dissertacao.pdf

It is straightforward that, when we input a prime number, the output will
always be prime: in steps 1 and 3 it cannot be considered composite, and in
step 6, because of the previous theorem, it will also not be considered composite
in any of the iterations of the cycle. So it will be considered prime, either in
step 5 or step 7. For a proof that a composite integer is never considered prime
one can read the original AKS article [29]. In [41] an implementation of the
algorithm is described.
There is one important conjecture that, if proven, would improve the effi-
ciency of this algorithm.
Conjecture 2. Let r be a prime that does not divide the positive integer N
and such that (X − 1) N ∼ = X N − 1 (mod n, X r − 1). Then
(N is prime N 2 ∼ = 1 (mod r)) (37)
If this conjecture could be proved, Lenstra and Pomerance’s variant of the
algorithm would have complexity O(log 3 n)[41].
2.2 Factoring Algorithms
Like we said before, the security of RSA depends strongly on the difficulty of
factoring the modulus N. For this reason, special attention should be given to
the state of the art of integer factoring algorithms when implementing RSA.
There are published standards with recommendations 2 for the size of N and
the primes’ size depending mostly on how long we wish to keep the encrypted
data safe. This recommendations depend on some of the algorithms presented
in this section.
Factoring algorithms fall into two categories. One is the General Purpose
Factoring Algorithms which behave in approximately the same way for inte-
gers with the same size. The second category is that of the Special Purpose
Factoring Algorithms which behave better for integers N with certain spe-
cific characteristics, like the size of the smallest factor of N or the divisors of
(N − 1) or (N + 1). To attack an RSA session with balanced primes, a general
purpose algorithm is a priori more suitable, since the factors of N are randomly
chosen and of the same size.
2 http://www.rsa.com/rsalabs/node.asp?id=2218
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