Research on θ -Intuitionistic Fuzzy Homomorphism Theory

Fuzzy homomorphism is an important research content of fuzzy group theory, different fuzzy mappings will produce different fuzzy homomorphisms. In this paper, the fuzzy homomorphism of groups is generalized. Firstly, the θ-intuitionistic fuzzy mapping is defined, and the θ-intuitionistic fuzzy homomorphism of groups is obtained. The properties of intuitionistic fuzzy subgroups and intuitionistic fuzzy normal subgroups are studied under the θ-intuitionistic fuzzy homomorphism of groups, and the fundamental theorem of θ-intuitionistic fuzzy homomorphism is proved.

Research on Data Privacy Protection Algorithm with Homomorphism Mechanism Based on Redundant Slice Technology in Wireless Sensor Networks

Wireless transmission method in wireless sensor networks has put forward higher requirements for private protection technology. According to the packet loss problem of private protection algorithm based on slice technology, this paper proposes the data private protection algorithm with redundancy mechanism, which ensures privacy by privacy homomorphism mechanism and guarantees redundancy by carrying hidden data. Moreover,it selects the routing tree generated by CTP(Collection Tree Protocol) as routing path for data transmission. By dividing at the source node, it adds the hidden information and also the privacy homomorphism. At the same time,the information feedback tree is established between the destination node and the source node. In addition, the destination node immediately sends the packet loss information and the encryption key via the information feedback tree to the source node. As a result,it improves the reliability and privacy of data transmission and ensures the data redundancy.

The Homomorphism and Isomorphism on Hypergroup

Basing on the papers from [1] to [4], this paper gives some further research,mainly solves the following problems:(1) It proved that several theorems of subgroup which have been raised to the hypergroup are still true.(2) It proved that the isomorphous relationship of the bottomgroups which guid to the hypergroup can still keep such relationship.(3) It proved that the basic isomorphous Theorem and the homogenous Theorem of the bottomgroup which raise to the hypergroup are still true.(4) It point out that when the isomorphous bottomgroups have been raised to the hypergroup,the conditions which form the isomorphism should be lessened.

Exact Sequence and Commutative Diagrams of Semimodule Homomorphism

Throught the introduction of the concept * exact sequence,we set up several satisfied results of commutative diagrams of semimodule homomorphism.

ON INDUCED GROUP HOMOMORPHISMS

In this paper, the induced group homomorphism was studied. It is proved that for any ideal I of a ring R contained in J(R), K 0(π):K 0(R)→K 0(R/I) is isomorphic if and only if K 0(π) + is a semigroup isomorphism; characterizations are given for the semilocal rings being semiperfect.

HOMOMORPHISMS BETWEEN MULTIPLICATIVE SEMIGROUPS OF MATRICES OVER FIELDS

Suppose F is a field, and n, p are integers with 1 ≤ p 〈 n. Let Mn（F） be the multiplicative semigroup of all n × n matrices over F, and let M^Pn（F） be its subsemigroup consisting of all matrices with rank p at most. Assume that F and R are subsemigroups of Mn（F） such that F M^Pn（F）. A map f ： F→R is called a homomorphism if f（AB） = f（A）f（B） for any A, B ∈F. In particular, f is called an endomorphism if F = R. The structure of all homomorphisms from F to R （respectively, all endomorphisms of Mn（F）） is described.

ALMOST HOMOMORPHISMS BETWEEN UNITAL C^＊-ALGEBRAS： A FIXED POINT APPROACH

Let A, B be two unital C^＊-algebras. By using fixed pint methods, we prove that every almost unital almost linear mapping h ： A →B which satisfies h（2^nuy） = h（2^nu）h（y） for all u ∈ U（A）, all y ∈ A, and all n = 0,1,2,..., is a homomorphism. Also, we establish the generalized Hyers-Ulam-Rassias stability of ,-homomorphisms on unital C^＊-algebras.

On Homomorphism of Valuation Algebras

In this paper, firstly, a necessary condition and a sufficient condition for an isomorphism between two semiring-induced valuation algebras to exist are presented respectively. Then a general valuation homomorphism based on different domains is defined, and the corresponding homomorphism theorem of valuation algebra is proved.

Homomorphism Key Agreement Based on RSA

This paper reviews the basic content of key management, points out the limitation of the existing key agreement protocols. It introduces RSA public key cryptosystems in detail and homomorphie encryption scheme in the integer ring. The homomorphism key agreement based on RSA is presented. Compared with Diffie-Hellman protocol, its operation speed is more quickly; Compared with key agreement protocol based on password, it has better security. The protocol can use the public key of RSA to create the contributory, forward secrecy the session key. Finally, the security of the protocol is proved by BAN logic.

STABILITY AND SUPERSTABILITY OF JORDAN HOMOMORPHISMS AND JORDAN DERIVATIONS ON BANACH ALGEBRAS AND C~＊-ALGEBRAS： A FIXED POINT APPROACH

Using fixed point methods, we prove the Hyers–Ulam–Rassias stability and superstability of Jordan homomorphisms （Jordan ＊-homomorphisms）, and Jordan derivations （Jordan ＊-derivations） on Banach algebras （C＊-algebras） for the generalized Jensen–type functional equationwhere r is a fixed positive real number in （1, ∞）.

A Local Characterization of Lie Homomorphisms of Nest Algebras

In this paper,linear maps preserving Lie products at zero points on nest algebras are studied.It is proved that every linear map preserving Lie products at zero points on any finite nest algebra is a Lie homomorphism.As an application,the form of a linear bijection preserving Lie products at zero points between two finite nest algebras is obtained.

A HYBRID FIXED POINT RESULT FOR LIPSCHITZ HOMOMORPHISMS ON QUASI-BANACH ALGEBRAS

We shall generalize the results of [9] about characterization of isomorphisms on quasi-Banach algebras by providing integral type conditions. Also, we shall give some new results in this way and finally, give a result about hybrid fixed point of two homomorphisms on quasi-Banach algebras.

The Maximum Size of an Edge Cut and Graph Homomorphisms

For a graph G, let b(G)=max﹛|D|: Dis an edge cut of G﹜ . For graphs G and H, a map Ψ: V(G)→V(H) is a graph homomorphism if for each e=uv∈E(G), Ψ(u)Ψ(v)∈E(H). In 1979, Erd?s proved by probabilistic methods that for p ≥ 2 with if there is a graph homomorphism from G onto Kp then b(G)≥f(p)|E(G)| In this paper, we obtained the best possible lower bounds of b(G) for graphs G with a graph homomorphism onto a Kneser graph or a circulant graph and we characterized the graphs G reaching the lower bounds when G is an edge maximal graph with a graph homomorphism onto a complete graph, or onto an odd cycle.

Applications of Homomorphism on the Structure of Semigroups

By utilizing homomorphisms and -strong semilattice of semigroups, we show that the Green (*,~)-relation H*,~ is a regular band congruence on a r-ample semigroup if and only if it is a G-strong semilattice of completely J*,^-simple semigroups. The result generalizes Petrich’s result on completely regular semigroups with Green’s relation H a normal band congruence or a regular band congruence from the round of regular semigroups to the round of r-ample semigroups.

Enhancing IoT Data Security with Lightweight Blockchain and Okamoto Uchiyama Homomorphic Encryption

Blockchain technology has garnered significant attention from global organizations and researchers due to its potential as a solution for centralized system challenges.Concurrently,the Internet of Things(IoT)has revolutionized the Fourth Industrial Revolution by enabling interconnected devices to offer innovative services,ultimately enhancing human lives.This paper presents a new approach utilizing lightweight blockchain technology,effectively reducing the computational burden typically associated with conventional blockchain systems.By integrating this lightweight blockchain with IoT systems,substantial reductions in implementation time and computational complexity can be achieved.Moreover,the paper proposes the utilization of the Okamoto Uchiyama encryption algorithm,renowned for its homomorphic characteristics,to reinforce the privacy and security of IoT-generated data.The integration of homomorphic encryption and blockchain technology establishes a secure and decentralized platformfor storing and analyzing sensitive data of the supply chain data.This platformfacilitates the development of some business models and empowers decentralized applications to perform computations on encrypted data while maintaining data privacy.The results validate the robust security of the proposed system,comparable to standard blockchain implementations,leveraging the distinctive homomorphic attributes of the Okamoto Uchiyama algorithm and the lightweight blockchain paradigm.

Complexity of Injective Homomorphisms to Small Tournaments, and of Injective Oriented Colourings

Several possible definitions of local injectivity for a homomorphism of an oriented graph G to an oriented graph H are considered. In each case, we determine the complexity of deciding whether there exists such a homomorphism when G is given and H is a fixed tournament on three or fewer vertices. Each possible definition leads to a locally-injective oriented colouring problem. A dichotomy theorem is proved in each case.

FL-EASGD:Federated Learning Privacy Security Method Based on Homomorphic Encryption

Federated learning ensures data privacy and security by sharing models among multiple computing nodes instead of plaintext data.However,there is still a potential risk of privacy leakage,for example,attackers can obtain the original data through model inference attacks.Therefore,safeguarding the privacy of model parameters becomes crucial.One proposed solution involves incorporating homomorphic encryption algorithms into the federated learning process.However,the existing federated learning privacy protection scheme based on homomorphic encryption will greatly reduce the efficiency and robustness when there are performance differences between parties or abnormal nodes.To solve the above problems,this paper proposes a privacy protection scheme named Federated Learning-Elastic Averaging Stochastic Gradient Descent(FL-EASGD)based on a fully homomorphic encryption algorithm.First,this paper introduces the homomorphic encryption algorithm into the FL-EASGD scheme to preventmodel plaintext leakage and realize privacy security in the process ofmodel aggregation.Second,this paper designs a robust model aggregation algorithm by adding time variables and constraint coefficients,which ensures the accuracy of model prediction while solving performance differences such as computation speed and node anomalies such as downtime of each participant.In addition,the scheme in this paper preserves the independent exploration of the local model by the nodes of each party,making the model more applicable to the local data distribution.Finally,experimental analysis shows that when there are abnormalities in the participants,the efficiency and accuracy of the whole protocol are not significantly affected.

A Method of Homomorphic Encryption

The existing homomorphie eneryption scheme is based on ring of the integer, and the possible operators are restricted to addition and multiplication only. In this paper, a new operation is defined Similar Modul. Base on the Similar Modul, the number sets of the homomorphic encryption scheme is extended to real number, and the possible operators are extended to addition, subtraction, multiplication and division. Our new approach provides a practical ways of implementation because of the extension of the operators and the number sets.

The Homomorphic Key Agreement

There are various challenges that are faced in group communication, so it is necessary to ensure session key. Key agreement is the fundamental cryptographic primitive for establishing a secure communication. It is a process of computing a shared secret contributed by two or more entities such that no single node can predetermine the resulting value. An authenticated key agreement is attained by combining the key agreement protocol with digital signatures. After a brief introduction to existing key agreement in group communication, Making use of the additive-multiplicative homomorphism in the integer ring defined by Sander and Tschudin: A new protocols, called the homomorphism key agreement, was designed, which can be self-contributory, robust, scalable and applicable in group communication.

A Fully Homomorphic Encryption Scheme with Better Key Size

Fully homomorphic encryption is faced with two problems now. One is candidate fully homomorphic encryption schemes are few. Another is that the efficiency of fully homomorphic encryption is a big question. In this paper, we propose a fully homomorphic encryption scheme based on LWE, which has better key size. Our main contributions are： （1） According to the binary-LWE recently, we choose secret key from binary set and modify the basic encryption scheme proposed in Linder and Peikert in 2010. We propose a fully homomorphic encryption scheme based on the new basic encryption scheme. We analyze the correctness and give the proof of the security of our scheme. The public key, evaluation keys and tensored ciphertext have better size in our scheme. （2） Estimating parameters for fully homomorphic encryption scheme is an important work. We estimate the concert parameters for our scheme. We compare these parameters between our scheme and Bral2 scheme. Our scheme have public key and private key that smaller by a factor of about logq than in Bral2 scheme. Tensored ciphertext in our scheme is smaller by a factor of about log2q than in Bral2 scheme. Key switching matrix in our scheme is smaller by a factor of about log3q than in Bra12 scheme.