𝗖𝗮𝗹𝗰𝘂𝗹𝗮𝘁𝗶𝗼𝗻𝘀

If an attacker strives to create an alternate blockchain that’s longer than the honest one, it's essential to understand that Bitcoin's security and integrity remain intact. The system has protections against arbitrary alterations, such as generating invalid bitcoin or seizing someone else’s funds.

The main way for an attacker to exploit #Bitcoin is by modifying one of their recent transactions. In other words, they could buy something with bitcoin, and then undo the transaction. But even this would need to follow the system's rules and withstand scrutiny from honest nodes.

In the realm of Bitcoin's security, there's an ongoing competition between the honest chain (the legitimate blockchain) and an attacker's chain (a potentially malicious blockchain). Satoshi likens this contest to a "Binomial Random Walk", which is a statistical model where something moves unpredictably from one state to another.

Essentially, this description portrays the race between the honest and attacker's chains as a series of random events, with each additional block being a “step” in this unpredictable journey. Seeing it this way helps us grasp the probabilistic nature of their competition, and the factors influencing their progress.

It’s possible to calculate the probability of an attacker catching up to the legitimate chain. Satoshi compared this to a "Gambler's Ruin" problem, in which a hypothetical gambler begins with a deficit and has unlimited resources to continue playing. This mirrors an attacker attempting to catch up to the honest blockchain when the attacker's chain is initially behind.

The probability being calculated is similar to assessing the likelihood of the gambler ever reaching the point of breaking even in their betting game. This provides a formal method to gauge the likelihood of these events happening over time.

As we calculate the probability of an attacker catching up to the honest blockchain, we see that 𝘵𝘩𝘦𝘪𝘳 𝘤𝘩𝘢𝘯𝘤𝘦𝘴 𝘥𝘪𝘮𝘪𝘯𝘪𝘴𝘩 𝘦𝘹𝘱𝘰𝘯𝘦𝘯𝘵𝘪𝘢𝘭𝘭𝘺 as the number of blocks they need to catch up with increases.

This analysis even gives the attacker the benefit of the doubt, assuming he has greater computational power than the honest nodes. But even then, 𝗶𝗳 𝘁𝗵𝗲 𝗮𝘁𝘁𝗮𝗰𝗸𝗲𝗿 𝗱𝗼𝗲𝘀𝗻'𝘁 𝗴𝗮𝗶𝗻 𝗮 𝘀𝘂𝗯𝘀𝘁𝗮𝗻𝘁𝗶𝗮𝗹 𝗹𝗲𝗮𝗱 𝗮𝘁 𝘁𝗵𝗲 𝗼𝘂𝘁𝘀𝗲𝘁, 𝘁𝗵𝗲𝗶𝗿 𝗰𝗵𝗮𝗻𝗰𝗲𝘀 𝗼𝗳 𝘀𝘂𝗰𝗰𝗲𝘀𝘀 𝗳𝗮𝗹𝗹 𝗼𝘃𝗲𝗿 𝘁𝗶𝗺𝗲.

The expanding computational power of the honest network makes it progressively harder for the attacker to overtake it. This beautifully demonstrates the security and robustness of the Bitcoin network.

Next, Satoshi delves into the question of how long a bitcoin recipient may wish to wait, in order to have enough confidence that the transaction can never be altered. He describes a scenario that involves a cautious recipient who wants to guard against a possibly malicious sender who intends to alter the transaction.

As a countermeasure to prevent a dishonest sender from preparing a fraudulent transaction in advance, the recipient can generate a new pair of cryptographic keys and provide the sender with the new public key just before signing the transaction. This approach reduces the window of opportunity for the sender to work on an attacking chain ahead of time.

Then Satoshi explains how waiting for Bitcoin transactions to be confirmed is crucial for transaction security. He estimates the potential progress an attacker might make during this waiting period, using statistical calculations, including Poisson distributions (𝘪.𝘦. 𝘢 𝘮𝘦𝘵𝘩𝘰𝘥 𝘧𝘰𝘳 𝘱𝘳𝘦𝘥𝘪𝘤𝘵𝘪𝘯𝘨 𝘳𝘢𝘯𝘥𝘰𝘮 𝘰𝘤𝘤𝘶𝘳𝘳𝘦𝘯𝘤𝘦𝘴 𝘸𝘪𝘵𝘩𝘪𝘯 𝘢 𝘴𝘱𝘦𝘤𝘪𝘧𝘪𝘤 𝘱𝘦𝘳𝘪𝘰𝘥 𝘰𝘧 𝘵𝘪𝘮𝘦, 𝘶𝘴𝘪𝘯𝘨 𝘢𝘷𝘦𝘳𝘢𝘨𝘦𝘴), to determine the expected value of the attacker's progress.

When all the mathematical probabilities are taken into account, let’s just say the attacker would be better off just using Bitcoin honestly. 😉

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Are digital signatures also public?