dissertacao.pdf

2.1.2 Solovay-Strassen Test
This is the primality test suggested by the RSA team for generating the primes
p and q. It is based on the following theorem, due to Euler 1 :
Theorem 9. If p is an odd prime number, and b ∈ {1, ..., p − 1} such that
(b, p) = 1 then:
b
∼= b
p
p−1
2 (mod p) (29)
So the idea of the test goes as follows: given an integer N whose primality
we want to check, we choose a random integer b such that 0 < b < N and check
(b, N). If the greatest common divisor is different from one, we found a factor
of N so N is composite. If it is 1, we verify the congruence: if it fails, then we
know that N is composite. If it is true, then there is a positive probability that
N is prime.
Like the FPT, this test only gives a reliable output when it proves composite-
ness because like with the FPT, there are composite numbers which will satisfy
the congruence for some bases. However, there are no composite numbers which
will satisfy the congruences for all the bases like the Carmichael Numbers do
for the FPT. In this way, the Solovay-Strassen test is a much better test than
FPT. The following result states this:
Theorem 10. Let N be an odd composite. Then there is
an element b ∈ ZN : (N, b) = 1 such that:
b
≇ b
N
N−1
2 (mod N) (30)
So a composite number N can be a pseudoprime for some bases b, but it
will never be for all of them. Therefore, if we run the test using all integers up
to N as bases, we will be sure about N’s primality. But do we really need to
use all of them? The next theorem tells us about the number of bases that, for
a given composite N, actually satisfy the congruence:
Theorem 11. Let N be an odd composite. Then at least half of the integers b
co-prime to N in {1, ..., N − 1} satisfy:
b
≇ b
N
N−1
2 (mod N) (31)
1 a
In this section ( ) refers to the Legendre Symbol
b
23