Fast polynomial arithmetic for Somewhat Homomorphic Encryption operations in hardware with Karatsuba algorithm

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.

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.)

We present a computationally secure MPC protocol for threshold adversaries which is parametrized by a value L. When L = 2 we obtain a classical form of MPC protocol in which interaction is required for multiplications, as L increases interaction is reduced, in that one requires interaction only after computing a higher degree function. When L approaches infinity one obtains the FHE based protocol of Gentry, which requires no interaction. Thus one can trade communication for computation in a simple way. Our protocol is based on an interactive protocol for “bootstrapping” a somewhat homomorphic encryption (SHE) scheme. The key contribution is that our presented protocol is highly communication efficient enabling us to obtain reduced communication when compared to traditional MPC protocols for relatively small values of L.

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.

Deep learning (DL) has shown outstanding performances in many fields, including remote sensing (RS). DL is turning into an essential tool for the RS research community. Recently, many cloud platforms have been developed to provide access to large-scale computing capacity, consequently permitting the usage of DL architectures as a service. However, this opened the door to new challenges associated with the privacy and security of data. The RS data used to train the DL algorithms have several privacy requirements. Some of them need a high level of confidentiality, such as satellite images related to public security with high spatial resolutions. Moreover, satellite images are usually protected by copyright, and the owner may strictly refuse to share them. Therefore, privacy-preserving deep learning (PPDL) techniques are a possible solution to this problem. PPDL enables training DL on encrypted data without revealing the original plaintext. This study proposes a hybrid PPDL approach for object classification for very-high-resolution satellite images. The proposed encryption scheme combines Paillier homomorphic encryption (PHE) and somewhat homomorphic encryption (SHE). This combination aims to enhance the encryption of satellite images while ensuring a good runtime and high object classification accuracy. The method proposed to encrypt images is maintained through the public keys of PHE and SHE. Experiments were conducted on real-world high-resolution satellite images acquired using the SPOT6 and SPOT7 satellites. Four different CNN architectures were considered, namely ResNet50, InceptionV3, DenseNet169, and MobileNetV2. The results showed that the loss in classification accuracy after applying the proposed encryption algorithm ranges from 2% to 3.5%, with the best validation accuracy on the encrypted dataset reaching 92%.

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.

Outsourcing encrypted data to the cloud demands efficient and confidentiality-preserving search techniques, yet existing Searchable Symmetric Encryption (SSE) schemes suffer from severe access and search pattern leakage, and the problem is more intractable in multi-data-owner scenarios where cross-dataset privacy-preserving search remains unsolved. This paper proposes PSSE-MDO (Privacy-Preserving SSE with Pattern Obfuscation for Multiple Data Owners), the first SSE scheme that achieves robust access and search pattern hiding for multi-data-owner settings with constant client-side overhead. Our core innovation is a cryptographic protocol enabling users to issue a single constant-sized search token (trapdoor) for searching across all data owners' encrypted indexes. The scheme integrates Inner-Product Predicate Encryption (IPE) and Somewhat Homomorphic Encryption (SHE) to probabilistically obfuscate search results for each query. A key experimental result shows that the client-side cost approximately 120 ms for search token generation—remains constant regardless of the number of data owners, in contrast to existing pattern-hiding schemes with linearly scaling costs. This validates PSSE-MDO's practicality for real-world large-scale collaborative cloud systems, marking a significant step toward practical privacy-preserving search in federated cloud environments.

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.

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.

With the increasingly powerful and extensive deployment of edge devices, edge/fog computing enables customers to manage and analyze data locally, and extends computing power and data analysis applications to network edges. Meanwhile, as the next generation of the power grid, the smart grid can achieve the goal of efficiency, economy, security, reliability, use safety and environmental friendliness for the power grid. However, privacy and secure issues in fog-based smart grid communications are challenging. Without proper protection, customers’ privacy will be readily violated. This article presents a smart and practical Privacy-preserving Data Aggregation (PDA) scheme with smart pricing and packing method for fog-based smart grids, which achieves diversified tariffs, multifunctional statistics and efficiency. Especially, we first propose a smart PDA scheme with Smart Pricing (PDA-SP). With PDA-SP, the Control Center (CC) can compute more complex and higher-order aggregation statistics to provide various services, provide diversiform pricing strategies and choose a double-winning strategy. Subsequently, we put forward a practical PDA scheme with Packing Method (PDA-PM), which is able to reduce the size of encrypted data and improve performance in performing various secure computations. Moreover, we extend our original packing method and present a more useful packing method, which can handle general vectors with large entries. The security analysis shows that our proposed scheme is secure against many threats. The performance evaluation reveals that the computation and communication overheads of our proposed scheme are effectively reduced by employing the Somewhat Homomorphic Encryption (SHE), and our packing method can further significantly reduce these overheads.

The research presented in this paper aims to establish functional mockup units (FMU) co-simulation methods to simulate and evaluate encrypted dynamic systems using some-what homomorphic encryption (SHE). The proposed approach encrypts the entire dynamic system expressions, including: model parameters, state variables, feedback gains, and sensor signals, and perform computation in the ciphertext space to simulate dynamic behaviors or generate motion commands to servo systems. The developed FMU co-simulation helps analyze the relationship between security parameters and performance. Two illustrative examples are presented and analyzed: 1) encrypted Duffing oscillator and 2) encrypted teleoperation. How the time delay due to FMU co-simulation affects the refresh rate is also reported.

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.

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.

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.

Emotion recognition has been an actively researched topic in the field of HCI. However, multimodal datasets used for emotion recognition often contain sensitive personal information, such as physiological signals, facial images, and behavioral patterns, raising significant privacy concerns. In particular, the privacy issues become crucial in workplace settings because of the risks such as surveillance and unauthorized data usage caused by the misuse of collected datasets. To address this issue, we propose an Encrypted Emotion Recognition (EER) framework that performs real-time inference on encrypted data using the CKKS homomorphic encryption (HE) scheme. We evaluated the proposed framework using publicly available WESAD and Hide-and-seek datasets, demonstrating successful stress/emotion recognition under encryption. The results demonstrated that encrypted inference achieved similar accuracy to plaintext inference, with accuracy of 0.966 (plaintext) vs. 0.967 (ciphertext) on the WESAD dataset, and 0.868 for both cases on the Hide-and-Seek dataset. Encrypted inference was performed on a GPU, with average inference times of 333 milliseconds for the general model and 455 milliseconds for the personalized model. Furthermore, we validated the feasibility of semi-supervised learning and model personalization in encrypted environments, enhancing the framework's real-world applicability. Our findings suggest that the EER framework provides a scalable, privacy-preserving solution for emotion recognition in domains such as healthcare and workplace settings, where securing sensitive data is of critical importance.