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Comment by stingraycharles | Hacker News Reader

While I understand where your argument is coming from, it is important to ask the question: how many times a day is this piece of code executed on all computers worldwide?

Surely the chance of one person running into this bug is extremely small, but what about all the people on earth combined?

Nevertheless, I think we can classify this method of testing as some sort of hybrid between brute force testing and auditing; I don't think the authors are dismissing that claim this either, but are merely stating that there are ways to make very informed test cases without looking at the code.

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Caveat that I might be completely off, but if, say, 10 bn people tried once a nanosecond since the inception of the universe, you'd still have only 10^10 * 2^86 ≈ 2^33 * 2^86 = 2^119 attempts, that still only cover's 2^-9 (i.e. 1/512th) of the total addressable 2^128 space, i.e. still fairly unlikely that you'll have hit that specific number.

Reasoning about large numbers is hard. 2^128 is considered enough security in any crypto scheme to be "impossible" to break. Consider that Bitcoin uses ECDSA which gives 128 bits of security. This is not brute forceable as there isn't enough energy around to actually flip the bits that many times, let alone do the calculations.

OB: pedantry, an attacker who does one test with 256-bit ECC has a probability of success of ~2^-256. 128-bit security comes from a rho attack which starts off unlikely (2^-256) to be successful and becomes sure to be successful at around 2^128 operations: e.g. it makes progress unlike normal "guess and check" brute force.

I just mention it because this fact means that the sufficiency of ECC security by itself doesn't mean that 2^-128 is "sufficient" against random chance.

But, course, 2^-128 is unfathomably low probability, and it's generally sufficient against 'chance'. Though chance is usually the wrong way to think about attack. For example: If I create software which takes a 256 bit input and does a "if (input == 8675309) exec_shell();" and expose it to the Internet what is the probability of that input? ... probably 1. :)

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