Due to the fast progress of the internet technologies, electronic commerce becomes more and more popular. Many people and businesses deal with their transactions via the internet. The technologies of credit cards, electronic tickets, e-cash, and other advanced services have realized the vision of electronic commerce. In this thesis, we propose an off-line e-cash scheme with anonymity, untraceability, double-spending checking, and traceability. Anonymity and untraceability must be possessed in any e-cash scheme. In an off-line e-cash scheme, the bank or the third party (TTP) must be able to revoke the anonymity of a user who doubly spent her/his e-cash(s). In our proposed e-cash scheme, the bank can fast derive the identity of the user who doubly spent her/his e-cash(s) without the participation of TTP. If some illegal transactions are reported, TTP can also directly revoke the anonymity of the user who spent her/his e-cash(s) in the illegal transactions. In addition, the police needs to trace a specific user in some situation, and we propose a process to satisfy this requirement,called traceability.

A digital signature technique has evolved into varies digital signature schemes in different application environments. In general, a digital signature consists of a random number and a hash function in addition to signing function. The random number can be used to provide the randomization of digital signatures. The hash function can be used for generating a message digest that has a fix length and is convenient for signing. The random number that hides in the digital signature is a useful factor. If we can use this factor well, then the digital signature can carry the other secret messages. On the basis of the concept of a subliminal channel proposed by Simmon, we have proposed multiple subliminal channels that can carry more than one subliminal message to different subliminal receivers. Furthermore, by using the concept of a subliminal channel, we can use the random number as another secure parameter of the digital signature. This concept leads to a forward-secure digital signature with backward-secure detection when the subliminal channel is embedded in the signature. We have proposed a forward-backward secure digital signature. A hash function is an important tool for generating a message digest. The hash function used in a signature must be one-way and collision resistant. A signing message will map to a message digest via a hash function. In recent years, several chameleon hash functions have been proposed. A chameleon hash function is a trapdoor one-way hash function that prevents everyone except the holder of the trapdoor key from computing the collisions for a randomly given input. There are various studies that apply the chameleon hash function to online/offline digital signatures and sterilization signatures. In this thesis, we apply this concept to a network secure gateway. We have achieved fast blind verification for an application gateway, such as a firewall. Further, we propose triple-trapdoor chameleon hash function and apply to vehicle owenship identification scheme. We have achieved the fast identification for vehicle ownership without connect to online database. We also have proposed threshold chameleon hash function and achieved that the collision will control under the threshold value. The trapdoor information will be exposed after the number of collision has accomplished.