A system for bankcard payments, wherein payments are peer-to-peer transactions between the cardholder and merchant without participation from any third party. The system uses standard bankcards and supports both debit and credit card payments. For authorization of payments the system uses a special, global, distributed, append-only, and secure public bankcard ledger. The entries in the ledger are virtual accounts used for bankcard payments, organized into account chains. The system provides security, privacy, anonymity, and untraceability for cardholders and their transactions. The system allows for various users of the system to participate in transactions using preexisting payment systems. In one embodiment, specially designed mobile software modules and an innovative payment protocol all for transactions to be made between users.