I-Hop Homomorphic Encryption and Rerandomizable Yao Circuits

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.)

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.

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.

Homomorphic encryption is a novel encryption method for it enables computing over encrypted data. This has a wide range of real world ramifications such as being able to blindly compute a search result sent to a remote server without revealing it's content. In this paper we summarize how SQL queries can be made secure using a homomorphic encryption scheme based on the ideas of Gahi, et al [1]. Gahi's model is based on the DGHV scheme which is a homomorphic encryption scheme which acts on plaintext bits and produce corresponding ciphertext. We use Gahi's blueprint to propose an improved model which is based on the more recent Ring based homomorphic encryption scheme by Braserski, et al [2]. Our method is more general in the sense that it can be extended to use with any fully homomorphic encryption scheme rather than being restricted to the DGHV scheme as in Gahi's method.

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.

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.

The ability to perform computations on ciphertexts without knowing any information about the plaintexts makes homomorphic encryption technique useful in a wide variety of confidentiality preserving protocols (e.g., e-voting, e-health, etc.). Unfortunately, the traditional encyption schemes support a limited number of homomorphic operations (usually addition or multiplication). Indeed, in 2009, Gentry proposed a fully homomorphic encryption scheme which supports both multiplicative and additive homomorphic operations. Since then, several fully homomorphic encryption schemes have been proposed. However, the fully homomorphic encryption schemes are still undergoing experimentation and improvement. The hybridization of homomorphic encryption schemes seems to be an effective way to overcome their limitations and to benefit from their resistance against the confidentiality attacks. In this paper, we will study the possibility to hybridize the homomorphic encryption schemes so as to support all homomorphic properties.

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.

When users store their data on a cloud, they may concern on whether their data is stored correctly and can be fully retrieved. Proofs of Retrivability (PoR) is a cryptographic concept that allows users to remotely check the integrity of their data without downloading. This check is usually done by attaching data with message authenticators that contain data integrity information. The existing PoR schemes consider only the retrievability of unencrypted data and their message authenticators are usually deterministic. In this paper, we propose a PoR scheme that is built over homomorphic encryption schemes. Our PoR scheme can prove the retrievability of homomorphically encrypted data by generating probabilistic and homomorphic message authenticators. Moreover, the homomorphically encrypted data can be processed by the cloud directly and our PoR scheme can verify the integrity of such outsourced computations over ciphertexts. A prototype of our scheme is implemented to evaluate its performance.

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.

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.

We consider the setting in which an untrusted server stores a collection of data and is asked to compute a function over it. In this scenario, we aim for solutions where the untrusted server does not learn information about the data and is prevented from cheating. This problem is addressed by verifiable and private delegation of computation, proposed by Gennaro, Gentry and Parno (CRYPTO’10), a notion that is close to both the active areas of homomorphic encryption and verifiable computation (VC). However, in spite of the efficiency advances in the respective areas, VC protocols that guarantee privacy of the inputs are still expensive. The only exception is a protocol by Fiore, Gennaro and Pastro (CCS’14) that supports arithmetic circuits of degree at most 2. In this paper we propose new efficient protocols for VC on encrypted data that improve over the state of the art solution of Fiore et al. in multiple aspects. First, we can support computations of degree higher than 2. Second, we achieve public delegatability and public verifiability whereas Fiore et al. need the same secret key to encode inputs and verify outputs. Third, we achieve a new property that guarantees that verifiers can be convinced about the correctness of the outputs without learning information on the inputs. The key tool to obtain our new protocols is a new SNARK that can efficiently handle computations over a quotient polynomial ring, such as the one used by Ring-LWE somewhat homomorphic encryption schemes. This SNARK in turn relies on a new commit-and-prove SNARK for proving evaluations on the same point of several committed polynomials. We propose a construction of this scheme under an extractability assumption over bilinear groups in the random oracle model.

Homomorphic encryption schemes allow to perform computations over encrypted data. In schemes based on RLWE assumption the plaintext data is a ring polynomial. In many use cases of homomorphic encryption only the \(\text {degree-}0\) coefficient of this polynomial is used to encrypt data. In this context any computation on encrypted data can be performed. It is trickier to perform generic computations when more than one coefficient per ciphertext is used.

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.

RETRACTED: A public key size homomorphic encryption scheme based on the sum of sparse subsets and integers