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

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.

attack

Maciej Żurad

Posted 2015-11-30T09:39:25.233

Reputation: 41

2The attacker is only guaranteed to eventually get the longer chain when they have more than 50% of hashpower. – Nick ODell – 2015-11-30T17:41:52.777

@Nick Yes, I know that, but my question is: If the attacker has q hashing power and transaction requires 6 confirmation, then what is the probability, that he will succeed with the attack if he has some time t to perform attack. Let's say he has only control of q hashing power for t time. I think this can be seen as Gambler's Ruin, but trying to calculate the probability of getting x coins with an unfair coin of prob. q – Maciej Żurad – 2015-11-30T18:44:26.747

Answers

So there are two ways to launch this attack.

(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.
(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.

Jimmy Song

Posted 2015-11-30T09:39:25.233

Reputation: 7 067