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I'm new to bitcoin and was wondering: As far as I know, bitcoin relies that the majority of the users is to be trusted.
Why is it not a problem then if someone starts up billions of clients, eg. a data center owner? Seemingly a single entity could take over the whole network. What is the bitcoin feature that is the counter-measure here?
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The majority needs to be trusted, but not weighted by their number of software instances, but by either computational power or economic power, depending on the context.
The main technical assumption is that the majority of hashrate is not trying to attack the network. So an attacker would need to purchase and operate more hashrate than the rest of the network combined; this isn't easy.
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You slightly misunderstood the concept of majority here. Bitcoin relies on the majority of trusted computational power. So it does not matter if there will be billions of small clients controlled by the same owner if they are not powerful enough.
The main idea in the beginning was that each client is running independently. So if there will be a lot of independent clients, then for any person who would try to harm the system by combining a lot of computational power (and thus manipulating the results) it would be really hard, because he would be outnumbered by a trusted majority.
But with the idea of mining pools a lot of things has changed. If you will look here: you will see that 3 major mining pools control more then 51% of hashrate.
3The general purpose computing equipment you would find in a data center isn't suitable for such a specialized application. You'd need to invest millions in a data center filled with hardware specifically designed for this task. And if you succeeded, you'd make those millions of dollars worth of equipment worthless. – David Schwartz – 2013-10-23T19:13:34.380