Anonymous internet banking

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Anonymous Internet Banking is the name given to 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.

1 The Concept
2 The underlying mathematics
3 See also
4 External links

[edit] The Concept

Abraham wishes to send Zelda the secret documents. Abraham requests a padlock from Zelda. Zelda sends an open padlock but keeps the key. Abraham padlocks the secret docs in a box and sends them to her.

[edit] The underlying mathematics

Let N=PQ where P, Q are large primes with a few additional properties, and thus N is a composite number. Computing the cube root of a number, A, modulo some large composite N is hard, at least compared to computing the cube of A modulo N.

Example: Suppose we're mod 55. Calculating the cube of something is easy, e.g.,

$73 (mod 55) = 343(mod 55) = 343 - (6 * 55) = 13$ .

But finding the cube root is tough; solve for x:

$x 3 (mod 55) = 6$

However if the factorization of N into two primes is known then both can be done with ease. In the following example all arithmetric is done modulo N.

We have two people, Alice and Bob. For simplicity we shall assume they are using the same bank. Alice wants to transfer $100 to Bob anonymously.

Only the bank has the ability to calculate cube roots, so the bank charges $100 for the service of finding the cube root of a number. The bank pays anyone who can calculate the cube root of an arbitrary number $100.

Bob asks the bank for several random numbers in anticipation that someone will want to pay him money. The bank remembers which numbers he was given so that it knows when Bob manages to calculate a cube root of one of them.

When Alice wants to pay Bob $100 she asks for one of these numbers R from Bob. She multiplies it by w cubed (W) and pays the bank $100 to cube root it. The bank returns the answer which is wr (where r is the cube root of R) so Alice divides it by w and sends Bob r. Bob now knows the cube root of R and can tell the bank what it is. The bank pays him $100.

Different composites N 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.