Due to ubiquity of internet, the development of electronic commerce is growing up rapidly. Many payment mechanisms of electronic transactions are proposed, such as electronic cash, credit cards, and electronic wallets. Electronic cash (e-cash) is like the digital version of paper cash, which possesses some key features such as anonymity and untraceability, and it can be used on the Internet for various electronic transactions. In general, electronic cash is classified into two types which are on-line electronic cash and off-line electronic cash. Each of them is suitable for different applications and environments. All of the proposed electronic cash schemes only focus on one of two types. In these schemes, users must decide which type of electronic cash they will use later before withdrawing. However, in this thesis we propose a novel electronic cash scheme which supports a user to withdraw a generic e-cash and then decide to spend it as an on-line e-cash or an off-line e-cash. Our proposed scheme is more convenient for users and more flexible for the bank and shops. It not only satisfies the necessary properties of electronic cash but also provides flexibility.

Mobile communication is very mature today due to the powerful computation and communication capabilities of mobile devices, the flourishing of mobile networks, the popularity of electronic commerce, and the completeness of e-payment mechanisms. It is a pleasure for mobile users to roam around the mobile networks and enjoy the mobile network services. However, there are a lot of security threats in the mobile networks, and thus we need an anonymous mutual authentication and key exchange scheme to guarantee the security and privacy for mobile users in the networks. A payment protocol is also required for charging the mobile users after using the mobile services. However, the existing payment schemes do not support anonymity and credit-based chargeability at the same time. In this dissertation, we propose a secure authentication scheme such that the mobile users can be anonymously authenticated by the system and the system can still make correct charge to these anonymous mobile users via a credit-based way simultaneously. We also propose a novel e-cash scheme which can support each mobile user to withdraw a generic e-cash and decide to spend it as an on-line e-cash or an off-line e-cash according to the payment requirement of the anonymous authentication scheme. Our proposed schemes are convenient and flexible for the mobile users, the system operator, and the bank. Besides, full privacy can be achieved for mobile users owing to the combination of our proposed schemes, which can be performed in current mobile devices efficiently with few battery energy consumptions. Furthermore, we provide anonymity control, no swindling, tamper resistance, secure mutual authentication, secure key exchange, and secure forward secrecy in the proposed anonymous authentication scheme and the e-cash scheme, where these security features are demonstrated by formal security models and theoretical proofs.