## 3 Materials and Methods
Both algorithms have similar capabilities because they support only integer arithmetic operations as specified by [53], and are based on the generation of two types of keys for biometric data encryption: a public key, which is publicly released, and a private decryption key that is kept secret and shares the same plaintext space \(\mathcal{R}_{t}\) for an integer \(t>1\). The mathematical description of the considered homomorphic algorithms is given in [54,55], which are characterized by three main operations: key generation, encryption, and decryption. The first phase comprises key generation (secret key, sk; and public key, pk) based on the initial parameters \(\lambda\) and \(L\), where \(\lambda\) represents the security level of the HE scheme, and \(L\) is the highest depth of the homomorphic circuits. The second phase, which consists of text encryption, generates ciphertext (ct) by encrypting plaintext (pt) using public key pk, such that \(\big{(}\text{pt}\in\mathfrak{R}_{p},\text{pk}\big{)}\rightarrow\text{ct}\). The plaintext is a ring expressed by \(\mathcal{R}_{p}=Z_{p}[x]/(x^{n}+1)\), while the ciphertexts are considered to be pairs of polynomials in \(\mathcal{R}_{p}=Z_{p}[x]/(x^{n}+1)\). In the last phase, decryption implies the existence of a ciphertext (ct) that needs to be decrypted, using a private key (sk) from which the original message is obtained: \(\text{Decrypt}(\text{ct},\text{sk})\rightarrow\text{pt}\). Another important step described by [56] includes the evaluation of the homomorphic circuit that encompasses the development of a supplementary function over the ciphertexts (ct1, \(\text{ct2}\)) without seeing the messages (\(\text{pt}_{1},\text{pt2}\)), which takes ciphertexts as input points from which the evaluated ciphertexts result. The encryption scheme comprises four main operations, namely KeyGen, Encrypt, Decrypt, Evaluate, for circuit C, considering the input parameter \(t\), the encryption key pair (sk,pk) generated via the KeyGen function, the plaintext messages \(\text{pt}=(\text{pt}_{1},\text{pt2}\ldots\text{pt}_{t})\), and the encrypted texts \(\text{ct}=(\text{ct}_{1},\text{ct2}\ldots\text{ct}_{t})\), so that the result can be concretized using the following formulas:  
\[\text{ct}_{i}=\text{Encryption}(\text{pk},\text{pt}_{i}) \tag{1}\]  
\[\text{Decryption}(\text{sk},\text{Evaluation}(\text{pk},\text{C},\text{ct})) =C(\text{pt}_{1},\text{pt}_{2}\ldots\text{pt}_{t}) \tag{2}\]  
We decided to test these two emerging encryption techniques and appealing solutions for the research community, to evaluate their performance and efficiency for protecting confidential information deployed in potentially dangerous environments where attack vectors are continuously evolving and to overcome the security and privacy issues from real-world applications. In this study, the optimization of the authentication process is achieved by combining different architectural approaches, implementing an efficient solution to improve existing technologies, overcoming the drawbacks of traditional authentication methods, and bringing innovation in terms of security and resilience. The evaluation performed on the biometric database using several representative statistical parameters specific to image processing has the main objective of minimizing encryption time, increasing encryption speed, and providing a higher level of security.