# Efficient and HRA Secure Universal Conditional Proxy Re-Encryption for Cloud-Based Data Sharing
## 6 Efficiency Analysis  
We first define the notations used in Table 1. Notations \(t_{p}\) and \(t_{e}\), and \(t_{e}\) denote the times consumed for a pairing operation, and a modular exponentiation in \(\mathbb{G}\), and a modular exponentiation in \(\mathbb{G}_{T}\), separately. Notations \(\text{Dec}(\text{Or})\) and \(\text{Dec}(\text{Re})\) denote the decryption execution for the original ciphertext and the re-encryption ciphertext, respectively. Here, we omit the computing time of addition, multiplication, and hash function operations because these operations are much less modular exponentiation and pairing operations. As shown in Table 1, the computation overhead of the instantiation scheme for UCPRE in each algorithm is compared to other works  
* Enc. The instantiation scheme for UCPRE has lower computing cost to set the original ciphertext. Nevertheless, the delegator in works [24; 25; 26; 27] have to undertake the amount of computing overhead in the encryption phase. For exzample, the delegator in work [27] needs to undertake 8 modular exponentiation operations in \(\mathbb{G}_{T}\) and a pairing operation for setting ciphertext.
* RKeyGen. Table 1 shows that scheme [27] and our scheme have lower computation overhead to generate the re-encryption key. However, the delegator in work [26] needs abundant computing overhead to set the re-encryption key.
* ReEnc. Table 1 shows that our instantiation scheme has less computing cost to execute the re-encryption algorithm for setting re-encryption ciphertext compared with the works [24; 26]. While our scheme is less efficient compared with works [25; 27] in the re-encryption phase. However, it cannot support the ciphertext switching functionality.
* Dec(Or). In the decryption algorithm for the original ciphertext, the delegator in our instantiation scheme and work [27] only executes lower computing costs to decrypt the original ciphertext. However, there are heavy computing overhead in works [24; 25; 26].
* Dec(Re). Table 1 shows that our instantiation scheme has less computing cost to execute the decryption algorithm for the re-encryption ciphertext compared with CPRE schemes [24; 25; 26; 27].  
\begin{table}
\begin{tabular}{l l l l l l} \hline \hline
**Schemes** & **Enc** & **RKeyGen** & **ReEnc** & **Dec(Or)** & **Dec(Re)** \\ \hline
[24] & \(2t_{e}+5t_{e}+5t_{p}\) & \(8t_{e}\) & \(2t_{e}+2t_{e}+2t_{p}\) & \(t_{e}+2t_{e}+t_{p}\) & \(t_{e}+t_{e}+t_{p}\) \\
[25] & \(4t_{e}+t_{o}+t_{p}\) & \(8t_{e}\) & \(t_{e}\) & \(2t_{e}+1t_{o}+3t_{p}\) & \(t_{e}+t_{e}+4t_{p}\) \\
[26] & \(4t_{e}+t_{o}+t_{p}\) & \(8t_{e}+t_{e}+t_{p}\) & \(5t_{e}+2t_{p}\) & \(2t_{e}+t_{e}+3t_{p}\) & \(4t_{e}+t_{e}+12t_{p}\) \\
[27] & \(8t_{e}+t_{p}\) & \(t_{e}+2t_{p}\) & \(6t_{e}\) & \(t_{e}+t_{p}\) & \(t_{e}+t_{p}\) \\ Ours & \(2t_{e}\) & \(3t_{e}\) & \(6t_{e}\) & \(t_{e}\) & \(2t_{e}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Computing cost comparison.  
The comparison results displayed in Table 1 clearly show that our UCPRE has the least computation overhead compared to related works. Meanwhile, our construction UCPRE adds the property of ciphertext transformation between two (different) homormorphic encryption schemes compared with conditional proxy re-encryption schemes. This function is important for a CPRE scheme that may be used in actual applications (e.g., data sharing in clouds).