Primality Proving via One round in Ecpp and One Iteration

On August 2002, Agrawal, Kayal and Saxena announced the first deterministic and polynomial time primality testing algorithm. For an input n, the AKS algorithm runs in heuristic time Õ(log n). Verification takes roughly the same amount of time. On the other hand, the Elliptic Curve Primality Proving algorithm (ECPP) runs in random heuristic time Õ(log n) ( Õ(log n) if the fast multiplication is used), and generates certificates which can be easily verified. More recently, Berrizbeitia gave a variant of the AKS algorithm, in which some primes cost much less time to prove than a general prime does. Building on these celebrated results, this paper explores the possibility of designing a more efficient algorithm. A random primality proving algorithm with heuristic time complexity Õ(log n) is presented. It generates a certificate of primality which is O(logn) bits long and can be verified in deterministic time Õ(log n). The reduction in time complexity is achieved by first generalizing Berrizbeitia’s algorithm to one which has higher density of easily-proved primes. For a general prime, one round of ECPP is deployed to reduce its primality proof to the proof of a random easily-proved prime.