dissertacao.pdf

2.1.4 AKS Test
In 2004, a major breakthrough was achieved by a team of computer scientists
from the Indian Institute of Technology Kanpur. In an email sent worldwide,
Manindra Agrawal, Neeraj Kayal and Nitin Saxena (AKS) announced the world
they had found a deterministic polynomial time algorithm to verify an integer’s
primality.
Although the running time of the original algorithm was O (log 21
2 N)[29],
some recent efforts by Hendrik Lenstra and Carl Pomerance have created a
variation of the algorithm whose time complexity is O (log 6 N) [29] (which is
still slower than the Miller-Rabin Deterministic Test under the GRH).
The algorithm is based on the following result:
Theorem 17. Let a ∈ Z, N ∈ N, N ≥ 2 and (a, N) = 1. Then N is prime if
and only if:
(X + a) N = X N + a (mod N). (35)
This criteria, which is stronger than those presented before as it is both
necessary and sufficient for an integer to be prime, cannot be applied straight-
forwardly to check an integer’s primality because, for large N, there are too
many coefficients to calculate in the binomial. So the great development in
2004 was to, based on this criteria, create a deterministic algorithm for check-
ing primality. The original AKS algorithm is presented next:
Algorithm 4. The AKS algorithm. Given a positive integer N > 1:
1. Choose a ∈ Z and b > 1. If N = a b , then STOP and output COMPOS-
ITE.
2. Find the smallest r such that or(N) > log 2 (N).
3. If 1 < (a, N) < N, for some a ≤ r, output COMPOSITE.
4. If N ≤ r, output PRIME.
5. For a = 1, 2, ..., ⌊ φ(r) log(N)⌋, do:
if ((X +a) N ≇ X N +a (mod X r −1, N))), output COMPOSITE. (36)
6. Output PRIME
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