# Homomorphic Encryption Based Privacy-Preservation for IoMT
## 2 Related Works
### Existing Research  
Homomorphic encryption is based on the principle of processing data without the requirement of decrypting it [15; 16; 17]. Data confidentiality is one of the primary principles of implementing the encryption system in untrusted server systems. Compared with other encryption protocols, homomorphic encryption allows algebraic computations such as addition and multiplication directly on encrypted data and treats it as plaintext. This is expressed as,  
\[v=f(u)\ \Leftrightarrow\ Enc(v)=g(Enc(u)) \tag{1}\]  
Here \(v\) and \(u\) represent unencrypted vectors, while \(Enc\) behaves as the encryption operation performed on \(v\) and \(u\) vectors. We consider a cryptosystem as \(c\) with \(Enc\) (encryption function), plaintext \((p_{i})\), and ciphertext \(\big{(}c_{j}\big{)}\), where:  
\[Enc(p_{i})=\big{(}c_{j}\big{)} \tag{2}\]  
We perform additive (3) and multiplicative (4) operations on \(p_{i}\) such as:  
\[\Delta:\ Enc(x_{1})\ \Delta\ Enc\ (x_{2})=Enc\ (x_{1}+x_{2}) \tag{3}\]  
\[\Delta:\ Enc(x_{1})\ \Delta\ Enc\ (x_{2})=Enc\ (x_{1}*x_{2}) \tag{4}\]  
An encryption process satisfying both additive and multiplicative properties is considered homomorphic. A combination of XOR and AND Boolean functions are used in the encryption scheme. The homomorphic encryption \((H(Enc))\) scheme is further divided into four algorithms, which are as follows:  
\[H(Enc)=(key\ generate,\ encrypt,\ decrypt,\ evaluate) \tag{5}\]  
* Key generation is based on security parameters (\(\sigma\)) taken as input, and the algorithm outputs an encryption \((k_{e})/\)decryption \((k_{d})\) key pair. * The encryption algorithm takes input \((p_{1},\ p_{2},\ldotsldots,p_{i})\) and is divided into bits \((b_{1},\ b_{2},\ldotsldots b_{i})\) of 0 and 1s and ciphertext \((c_{1},\ c_{2},\ldotsldots c_{j})\). * The decryption algorithm reveals data by receiving input \(p_{i}\) to provide output \(c_{1},\ c_{2},\ldotsldots c_{j}\). * Evaluation of the algorithm takes input \(p_{i}\) with function \((f)\) and reveals output \(c_{1},\ c_{2},\ldotsldots c_{j}\). Homomorphic encryption has three types of encryption schemes: partially homomorphic, somewhat homomorphic, and fully homomorphic encryption [18]. * Partially homomorphic: A single type of mathematical operation is permissible on \(Enc(p_{i})\)--either addition or multiplication performed without any limitation in execution. * Somewhat homomorphic: There is a restriction on the execution limit, with only a certain number of times that either addition or multiplication are permissible. * Fully homomorphic: Enables a larger subset of mathematical operations on the \(p_{i}\) without any limit on the number of executions. There is a limitation on the number of users it supports; its implementation results in a significant computation overhead and increases network latency due to slow runtimes. [PERSON] [19] proposed the concept of secret sharing [20] by dividing data \((x)\) into multiple and reconstructible different pieces \((k)\), where no piece of information \((k-1)\) reveals \(d\). We assume that the secret data \(d\) is confidential information, and break \(d\) into an \(n\) number of pieces \((d_{1},\ d_{2},\ldotsldots,\ d_{n})\) in a way that satisfies two conditions:  
* A third entity requires a certain pre-fixed number of \(k\) pieces of \(d_{1},\ d_{2},\ldotsldots,\ d_{n}\) to make \(d\) accessible. * The possession of \(k-1\) makes \(d\) unaccessible, with the condition that each element of \(d_{i}\) is by itself insufficient to reveal the message. The determination of selection of \(d_{1}\), \(d_{2}\),......., \(d_{n}\) is known as the threshold value, and is especially helpful in securing keys used to encrypt plaintexts. The reconstruction of the key \((k)\) requires all threshold members to announce their individual shares to reconstruct \(k\) and gain access to the secret \(d\). Recent research for securing IoMT data focuses on encrypting using homomorphic encryption for ensuring data privacy. [PERSON] et al.