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.

Rings Over Which Every Simple Module is Rationally Complete

In 1958, G. D. Findlay and J. Lambek defined a relationship between three R-modules, A ≦ B(C), to mean that A ⫅ B and every R-homomorphism from A into C can be uniquely extended to an irreducible partial homomorphism from B into C. If A ≦ B(B), then B is called a rational extension of A and in [5] it is shown that every module has a maximal rational extension in its injective hull which is unique up to isomorphism. A module is called rationally complete provided it has no proper rational extension.

The translational hull of an inverse semigroup

An ideal extension of one semigroup by another is determined by a partial homomorphism into the translational hull of the first semigroup [3, §2, Theorem 5]. In most ins tances, the development of the theory of ideal extensions has been hindered by inadequate knowledge of the translational hull; it is our purpose here to characterize certain basic structures in the translational hull of an arbitrary inverse semigroup.

The translational hull of a completely 0-simple semigroup

The translational hull Ω(S) of a semigroup S plays an important role in the theory of ideal extensions of semigroups. In fact, every ideal extension of S by a semigroup T with zero can be constructed using a certain partial homomorphism of T\0 into Ω(S); a particular case of interest is when S is weakly reductive (see §4.4 of [3], [2], [7]). A theorem of Gluskin [6, 1.7.1] states that if S is a weakly reductive semigroup and a densely embedded ideal of a semigroup Q, then Q and Ω(S) are isomorphic. A number of papers of Soviet mathematicians deal with the abstract characteristic (abstract semigroup, satisfying certain conditions, isomorphic to the given semigroup) of various classes of (partial) transformation semigroups in terms of densely embedded ideals (see, e.g., [4]). In many of the cases studied, the densely embedded ideal in question is a completely 0-simple semigroup, so that Gluskin's theorem mentioned above applies. This enhances the importance of the translational hull of a weakly reductive, and in particular of a completely 0-simple semigroup. Gluskin [5] applied the theory of densely embedded ideals (which are completely 0-simple semigroups) also to semigroups and rings of endomorphisms of a linear manifold and to certain classes of abstract rings.

A Generalized Ring of Quotients II

A partial homomorphism ϕ from B into A will be called a fractional homomorphism if dom ϕ ≤ B (A). Every extension of a fractional homomorphism is again fractional, and each fractional homomorphism has a unique irreducible extension.