## 6 Security Proof  
In this section, we continue to demonstrate what an adversary is and its probability. We model the _Game_, our scheme simulation steps, and the related oracle responses. An adversary (call it \(\mathcal{S}\)) can control all communication messages in this scheme. The adversary can obtain related information by sending oracles. A _Game_ is the simulation of our proposed scheme, which can equip all kinds of oracles, and oracles can reply back according to the adversary's questions. There is also another adversary (call it \(\mathcal{S}\)) that controls the simulation and takes \(\mathcal{A}\)'s ability to break the hard problem defined in the security definition. Let _Game_ be a \"game\", the simulation of our scheme, where the adversary \(\mathcal{A}\) can ask queries to the oracles, and the oracles can answer back to the adversary. The following are query types that an adversary can make in the game. * Send query \(\mathit{Send}(i,k,M)\)(or \(\mathit{Send}(j,k^{\prime},M)\)): this query models an adversary that can send message \(M\) to the authentication party \(i\)(or \(j\)), where \(i,j\in\{U,V\}\) in the \(k\)- (or \(k^{\prime}\))-th session, where \(k\) and \(k^{\prime}\) are two different session numbers in \(N\). * Reveal query \(\mathit{Reveal}(i,k)\)(or \(\mathit{Reveal}(j,k)\)): this query is used to model a situation that exposes a session key of \(\Pi^{k}_{i}\)(or \(\Pi^{k^{\prime}}_{j}\)) to an adversary, where \(i,j\in\{U,V\}\) and \(k,k^{\prime}\in N\) in the \(k\)(or \(k^{\prime}\))-the session. * Encryption query: this query is used to model that an adversary can obtain a ciphertext \(C^{\prime}\) on the input of a chosen message \(M^{\prime}\). * Decryption query: this query is used for modelling an adversary that can obtain a plain-text \(M^{\prime}\) on the input of a cipher-text \(C^{\prime}\). * Corrupt query \(\mathit{Corrupt}(i)\)(or \(\mathit{Corrupt}(j)\)): this query is used to expose the private key of the player \(p_{i}\)(or \(p_{j}\)) to the adversary, where \(i,j\in\{U,V\}\). * Hash query: this query depends on what the input is; the simulator then returns the related output to the attacker. * Test query \(\mathit{Test}(i,k)\): this query is used to define the advantage of an adversary. When the adversary \(\mathcal{A}\) has finished all of the above queries to the oracle, they can make this test query on an instance \(\Pi^{k}_{i}\)(or \(\Pi^{k^{\prime}}_{j}\)) to the simulator. At this time, the simulator will flip a coin \(b\). If \(b\) is \(1\), then the real session key is \(\mathit{ssk  
in the polynomial time bound \(t^{\prime}\) under the above Ind-CCA security definition with \(q_{h}\) hash queries, at most \(q_{e}\) encryption queries, and at most \(q_{s}\) decryption queries, respectively._  
Proof of Lemma 4.: We assume that \(Pr[dis]\) is a non-negligible probability in the simulation game. We can then construct an attacker \(\mathcal{D}\) whose work is to distinguish the cipher-text under the Ind-CCA encryption/decryption scheme. There is also an attacker \(\mathcal{F}\) whose goal is to break the encryption/decryption of our proposed scheme \(SE\). Next, we construct \(\mathcal{D}\) as the simulator that simulates the attacking environment in which \(\mathcal{F}\) can mount its attack. First, \(\mathcal{D}\) simulates an encryption oracle \(SE_{pk_{i}}(\cdot,\theta)\), where \(i\in\{U,V\}\), and generates the \(C^{\prime}\) to the attacker \(\mathcal{F}\) on the plain-texts (\(M^{\prime}\)) chosen by the attacker \(\mathcal{D}\) in the selected instance \(\Pi^{k}_{i^{*}}\), where the partner of \(\Pi^{k}_{i^{*}}\) is \(p_{j^{*}}\). In addition, \(\mathcal{D}\) also simulates the decryption oracle to answer the decryption query issued by the attacker \(\mathcal{D}\). We consider the following steps. First, \(\mathcal{D}\) prepares all hash functions, including \(H_{1}\) and \(H_{2}\), two hash functions with collision-resistance. It also generates the instances \(i^{*},j^{*}\longleftarrow[1,...,I-1]\) of each player \(i\), where \(i\in\{U,V\}\).