## 5 Conclusions

In conclusion, we presented the first unidirectional PRE scheme based on asymmetric pairings. We started with the application of the automated black-box reduction technique by [PERSON] et al. [20] to transform the base scheme and its security proof from the Type 1 to the Type 3 setting. Our findings reveal that relying solely on this automated technique does not necessarily produce a scheme suitable for practical use, as it introduced flaws that rendered the resulting scheme impractical. To address these issues, we proposed an adapted manually transformed scheme. The scheme retains the properties of the base scheme, achieving CCA security under adaptive corruptions in the standard model. We refined the hardness assumption for the Type 3 setting, ensuring that it is at least as hard as in the base scheme, and outlined the necessary adjustments to the security proof. The transformed scheme enables the use of superior Type 3 pairing-friendly curves, thereby superseding the obsolete Type 1 setting of the base scheme. The manual transformation is essential as it fixes the flaws present in the automatically transformed scheme, as discussed in Section 3.3, making it suitable for practical use. While this comes with slight computational overhead associated with encryption performance and increased ciphertext size, it also results in faster re-encryption and decryption, contributing to the scheme's overall practicality. Therefore, we argue that our manually transformed \begin{table}
\begin{tabular}{l c c} \hline \hline
**Method/Scheme** & **Automated** & **Manual** \\ \hline KeyGen & 1.5 ms & 1.5 ms \\ ReKeyGen & 1.0 ms & 1.0 ms \\ Encrypt2 & 4.3 ms & 4.7 ms \\ Encrypt1 & 9.3 ms & 9.7 ms \\ ReEncrypt & 20.1 ms & 17.8 ms \\ Decrypt2 & 21.6 ms & 18.8 ms \\ Decrypt1 & 13.7 ms & 12.5 ms \\ Second-Level Ciphertext Size & 208 B & 256 B \\ First-Level Ciphertext Size & 736 B & 784 B \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Ciphertext sizes and benchmark of the automatically and manually transformed schemes on an Intel Core i7-6600U CPU with 16 GB of RAM. scheme represents a preferable choice for the practical instantiation of a Type 3 pairing-based PRE scheme, while still being faster and more secure overall than the base scheme. Our scheme offers an alternative to more complex pairing-free PRE schemes, providing a practical and efficient solution for real-world applications. Future work will focus on leveraging the distinctive properties of Type 3 pairings to construct an even more efficient asymmetric pairing-based PRE scheme or create a PRE scheme with advanced properties that is CCA-secure in the standard model. Conceptualization, B.Z. and N.B. ; methodology, B.Z. ; software, B.Z. ; investigation, B.Z. ; writing--original draft preparation, B.Z., N.B., P.D. and M.M. ; writing--review and editing, B.Z., N.B., P.D. and M.M., visualization, B.Z. ; supervision, N.B. ; project administration, B.Z. and N.B. ; funding acquisition, M.M. All authors have read and agreed to the published version of the manuscript. This research received no external funding. Not applicable. The original contributions presented in the study are included in the article, further inquiries can be directed to the corresponding author. The authors declare no conflicts of interest. ## References

* [PERSON] et al. (1998) [PERSON]; [PERSON]; [PERSON] Divertible Protocols and Atomic Proxy Cryptography. In Proceedings of the EUROCRYPT, Espoo, Finland, 31 May-4 June 1998; [PERSON]; LNCS; Springer: Berlin/Heidelberg, Germany, 1998; Volume 1403, pp. 127-144. [CrossRef]
* [PERSON] et al. (2006) [PERSON]; [PERSON]; [PERSON]; [PERSON] Improved proxy re-encryption schemes with applications to secure distributed storage. _ACM Trans. Inf. Syst. Secur._**2006**, 9, 1-30. [CrossRef]
* [PERSON] and [PERSON] (2008) [PERSON]; [PERSON] Unidirectional Chosen-Ciphertext Secure Proxy Re-encryption. In Proceedings of the PKC 2008, Barcelona, Spain, 9-12 March 2008; [PERSON], [PERSON]; LNCS; Springer: Berlin/Heidelberg, Germany, 2008; Volume 4939, pp. 360-379.