Public-Key Cryptosystems Based on Composite Degree Residuosity Classes

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.

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.)

Identity‐based fully homomorphic encryption (IBFHE) provided a fundamental solution to the problem of huge public key size that exposed in fully homomorphic encryption (FHE) schemes, thus it is significant to make FHE become more practical. In recent years, the construction of IBFHE schemes were mainly based on lattices due to their conjectured resistance against quantum cryptanalysis, however, which makes these cryptosystems further unpractical. The main reason is the trapdoor function on which the scheme was based is rather complex for practical and the ciphertext size is too large. In this study, the authors propose an efficient batch IBFHE scheme, which can be proven secure from the standard LWE assumption in the standard model. The first contribution of this work is that the authors construct an efficient batch version of MP12 preimage sampling algorithm, which can efficiently generate identity keys for multi‐bit IBE schemes. Based on that, the authors construct an asymptotically‐faster multi‐bit IBE scheme as the second contribution. The third contribution is that the authors transform the multi‐bit IBE scheme to batch IBFHE scheme which supports to encrypt any message in . Compared with the similar schemes, the authors show their schemes are essentially improved.

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.

Aiming at the problem that the fully homomorphic encryption scheme based on single identity cannot satisfy the homomorphic operation of ciphertext under different identities, as well as the inefficiency of trapdoor function and the complexity of sampling algorithm, an improved lattice MIBFHE scheme was proposed. Firstly, we combined MP12 trapdoor function with dual LWE algorithm to construct a new IBE scheme under the standard model, and prove that the scheme is IND-sID-CPA security under the selective identity. Secondly, we used the eigenvector method to eliminate the evaluation key, and transform the above efficient IBE scheme into a single identity IBFHE scheme to satisfy the homomorphic operation. Finally, we improved the ciphertext extension method of CM15 and constructed a new Link-mask system that supports the transformation of IBFHE scheme under the standard model, and then, converted the above IBFHE scheme into MIBFHE scheme based on this system. The comparative analysis results showed that the efficiency of this scheme is improved compared with similar schemes in the trapdoor generation and preimage sampling, and the dimension of lattice and ciphertext size are significantly shortened.

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.

Lossy trapdoor functions (LTDF) and all-but-one trapdoor functions (ABO-TDF) are fundamental cryptographic primitives. And given the recent advances in quantum computing, it would be much desirable to develop new and improved lattice-based LTDF and ABO-TDF. In this work, we provide more compact constructions of LTDF and ABO-TDF based on the learning with errors (LWE) problem. In addition, our LWE-based ABO-TDF can allow smaller system parameters to support super-polynomially many injective branches in the construction of CCA secure public key encryption. As a core building tool, we provide a more compact homomorphic symmetric encryption schemes based on LWE, which might be of independent interest. To further optimize the ABO-TDF construction, we employ the full rank difference encoding technique. As a consequence, the results presented in this work can substantially improve the performance of all the previous LWE-based cryptographic constructions based upon LTDF and ABO-TDF.

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.

Lossy trapdoor functions, introduced by Peikert and Waters (STOC’08), have received a lot of attention in the last years, because of their wide range of applications. The notion has been recently extended to the identity-based setting by Bellare et al. (Eurocrypt’12). An identity-based trapdoor function (IB-TDF) satisfying the lossy property introduced by Bellare et al. can be used to construct other cryptographic primitives in the identity-based setting: encryption schemes with semantic security under chosen-plaintext attacks, deterministic encryption schemes, and hedged encryption schemes that maintain some security when messages are encrypted using randomness of poor quality. However, the constructed primitives can be proved secure only against selective adversaries who select the target identity upfront.Our first contribution is an alternative definition for the lossiness of an identity-based trapdoor function. We prove that an IB-TDF satisfying the new property can be used to construct all the aforementioned primitives, in the identity-based setting, with security against adaptive adversaries. We further consider the new definition and its implications in the more general scenario of hierarchical identity-based cryptography, which has proved very useful both for practical applications and to establish theoretical relations with other cryptographic primitives (including encryption with chosen-ciphertext security or with forward-security).As a second contribution, we describe a pairing-based hierarchical IB-TDF satisfying the new definition of lossiness against either selective or, for hierarchies of constant depth, adaptive adversaries. This is also the first example of hierarchical trapdoor functions based on traditional (i.e., non-lattice-related) number theoretic assumptions. As a direct consequence of our two contributions, we obtain a hierarchical identity-based (HIB) encryption scheme with chosen-plaintext security, a HIB deterministic encryption scheme and a HIB hedged encryption scheme, all of them with security against adaptive adversaries.

In this paper we consider the problem of securely instantiating Probabilistic Signature Scheme (PSS) in the standard model. PSS, proposed by Bellare and Rogaway [3] is a widely deployed randomized signature scheme, provably secure (unforgeable under adaptively chosen message attacks) in Random Oracle Model.Our main result is a black-box impossibility result showing that one can not prove unforgeability of PSS against chosen message attacks using blackbox techniques even assuming existence of ideal trapdoor permutations (a strong abstraction of trapdoor permutations which inherits all security properties of a random permutation, introduced by Kiltz and Pietrzak in Eurocrypt 2009) or the recently proposed lossy trapdoor permutations [20]. Moreover, we show onewayness, the most common security property of a trapdoor permutation does not suffice to prove even the weakest security criteria, namely unforgeability under zero message attack. Our negative results can easily be extended to any randomized signature scheme where one can recover the random string from a valid signature.KeywordsPSSBlackbox ReductionsRandomized SignatureStandard Model

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.

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 cryptography has been one of the most interesting topics of mathematics and computer security since Gentry presented the first construction of a fully homomorphic encryption (FHE) scheme in 2009. Since then, a number of different schemes have been found, that follow the approach of bootstrapping a fully homomorphic scheme from a somewhat homomorphic foundation. All existing implementations of these systems clearly proved, that fully homomorphic encryption is not yet practical, due to significant performance limitations. However, there are many applications in the area of secure methods for cloud computing, distributed computing and delegation of computation in general, that can be implemented with homomorphic encryption schemes of limited depth. We discuss a simple algebraically homomorphic scheme over the integers that is based on the factorization of an approximate semiprime integer. We analyze the properties of the scheme and provide a couple of known protocols that can be implemented with it. We also provide a detailed discussion on searching with encrypted search terms and present implementations and performance figures for the solutions discussed in this paper.

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.