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

Irreducibility of Automorphic Galois Representations of Low Dimensions

Let $\pi$ be a polarizable, regular algebraic, cuspidal automorphic representation of \Text{GL}_n(\mathbb{A}_F), where $F$ is a CM field. We show that for $n\leq 6$, there is a Dirichlet density 1 set $\mathfrak{L}$ of rational primes, such that for all $l\in\mathfrak{L}$, the $l$-adic Galois representations associated to $\pi$ are irreducible. We also show that for any integer $n\geq 1$, in order to show the existence of the aforementioned set $\mathfrak{L}$, it suffices to show that for all but finitely many finite primes $\lambda$ in a number field determined by $\pi$, all the irreducible constituents of the restriction of the corresponding Galois representation \rep to the derived subgroup of the identity component of the Zariski closure of the image, are conjugate self-dual

Patching over Fields and its Applications

Patching over fields is a method of studying algebraic structures over a field by studying corresponding objects over extension fields. The method was developed\ud by David Harbater and Julia Hartmann based on the older form of patching where the base objects were rings rather than fields. We will define patching of vector\ud spaces over a given field and discuss how to get patching of algebraic structures other than vector spaces. In later sections of the thesis, we will present applications of patching over fields to inverse Galois theory, the study of the u-invariant of a field and the period-index problem of central simple algebras in the situation where the\ud field of interest is an one-variable function field over a complete discretely valued field

FRICTION PROPERTIES AND DISTRIBUTION RULE OF LUBRICANT FILM OF FULL CERAMIC BALL BEARING UNDER DIFFERENT SERVICE CONDITION

Full ceramic ball bearings are widely used in extreme and complex conditions, such as ultra-high/low temperature, ultra-high speed, corrosion and insulation, because of their material specificity. In order to reveal the friction and lubrication properties, improve the service performance life of all ceramic ball bearings, a mathematical model of the oil lubrication for full ceramic ball bearings has been established in this paper. The distribution rule of the lubricant film in the contact area under different speeds and loads were analysed. The main factors influencing the peak mutation of the lubricant film pressure are clearly defined. The results are compared and analysed by using a ball-disc rolling lubricant film test machine. The study found that the thickness of the lubricant film of a full ceramic ball bearing is positively correlated with the bearing speed and negatively correlated with the bearing loads. The bearing speed has a relatively large effect on the change in the lubricant film thickness. The pressure of the lubricant film in the contact area is positively correlated with the bearing speed, but it is not affected by the bearing load. Unlike metal ball bearings, the thickness and pressure of the lubricant film have a greater relative rate of change in different positions in the contact area of full ceramic ball bearings. With an increase in the bearing speed, the necking-down effect has a greater influence on the peak mutation of the oil pressure. Only one pressure peak occurs in the oil film in the contact area. The results of this paper play an important role in revealing the friction and lubrication properties of full ceramic ball bearings and improving their service performance and life under oil lubrication conditions

Secure Logistic Regression Based on Homomorphic Encryption: Design and Evaluation

Background: Learning a model without accessing raw data has been an intriguing idea to security and machine learning researchers for years. In an ideal setting, we want to encrypt sensitive data to store them on a commercial cloud and run certain analyses without ever decrypting the data to preserve privacy. Homomorphic encryption technique is a promising candidate for secure data outsourcing, but it is a very challenging task to support real-world machine learning tasks. Existing frameworks can only handle simplified cases with low-degree polynomials such as linear means classifier and linear discriminative analysis. Objective: The goal of this study is to provide a practical support to the mainstream learning models (eg, logistic regression). Methods: We adapted a novel homomorphic encryption scheme optimized for real numbers computation. We devised (1) the least squares approximation of the logistic function for accuracy and efficiency (ie, reduce computation cost) and (2) new packing and parallelization techniques. Results: Using real-world datasets, we evaluated the performance of our model and demonstrated its feasibility in speed and memory consumption. For example, it took approximately 116 minutes to obtain the training model from the homomorphically encrypted Edinburgh dataset. In addition, it gives fairly accurate predictions on the testing dataset. Conclusions: We present the first homomorphically encrypted logistic regression outsourcing model based on the critical observation that the precision loss of classification models is sufficiently small so that the decision plan stays still

Private queries on encrypted genomic data

Abstract Background One of the tasks in the iDASH Secure Genome Analysis Competition in 2016 was to demonstrate the feasibility of privacy-preserving queries on homomorphically encrypted genomic data. More precisely, given a list of up to 100,000 mutations, the task was to encrypt the data using homomorphic encryption in a way that allows it to be stored securely in the cloud, and enables the data owner to query the dataset for the presence of specific mutations, without revealing any information about the dataset or the queries to the cloud. Methods We devise a novel string matching protocol to enable privacy-preserving queries on homomorphically encrypted data. Our protocol combines state-of-the-art techniques from homomorphic encryption and private set intersection protocols to minimize the computational and communication cost. Results We implemented our protocol using the homomorphic encryption library SEAL v2.1, and applied it to obtain an efficient solution to the iDASH competition task. For example, using 8 threads, our protocol achieves a running time of only 4 s, and a communication cost of 2 MB, when querying for the presence of 5 mutations from an encrypted dataset of 100,000 mutations. Conclusions We demonstrate that homomorphic encryption can be used to enable an efficient privacy-preserving mechanism for querying the presence of particular mutations in realistic size datasets. Beyond its applications to genomics, our protocol can just as well be applied to any kind of data, and is therefore of independent interest to the homomorphic encryption community

Study on Distribution of Lubricating Oil Film in Contact Micro-Zone of Full Ceramic Ball Bearings and the Influence Mechanism on Service Performance

Compared with metal ball bearings, full ceramic ball bearings have more outstanding service performance under extreme working conditions. In order to reveal the lubrication mechanism and improve the operation performance and service life of full ceramic ball bearings, in this paper, the friction, vibration, and temperature rise characteristics of 6208 silicon nitride full ceramic deep groove ball bearing, under the condition of oil lubrication, are studied experimentally. Based on the test results, and through theoretical calculation and simulation analysis, the distribution of the lubricating oil film in bearing contact micro-zone under different working conditions was simulated. After that, the surface of contact micro-zone of full ceramic ball bearing was analyzed. It was found that there is an optimal oil supply for full ceramic ball bearing oil lubrication in service. Under the optimal oil supply lubrication, full film lubrication can be achieved, and the bearing exhibits the best characteristics of friction, vibration, and temperature rise. Compared with the load, the rotational speed of the bearing has a decisive influence on the optimal oil supply. When the rotational speed and load are constant, the minimum oil film thickness and oil film pressure in the contact area of the rolling body decrease with the increase of angle ψ from the minimum stress point of the rolling body. Under the action of high contact stress, thin oil film will be formed in the bearing outer ring raceway. In the field of full ceramic ball bearings, the research content of this paper is innovative. The research results of this paper have an important guiding significance for revealing the oil lubrication mechanism of full ceramic ball bearing and enriching its lubrication theory and methods