dissertacao.pdf

2 Primality Tests and Factorization Algorithms
2.1 Primality Tests
Primality tests are algorithms that receive as input a positive integer N and
output some statement about its primality. In the best scenario, they guarantee
that either N is prime or that N is composite. Such tests are called determinis-
tic. Other tests either guarantee that N is composite, or that N is prime with
a certain probability. These tests are called probabilistic.
The most obvious primality test for an integer N, known for thousands
of years, consists in simple trial division by all integers up to √ N, since any
composite number must have a divisor p that satisfies p ≤ √ N. If a factor
is found, we know that the number is composite, otherwise we prove that N
is prime. This deterministic algorithm has efficiency O( √ N) and therefore is
extremely inefficient for large integers, say with 1024 bits like often used in RSA.
The probabilistic primality tests we will explain, most of them, simply aim
to prove that N is a composite number. If they do not prove it, then there is a
chance that they are prime. Repeating this test with different parameters will
lead to a bigger certainty that we are in the presence of a prime number.
2.1.1 Fermat’s Primality Test
The first primality test presented in this section derives from a theorem discov-
ered by Fermat in 1640. The proof is due to Leibniz, who found it 40 years
later. It states the following:
Theorem 7. For any prime number p and for any integer a such that 0 < a < p,
we have:
a (p−1) ≡ 1 (mod p) (24)
Proof. a, 2a, 3a, ..., (p − 1)a are all distinct mod p, therefore each of them is
congruent to one of 1, 2, 3, ..., p − 1. Because congruences preserve multiplicity,
we have:
a2a3a...(p − 1)a ∼ = 1 ∗ 2 ∗ 3 ∗ ... ∗ (p − 1) (mod p) ⇔ (25)
(p − 1)!a p−1 ∼ = (p − 1)! (mod p) (26)
Diving both sides by (p − 1)! we get the equality.
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