## 3 Preliminaries
### Elliptic Curve Cryptography (ECC) and Hash-Chain  
**Elliptic Curve Cryptography (ECC):** ECC [21] is a public-key encryption technique based on the elliptic curves' mathematical properties. Compared to traditional cryptographic methods such as RSA, ECC offers the same level of security with much smaller key sizes, leading to increased computational efficiency and reduced storage and bandwidth requirements.  
* Elliptic Curve Overview: An elliptic curve is defined by the equation \(y^{2}=x^{3}+ax+b\ mod\ q\), where \(a\) and \(b\) are parameters and \(q\) is a large prime number. The set of points on the curve, along with the point at infinity \(O\), forms a group. For the curve to have a valid group structure, the parameters must satisfy \(4a^{3}+27b^{3}\cong 0\ (mod\ q)\).
* Key Generation: In ECC, the private key is a randomly selected integer \(d\). The corresponding public key is obtained by multiplying the base point \(P\) on the curve by the private key, resulting in \(Q=d\cdot P\). This multiplication is effectively repeated addition of the point \(P\ d\) times.
* Group Operations: The addition of two points \(P\) and \(Q\) on the elliptic curve results in another point \(R=P+Q\). Scalar multiplication, denoted as \(k\cdot P\), involves adding the point \(P\) to itself \(k\) times, which is fundamental to ECC operations.
* Security Foundation: The security of ECC is based on the elliptic curve discrete logarithm problem (ECDLP). Given two points \(P\) and \(Q=k\cdot P\), it is computationally infeasible to determine the scalar \(k\), especially when \(q\) is large. This difficulty ensures the robustness of ECC against various cryptographic attacks.
* Encryption and Decryption: ECC encryption transforms a message into a point on the elliptic curve using the recipient's public key. The sender selects a random integer \(k\) and computes \(C_{1}=k\cdot P\) and \(C_{2}=M+k\cdot Q\), where \(M\) is the message point and \(Q\) is the recipient's public key. The recipient decrypts the message by calculating \(M=C_{2}-d\cdot C_{1}\) using their private key \(dd\).
* Computational Efficiency: ECC achieves high levels of security with shorter key lengths, resulting in faster computations and lower resource usage. This makes ECC particularly suitable for environments with limited computational power, such as mobile devices and embedded systems.  
**Hash-chain:** The Hash-chain involves the repeated application of a cryptographic hash function to produce a sequence of hash values.  
* Construction: A Hash-chain starts with a seed value \(x\), and each subsequent value is generated by hashing the previous value. For example, starting with \(x\), the chain progresses as \(h(x),h(h(x)),h(h(h(x)))\), and so on. Originally proposed by [PERSON] [22], a Hash-chain is a fundamental cryptographic method used to prevent the theft of user passwords over public channels.
* Features: One of the key characteristics of a Hash-chain is its one-way property: it is computationally feasible to compute the chain in the forward direction but infeasible  
Figure 1: System model of 2 FAKA-C/S protocol.  
to reverse the process. This means that given \(h^{N}(x)\), it is practically impossible to determine \(h^{N-1}(x)\). However, given \(h^{N-1}(x)\), it is straightforward to verify \(h^{N}(x)\). This property makes Hash-chain ideal for resisting replay attacks [23].