So say they did all of the malicious things you've stated, is that illegal?

> The way they could take over the network is by crafting a new blockchain, forging fake transactions or even claiming that they own all bitcoins. If they solve blocks faster than the rest of the network, soon their malicious chain would become longer and therefore accepted by all clients! Clients always trust the longest chain because it takes the most work to make, so you would have to control most of the mining power to fake anything for long.

This is incorrect. Having 51% or more of the computing power of the network doesn't allow you to replace the whole blockchain (there are checkpoints, and it gets more difficult as you want to replace a bigger part of it). You also can't forge new transactions, since transactions need to be signed by the sender.

Basically the only practical attack (assuming no serious bugs in the clients) is a double spend: they could pay out of their own wallet a sum X to a third party, the third party checks that the transaction has Y confirmations (remember, as Y grows, it gets more difficult to pull this off), they receive whatever they've bough from the third party, and then they start pushing a different and longer block not including the transaction.

It's basically a computer with specialized hardware whose sole purpose is to mine Bitcoins, see http://www.butterflylabs.com/ for an example

An ASIC is an application-specific integrated circuit. Basically, dedicated hardware like this: http://www.butterflylabs.com/

Application Specific Integrated Circuit

in the bitcoin context they chips created for the specific purpose of hashing, allowing you to mine for bitcoins quickly and effficiently.

Here's an explanation:

http://cryptojunky.com/blog/2013/02/17/bitcoin-asic-roundup/

ASIC is an implementation of a circuit. They will be the fastest implementation of the logic, however they are the most expensive implementation of that logic by far.

Unlike FPGAs once logic is implemented in an ASIC the implementation is set in stone, so if there is a bug in the hardware fixing it will require producing an additional run and incur the expensive cost again.

The high cost is why FPGAs are popular for both prototyping and production units when possible as it is much cheaper for smaller quantities.

ASIC having exactly the logic necessary to target a task is why they are the most efficient at the task compared to general purpose hardware.

https://en.wikipedia.org/wiki/Application-specific_integrate...