## 6 Security and Performance Analysis  
### Security Analysis  
The following complexity assumptions serve as the foundation for our security proofs: Subgroup Decision Problem for Three Primes [38; 50]Three assumptions are contained in this SDP assumption. Here Pr denotes the probability function. **Assumption 1:**: _Let us take a group generator \(\tilde{\mathbf{g}}\) and consider the distribution: \(\mathbb{G}\)=(\(O=p_{1}p_{2}p_{3},G,\)\(G_{T},e\)) \(\overset{R}{\leftarrow}\)\(\tilde{\mathbf{g}}\), \(g\overset{R}{\leftarrow}\)\(G_{p_{1}},X_{3}\overset{R}{\leftarrow}\)\(G_{p_{3}}\), \(D\) = (\(G,g,X_{3}\)), \(T_{1}\overset{R}{\leftarrow}\)\(G_{p_{1}}\), \(T_{2}\overset{R}{\leftarrow}\)\(G_{p_{1}}\). The advantage of an algorithm \(A\) in breaking the mentioned assumption is:_  
\[SDP\_Adv1_{\mathcal{G},\mathrm{A}}(\lambda)=\mid\Pr[\ \mathrm{A}(D,T_{1})=1]-\Pr[ \ \mathrm{A}(D,T_{2})=1] \tag{15}\]  
**Definition 7**: _If for any polynomial time (PT) algorithm \(A\), \(SDP\_Adv1_{\mathcal{G},\mathrm{A}}(\lambda)\) is a negligible function of \(\lambda\), then \(\hat{\mathbf{g}}\) is said to satisfy the above-mentioned assumption 1._  
**Assumption 2:**: _Let us take a group generator \(\tilde{\mathbf{g}}\) and consider the distribution: \(\mathbb{G}\)= (\(O=p_{1}p_{2}p_{3},G,\)\(G_{T},e\)) \(\overset{R}{\leftarrow}\)\(\tilde{\mathbf{g}},g,X_{1}\overset{R}{\leftarrow}\)\(G_{p_{1}},X_{2},Y_{2}\overset{R}{\leftarrow}\)\(G_{p_{2}},X_{3},Y_{3}\overset{R}{\leftarrow}\)\(G_{p_{3}}\), \(D\) = (\(G,g,X_{1}X_{2},X_{3},Y_{2}Y_{3}\)), \(T_{1}\overset{R}{\leftarrow}\)\(G,T_{2}\overset{R}{\leftarrow}\)\(G_{p_{2}p_{3}}\). The advantage of an algorithm \(A\) in breaking the mentioned assumption is:_  
\[SDP\_Adv2_{\mathcal{G},\mathrm{A}}(\lambda)=\mid\Pr[\mathrm{A}(D,T_{1})=1]-\Pr[ \mathrm{A}(D,T_{2})=1]\mid \tag{16}\]  
**Definition 8**: _If for any polynomial-time (PT) algorithm \(A\), \(SDP\_Adv2_{\mathcal{G},\mathrm{A}}(\lambda)\) is a negligible function of \(\lambda\), then \(\mathbf{g}\) is said to satisfy the above-mentioned assumption 2._  
**Assumption 3:**: _Let us take a group generator \(\tilde{\mathbf{g}}\) and consider the distribution: \(\mathbb{G}\)=(\(O=p_{1}p_{2}p_{3},G,\)\(G_{T},e\)) \(\overset{R}{\leftarrow}\)\(\mathcal{G},\alpha,s\overset{R}{\leftarrow}\)\(\mathbb{Z}_{N},g\overset{R}{\leftarrow}\)\(G_{p_{1}},X_{2},Y_{2},Z_{2}\overset{R}{\leftarrow}\)\(G_{p_{2}},X_{3}\overset{R}{\leftarrow}\)\(G_{p_{3}}\), \(D\)=(\(G,g,g^{\mathrm{A}}X_{2},X_{3},g^{\mathrm{A}}Y_{2},Z_{2}\)), \(T_{1}=e(g,g)^{\mathrm{A}}s,T_{2}\overset{R}{\leftarrow}\)\(G_{T}\). The advantage of an algorithm \(A\) in breaking the mentioned assumption is:_  
\[SDP\_Adv3_{\mathcal{G},\mathrm{A}}(\lambda)=\mid\Pr[\mathrm{A}(D,T_{1})=1]- \Pr[\mathrm{A}(D,T_{2})=1]\mid \tag{17}\]  
**Definition 9**: _If for any polynomial-time (PT) algorithm \(A\), \(SDP\_Adv3_{\mathcal{G},\mathrm{A}}(\lambda)\) is a negligible function of \(\lambda\), then \(\tilde{\mathbf{g}}\) is said to satisfy the above-mentioned Assumption 3._  
**Theorem 1**: _If the above three assumptions are true, then the proposed SP-MAACS scheme is fully secure according to the model presented in Section 4.3--Definition 6._  
**Proof. ** We will use two terms here: ciphertext in semi-functional form (SF-CT) and key in semi-functional form (SF-Key). The terms are used in proof [16] and are not used in the construction of the scheme. We chose a random exponent \(z_{\mathrm{k,i}}\in Z_{\mathrm{N}}\) for each attribute i \(\in\mathrm{U_{k}}\). \(\square\)  
Semi-functional ciphertext. To make an SF-CT, perform the following:  
Let \(g_{2}\) be a generator of \(G_{p_{2}}\), \(d\overset{R}{\leftarrow}\)\(Z_{N}\). \(\forall\)row \(xe[l]\), let us randomly select \(\xi_{\mathrm{X}}\in Z_{\mathrm{N}}\). In addition, choose a random vector \(\overset{\rightarrow}{\mathcal{Y}}\in Z_{N}^{n}\).