# SEAIS: Secure and Efficient Agricultural Image Storage Combining Blockchain and Satellite Networks
## 5 Security Analysis  
To prove the security of the SEAIS, this section proves it through the reductive approach. We verify the security of each key component step by step, including homomorphic encryption, zero-knowledge proofs, and blockchain recording, and finally prove the security of the whole system. Theorem 5.1: _The Paillier homomorphic encryption scheme is secure against chosen plaintext attacks (IND-CPA) in the random oracle model and possesses additive and scalar homomorphic properties, thereby ensuring data confidentiality and integrity._  
Proof of Theorem 5.1.: Given \(z=p\cdot q\), we assume the existence of an attacker A; if A can distinguish encrypted ciphertexts from random ciphertexts in polynomial time, this attacker can be used to factor \(z\). However, this contradicts the difficulty of the large number decomposition problem. In the random oracle model, the attacker gains no information about \(p\) and \(q\), thus making it impossible to factor \(z\), which ensures data confidentiality. Meanwhile, Paillier homomorphic encryption has additive and scalar homomorphic properties. The additive homomorphism is as follows:  
\[ \begin{array}{l}{Encrypt(I_{1})\cdot Encrypt(I_{2})=(g^{I_{1}}\cdot r_{1}^ {z})\cdot(g^{I_{2}}\cdot r_{2}^{z})\bmod z^{2}}\\ =g^{I_{1}+I_{2}}\cdot(r_{1}r_{2})^{z}\bmod z^{2}=Encrypt(I_{1}+I_{2})\end{array} , \tag{23}\]  
\[Encrypt(I_{1})^{k}=(g^{I_{1}}\cdot r_{1}^{z})^{k}\bmod z^{2}=g^{k\cdot I_{1}} \cdot(r_{1}^{k})^{z}\bmod z^{2}=Encrypt(k\cdot I_{1}) \tag{24}\]  
These homomorphic properties allow specific operations on the encrypted data without decrypting it. By verifying the additive and scalar homomorphic properties, we ensure that the integrity of the encrypted data is maintained during processing. That is, any operations performed on the ciphertext correspond to equivalent operations on the plaintext, thus preserving the correctness of the data. Theorem 5.2: _Zero-knowledge-proof protocol ensures the authenticity and integrity of encrypted image data. It allows a verifier to confirm the validity of the data without revealing the data itself._  
Proof of Theorem 5.2.: Assume there exists an adversary \(B\) that can produce a valid proof \(\pi=(c,s,hash_{c})\) for a statement that is false. Given a challenge to solve the discrete logarithm problem (DLP) for \(g\) and \(y=g^{x}\text{mod}p\), where \(x\) is unknown, if \(B\) can generate a proof for a false statement, it must mean that \(B\) can find \(c\), \(s\), and \(hash_{c}\), such that \(c^{\prime}=g^{s}\cdot(Encrypt(I_{enhanced}(u,v)))^{-hash_{c}}\text{mod}q\), which means \(B\) can solve the DLP by finding \(x\) in polynomial time. This contradicts the DLP assumption. Moreover, given a challenge to find a collision in the hash function \(H\), if \(B\) can modify the ciphertext \(Encrypt(I_{enhanced}(u,v))\) to \(Encrypt(I^{\prime}_{enhanced}(u,v))\) and produces a valid proof for \(Encrypt(I^{\prime}_{enhanced}(u,v))\), it must mean that \(B\) can find the two distinct inputs \((c,hash_{d})\) and \((c^{\prime},hash^{\prime}_{d})\) such that \(H(c\big{|}|hash_{d})=H(c^{\prime}\big{|}|hash^{\prime}_{d})\), which directly contradicts the collision resistance of \(H\). Therefore, zero-knowledge proof ensures the authenticity and integrity of encrypted image data. **Theorem 3**. : _Blockchain ensures the immutability and transparency of data records by using cryptographic hash functions and a distributed consensus mechanism._  
Proof of Theorem 3.: Assume there exists an adversary \(\mathbb{C}\) that can alter the data in a block without being detected. Let \(B_{l}\) be the l-th block in the blockchain and \(H(B_{l})\) be its hash. If \(\mathbb{C}\) modifies the image data in \(B_{l}\) to \(B_{l}^{\prime}\) and recomputes the hash \(H(B_{l}^{\prime})\) to match the hash stored in the next block \(B_{l+1}\), this implies that \(\mathbb{C}\) can break the cryptographic hash function or compromise the consensus protocol, both of which are assumed to be secure. This contradiction proves the immutability of blockchain records.