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...

In reply to Rabin Miller may be deterministic by tilly
in thread Rabin Miller by Adam

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