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tool ([https://github.com/JHUUSI/auto-tools/tree/88d20b0/auto_group](https://github.com/JHUUSI/auto-tools/tree/88d20b0/auto_group), (accessed on 31 January 2024)). The application of the transformation resulted in the following scheme. Differences from the base scheme are highlighted in gray (Algorithms 9-16). ```
1:Choose \(\mathsf{Type}\)3 bilinear groups \(\mathbb{G}_{1}\), \(\mathbb{G}_{2}\), \(\mathbb{G}_{T}\) of prime order \(p>2^{k}\)
2:\(g\xleftarrow{\xi}\mathbb{G}_{1}\), \(\xleftarrow{\xi}\mathbb{G}_{2}\)
3:\(\xleftarrow{\xi}\mathbb{Z}_{p^{\prime}}^{*}\mathbb{G}_{1}=g^{a}\), \(\xleftarrow{\xi}\mathbb{G}_{p^{\prime}}^{*}\mathbb{G}_{1}=g^{a}\)
4:\(\xleftarrow{\xi}\mathbb{Z}_{p^{\prime}}^{*}u=g^{b}\), \(u=g^{b}\)
5:\(\xleftarrow{\xi}\mathbb{Z}_{p^{\prime}}^{*}v=g^{c}\), \(\xleftarrow{\xi}\mathbb{G}\)
6:\(\xleftarrow{\xi}\mathbb{Z}_{p^{\prime}}^{*}w=g^{d}\), \(\xleftarrow{\xi}\mathbb{Z}_{p}^{*}w=g^{d}\)
7:\(Z=e(g,\xleftarrow{\xi})\)
8:Choose a \(\mathsf{TCR}\) hash function \(H:\mathbb{G}_{1}\times\{0,1\}^{\ell}\to\mathbb{Z}_{p}^{*}\)
9:Choose a \(\mathsf{PRF}\)
10:return\(param=(p,\mathbb{G}_{1},\xleftarrow{\xi}\mathbb{G}_{T},g,g_{1},u,v,w, \xleftarrow{\xi}\mathbb{G}_{1},\xleftarrow{\xi}\mathbb{G}_{1},\xleftarrow{ \xi}\mathbb{G}_{2},H,F,\ell_{1},\ell)\)
```  
**Algorithm 10**\(\mathsf{KeyGen}(1^{k})\)  
```
1:\(x_{i}\xleftarrow{\xi}\mathbb{Z}_{p}^{*}\)
2:\(pk_{i}=(pk_{i1},\xleftarrow{\xi}\mathbb{Z}_{p})=(g^{x_{i}},\xleftarrow{\xi} \mathbb{Z})\)
3:\(sk_{i}=x_{i}\)
4:return\((pk_{i},sk_{i})\)
```  
**Algorithm 11**\(\mathsf{KeyGen}(sk_{i},pk_{j})\)  
```
1:\(rk_{i\to j}=\xleftarrow{\xi}\mathbb{Z}_{p}^{*}\)
2:\(\mathsf{C}_{1}=g^{r}\)
3:\(\mathsf{C}_{2}=pk_{i1}^{r}\)
4:\(K=Z^{r}\)
5:\(\mathsf{C}_{3}=[F(K,\mathsf{C}_{1})]_{\ell-\ell_{1}}||([F(K,\mathsf{C}_{1})]^{ \ell_{1}}\oplus m)\)
6:\(t\xleftarrow{\xi}\mathbb{Z}_{p}^{*}\)
7:\(h=H(\mathsf{C}_{1},\mathsf{C}_{3})\)
8:\(\mathsf{C}_{4}=(u^{h}v^{t}w)^{r}\)
9:return\(\mathsf{C}\mathsf{I}_{i}=(t,\mathsf{C}_{1},\mathsf{C}_{2},\mathsf{C}_{3}, \mathsf{C}_{4})\)
```  
**Algorithm 12**\(\mathsf{Enc}_{2}(pk_{i},m)\)  
```
1:\(r\xleftarrow{\xi}\mathbb{Z}_{p}^{*}\)
2:\(\mathsf{C}_{1}=g^{r}\)
3:\(\mathsf{C}_{2}=pk_{i1}^{r}\)
4:\(K=Z^{r}\)
5:\(\mathsf{C}_{3}=[F(K,\mathsf{C}_{1})]_{\ell-\ell_{1}}||([F(K,\mathsf{C}_{1})]^{ \ell_{1}}\oplus m)\)
6:\(t\xleftarrow{\xi}\mathbb{Z}_{p}^{*}\)
7:\(h=H(\mathsf{C}_{1},\mathsf{C}_{3})\)
8:\(\mathsf{C}_{4}=(u^{h}v^{t}w)^{r}\)
9:return\(\mathsf{C}\mathsf{I}_{i}=(t,\mathsf{C}_{1},\mathsf{C}_{2},\mathsf{C}_{3}, \mathsf{C}_{4})\)
```  
**Algorithm 13**\(\mathsf{KeyGen}(1^{k})\)  
```
1:\(\mathsf{C}_{1}=g^{r}\)
2:\(\mathsf{C}_{2}=g^{r}\)
3:\(\mathsf{C}_{3}=[F(K,\mathsf{C}_{1})]_{\ell-\ell_{1}}||([F(K,\mathsf{C}_{1})]^{ \ell_{1}}\oplus m)\)
4:\(t\xleftarrow{\xi}\mathbb{Z}_{p}^{*}\)
5:\(h=H(\mathsf{C}_{1},\mathsf{C}_{3})\)
6:\(\mathsf{C}_{4}=(u^{h}v^{t}w)^{r}\)
7:return\(\mathsf{C}\mathsf{I}_{i}=(t,\mathsf{C}_{1},\mathsf{C}_{2},\mathsf{C}_{3}, \mathsf{C}_{4})\)
```  
**Algorithm 14**\(\mathsf{KeyGen}(1^{k})\)  
```
1:\(\mathsf{C}_{1}=g^{r}\)
2:\(\mathsf{C}_{2}=g^{r}\)
3:\(\mathsf{C}_{3}=[F(K,\mathsf{C}_{1})]_{\ell-\ell_{1}}||([F(K,\mathsf{C}_{1})]^{ \ell_{1}}\oplus m)\)
4:\(t\xleftarrow{\xi}\mathbb{Z}_{p}^{*}\)
5:\(h=H(\mathsf{C}_{1},\mathsf{C}_{3})\)
6:\(\mathsf{C}_{4}=(u^{h}v^{t}w)^{r}\)
7:return\(\mathsf{C}\mathsf{I}_{i}=(t,\mathsf{C}_{1},\mathsf{C}_{2},\mathsf{C}_{3}, \mathsf{C}_{4})\)
```  
**Algorithm 15**\(\mathsf{KeyGen}(1^{k})\)  
```
1:\(\mathsf{C}_{1}=g^{r}\)
2:\(\mathsf{C}_{2}=g^{r}\)
3:\(\mathsf{C}_{3}=[F(K,\mathsf{C}_{1})]_{\ell-\ell_{1}}||([F(K,\mathsf{C}_{1})]^{ \ell_{1}}\oplus m)
+++



### Analysis of the Transformed Scheme

The scheme resulting from the automated transform retains the ciphertext size of the original scheme and adds a second group element to the public key. The main portion of the transformation to the asymmetric setting is realized by duplicating the generators from the public parameters into both source groups.