Company A allows people to rent out the spare processor power on their home PCs. A bit like seti@home, only done commercially. Users download a client that runs in the background and get paid a few dollars every week in return (I'll call this the open mode). The company sells the combined users' teraflops
in bulk to anyone who wants it (eg. drug companies who need to compute protein structures). For security reasons, all data and algorithms are encrypted before they reach the user.
In order to be able to trade the processing power like a commodity, it gets broken down into chunks (I'll call them units). Every day a new set of units is issued according to projected processing power of the network for the next 30 days, minus a buffer of 20% to account for errors and users who cancel their subscription. The units are sold on the same day and expire 30 days after they have been issued. Each unit has two hash sums associated with it: one private and one public. Both are sent to the buyer over an encrypted line.
Let's say a customer (eg. the drug company) wants to perform a computation worth 100 units. They simply send 100 private/public hash sum pairs, which they have purchased previously, to company A. Once the calculation is completed, the hash sums are rendered unusable. The public hash sum only serves one purpose: To query company A if the associated private hash exists and has been consumed yet.
So far so good. Now comes the "anonymous currency" bit. As well as "open mode" users can also run the client in "stealth mode." In this scenario users are sent a hash pair for every unit of computation they perform, instead of cash. Users in stealth mode are not connected to company A directly, but via several hops over other users' computers and layers of encryption, so that it's very hard to trace them, even by company A itself (like i2p and tor).
Why all this trouble? Because these hash pairs can now be used like an anonymous currency. Another company, B, runs an online marketplace like ebay. Except it's not run on a traditonal website, but over company A's encrypted anonymous network (or on similar networks like Freenet and i2p). Company B also allows users to turn their hash pairs back into cash by acting as an agent between home users and buyers of processing power.
Let's say I meet a hacker on some anonymous chat who says he can give me information on how to reverse engineer a program, but he wants $20 for that information. The trouble is, how can I send him $20 without knowing his identity? Simple: I run my client for a few weeks until I have collected $20 worth of hashes, and send him the hashes. He can simply trade his hashes for cash without fear of being revealed - remeber - company A does not know WHO they've sent those hashes to, only the date. What if I want to send larger sums? I could even image that neighbourhood server farms emerge where people can buy hashes for cash.