We suggest a method to construct a homomorphic encryption scheme for approximate arithmetic. It supports an approximate addition and multiplication of encrypted messages, together with a new rescaling procedure for managing the magnitude of plaintext. This procedure truncates a ciphertext into a smaller modulus, which leads to rounding of plaintext. The main idea is to add a noise following significant figures which contain a main message. This noise is originally added to the plaintext for security, but considered to be a part of error occurring during approximate computations that is reduced along with plaintext by rescaling. As a result, our decryption structure outputs an approximate value of plaintext with a predetermined precision.

/pdf/homomorphic-encryption-for-arithmetic-of-approximate-numbers-2id1fgj9hk.pdf

Homomorphic Encryption for Arithmetic of Approximate Numbers

The paradigm of Internet of Things (IoT) is paving the way for a world, where many of our daily objects will be interconnected and will interact with their environment in order to collect information and automate certain tasks. Such a vision requires, among other things, seamless authentication, data privacy, security, robustness against attacks, easy deployment, and self-maintenance. Such features can be brought by blockchain, a technology born with a cryptocurrency called Bitcoin. In this paper, a thorough review on how to adapt blockchain to the specific needs of IoT in order to develop Blockchain-based IoT (BIoT) applications is presented. After describing the basics of blockchain, the most relevant BIoT applications are described with the objective of emphasizing how blockchain can impact traditional cloud-centered IoT applications. Then, the current challenges and possible optimizations are detailed regarding many aspects that affect the design, development, and deployment of a BIoT application. Finally, some recommendations are enumerated with the aim of guiding future BIoT researchers and developers on some of the issues that will have to be tackled before deploying the next generation of BIoT applications.

A Review on the Use of Blockchain for the Internet of Things

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. The users or service providers with the key have exclusive rights on the data. Especially with popular cloud services, 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. Therefore, this survey focuses on HE and FHE schemes. 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 for 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, and extending the state-of-the-art HE, PHE, SWHE, and FHE systems.

https://dl.acm.org/doi/pdf/10.1145/3214303

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.

Designing an efficient cipher was always a delicate balance between linear and non-linear operations. This goes back to the design of DES, and in fact all the way back to the seminal work of Shannon.

/pdf/ciphers-for-mpc-and-fhe-whyo8mkwpv.pdf

Ciphers for MPC and FHE

We present a custom architecture for realizing the Gentry-Halevi fully homomorphic encryption (FHE) scheme. This contribution presents the first full realization of FHE in hardware. The architecture features an optimized multi-million bit multiplier based on the Schonhage Strassen multiplication algorithm. Moreover, a number of optimizations including spectral techniques as well as a precomputation strategy is used to significantly improve the performance of the overall design. When synthesized using 90 nm technology, the presented architecture achieves to realize the encryption, decryption, and recryption operations in 18.1 msec, 16.1 msec, and 3.1 sec, respectively, and occupies a footprint of less than 30 million gates.

Accelerating Fully Homomorphic Encryption in Hardware

Cloud computing technology has rapidly evolved over the last decade, offering an alternative way to store and work with large amounts of data. However data security remains an important issue particularly when using a public cloud service provider. The recent area of homomorphic cryptography allows computation on encrypted data, which would allow users to ensure data privacy on the cloud and increase the potential market for cloud computing. A significant amount of research on homomorphic cryptography appeared in the literature over the last few years; yet the performance of existing implementations of encryption schemes remains unsuitable for real time applications. One way this limitation is being addressed is through the use of graphics processing units (GPUs) and field programmable gate arrays (FPGAs) for implementations of homomorphic encryption schemes. This review presents the current state of the art in this promising new area of research and highlights the interesting remaining open problems.

/pdf/practical-homomorphic-encryption-a-survey-51wiytw28z.pdf

Practical homomorphic encryption: A survey

We introduce a large polynomial arithmetic library optimized for Nvidia GPUs to support fully homomorphic encryption schemes. To realize the large polynomial arithmetic library we convert polynomials with large coefficients using the Chinese Remainder Theorem into many polynomials with small coefficients, and then carry out modular multiplications in the residue space using a custom developed discrete Fourier transform library. We further extend the library to support the homomorphic evaluation operations, i.e. addition, multiplication, and relinearization, in an NTRU based somewhat homomorphic encryption library. Finally, we put the library to use to evaluate homomorphic evaluation of two block ciphers: Prince and AES, which show 2.57 times and 7.6 times speedup, respectively, over an Intel Xeon software implementation.

/pdf/accelerating-ntru-based-homomorphic-encryption-using-gpus-2dvvasa00w.pdf

Accelerating NTRU based homomorphic encryption using GPUs

We present the homomorphic evaluation of the Prince block cipher. Our leveled implementation is based on a generalization of NTRU. We are motivated by the drastic bandwidth savings that may be achieved by scheme conversion. To unlock this advantage we turn to lightweight ciphers such as Prince. These ciphers were designed from scratch to yield fast and compact implementations on resource-constrained embedded platforms. We show that some of these ciphers have the potential to enable near practical homomorphic evaluation of block ciphers. Indeed, our analysis shows that Prince can be implemented using only a 24 level deep circuit. Using an NTRU based implementation we achieve an evaluation time of 3.3 s per Prince block – one and two orders of magnitude improvement over homomorphic AES implementations achieved using NTRU, and BGV-style homomorphic encryption libraries, respectively.

/pdf/toward-practical-homomorphic-evaluation-of-block-ciphers-3z1ltuoqlv.pdf

Toward Practical Homomorphic Evaluation of Block Ciphers Using Prince

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 Lopez-Alt, Tromer and Vaikuntanathan (LTV). 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 LTV. First parameters are selected to yield an efficient and yet secure LTV 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 LTV scheme that embodies a number of optimizations. To assess the performance of the scheme we homomorphically evaluate the full 10 round AES circuit in 29 h with 2048 message slots resulting in 51 s per AES block evaluation time.

Yarkin Doröz

Papers

Accelerating Fully Homomorphic Encryption in Hardware

Practical homomorphic encryption: A survey

Accelerating NTRU based homomorphic encryption using GPUs

Toward Practical Homomorphic Evaluation of Block Ciphers Using Prince

Homomorphic AES evaluation using the modified LTV scheme