## 7 Security Analysis and Cryptographic Guarantees
### Formal Security Proof in the Canetti-Krawczyk Model  
In this section, we provide a formal security analysis of the refined key agreement protocol using the Canetti-Krawczyk (CK) model, which captures realistic adversarial behavior and session-level freshness. The goal is to prove that our construction satisfies _authenticated key exchange_ (AKE) security under standard cryptographic assumptions, namely the hardness of the Elliptic Curve Discrete Logarithm Problem (ECDLP), the collision resistance of the employed hash function, and the semantic security of the encryption scheme used in the protocol. Security Model  
The CK model considers a probabilistic polynomial-time (PPT) adversary \(\mathcal{A}\) interacting with multiple protocol instances, or sessions, associated with honest parties. Each party can participate in multiple concurrent sessions, modeled as stateful oracles. The adversary controls the communication channel and can:  
* Send: Deliver arbitrary messages to protocol instances. * Reveal: Obtain session keys from completed sessions. * Corrupt: Reveal long-term keys of a party. * Test: Challenge a session with a random key or the real session key. A protocol is considered AKE-secure if the adversary cannot distinguish the real session key from a random value, except with negligible advantage, provided the session is _fresh_ (i.e., it has not been subject to both corruption and key reveal). Let \(E\) be an elliptic curve defined over a binary field \(\mathbb{F}_{2^{m}}\), and assume that:  
* The ECDLP over \(E\) is hard;
* The hash function \(H(\cdot)\) is modeled as a random oracle;
* The encryption scheme \(\mathsf{Enc}_{\mathsf{sk}}(\cdot)\) is IND-CPA secure. Then, the refined GN-AK protocol using double-base scalar multiplication and point halving is AKE-secure in the CK model. We outline the proof via a sequence of security scenarios and reduction arguments, where \(\mathcal{E}\) is a potential attacker. Security Scenario 0: \(\mathsf{Exp}_{\mathcal{E}}^{\Pi^{\prime},0}(k)\)--Real Execution of the Protocol  
This experiment models the real-world execution of the protocol \(\Pi^{\prime}\) under security parameter \(k\), in the presence of a probabilistic polynomial-time adversary \(\mathcal{E}\). It serves as the baseline experiment against which all subsequent transformations will be compared in the sequence-of-games approach. In \(\mathsf{Exp}_{\mathcal{E}}^{\Pi^{\prime},0}(k)\), all cryptographic components are instantiated as specified in the protocol design:  
* The hash function \(H(\cdot)\) is treated as a standard deterministic function;
* The encryption function \(\mathsf{Enc}_{\mathsf{sk}}(\cdot)\) is evaluated using the real asymmetric encryption scheme;
* Session keys are derived as \[K=[\kappa_{\mathcal{A}}][\kappa_{B}]\Phi,\] where \(\kappa_{\mathcal{A}},\kappa_{B}\leftarrow\mathbb{Z}_{n}\) are independently and uniformly sampled ephemeral scalars, and \(\Phi\) is the fixed public base point. The adversary \(\mathcal{E}\) interacts with the protocol by activating multiple instances (sessions) of honest parties \(\mathcal{A}\) and \(\mathcal{B}\) using the following oracle queries:  
* Send: deliver messages to protocol participants and receive their outputs;
* Reveal: obtain the session key of a completed session;
* Corrupt: learn the long-term private key of a party;
* Test: challenge a session with either the real session key or a uniformly random value. The Test query is issued on a session that is _fresh_, i.e., it satisfies the following freshness conditions:  
* Neither the session nor its partner has been subjected to a Reveal query;
* The long-term private key of the owner has not been revealed via Corrupt. The experiment outputs 1 if the adversary \(\mathcal{E}\) correctly guesses whether the Test query returned the real session key or a random value. Otherwise, it outputs 0.