Reversing network hashpower I'm reading this article by Vitalik Buterin on Problems in Cryptocurrency and don't quite understand what this means:


  Ideally, after some number of blocks (perhaps logarithmic in the total size of the network) every transaction should require 51% of network hashpower to reverse. However, solutions where transactions can pay very small fees for a lower "level" of security are acceptable, though one should take care to avoid situations where an attacker can profit by performing one attack to reverse very many small transactions at the same time


I understand that Sybil attack can happen if more than 51% of the network is under one person's control, but I don't understand what it means to reverse network hashpower, and why 51% of it should be reversed.
In theory, this attacker owns enough computing power that they could execute a "double spend" attack. They could spend coins in one place, allow the coins to enter the block chain as normal until the required confirmations are met, then fire up their 51% of the miners to craft a fraudulent fork of the block chain in which those coins were never spent, allowing them to re-spend the coins. This could theoretically be repeated for as long as the attacker maintained control of 51% or more of the hashrate.

Realistically, 51% is only the point at which this becomes possible not the point at which it becomes likely or easy. An attacker would probably need something like 65% to actually execute such an attack.

What can an attacker with 51% of hash power do?