Proving primality in essentially quartic random time

Proving Primality in Essentially Quartic

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 ...

Fast Primality Proving on Cullen Numbers

We present a unconditional deterministic primality proving algorithm for Cullen numbers. The expected running time and the worst case running time of the algorithm are Õ(logN) bit operations and Õ(logN) bit operations, respectively.

Cyclotomy Primality Proving

Two rational primes p, q are called dual elliptic if there is an elliptic curve E mod p with q points. They were introduced as an interesting means for combining the strengths of the elliptic curve and cyclotomy primality proving algorithms. By extending to elliptic curves some notions of galois theory of rings used in the cyclotomy primality tests, one obtains a new algorithm which has heurist...

Implementing the Asymptotically Fast Version of the Elliptic Curve Primality Proving Algorithm Less Prelimimary Version 040825

The elliptic curve primality proving algorithm is one of the fastest practical algorithms for proving the primality of large numbers. Its running time cannot be proven rigorously, but heuristic arguments show that it should run in time Õ((logN)) to prove the primality of N . An asymptotically fast version of it, attributed to J. O. Shallit, runs in time Õ((logN)). The aim of this article is to ...