## 5 The Proposed Scheme
### The Designed Fast Searchable Decryption Algorithm
 \tag{18}\]\[\prod_{j\in\mathcal{I}}\left(e(K_{5},C_{4,j})\cdot e(K_{2,\zeta(j)},C_{5,j})\right)^{-c_{j}} \tag{19}\] \[= \prod_{j\in\mathcal{I}}\left(e(g^{t}R^{\prime\prime\prime},\frac{g^ {\alpha A_{j}\cdot\mathbf{v}}}{Q_{\zeta(j)}^{j}})\cdot e(Q_{\zeta(j)}^{t}R_{ \zeta(j)},g^{\prime j})\right)^{-c_{j}}\] \[= \prod_{j\in\mathcal{I}}\left(\frac{e(g,g)^{\alpha tA_{j}\cdot \mathbf{v}}}{e(g,Q_{\zeta(j)})^{tr_{j}}}\cdot e(g,Q_{\zeta(j)})^{tr_{j}}\right) ^{-c_{j}}\] \[= e(g,g)^{\sum_{j\in\mathcal{I}}-\alpha t_{j}A_{j}\cdot\mathbf{v}}\] \[= e(g,g)^{-\alpha ts}.\]

### Security Analysis

In this section, we analyze the security of ABE-RS against chosen plaintext attacks (CPAs) and chosen keyword attacks (CKAs). Theorem 6.1: _The ABE-RS scheme is secure under IND-CPA conditions._

Proof: To establish the IND-CPA security of our ABE-RS framework, we utilize the encryption technique detailed in [41]. Here, we focus on the definitions of keys and ciphertexts. Key generation:The KeyGen\((GP,MSK,S,uid)\) algorithm is executed to generate \(SK_{S,uid}\), which consists of \(\{K_{1,y_{x}}\}_{y_{x}\in path(uid)\), \(K_{2}\), \(K_{3}\), and \(\{K_{4,x}\}_{x\in S}\). Additionally, a random value \(W\in\mathbb{G}\) is selected. The key generation process is described as follows:

For every user identified by \(uid\) and associated with an attribute set \(S\), the process KeyGen\((GP,MSK,S,uid)\) generates a key \(SK_{S,uid}\). This includes components \(\{K_{1,y_{x}}\}_{y_{x}\in path(uid)}\), \(K_{2}\), \(K_{3}\), and \(\{K_{4,x}\}_{x\in S}\). Additionally, a random element \(W\) from group \(\mathbb{G}\) is integrated into each component to enhance the key's complexity as follows:

\[\{K_{1,y_{x}}\cdot W\}_{y_{x}\in path(uid)},K_{2},K_{3},\{K_{4,x}\}_{x\in S}. \tag{20}\]

Ciphertexts ConstructionExecuting Encrypt\((GP,m,\mathbb{M},L)\), we derive a ciphertext \(CT_{\mathbb{M},L}^{(up)}\) that encapsulates \(C_{0}\), \(C_{1}\), and \(C_{2}\) and structured data \(\{C_{3,j},C_{4,j}\}_{j\in[|]}\) and \(\{C_{5,y_{x}}\}_{y_{x}\in cover(L)}\). Random coefficients \(\alpha^{\prime},\beta^{\prime},\gamma^{\prime}\in\mathbb{Z}_{p}^{*}\), along with vectors \(\mathbf{v}^{\prime}=(s^{\prime},v_{2}^{\prime},\ldots,v_{n}^{\prime})\) and sets \(\{\eta_{x}\}_{x\in S_{A}}\), \(\{\varepsilon_{j}\}_{j\in[|]}\) are chosen to form each encrypted component as follows:

\[C_{0},C_{1}\cdot g^{\alpha^{\prime}},C_{2}\cdot g^{\alpha^{ \prime}\beta^{\prime}},\{C_{3,y_{x}}\cdot g^{\epsilon^{\prime}H_{2}(y_{x})s^{ \prime}}\}_{y_{x}\in cover(L)}, \tag{21}\] \[\{C_{4,j}\cdot g^{\alpha^{\prime}A_{j}\cdot\mathbf{v}}g^{-\eta_{ \beta(j)}\varepsilon_{j}},C_{5,j}\cdot g_{2}^{\varepsilon_{j}}\}_{j\in[|]}.\]

Limiting the adversary to \(\mathcal{U}\) key queries, a sequence of games from \(Game_{real}\) to \(Game_{final}\) is set. The initial \(k\) games involve specialized keys and ciphertexts, shifting gradually to standard forms until \(\mathcal{U}\), where the ciphertext is encrypted with an arbitrary message. This structured approach of transitioning through games illustrates the robustness of the encryption against distinguishing attacks, mimicking the methods established in [PERSON] et al. [41]. Theorem 6.2: _Under the random oracle model and assuming the BDH assumption holds, the ABE-RS scheme is semantically secure against chosen keyword attacks._

Proof: \(\mathcal{A}\), as the PPT adversary, and \(\mathcal{C}\), as the challenger, are set to tackle the BDH problem with a determined advantage \(\epsilon^{\prime}=\frac{2\epsilon}{\alpha\mu_{\beta}\delta^{T}}\). Here, \(\epsilon\) represents \(\mathcal{A}\)'s advantage, \(e\) is the natural logarithm base, and \(q_{H_{3}}\) and \(q_{T}\) denote the maximum number of queries to the hash function \(H_{3}\) and the trapdoor, respectively. * _Setup_: \(\mathcal{C}\) is given a BDH challenge consisting of the tuple \((u_{1}=g^{a},u_{2}=g^{b}\), \(u_{3}=g^{d}\in\mathbb{G})\) and is tasked with computing the value \(e(g,g)^{abd}\in\mathbb{G}_{T}\).