Credit card systems are one of the most effective and efficient means for remote payments. For example, when the Internet and credit card systems are combined, people can buy anything without going out from their homes or offices, and as a consequence, development of highly efficient, convenient and environmentally friendly society becomes possible. However, all privacies of customers will be revealed, when credit card companies and Internet providers exchange information about their customers. Therefore, credit card systems must be endowed with anonymous features so that more people accept the systems and fully exploit their benefits.