English

Abstract: Encryption is essential in preventing unauthorized access to sensitive data in light of the growing concerns about data security in cloud computing. Homomorphic encryption promises to enable secure calculations on encrypted data without the need for decryption, particularly for cloud-based operations. To evaluate the effectiveness and applicability of several homomorphic encryption algorithms for safe cloud computing, we compare and contrast them in this research paper. Partially homomorphic encryption (PHE), somewhat homomorphic encryption (SHE), and fully homomorphic encryption (FHE) are the three basic homomorphic encryption subtypes that we examine. The implications of this study can aid cloud service providers and organizations in selecting the most appropriate homomorphic encryption scheme based on their specific security requirements and performance considerations. The research contributes to the ongoing efforts to enhance data privacy in cloud computing environments, opening new possibilities for secure data processing in an increasingly connected digital world. The exploration of homomorphic encryption schemes in this study opens new avenues for research and development in the field of cryptographic techniques. As technology continues to evolve, so too must our approaches to safeguarding data. This research serves as a catalyst for further innovations in homomorphic encryption algorithms, enabling even more efficient and robust methods for secure data processing in cloud environments and beyond. The insights derived from this research paper not only empower cloud service providers and organizations to make informed decisions about selecting the most appropriate homomorphic encryption scheme but also contribute to the broader mission of fortifying data privacy and security in cloud computing.

With the wide adoption of cloud computing paradigm, it is important to develop appropriate techniques to protect client data privacy in the cloud. Encryption is one of the major techniques that could be used to achieve this goal. However, data encryption at the rest alone is insufficient for secure cloud computation environments. There is also the need for efficient techniques to carry out computation over encrypted data. Fully homomorphic encryption (FHE) and garbled circuits are naturally used to process encrypted data without leaking any information about the data. However, existing FHE schemes are inefficient for processing large amount of data in cloud and garbled circuits are one time programs and cannot be reused. Based on quaternion/octonion algebra and Jordan algebra over finite rings \(\mathbb {Z}_q\), this paper designs efficient fully homomorphic symmetric key encryption (FHE) schemes without bootstrapping (that is, noise-free FHE schemes) that are secure in the weak ciphertext-only security model assuming the hardness of solving multivariate quadratic equation systems and solving univariate high degree polynomial equation systems in \(\mathbb {Z}_q\). The FHE scheme designed in this paper is sufficient for privacy preserving computation in cloud.

We present new constructions of multi-party homomorphic secret sharing (HSS) based on a new primitive that we call homomorphic encryption with decryption to shares (HEDS). Our first construction, which we call \(\mathsf {Scooby} \), is based on many popular fully homomorphic encryption (FHE) schemes with a linear decryption property. \(\mathsf {Scooby} \) achieves an n-party HSS for general circuits with complexity \(O(|F| + \log n)\), as opposed to \(O(n^2 \cdot |F|)\) for the prior best construction based on multi-key FHE. \(\mathsf {Scooby} \) can be based on (ring)-LWE with a super-polynomial modulus-to-noise ratio. In our second construction, \(\mathsf {Scrappy} \), assuming any generic FHE plus HSS for NC1-circuits, we obtain a HEDS scheme which does not require a super-polynomial modulus. While these schemes all require FHE, in another instantiation, \(\mathsf {Shaggy} \), we show how in some cases it is possible to obtain multi-party HSS without FHE, for a small number of parties and constant-degree polynomials. Finally, we show that our \(\mathsf {Scooby} \) scheme can be adapted to use multi-key fully homomorphic encryption, giving more efficient spooky encryption and setup-free HSS. This latter scheme, \(\mathsf {Casper} \), if concretely instantiated with a B/FV-style multi-key FHE scheme, for functions F which do not require bootstrapping, gives an HSS complexity of \(O(n \cdot |F| + n^2 \cdot \log n)\).

Accelerating the development of a practical Fully Homomorphic Encryption (FHE) scheme is the goal of the DARPA PROCEED program. For the past year, this program has had as its focus the acceleration of various aspects of the FHE concept toward practical implementation and use. FHE would be a game-changing technology to enable secure, general computation on encrypted data, e.g., on untrusted off-site hardware. However, FHE will still require several orders of magnitude improvement in computation before it will be practical for widespread use. Recent theoretical breakthroughs demonstrated the existence of FHE schemes [1, 2], and to date much progress has been made in both algorithmic and implementation improvements. Specifically our contribution to the Proceed program has been the development of FPGA based hardware primitives to accelerate the computation on encrypted data using FHE based on lattice techniques [3]. Our project, SIPHER, has been using a state of the art tool-chain developed by Mathworks to implement VHDL code for FPGA circuits directly from Simulink models. Our baseline Homomorphic Encryption prototypes are developed directly in Matlab using the fixed point toolbox to perform the required integer arithmetic. Constant improvements in algorithms require us to be able to quickly implement them in a high level language such as Matlab. We reported on our initial results at HPEC 2011 [4]. In the past year, increases in algorithm complexity have introduced several new design requirements for our FPGA implementation. This report presents new Simulink primitives that had to be developed to deal with these new requirements.

In cryptography, there has been tremendous success in building various two-party protocols with small communication complexity out of homomorphic semantically-secure encryption schemes, using their homomorphic properties in a black-box way. A few notable examples of such primitives include items like single database Private Information Retrieval (PIR) schemes (introduced in [15]) and private database update with small communication (introduced in [5]). In this paper, we illustrate a general methodology for determining what types of protocols can and cannot be implemented with small communication by using homomorphic encryption in a black-box way. We hope that this work will provide a simple litmus test of feasibility for black-box use of known homomorphic encryption schemes by other cryptographic researchers attempting to develop new protocols with low communication. Additionally, a precise mathematical language for reasoning about such problems is developed in this work, which may be of independent interest. We stress that the class of algebraic structures for which we prove communication complexity lower bounds is large, and covers practically all known semantically-secure homomorphic cryptosystems (including those based upon bilinear maps). Finally, we show the following equivalence which relates group homomorphic encryption and a major open question of designing a so-called fully-homomorphic cryptosystem: a fully homomorphic encryption scheme (over a non-zero ring) exists if and only if there exists homomorphic encryption over any finite non-abelian simple group. This result somewhat generalizes results of Barrington [1] (to any group containing a finite non-abelian simple subgroup) and of Maurer and Rhodes [18], and in fact gives a constructiveproof of the 1974 result Werner [28]. (This also answers an open question posed by Rappe in [23], who in 2004 proved a special case of this result.)

The Shell sort algorithm is one of the most practically effective in-place sorting algorithms. However, it is difficult to execute this algorithm with its intended running time complexity on data encrypted using fully homomorphic encryption (FHE), because the insertion sort in Shell sort has to be performed by considering the worst-case input data. In this paper, in order for the sorting algorithm to be used on the FHE data, we modify the Shell sort with an additional parameter α, allowing exponentially small sorting failure probability. For a gap sequence of powers of two, the modified Shell sort with input array length n is found to have the trade-off between the running time complexity of O(n3/2√α+loglogn) and the sorting failure probability of 2-α. Its running time complexity is close to the intended running time complexity of O(n3/2) and the sorting failure probability can be made very low with slightly increased running time. Further, the near-optimal window length of the modified Shell sort is also derived via convex optimization. The proposed analysis of the modified Shell sort is numerically confirmed by using randomly generated arrays. For the practical aspect, our modification can be applied to any gap sequence, and we show that Ciura’s gap sequence, which is known to have good practical performance, is also practically effective when our modified Shell sort is applied. We compare our modified Shell sort with other sorting algorithms with the FHE over the torus (TFHE) library, and it is shown that this modified Shell sort has the best performance in running time among in-place sorting algorithms on homomorphic encryption scheme.

Identity‐based fully homomorphic encryption (IBFHE) provided a fundamental solution to the problem of huge public key size that exposed in fully homomorphic encryption (FHE) schemes, thus it is significant to make FHE become more practical. In recent years, the construction of IBFHE schemes were mainly based on lattices due to their conjectured resistance against quantum cryptanalysis, however, which makes these cryptosystems further unpractical. The main reason is the trapdoor function on which the scheme was based is rather complex for practical and the ciphertext size is too large. In this study, the authors propose an efficient batch IBFHE scheme, which can be proven secure from the standard LWE assumption in the standard model. The first contribution of this work is that the authors construct an efficient batch version of MP12 preimage sampling algorithm, which can efficiently generate identity keys for multi‐bit IBE schemes. Based on that, the authors construct an asymptotically‐faster multi‐bit IBE scheme as the second contribution. The third contribution is that the authors transform the multi‐bit IBE scheme to batch IBFHE scheme which supports to encrypt any message in . Compared with the similar schemes, the authors show their schemes are essentially improved.

Homomorphic encryption (HE) schemes enable computing functions on encrypted data, by means of a public Eval procedure that can be applied to ciphertexts. But the evaluated ciphertexts so generated may differ from freshly encrypted ones. This brings up the question of whether one can keep computing on evaluated ciphertexts. An i-hop homomorphic encryption scheme is one where Eval can be called on its own output up to i times, while still being able to decrypt the result. A multi-hop homomorphic encryption is a scheme which is i-hop for all i. In this work we study i-hop and multi-hop schemes in conjunction with the properties of function-privacy (i.e., Eval's output hides the function) and compactness (i.e., the output of Eval is short). We provide formal definitions and describe several constructions. First, we observe that bootstrapping techniques can be used to convert any (1-hop) homomorphic encryption scheme into an i-hop scheme for any i, and the result inherits the function-privacy and/or compactness of the underlying scheme. However, if the underlying scheme is not compact (such as schemes derived from Yao circuits) then the complexity of the resulting i-hop scheme can be as high as nO(i). We then describe a specific DDH-based multi-hop homomorphic encryption scheme that does not suffer from this exponential blowup. Although not compact, this scheme has complexity linear in the size of the composed function, independently of the number of hops. The main technical ingredient in this solution is a re-randomizable variant of the Yao circuits. Namely, given a garbled circuit, anyone can re-garble it in such a way that even the party that generated the original garbled circuit cannot recognize it. This construction may be of independent interest.

In this paper we report on our work to design, implement and evaluate a Fully Homomorphic Encryption (FHE) scheme. Our FHE scheme is an NTRU-like cryptosystem, with additional support for efficient key switching and modulus reduction operations to reduce the frequency of bootstrapping operations. Ciphertexts in our scheme are represented as matrices of 64-bit integers. The basis of our design is a layered software services stack to provide high-level FHE operations supported by lower-level lattice-based primitive implementations running on a computing substrate. We implement and evaluate our FHE scheme to run on a commodity CPU-based computing environment. We implemented our FHE scheme to run in a compiled C environment and use parallelism to take advantage of multi-core processors. We provide experimental results which show that our FHE implementation provides at least an order of magnitude improvement in runtime as compared to recent publicly known evaluation results of other FHE software implementations.

Multi‐key fully homomorphic encryption (MKFHE) allows computations on ciphertexts encrypted by different users, which can be applied to implement secure multi‐party computing (MPC). The current NTRU‐based MKFHE has the following two drawbacks: One is that the relinearisation process during homomorphic evaluation is so complicated that the corresponding computation time is costly. The other is that a class of subfield attacks are proposed and affects the security of NTRU schemes over power‐of‐2 cyclotomic rings for large moduli q, especially for the NTRU‐based fully homomorphic encryption (FHE) schemes. In this work, an efficient MKFHE scheme is proposed over prime cyclotomic rings with fewer relinearisations, which seems a good choice because of its potential to resist a subfield attack. More specifically, the time of the relinearisation process is reduced by half in homomorphic evaluations by separating the homomorphic multiplication and the relinearisation process (implementing two homomorphic multiplication operations together before relinearisation), while in current NTRU‐type MKFHE schemes, these two processes are usually performed together. The error bound of the basic function components is re‐analysed over prime cyclotomic rings in the average case, which can be used in the error analysis of our scheme. We construct an efficient NTRU‐based single‐key FHE scheme and an efficient MKFHE scheme over prime cyclotomic rings through relinearisation and modulus‐switching techniques. The MKFHE scheme proposed has the on‐the‐fly property and has a tight ciphertext size compared with the GSW‐type and BGV‐type MKFHE schemes. An experiment shows that the homomorphic evaluation of the optimised single‐key FHE scheme proposed is 1.9 times faster than an efficient NTRU‐type MKFHE DHS16 proposed at DCC 2016.

As a major breakthrough, in 2009 Gentry introduced the first plausible construction of a fully homomorphic encryption (FHE) scheme. FHE allows the evaluation of arbitrary functions directly on encrypted data on untwisted servers. In 2010, Gentry and Halevi presented the first FHE implementation on an IBM x3500 server. However, this implementation remains impractical due to the high latency of encryption and recryption. The Gentry-Halevi (GH) FHE primitives utilize multi-million-bit modular multiplications and additions which are time-consuming tasks for a general purpose computer. In the GH-FHE implementation, the most computationally intensive arithmetic operation is modular multiplication. In this paper, the million-bit modular multiplication is computed in two steps. For large number multiplication, Strassen's FFT based algorithm is employed and accelerated on a graphics processing unit (GPU) through its massive parallelism. Subsequently, Barrett modular reduction algorithm is applied to implement modular reduction. As an experimental study, we implement the GH-FHE primitives for the small setting with a dimension of 2048 on NVIDIA C2050 GPU. The experimental results show the speedup factors of 7.68, 7.4 and 6.59 for encryption, decryption and recrypt respectively, when compared with the existing CPU implementation.

Statistical confidentiality focuses on protecting data to preserve its analytical value while preventing identity exposure, ensuring privacy and security in any system handling sensitive information. Homomorphic encryption allows computations on encrypted data without revealing it to anyone other than an owner or an authorized collector. When combined with other techniques, homomorphic encryption offers an ideal solution for ensuring statistical confidentiality. TFHE (Fast Fully Homomorphic Encryption over the Torus) is a fully homomorphic encryption scheme that supports efficient homomorphic operations on Booleans and integers. Building on TFHE, Zama’s Concrete project offers an open-source compiler that translates high-level Python code (version 3.9 or higher) into secure homomorphic computations. This study examines the feasibility of the Concrete compiler to perform core statistical analyses on encrypted data. We implement traditional algorithms for core statistical measures including the mean, variance, and five-point summary on encrypted datasets. Additionally, we develop a bitonic sort implementation to support the five-point summary. All implementations are executed within the Concrete framework, leveraging its built-in optimizations. Their performance is systematically evaluated by measuring circuit complexity, programmable bootstrapping count (PBS), compilation time, and execution time. We compare these results to findings from previous studies wherever possible. The results show that the complexity of sorting and statistical computations on encrypted data with the Concrete implementation of TFHE increases rapidly, and the size and range of data that can be accommodated is small for most applications. Nevertheless, this work reinforces the theoretical promise of Fully Homomorphic Encryption (FHE) for statistical analysis and highlights a clear path forward: the development of optimized, FHE-compatible algorithms.

The main bottleneck affecting the efficiency of all known fully homomorphic encryption (FHE) schemes is Gentry’s bootstrapping procedure, which is required to refresh noisy ciphertexts and keep computing on encrypted data. Bootstrapping in the latest implementation of FHE, the HElib library of Halevi and Shoup (Crypto 2014), requires about six minutes. We present a new method to homomorphically compute simple bit operations, and refresh (bootstrap) the resulting output, which runs on a personal computer in just about half a second. We present a detailed technical analysis of the scheme (based on the worst-case hardness of standard lattice problems) and report on the performance of our prototype implementation.KeywordsEncryption SchemeHomomorphic EncryptionNAND GateMessage SpaceRing ElementThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We present a multi-GPU design, implementation and performance evaluation of the Halevi-Polyakov-Shoup (HPS) variant of the Fan-Vercauteren (FV) levelled Fully Homomorphic Encryption (FHE) scheme. Our design follows a data parallelism approach and uses partitioning methods to distribute the workload in FV primitives evenly across available GPUs. The design is put to address space and runtime requirements of FHE computations. It is also suitable for distributed-memory architectures, and includes efficient GPU-to-GPU data exchange protocols. Moreover, it is user-friendly as user intervention is not required for task decomposition, scheduling or load balancing. We implement and evaluate the performance of our design on two homogeneous and heterogeneous NVIDIA GPU clusters: K80, and a customized P100. We also provide a comparison with a recent shared-memory-based multi-core CPU implementation using two homomorphic circuits as workloads: vector addition and multiplication. Moreover, we use our multi-GPU Levelled-FHE to implement the inference circuit of two Convolutional Neural Networks (CNNs) to perform homomorphically image classification on encrypted images from the MNIST and CIFAR - 10 datasets. Our implementation provides 1 to 3 orders of magnitude speedup compared with the CPU implementation on vector operations. In terms of scalability, our design shows reasonable scalability curves when the GPUs are fully connected.

Privacy preserving service selection using fully homomorphic encryption scheme on untrusted cloud service platform