Fully Homomorphic Encryption for Secure Cloud Computation

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.

Fully Homomorphic Encryption (FHE) is the holy grail of encrypted communications. It opens the door to several advanced functionalities to overcome the security and trust issues of the IT world. After 2009, once Craig Gentry had shown that FHE could be achieved, a study in this field boomed, and significant improvement was made in identifying more efficient and realistic programs. FHE is primitive cryptography that enables arbitrary functions to be calculated via encrypted data. These systems are applicable in different ways since they permit users to encrypt their private information securely while still outsourcing the processing of protected data without fearing disclosing the real data. In 2012, LTV12 presented the first multi-key FHE system and demonstrated the possibility of using multi-key systems in somewhat homomorphic encryption (SHE). Like in the one key context, there have been many advances in the field, but no effort has been made to develop the multi-key methods. This paper presents a discussion of FHE and MKFHE with a specific focus on the current techniques and three implementations, comprising the first in the multi-key setups, to the extent of our understanding.

Fully Homomorphic Encryption (FHE) is pivotal for secure computation on encrypted data, crucial in privacy-preserving data analysis. However, efficiently processing high-dimensional data in FHE, especially for machine learning and statistical (ML/STAT) algorithms, poses a challenge. In this paper, we present an effective acceleration method using the kernel method for FHE schemes, enhancing time performance in ML/STAT algorithms within encrypted domains. This technique, independent of underlying FHE mechanisms and complementing existing optimizations, notably reduces costly FHE multiplications, offering near-constant time complexity relative to data dimension. Aimed at accessibility, this method is tailored for data scientists and developers with limited cryptography background, facilitating advanced data analysis in secure environments.

Fully homomorphic encryption (FHE) allows us to perform computations directly over encrypted data and can be widely used in some highly regulated industries. Gentry’s bootstrapping procedure is used to refresh noisy ciphertexts and is the only way to achieve the goal of FHE up to now. In this paper, we optimize the LWE-based GSW-type bootstrapping procedure. Our optimization decreases the lattice approximation factor for the underlying worst-case lattice assumption from \(\tilde{O}(N^{2.5})\) to \(\tilde{O}(N^{2})\), and is time-efficient by a \(O(\lambda )\) factor. Our scheme can also achieve the best factor in prior works on bootstrapping of standard lattice-based FHE by taking a larger lattice dimension, which makes our scheme as secure as the standard lattice-based PKE. Furthermore, in this work we present a technique to perform more operations per bootstrapping in the LWE-based FHE scheme. Although there have been studies to evaluate large FHE gates using schemes over ideal lattices, (i.e. using FHEW or TFHE), we are the first to study how to perform complex functions homomorphically over standard lattices.

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.

Fully homomorphic encryption (FHE) allows computing on encrypted data, enabling secure offloading of computation to untrusted servers. Though it provides ideal security, FHE is prohibitively expensive when executed in software. These overheads are a major barrier to FHE's widespread adoption. We present F1, the first FHE accelerator that is capable of executing full FHE programs. F1 builds on an in-depth architectural analysis of the characteristics of FHE computations that reveals acceleration opportunities. F1 is a wide-vector processor with novel functional units deeply specialized to FHE primitives, such as modular arithmetic, number-theoretic transforms, and structured permutations. This organization provides so much compute throughput that data movement becomes the bottleneck. Thus, F1 is primarily designed to minimize data movement. F1 is the first system to accelerate complete FHE programs, and outperforms state-of-the-art software implementations by gmean 5,400x. These speedups counter FHE's overheads and enable new applications, like real-time private deep learning in the cloud.

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.

Secure report is the problem of a client that retrieves all records matching specified attributes from a database table at the server (e.g. cloud), as in SQL SELECT queries, but where the query and the database are encrypted. Here, only the client has the secret key, but still the server is expected to compute and return the encrypted result. Secure report is theoretically possible with Fully Homomorphic Encryption (FHE). However, the current state-of-the-art solutions are realized by a polynomial of degree that is at least linear in the number m of records, which is too slow in practice even for very small databases. We present the first solution that is realized by a polynomial that attains degree independent of the number of records m, as well as the first implementation of an FHE solution to Secure report. This is by suggesting a novel paradigm that forges a link between cryptography and modern data summarization techniques known as coresets (core-sets), and sketches in particular. The key idea is to compute only a coreset of the desired report. Since the coreset is small, the client can quickly decode the desired report that the server computes after decrypting the coreset. We implemented our main reporting system in an open source library. This is the first implemented system that can answer such database queries when processing only FHE encrypted data and queries. As our analysis promises, the experimental results show that we can run Secure report queries on billions records in minutes on an Amazon EC2 server, compared to less than a hundred-thousands in previous FHE based solutions.

Garbled computation in cloud

A KPU is a replacement for a standard RISC processor that natively runs encrypted machine code on encrypted data in registers and memory — a 'general-purpose crypto-processor', in other words. It works because the processor's arithmetic is customised to make the chosen encryption into a mathematical homomorphism, resulting in what is called a 'fully-homomorphic encryption' design. This paper discusses the problems and solutions associated with implementing a KPU in hardware.

Предметом дослідження є шифрування інформації в хмарних обчисленнях і сховищах даних. Хмарні технології дозволяють значно скоротити витрати на ІТ-інфраструктуру і гнучко реагувати на зміни обчислювальних потреб. В такому випадку має бути забезпечено можливість проведення обчислень над зашифрованими даними без їх дешифрування. Таку властивість має повністю гомоморфне шифрування. Метою даної статті є підвищення ефективності повністю гомоморфного шифрування (ПГШ) на основі матричних поліномів за допомогою методу пакетного шифрування в один шифротекст декількох відкритих текстів з наступною комплексною обробкою зашифрованих даних. Пакетне шифрування зводиться до того, що при одній операції над двома шифротекстами відбувається одночасне виконання операцій покоординатно над усіма даними, що містяться в цих шифротекстах у вигляді відкритих текстів (SIMD). Завданнями визначено побудову алгоритмів повністю гомоморфного шифрування даних за допомогою матричних поліномів. У статті використано методи шифрування: з використанням китайської теореми про залишки; шляхом запису в одній матриці декількох різних власних значень при різних власних векторах; за допомогою інтерполяції матричних поліномів. В результаті описано та проаналізовано можливі підходи до побудови пакетних ПГШ на підставі матричних поліномів, а також представлено набір алгоритмів, що реалізують криптосхему ПГШ з інтерполяцією матричних поліномів. Наведені алгоритми і криптосхеми дозволяють передавати інформацію в повідомленнях і дані в запитах як відкритий текст, бо над шифрованими даними можна здійснювати необмежену кількість складних алгебраїчних операцій. Це, у свою чергу, ускладнює можливість дешифрування і зчитування даних без знання всього алгоритму. Було показано, що побудовані криптосхеми перевершують аналоги по ефективності, які розроблені дослідниками з IBM. Можна зробити наступний висновок: пакетне повністю е шифрування на основі матричних поліномів здатне виключити необхідність хоча б часткового дешифрування даних для несанкціонованих обчислень над зашифрованими масивами даних у хмарних сховищах.

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 \(O(n \cdot |F| + n^2 \cdot \log n)\).

Evaluating arbitrary functions on encrypted data is one of the holy grails of cryptography, with Fully Homomorphic Encryption (FHE) being probably the most prominent and powerful example. FHE, in its current state is, however, not efficient enough for practical applications. On the other hand, simple homomorphic and somewhat homomorphic approaches are not powerful enough to support arbitrary computations.We propose a new approach towards a practicable system for evaluating functions on encrypted data. Our approach allows to chain an arbitrary number of computations, which makes it more powerful than existing efficient schemes. As with basic FHE we do not encrypt or in any way hide the function, that is evaluated on the encrypted data. It is, however, sufficient that the function description is known only to the evaluator. This situation arises in practice for software as a Software as a Service (SaaS)-scenarios, where an evaluator provides a function only known to him and the user wants to protect his data. Another application might be the analysis of sensitive data, such as medical records.In this paper we restrict ourselves to functions with only one input parameter, which allow arbitrary transformations on encrypted data.

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.

Fully homomorphic encryption (FHE) enables direct computation on encrypted data without decryption, ensuring data privacy in cloud computing scenarios and preventing the leakage of sensitive information. However, the computational overhead of HE typically exceeds that of plaintext computation by 4 to 5 orders of magnitude, while energy consumption is 5 to 6 orders of magnitude higher. These substantial performance and energy overheads significantly hinder the widespread adoption of FHE. This paper proposed LP-HENN, a novel low-power and energy-efficient FHE accelerator architecture that leverages a RISC-V vector coprocessor and ReRAM crossbar arrays. LP-HENN targets power-constrained application scenarios such as edge devices, aiming to provide highly energy-efficient acceleration support for FHE applications. LP-HENN leverages the collaborative work of the vector processor and ReRAM crossbars, employing optimization strategies to achieve full pipelining and minimize memory access. Furthermore, this paper proposed a parameter selection model for early-stage architecture design, which achieves an optimal balance between performance and energy consumption through the collaborative optimization of multiple parameters. Experimental results show that, for an FHE-based convolutional neural network (HE-CNN) inference application, LP-HENN achieves a 31.82Ã- and 11920.56Ã- improvement in performance and energy efficiency, respectively, compared to CPU. Compared to FxHENN, the state-of-the-art FPGA-based FHE accelerator with high energy efficiency for edge devices, LP-HENN achieves a 2.36Ã- and 10.04Ã- improvement in performance and energy efficiency, respectively. The energy efficiency of LP-HENN is comparable to that of F1, the state-of-the-art ASIC FHE accelerator, while featuring a low power design suitable for edge computing.