An Experimental Analysis on Lattice Attacks against Ring-LWE over Decomposition Fields

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.)

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.

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.

Healthcare data has risen as a top target for cyberattacks due to the rich amount of sensitive patient information. This negatively affects the potential of advanced analytics and collaborative research in healthcare. Homomorphic encryption (HE) has emerged as a promising technology for securing sensitive healthcare data while enabling computations on encrypted information. This paper conducts a background survey of HE and its various types. It discusses Partially Homomorphic Encryption (PHE), Somewhat Homomorphic Encryption (SHE), Fully Homomorphic Encryption (FHE) and Fully Leveled Homomorphic Encryption (FLHE). A critical analysis of these encryption paradigms’ theoretical foundations, implementation schemes, and practical applications in healthcare contexts is presented. The survey encompasses diverse healthcare domains. It demonstrates HE’s versatility in securing electronic health records (EHRs), enabling privacy-preserving genomic data analysis, protecting medical imaging, facilitating privacy-preserving machine learning (ML), supporting secure federated learning, ensuring confidentiality in clinical trials, and enhancing remote monitoring and telehealth services. A comprehensive examination of potential vulnerabilities in HE systems is conducted. The research systematically investigates various attack vectors, including side-channel attacks, key recovery attacks, chosen plaintext attacks (CPA), chosen ciphertext attacks (CCA), known plaintext attacks (KPA), fault injection attacks (FIA), and lattice attacks. A detailed analysis of potential defense mechanisms and mitigation strategies is provided for each identified threat. The analysis underscores the importance of HE for long-term security and sustainability in healthcare systems.

An encryption scheme is homomorphic if it supports operations on encrypted data. Homomorphic encryption allows a device to perform arbitrary computations on encrypted data without user secret key. Recently it is introduced new homomorphic encryption schemes with improved performance that can be implemented in IoT device in production environments. The IoT concept encompasses devices, sensors, and services existing within an interconnected infrastructure with an efficient access to sample computational facilities. In this paper we evaluated features of exact arithmetic homomorphic encryption mechanisms: BFV and BGV and approximate homomorphic encryption scheme: CKKS. In the paper we measured performances of operations of homomorphic encryption schemes: BGV, BFV and CKKS that are implemented in Raspberry Pi 4 IoT device.

Public key encryption schemes developed based on learning with error problem became popular for homomorphic encryption, and are proved as secured schemes based on the worst-case hardness of short vector problems. Homomorphic encryption allows computations over the cipher text without decryption. Implementation of the scheme is not considered to be practical because of its larger public keys and larger cipher texts. Large public key and cipher texts require huge space in the cloud storages. However, there are some approaches proposed to shorten the cipher texts and public keys of LWE-based homomorphic encryption scheme. The objective of the paper is to introduce an idea to shorten the public keys and cipher texts in the storage. Also, support homomorphic addition operation on reduced cipher texts. The aim of the paper is not only to give the practical implementation of standard LWE-based homomorphic encryption operation (Addition) on the reduced cipher texts and also the performance of the proposed scheme.

ABSTRACTIn witness encryption (WE), an instance of an NP problem is allowed to be used to encrypt a message, and who holding a witness of the problem can efficiently decrypt the ciphertext. In this work, we put forth the concept of homomorphic witness encryption (HWE), where one can evaluate functions over ciphertexts of the same instance without decrypting them, that is, one can manipulate a set of ciphertexts with messages to obtain the evaluation of , for any function . We declare that such homomorphic witness encryption schemes can be generically constructed from indistinguishable obfuscation () for any classes of functions. Then we propose the instantiate of multiplicatively homomorphic witness encryption (MHWE) and linearly homomorphic witness encryption (LHWE) using an , homomorphic encryption for NP problems such as Subset‐Sum and a batch‐processed GS‐proof system, which enables us to evaluate multiplication operations and linear operations over ciphertext. Furthermore, we show the practicality of homomorphic witness encryption by proposing new protocols for applications of interest, such as homomorphic time‐lock encryption, multi‐party contract signing, and e‐voting.

The objective of this research is to develop a novel image encryption method that can be used to considerably increase the security of encrypted images. To solve this image security problem, we propose a distributed homomorphic image encryption scheme where the images of interest are those in the visible electromagnetic spectrum. In our encryption phase, a red green blue (RGB) image is first separated into its constituent channel images, and then the numerical intensity value of a pixel from each channel is written as a sum of smaller pixel intensity sub-values, leading to having several component images for each of the R, G, and B-channel images. A homomorphic encryption function is used to separately encrypted each of the pixel intensity sub-values in each component image using an encryption key, leading to a distributed image encryption approach. Each of the encrypted component images can be compressed before transmission and/or storage. In our decryption phase, each encrypted component image is decompressed if necessary, and then the homomorphic property of the encryption function is used to transform the product of individually encrypted pixel intensity sub-values in each encrypted component images, to the encryption of their sum, before applying the corresponding decryption function with a decryption key to recover the original pixel's intensity values for each channel image, and then recovering the original RGB image. Furthermore, a special case of an RGB image encryption and decryption where a pixel's intensity value from each channel is written as a sum of only two sub-values is implemented and simulated with a software. The resulting cipher-images are subject to a range of security tests and analyses. Results from these tests shown that our proposed homomorphic image encryption scheme is robust and can resist security attacks, as well as increases the security of the associated encrypted images. Our proposed homomorphic image encryption scheme has produced highly secure encrypted images.

Somewhat Homomorphic Encryption (SHE) schemes allow to carry out operations on data in the cipher domain. In a cloud computing scenario, personal information can be processed secretly, inferring a high level of confidentiality. Most practical Somewhat Homomorphic Encryption (SHE) schemes require the implementation of fast polynomial arithmetic, that is why hardware accelerators usually target the FFT/NTT algorithm. This paper proposes a co-design hardware/software approach to accelerate SHE using Karatsuba algorithm. Depending on the needs, Karatsuba algorithm allows to implement additional computations to the hardware in order to reduce software computation time. Our accelerator is designed to speed up arithmetic on degree 2560 polynomials with 125 bits coefficients. We provide 3 different approaches: An area efficient design, a balanced design, and a performance-oriented design. Our accelerator performs a polynomial multiplication in respectively 2.46 ms, 1.70 ms and 1.24 ms, and a relinearization operation in 2.28 ms, 1.53 ms and 1.1 ms, while a functionally equivalent design using the FFT [1] performs the multiplication in 1.96 ms and the relinearization in 4.79 ms for hardware resources consumption equivalent to the balanced design.

Multi-key Homomorphic Encryption (MKHE) scheme can homomorphically evaluate ciphertexts encrypted by different keys, which can effectively protect the privacy information of data holders in the joint computing of cloud services. Since the first full Homomorphic encryption scheme was proposed, bootstrapping is the only way to realize the arbitrary depth homomorphic computation of MKHE schemes. But bootstrap operation is quite expensive. In order to implement fast bootstrapping in MKHE schemes, previous works proposed multi-key TFHE schemes to implement low-latency bootstrapping and output a univariate function of messages after bootstrapping, called Programmable Bootstrapping (PBS). However, these schemes can only encrypt single-bit messages. PBS only outputs a function. And after a homomorphic operation, a bootstrap is required, which undoubtedly results in an increase in the cost of the whole multi-key homomorphic encryption operation. In this paper, we propose a MKHE scheme for multi-output PBS. For this purpose, we study the encryption method and homomorphic operation steps of MKHE, and add BFV homomorphic encryption multiplication and multi-key ciphertext relinearization. We separate the homomorphic operation from bootstrapping. We homomorphically evaluate test polynomials for multiple functions. In contrast to previous MKHE schemes, we support the output of multiple message-related functions with a single bootstrapping operation on the ciphertext. It is no longer limited to encrypting single-bit plaintext, and an effective ciphertext packaging technology is added. According to the analysis given in this paper, it is known that in the scenario of multi-party joint computation, the proposed scheme can be implemented with less bootstrapping when the same number of functions are homomorphically operated. This will effectively reduce the computational overhead.

This paper discusses how to implement Paillier homomorphic encryption (HE) scheme in Java as an API. We first analyze existing Pailler HE libraries and discuss their limitations. We then design a comparatively accomplished and efficient Pailler HE Java library. As a proof of concept, we applied our Pailler HE library in an electronic voting system that allows the voting server to sum up the candidates' votes in the encrypted form with voters remain anonymous. Our library records an average of only 2766ms for each vote placement through HTTP POST request.

Hierarchical identity-based fully homomorphic encryption (HIBFHE) scheme is a powerful scheme, as it aggregates the advantages of both fully homomorphic encryption and hierarchical identity-based encryption systems. In recent years, the construction of HIBFHE schemes were mainly based on lattices due to their conjectured resistance against quantum cryptanalysis, however, which makes these cryptosystems further unpractical. The first hierarchical identity-based fully homomorphic encryption scheme was presented by Gentry, Sahai and Waters (CRYPTO 2013). Their scheme however works with a not well performed trapdoor and delegation algorithm; that is, the trapdoor is conceptually and algorithmically complex, and the delegation algorithm’s performance is sensitive with the lattice dimension. In this work, we substantially improve their work by using a novel trapdoor function and its relevant algorithms. Specifically, we first use that construct an efficient algorithm for sampling-invertible matrix, based on this we construct a novel delegation algorithm which can keep the lattice dimension unchanged upon delegation. Building on this result, we first construct a more efficient hierarchical identity-based encryption scheme, and then transform it to HIBFHE scheme by using eigenvector method. Under the hardness of Learning with Errors problem, the resulting scheme can be proven secure in the standard model. To the best of our knowledge, this is the first HIBFHE scheme in fixed dimension.

Many applications deployed to public clouds are concerned about the confidentiality of their outsourced data, such as financial services and electronic patient records. A plausible solution to this problem is homomorphic encryption (HE), which supports certain algebraic operations directly over the ciphertexts. The downside of HE schemes is their significant, if not prohibitive, performance overhead for data-intensive workloads that are very common for outsourced databases, or database-as-a-serve in cloud computing. The objective of this work is to mitigate the performance overhead incurred by the HE module in outsourced databases. To that end, this paper proposes a radix-based parallel caching optimization for accelerating the performance of homomorphic encryption (HE) of outsourced databases in cloud computing. The key insight of the proposed optimization is caching selected radix-ciphertexts in parallel without violating existing security guarantees of the primitive/base HE scheme. We design the radix HE algorithm and apply it to both batch- and incremental-HE schemes; we demonstrate the security of those radix-based HE schemes by showing that the problem of breaking them can be reduced to the problem of breaking their base HE schemes that are known IND-CPA (i.e. Indistinguishability under Chosen-Plaintext Attack). We implement the radix-based schemes as middleware of a 10-node Cassandra cluster on CloudLab; experiments on six workloads show that the proposed caching can boost state-of-the-art HE schemes, such as Paillier and Symmetria, by up to five orders of magnitude.

Homomorphic encryption was introduced to enable untrusted parties to apply computation over encrypted data, without decrypting the message. It makes a promising future to solve security challenges with Cloud Computing. Still, because of complex problems, HE is not yet usable in real-world applications. However homomorphic encryption suffers from many complex issues like large-time computation, which is difficult to implement, and huge ciphertext size. In this paper, we take into account many vital homomorphic encryption schemes, all of them is depend on computing over integers, by converting the message to a binary format, then the ciphertext is calculated to every single bit of plaintext. So, we suggested another symmetric Fully Homomorphic encryption that depends on an integer number without converting it to binary format, it encrypts the message using a prime secret key. The proposed system is considered a great improvement to the DGHV system. Where it reduced the ciphertext size and the execution time significantly. The present proposed system shows an improvement of DGHV where it is faster about 23.8 times from DGHV, and we gain one ciphertext for the whole message instead of ciphertext for every bit of the message. The scheme's hardness depends on the greatest common divisor (GCD) problem. We present a detailed explanation of these schemes showing their efficient techniques and compare it with the proposed algorithm in terms of ciphertext size of algorithm, and execution time and analyze their security them finally draw several conclusions.

At present study, homomorphic encryption scheme is most focusing on algorithm efficiency and security and the rare for homomorphic encryption simulation research. This paper for the Gentry's Somewhat Homomorphic Encryption scheme for the simulation research, in clear text size within a certain range simulation was Somewhat Homomorphic Encryption scheme, and presents the relationship between the length of plaintext and ciphertext size.