I randomly ran across an
interesting note
that if the Extended Riemann Hypothesis is true then with
the exception of the easily recognized Carmichael numbers
there is always a witness of size at most
(log2n)2. This is a
polynomial primality test.
Another interesting point is that Rabin-Miller raises some
philosophical questions about mathematics. The fact is
that no mathematical operation can be carried out without
error. Indeed this is why mathematics, often popularly
supposed to be a bastion of pure truth, is in fact a
subject like any other where people have to make informed
decisions about what results they will and will not
believe. (You can think of mathematical proof as being like
trying to write and verify a computer program without
ever having the opportunity to run it and see if it works.
This is not far wrong.)
So the question is at what point does showing that a
result is likely true an acceptable substitute for showing
that it is true? This question is not hypothetical.
Suppose that you want to use RSA in practice. Well first
you need to have 2 big primes. Well from the prime number
theorem, primes are not hard to find. The density of the
primes near n is 1/log(n). But it is not hard to just look
at numbers that you know are congruent to 1 or 5 mod 6,
which drops 2/3 of the composites from your net. If you
are, say, interested in 100 digit numbers this means that
about 1% of the numbers you look at are prime. Grab
enough numbers and you probably have quite a few primes in
there, the issue is figuring out which ones they are. Well
the answer to that is to apply something like Rabin-Miller.
If a number survives a dozen rounds of Rabin-Miller then the
conditional odds of it being prime pass 99.999%. At some
point most people will trust their credit card to it...
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