Every Bitcoin address is based on a secret key, from which the public key (associated to a Bitcoin address) is calculated. Once you have the private key for an address, you have the control of that address and can use it to transfer funds.

This secret key is a 32-bytes unsigned integer. You can generate a lot of secret keys, calculate the public keys associated to them and see if they contain bitcoins. If it’s the case, you can transfer the money to an address you control, because you have the secret key.

Such an attack is completely infeasible, because the private key space is really, really huge. There are 115792089237316195423570985008687907853269984665640564039457584007913129639936 secret keys available (1077).

Oh, and they are all listed on directory.io ! Of course, this website is kind of a joke, and all is calculated on the fly when you request a specific page. It also shows the danger of entering your secret key on an unknown website, for example to see if it was compromised...

However, we can bruteforce only a tiny fraction of this space, concentrating on secret keys with some distinctive features. This is what I will explain.
Private keys are numbers...

So, why not try really tiny numbers ?

I have made a script that tries every secret key, counting from 1. After some seconds, I found dozens of already used addresses, with private key smaller than 100 000 !

In particular, the 1EHNa6Q4Jz2uvNExL497mE43ikXhwF6kZm address (corresponding to the private key 1) was already used quite a lot, as 4 bitcoins already flowed through it.
Brainwallet

Brainwallet is a website that allow people to create private keys from a passphrase. It calculates the private key from the sha256 of the passphrase.

By using a password dictionary, we can search for private keys corresponding to classic password that were already used. A search allowed me to find nearly 10 000 addresses that have contained Bitcoins at some point in time ! I was never able to find any address containing money, and nearly every time they had contained only really small amounts of money, but here is an interesting sample :

asd - 1G4Mt5JLtrdj4hM6MkyaQpHmZzVoojLFX3
cat - 162TRPRZvdgLVNksMoMyGJsYBfYtB4Q8tM
hello - 1HoSFymoqteYrmmr7s3jDDqmggoxacbk37
password - 16ga2uqnF1NqpAuQeeg7sTCAdtDUwDyJav
test - 1HKqKTMpBTZZ8H5zcqYEWYBaaWELrDEXeE
fuckyou - 1LdgTMX2MEqdfT3VcDpX4GyD1mqCP8LkYe
1 - 12AKRNHpFhDSBDD9rSn74VAzZSL3774PxQ
icecream - 1CwjHYsPUc4Du8dx7AkdBJj4ebWC8bxkF3
alfanumerico - 19JsLFDRxuTsAjapE79FgoVNdNdB2hNU5M
[empty string] - 1HZwkjkeaoZfTSaJxDw6aKkxp45agDiEzN
correct horse battery staple - 1JwSSubhmg6iPtRjtyqhUYYH7bZg3Lfy1T

If we look at the history of transactions on these addresses we can see that in a lot of cases, a few seconds after money was deposed in them, it was withdrew to another account.

In particular, it was often sent to one of these accounts :

    1brain7kAZxPagLt2HRLxqyc3VgGSa1GR
    15LPodNPGv5qsKPc4PUfyeJKdDReWjCknS
    1mineraceNChe36ftTXU5Y8EvypyM3iu8

We can see that the first address already stole 0.36 Bitcoins, 2 seconds after they were transferred to the address corresponding to the key « alfanumerico » ! This address seems to contain more than 1 Bitcoin, probably stolen all automatically... Notice that the address starts with « brain », maybe it could have something to do with Brainwallet, no ?

Another interesting fact, the second address stole very tiny amounts of money from classical addresses, and we can even see that a comment was left about one of these transactions :

Stop taking this dust! You must wait until there is more sent to complex ones! Set the trap...

A Bitcoin theft that left a comment to another, advising him to stop stealing tiny amounts of money and wait for bigger amount on more difficult addresses...
Technical considerations

To bruteforce Bitcoins like that, you need to find the address associated to a private key, as fast as possible. But, you will also need to know if each of the address was already used in a transaction.

For that, you will need to iterate trough the entire blockchain. You can then fill a Bloom filter with every address seen in it. Once it is built, you will be able to know if an address was used in the network with a really small lookup time.

As we have seen, it seems that some people even built a gigantic database, mapping from “classic” addresses to their associated private keys. They seems to be able to steal money only seconds after it was deposited...
Conclusion

Well, it seems that there is money to make, and here is the proof that some people are already on it...

This also demonstrate that you can use a service like Brainwallet, but you need to choose a really strong passphrase, that will resist bruteforce. If any computer or any other human can think of it, you are doomed !

Instead of attacking every potential transaction that used a weak private key, it’s also possible to focus on weak random number generators, or implementation problems in ECDSA. This is also a reality, and you can check out this blog post for more.