Homomorphism on bipolar-valued fuzzy sub-bigroup of a bigroup for secured data transmission over WSN

Cryptography field, which enables secure communication between civilians, governmental organizations, military forces, and many more, addresses security, confidentiality, and integrity of information being conveyed regardless of the medium used. Protection of priceless information resources on intranets, the Internet, and the cloud has become a vital demand of contemporary electronic security systems. In this paper, we deploy homomorphic data encryption (HDE) technique on Bipolar-valued Fuzzy Sub-Bigroups (BVFSB). Intuitionistic Fuzzy Sets were presented by Atanassov in 1986, as a modification of Fuzzy Sets, by taking into consideration the grade of membership value and non-membership value for each element in the universe. A fuzzy algebraic structure known as Bipolar-valued Fuzzy sub-bigroup of a bigroup has been established by applying the concept of bipolar-valued Fuzzy sets to bigroups. Some of the theorems related to homomorphism and anti-homomorphism, are stated and proved.


I Introduction
The cyber-physical systems are used frequently in several applications like smart home systems, equipment diagnostics during production, security systems, health care, military applications, etc.The development of IT technology is accelerating over time.Often, wireless sensor networks serve as the systems' obvious base.Sensor networks are wireless networks made up of multiple tiny sensors that are dispersed throughout space.Because wireless sensor networks cannot be protected using standard computer network security measures, unique security measures that are appropriate for sensor networks must be developed.One such measure is the use of homomorphic encryption algorithms.Digital communication is now more susceptible than ever to surveillance or hostile interference, such as hacking or eavesdropping.New techniques for secure transmission via unsecure channels are needed to ensure the security of sensitive data in applications including copyright protection, distant military communication, safe storage, authentication, and secure video conferencing.A unique class of cryptosystems called homomorphic cryptosystems keeps track of group operations carried out on the ciphertexts.The homomorphic characteristic of public key cryptosystems has been utilized in a large number of data security protocols, including electronic voting systems, bidding protocols, cashing systems, and asymmetric photo fingerprinting [1].
As an extension of Fuzzy sets [2], in 1994, W.R.Zhang [3,4] introduced 'Bipolar-valued Fuzzy sets' and was further developed by Lee [5,6].In a bipolar-valued Fuzzy set items with a membership degree of 0 are irrelevant to the corresponding property whereas those with a membership degree of 0 1 and −1 0] partially meet the property and the implicit counter property, respectively.Vasantha kandasamy.W.B [7] introduced the basic idea of fuzzy bigroup.M.S.Anitha et al. [8] introduced the bipolar-valued Fuzzy subgroup and A.Balasubramanian et al. [9] introduced intuitionistic fuzzy sub-bigroup of a bigroup.Justin Prabu.T and K.Arjunan [10] introduced Q-fuzzy subbigroup of a bigroup.In continuation with the works related to Bipolar-valued Fuzzy Sets, Sheena.K. P and K.Uma Devi [11] familiarized bipolar-valued Fuzzy sub-bigroup ( BVF SBG) of a bigroup by applying the concept of bipolar-valued Fuzzy sets, on bigroups.Bigroup is an algebraic structure consisting of two groups with respective group operations.
Abdulatif Alabdulati et al. [12] presented the lightweight homomorphic encryption algorithm in privacy-preserving cloud-based practical and secure billing system.The effectiveness, reliability, and adaptability of operating smart infrastructure could be considerably increased if sensors could be integrated with cloud-based data storage and processing.One of the most valuable features of this privacy-preserving system has the potential for secure transfer of billing management into the cloud, with on-demand data retrieval and statistical analyses.A scrambled image can then be safely obtained by the dealer.In fact, using this method, each player merely needs to deduct their own key image from the scrambled image to extract the secret image.

Objective
The objectives of this research are as follows: (1) The extracted shared secret data is extracted once the encryption and decryption processes have been completed in the proposed system.(2) With the help of the encryption technique known as homomorphic encryption, you can add and multiply cipher texts in order to get this output which match an outcome of similar one.(3) Encoding messages or information so that only authorized parties can read it is known as encryption in cryptography.(4) A message or piece of information, known as plaintext in an encryption scheme, is encrypted using an encryption algorithm to create cipher text that can only be decoded and read.(5) An encryption technique typically employs a pseudo-random encryption key produced by an algorithm for technical reasons.

Preliminaries
If there are two bipolar-valued Fuzzy subsets A 2 is a bipolar-valued Fuzzy subgroup of (G 2 , •), then the bipolar-valued Fuzzy set A of G is said to be a bipolar-valued Fuzzy sub-bigroup.
]): Let f : X → X be an onto function defined on X.Let A represent a fuzzy bipolar subset of X.Then, the image of A under f is indicated by f (A) = V, which is a bipolarvalued Fuzzy subset on f (X) = X and is defined by A − m (p) for all q in X .Let B be a fuzzy bi-polar valued subset of X .The pre-image of B which is denoted as

Theorems Theorem 2.1: Given two bigroups
Let f (a 1 ) and f (b 1 ) be in

Theorem 2.2: Given two bigroups
Let a 1 and b 1 be in Similarly, if a 1 and b 1 are in F 1 , then,

•) be any two bigroups. Then the anti-homomorphic preimage of a bipolar valued fuzzy sub-bigroup of G 2 is a bipolar valued fuzzy sub-bigroup of G
Let a 1 and b 1 be in Similarly, if a 1 and b 1 are in F 1 , then,

Theorem 2.5:
•) be any two bigroups.The isomorphic image of a bipolar valued fuzzy normal sub-bigroup of G 1 is a bipolar valued fuzzy normal sub-bigroup of G 2 .

Proof:
Hence f (A) is a bipolar valued fuzzy normal subbigroup of G 2 .

Theorem 2.6:
•) be any two bigroups.The homomorphic preimage of a bipolar valued fuzzy normal sub-bigroup of G 2 is a bipolar valued fuzzy normal sub-bigroup of G 1 .

Proof: Let
− m be a bipolar valued fuzzy sub-bigroup of G 2 .Then, by theorem 2.2, its homomorphic preimage Let a 1 and b 1 be in

•) be any two bigroups. The anti-isomorphic image of a bipolar valued fuzzy normal sub-bigroup of G 1 is a bipolar valued fuzzy normal sub-bigroup of G 2 .
Proof:

Theorem 2.9:
Proof: Let a 1 and b 1 be in G 1 and A be a bipolar valued fuzzy sub-bigroup of the bigroup G 2 .If a 1 and b 1 in H 1 , then,

Theorem 2.10:
Proof: Let a 1 and b 1 be in G 1 and A be a bipolar valued fuzzy sub-bigroup of the bigroup If a 1 and b 1 in H 1 , then, (M

Theorem 2.11:
Then we have the following: Proof: (i) Let A be a generalized characteristic bipolar valued fuzzy sub-bigroup of G 2 .By Theorem 2.9, Then we have the following: If a 1 and b 1 in Hence A • f is a generalized characteristic bipolar valued fuzzy sub-bigroup of G 1 .

The proposed encryption method
The extracted shared secret data is extracted once the encryption and decryption processes have been completed in the proposed system.With the help of the encryption technique known as homomorphic encryption, you can add and multiply ciphertexts in order to get this output which match an outcome of similar one.Without disclosing private information for each action, homomorphic encryption enables you to carry out a variety of tasks in an untrusted environment [14].Encoding messages or information so that only authorized parties can read it is known as encryption in cryptography.Although encryption does not by itself stop interceptions, it does prevent the interceptor from seeing the message's contents.A message or piece of information, known as plaintext in an encryption scheme, is encrypted using an encryption algorithm to create cipher text that can only be decoded and read.An encryption technique typically employs a pseudorandom encryption key produced by an algorithm for technical reasons.
In general, a cryptosystem provides a mechanism to use a secret key to convert one message, known as the plaintext, into another, known as the ciphertext.If the cryptosystem is trustworthy, the plaintext cannot be deciphered by anybody without the secret key, and the ciphertext can be safely made public.The Paillier Cryptosystem was found as a result of further investigation into trapdoor discrete logarithmbased cryptosystems that was prompted by the Okamoto-Uchiyama system.The Paillier cryptosystem uses a logarithm function L to decrypt ciphertext and is analogous to the Okamoto-Uchiyama cryptosystem, but it is implemented significantly differently.The Paillier cryptosystem enhances the Benaloh cryptosystem by taking advantage of the difficulty of selecting higher order residues modulo a composite n 2 where n = pq.
Then [w 1 ] g = x 1 and [w 2 ] g = x 2 and there exists y 1 and y 2 with w 1 = g x 1 y n 1 and and when the order of g is α n for α ∈ {1, ..., λ = lcm(p − 1, q − 1)}, g decides x for a given g x y n .So, using y as a randomizer, ε g can accept a message x and determine w such that [w] g = x.The class function serves as the Paillier cryptosystem's decryption function, whereas ε g serves as its encryption function.

Encryption step
When the original data, D 1 , . . ...D l , and secret data D x are encrypted will produce l + 1 encrypted data, C 1 , . . ., C l , and C x respectively.A new encrypted data is  created by incrementally multiplying one encrypted data by another encrypted data.Once the secret encrypted data D x is added, this new encrypted data is put into the homomorphic multiplication procedure to create the final encrypted data C y .All of the individually encrypted data must be used in order to safeguard the secret data [15][16][17][18][19].The final encrypted data may have two different sized blocks because two distinct encryption techniques are possible.The Paillier cryptosystem is used with the suggested strategy.It is given by The Paillier cryptosystem uses the n 2 basis for the modulo operation.The blocks C y (i) are used to create the encrypted image C y , and the decrypted data D x is the information we want to transport or exchange across an unsecure channel.

Decryption part
Our goal is to extract The process is entirely reversible using the Paillier cryptosystem; we can obtain the shared secret data without suffering any loss.

Results and discussion
These strategies are put into practice in a simulator designed specifically for WSNs.Because of the message overhead associated with cryptography, the network lifetime is decreased.Each cryptographic primitive has a unique CPU cycle time requirement, which affects how much energy is used during execution.
We contrast the communication costs between our plan and the other two plans.The outcomes are displayed in Figure 1.As a result, we can see that even in the absence of an attack, existing systems incur a very large overhead in transmission and calculation compared to our approach.
Figure 2 shows the energy used by our strategy as well as the energy used by each networked sensor node.

Conclusion
WSNs connect the processing and storage of the data provided by an application process on the appropriate servers.In this work, we looked at how to share a secret data by key-data with exploiting an extra property of homomorphic Pailier crypto system on a bipolarvaluedfuzzy sub-bigroup and the bipolarvaluedfuzzy normal sub-bi-group.The impact of homomorphism and isomorphism over various fuzzy algebraic structures has its own scope for the research work in this area.Still there are so many aspects of homomorphism and isomorphism which can be applied on Bipolar valued fuzzy structures.Here, we will be beneficial for the researchers for their future work in this area.Future work will include investigating and putting various Secured Data Transmission into use, as well as the impact of applying stream cyphers, and making a thorough comparison of the outcomes over WSNs.

Disclosure statement
No potential conflict of interest was reported by the authors.

Definition 1 . 3 ([ 8 ]Definition 1 . 4 :
): Let (G, * ) be a group.If N + m (a * b) = N + m (b * a), and N − m (a * b) = N − m (b * a) for every a and b in G, then the bipolar-valued Fuzzy subgroup N of G is said to be a bipolar-valued Fuzzy normal subgroup of G. Let (G = G 1 ∪ G 2 , +, •) be a bigroup.If there are two bipolar-valued Fuzzy subsets A 1 of G 1 and

Definition 3 . 1 :
A cryptosystem is a five-tuple (B, R, H, E, M), that meets the requirements listed below:• B -finite set of possible plaintexts; • R -finite set of ciphertexts; • H -finite set of key combinations; • Encryption rule is defined as e k1 ∈ E & a corresponding decryption rule d k1 ∈ M for every K ∈ H.There are functions for each e k : B → R and d k : R → B such that d k (e k (x)) = x for each plaintext x ∈ B.
D x from the D 1 , . . . . . .D l and D y present at the receiving end.According to Paillier's additive homomorphic characteristic,D y (i) = L l=1 D l (i) + D x (i) |n|(2)

Theorem 1.7 ([10]):
(i) Let A be a generalized characteristic bipolar valued fuzzy sub-bigroup of G 2 .By Theorem 2.10, A • f is a bipolar valued fuzzy sub-bigroup of G 1 .Let a 1 and b 1 be in G 1 .If a 1 and b 1 in H 1 and Proof: