Satoshi Nakamoto is not good at mathematics? Two scholars publish a paper claiming to have fixed the flaws in the Bitcoin white paper

The mathematical achievements of Bitcoin founder Satoshi Nakamoto have been questioned by two mathematicians.

In Chapter 11 of the Bitcoin white paper, Satoshi Nakamoto explains that the majority of honest miners can always outperform the minority of dishonest miners. As a key innovation of digital currency, this ensures that once transactions have enough confirmations, they are almost irreversible, which effectively solves the double-spending problem.

However, as Israeli mathematician Meni Rosenfeld explained in 2012, Nakamoto was only making some simplifying assumptions. While Bitcoin mining is a random process, Nakamoto did not fully take into account that both honest and dishonest miners can be lucky or unlucky.

Cyril Grunspan, a mathematician at Leonardo da Vinci School of Engineering (ESILV) in Leonardo, and Ricardo Pérez-Marco, a mathematician at the French National Center for Scientific Research, have taken this randomness into account and published a new paper (full text download) that finally corrects Nakamoto's "error".

“Satoshi incorrectly assumed that it would take exactly the same amount of time on average for honest miners to find a block,” Grunspan told reporters. “However, this is actually a rough approximation because the time it takes for honest miners to find a block is non-deterministic. Therefore, the distribution of the number of blocks mined by the attacker is actually what is called a ‘负二项分布’, rather than the assumed ‘泊松分布’.”

Essentially, the Bitcoin whitepaper assumes that two factors are needed to calculate the irreversibility of a transaction. Satoshi correctly assumed that the total computing power of the attacker is one of the factors: as the attacker controls more computing power, more confirmations are required. Then, Satoshi correctly assumed that the number of transaction confirmations is another factor: the more confirmations a transaction has, the more secure it is.

Grunspan and Pérez-Marco now show that a third factor also comes into play: the deviation from the average mining time of honest miners when they find a block, i.e. their "luck". If they are very lucky and find a block faster than average, their chain may be further ahead, and attackers will have less time to secretly mine an alternative chain. On the other hand, if honest miners are unlucky and find blocks slower than average, they may be at a disadvantage. As a result, attackers will have more time to mine an alternative chain.

what does that mean?

The good news is that Bitcoin is still working as Satoshi intended.

“In this paper, we show that as the majority of honest miners find more blocks, the probability of a double spend drops to zero,” Grunspan said. In other words, it becomes increasingly difficult for a small number of attackers to catch up and surpass the majority of honest miners.

In other words, the security assumptions in Satoshi’s white paper need to be slightly modified. Not only the attacker’s computing power and the number of blocks behind the attacker must be taken into account, but a third factor must also be taken into account. In their paper, Grunspan and Pérez-Marco explain the importance of this issue.

“This is interesting information that merchants can use to monitor risk,” Grunspan said. “Let’s say a merchant always waits for six confirmations before shipping his product to a customer, his risk is low. On average, it takes 60 minutes, but sometimes he has to wait two hours to get six confirmations, and if that happens, the risk of double spending is higher. So for the same level of security, he’s actually waiting for the seventh confirmation. If the confirmations come faster, he may even have no problem waiting for five confirmations.”

Since double-spending prevention is the core innovation of Bitcoin, the simplification of mathematics in Satoshi’s white paper is remarkable, especially for mathematicians. Therefore, Grunspan does not allow such errors to happen, but from another perspective, it is understandable to simplify assumptions in the white paper.

Perhaps, it reveals another hint about the origins of Bitcoin.

“Satoshi is a genius,” Grunspan concluded, “but he’s not a mathematician.”