Why does a malicious miner need more hashing power than the rest of the network combined? I don’t quite understand why a malicious miner needs more hashing power than the rest of the network combined. 

When miners are solving a hash puzzle, aren’t they solving the puzzle individually? If so, a malicious miner merely needs to be the most powerful node in terms of her hashing power. But I guess miners are actually solving the puzzle together? In this case, it makes sense that the malicious miner needs to have more than 51% of the hashing power.

edited: November 21, 2018

My question is not how to pull of the 51% attack, but why a malicious miner needs more hashing power than the rest of the network combined.
I don’t quite understand why a malicious miner needs more hashing power than the rest of the network combined


Generally speaking, a malicious miner is one attempting to perform a double-spend or reverse the block-chain.

A miner can attempt to broadcast any block, but it will get rejected if the receiving nodes do not agree with it (checking against consensus rules).

It is important to remember that Bitcoin nodes will always favor the chain with the most amount of proof-of-work (chainwork); this is specified in the protocol's node implementation. The malicious miner needs sufficient hash power to recalculate more blocks than the current chain such that it will have more proof-of-work. Once the attacker's chain has greater chainwork, nodes will discard the existing chain in favor of the attackers.

The miner is indeed required to have more power than the rest (the "honest" part) of the network combined. The miner would need a majority of the hashing power. You will often hear this referred to as a 51% attack.

Additionally as Learn Cryptography suggests:


  Hitting 51% network control is not a guarantee of success, just the point where success is likely. In fact, you could attempt this sort of attack with much less network control, but your odds of success would be very low. 


This is also why it is recommended for users to wait until a transaction has over 6 confirmations as it gets exponentially more difficult for an attacker to double-spend as confirmations build on top of the transactions current block. With 51% hash power and 1 confirmation, the attacker is practically guaranteed to be successful in a double-spend.

You can calculate your own attack probabilities for this here


  When miners are solving a hash puzzle, aren’t they solving the puzzle individually?


Miners do their hashing independently, yes. Each miner will check many nonces for the current block until either a solution is found that meets or is below the target, or the block has been broadcast by someone else.