Probabilty of a Race Attack in time t with hashing power q

Question

I read the original Satoshi's paper on bitcoin as well as Rosenfeld's paper on "Analysis of hashrate-based double-spending". However, they don't answer my question, which is as follows.

Let's assume attacker has q hash power (less or more than 50%) and that the merchant is waiting k confirmations (e.g. 6). What is the probability that after time t (e.g. 12 hours), the attacker will produce longer chain in order to double-spend money?

I have seen formulas to calculate the probability, that the attacker will eventually produce the chain, but what I am looking for is, what happens if the attacker is constrainted by time t. I know that the attacker has 100% chance on eventually getting the longer chain, but I also know that when the hashing power q is around 50%, it would take a lot of time and you would normally need around 60-70% of hashing power.

Answer 1

So there are two ways to launch this attack.

1. (stupid way) get 6 confirmations on a transaction, the go back to the block before the transaction and mine 6 blocks in a private chain before catching up to the mainline and eventually beating it. I would describe this as a race between main chain and private chain where the main chain has a 6-block lead.

2. (smart way) immediately after the transaction is in the mempool of the main chain, start a private chain with the double spend tx and keep the chain private until there are 6 confirmations on the main chain and release the longer chain into the world to win. I would describe this as a race between the main chain and private chain where they start from the same place.

I think you're talking about doing (1) where (2) is the more likely to succeed attack. With (1), yes, this will take a long time since you have 6 more blocks to catch up. With (2), you're likely to beat (1) soon after however many confirmations. Since the length of the chain is determined by how much proof-of-work is in it, you might be able to win with a shorter chain (!). In any case, calculating the probability of doing so by time T is an interesting problem for (2) but it's at or above your percentage of the hashing power at any length as long as the second method of attack is used.