Chosen Ciphertext Secure Keyed-Homomorphic Public-Key Encryption

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 has become a viable method for privacy preservation. It basically deals with the computations of encoded data. Homomorphic encryption scheme allows operations on encrypted data which, when decrypted will provide the same results as we performed directly on the raw data. Certain cryptographic algorithms having the homomorphic property. A detailed survey is carried out on the various homomorphic encryption scheme based on the parameter involved, the encryption decryption mechanisms, their hammerphic properties, security considerations etc. The encryption methods with homomorphic property are much suitable for applications require data privacy and security.

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.

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.

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.

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.

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.

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.

To simultaneously achieve CCA security and homomorphic property for encryption, Emura et al. introduced a new cryptographic primitive named keyed-homomorphic encryption, in which homomorphic ciphertext manipulations can only be performed by someone holding a devoted evaluation key which, by itself, does not enable decryption. A keyed-homomorphic encryption scheme should provide CCA2 security when the evaluation key is unavailable to the adversary and remain CCA1-secure when the evaluation key is exposed. While existing keyed-homomorphic encryption schemes only allow simple computations on encrypted data, our goal is to construct CCA-secure keyed-fully homomorphic encryption keyed-FHE capable of evaluating any functions on encrypted data with an evaluation key. In this paper, we first introduce a new primitive called convertible identity-based fully homomorphic encryption IBFHE, which is an IBFHE with an additional transformation functionality, and define its security notions. Then, we present a generic construction of CCA-secure keyed-FHE from IND-sID-CPA-secure convertible IBFHE and strongly EUF-CMA-secure signature. Finally, we propose a concrete construction of IND-sID-CPA-secure convertible IBFHE, resulting in the first CCA-secure keyed-FHE scheme in the standard model.

This chapter discusses two popular homomorphic public key encryption (PKE) techniques, i.e., Paillier PKE and Boneh-Goh-Nissim (BGN) PKE, which serve as the preliminary for building most privacy-enhancing aggregation techniques in the rest chapters of this monograph. In addition, the Java source codes of the two homomorphic PKEs are also provided for the interested readers to better understand and implement them. In order to attain privacy-preserving data aggregation in smart grid communications, we need to first understand the homomorphic public key encryption (HPKE) techniques [1, 2, 3, 4, 5, 6, 7]. Different from the general public key encryption algorithms [8, 9], HPKE further holds an additional “homomorphic” property, which makes the privacy-preserving data aggregation possible. As shown in Fig. 2.1, when we directly operate over two encrypted data E(x) and E(y) with some operation “•”, we can gain E(x ∘ y) = E(x)•E(y). In most HPKE cases, the operations “•” and “∘” are respectively referred to the common multiplication “×” and addition “+” operations. Because most of privacy-enhancing aggregation techniques illustrated in this monograph are based on Paillier public key encryption [2] and Boneh-Goh-Nissim (BGN) public key encryption [6], in this chapter, we first take a close look at these two popular homomorphic encryption techniques. Note that, both of them are randomized encryption algorithms [1], i.e., in addition to the plaintext as the input of encryption, a random number is also input for achieving semantic security.

KDM $$[\mathcal {F}]$$ -CCA secure public-key encryption PKE protects the security of message fsk, with $$f\in \mathcal {F}$$ , that is computed directly from the secret key, even if the adversary has access to a decryption oracle. An efficient KDM $$[\mathcal {F}_{\text {aff}}]$$ -CCA secure PKE scheme for affine functions was proposed by Lu, Li and Jia LLJ, EuroCrypt2015. We point out that their security proof cannot go through based on the DDH assumption. In this paper, we introduce a new concept Authenticated Encryption with Auxiliary-Input $$\mathsf {AIAE}$$ and define for it new security notions dealing with related-key attacks, namely IND-RKA security and weak INT-RKA security. We also construct such an $$\mathsf {AIAE}$$ w.r.t. a set of restricted affine functions from the DDH assumption. With our $$\mathsf {AIAE}$$ ,we construct the first efficient KDM $$[\mathcal {F}_{\text {aff}}]$$ -CCA secure PKE w.r.t. affine functions with compact ciphertexts, which consist only of a constant number of group elements;we construct the first efficient KDM $$[\mathcal {F}_{\text {poly}}^d]$$ -CCA secure PKE w.r.t. polynomial functions of bounded degree d with almost compact ciphertexts, and the number of group elements in a ciphertext is polynomial in d, independent of the security parameter. Our PKEs are both based on the DDH & DCR assumptions, free of NIZK and free of pairing.

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.

Homomorphic encryption technique is a form of encryption which allows specific types of computations to be performed on encrypted data and generates an encrypted result. The decrypted result of any operation is the same such as working directly on raw data. The ability to perform computations on ciphertexts makes homomorphic encryption a widely adopted technique for ensuring confidentiality of both storage and treatment of outsourced data into a third party. Indeed, the RSA is the first asymmetric-key encryption scheme with a homomorphic property. Unfortunately, many attacks can threaten the encrypted data confidentiality of this scheme. In this paper, we will address the standard RSA drawbacks by suggesting an enhanced encryption scheme of it, which keeps a homomorphic property and resists more the well-known attacks against RSA.

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

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.