## 6 Security Analysis
In considering chosen ciphertext attacks (CCAs) or other forms of adaptive attacks, the dependency of \(m\) on both \(a\) and \(s\) complicates the decryption process. Without access to the correct symmetric key, \(s\), or signature, \(a\), decryption attempts are unlikely to yield meaningful results. The integration of asymmetric and symmetric encryption components thus creates a robust security framework that resists such sophisticated attacks. Next, let us turn to the threat posed by an internal attacker who has access to the secret key, \(s\), but lacks knowledge of the private key, \(k\). This attacker might attempt to forge a signature or craft a counterfeit message that is verified as authentic. However, the security properties inherent to the modified ElGamal signature mechanism effectively prevent these actions. Since the computation of \(k\) requires solving discrete logarithm, the inside attacker cannot generate a valid signature for any message. Additionally, the hash function's integrity ensures that the signature is specific to the content of the message. Thus, even if the internal attacker attempts to replay a previously signed message, \(m\), or construct a similar message, \(m^{\prime}\), the hash value will differ unless \(m\) is identical to \(m^{\prime}\). This characteristic prevents successful replay or tampering attacks. The combined use of ElGamal-based encryption signatures and symmetric encryption enhances the overall security posture of the protocol by addressing potential weaknesses in isolation. The digital signature ensures message integrity and authenticity, while the symmetric encryption protects the confidentiality of the message content. This hybrid approach benefits from the complementary strengths of both cryptographic techniques, making it challenging for an adversary to compromise the system. Nevertheless, potential weaknesses must be acknowledged and mitigated to ensure comprehensive security. For instance, if either the signature, \(\sigma\), or the symmetric key, \(s\), is compromised, the confidentiality of the message, \(m\), is at risk. Thus, secure key management practices are crucial to prevent key leakage or unauthorized access. One effective mitigation strategy is to use unique keys for each message exchange, thereby reducing the risk of key reuse and subsequent pattern recognition in the ciphertext. Furthermore, ensuring that the hash function used in the signature generation process remains secure against emerging cryptographic attacks is essential. The selection of a hash function resistant to both classical and quantum computing attacks can future-proof the protocol against evolving threats. Key length considerations for ElGamal modulus, \(p\), also play a critical role in maintaining the security of the signature scheme. As computational power increases, longer key lengths are necessary to ensure that obtaining \(k\) remains computationally infeasible, especially with hiding \(p\). It it important to avoid extracting secret information due to a pattern in encrypted messages; this can be illustrated as follows: If \(c_{1}=a_{1}\times s+m_{1}\) and \(c_{2}=a_{2}\times s+m_{2}\), by computing the difference \(c_{1}-c_{2}\), the attacker can obtain a multiple of the secret key, \(s\), if \(m\) has a pattern. Since the security of this linear encryption is dependent on message randomness, we can modify the ciphertext calculation and add randomness not related to the message, \(m\), as follows: \(c=a\times s_{1}+m\times s_{2}+r\), where \(r\) is a random number for each message and the new secret key of the system is \((s_{1},s_{2})\) instead of \(s\). \[c=a\times s_{1}+m\times s_{2}+r \tag{7}\]  
By using Equation (7), \(c_{1}-c_{2}\) would be equal to \(x\times s_{1}+t\times s_{2}+r_{3}\), where \(x\) is a multiple of \(s_{1}\), \(t\) is a pattern into \(m\), and \(r_{3}\) is a new random number with \(r_{3}=(r_{1}-r_{2})\). As a result, the decryption of \(c\) should be as follows: \(a=c/s_{1}\), \(m=(c-a\times s_{1})/s_{2}\).