A Potentially Fast Primality Test

The running time is O(r log n). It can be shown by elementary means that the required r exists in O(log n). So the running time is O(log n). Moreover, by Fouvry’s Theorem [8], such r exists in O(log n), so the running time becomes O(log n). In [10], Lenstra and Pomerance showed that the AKS primality test can be improved by replacing the polynomial x − 1 in equation (1.1) with a specially constructed polynomial f(x), so that the degree of f(x) is O(log n). The overall running time of their algorithm is O(log n). With an extra input integer a, Berrizbeitia [6] has provided a deterministic primality test with time complexity 2−min(k,b2 log log nc)O(log6 n), where 2k||n−1 if n ≡ 1 (mod 4) and 2k||n + 1 if n ≡ 3 (mod 4). If k ≥ b2 log log nc, this algorithm runs in O(log n). The algorithm is also a modification of AKS by verifying the congruent equation