2^-80 is an incomprehensibly tiny number. Malice or incompetence are both FAR more likely.
If on the other hand you are looking for a number that is_prime says is prime, and you are iterating through candidates, you need to know how likely it is to find a prime number in the first place to tell you how unlikely this is. In most cases the chance of a false positive will be much, much higher than 2^-80.
If for example, you only expected to find a true prime in 1 every 2^80 numbers anyway, there would be a 50% chance that a number you found is prime and a 50% chance it would not actually be.
This isn't rocket science.
So Bayes' theorem doesn't save you here: it is still statistically unlikely that you will stumble upon a 1024-bit pseudoprime by mistake.
This is the count of 1023 bits numbers [(x - x/2) = x/2]. 0 is a valid leftmost digit.
It matches the common-English usage of a phrase like "a six-figure salary." A salary of $020,000 isn't what's meant by the phrase.
You can verify this by, say, running `openssl genrsa 1024 | openssl rsa -noout -text` a few times and looking at the generated prime1 and prime2. They each have the 512th bit set. (They seem to be printed with a leading hex "00:", but there are 512/8 = 64 bytes afterwards, and the first byte always has the high bit set.)
Q: My friend got a perfect hand in bridge. What are the odds of this happening?
A: Your friend is a liar.
I wonder if someone has compiled a list of very improbable events that have been observed.
You say you want to see it twice, but one data point with an error rate of 1 in 2^80 is, statistically, about a billion times more convincing than a million observations with an error rate of 1 in a million.
(That being said, there are a lot of of explanations that don't involve malice, including honest error, bugs in any of the software products involved in the process, etc. But no, I don't believe that this was a 1 in 2^80 fluke.)
It's a practical impossibility, a philosophical exercise.
However, if you picked a single grain from Sahara desert, you'll be only 2 or 3 orders of magnitude off, so one could say that 2^-80 is only slightly easier than finding a particular grain in a Sahara desert.
[1] http://www.telegraph.co.uk/news/science/space/5511619/14-yea...
2^-80 is absurdly small. But that only means something useful if the number of attempts isn't absurdly large.
I guess the harsh reaction came from the fact that I didn't define the scope very well: My question wasn't in reference to M-R specifically, just in general. I understand that in this case it makes sense to look at likelier causes (see Sharlin's response).
My point was that it's interesting to look at what happens (or what our reaction is) when very very very improbable events do happen. It seems weird to go with the assumption that because something is extremely unlikely that it won't happen.
When I roll a dice 20 times, I get a particular arrangement of numbers. Given the total number of arrangements possible, that particular arrangement is extremely unlikely, yet I just got it.
A guy got struck by lightning 7 times (https://en.wikipedia.org/wiki/Roy_Sullivan). The odds of any person getting struck by lightning is 1 in 10000. Seven times in a row is 1 in 2^93. But then when you start drilling down, you see that he's a park ranger, and that he's out while lightning happens, which makes the probability that he'll get struck much higher.
If I had phrased the question to you asking what the likelihood of any given person in the world being struck seven times was, you could have calculated the former and said 2^-93 is such a small probability that it's not worth thinking about - and yet here is Roy Sullivan, so there's some sort of conflict in my logic. What's wrong with the former calculation?
Why is it that for any given person the probability is 2^-93 but for Roy it's somehow different, even though he is a "given person"? Is it that the 1 in 10000 number was wrong? But then if we look at all the people who never once got struck, it seems about right. If we inflate that number to 1 in 100 to make Roy likelier to get 7 in a row, then it seems everyone also should be getting shocked more often at least once or twice.
Or maybe it's that somehow the probability changes when we have more information and those two numbers and situations are not comparable on an absolute scale. Maybe if you get hit twice then you're much likelier to get hit again because you're probably in some dangerous location - but how was I to know to factor this in? It seems that it's very much about how you calculate the probability. Who knows what other hidden factors could be wildly affecting the true value of the probability?
That also makes me think - is there even such a thing as the "true" or inherent probability of an event happening?
edit: Or maybe it's the law of large numbers - given enough "trials" or in this case lightning events with people around, even something with an absurdly small probability is bound to happen eventually. But then why do we never factor that in and always just call it a day with 10000^7?
"The probability of a randomly-selected 1024-bit number passing this primarily test without being prime is 2^-80" is a much more precise statement, because you have selected that number randomly. Obviously, if you have a specific number in mind, the probability of that number being a false positive is either 0 or 1. It either is prime, or it isn't.
Remember that there is no such thing as a random number; there is only a randomized process for selecting numbers.
No need to spam links to that cult site.
Because it isn't true that the probability of any given person being struck by lightning is 1/10000. For instance, the internet says that men are around 4 times as likely to be struck by lightning as women. Rather, what's true is that the likelihood of a randomly selected person being struck by lightning is 1/10000. Each individual person has a different likelihood to be struck. I don't know how to calculate it for Roy, but given that he's been struck 7 times I'm sure it's way higher than 1/10000.
> That also makes me think - is there even such a thing as the "true" or inherent probability of an event happening?
I love when people email me after reaching an enlightenment.
There was a probability question that drove me nuts until I figured out what was going on. The question is "A woman has two children. One of her children is a boy. What is the probability that she has two boys?".
The question isn't well defined, because it matters how you learned that one of her children is a boy.
If you asked her "what is the gender of your oldest child?", and she says it's a boy, then the probability that she has two boys is 1/2.
But if you asked her "do you have at least one boy?", and she says yes, then the probability that she has two boys is 1/3.
I don't know what the lesson here is. Probability is hard? Always ask about the experiment?
> is there even such a thing as the "true" or inherent probability of an event happening?
The probability of something depends on what you know. What's the chance that Roy will be struck by lightning tomorrow? If you don't know who Roy is, you might say 1/(10000 * 28000) (where 28000 days is the average human lifespan.). If you do know Roy's history, you'll probably bump that estimate up quite a bit. But if you look up the weather in his park tomorrow and see that it will be sunny all day, the probability will go back down close to zero.
Some things are so hard to know they might as well have an inherent probability, though. If you roll a die, no one's going to predict the outcome of the roll, so we might as well say it's inherently got a 1/6 chance of rolling a 5. And if you shine a photon at a half-silvered mirror, it's actually impossible to predict which way it will go, so I guess that really does have in inherent probability.
You've got to watch out for your probability estimates being wrong, though. I thought the probability of my friend flipping a coin and getting heads was 1/2, until he demonstrated that he could flip heads 10 times in a row.
> edit: Or maybe it's the law of large numbers - given enough "trials" or in this case lightning events with people around, even something with an absurdly small probability is bound to happen eventually. But then why do we never factor that in and always just call it a day with 10000^7?
Because even with 10 billion people, each living a billion days, and having a billion things happen to them in a day... well coincidentally that's exactly 10^28=10000^7, so never mind.
No, it's rational.
As for the rest of your post, you are confused between a priori and a posteriori probability.
Um, yes, that's all that probability is. See Bayes Theorem.
https://blogs.msdn.microsoft.com/oldnewthing/20160114-00/?p=...
When one wants a 128-bit security margin, 2^-80 is 2^48 times too big.
An attacker who can bruteforce, say, 2^80 128-bit keys (approximate limit of the computational power of the largest adversaries) has 1 chance out of 2^48 to break the security.
But an attacker has only 1 chance out of 2^80 that this parameter is a non-prime.
2^80 is much larger than 2^48, therefore it is not a problem.
If I want a 128-bit security level, I'm willing for a random guess of my key to have a 2^-128 chance of being correct. I'm likewise willing for a prime I generate to have a 2^-128/uses chance of not being prime (where uses is equal to the number of times I'll actually be using it).
I can't be 100% certain that a large number is actually prime. But I can be certain enough. Having a 2^-80 chance of being wrong isn't good enough when I want a 2^-128 security level.
You can't compare these 2 scenarios directly, because in practice:
- attackers can take as many guesses as they want against a piece of data protected by a 128-bit key, but
- attackers get only 1 chance that a particular piece of data protected by a supposed prime is not a prime.
If the defender is somehow put in a situation to generate 2^80 primes, then he's in trouble.
But it also implies that e.g. the attacker has a 2^-128 chance of randomly guessing a key.