dissertacao.pdf

To apply the Miller-Rabin test, we need the following theorem:
Theorem 13. Given an integer N, let
N − 1 = 2 s d
with d odd, and s 0. Then N is probably prime if:
or
for some r less than s.
a d ∼ = 1 (mod N) (33)
(a d ) 2r ∼ = (N − 1) (mod N) (34)
The running time of this probabilistic algorithm is O(k log 3 N), where k is
the number of times we run the test with different bases a. Therefore the Miller-
Rabin Primality test is a probabilistic primality test running in polynomial time.
Could we make this a deterministic primality test? The answer is yes, we can,
but we lose its efficiency.
Theorem 14. Let N > 1 be an odd composite integer. Then N passes the
Miller-Rabin Primality Test for at most N−1
4
bases a with 1 < a < N.
So here is yet another deterministic primality test: given that an integer N
passes k > N−1
4 tests, we are sure it is prime. However, the running time of this
algorithm is O( N
4 log3 (N)), still infeasible. There is one interesting result that
would permit this test to become both deterministic and efficient:
Theorem 15. If the Generalized Riemann Hypothesis (GRH) is true and N
passes the Miller-Rabin Primality Test for all bases a : 1 < a < 2(log N) 2 , then
N is prime.
So the proof of the GRH would lead to a deterministic version of the Miller-
Rabin Primality Test with running time O(log 5 N). Though this is only a
conjecture and therefore the test can be applied reliably only in its probabilistic
version, it is one of the most used primality tests within programs such as
Mathematica. The reason follows:
Theorem 16. The probability that a composite N passes the Miller-Rabin Pri-
mality Test for a random base a is 1
4
at most.
This means that, if we test for 100 different random bases, the probability
that all the tests are wrong is less than 1
10 40 !
25