Towards the AlexNet Moment for Homomorphic Encryption: HCNN, theFirst Homomorphic CNN on Encrypted Data with GPUs

Deep Learning as a Service (DLaaS) stands as a promising solution for cloud-based inference applications. In this setting, the cloud has a pre-learned model whereas the user has samples on which she wants to run the model. The biggest concern with DLaaS is user privacy if the input samples are sensitive data. We provide here an efficient privacy-preserving system by employing high-end technologies such as Fully Homomorphic Encryption (FHE), Convolutional Neural Networks (CNNs) and Graphics Processing Units (GPUs). FHE, with its widely-known feature of computing on encrypted data, empowers a wide range of privacy-concerned applications. This comes at high cost as it requires enormous computing power. In this paper, we show how to accelerate the performance of running CNNs on encrypted data with GPUs. We evaluated two CNNs to classify homomorphically the MNIST and CIFAR-10 datasets. Our solution achieved a sufficient security level (> 80 bit) and reasonable classification accuracy (99%) and (77.55%) for MNIST and CIFAR-10, respectively. In terms of latency, we could classify an image in 5.16 seconds and 304.43 seconds for MNIST and CIFAR-10, respectively. Our system can also classify a batch of images (> 8,000) without extra overhead

Conditionals in Homomorphic Encryption and Machine Learning Applications

Homomorphic encryption aims at allowing computations on encrypted data without decryption other than that of the final result. This could provide an elegant solution to the issue of privacy preservation in data-based applications, such as those using machine learning, but several open issues hamper this plan. In this work we assess the possibility for homomorphic encryption to fully implement its program without relying on other techniques, such as multiparty computation (SMPC), which may be impossible in many use cases (for instance due to the high level of communication required). We proceed in two steps: i) on the basis of the structured program theorem (Bohm-Jacopini theorem) we identify the relevant minimal set of operations homomorphic encryption must be able to perform to implement any algorithm; and ii) we analyse the possibility to solve -- and propose an implementation for -- the most fundamentally relevant issue as it emerges from our analysis, that is, the implementation of conditionals (requiring comparison and selection/jump operations). We show how this issue clashes with the fundamental requirements of homomorphic encryption and could represent a drawback for its use as a complete solution for privacy preservation in data-based applications, in particular machine learning ones. Our approach for comparisons is novel and entirely embedded in homomorphic encryption, while previous studies relied on other techniques, such as SMPC, demanding high level of communication among parties, and decryption of intermediate results from data-owners. Our protocol is also provably safe (sharing the same safety as the homomorphic encryption schemes), differently from other techniques such as Order-Preserving/Revealing-Encryption (OPE/ORE).Comment: 14 pages, 1 figure, corrected typos, added introductory pedagogical section on polynomial approximatio

High-Precision Arithmetic in Homomorphic Encryption

In most RLWE-based homomorphic encryption schemes the native plaintext elements are polynomials in a ring $\mathbb{Z}_t[x]/(x^n+1)$, where $n$ is a power of $2$, and $t$ an integer modulus. For performing integer or rational number arithmetic one typically uses an encoding scheme, which converts the inputs to polynomials, and allows the result of the homomorphic computation to be decoded to recover the result as an integer or rational number respectively. The problem is that the modulus $t$ often needs to be extremely large to prevent the plaintext polynomial coefficients from being reduced modulo~$t$ during the computation, which is a requirement for the decoding operation to work correctly. This results in larger noise growth, and prevents the evaluation of deep circuits, unless the encryption parameters are significantly increased. We combine a trick of Hoffstein and Silverman, where the modulus $t$ is replaced by a polynomial $x-b$, with the Fan-Vercauteren homomorphic encryption scheme. This yields a new scheme with a very convenient plaintext space $\mathbb{Z}/(b^n+1)\mathbb{Z}$. We then show how rational numbers can be encoded as elements of this plaintext space, enabling homomorphic evaluation of deep circuits with high-precision rational number inputs. We perform a fair and detailed comparison to the Fan-Vercauteren scheme with the Non-Adjacent Form encoder, and find that the new scheme significantly outperforms this approach. For example, when the new scheme allows us to evaluate circuits of depth $9$ with $32$-bit integer inputs, in the same parameter setting the Fan-Vercauteren scheme only allows us to go up to depth $2$. We conclude by discussing how known applications can benefit from the new scheme

Efficiently processing complex-valued data in homomorphic encryption

We introduce a new homomorphic encryption scheme that is natively capable of computing with complex numbers. This is done by generalizing recent work of Chen, Laine, Player and Xia, who modified the Fan–Vercauteren scheme by replacing the integral plaintext modulus t by a linear polynomial X − b. Our generalization studies plaintext moduli of the form Xm + b. Our construction significantly reduces the noise growth in comparison to the original FV scheme, so much deeper arithmetic circuits can be homomorphically executed

동형암호 재부팅 기법에 관한 연구

학위논문 (박사)-- 서울대학교 대학원 : 자연과학대학 수리과학부, 2019. 2. 천정희.2009년 Gentry에 의해서 완전동형암호가 처음 설계된 이후로 최적화와 고속화를 위해서 다양한 기법들과 스킴들이 설계되어 왔다. 하지만 동형암호의 연산횟수를 무제한으로 늘리기 위해서 필수적인 재부팅 기법의 효율성 문제로 실제 응용에 적용하기에는 부적합하다는 평가를 많이 받아왔다. 본 논문에서는 재부팅 기법의 고속화를 위한 다양한 기법을 제시하고 이를 실제로 응용분야에 적용하였다. 본 논문에서는 대표적인 동형암호 스킴들에 대한 재부팅 기법에 대한 연구를 수행하였는데, 첫 번째로는 Microsoft Research와 IMB에서 만든 동형암호 라이브러리인 SEAL과 HElib에 적용가능한 재부팅 기법에 대한 연구를 수행하였다. 해당 재부팅 기법에서 핵심적이 과정은 암호화된 상태에서 복호화 함수를 계산하는 부분이다. 암호된 상태에서 최하위 비트를 추출하는 새로운 방법을 제시하여 재부팅 과정에서 소모되는 계산량과 표현되는 다항식의 차수를 줄이는데에 성공하였다. 두 번째로는, 비교적 최근에 개발된 근사계산 동형암호인 HEAAN 스킴의 재부팅 기법을 개선하는 연구를 수행하였다. 2018년에 삼각함수를 이용한 근사법을 통해서 처음 해당 스킴에 대한 재부팅 기법이 제시되었는데, 많은 데이터를 담고있는 암호문에 대해서는 전처리, 후처리 과정이 계산량의 대부분을 차지하는 문제가 있었다. 해당 과정들을 여러 단계로 재귀적인 함수들로 표현하여 계산량이 데이터 사이즈에 대해서 로그적으로 줄이는 것에 성공하였다. 추가로, 다른 스킴들에 비해서 많이 사용되지는 않지만, 정수기반 동형암호들에 대해서도 재부팅 기법을 개선하는 연구를 수행하였고 그 결과 계산량을 로그적으로 줄이는 것에 성공하였다. 마지막으로, 재부팅 기법의 활용성과 사용 가능성을 보이기 위해 실제 데이터 보안을 필요로 하는 기계학습 분야에 적용해보았다. 실제로 400,000건의 금융 데이터를 이용한 회귀분석을 암호화된 데이터를 이용해서 수행하였다. 그 결과 약 16시간 안에 80\% 이상의 정확도와 0.8 정도의 AUROC 값을 가지는 유의미한 분석 모델을 얻을 수 있었다.After Gentry's blueprint on homomorphic encryption (HE) scheme, various efficient schemes have been suggested. For unlimited number of operations between encrypted data, the bootstrapping process is necessary. There are only few works on bootstrapping procedure because of the complexity and inefficiency of bootstrapping. In this paper, we propose various method and techniques for improved bootstrapping algorithm, and we apply it to logistic regression on large scale encrypted data. The bootstrapping process depends on based homomorphic encryption scheme. For various schemes such as BGV, BFV, HEAAN, and integer-based scheme, we improve bootstrapping algorithm. First, we improved bootstrapping for BGV (HElib) and FV (SEAL) schemes which is implemented by Microsoft Research and IMB respectively. The key process for bootstrapping in those two scheme is extracting lower digits of plaintext in encrypted state. We suggest new polynomial that removes lowest digit of input, and we apply it to bootstrapping with previous method. As a result, both the complexity and the consumed depth are reduced. Second, bootstrapping for multiple data needs homomorphic linear transformation. The complexity of this part is O(n) for number of slot n, and this part becomes a bottleneck when we use large n. We use the structure of linear transformation which is used in bootstrapping, and we decompose the matrix which is corresponding to the transformation. By applying recursive strategy, we reduce the complexity to O(log n). Furthermore, we suggest new bootstrapping method for integer-based HE schemes which are based on approximate greatest common divisor problem. By using digit extraction instead of previous bit-wise approach, the complexity of bootstrapping algorithm reduced from O(poly(lambda)) to O(log^2(lambda)). Our implementation for this process shows 6 seconds which was about 3 minutes. To show that bootstrapping can be used for practical application, we implement logistic regression on encrypted data with large scale. Our target data has 400,000 samples, and each sample has 200 features. Because of the size of the data, direct application of homomorphic encryption scheme is almost impossible. Therefore, we decide the method for encryption to maximize the effect of multi-threading and SIMD operations in HE scheme. As a result, our homomorphic logistic regression takes about 16 hours for the target data. The output model has 0.8 AUROC with about 80% accuracy. Another experiment on MNIST dataset shows correctness of our implementation and method.Abstract 1 Introduction 1.1 Homomorphic Encryption 1.2 Machine Learning on Encrypted Data 1.3 List of Papers 2 Background 2.1 Notation 2.2 Homomorphic Encryption 2.3 Ring Learning with Errors 2.4 Approximate GCD 3 Lower Digit Removal and Improved Bootstrapping 3.1 Basis of BGV and BFV scheme 3.2 Improved Digit Extraction Algorithm 3.3 Bootstrapping for BGV and BFV Scheme 3.3.1 Our modications 3.4 Slim Bootstrapping Algorithm 3.5 Implementation Result 4 Faster Homomorphic DFT and Improved Bootstrapping 4.1 Basis of HEAAN scheme 4.2 Homomorphic DFT 4.2.1 Previous Approach 4.2.2 Our method 4.2.3 Hybrid method 4.2.4 Implementation Result 4.3 Improved Bootstrapping for HEAAN 4.3.1 Linear Transformation in Bootstrapping 4.3.2 Improved CoeToSlot and SlotToCoe 4.3.3 Implementation Result 5 Faster Bootstrapping for FHE over the integers 5.1 Basis of FHE over the integers 5.2 Decryption Function via Digit Extraction 5.2.1 Squashed Decryption Function 5.2.2 Digit extraction Technique 5.2.3 Homomorphic Digit Extraction in FHE over the integers 5.3 Bootstrapping for FHE over the integers 5.3.1 CLT scheme with M Z_t 5.3.2 Homomorphic Operations with M Z_t^a 5.3.3 Homomorphic Digit Extraction for CLT scheme 5.3.4 Our Method on the CLT scheme 5.3.5 Analysis of Proposed Bootstrapping Method 5.4 Implementation Result 6 Logistic Regression on Large Encrypted Data 6.1 Basis of Logistic Regression 6.2 Logistic Regression on Encrypted Data 6.2.1 HE-friendly Logistic Regression Algorithm 6.2.2 HE-Optimized Logistic Regression Algorithm 6.2.3 Further Optimization 6.3 Evaluation 6.3.1 Logistic Regression on Encrypted Financial Dataset 6.3.2 Logistic Regression on Encrypted MNIST Dataset 6.3.3 Discussion 7 Conclusions Abstract (in Korean)Docto

Harnessing the Power of Distributed Computing: Advancements in Scientific Applications, Homomorphic Encryption, and Federated Learning Security

Data explosion poses lot of challenges to the state-of-the art systems, applications, and methodologies. It has been reported that 181 zettabytes of data are expected to be generated in 2025 which is over 150\% increase compared to the data that is expected to be generated in 2023. However, while system manufacturers are consistently developing devices with larger storage spaces and providing alternative storage capacities in the cloud at affordable rates, another key challenge experienced is how to effectively process the fraction of large scale of stored data in time-critical conventional systems. One transformative paradigm revolutionizing the processing and management of these large data is distributed computing whose application requires deep understanding. This dissertation focuses on exploring the potential impact of applying efficient distributed computing concepts to long existing challenges or issues in (i) a widely data-intensive scientific application (ii) applying homomorphic encryption to data intensive workloads found in outsourced databases and (iii) security of tokenized incentive mechanism for Federated learning (FL) systems.The first part of the dissertation tackles the Microelectrode arrays (MEAs) parameterization problem from an orthogonal viewpoint enlightened by algebraic topology, which allows us to algebraically parametrize MEAs whose structure and intrinsic parallelism are hard to identify otherwise. We implement a new paradigm, namely Parma, to demonstrate the effectiveness of the proposed approach and report how it outperforms the state-of-the-practice in time, scalability, and memory usage.The second part discusses our work on introducing the concept of parallel caching of secure aggregation to mitigate the performance overhead incurred by the HE module in outsourced databases. The key idea of this optimization approach is caching selected radix-ciphertexts in parallel without violating existing security guarantees of the primitive/base HE scheme. A new radix HE algorithm was designed and applied to both batch and incremental HE schemes, and experiments carried out on six workloads show that the proposed caching boost state-of-the-art HE schemes by high orders of magnitudes.In the third part, I will discuss our work on leveraging the security benefit of blockchains to enhance or protect the fairness and reliability of tokenized incentive mechanism for FL systems. We designed a blockchain-based auditing protocol to mitigate Gaussian attacks and carried out experiments with multiple FL aggregation algorithms, popular data sets and a variety of scales to validate its effectiveness

Cloud-based homomorphic encryption for privacy-preserving machine learning in clinical decision support

While privacy and security concerns dominate public cloud services, Homomorphic Encryption (HE) is seen as an emerging solution that ensures secure processing of sensitive data via untrusted networks in the public cloud or by third-party cloud vendors. It relies on the fact that some encryption algorithms display the property of homomorphism, which allows them to manipulate data meaningfully while still in encrypted form; although there are major stumbling blocks to overcome before the technology is considered mature for production cloud environments. Such a framework would find particular relevance in Clinical Decision Support (CDS) applications deployed in the public cloud. CDS applications have an important computational and analytical role over confidential healthcare information with the aim of supporting decision-making in clinical practice. Machine Learning (ML) is employed in CDS applications that typically learn and can personalise actions based on individual behaviour. A relatively simple-to-implement, common and consistent framework is sought that can overcome most limitations of Fully Homomorphic Encryption (FHE) in order to offer an expanded and flexible set of HE capabilities. In the absence of a significant breakthrough in FHE efficiency and practical use, it would appear that a solution relying on client interactions is the best known entity for meeting the requirements of private CDS-based computation, so long as security is not significantly compromised. A hybrid solution is introduced, that intersperses limited two-party interactions amongst the main homomorphic computations, allowing exchange of both numerical and logical cryptographic contexts in addition to resolving other major FHE limitations. Interactions involve the use of client-based ciphertext decryptions blinded by data obfuscation techniques, to maintain privacy. This thesis explores the middle ground whereby HE schemes can provide improved and efficient arbitrary computational functionality over a significantly reduced two-party network interaction model involving data obfuscation techniques. This compromise allows for the powerful capabilities of HE to be leveraged, providing a more uniform, flexible and general approach to privacy-preserving system integration, which is suitable for cloud deployment. The proposed platform is uniquely designed to make HE more practical for mainstream clinical application use, equipped with a rich set of capabilities and potentially very complex depth of HE operations. Such a solution would be suitable for the long-term privacy preserving-processing requirements of a cloud-based CDS system, which would typically require complex combinatorial logic, workflow and ML capabilities

Vers une arithmétique efficace pour le chiffrement homomorphe basé sur le Ring-LWE

Fully homomorphic encryption is a kind of encryption offering the ability to manipulate encrypted data directly through their ciphertexts. In this way it is possible to process sensitive data without having to decrypt them beforehand, ensuring therefore the datas' confidentiality. At the numeric and cloud computing era this kind of encryption has the potential to considerably enhance privacy protection. However, because of its recent discovery by Gentry in 2009, we do not have enough hindsight about it yet. Therefore several uncertainties remain, in particular concerning its security and efficiency in practice, and should be clarified before an eventual widespread use. This thesis deals with this issue and focus on performance enhancement of this kind of encryption in practice. In this perspective we have been interested in the optimization of the arithmetic used by these schemes, either the arithmetic underlying the Ring Learning With Errors problem on which the security of these schemes is based on, or the arithmetic specific to the computations required by the procedures of some of these schemes. We have also considered the optimization of the computations required by some specific applications of homomorphic encryption, and in particular for the classification of private data, and we propose methods and innovative technics in order to perform these computations efficiently. We illustrate the efficiency of our different methods through different software implementations and comparisons to the related art.Le chiffrement totalement homomorphe est un type de chiffrement qui permet de manipuler directement des données chiffrées. De cette manière, il est possible de traiter des données sensibles sans avoir à les déchiffrer au préalable, permettant ainsi de préserver la confidentialité des données traitées. À l'époque du numérique à outrance et du "cloud computing" ce genre de chiffrement a le potentiel pour impacter considérablement la protection de la vie privée. Cependant, du fait de sa découverte récente par Gentry en 2009, nous manquons encore de recul à son propos. C'est pourquoi de nombreuses incertitudes demeurent, notamment concernant sa sécurité et son efficacité en pratique, et devront être éclaircies avant une éventuelle utilisation à large échelle.Cette thèse s'inscrit dans cette problématique et se concentre sur l'amélioration des performances de ce genre de chiffrement en pratique. Pour cela nous nous sommes intéressés à l'optimisation de l'arithmétique utilisée par ces schémas, qu'elle soit sous-jacente au problème du "Ring-Learning With Errors" sur lequel la sécurité des schémas considérés est basée, ou bien spécifique aux procédures de calculs requises par certains de ces schémas. Nous considérons également l'optimisation des calculs nécessaires à certaines applications possibles du chiffrement homomorphe, et en particulier la classification de données privées, de sorte à proposer des techniques de calculs innovantes ainsi que des méthodes pour effectuer ces calculs de manière efficace. L'efficacité de nos différentes méthodes est illustrée à travers des implémentations logicielles et des comparaisons aux techniques de l'état de l'art

Large FHE Gates from tensored homomorphic accumulator

The main bottleneck of all known Fully Homomorphic Encryption schemes lies in the bootstrapping procedure invented by Gentry (STOC’09). The cost of this procedure can be mitigated either using Homomorphic SIMD techniques, or by performing larger computation per bootstrapping procedure.In this work, we propose new techniques allowing to perform more operations per bootstrapping in FHEW-type schemes (EUROCRYPT’13). While maintaining the quasi-quadratic Õ(n2) complexity of the whole cycle, our new scheme allows to evaluate gates with Ω(log n) input bits, which constitutes a quasi-linear speed-up. Our scheme is also very well adapted to large threshold gates, natively admitting up to Ω(n) inputs. This could be helpful for homomorphic evaluation of neural networks.Our theoretical contribution is backed by a preliminary prototype implementation, which can perform 6-to-6 bit gates in less than 10s on a single core, as well as threshold gates over 63 input bits even faster.<p

Towards Improved Homomorphic Encryption for Privacy-Preserving Deep Learning

Mención Internacional en el título de doctorDeep Learning (DL) has supposed a remarkable transformation for many fields, heralded by some as a new technological revolution. The advent of large scale models has increased the demands for data and computing platforms, for which cloud computing has become the go-to solution. However, the permeability of DL and cloud computing are reduced in privacy-enforcing areas that deal with sensitive data. These areas imperatively call for privacy-enhancing technologies that enable responsible, ethical, and privacy-compliant use of data in potentially hostile environments. To this end, the cryptography community has addressed these concerns with what is known as Privacy-Preserving Computation Techniques (PPCTs), a set of tools that enable privacy-enhancing protocols where cleartext access to information is no longer tenable. Of these techniques, Homomorphic Encryption (HE) stands out for its ability to perform operations over encrypted data without compromising data confidentiality or privacy. However, despite its promise, HE is still a relatively nascent solution with efficiency and usability limitations. Improving the efficiency of HE has been a longstanding challenge in the field of cryptography, and with improvements, the complexity of the techniques has increased, especially for non-experts. In this thesis, we address the problem of the complexity of HE when applied to DL. We begin by systematizing existing knowledge in the field through an in-depth analysis of state-of-the-art for privacy-preserving deep learning, identifying key trends, research gaps, and issues associated with current approaches. One such identified gap lies in the necessity for using vectorized algorithms with Packed Homomorphic Encryption (PaHE), a state-of-the-art technique to reduce the overhead of HE in complex areas. This thesis comprehensively analyzes existing algorithms and proposes new ones for using DL with PaHE, presenting a formal analysis and usage guidelines for their implementation. Parameter selection of HE schemes is another recurring challenge in the literature, given that it plays a critical role in determining not only the security of the instantiation but also the precision, performance, and degree of security of the scheme. To address this challenge, this thesis proposes a novel system combining fuzzy logic with linear programming tasks to produce secure parametrizations based on high-level user input arguments without requiring low-level knowledge of the underlying primitives. Finally, this thesis describes HEFactory, a symbolic execution compiler designed to streamline the process of producing HE code and integrating it with Python. HEFactory implements the previous proposals presented in this thesis in an easy-to-use tool. It provides a unique architecture that layers the challenges associated with HE and produces simplified operations interpretable by low-level HE libraries. HEFactory significantly reduces the overall complexity to code DL applications using HE, resulting in an 80% length reduction from expert-written code while maintaining equivalent accuracy and efficiency.El aprendizaje profundo ha supuesto una notable transformación para muchos campos que algunos han calificado como una nueva revolución tecnológica. La aparición de modelos masivos ha aumentado la demanda de datos y plataformas informáticas, para lo cual, la computación en la nube se ha convertido en la solución a la que recurrir. Sin embargo, la permeabilidad del aprendizaje profundo y la computación en la nube se reduce en los ámbitos de la privacidad que manejan con datos sensibles. Estas áreas exigen imperativamente el uso de tecnologías de mejora de la privacidad que permitan un uso responsable, ético y respetuoso con la privacidad de los datos en entornos potencialmente hostiles. Con este fin, la comunidad criptográfica ha abordado estas preocupaciones con las denominadas técnicas de la preservación de la privacidad en el cómputo, un conjunto de herramientas que permiten protocolos de mejora de la privacidad donde el acceso a la información en texto claro ya no es sostenible. Entre estas técnicas, el cifrado homomórfico destaca por su capacidad para realizar operaciones sobre datos cifrados sin comprometer la confidencialidad o privacidad de la información. Sin embargo, a pesar de lo prometedor de esta técnica, sigue siendo una solución relativamente incipiente con limitaciones de eficiencia y usabilidad. La mejora de la eficiencia del cifrado homomórfico en la criptografía ha sido todo un reto, y, con las mejoras, la complejidad de las técnicas ha aumentado, especialmente para los usuarios no expertos. En esta tesis, abordamos el problema de la complejidad del cifrado homomórfico cuando se aplica al aprendizaje profundo. Comenzamos sistematizando el conocimiento existente en el campo a través de un análisis exhaustivo del estado del arte para el aprendizaje profundo que preserva la privacidad, identificando las tendencias clave, las lagunas de investigación y los problemas asociados con los enfoques actuales. Una de las lagunas identificadas radica en el uso de algoritmos vectorizados con cifrado homomórfico empaquetado, que es una técnica del estado del arte que reduce el coste del cifrado homomórfico en áreas complejas. Esta tesis analiza exhaustivamente los algoritmos existentes y propone nuevos algoritmos para el uso de aprendizaje profundo utilizando cifrado homomórfico empaquetado, presentando un análisis formal y unas pautas de uso para su implementación. La selección de parámetros de los esquemas del cifrado homomórfico es otro reto recurrente en la literatura, dado que juega un papel crítico a la hora de determinar no sólo la seguridad de la instanciación, sino también la precisión, el rendimiento y el grado de seguridad del esquema. Para abordar este reto, esta tesis propone un sistema innovador que combina la lógica difusa con tareas de programación lineal para producir parametrizaciones seguras basadas en argumentos de entrada de alto nivel sin requerir conocimientos de bajo nivel de las primitivas subyacentes. Por último, esta tesis propone HEFactory, un compilador de ejecución simbólica diseñado para agilizar el proceso de producción de código de cifrado homomórfico e integrarlo con Python. HEFactory es la culminación de las propuestas presentadas en esta tesis, proporcionando una arquitectura única que estratifica los retos asociados con el cifrado homomórfico, produciendo operaciones simplificadas que pueden ser interpretadas por bibliotecas de bajo nivel. Este enfoque permite a HEFactory reducir significativamente la longitud total del código, lo que supone una reducción del 80% en la complejidad de programación de aplicaciones de aprendizaje profundo que usan cifrado homomórfico en comparación con el código escrito por expertos, manteniendo una precisión equivalente.Programa de Doctorado en Ciencia y Tecnología Informática por la Universidad Carlos III de MadridPresidenta: María Isabel González Vasco.- Secretario: David Arroyo Guardeño.- Vocal: Antonis Michala