavior and voltage transitions.\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-7249561/v1/5b2a557f625a5f9f6ee12bf6.png"},{"id":89837618,"identity":"0fdaf2e5-11aa-458d-a3ec-55038e0cb8f8","added_by":"auto","created_at":"2025-08-25 14:58:05","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":358128,"visible":true,"origin":"","legend":"\u003cp\u003eImplementation of SRAM-based Gaussian noise generatorand Hamming weight extraction. (a) Operational workflow of generator, where SRAM raw data undergoes Hamming weight extraction to produce Gaussian-distributed integer. (b) Experimental setup, using Open103Z development board to interface with SRAM chip and extract noise data. (c) Two approaches for implementing Hamming weight extractor: (top) software-based method using C programming language and (bottom) hardware-based method using hardware description language (HDL), both demonstrating efficient extraction of Gaussian-distributed noise with minimal overhead.\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-7249561/v1/17c44d1795f487ea44ae41e9.png"},{"id":89838716,"identity":"4ae7b10f-4b80-41e9-8e81-eb20028eee85","added_by":"auto","created_at":"2025-08-25 15:06:06","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":385273,"visible":true,"origin":"","legend":"\u003cp\u003eStatistical evaluation of Gaussian characteristics of SRAM-based noise. (a) Comparison of three datasets: (A group) SRAM experimental data, (B group) ideal Gaussian simulation, and (C group) fair coin toss simulation, confirming Gaussian properties. (b) Empirical distributions of Hamming weight values for each dataset, overlaid with theoretical Gaussian curves (red lines). (c) Quantile–quantile (Q–Q) plot, demonstrating alignment of all datasets with theoretical Gaussian distribution. (d) Statistical normality metrics, including Shapiro–Wilk (SW) test, Kolmogorov–Smirnov (KS) test, and Kullback–Leibler (KL) divergence, confirming strong normality of SRAM-based noise.\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-7249561/v1/36047ecaad173335c13e9375.png"},{"id":89837620,"identity":"be1b1b1c-d714-479d-8db3-686bbcecd331","added_by":"auto","created_at":"2025-08-25 14:58:05","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":317495,"visible":true,"origin":"","legend":"\u003cp\u003eEvaluation of Hamming weight distributions based on different cell group sizes. (a) Illustration of cell grouping methodology, where varying numbers of SRAM cells are combined to compute Hamming weight, revealing Gaussian properties. (b) Average Hamming weight values as function of group size, showing linear increase, consistent with theoretical fair coin toss models. (c) Standard deviation trends, where SRAM-based noise closely follows ideal binomial distribution. (d) Normalized standard deviation (NSD), confirming that SRAM-generated noise retains unbiased Bernoulli process characteristics, validating its applicability in PQC.\u003c/p\u003e","description":"","filename":"5.png","url":"https://assets-eu.researchsquare.com/files/rs-7249561/v1/35fe8a0b5c798d043440762a.png"},{"id":89837627,"identity":"9228a234-f03f-448a-b373-2ecd32ef1b48","added_by":"auto","created_at":"2025-08-25 14:58:06","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":289372,"visible":true,"origin":"","legend":"\u003cp\u003eQuantitative assessment of Gaussian normality across different cell group sizes. (a) KL divergence analysis demonstrates that SRAM-generated noise achieves near-Gaussian distribution with KL values well below 10⁻², ensuring statistical normality. (b) SW test pass rates show increasing normality with larger cell group sizes, with pass rates exceeding 95% for 32 bits and above. (c) KS test pass ratesfurther support strong Gaussian properties for larger cell group sizes, aligning with SW test results. The findings confirm that SRAM-based Gaussian noise generator is highly suited for PQCapplications.\u003c/p\u003e","description":"","filename":"6.png","url":"https://assets-eu.researchsquare.com/files/rs-7249561/v1/149e438dcbb25b4a140fbe91.png"},{"id":89837635,"identity":"b83cda01-d3fe-42d3-9194-45d217a0887e","added_by":"auto","created_at":"2025-08-25 14:58:06","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":237829,"visible":true,"origin":"","legend":"\u003cp\u003eImpact of temperature variations on statistical properties of SRAM-based Gaussian noise. (a) Hamming weight distribution at –20 °Cshows Gaussian shape with mean (μ = 131.6) and standard deviation (σ = 12.1). (b) Hamming weight distribution at 100 °C maintains Gaussian characteristics with reduced standard deviation (σ = 8.9). (c) Temperature dependence of mean and standard deviation reveals that mean remains stable across temperatures, while standard deviation increases at lower temperatures. (d) Q–Q plots at –20 °C, 20 °C, and 100 °C demonstrate consistent Gaussian characteristics across temperature variations. (e) KL divergence analysis shows low values across all temperatures, confirming statistical similarity to ideal Gaussian distribution. (f) SW and KS test pass rates remain above 95%at all temperatures, validating robustness of SRAM-based noise under environmental fluctuations. These results confirm the temperature resilience of the SRAM-based Gaussian noise generator, ensuring reliable cryptographic security across diverse operating conditions.\u003c/p\u003e","description":"","filename":"7.png","url":"https://assets-eu.researchsquare.com/files/rs-7249561/v1/b5611a5c611e35a5904a61f2.png"},{"id":98244209,"identity":"ed9e968d-7e56-4e4e-914c-6d3bf8d20fda","added_by":"auto","created_at":"2025-12-15 16:13:42","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":2389047,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7249561/v1/576c1c39-f010-464f-b19d-aaa3634d508c.pdf"},{"id":89837621,"identity":"df631c1c-2ddb-47cc-b492-487b401665c6","added_by":"auto","created_at":"2025-08-25 14:58:05","extension":"docx","order_by":0,"title":"","display":"","copyAsset":false,"role":"supplement","size":2760624,"visible":true,"origin":"","legend":"","description":"","filename":"Supportinginformationver00.docx","url":"https://assets-eu.researchsquare.com/files/rs-7249561/v1/1bffdc33ebed7cfeed4baa8b.docx"}],"financialInterests":"No competing interests reported.","formattedTitle":"SRAM-based Gaussian Noise Generation for Post- Quantum Cryptography: A Lightweight Hardware Approach","fulltext":[{"header":"INTRODUCTION","content":"\u003cp\u003eThe emergence of quantum computing technology poses a substantial threat to conventional cryptographic systems, such as Rivest\u0026ndash;Shamir\u0026ndash;Adleman (RSA) and elliptic curve cryptography (ECC), which rely on the computational hardness of mathematical problems for security [1]\u0026ndash;[3]. These classical cryptographic protocols, foundational to modern digital security, are vulnerable to quantum algorithms that can efficiently factor large integers and solve discrete logarithm problems [4]\u0026ndash;[5]. Consequently, post-quantum cryptography (PQC) has gained attention as a promising alternative, ensuring resilience against quantum adversaries [6]\u0026ndash;[7].\u003c/p\u003e\n\u003cp\u003eA fundamental principle of PQC is the learning with errors (LWE) problem, which introduces intentional noise in encryption and public-key generation operations to enhance security [7]\u0026ndash;[9]. The LWE problem renders cryptanalysis infeasible for adversaries by introducing a Gaussian-distributed error term [7]\u0026ndash;[9]. Thus, the ability to generate Gaussian-distributed signals is crucial for implementing PQC, particularly in lattice-based and code-based cryptographic schemes [10]\u0026ndash;[11]. However, existing Gaussian noise generation techniques, such as Box\u0026ndash;Muller transformation, rejection sampling, and Ziggurat sampling, impose significant computational and memory overheads, requiring hardware-intensive arithmetic operations or large precomputed lookup tables [12]\u0026ndash;[15].\u003c/p\u003e\n\u003cp\u003eTo address these limitations, we propose a hardware-based Gaussian noise generator\u0026nbsp;that leverages\u0026nbsp;the intrinsic power-on initialization characteristics of static random-access memory (SRAM). Unlike traditional Gaussian samplers that require dedicated analog-to-digital converters (ADCs), precomputed cumulative distribution tables (CDTs), or random number generators (RNGs), the proposed approach extracts Gaussian noise directly from SRAM initialization data, eliminating additional computational burdens. To our knowledge, this is the first demonstration of a fully digital Gaussian noise generator for PQC that requires no additional entropy source, CDTs, RNGs, or analog circuitry.\u0026nbsp;The key idea behind this technique is the inherent randomness of SRAM power-on states, which result from process variations and intrinsic device-level fluctuations [16]\u0026ndash;[17]. When an SRAM module is powered on, each memory cell unpredictably stabilizes in either logic \u0026ldquo;0\u0026rdquo; or logic \u0026ldquo;1\u0026rdquo; [16]\u0026ndash;[17]. By aggregating the start-up states of multiple SRAM cells and computing their Hamming weight, the proposed module efficiently generates integer values that follow a Gaussian normal distribution. The contributions of this study are as follows:\u003c/p\u003e\n\u003cp\u003e1.\u0026nbsp; \u0026nbsp;We propose a novel SRAM-based Gaussian noise generator for PQC applications, leveraging power-on start-up variations in SRAM cells.\u003c/p\u003e\n\u003cp\u003e2.\u0026nbsp; \u0026nbsp;We demonstrate that computing the Hamming weight of SRAM power-on bits results in a distribution\u0026nbsp;that follows\u0026nbsp;a Gaussian shape,\u0026nbsp;similar to a large-scale fair coin toss experiment governed by the binomial distribution.\u003c/p\u003e\n\u003cp\u003e3. \u0026nbsp; We validate the statistical normality of the generated noise under different environmental conditions and operating temperatures, ensuring robustness for practical cryptographic implementations.\u003c/p\u003e"},{"header":"MATERIALS AND METHODS","content":"\u003cp\u003e\u003cstrong\u003eExperimental setup\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe experimental setup is designed to generate and analyze SRAM-based Gaussian noise for PQC applications. SRAM initialization data serves as a physical entropy source, leveraging power-up state initialization variations to produce randomness. The SRAM chip model was selected based on the following criteria:\u003c/p\u003e\n\u003cp\u003e1. Models that allow the extraction of SRAM data before executing SRAM writing operations.\u003c/p\u003e\n\u003cp\u003e2. Industrial models that guarantee reliable operation within a tolerable temperature range of −20 °C to 80 °C.\u003c/p\u003e\n\u003cp\u003eBased on these criteria, IS62WV51216 (Integrated Silicon Solution Inc.: ISSI) models were selected [18]. The Open103Z development board, built around an STM32 microcontroller, serves as the primary hardware interface, facilitating real-time communication with the SRAM module and executing Hamming weight extraction algorithms [19]. A host personal computer (PC) communicates with the Open103Z board to control power cycling and data extraction. The data acquisition process consists of the following steps:\u003c/p\u003e\n\u003cp\u003e1. Power cycling: The SRAM chips are powered on and off under controlled conditions.\u003c/p\u003e\n\u003cp\u003e2. Data extraction: After each power-up, the host PC extracts the SRAM initialization data.\u003c/p\u003e\n\u003cp\u003e3. Gaussian conversion: The extracted SRAM initialization data is processed to compute the Hamming weight, which is subsequently utilized in PQC applications as a Gaussian-distributed integer.\u003c/p\u003e\n\u003cp\u003eTo evaluate the temperature dependence of the SRAM-based Gaussian noise generator, a CTHC-65N temperature chamber was utilized. The chamber supports an operational temperature range of −30 °C to 150 °C. The SRAM chip and test boards were placed inside the chamber, while the host PC remained external for data extraction and system control. The target test temperatures were set at −20 °C, 0 °C, 20 °C, 40 °C, 60 °C, and 80 °C to ensure a comprehensive analysis of thermal effects.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eSimulation setup\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eTo analyze SRAM initialization dynamics and logic state transitions, simulations were conducted using LTSPICE (SPICE: Simulation Program with Integrated Circuit Emphasis), an analog circuit simulator from Analog Devices [20]. The Si7540DP_N and Si7540DP_P transistor models were used to represent NMOS and PMOS devices, respectively.\u003c/p\u003e\n\u003cp\u003eTo simulate ideal Gaussian noise and ideal fair coin toss behavior, MATLAB software was employed. Specifically:\u003c/p\u003e\n\u003cp\u003e1. The randn() function generated an ideal Gaussian-distributed dataset, serving as a statistically valid reference for comparison [21]. \u003c/p\u003e\n\u003cp\u003e2. The randi() function simulated an unbiased Bernoulli process, where each bit follows a fair coin toss distribution with equal probabilities of “0” and “1” [21].\u003c/p\u003e\n\u003cp\u003eThese simulations provide a theoretical baseline for quantitative comparisons between SRAM-generated data, ideal Gaussian data, and unbiased binary data models.\u003c/p\u003e"},{"header":"RESULTS AND DISCUSSION","content":"\u003cp\u003eFigure 1 illustrates the overall architecture of a post-quantum cryptography (PQC) system designed to leverage an SRAM-based Gaussian noise generator. Figure 1(a) presents the schematic of a PQC system integrating an SRAM module as a Gaussian noise source, detailing the functional connections between the system bus, PQC engine, and the SRAM-based Gaussian noise generator. The system bus transmits the Gaussian noise signal from the SRAM module to the PQC engine. The SRAM module is a key component in ensuring cryptographic security by generating Gaussian noise, which mitigates adversarial decryption attempts, even in the presence of quantum computing capabilities. Figure 1(b) highlights the internal structure of the SRAM-based Gaussian noise generator. Raw SRAM power-on data is retrieved and processed through a Hamming weight extractor, which transforms the raw bits into a statistically Gaussian-distributed noise signal. The resulting noise, visualized in the schematic as a bell-shaped histogram, is critical for LWE or related lattice-based and code-based encryption schemes. This hardware-driven noise sampling approach significantly reduces the computational overhead associated with traditional software-based Gaussian samplers while leveraging the intrinsic stochastic behavior of SRAM devices to enhance security. Figure 1(c) illustrates the PQC encryption process using Gaussian noise errors. A plain image (left) is input into the PQC engine, which employs the SRAM-generated Gaussian noise to produce a ciphered image (right). Without the aid of SRAM-generated noise, quantum computing algorithms can perfectly decrypt the ciphered output back to the original image [4]\u0026ndash;[5]. In contrast, incorporating SRAM-generated noise prevents quantum adversaries from achieving decryption. Thus, the SRAM-based Gaussian noise generator plays a vital role in PQC implementations, providing resilience against quantum computing threats [10]\u0026ndash;[11].\u003c/p\u003e\n\u003cp\u003eFigure 2 provides a detailed analysis of SRAM initialization and logic state transitions. Figure 2(a) shows simulated voltage transient curves during SRAM initialization, illustrating the time-dependent variations of \u003cem\u003eV\u003c/em\u003e\u003csub\u003eS\u003c/sub\u003e, \u003cem\u003eV\u003c/em\u003e\u003csub\u003eA\u003c/sub\u003e, and \u003cem\u003eV\u003c/em\u003e\u003csub\u003eB\u003c/sub\u003e. Figure 2(b) depicts the circuit structure of an SRAM unit cell, comprising PMOS transistors (M1 and M2) and NMOS transistors (M3 and M4). \u003cem\u003eV\u003c/em\u003e\u003csub\u003eS\u003c/sub\u003e acts as the primary voltage supply, while \u003cem\u003eV\u003c/em\u003e\u003csub\u003eA\u003c/sub\u003e and \u003cem\u003eV\u003c/em\u003e\u003csub\u003eB\u003c/sub\u003e serve as internal node voltages between PMOS transistors M1 and M2, and NMOS transistors M3 and M4, respectively. The voltage levels at \u003cem\u003eV\u003c/em\u003e\u003csub\u003eA\u003c/sub\u003e and \u003cem\u003eV\u003c/em\u003e\u003csub\u003eB\u003c/sub\u003e dynamically adjust based on the threshold voltage (\u003cem\u003eV\u003c/em\u003e\u003csub\u003eTH\u003c/sub\u003e) conditions of the transistors in the SRAM unit cell (Threshold voltages: \u003cem\u003eV\u003c/em\u003e\u003csub\u003eTH1\u003c/sub\u003e for M1, \u003cem\u003eV\u003c/em\u003e\u003csub\u003eTH2\u003c/sub\u003e for M2, \u003cem\u003eV\u003c/em\u003e\u003csub\u003eTH3\u003c/sub\u003e for M3, and \u003cem\u003eV\u003c/em\u003e\u003csub\u003eTH4\u003c/sub\u003e for M4). The simulated transient curve in Figure 2(a) represents time-dependent voltage variations when the threshold voltages are intentionally set to |\u003cem\u003eV\u003c/em\u003e\u003csub\u003eTH1\u003c/sub\u003e| \u0026lt; |\u003cem\u003eV\u003c/em\u003e\u003csub\u003eTH2\u003c/sub\u003e|, where absolute values are considered due to the negative \u003cem\u003eV\u003c/em\u003e\u003csub\u003eTH\u003c/sub\u003e characteristics of typical PMOS transistors.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eFigures 2(c) and 2(d) describe the sequential transition process for initializing the SRAM cell into logic \u0026ldquo;1\u0026rdquo; and logic \u0026ldquo;0\u0026rdquo; states, respectively. The logic \u0026ldquo;1\u0026rdquo; state follows a four-step process: (1) power-off state, (2) activation of PMOS M1, which causes \u003cem\u003eV\u003c/em\u003e\u003csub\u003eA\u003c/sub\u003e to track \u003cem\u003eV\u003c/em\u003e\u003csub\u003eS\u003c/sub\u003e, (3) activation of NMOS M4, pulling \u003cem\u003eV\u003c/em\u003e\u003csub\u003eB\u003c/sub\u003e to \u003cem\u003eV\u003c/em\u003e\u003csub\u003eGND\u003c/sub\u003e, and (4) stabilization with \u003cem\u003eV\u003c/em\u003e\u003csub\u003eA\u003c/sub\u003e = \u003cem\u003eV\u003c/em\u003e\u003csub\u003eDD\u003c/sub\u003e and \u003cem\u003eV\u003c/em\u003e\u003csub\u003eB\u003c/sub\u003e = \u003cem\u003eV\u003c/em\u003e\u003csub\u003eGND\u003c/sub\u003e [22]\u0026ndash;[23]. Conversely, the logic \u0026ldquo;0\u0026rdquo; transition sequence is as follows: (1) power-off, (2) activation of PMOS M2, causing \u003cem\u003eV\u003c/em\u003e\u003csub\u003eB\u003c/sub\u003e to track \u003cem\u003eV\u003c/em\u003e\u003csub\u003eS\u003c/sub\u003e, (3) activation of NMOS M3, pulling \u003cem\u003eV\u003c/em\u003e\u003csub\u003eA\u003c/sub\u003e to \u003cem\u003eV\u003c/em\u003e\u003csub\u003eGND\u003c/sub\u003e, and (4) final stabilization with \u003cem\u003eV\u003c/em\u003e\u003csub\u003eB\u003c/sub\u003e = \u003cem\u003eV\u003c/em\u003e\u003csub\u003eDD\u003c/sub\u003e and \u003cem\u003eV\u003c/em\u003e\u003csub\u003eA\u003c/sub\u003e = \u003cem\u003eV\u003c/em\u003e\u003csub\u003eGND\u003c/sub\u003e. Process variations influence whether the SRAM unit cell stabilizes in the logic \u0026ldquo;0\u0026rdquo; or \u0026ldquo;1\u0026rdquo; state, resulting in a statistical distribution that exhibits inherent unpredictability and unbiased characteristics, akin to a fair coin toss.\u003c/p\u003e\n\u003cp\u003eFigure 3 provides an in-depth view of the implementation of the SRAM-based Gaussian noise generator, focusing on the extraction of Gaussian-distributed data. Figure 3(a) illustrates the operational workflow of the SRAM-based Gaussian noise generator, where raw SRAM initialization data is collected and processed to extract Gaussian-distributed integers. The Hamming weight extractor converts raw binary data into Gaussian-distributed integers by counting the number of logic \u0026ldquo;1\u0026rdquo; values within a defined bit group. These Gaussian-distributed integers are critical for PQC applications, particularly in the implementation of the LWE. Figure 3(b) presents the experimental setup for the SRAM-based Gaussian noise generator. An Open103Z development board interfaces with the SRAM chip, facilitating communication and executing the Hamming weight extraction process. Figure 3(c) compares two implementations of the Hamming weight extractor: one using the C programming language for processor-based applications and the other utilizing hardware description language (HDL) for FPGA and SoC-based cryptographic accelerators [24]\u0026ndash;[25]. Both implementations demonstrate low overhead, making them well-suited for practical cryptographic systems.\u003c/p\u003e\n\u003cp\u003eFigure 4 presents a comprehensive statistical analysis to validate the Gaussian characteristics of the SRAM-based noise generator. To ensure the reliability of the generated noise, three datasets are compared: (A group) experimentally obtained SRAM data, (B group) an ideal Gaussian distribution generated by simulation, and (C group) a fair coin toss simulation. The results confirm that the SRAM-generated noise closely follows a Gaussian distribution, reinforcing its applicability in PQC.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eFigure 4(a) illustrates the methodology for generating the experimental and simulated datasets. The (A group) dataset represents the raw Hamming weight data extracted from the SRAM module, following the methodology described in Figure 3. First, 256 SRAM power-on bits are collected, and the Hamming weight is computed, yielding a Gaussian-distributed integer ranging from 0 to 256. (B group) serves as an ideal Gaussian reference with the same mean (\u003cem\u003e\u0026mu;\u003c/em\u003e = 129.8) and standard deviation (\u003cem\u003e\u0026sigma;\u003c/em\u003e = 9.2) as the SRAM dataset. This reference dataset is generated using MATLAB\u0026rsquo;s randn() function, which produces normally distributed random numbers [21]. (C group) is derived from an independent and identically distributed (IID) Bernoulli process, simulating a fair coin toss.\u003c/p\u003e\n\u003cp\u003eFigure 4(b) compares the empirical distributions of the three datasets using histograms overlaid with theoretical Gaussian curves (red lines). Visually, all three datasets exhibit near-Gaussian distributions; however, slight deviations in the mean (\u003cem\u003e\u0026mu;\u003c/em\u003e) and standard deviation (\u003cem\u003e\u0026sigma;\u003c/em\u003e) are observed between the SRAM-generated data (A group) and the IID binary sequence (C group). These deviations suggest that the SRAM experimental data (A group) has some intrinsic correlations among SRAM cells [26]\u0026ndash;[27].\u003c/p\u003e\n\u003cp\u003eTo rigorously validate the normality of these datasets, several statistical metrics are employed, including the quantile\u0026ndash;quantile (Q\u0026ndash;Q) plot, Shapiro\u0026ndash;Wilk (SW) test, Kolmogorov\u0026ndash;Smirnov (KS) test, and Kullback\u0026ndash;Leibler (KL) divergence. Figure 4(c) presents Q\u0026ndash;Q plots for each dataset, comparing the empirical quantiles to theoretical Gaussian quantiles. If the data follows a Gaussian distribution, the points should align closely with the \u003cem\u003ey\u003c/em\u003e = \u003cem\u003ex\u003c/em\u003e reference line [28]\u0026ndash;[29]. All Q\u0026ndash;Q plots in Figure 4(c) show straight-line alignment, confirming that the SRAM-generated noise (A group) adheres to a Gaussian distribution, making it suitable for PQC applications.\u003c/p\u003e\n\u003cp\u003eFigure 4(d) presents the results of the statistical tests. The SW test assesses whether a dataset follows a normal distribution, while the KS test determines whether two datasets originate from the same probability distribution [30]\u0026ndash;[31]. The pass rates for the SW and KS tests exceed 95%, demonstrating that the SRAM experimental data exhibits strong normality. Additionally, KL divergence quantifies the difference between a probability distribution and a reference distribution. A KL divergence value of 0 indicates a perfect match, while values below 0.01 suggest near-identical distributions [32]-[33]. In this study, KL divergence values remain well below 0.01, confirming that the SRAM-based Gaussian noise generator produces statistically valid Gaussian noise.\u003c/p\u003e\n\u003cp\u003eOverall, Figure 4 provides strong empirical evidence that the SRAM-based noise generator effectively produces Gaussian-distributed integers. While the SRAM-based integer data shows slight deviations from ideal IID Bernoulli distributions, the observed statistical properties remain sufficiently close to a true Gaussian distribution to support its application in PQC. By employing multiple statistical validation methods, including Q\u0026ndash;Q plots, SW tests, KS tests, and KL divergence, these results confirm the suitability of the SRAM-based noise generator for PQC.\u003c/p\u003e\n\u003cp\u003eFigure 5 presents a detailed analysis of the Hamming weight distribution as a function of cell group size. This analysis evaluates whether the Gaussian characteristics of SRAM-generated noise are maintained across different grouping configurations.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eFigure 5(a) illustrates the process of computing the Hamming weight for different cell group sizes, where the cell group size represents the number of SRAM bits used in each calculation. For example, with an 8-bit cell group, the Hamming weight ranges from 0 to 8. The extracted SRAM experimental data exhibits a near-Gaussian distribution, closely following the theoretical Gaussian curve (\u003cem\u003e\u0026mu;\u003c/em\u003e = 4.1 and \u003cem\u003e\u0026sigma;\u003c/em\u003e = 1.7), as shown by the red lines.\u003c/p\u003e\n\u003cp\u003eFigures 5(b), 5(c), and 5(d) provide a comparative statistical evaluation between the SRAM-based experimental data (shaded bars) and the fair coin toss simulation (solid red lines) across various cell group sizes. Figure 5(b) presents the average Hamming weight for each group size, plotted on a logarithmic scale. Both the SRAM data and the coin toss simulation show a linear increase in the mean Hamming weight with increasing group size, consistent with theoretical expectations.\u003c/p\u003e\n\u003cp\u003eFigure 5(c) shows the standard deviation of the Hamming weight distribution as a function of group size, while Figure 5(d) presents the normalized standard deviation (NSD), defined as the standard deviation divided by the square root of the group size. Theoretically, for an ideal Bernoulli process, the NSD converges to a constant value of 0.5 [34]\u0026ndash;[35]. The NSD values obtained from the SRAM experimental data in Figure 5(d) closely follow this theoretical trend, aligning consistently with the values derived from the fair coin toss simulation. This indicates that the statistical behavior of SRAM power-on states approximates that of an ideal fair Bernoulli process across different group sizes. Minor deviations from the theoretical model are observed, possibly due to intrinsic correlations among SRAM cells during the power-on process [26]\u0026ndash;[27]. However, Figure 5 as a whole demonstrates that the SRAM-based noise follows a statistical trend remarkably similar to that of the fair coin toss model. Furthermore, the NSD results suggest suitability for application as a Gaussian error sampler in lattice-based and code-based PQC algorithms [36]\u0026ndash;[37].\u003c/p\u003e\n\u003cp\u003eOverall, the results provide strong evidence that the SRAM-based data exhibits stable and reliable statistical properties, supporting its applicability for PQC, particularly in Gaussian error sampling for Kyber and NTRU algorithms [36]\u0026ndash;[37].\u003c/p\u003e\n\u003cp\u003eFigure 6 presents a statistical assessment of the normality of SRAM-based Gaussian noise across different cell group sizes. The analysis includes three key metrics: (a) KL divergence, (b) SW test pass rate, and (c) KS test pass rate. The evaluation is conducted for three datasets: (A group) SRAM experimental data, (B group) an ideal Gaussian distribution, and (C group) a fair coin toss simulation. The objective is to validate the Gaussian characteristics of SRAM-generated noise and its applicability to PQC.\u003c/p\u003e\n\u003cp\u003eFigure 6(a) presents the KL divergence values comparing the SRAM experimental data (A group) to the ideal Gaussian distribution (B group) and the fair coin toss dataset (C group) to the same Gaussian reference. The results indicate that KL divergence values remain below 10⁻\u0026sup2; for all datasets, confirming that the SRAM-based data successfully produces Gaussian-distributed noise with a high degree of statistical fidelity [32]\u0026ndash;[33]. Notably, the KL divergence reaches its lowest value for the SRAM data when the group size is 64 bits, suggesting optimal Gaussian conformity at this configuration.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eFigure 6(b) shows the SW test pass rates for each dataset. For smaller cell group sizes (\u003cem\u003ee.g.\u003c/em\u003e, 8 and 16 bits), the pass rates are lower due to the inherent variability associated with small sample sizes. However, as the group size increases to 32 bits and beyond, the pass rates for all three datasets exceed 95%, indicating strong normality. This suggests that SRAM-generated noise demonstrates Gaussian characteristics when the cell group size is sufficiently large, \u003cem\u003ee.g.\u003c/em\u003e, 32 bits or more.\u003c/p\u003e\n\u003cp\u003eFigure 6(c) displays the KS test pass rates across varying group sizes for all datasets. As the group size increases to 16 bits and above, the pass rates for the SRAM experimental data (A group) consistently exceed 95%, further confirming normality. These results align with those observed in the SW test, as shown in Figure 6(b).\u003c/p\u003e\n\u003cp\u003eIn summary, Figure 6 provides robust statistical validation that SRAM-based noise exhibits Gaussian-distributed properties, particularly for cell group sizes of 32 bits and above. The low KL divergence values (Figure 6a), high pass rates in the SW (Figure 6b), and KS (Figure 6c) tests confirm the suitability of SRAM-generated noise for PQC applications.\u003c/p\u003e\n\u003cp\u003eFigure 7 presents a comprehensive evaluation of the impact of temperature variations on the statistical properties of SRAM-based Gaussian noise. The analysis spans a temperature range from \u0026ndash;20 \u0026deg;C to 100 \u0026deg;C and examines its effects on the Hamming weight distribution, mean, standard deviation, and normality metrics, including Q\u0026ndash;Q plots, KL divergence, SW test, and KS test. This investigation aims to validate the robustness and environmental stability of SRAM-generated noise, ensuring its reliability for PQC applications.\u003c/p\u003e\n\u003cp\u003eFigures 7(a) and 7(b) illustrate the Hamming weight distributions obtained from SRAM data at \u0026ndash;20 \u0026deg;C and 100 \u0026deg;C, respectively. Both distributions exhibit near-Gaussian shapes, with the annotated mean and standard deviation values demonstrating temperature dependence. Notably, the mean remains relatively stable (\u003cem\u003e\u0026mu;\u003c/em\u003e = 131.6 at \u0026ndash;20 \u0026deg;C and \u003cem\u003e\u0026mu;\u003c/em\u003e = 129.8 at 100 \u0026deg;C), while the standard deviation decreases from \u003cem\u003e\u0026sigma;\u003c/em\u003e = 12.1 to \u003cem\u003e\u0026sigma;\u003c/em\u003e = 8.9 as the temperature increases.\u003c/p\u003e\n\u003cp\u003eFigure 7(c) provides a detailed analysis of the temperature-dependent trends in the mean and standard deviation of the Hamming weight distribution. The mean (gray bars) remains consistently stable across the full temperature range, confirming the temperature-invariant nature of the mean for SRAM-based Gaussian noise. In contrast, the standard deviation (red line) shows an increasing trend at lower temperatures, indicating broader distribution spreads in colder conditions. This behavior is attributed to temperature-induced changes in SRAM cell characteristics, such as data remanence and static noise margin [23], [38], [39].\u003c/p\u003e\n\u003cp\u003eFigure 7(d) presents Q\u0026ndash;Q plots at three representative temperatures (\u0026ndash;20 \u0026deg;C, 20 \u0026deg;C, and 100 \u0026deg;C), visually assessing the normality of the SRAM-generated noise. Across all temperature conditions, the data points align closely with the \u003cem\u003ey\u003c/em\u003e = \u003cem\u003ex\u003c/em\u003e reference line, indicating that Gaussian characteristics are preserved despite environmental variations.\u003c/p\u003e\n\u003cp\u003eFigure 7(e) shows KL divergence values at each temperature point, quantifying the similarity between the SRAM data and an ideal Gaussian distribution. In all cases, KL divergence values remain well below 10⁻\u0026sup2;, indicating a high degree of statistical similarity between the measured and reference distributions [32]\u0026ndash;[33].\u003c/p\u003e\n\u003cp\u003eFigure 7(f) displays the pass rates of the SW and KS tests across the tested temperature range. The pass rates exceed 95% in all cases, confirming strong normality and reinforcing the temperature resilience of the SRAM-based noise generator.\u003c/p\u003e\n\u003cp\u003eIn summary, Figure 7 provides compelling empirical evidence that the statistical properties of SRAM-generated Gaussian noise remain stable across a wide temperature range (\u0026ndash;20 \u0026deg;C to 100 \u0026deg;C). Although slight increases in standard deviation occur at lower temperatures, overall Gaussian behavior is preserved, as confirmed by Q\u0026ndash;Q plots (Figure 7d), KL divergence (Figure 7e), and SW/KS test results (Figure 7f). These findings validate the environmental robustness of the SRAM-based Gaussian noise generator, ensuring its suitability for secure and reliable PQC implementations across diverse operating conditions.\u003c/p\u003e"},{"header":"CONCLUSION","content":"\u003cp\u003eThis study presents a hardware-efficient Gaussian noise generator based on static random access memory (SRAM) power-on behavior for secure post-quantum cryptography (PQC) implementations. By leveraging the inherent unpredictability in SRAM start-up states, the proposed system extracts Gaussian-distributed integers using a lightweight Hamming weight extraction method. Compared to conventional software-based Gaussian samplers, this approach significantly reduces computational overhead and is well-suited for resource-constrained PQC hardware platforms.\u003c/p\u003e\n\u003cp\u003eExperimental validation confirms that the data generated from SRAM start-up sequences closely follows a Gaussian distribution across varying cell group sizes and environmental conditions. Statistical tests further support that the SRAM-based Gaussian integer generator is highly suitable for PQC applications, particularly lattice-based and code-based PQC algorithms. Kullback\u0026ndash;Leibler (KL)\u0026nbsp;divergence values consistently remain below 0.01, and both the Shapiro\u0026ndash;Wilk (SW) and Kolmogorov\u0026ndash;Smirnov (KS) tests report pass rates exceeding 95% across all cell group sizes.\u003c/p\u003e\n\u003cp\u003eRobustness under environmental variation was also demonstrated. Across a temperature range from \u0026ndash;20 \u0026deg;C to 100 \u0026deg;C, the mean Hamming weight remains nearly constant, while the standard deviation shifts modestly from \u003cem\u003e\u0026sigma;\u003c/em\u003e = 12.1 to \u003cem\u003e\u0026sigma;\u003c/em\u003e = 8.9. Despite these slight deviations, all statistical normality metrics remain stable, indicating strong environmental resilience.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eIn conclusion, the proposed SRAM-based Gaussian noise generator offers a lightweight, statistically validated, and thermally robust solution for PQC systems, paving the way for practical and secure cryptographic implementations in system on chip (SoC) or microcontrollers in the post-quantum era.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eData availability Statement\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe data that support the findings of this study are available from the corresponding author upon reasonable request.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eDeclaration of\u0026nbsp;\u003c/strong\u003e\u003cstrong\u003eCompeting\u0026nbsp;Interests\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe authors declare no competing financial interests.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAcknowledgments\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThis work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (RS-2024-00449412).\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eDeclaration of competing interests\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe authors declare no competing financial interest.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAuthor contribution declaration\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eM. S. Kim performed conceptualization, investigation, writing-original draft, software, and visualization.\u003c/p\u003e\n\u003cp\u003eS. B. Jeon performed data curation, and methodology.\u003c/p\u003e\n\u003cp\u003eS. Kim performed Writing-Review \u0026amp; Editing, supervision, project administration, and funding acquisition.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eFunding declaration\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThis work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (RS-2024-00449412).\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eW. Buchanan and A. Woodward, \u0026quot;Will quantum computers be the end of public key encryption?,\u0026quot; \u003cem\u003eJ. Cyber Secur. Technol.\u003c/em\u003e, vol. 1, no. 1, pp. 1\u0026ndash;22, 2017.\u003c/li\u003e\n\u003cli\u003eR. Azhari and A. N. 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(ISQED)\u003c/em\u003e, 2012, pp. 516\u0026ndash;519.\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true},"keywords":"Gaussian error sampler, post-quantum cryptography (PQC), learning with error (LWE), SRAM (Static Random Access Memory)","lastPublishedDoi":"10.21203/rs.3.rs-7249561/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7249561/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eAs quantum computing progresses, conventional public-key cryptographic schemes such as RSA and ECC face increasing vulnerability to quantum attacks. Post-quantum cryptography (PQC), especially schemes based on the learning with errors (LWE) problem, depends on Gaussian-distributed noise for security. However, traditional Gaussian noise generation methods—such as Box–Muller, rejection sampling, and Ziggurat—incur high computational and memory costs, making them unsuitable for lightweight or embedded systems. This paper proposes a hardware-based Gaussian noise generator that uses the inherent randomness of static random access memory (SRAM) power-on states. The method aggregates SRAM start-up bits and computes their Hamming weight to efficiently generate Gaussian-distributed integers without analog components, large lookup tables, or external random number generators. Experimental results show that the output closely matches a Gaussian distribution under various group sizes and environmental conditions. Statistical tests, including Shapiro–Wilk and Kolmogorov–Smirnov, achieve over 95% pass rates, while Kullback–Leibler divergence remains below 0.01. The generator also maintains Gaussian properties across a wide thermal range (−20 °C to 100 °C). These results demonstrate that the proposed SRAM-based generator offers a practical, lightweight, and thermally robust solution for PQC, particularly in lattice- and code-based cryptographic schemes.\u003c/p\u003e","manuscriptTitle":"SRAM-based Gaussian Noise Generation for Post- Quantum Cryptography: A Lightweight Hardware Approach","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-08-25 14:58:00","doi":"10.21203/rs.3.rs-7249561/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2025-10-01T14:27:27+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-09-30T14:44:18+00:00","index":"hide","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-09-23T19:37:10+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"56860177780645913729086310884949682596","date":"2025-09-09T02:00:19+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"328970015508707318960344694694451911744","date":"2025-09-02T15:00:33+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"28908898044577935449021693321708637084","date":"2025-09-02T09:29:14+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"100431607346046429552841130073885696071","date":"2025-08-23T05:23:28+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2025-08-18T03:37:45+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2025-08-18T02:15:18+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2025-08-11T06:53:38+00:00","index":"","fulltext":""},{"type":"submitted","content":"Scientific Reports","date":"2025-08-11T06:50:17+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"2d89be4c-df56-4996-931b-bfe5bb4a86d3","owner":[],"postedDate":"August 25th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[{"id":53598442,"name":"Physical sciences/Engineering"},{"id":53598443,"name":"Physical sciences/Mathematics and computing"},{"id":53598444,"name":"Physical sciences/Physics"}],"tags":[],"updatedAt":"2025-12-15T16:07:14+00:00","versionOfRecord":{"articleIdentity":"rs-7249561","link":"https://doi.org/10.1038/s41598-025-27929-3","journal":{"identity":"scientific-reports","isVorOnly":false,"title":"Scientific Reports"},"publishedOn":"2025-12-11 15:58:09","publishedOnDateReadable":"December 11th, 2025"},"versionCreatedAt":"2025-08-25 14:58:00","video":"","vorDoi":"10.1038/s41598-025-27929-3","vorDoiUrl":"https://doi.org/10.1038/s41598-025-27929-3","workflowStages":[]},"version":"v1","identity":"rs-7249561","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-7249561","identity":"rs-7249561","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}