Faster Bootstrapping with Polynomial Error

\emph{Bootstrapping} is a technique, originally due to Gentry (STOC 2009), for ``refreshing\u27\u27 ciphertexts of a somewhat homomorphic encryption scheme so that they can support further homomorphic operations. To date, bootstrapping remains the only known way of obtaining fully homomorphic encryption for arbitrary unbounded computations. Over the past few years, several works have dramatically improved the efficiency of bootstrapping and the hardness assumptions needed to implement it. Recently, Brakerski and Vaikuntanathan~(ITCS~2014) reached the major milestone of a bootstrapping algorithm based on Learning With Errors for \emph{polynomial} approximation factors. Their method uses the Gentry-Sahai-Waters~(GSW) cryptosystem~(CRYPTO~2013) in conjunction with Barrington\u27s ``circuit sequentialization\u27\u27 theorem~(STOC~1986). This approach, however, results in \emph{very large} polynomial runtimes and approximation factors. (The approximation factors can be improved, but at even greater costs in runtime and space.) In this work we give a new bootstrapping algorithm whose runtime and associated approximation factor are both \emph{small} polynomials. Unlike most previous methods, ours implements an elementary and efficient \emph{arithmetic} procedure, thereby avoiding the inefficiencies inherent to the use of boolean circuits and Barrington\u27s Theorem. For $2^{\lambda}$ security under conventional lattice assumptions, our method requires only a \emph{quasi-linear} \Otil(\lambda) number of homomorphic operations on GSW ciphertexts, which is optimal (up to polylogarithmic factors) for schemes that encrypt just one bit per ciphertext. As a contribution of independent interest, we also give a technically simpler variant of the GSW system and a tighter error analysis for its homomorphic operations

A Survey on Homomorphic Encryption Schemes: Theory and Implementation

Legacy encryption systems depend on sharing a key (public or private) among the peers involved in exchanging an encrypted message. However, this approach poses privacy concerns. Especially with popular cloud services, the control over the privacy of the sensitive data is lost. Even when the keys are not shared, the encrypted material is shared with a third party that does not necessarily need to access the content. Moreover, untrusted servers, providers, and cloud operators can keep identifying elements of users long after users end the relationship with the services. Indeed, Homomorphic Encryption (HE), a special kind of encryption scheme, can address these concerns as it allows any third party to operate on the encrypted data without decrypting it in advance. Although this extremely useful feature of the HE scheme has been known for over 30 years, the first plausible and achievable Fully Homomorphic Encryption (FHE) scheme, which allows any computable function to perform on the encrypted data, was introduced by Craig Gentry in 2009. Even though this was a major achievement, different implementations so far demonstrated that FHE still needs to be improved significantly to be practical on every platform. First, we present the basics of HE and the details of the well-known Partially Homomorphic Encryption (PHE) and Somewhat Homomorphic Encryption (SWHE), which are important pillars of achieving FHE. Then, the main FHE families, which have become the base for the other follow-up FHE schemes are presented. Furthermore, the implementations and recent improvements in Gentry-type FHE schemes are also surveyed. Finally, further research directions are discussed. This survey is intended to give a clear knowledge and foundation to researchers and practitioners interested in knowing, applying, as well as extending the state of the art HE, PHE, SWHE, and FHE systems.Comment: - Updated. (October 6, 2017) - This paper is an early draft of the survey that is being submitted to ACM CSUR and has been uploaded to arXiv for feedback from stakeholder

Nonparametric Inference via Bootstrapping the Debiased Estimator

In this paper, we propose to construct confidence bands by bootstrapping the debiased kernel density estimator (for density estimation) and the debiased local polynomial regression estimator (for regression analysis). The idea of using a debiased estimator was recently employed by Calonico et al. (2018b) to construct a confidence interval of the density function (and regression function) at a given point by explicitly estimating stochastic variations. We extend their ideas of using the debiased estimator and further propose a bootstrap approach for constructing simultaneous confidence bands. This modified method has an advantage that we can easily choose the smoothing bandwidth from conventional bandwidth selectors and the confidence band will be asymptotically valid. We prove the validity of the bootstrap confidence band and generalize it to density level sets and inverse regression problems. Simulation studies confirm the validity of the proposed confidence bands/sets. We apply our approach to an Astronomy dataset to show its applicabilityComment: Accepted to the Electronic Journal of Statistics. 64 pages, 6 tables, 11 figure

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

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

Learning Generalized Reactive Policies using Deep Neural Networks

We present a new approach to learning for planning, where knowledge acquired while solving a given set of planning problems is used to plan faster in related, but new problem instances. We show that a deep neural network can be used to learn and represent a \emph{generalized reactive policy} (GRP) that maps a problem instance and a state to an action, and that the learned GRPs efficiently solve large classes of challenging problem instances. In contrast to prior efforts in this direction, our approach significantly reduces the dependence of learning on handcrafted domain knowledge or feature selection. Instead, the GRP is trained from scratch using a set of successful execution traces. We show that our approach can also be used to automatically learn a heuristic function that can be used in directed search algorithms. We evaluate our approach using an extensive suite of experiments on two challenging planning problem domains and show that our approach facilitates learning complex decision making policies and powerful heuristic functions with minimal human input. Videos of our results are available at goo.gl/Hpy4e3

Privately Connecting Mobility to Infectious Diseases via Applied Cryptography

Human mobility is undisputedly one of the critical factors in infectious disease dynamics. Until a few years ago, researchers had to rely on static data to model human mobility, which was then combined with a transmission model of a particular disease resulting in an epidemiological model. Recent works have consistently been showing that substituting the static mobility data with mobile phone data leads to significantly more accurate models. While prior studies have exclusively relied on a mobile network operator's subscribers' aggregated data, it may be preferable to contemplate aggregated mobility data of infected individuals only. Clearly, naively linking mobile phone data with infected individuals would massively intrude privacy. This research aims to develop a solution that reports the aggregated mobile phone location data of infected individuals while still maintaining compliance with privacy expectations. To achieve privacy, we use homomorphic encryption, zero-knowledge proof techniques, and differential privacy. Our protocol's open-source implementation can process eight million subscribers in one and a half hours. Additionally, we provide a legal analysis of our solution with regards to the EU General Data Protection Regulation.Comment: Added differentlial privacy experiments and new benchmark