Anonymous Internet banking is the proposed use of strong financial cryptography to make electronic bank secrecy (or more precisely pseudonymous banking) possible. The bank issues currency in the form of electronic tokens that can be converted on presentation to the bank to some other currency. This concept has a long history in which free banking institutions have issued their own paper currency often backed by a physical commodity.
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History
While the academic study of trust relationships and systems has long been the domain of intelligence services such as the NSA, the growth of the Internet in the 1990s and the contemporary declassification of related knowledge allowed for greater public discussion of the potential for anonymous banking services by groups such as the cryptoanarchists and cypherpunks.
Example system implementations
For a full list see list of anonymous internet banking systems
The underlying mathematics
Anonymous internet banking depends on the mathematics of public key cryptography and blind signature algorithms. In this simple example we have Alice and Bob and a banker. The banker generates an RSA public key with modulus
Bob asks the banker for a $100 deposit slip in anticipation of Alice wanting to transfer money to him. To generate a deposit slip the bank selects a large, globally unique random number
This encrypted value
When Alice wants to pay Bob $100 she asks for the deposit slip and Bob sends her
Because of the blinding process, the Bank is not able to associate
Different public keys can be used for different denominations of currency so this system doesn't take appreciably longer for large transactions.
Note that if neither Alice nor Bob wishes the bank to know that they performed a transaction with each other, then it is hard for the bank to find out. However, in order to ensure this is the case many people need to be making transactions at the same time. Otherwise the bank can figure it out by the timing of the transactions, using traffic analysis.