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