Homomorphic Property Research Articles

Homomorphic extensions of CRT-based secret sharing

In this work, we explore the homomorphic aspect of CRT-based secret sharing schemes. Secret sharing homomorphism is the notion of operating on multiple secrets by direct computation on shares. There are schemes based on polynomial interpolation which have partially homomorphic properties, whereas CRT-based secret sharing homomorphism is an open area.Because of the disparate structure of the CRT-based schemes, we introduce advanced security notion and analyze the existing schemes. We formulate homomorphic inadequacy caused by the overflow problem and present sufficient and necessary homomorphism conditions. Then, we show the impossibility of homomorphic and secure threshold Asmuth–Bloom scheme while keeping the original structure. Accordingly, we propose possible homomorphic extensions to the Asmuth–Bloom SSS.Our first extension, additively homomorphic ramp scheme, can attain arbitrarily large information rate. Besides, it is the only CRT-based scheme possessing perfect secrecy. The second scheme allows homomorphic addition and also multiplication to some point. The bound on addition operations is correlated with the share size of the scheme, whereas multiplication bound is inversely proportional to the secrecy threshold.We give detailed analyses of the extensions and their security proofs as well as their properties like information rate, security characteristic, and homomorphic capabilities.

On Commutators of Fuzzy Multigroups

Fuzzy multigroup is an application of fuzzy multiset to group theory. Although, a lots have been done on the theory of fuzzy multigroups, some group's theoretic notions could still be investigated in fuzzy multigroup context. Certainly, the idea of commutator is one of such group's theoretic notions yet to be studied in the environment of fuzzy multigroups. Hence, the aim of this article is to establish the notion of commutator in fuzzy multigroup setting. A number of some related results are obtained and characterized. Among several results that are obtained, it is established that, if $A$ and $B$ are fuzzy submultigroups of a fuzzy multigroup $C$, then $[A, B]\subseteq A\cup B$ holds. Some homomorphic properties of commutator in fuzzy multigroup context are discussed. The notion of admissible fuzzy submultisets $A$ and $B$ of $C\in FMG(X)$ under an operator domain $\mathcal{D}$ is explicated, and it is shown that $(A,B)$ and $[A,B]$ are $\mathcal{D}$-admissible.

Fuzzy BCK-module on t-norm

In this paper, the concept of fuzzy BCK-module on t-norm is introduced. We consider properties of intersection, sum and homomorphisms for fuzzy BCK-modules.

Lightweight privacy-Preserving data classification

Internal attacks are of a huge concern, because they are usually delicately masqueraded under harmless-looking activities, which are very difficult to detect. Machine learning techniques have been successfully applied to identify insider threats. However, they may violate user privacy since they can legally access user’s sensitive information. To preserve user privacy, encryption algorithms have been lately exploited as a powerful tool, to hide private data in a multiple-party collaboration. A combination of encryption and data mining techniques raises high computational complexity. Hence, in order to improve the system’s performance while securing both user’s private data and the classifier, we propose a new secure data analysis protocol, namely SmartClass, by adopting the garbled circuit technique to speed-up the system performance. We developed an efficient encryption step that exploits the additive homomorphism and best properties of the binary Elliptic Curve Cryptography (ECC) algorithm, while keeping the protocol highly secure. We implemented the proposed system and study its effectiveness. Experimental results show the proposed approach is very promising.

Fully Homomorphic Symmetric Key Encryption-Based Access Control over Outsourced Cloud Data Using Smith Normal Form

Cloud computing is an innovative technology to support outsourcing of computing resources on-demand. But it is lack of security, confidentiality, and privacy. This paper attempts to propose a solution to the above-said problems by a secure, confidential and Fully Homomorphic Encryption technique with the desired level of access control. The present scheme is based on symmetric keys and Smith Normal Form assures the homomorphic property. All the operations are simple matrix multiplications and consequently, the present scheme sustains the owners of the cloud to secure their private data from its uncertain clients.

Elliptic Curve and Homomorphic ElGamal Hybrid Based Mutual Authentication Framework for Security in Cloud Computing

Cloud computing is an extensive technology from which the client could access several services through a remote server. Authentication of the remote services needs a common key between the client and the server in a secured manner. The existing key agreement protocols utilized various techniques for the preservation of the data security. The proposed work attempted to provide a high secure data by optimizing the key generation for encryption and decryption process. Hybridization of a novel ECC (Elliptic Curve Cryptography) algorithm and homomorphic ElGamal algorithm for encryption and decryption and also employment of a bio-inspired Genetic algorithm for the generation of maximum secured key is performed. A random 128 bit hash have been developed for the generation of two input points to run the homomorphic property in the ElGamal algorithm secures the key and make it as a non breakable one. The integrated property of ECC and the ElGamal associated with the Genetic algorithm makes the key more stabilized and a confidential one that is more resistant to the various kinds of attacks. The performance analysis depicted that the proposed technique outperform the existing with respect to computational time.

Highly Secure Privacy-Preserving Outsourced k-Means Clustering under Multiple Keys in Cloud Computing

Data clustering is the unsupervised classification of data records into groups. As one of the steps in data analysis, it has been widely researched and applied in practical life, such as pattern recognition, image processing, information retrieval, geography, and marketing. In addition, the rapid increase of data volume in recent years poses a huge challenge for resource-constrained data owners to perform computation on their data. This leads to a trend that users authorize the cloud to perform computation on stored data, such as keyword search, equality test, and outsourced data clustering. In outsourced data clustering, the cloud classifies users’ data into groups according to their similarities. Considering the sensitive information in outsourced data and multiple data owners in practical application, it is necessary to develop a privacy-preserving outsourced clustering scheme under multiple keys. Recently, Rong et al. proposed a privacy-preserving outsourced k-means clustering scheme under multiple keys. However, in their scheme, the assistant server (AS) is able to extract the ratio of two underlying data records, and key management server (KMS) can decrypt the ciphertexts of owners’ data records, which break the privacy security. AS can even reduce all data records if it knows one of the data records. To solve the aforementioned problem, we propose a highly secure privacy-preserving outsourced k-means clustering scheme under multiple keys in cloud computing. In this paper, noncolluded cloud computing service (CCS) and KMS jointly perform clustering over the encrypted data records without exposing data privacy. Specifically, we use BCP encryption which has additive homomorphic property and AES encryption to double encrypt data records, where the former cryptosystem prevents CCS from obtaining any useful information from received ciphertexts and the latter one protects data records from being decrypted by KMS. We first define five protocols to realize different functions and then present our scheme based on these protocols. Finally, we give the security and performance analyses which show that our scheme is comparable with the existing schemes on functionality and security.

Continuous Homomorphisms Defined on (Dense) Submonoids of Products of Topological Monoids

We study the factorization properties of continuous homomorphisms defined on a (dense) submonoid S of a Tychonoff product D = ∏ i ∈ I D i of topological or even topologized monoids. In a number of different situations, we establish that every continuous homomorphism f : S → K to a topological monoid (or group) K depends on at most finitely many coordinates. For example, this is the case if S is a subgroup of D and K is a first countable left topological group without small subgroups (i.e., K is an NSS group). A stronger conclusion is valid if S is a finitely retractable submonoid of D and K is a regular quasitopological NSS group of a countable pseudocharacter. In this case, every continuous homomorphism f of S to K has a finite type, which means that f admits a continuous factorization through a finite subproduct of D. A similar conclusion is obtained for continuous homomorphisms of submonoids (or subgroups) of products of topological monoids to Lie groups. Furthermore, we formulate a number of open problems intended to delimit the validity of our results.

Storage and Communication Security in Cloud Computing Using a Homomorphic Encryption Scheme Based Weil Pairing

With introduction of smart things into our lives, cloud computing is used in many different areas and changes the communication method. However, cloud computing should guarantee the complete security assurance in terms of privacy protection, confidentiality, and integrity. In this paper, a Homomorphic Encryption Scheme based on Elliptic Curve Cryptography (HES-ECC) is proposed for secure data transfer and storage. The scheme stores the data in the cloud after encrypting them. While calculations, such as addition or multiplication, are applied to encrypted data on cloud, these calculations are transmitted to the original data without any decryption process. Thus, the cloud server has only ability of accessing the encrypted data for performing the required computations and for fulfilling requested actions by the user. Hence, storage and transmission security of data are ensured. The proposed public key HES-ECC is designed using modified Weil-pairing for encryption and additional homomorphic property. HES-ECC also uses bilinear pairing for multiplicative homomorphic property. Security of encryption scheme and its homomorphic aspects are based on the hardness of Elliptic Curve Discrete Logarithm Problem (ECDLP), Weil Diffie-Hellman Problem (WDHP), and Bilinear Diffie-Helman Problem (BDHP).

Achieving Searchable Encryption Scheme With Search Pattern Hidden

Searchable Encryption (SE) enables a data owner to outsource encrypted data to an untrusted server while preserving the keyword search functionality. Typically, the server learns whether or not a query has been performed more than once, which is usually called the search pattern. However, such kind of information leakage might be leveraged to break query privacy. To further reduce such type of leakage and provide strong privacy guarantee, Wang <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al.</i> proposed a novel SE scheme based on the Paillier encryption scheme in INFOCOM’15. Unfortunately, their scheme cannot perform keyword search successfully, because the additive homomorphic property is not sufficient for their construction. In this article, we first show that why their scheme fails to return the correct search result, and then propose a new SE scheme by adopting a special additive homomorphic encryption scheme to achieve the multiplicative homomorphic property efficiently. Furthermore, we enhance the security on the user side. Specifically, we use random polynomials with an appropriate degree to guarantee that the user cannot learn anything other than the desired search result. Finally, we present a formal security analysis and implement our scheme on a real-world database, which demonstrates that our construction can achieve the desired security properties with good performance.

Axiomatic Approach in the Analytic Theory of Singular Perturbations

Introduced by S.A. Lomov, the concept of a pseudoanalytic (pseudoholomorphic) solution laid the foundation for the development of the singular perturbation analytical theory. In order for this concept to work in case of linear problems, an apparatus for the theory of exponential type vector spaces was developed. When considering nonlinear singularly perturbed problems, an algebraic approach is currently used. This approval is based on the properties of algebra homomorphisms for holomorphic functions with various numbers of variables, as a result of which it is possible to obtain pseudoholomorphic solutions. In this paper, formally singularly perturbed equations are considered in topological algebras, which allows the authors to formulate the main concepts of the singular perturbation analytical theory from the standpoint of maximal generality.

Homomorphic Encryption-Based LSB Substitution for High Capacity Data Hiding in the Encrypted Domain

During the last few decades, multimedia security over the cloud has become a major issue. Public-key homomorphism is an efficient approach for data hiding in encrypted images (DHEI). An original image is encrypted using a public key and sent across a network and then processed to embed a secret message directly in the encrypted domain. During the decoding step, a private key is used to obtain a marked reconstructed image, where the secret message can be extracted. In this paper, we propose an efficient method of DHEI based on the Paillier cryptosystem. Using its homomorphic properties, pixel blocks and bits of the message are multiplied in the encrypted domain resulting in an addition in the clear domain. By applying a pre-processing step on the original image before encryption, this addition becomes a least significant bits (LSB) substitution. Experimental results show that using our proposed scheme, we obtain a high payload value ( $1~bpp$ ) without expanding a lot the original image size contrary to current state-of-the-art methods. Indeed, whatever the key size, the expansion rate is always equal to 2. In addition, the original image and the marked reconstructed image are very similar.

Secure hash algorithm-based multiple tenant user security over green cloud environment

This paper proposes a green cloud multi-tenant trust authentication with secure hash algorithm-3 (GreenCloud-MTASHA3) scheme to eliminate the unauthorised tenant access. GreenCloud-MTASHA3 scheme provide security over the multiple tenant requests by referring the confidentiality, integrity and availability rate. Confidentiality refers to limiting the unauthorised tenant's green cloud data access using the additive homomorphic privacy property in proposed scheme. Additive homomorphic privacy property-based encryption function is developed to improve the privacy preserving level. To attain the integrity level between the tenant requests and green cloud server machine in GreenCloud-MTASHA3 scheme an encrypted trust data management process is carried out. Trustworthiness of tenant request is measured to maintain the consistency level on security with minimal computational time. The proposed scheme attains the confidentiality, integrity and availability rate on communicating task. Experiment is conducted on factors such as secure computation confidence, authorised tenant computational time and space taken on storing encrypted data.

Properties of Discrete Dynamical System in BCI-Algebra

In the present manuscript we introduce the concept of some notions such as fixed points, periodic points, invariant set and strongly invariant or S-invariant set of discrete dynamical system (Z, Ψ) in BCI-algebra where in (Z, Ψ), Z is a non-empty set and supposed to be a BCI-algebra and the mapping Ψ is a homomorphism from Z to Z and establish some new homomorphic properties of BCI-algebra based on these notions. We also prove some new results related to the set that contains the all fixed points and to the set that contains all periodic points in Z such that we prove that the set of all fixed points and the set of all periodic points in BCI-algebra Z are the BCI-sub algebras. We show that when a sub set of BCI-algebra Z is an invariant set with respect to Ψ. We prove that the set of all fixed points and the set of all periodic points in p-semisimple BCI-algebra Z are the ideals of Z. We also prove that the set of all fixed points in Z is an S-invariant subset of a BCI-algebra Z. We have no doubt that the research along this line can be kept up, and indeed, some results in this manuscript have already made up a foundation for further exploration concerning the further progression of a discrete dynamical system in BCI-algebra and their applications in other disciplines of algebra.

Homomorphic Cryptosystems for Data Security in Cloud Storage

Cloud Computing enables users to use remote resources thus reduces the burden on local storage. However, the use of such services gives rise to new set of problems. The users have no control over the data which they have stored on those storages so to achieve data authentication with confidentiality is utmost important. As every user may not have that expertise so they can request for data verification task to Trusted Verifier (TV) which will be an authorized party to check the intactness of outsourced data. Since the data owner stores the data on the cloud in an encrypted format, it becomes difficult to check the integrity of the data without decrypting. But by using homomorphic encryption schemes the integrity checking can be made possible without original copy. In this paper, we have given implementation and performance details of two homomorphic encryption schemes, Rivest Shamir Adleman (RSA) and Paillier. The RSA is multiplicative homomorphic scheme where the Paillier is additive homomorphic scheme. Both the algorithms are partially homomorphic thus limited in their functions. Due to homomorphic property of these algorithms, original contents will not get revealed in the verification process. This framework will achieve authentication of data by maintaining confidentiality.

Характеристические теории и обращение подстановки

&#x0D; Матрицей Линденбаума некоторой логики называют логическую матрицу, заданную на алгебре всех формул этой логики и имеющую в качестве выделенного множества некоторую ее теорию. Множество всех таких матриц называют расслоением Линденбаума. Расслоение Линденбаума некоторой логики образует характеристическое для этой логики множество логических матриц. Однако известно достаточно много логик, например классическая, которые характеризуются одной матрицей.&#x0D; &#x0D; В работе получен ряд критериев того, что логика имеет характеристическую матрицу. Показано, что вопрос наличия такой матрицы связан с операцией обращения подстановки для определенного вида подстановок. Здесь под операцией обращения подстановки подразумеваем взятие прообраза некоторого множества формул для некоторой подстановки.&#x0D; &#x0D; Получен следующий критерий наличия характеристической матрицы для данной логики: для некоторой логики существует не более чем счетная характеристическая матрица тогда и только тогда, когда существует такая теория, для которой матрица Линденбаума является характеристической. На основании этого факта введено понятие характеристической теории.&#x0D; &#x0D; Как следует из Леммы Сушко, для любой логики результат обращения подстановки некоторой теории является теорией. В данной работе показано, что теория $T$ является характеристической тогда и только тогда, когда любая непротиворечивая теория является пересечением прообразов теории $T$, взятых по всем подстановкам, которые вкладывают эту теорию в $T$.&#x0D; Таким образом, получается, что логика имеет характеристическую матрицу тогда и только тогда, когда существует такая теория $T$, что любая непротиворечивая теория является пересечением прообразов теории $T$, взятых по всем вкладывающим подстановкам.&#x0D; &#x0D; В качестве иллюстрации рассматривается вопрос о характеристических теориях классической логики. Доказано, что любая полная теория классической логики является характеристической.&#x0D; Замечено, что в классической логике любая полная теория является результатом обращения некоторой подстановки, примененной к некоторой характеристической теории. Приведен пример такой подстановки для случая, когда характеристическая теория является полной.&#x0D; &#x0D; Доказано, что минимальная матрица Линденбаума классической логики является характеристической, то есть множество всех тавтологий является характеристической теорией классической логики.&#x0D; &#x0D;

Fuzzy filters of pseudo-BE algebras

The theory of fuzzy filters in pseudo-BE algebras is developed. Various characterizations of fuzzy filters are given. It is proved that the set of all fuzzy filters of a pseudo-BE algebra is a complete lattice. Some characterizations of Noetherian pseudo-BE algebras by fuzzy filters are obtained. Finally, fuzzy commutative filters are defined and studied. Moreover, the homomorphic properties of fuzzy (commutative) filters are provided.

Reversible data hiding in homomorphically encrypted image using interpolation technique

In this paper, a reversible data hiding scheme based on interpolation technique for encrypted images by using homomorphic and probabilistic properties of Paillier cryptosystem is presented. At first, the image owner generates a location map by using an interpolation technique to estimate the Most Significant Bits (MSBs) of pixel to find whether a pixel can be used for embedding or not. Next, with the help of the location map, the original image is preprocessed to create some spare space for data embedding. Meanwhile, the location map is compressed losslessly and the information of compressed location map is substituted with Least Significant Bits (LSBs) of border pixels. Furthermore, the preprocessed image is encrypted by Paillier cryptosystem and sent to the data hider along with location map and original LSBs of border pixels. At the data hiding phase (note that original image cannot be accessed), the additional data and LSBs of border pixels are embedded into homomorphic encrypted image using location map. At the receiver side, the embedded additional data and original image are recovered losslessly in non-separable manner (i.e., from directly decrypted image). Experimental results demonstrate feasibility and efficiency of the proposed scheme, particularly in embedding, image recovery and performance on different security key size.

Factoring Continuous Homomorphisms Defined on Submonoids of Products of Topologized Monoids

We study factorization properties of continuous homomorphisms defined on submonoids of products of topologized monoids. We prove that if S is an ω-retractable submonoid of a product D = ∏ i ∈ I D i of topologized monoids and f : S → H is a continuous homomorphism to a topologized semigroup H with ψ ( H ) ≤ ω , then one can find a countable subset E of I and a continuous homomorphism g : p E ( S ) → H satisfying f = g ∘ p E ↾ S , where p E is the projection of D to ∏ i ∈ E D i . The same conclusion is valid if S contains the Σ -product Σ D ⊂ D . Furthermore, we show that in both cases, there exists the smallest by inclusion subset E ⊂ I with the aforementioned properties.

On Companion B-algebras

This study introduces the concept of companion \emph{B}-algebra and establishes some of its properties. Also, this paper introduces the notions of $\odot$-subalgebra and $\odot$-ideal of a companion \emph{B}-algebra and investigates their relationship. Furthermore, this study establishes some homomorphic properties of $\odot$-subalgebra and $\odot$-ideal.