Abstract
Physical unclonable functions (PUFs) are complex physical objects that aim at overcoming the vulnerabilities of traditional cryptographic keys, promising a robust class of security primitives for different applications. Optical PUFs present advantages over traditional electronic realizations, namely, a stronger unclonability, but suffer from problems of reliability and weak unpredictability of the key. We here develop a two-step PUF generation strategy based on deep learning, which associates reliable keys verified against the National Institute of Standards and Technology (NIST) certification standards of true random generators for cryptography. The idea explored in this work is to decouple the design of the PUFs from the key generation and train a neural architecture to learn the mapping algorithm between the key and the PUF. We report experimental results with all-optical PUFs realized in silica aerogels and analyzed a population of 100 generated keys, each of 10,000 bit length. The key generated passed all tests required by the NIST standard, with proportion outcomes well beyond the NIST’s recommended threshold. The two-step key generation strategy studied in this work can be generalized to any PUF based on either optical or electronic implementations. It can help the design of robust PUFs for both secure authentications and encrypted communications.
1 Introduction
The modern digital society relies on mobile and ubiquitous optoelectronic devices whose software and hardware security is becoming a global concern owing to the increasing number of disclosed attacks every day [1], [2], [3], [4]. The emergence of smart cities, the Internet of things, cloud computing, and big data will generate more challenges in this field [5], [6], [7], [8], calling for new opportunities in research. Current cryptography methods for addressing security issues center on the idea of having a digital key, which is safely stored and whose information remains unknown to an adversary. However, implementing this simple concept turns out to be a difficult task: software such as Trojan horses and malware, and side-channel attacks carried out by enemies with single access to the device, can expose the key and lead to security breaches [4], [9], [10], [11], [12]. As Tim Cook (Apple CEO) emphasized in a recent interview [13]:
“If you put a key under the mat for the cops, a burglar can find it, too. Criminals are using every technology tool at their disposal to hack into people’s accounts. If they know there’s a key hidden somewhere, they will not stop until they find it.”
These considerations fueled the development of physical unclonable functions (PUFs) [14], [15], [16]. A PUF is an object composed of a disordered structure, such as, e.g., a light scatterer, which stores a physical key inside a material layer with no mathematical description. In these systems, a digital key is typically generated by first challenging the PUF with an input signal and then converting into a binary sequence the analog response measured in either time, space, or frequency. The main assumption is that the physical disorder of the PUF cannot be reverse engineered, not even by the original manufacturer. If the PUF is safely stored, an adversary who wants to recreate the key has the only possibility of performing a brute force attack, which is practically unfeasible owing to the exponentially large complexity of a PUF [14].
In this field of research, photonics is pioneering technologies for different lines of applications, including authentication [17], [, 18], secure communications [19], [20], [21], and classical equivalent schemes to quantum key distribution with perfect secrecy [22]. The main advantage of photonics PUFs is strong device unclonability: while cloning electronic PUF implementations has been reported [23], no one was ever able to replicate an optical PUF. The main challenge in photonics is the development of general algorithms that transform the response of PUFs into digital keys that appear as unpredictable random sequences. The issue is the local correlations that are present in the PUF response: when transformed into a binary string with conventional techniques, a certain degree of correlation remains in the key and between different keys [24]. To the best of the authors’ knowledge, with the exception of the study by Di Falco et al. [22], no optical PUFs has been verified against certification standards that guarantee the genuine unpredictability and uncorrelation of the keys, and no technique has been devised to address this problem controllably for optical PUFs.
Another difficulty originates from the fact that the complex PUFs are strongly sensitive to input conditions. When traditional analog-digital conversion methods are applied to generate the key, such sensibility can generate different keys for apparently identical input conditions [25]. The issue lies in the impossibility of reproducing the same input conditions in different experiments. In a strongly chaotic system such as a PUF, even a small variation in the input parameters can strongly affect the security primitive’s reliability. If this problem is addressed, it could also open to new PUFs generated via, e.g., soft-like materials, including gels (e.g., hydrogel, aerogels) and foams. These materials are more input sensitive than solid-state counterparts and are currently not employed as security tokens. However, soft-like structures offer security advantages because their nanoscale disorder can reach a higher entropy than artificial human-made PUFs, which are intrinsically limited by the cost, resolution, and scalability of the present nanofabrication technology [26]. In this article, we propose to address the issues mentioned above by combining PUF with deep learning [27], [, 28]. We develop a general and versatile two-step key generation strategy, which guarantees the generation of truly random keys verified against the National Institute of Standards and Technology (NIST) standards for cryptographic applications [29], with each key entirely uncorrelated with the others and reliable. We experimentally demonstrate these results with a new class of nonlinear PUFs implemented with silica aerogels (SAs).
2 Results
2.1 All-optical PUFs with aerogels
SA is a material composed of an ultraporous network of sparse silica aggregates. The SA optical response can be adjusted from complete transparency to strongly chaotic scattering by controlling the silica inclusions’ size and distribution by either mechanical or optical effects [30]. Owing to a low thermal conductivity, SA exhibits a very strong optothermal nonlinearity [31], [, 32], which is associated with large and reversible structural deformations, making SA a nonlinear controllable random material that can be employed in different lines of applications [33], [, 34].
The SA produced in this work is manufactured by a base-catalyzed polymerization process [35], [, 36], which starts by mixing tetramethyl orthosilicate, methanol, and ammonium hydroxide in a 2:4:1 ratio, producing a gel of good clarity and with minimal defects [37]. The mixture is then poured into a Teflon mold, producing a cuboid-shaped gel of 1 cm side. The gel is subsequently removed from the mold and then washed in a series of acetone baths, each lasting 24 h. The transition from wet gel to aerogel happens by using a low-temperature supercritical CO2 drying process [38], with a custom setup assembled in our lab. Figure 1a illustrates the setup used to generate all-optical PUFs, acquired as speckle patterns obtained by illuminating the SA sample with a pump-probe configuration. The setup comprises an expanded monochromatic laser probe (wavelength, λ = 632 nm) and a collimated beam with λ = 488 nm waist of approximately 200 μm. The speckle patterns are converted into digital PUFs using a CCD camera placed after the sample (Figure 1a).
Experimental PUF setup.
(a) All-optical PUF aerogel setup and configuration. (b) PUFs collected by the CCD as speckle patterns for t = 0 s (i), t = 20 s (ii), t = 40 s (iii), and t = 60 s (iv). (c) Picture of a real aerogel sample with (d) the corresponding scanning electron microscope (SEM) image. PUF, physical unclonable function.
In the mapping procedure introduced in this work, it is possible to associate different keys with any class of PUFs that differ in at least a characteristic feature (e.g., distribution or shape of a speckle pattern) that we train the network to resolve. The universal approximation theorem of neural networks [39] guarantees, at least theoretically, that a single neural network that could address this problem exists. In the PUF image acquisition setup illustrated in Figure 1a, it is possible to create PUF images with different speckle features by changing either the laser pump power or the acquisition time. While different pump powers generate diverse characteristic speckles, each pump power triggers a slow dynamical evolution of the speckles over characteristic times of the order of seconds, generating different PUFs in the CCD. Figure 1b shows a typical class of different PUFs that can be acquired at constant pump power (P = 200 mW) and at different times within 1 min of laser illumination. The speckles are observed to be repeatable owing to the good stability properties of SAs [40]. The primary source of entropy that triggers the generation of different speckle patterns in Figure 1b is the spatial fluctuation of the scattering centers of nanoparticles composing the aerogels. These depend on the thermodynamic condition (e.g., temperature, pressure, volume) of the aerogel.
While cloning the soft porous network of SA is hardly imaginable to be feasible now and in the long run owing to the ultradense packing of nanostructured silica components, employing this medium as a PUF generator is also challenging owing to the noticeable spatial fluctuations of the silica nanoparticles, which are visible in the PUF images collected by the CCD (Figure 1b). In the next section, we illustrate a general strategy to address this problem controllably.
2.2 Two-step key generation via deep learning
Figure 2a, b illustrates a high-level schematic of the proposed concept. Ideally, we would like to have at disposal a mapping function
Two-step random key generation via deep learning.
(a–b) Overall process for the training (a) and prediction (b) of key classification and association with PUFs. (c) Detailed schematic workflow of the training procedure: a switch S selects a PUF Inm at the input belonging to single classes cn (red, green, and blue colors). Each class is mapped by a deep neural network to a different integer number n = 1, …, N at the output y, with each number identifying a binary noise (BN) key kn. The space of different keys is generated independently by first sampling binary noise (orange solid line) and then splitting the random sequence into consecutive keys k1, k2, …, kN of equal length. The prediction stage (b) employs the trained neural network of (a) in real-time to associate the keys with PUF images Jmn acquired in different experiments under the same input conditions of Imn and subjected to experimental fluctuations of input parameters. PUF, physical unclonable function.
To address this problem on a general ground, we use a deep neural network (DNN) architecture (Figure 2a, b), which we train to learn the mapping function
Once we set the DNN weights, the network predicts the key association in future experiments with different classes of PUFs of Jnm (Figure 2b), measured under the same input conditions of Imn but acquired in different experiments that differ by uncontrollable fluctuations of the input parameters. The main idea is to include a representative data set comprising a sufficiently large number of PUFs. The DNN learns the features of the experimental fluctuations associated with the different input conditions arising in each class cn, becoming able to predict the future trends Jmn correctly. We increase the data set size until the DNN predicts correctly the key associated with a representative set of PUFs Jmn that does not exist in the training data set. When this occurs, the prediction (b) cross-validates the training (a), and it implies that the DNN has learned the required mapping function
In this classification system, the error is the norm between the integer n identifying the key kn and the output y corresponding to the PUF Inm belonging to the class cn. While more complex classification strategies are possible, we chose this method for its implementation simplicity.
We generate keys satisfying conditions i) and ii) by using binarized physical white noise. The latter is noise obtained by transforming in binary sequence a stream of white noise generated from a physical object and then split the binary stream into diverse keys k1, k2, … , kn of predefined equal length (Figure 2b, orange binary signal). With the method proposed in this work, the generation of PUFs and cryptographic keys are two different problems that we address independently, overcoming the traditional issues that arise when mapping a complex PUF directly to a binary sequence. The two problems are then combined via machine learning, using a DNN that finds the desired mapping that associates each PUF with a cryptography key.
The mapping function learned by the DNN conserves the security advantages of PUFs: it relies on a PUF object that has no mathematical representation, and it ensures a mapping between an input condition and a random key that cannot be guessed or recreated without the PUF object. From a security perspective, the DNN of Figure 2 acts as an additional, two-step protective layer to the PUF. If the PUF falls in the adversary’s hand, the attacker cannot recreate the key without brute forcing all the DNN architecture weights. In a typical integrated electronic system, the space of the combination Sc that the enemy has to explore is
In the scheme of Figure 2, the DNN operation is typically evaluated by electronic CPU at gigahertz speed. It does not add overhead to the PUF key generation process, which is mainly limited by the camera’s acquisition time of the optical PUFs.
2.3 Experimental results on PUF key generation and NIST validation
Figure 3 shows the typical results of cross-validation for two representative classes c1 and c2 of PUFs. Experimentally, we observe that classes cn composed approximately of ≤10 PUFs are sufficient to train the DNN to perform accurate predictions. Figure 3a reports the learning rate obtained by the DNN when training on the learning data set composed of seven PUFs, with I11−I13 belonging to c1 and I21−I24 belonging to c2. These PUFs are acquired in the most fluctuating scenario in the setup of Figure 1a, in which we fix the input power (P = 200 mW) and acquire images at different times. Representative 100 × 100 pixel images of the PUF belonging to each class are shown in Figure 3b.
Deep neural network training and key association.
(a) Learning error in the training data set composed of seven PUFs belonging to classes c1 and c2, which are obtained with the same pumping power P = 200 mW and at different acquisition times. The PUFs I11−I13 are associated with the output integer y = 1, while I21−I24 are associated with y = 2. (b) CCD images of the PUFs. Panel (c–d) report the same analysis of (a–b) for the prediction data set, composed of six PUFs J11−J13 of class c1 and J21−J23 of c2. The PUFs in the prediction data set are generated in a different experiment and are not included in the training data set. PUF, physical unclonable function; DNN, deep neural network.
The results for Figure 3a illustrate that the DNN learns with great accuracy (learning errors below machine precision 10−15) to associate the correct key number n with each PUF in the data set. Figure 3c and d report the resulting performances of the DNN when predicting the key number associated with the prediction data set, composed of six PUFs J11−J13 of c1 and J21−J23 of class c2. Images Imn and Jmn are obtained in different experiments and have the same input conditions, i.e., pump power at P = 200 mW and same acquisition time. Although none of the Jmn PUFs exist in the learning data set used to train the DNN, the network correctly predicts the right index to each image, with prediction errors below 0.3. These results allow using a simple threshold filter n ± 0.5 to assess correctly the key associated with each PUF, with no error arising from the natural fluctuations present in the experimental measures. The ability of the DNN to learn the feature of each PUF and the required mapping function
The error, or overfitting, between the DNN prediction and the correct key number in Figure 3c can be reduced by either increasing the length of the training data set and the associated DNN size or by adding a larger number of output channels, with each channel associating the corresponding key with a particular class of PUFs. To create random keys, we sample in time with an analog microphone, the sound arising from the electric engine of a desktop fan (Figure 4a). The sample rate of the microphone is much lower than the fan speed, allowing us to collect a time-varying random stream (Figure 4a, solid blue line). The sequence is then converted to a binary signal (Figure 4a, solid orange line) by using a Gabor transform [14], which associates 1 to all inputs above a threshold value, here chosen as the analog noise mean value. The random binary sequence generated is then partitioned into N different keys k1, … , kN. We generate a set of N = 100 digital keys in our experiments, each of 10,000 bits.
Key generation and NIST certification results.
(a) Generation steps of the binary noise sequence, starting from sampling in time, with a sufficiently low acquisition time, the noise emitted from the electric engine of a desktop fan, and then converting it into a binary sequence with a Gabor transform. The binary stream is then partitioned into N = 100 k1, …, kN consecutive keys each of 10,000 bits. (b) NIST proportion results on the SP 800-22 statistical test suite applied to the keys generated in (a), with a minimum threshold (dashed red line) recommended by the NIST. (c) Cross-correlation matrix between the keys kn. NIST, National Institute of Standards and Technology.
Figure 4b reports the results of the NIST SP 800–22 test suite on the generated keys. The test comprises a suite of different statistical tests to assess if the key at the output looks like an unpredictable binary sequence in the input’s absence of knowledge. The tests analyze the proportion of zeros and ones in the sequence and the existence of harmonic peaks (frequency, block frequency, and FFT), the presence of sequences with identical bits (run, longest run), the occurrence of prespecified target strings (nonoverlapping templates), the rank of disjoint submatrices in the stream (rank), and the sufficient complexity of the sequence to be considered random (serial and linear complexity). Detailed information on each test is available in the NIST reference [43]. Each test results in a proportion, which measures each key’s probability to pass the statistical test (Figure 4b, dashed red line). The analysis is performed using the software STS distributed by the NIST [43]. The results of Figure 4b demonstrate that the binary keys pass all the NIST tests well above the minimum threshold, showing that the procedure used to acquire the noise and transform it into a binary sequence generates a truly unpredictable stream of data (Figure 4a). Figure 4c reports the cross-correlation between the keys kn. The keys generated are completely uncorrelated with each other, with average cross-correlation coefficients between the key ki and kj of the order of
Figures 2–4 demonstrate that the technique proposed in this works satisfies requirements i)–iii), with the reliable associations of the same key with experimental PUFs measured after the same challenge with no errors and with each key representing an unpredictable random sequence that is completely uncorrelated to the other.
3 Discussion
We discussed a two-step key generation strategy for PUFs based on deep learning, which can address the shortcomings of unreliability and weak unpredictability of cryptographic keys. The idea explored is to design the PUF independently from the problem of key generation and then use machine learning to train a neural network to find the complex mapping function that can reliably associate the features of PUFs with identical input conditions to a single key. Different binary keys were generated by sampling white noise, representing a physically unpredictable random sequence that passed all validation tests against NIST standards for cryptographic applications. We report experimental results in SAs, exploiting a classification strategy based on integer numbers n, with each number directly identifying a binary key kn. Despite the high sensitivity of the aerogel to different input conditions, our experiments report that a trained neural network predicts the correct key with no errors. The results of this work can be of help in the development of stronger PUFs for different applications, including authentication and secure communications. The research data supporting this publication can be accessed at https://doi.org/10.17630/50b2f96f-ab3a-4b6e-abcd-c5d14c784de9.
Funding source: H2020 QuantERA ERA-NET Co-fund, project Quomplex
Award Identifier / Grant number: 731473
Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
Research funding: C.C. acknowledge funding from Horizon 2020 Framework Programme QuantERA grant QUOMPLEX, by National Research Council (CNR), Grant agreement ID 731473.
Conflict of interest statement: The authors declare no conflicts of interest regarding this article.
References
[1] S. Sakhare and D. Sakhare, “A review—hardware security using puf (physical unclonable function),” in ICCCE 2019, A. Kumar, and S. Mozar, Eds., Singapore, Springer Singapore, 2020, pp. 373–377.10.1007/978-981-13-8715-9_45Search in Google Scholar
[2] D. Adam, “Cryptography on the front line,” Nature, vol. 413, pp. 766–767, 2001, https://doi.org/10.1038/35101752.Search in Google Scholar
[3] S. Chen, “Random number generators go public,” Science, vol. 360, pp. 1383–1384, 2018, https://doi.org/10.1126/science.360.6396.1383.Search in Google Scholar
[4] H. Wang, D. Forte, M. M. Tehranipoor, and Q. Shi, “Probing attacks on integrated circuits: challenges and research opportunities,” IEEE Design Test, vol. 34, pp. 63–71, 2017, https://doi.org/10.1109/mdat.2017.2729398.Search in Google Scholar
[5] C. Tankard, “The security issues of the Internet of things,” Comput. Fraud. Secur., vol. 2015, pp. 11–14, 2015. Available at: http://www.sciencedirect.com/science/article/pii/S1361372315300841, https://doi.org/10.1016/s1361-3723(15)30084-1.Search in Google Scholar
[6] F. S. Ferraz and C. A. G. Ferraz, “Smart city security issues: depicting information security issues in the role of an urban environment,” in 2014 IEEE/ACM 7th International Conference on Utility and Cloud Computing, 2014, pp. 842–847.10.1109/UCC.2014.137Search in Google Scholar
[7] V. Mayer-Schonberger and K. Cukier, Big Data: A Revolution That Will Transform How We Live, Work, and Think, Boston, Houghton Mifflin Harcourt, 2013. Available at: http://www.amazon.com/books/dp/0544002695.Search in Google Scholar
[8] A. AlDairi and L. Tawalbeh, “Cyber security attacks on smart cities and associated mobile technologies,” Procedia Comput. Sci., vol. 109, pp. 1086–1091, 2017. Available at: http://www.sciencedirect.com/science/article/pii/S1877050917310669. 8th International Conference on Ambient Systems, Networks and Technologies, ANT-2017 and the 7th International Conference on Sustainable Energy Information Technology, SEIT 2017, 16–19 May 2017, Madeira, Portugal, https://doi.org/10.1016/j.procs.2017.05.391.Search in Google Scholar
[9] K. Thompson, “Reflections on trusting trust,” Commun. ACM, vol. 27, pp. 761–763, 1984, https://doi.org/10.1145/358198.358210.Search in Google Scholar
[10] S. Chen, R. Wang, X. Wang, and K. Zhang, “Side-channel leaks in web applications: A reality today, a challenge tomorrow,” in Proceedings of the IEEE Symposium on Security and Privacy (Oakland), IEEE Computer Society, 2010. Available at: https://www.microsoft.com/en-us/research/publication/side-channel-leaks-in-web-applications-a-reality-today-a-challenge-tomorrow/.10.1109/SP.2010.20Search in Google Scholar
[11] C. Ashokkumar, R. P. Giri, and B. Menezes, “Highly efficient algorithms for AES key retrieval in cache access attacks,” in 2016 IEEE European Symposium on Security and Privacy, EuroS P, 2016, pp. 261–275.10.1109/EuroSP.2016.29Search in Google Scholar
[12] A. Golder, D. Das, J. Danial, et al., “Practical approaches toward deep-learning-based cross-device power side-channel attack,” IEEE Trans. Very Large Scale Integr. Syst., vol. 27, pp. 2720–2733, 2019, https://doi.org/10.1109/tvlsi.2019.2926324.Search in Google Scholar
[13] L. Kahney, Tim Cook: The Genius Who Took Apple to the Next Level, Penguin Books Limited, 2019. Available at: https://books.google.com.sa/books?id=A5xlDwAAQBAJ.Search in Google Scholar
[14] R. Pappu, B. Recht, J. Taylor, and N. Gershenfeld, “Physical one-way functions,” Science, vol. 297, pp. 2026–2030, 2002, https://doi.org/10.1126/science.1074376.Search in Google Scholar
[15] C. Herder, M.-D. Yu, F. Koushanfar, and S. Devadas, “Physical unclonable functions and applications: A tutorial,” Proc. IEEE, vol. 102, pp. 1126–1141, 2014, https://doi.org/10.1109/jproc.2014.2320516.Search in Google Scholar
[16] B. Škorić, P. Tuyls, and W. Ophey, “Robust key extraction from physical uncloneable functions,” in International Conference on Applied Cryptography and Network Security, Springer, 2005, pp. 407–422.10.1007/11496137_28Search in Google Scholar
[17] S. A. Goorden, M. Horstmann, A. P. Mosk, B. Škorić, and P. W. H. Pinkse, “Quantum-secure authentication of a physical unclonable key,” Optica, vol. 1, pp. 421–424, 2014, https://doi.org/10.1364/optica.1.000421.Search in Google Scholar
[18] G. Zhang and Q. Liu, “A novel image encryption method based on total shuffling scheme,” Optic Commun., vol. 284, pp. 2775–2780, 2011, https://doi.org/10.1016/j.optcom.2011.02.039.Search in Google Scholar
[19] R. Horstmeyer, B. Judkewitz, I. M. Vellekoop, S. Assawaworrarit, and C. Yang, “Physical key-protected one-time pad,” Sci. Rep., vol. 3, no. 6, p. 3543, 2013, https://doi.org/10.1038/srep03543.Search in Google Scholar
[20] M. Leonetti, S. Karbasi, A. Mafi, E. DelRe, and C. Conti, “Secure information transport by transverse localization of light,” Sci. Rep., vol. 6, p. 29918, 2016, https://doi.org/10.1038/srep29918.Search in Google Scholar
[21] B. C. Grubel, B. T. Bosworth, M. Kossey, et al., “Secure communications using nonlinear silicon photonic keys,” Opt. Express, vol. 26, pp. 4710–4722, 2018, https://doi.org/10.1364/oe.26.004710.Search in Google Scholar
[22] A. Di Falco, V. Mazzone, A. Cruz, and A. Fratalocchi, “Perfect secrecy cryptography via mixing of chaotic waves in irreversible time-varying silicon chips,” Nat. Commun., vol. 10, p. 5827, 2019, https://doi.org/10.1038/s41467-019-13740-y.Search in Google Scholar
[23] C. Helfmeier, C. Boit, D. Nedospasov, and J.-P. Seifert, “Cloning physically unclonable functions,” in Hardware-Oriented Security and Trust (HOST), 2013 IEEE International Symposium on 1–6, IEEE, 2013.10.1109/HST.2013.6581556Search in Google Scholar
[24] U. Rührmair, “Optical pufs reloaded,” Eprint.Iacr.Org, 2013, https://doi.org/10.1109/sp.2013.27.Search in Google Scholar
[25] J. Danger, “Physically unclonable functions: principle, advantages and limitations,” in 2019 International Conference on Advanced Technologies for Communications (ATC), 2019, pp. xxxii–xxxii.10.1109/ATC.2019.8924551Search in Google Scholar
[26] A. Wali, A. Dodda, Y. Wu, et al., “Biological physically unclonable function,” Commun. Phys., vol. 2, no. 39, 2019. Available at: https://doi.org/10.1038/s42005-019-0139-3.Search in Google Scholar
[27] I. Goodfellow, Y. Bengio, and A. Courville, Deep Learning, Cambridge, The MIT Press, 2016.Search in Google Scholar
[28] G. Marcucci, D. Pierangeli, P. W. H. Pinkse, M. Malik, and C. Conti, “Programming multi-level quantum gates in disordered computing reservoirs via machine learning,” Opt. Express, vol. 28, pp. 14018–14027, 2020. Available at: http://www.opticsexpress.org/abstract.cfm?URI=oe-28-9-14018.10.1364/OE.389432Search in Google Scholar PubMed
[29] Bassham, L. E., Andrew, R., Juan, S., et al., “Sp 800-22 rev. 1a. a statistical test suite for random and pseudorandom number generators for cryptographic applications,” Tech. Rep., 2010, https://doi.org/10.6028/nist.sp.800-22r1a.Search in Google Scholar
[30] S. M. Jones, “A method for producing gradient density aerogel,” J. Sol. Gel Sci. Technol., vol. 44, pp. 255–258, 2007, https://doi.org/10.1007/s10971-007-1618-7.Search in Google Scholar
[31] S. Gentilini, F. Ghajeri, N. Ghofraniha, A. Di Falco, and C. Conti, “Optical shock waves in silica aerogel,” Opt. Express, vol. 22, pp. 1667–1672, 2014, https://doi.org/10.1364/oe.22.001667.Search in Google Scholar
[32] M. C. Braidotti, S. Gentilini, A. Fleming, M. C. Samuels, A. Di Falco, and C. Conti, “Optothermal nonlinearity of silica aerogel,” Appl. Phys. Lett., vol. 109, p. 041104, 2016, https://doi.org/10.1063/1.4959240.Search in Google Scholar
[33] A. Fleming, C. Conti, and A. Di Falco, “Perturbation of transmission matrices in nonlinear random media,” Ann. Phys., vol. 531, p. 1900091, 2019, https://doi.org/10.1002/andp.201900091.Search in Google Scholar
[34] A. Fleming, C. Conti, T. Vettenburg, and A. Di Falco, “Nonlinear optical memory effect,” Opt. Lett., vol. 44, pp. 4841–4844, 2019, https://doi.org/10.1364/ol.44.004841.Search in Google Scholar
[35] G. Nicolaon and S. Teichner, “The preparation of silica aerogels from methylorthosilicate in an alcoholic medium and their properties,” 1975.Search in Google Scholar
[36] J. Livage, M. Henry, and C. Sanchez, “Sol-gel chemistry of transition metal oxides,” Prog. Solid State Chem., vol. 18, pp. 259–341, 1988, https://doi.org/10.1016/0079-6786(88)90005-2.Search in Google Scholar
[37] B. Lin, S. Cui, X. Liu, X. Shen, Y. Liu, and G. Han, “Preparation and characterization of hmds modified hydrophobic silica aerogel,” Curr. Nanosci., 2011.10.2174/1573413711107061042Search in Google Scholar
[38] P. H. Tewari, A. J. Hunt, and K. D. Lofftus, “Ambient-temperature supercritical drying of transparent silica aerogels,” Mater. Lett., vol. 3, pp. 363–367, 1985, https://doi.org/10.1016/0167-577x(85)90077-1.Search in Google Scholar
[39] M. Leshno, V. Y. Lin, A. Pinkus, and S. Schocken, “Multilayer feedforward networks with a nonpolynomial activation function can approximate any function,” Neural Netw., vol. 6, pp. 861–867, 1993. Available at: http://www.sciencedirect.com/science/article/pii/S0893608005801315.10.1016/S0893-6080(05)80131-5Search in Google Scholar
[40] E. Strobach, B. Bhatia, S. Yang, L. Zhao, and E. N. Wang, “High temperature stability of transparent silica aerogels for solar thermal applications,” APL Mater., vol. 7, p. 081104, 2019, https://doi.org/10.1063/1.5109433.Search in Google Scholar
[41] R. H. R. Hahnloser, R. Sarpeshkar, M. A. Mahowald, R. J. Douglas, and H. S. Seung, “Digital selection and analogue amplification coexist in a cortex-inspired silicon circuit,” Nature, vol. 405, pp. 947–951, 2000, https://doi.org/10.1038/35016072.Search in Google Scholar
[42] J. Schwartz, “U.s. Selects a new encryption technique,” 2000. Available at: https://www.nytimes.com/2000/10/03/business/technology-us-selects-a-new-encryption-technique.html.Search in Google Scholar
[43] Computer Security Division, I. T. L, “Nist sp 800-22: documentation and software – random bit generation: Csrc,”. Available at: https://csrc.nist.gov/projects/random-bit-generation/documentation-and-software.Search in Google Scholar
© 2020 Andrea Fratalocchi et al., published by De Gruyter, Berlin/Boston
This work is licensed under the Creative Commons Attribution 4.0 International License.
