## 3 Proposed Method and Mathematical Analysis
### Encryption and Decryption Methodology
#### 3.3.2 ElGamal Encryption Algorithm  
As Arnold is cyclic and the key space is too small and vulnerable to perform exhaustive attacks, it is not possible to use Arnold alone. Therefore, the ElGamal algorithm is relied upon to expand the key space of Arnold and improve the overall security of the algorithm.  
This paper uses traditional cryptographic methods with Arnold superimposed on the encryption method used to enhance the security of the algorithm by employing the discrete logarithm-based ElGamal Encryption Algorithm on top of Arnold encryption for secondary encryption and better preserving the privacy of biometric images. Following Figure 1, Encryption and Decryption sections, we use the ElGamal Encryption Algorithm to encrypt the result of the Arnold Transform.  
The ElGamal algorithm is a Public-Key Cryptosystem based on the discrete logarithm problem over a finite field, proposed by [PERSON] [11] in 1985. It is a secure encryption method that ensures the confidentiality and integrity of the data. Based on the difficulty of computing discrete logarithms, the method is resistant to brute-force attacks. Public-Key Cryptosystems have certain advantages over Private-Key Cryptosystems (e.g., DES, AES, etc.) in that they can separate encryption from decryption, such that messages encrypted by multiple users can only be interpreted by one user, or messages encrypted by one user can be interpreted by multiple users. The Public-Key Cryptosystem is therefore suitable not only for encryption but also for signature authentication. This model contains a signature authentication component that makes it more appropriate to choose a Public Key cryptographic algorithm. Figure 3 is a block diagram of the public key system encryption applied in this paper.  
The ElGamal Encryption Algorithm has a unique key generation process; it first chooses a prime number p and two random numbers g and x that are less than p. Next, compute Equation (6). Use (y, g, p) as the public key and x as the secret key.  
\[y\equiv g^{x}mod\ p \tag{6}\]  
The encryption process is as follows: Suppose you want to encrypt a plaintext message, M. Randomly select an integer k that is prime to p - 1, then calculate Equations (7) and (8).  
\[C_{1}\equiv g^{k}mod\ p \tag{7}\]  
\[C_{2}\equiv\ y^{k}M\ mod\ p \tag{8}\]  
Finally, the ciphertext is Equation (9).  
\[C=(C_{1},C_{2}) \tag{9}\]  
Figure 3: A block diagram of the public key encryption system.  
The decryption process follows Equation (10).  
\[M=\frac{C_{2}}{C_{1}^{x}}\ mod\ p \tag{10}\]  
The reason for the calculation of Equation (10) is Equation (11).  
\[\frac{C_{2}}{C_{1}^{x}}\ mod\ p=\frac{y^{k}M}{g^{kx}}\ mod\ p=\frac{y^{k}M}{y^{k} }\ mod\ p=M\ mod\ p \tag{11}\]  
After Equation (10), we employed the Arnold Transform to conduct the final decryption, ultimately using the Digital Signature to verify that the ciphertext had not been tampered with. At the end of all processes, M is sent to Receiver B (refer to Figure 1).  
The ElGamal Encryption Algorithm is scalable, which means that it can be used to encrypt large amounts of data efficiently. This is achieved by breaking the data into smaller blocks and encrypting each block separately. At the same time, this algorithm has the advantages of high computational complexity and low space consumption compared with the now more common public key encryption algorithm, RSA. Its security is also higher; it is known to attack discrete logarithmic problems over finite fields using an exponential integration method with an operational complexity of \(O\left(exp\sqrt[3]{(\log p)(\log\log p)^{2}}\right)\).