Efficient “quasi”-deterministic primality test improving AKS (original) (raw)

We combine ideas from the seminal paper of Agrawal, Kayal and Saxena [AKS] as improved by Lenstra [Le3] with the particular case sharpening of Berrizbeitia and introduce the cyclotomy of rings setting [Le2, BvdH, Mi3] for the latter. Thus we deduce a new variant of the AKS algorithm which: (i) has running time O`log 4+o(1) (n)´; (ii) works on all prime candidates n > e e 2e ; and (iii) is "quasi deterministic", in the sense that it is deterministic under the assumption that some roots of unity can be found in polynomial time, while failing to do so would raise an explicit contradiction to the GRH. The bottleneck of the algorithm are the space requirements.