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

Homomorphic AES Evaluation using NTRU

Since its introduction more than a decade ago the homomorphic properties of the NTRU encryption scheme have gone largely ignored. A variant of NTRU proposed by Stehle and Steinfeld was recently extended into a full fledged multi-key fully homomorphic encryption scheme by Alt-Lopez, Tromer and Vaikuntanathan (ATV). This NTRU based FHE presents a viable alternative to the currently dominant BGV style FHE schemes. While the scheme appears to be more efficient, a full implementation and comparison to BGV style implementations has been missing in the literature. In this work, we develop a customized implementation of the ATV scheme. First parameters are selected to yield an efficient and yet secure ATV instantiation. We present an analysis of the noise growth that allows us to formulate a modulus cutting strategy for arbitrary circuits. Furthermore, we introduce a specialization of the ring structure that allows us to drastically reduce the public key size making evaluation of deep circuits such as the AES block cipher viable on a standard computer with a reasonable amount of memory. Moreover, with the modulus specialization the need for key switching is eliminated. Finally, we present a generic bit-sliced implementation of the ATV scheme that embodies a number of optimizations. To assess the performance of the scheme we homomorphically evaluate the full 10 round AES circuit in 31 hours with 2048 message slots resulting in 55 sec per AES block evaluation time

Accelerating LTV based homomorphic encryption in reconfigurable hardware

After being introduced in 2009, the first fully homomorphic encryption (FHE) scheme has created significant excitement in academia and industry. Despite rapid advances in the last 6 years, FHE schemes are still not ready for deployment due to an efficiency bottleneck. Here we introduce a custom hardware accelerator optimized for a class of reconfigurable logic to bring LTV based somewhat homomorphic encryption (SWHE) schemes one step closer to deployment in real-life applications. The accelerator we present is connected via a fast PCIe interface to a CPU platform to provide homomorphic evaluation services to any application that needs to support blinded computations. Specifically we introduce a number theoretical transform based multiplier architecture capable of efficiently handling very large polynomials. When synthesized for the Xilinx Virtex 7 family the presented architecture can compute the product of large polynomials in under 6.25 msec making it the fastest multiplier design of its kind currently available in the literature and is more than 102 times faster than a software implementation. Using this multiplier we can compute a relinearization operation in 526 msec. When used as an accelerator, for instance, to evaluate the AES block cipher, we estimate a per block homomorphic evaluation performance of 442 msec yielding performance gains of 28.5 and 17 times over similar CPU and GPU implementations, respectively

Depth optimized efficient homomorphic sorting

We introduce a sorting scheme which is capable of efficiently sorting encrypted data without the secret key. The technique is obtained by focusing on the multiplicative depth of the sorting circuit alongside the more traditional metrics such as number of comparisons and number of iterations. The reduced depth allows much reduced noise growth and thereby makes it possible to select smaller parameter sizes in somewhat homomorphic encryption instantiations resulting in greater efficiency savings. We first consider a number of well known comparison based sorting algorithms as well as some sorting networks, and analyze their circuit implementations with respect to multiplicative depth. In what follows, we introduce a new ranking based sorting scheme and rigorously analyze the multiplicative depth complexity as O(log(N) + log(l)), where N is the size of the array to be sorted and l is the bit size of the array elements. Finally, we simulate our sorting scheme using a leveled/batched instantiation of a SWHE library. Our sorting scheme performs favorably over the analyzed classical sorting algorithms

A Survey on Implementation of Homomorphic Encryption Scheme in Cloud based Medical Analytical System

The privacy of sensitive personal information is more and more important topic as a result of the increased availability of cloud services. These privacy issues arise due to the legitimate concern of a) having a security breach on these cloud servers or b) the leakage of this sensitive information due to an honest but curious individual at the cloud service provider. Standard encryption schemes try to address the ?rst concern by devising encryption schemes that are harder to break, yet they don’t solve the possible misuse of this sensitive data by the cloud service providers. Homomorphic encryption presents a tool that can solve both types of privacy concerns. The clients are given the possibility of encrypting their sensitive information before sending it to the cloud. The cloud will then compute over their encrypted data without the need for the decryption key. By using homomorphic encryption, servers guarantee to the clients that their valuable information to have no problems after being in a difficult situation.

Improving the Efficiency of Homomorphic Encryption Schemes

In this dissertation, we explore different approaches to practical homomorphic encryption schemes. For partial homomorphic encryption schemes, we observe that the versatility is the main bottleneck. To solve this problem, we propose general approaches to improve versatility of them by either extending the range of supported circuits or extending the message space. These general approaches can be applied to a wide range of partial HE schemes and greatly increase the number of applications that they support. For fully homomorphic encryption schemes, the slow running speed and the large ciphertext are the main challenges. Therefore, we propose efficient implementations as well as methods to compress the ciphertext. In detail, the Gentry Halevi FHE scheme and the LTV FHE scheme are implemented and the resulting performance shows significant improvement over previous works. For ciphertext compression, the concept of scheme conversion is proposed. Given a scheme converter, we can convert between schemes with compact ciphertext for communication and homomorphic schemes for computation

Encriptação parcialmente homomórfica CCA1-segura

Orientadores: Ricardo Dahab, Diego de Freitas AranhaTese (doutorado) - Universidade Estadual de Campinas, Instituto de ComputaçãoResumo: Nesta tese nosso tema de pesquisa é a encriptação homomórfica, com foco em uma solução prática e segura para encriptação parcialmente homomórfica (somewhat homomorphic encryption - SHE), considerando o modelo de segurança conhecido como ataque de texto encriptado escolhido (chosen ciphertext attack - CCA). Este modelo pode ser subdividido em duas categorias, a saber, CCA1 e CCA2, sendo CCA2 o mais forte. Sabe-se que é impossível construir métodos de encriptação homomórfica que sejam CCA2-seguros. Por outro lado, é possível obter segurança CCA1, mas apenas um esquema foi proposto até hoje na literatura; assim, seria interessante haver outras construções oferecendo este tipo de segurança. Resumimos os principais resultados desta tese de doutorado em duas contribuições. A primeira é mostrar que a família NTRU de esquemas SHE é vulnerável a ataques de recuperação de chave privada, e portanto não são CCA1-seguros. A segunda é a utilização de computação verificável para obter esquemas SHE que são CCA1-seguros e que podem ser usados para avaliar polinômios multivariáveis quadráticos. Atualmente, métodos de encriptação homomórfica são construídos usando como substrato dois problemas de difícil solução: o MDC aproximado (approximate GCD problem - AGCD) e o problema de aprendizado com erros (learning with errors - LWE). O problema AGCD leva, em geral, a construções mais simples mas com desempenho inferior, enquanto que os esquemas baseados no problema LWE correspondem ao estado da arte nesta área de pesquisa. Recentemente, Cheon e Stehlé demonstraram que ambos problemas estão relacionados, e é uma questão interessante investigar se esquemas baseados no problema AGCD podem ser tão eficientes quanto esquemas baseados no problema LWE. Nós respondemos afirmativamente a esta questão para um cenário específico: estendemos o esquema de computação verificável proposto por Fiore, Gennaro e Pastro, de forma que use a suposição de que o problema AGCD é difícil, juntamente com o esquema DGHV adaptado para uso do Teorema Chinês dos Restos (Chinese remainder theorem - CRT) de forma a evitar ataques de recuperação de chave privadaAbstract: In this thesis we study homomorphic encryption with focus on practical and secure somewhat homomorphic encryption (SHE), under the chosen ciphertext attack (CCA) security model. This model is classified into two different main categories: CCA1 and CCA2, with CCA2 being the strongest. It is known that it is impossible to construct CCA2-secure homomorphic encryption schemes. On the other hand, CCA1-security is possible, but only one scheme is known to achieve it. It would thus be interesting to have other CCA1-secure constructions. The main results of this thesis are summarized in two contributions. The first is to show that the NTRU-family of SHE schemes is vulnerable to key recovery attacks, hence not CCA1-secure. The second is the utilization of verifiable computation to obtain a CCA1-secure SHE scheme that can be used to evaluate quadratic multivariate polynomials. Homomorphic encryption schemes are usually constructed under the assumption that two distinct problems are hard, namely the Approximate GCD (AGCD) Problem and the Learning with Errors (LWE) Problem. The AGCD problem leads, in general, to simpler constructions, but with worse performance, wheras LWE-based schemes correspond to the state-of-the-art in this research area. Recently, Cheon and Stehlé proved that both problems are related, and thus it is an interesting problem to investigate if AGCD-based SHE schemes can be made as efficient as their LWE counterparts. We answer this question positively for a specific scenario, extending the verifiable computation scheme proposed by Fiore, Gennaro and Pastro to work under the AGCD assumption, and using it together with the Chinese Remainder Theorem (CRT)-version of the DGHV scheme, in order to avoid key recovery attacksDoutoradoCiência da ComputaçãoDoutor em Ciência da Computação143484/2011-7CNPQCAPE

A custom accelerator for homomorphic encryption applications

After the introduction of first fully homomorphic encryption scheme in 2009, numerous research work has been published aiming at making fully homomorphic encryption practical for daily use. The first fully functional scheme and a few others that have been introduced has been proven difficult to be utilized in practical applications, due to efficiency reasons. Here, we propose a custom hardware accelerator, which is optimized for a class of reconfigurable logic, for Lopez-Alt, Tromer and Vaikuntanathan’s somewhat homomorphic encryption based schemes. Our design is working as a co-processor which enables the operating system to offload the most compute–heavy operations to this specialized hardware. The core of our design is an efficient hardware implementation of a polynomial multiplier as it is the most compute–heavy operation of our target scheme. The presented architecture can compute the product of very–large polynomials in under 6.25 ms which is 102 times faster than its software implementation. In case of accelerating homomorphic applications; we estimate the per block homomorphic AES as 442 ms which is 28.5 and 17 times faster than the CPU and GPU implementations, respectively. In evaluation of Prince block cipher homomorphically, we estimate the performance as 52 ms which is 66 times faster than the CPU implementation

Flattening NTRU for Evaluation Key Free Homomorphic Encryption

We propose a new FHE scheme {\sf F-NTRU} that adopts the flattening technique proposed in GSW to derive an NTRU based scheme that (similar to GSW) does not require evaluation keys or key switching. Our scheme eliminates the decision small polynomial ratio (DSPR) assumption but relies only on the standard R-LWE assumption. It uses wide key distributions, and hence is immune to the Subfield Lattice Attack. In practice, our scheme achieves competitive timings compared to the existing schemes. We are able to compute a homomorphic multiplication in $24.4$~msec and $34.3$~msec for $5$ and $30$ levels, respectively, without amortization. Furthermore, our scheme features small ciphertexts, e.g. $1152$~KB for $30$ levels, and eliminates the need for storing and managing costly evaluation keys. In addition, we present a slightly modified version of F-NTRU that is capable to support integer operations with a very large message space along with noise analysis for all cases. The assurance gained by using wide key distributions along with the message space flexibility of the scheme, i.e. bits, binary polynomials, and integers with a large message space, allows the use of the proposed scheme in a wide array of applications