If transactions in a block are valid, in order to add that block in the block chain, a proof of work needs to be found. I have read the bitcoin paper by Satoshi.
If the difficulty of the proof of work requires say 2^52 computations (13 hex zeros) on an average, and since every node on the network is working independently, why can't the powerful adversary surpass the length of current block chain and present his version of block chain to the network? Specifically, why does the attacker have to control 1 percent or x percent of the network's computational power, when honest nodes on the network are not working in collaboration to find the proof of work?
If the adversary can find a proof of work quicker than the most powerful honest peer, he can compute a longer block chain and broadcast it to the network.
Let us assume that there are 2^20 nodes on the network, each computing 2^40 hashes per second on an average. Each node would then require 68 minutes to find a proof of work (trying 2^52 hashes). The total computing power of the network is number of nodes * computing power of each node = 2^20*2^40 = 2^60.
If the adversary operates at the speed of 2^45 hashes per second, he requires just 2^7 = 2 min to find a proof of work (2^52 hashes).
Now, the computing power of the network is 2^60, however each node is trying to find a proof of work independently. The computing power of adversary compared to the network is 32,000 times smaller. The amount of computing power controlled by the adversary is 1/32000 = 0.00001%, but still he can compute the longer block chain.
Please help, if I am assuming something wrong here. The honest nodes on the network do not work in collaboration. So attacker need not control 50% of computing power on the network and has to expend computing power just more than average computing power of the honest nodes.