dissertacao.pdf

The consequence of this theorem is that for each test we apply to verify N’s
primality, the probability that the test is wrong is less than 1
2 . This means that,
if we apply k tests with different random bases, and N passes all the tests, then
there is a chance of less than 1
2 k that N is composite.
2.1.3 Miller-Rabin Primality Test
The primality test presented in this section represents an evolution of FPT as
for it there is no equivalent for the Carmichael Numbers, that is, there is no
composite numbers which will pass the test for every different base. It is also
more reliable than the Solovay-Strassen test as we will see. Again there is some
composite numbers which fool the test. These are called the strong pseudoprimes
but these numbers are, as we will see, slightly more understandable. The test
is based on the following theorem, due to Miller:
Theorem 12. Given an odd prime N, written as N = 1 + 2 s d, where d is odd,
then the sequence:
ends with 1. If
a d , a 2d , a 4d , ..., a 2s−1 d , a 2 s d
a d ≇ 1 (mod N)
then the value preceding the first appearance of 1 is N − 1.
(mod N) (32)
This condition, unfortunately, is also verified by some composite numbers,
for some base a.
Definition 16. A composite number N which satisfies the conditions described
by the theorem above, for an integer a, is said to be a strong pseudoprime
to the base a. If a number is either prime or strong pseudoprime, we call it
probably prime.
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