Full Homomorphism Research Articles

Ensuring privacy and confidentiality of data on the cloud using an enhanced homomorphism scheme

Cloud computing is one of the widest phenomena embraced in information technology. This result from numerous advantages associated with it making many organizations and individuals offload their data to the cloud. Encryption schemes restrict access to data from unauthorized clients, helping attain confidentiality and privacy. The modification of the ciphertext of clients’ data on the cloud demand downloading, deciphering, editing, and finally uploading back to the cloud by sharing their private key with the cloud service provider making it tedious. The application of homomorphism, allows computation to be performed on ciphertext with no decipher activity which helps to avoid the surfacing of sensitive client data stored on the cloud. In this paper, an Enhanced Homomorphism Scheme (EHS) is proposed based on Good Prime Numbers (GPN), Linear Congruential Generator (LCG), Fixed Sliding Window Algorithm (FSWA), and Gentry’s homomorphism scheme. A dataset from the Kaggle database was used to test the proposed algorithm. A variety of tests were conducted using the proposed algorithm such as the Uniqueness of ciphertext, addition and multiplication property of full homomorphism, and the execution times using 2^n (n ∈ 2,3,4,5) data sizes. A comparison of the execution time of the proposed EHS was conducted with the New Fully Homomorphism Scheme (NFHS), and the Enhanced Homomorphism Encryption Scheme (EHES). From the comparison, the proposed EHS algorithm had the lowest encryption time when a data size of 24kb was executed but with a higher decryption time of 567.6667 ± 96.38911when a data size of 8kb was used. On the other hand, with a data size of 32kb, EHES had the highest decryption time of 1274ms with the proposed EHS having the lowest decryption time of 551.2222 ± 82.68746 indicating a decryption percentage decrease of 56.73%. This confirms that execution times are dependent on the size of the encryption key but not on data size.

A fully homomorphism encryption scheme based on LWR

We believe that isomorphic encryption technology can provide strong technical support for users’ privacy protection in a distributed computing environment. There are three types of quasi-homomorphism encryption methods: partial homomorphism encryption, shallow homomorphism encryption, and full homomorphism encryption. homomorphism encryption methods have important applications for ciphertext data computing in distributed computing environments, such as secure cloud computing, fee computing, and remote file storage ciphertext retrieval. It is pointed out that the construction of the homomorphism encryption method is still in the theoretical stage and cannot be used for real high-density data calculation problems. How to design (natural) isomorphic encryption schemes according to algebraic systems is still a challenging research. This question discusses the problem of Learning With Rounding (LWR). Based on the difficulty of LWR, multiple IDs, and attribute categories, a fully homomorphism encryption method corresponding to an ID is proposed. In this paper, in order to reflect the effectiveness of the proposed method, we propose a homomorphism encryption technology based on the password search attribute.

Increasing System Security with Full Homo-morphic Encryption using Lagrange’s Functions

Homo-morphic systems present a novel way to encrypt data, via which operations performed on the encrypted data are fully/partially reflected in the decrypted data. For example, if the input data D is homo-morphically encrypted to E, and if we perform X=E+N, then while decrypting X, we are bound to get D+N. Such a kind of encryption helps data owners to safely share their data among different analysers, without the risk of any kind of data leakages. Data analysers usually perform mathematical operations on the data which include addition, subtraction, multiplication and division. Homo-morphic systems which support addition & subtraction but do not support multiplication and division are termed as partially homo-morphic systems. While systems which support all the operations are termed as fully homo-morphic systems. In this paper, we have implemented a fully homo-morphic system based on Lagrange’s functions. These functions help in improving the overall security of the system by adding stochastics to the input data, which ensures that the same input data has different cipher text, and full homo-morphism is achieved. Security of overall system increased by 40% by using FHE.

Gaps in full homomorphism order

We characterise gaps in the full homomorphism order of finite graphs.

Cryptoleq: A Heterogeneous Abstract Machine for Encrypted and Unencrypted Computation

The rapid expansion and increased popularity of cloud computing comes with no shortage of privacy concerns about outsourcing computation to semi-trusted parties. Leveraging the power of encryption, in this paper, we introduce Cryptoleq: an abstract machine based on the concept of one instruction set computer, capable of performing general-purpose computation on encrypted programs. The program operands are protected using the Paillier partially homomorphic cryptosystem, which supports addition on the encrypted domain. Full homomorphism over addition and multiplication, which is necessary for enabling general-purpose computation, is achieved by inventing a heuristically obfuscated software re-encryption module written using Cryptoleq instructions and blended into the executing program. Cryptoleq is heterogeneous, allowing mixing encrypted and unencrypted instruction operands in the same program memory space. Programming with Cryptoleq is facilitated using an enhanced assembly language that allows the development of any advanced algorithm on encrypted data sets. In our evaluation, we compare Cryptoleq's performance against a popular fully homomorphic encryption library, and demonstrate correctness using a typical private information retrieval problem.

A novel fully homomorphic encryption scheme bsed on LWE

Although the learning with errors(LWE)-based full homomorphic encryption scheme was the first example of deviation from the original Gentry’s blueprint, the scheme did not give detailed conversion process of circuit layer structure, and must rely on bootstrapping technique to achieve full homomorphism. Therefore, through modifying the re-linearization technique proposed by the above scheme, a technique called non-matrix key switching is presented, which includes key switching with re-linearization and pure key switching. The complex matrix operations of existing key switching technique are removed. Combining this technique with modulus switching, a (leveled) fully homomorphic encryption scheme without bootstrapping from LWE is constructed. In order to make circuit layer structure clear, the scheme gives detailed refresh door operation. Finally, we use bootstrapping to upgrade arithmetic circuit to any layer, and make the homomorphic computing capability of the scheme have nothing to circuit depth.

Point determining digraphs, [formula omitted]-matrix partitions, and dualities in full homomorphisms

A digraph D is point determining if for any two distinct vertices u,v there exists a vertex w which has an arc to (or from) exactly one of u,v. We prove that every point-determining digraph D contains a vertex v such that D−v is also point determining. We apply this result to show that for any {0,1}-matrix M, with k diagonal zeros and ℓ diagonal ones, the size of a minimal M-obstruction is at most (k+1)(ℓ+1). This is a best possible bound, and it extends the results of Sumner, and of Feder and Hell, from undirected graphs and symmetric matrices to digraphs and general matrices.

Connected obstructions to full graph homomorphisms

Minimal obstructions to full homomorphisms to a graph B have been proved to be of size at most |B|+1. This turns out to require that disconnected obstructions be allowed. In this paper we prove that the size of minimal connected obstructions is at most |B|+2. We also prove that achieving |B|+2 is rare and present a complete list of the exceptional cases. Finally, we compute the dualities associated with these exceptions.

Dualities in full homomorphisms

In this paper we study dualities of graphs and, more generally, relational structures with respect to full homomorphisms, that is, mappings that are both edge- and non-edge-preserving. The research was motivated, a.o., by results from logic (concerning first order definability) and Constraint Satisfaction Problems. We prove that for any finite set of objects B (finite relational structures) there is a finite duality with B to the left. It appears that the surprising richness of these dualities leads to interesting problems of Ramsey type; this is what we explicitly analyze in the simplest case of graphs.

Colored graphs without colorful cycles

A colored graph is a complete graph in which a color has been assigned to each edge, and a colorful cycle is a cycle in which each edge has a different color. We first show that a colored graph lacks colorful cycles iff it is Gallai, i.e., lacks colorful triangles. We then show that, under the operation mon ≡ m + n − 2, the omitted lengths of colorful cycles in a colored graph form a monoid isomorphic to a submonoid of the natural numbers which contains all integers past some point. We prove that several but not all such monoids are realized.We then characterize exact Gallai graphs, i.e., graphs in which every triangle has edges of exactly two colors. We show that these are precisely the graphs which can be iteratively built up from three simple colored graphs, having 2, 4, and 5 vertices, respectively. We then characterize in two different ways the monochromes, i.e., the connected components of maximal monochromatic subgraphs, of exact Gallai graphs. The first characterization is in terms of their reduced form, a notion which hinges on the important idea of a full homomorphism. The second characterization is by means of a homomorphism duality.

On realizations of point determining graphs, and obstructions to full homomorphisms

A graph is point determining if distinct vertices have distinct neighbourhoods. A realization of a point determining graph H is a point determining graph G such that each vertex-removed subgraph G - x which is point determining, is isomorphic to H. We study the fine structure of point determining graphs, and conclude that every point determining graph has at most two realizations. A full homomorphism of a graph G to a graph H is a vertex mapping f such that for distinct vertices u and v of G, we have uv an edge of G if and only if f ( u ) f ( v ) is an edge of H. For a fixed graph H, a full H-colouring of G is a full homomorphism of G to H. A minimal H-obstruction is a graph G which does not admit a full H-colouring, such that each proper induced subgraph of G admits a full H-colouring. We analyse minimal H-obstructions using our results on point determining graphs. We connect the two problems by proving that if H has k vertices, then a graph with k + 1 vertices is a minimal H-obstruction if and only if it is a realization of H. We conclude that every minimal H-obstruction has at most k + 1 vertices, and there are at most two minimal H-obstructions with k + 1 vertices. We also consider full homomorphisms to graphs H in which loops are allowed. If H has ℓ loops and k vertices without loops, then every minimal H-obstruction has at most ( k + 1 ) ( ℓ + 1 ) vertices, and, when both k and ℓ are positive, there is at most one minimal H-obstruction with ( k + 1 ) ( ℓ + 1 ) vertices. In particular, this yields a finite forbidden subgraph characterization of full H-colourability, for any graph H with loops allowed.

Homomorphisms of Moufang-Veldkamp planes

We give here a complete description of full homomorphisms between Moufang-Veldkamp planes in terms of the algebra of their coordinate rings. This generalizes known results for projective planes and projective ring planes.

A DESCRIPTION OF A SET OF HYPOTHESES FOR SYSTEM IDENTIFICATION PROBLEM*

Abstract This paper concerns the system identification process which is a specific form of the hypothetico-deductive process. More specifically, this paper deals with the inductive inference, i.e., with the process of generating a set of hypotheses σ(E) that explain a given finite set of input-output experiments performed on a finite sequential system being identified. It is shown that Seσ(E) if and only if there exists a homomorphism of a basic hypothesis explaining E, into S. Next, a set of hypotheses σ1(E). defined as follows: Seσ1(E) if E is structure-complete w.r.t. S, is considered. Then it is proved that Seσ1(E) if and only if there exists a full homomorphism of a basic hypothesis onto S. Some important methodological consequences of obtained results are derived. Finally, relationship linking the properties of a system identification algorithm is investigated.