## 3 Offline Payment Function after Types of Currency  
Research studies related to the offline payment function of digital payment methods have been gaining attention with the emergence of e-cash in the 1990s and early 2000s. [PERSON] [86] defined the offline payment function as follows:  
* A currency that can be exchanged even in the absence of a network connection;
* A transaction that can be established without a third party acting as an arbitrator.  
[PERSON] and [PERSON] [87] mentioned that an offline payment function without bank intervention has the advantage of cost efficiency and speed. Another advantage of the offline payment function can be observed in terms of capacity. Furthermore, [PERSON] and [PERSON] [88] mentioned the advantage of a database was significantly reduced in size for the data-related bank transactions in the offline payment function.  
Further research from the works based on the advantages has progressed, focusing on the implementation of offline payment functions in the field of e-cash. [PERSON] and [PERSON] [88] proposed an untraceable offline e-cash system using the ElGamal signature scheme to ensure user privacy. In this system, double spending can be prevented by revealing the identity information of double spenders to other users through the ElGamal signature scheme.  
On the other hand, [PERSON] et al. [89] argued that the system proposed by [PERSON] and [PERSON] is vulnerable to man-in-the-middle attacks and does not completely prevent double-spending attacks. A man-in-the-middle attack occurs when the payer sends money to the payee through a mechanism in which a man-in-the-middle intercepts the money by eavesdropping. [PERSON] et al. [89] conducted a study to solve the problems that may arise in existing offline payment function using discrete logarithm problems and cryptographic techniques, such as the Schnorr signature and blind signature.  
As many studies have dealt with topics related to the offline payment function, offline transaction methods, such as near field communication (NFC) [90], quick response (QR) code [91], and Bluetooth [92], are now diversified, and it is apparent that these studies are meaningful. Among studies related to e-cash, [PERSON] et al. [72] present an online payment protocol and an offline payment function for auxiliary means. Bitcoin emerged after studies related to e-cash. Although only a few studies have dealt with the offline payment function of Bitcoin, [PERSON] et al. [93] presented a solution to enable a secure offline payment function of Bitcoin. With the appearance of CBDC, [PERSON] and [PERSON] [94] proposed a solution to use the offline payment function in situations where online payment of CBDC is impossible, such as network malfunction or underground environment, by using dual offline payment. Here, dual offline payment refers to when a payment is completed when both the payer and the payee are offline. [PERSON] and [PERSON] [94] stated the necessity of offline payment function for CBDC by covering the problems that may arise from making transactions and payments possible through offline payment. Eventually, [PERSON] et al. [95] conducted a direct protocol study that implements the offline payment function of CBDC. This section analyzes the three studies above that dealt with the offline payment function for each of the different types of currencies, e-cash, Bitcoin, and CBDC.