Proving primality in essentially quartic random time

Proving primality in essentially quartic random time

This paper presents an algorithm that, given a prime n, finds and verifies a proof of the primality of n in random time (lg n)4+o(1). Several practical speedups are incorporated into the algorithm and discussed in detail.

Proving Primality in Essentially Quartic

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...

Pseudopowers and primality proving

The so-called pseudosquares can be employed in very powerful machinery for the primality testing of integers N . In fact, assuming reasonable heuristics (which have been confirmed for numbers to 2) they can be used to provide a deterministic primality test in time O(log N), which some believe to be best possible. In the 1980s D.H. Lehmer posed a question tantamount to whether this could be exte...

Primality Proving with Elliptic Curves

Elliptic curves are fascinating mathematical objects. In this paper, we present the way they have been represented inside the COQ system, and how we have proved that the classical composition law on the points is internal and gives them a group structure. We then describe how having elliptic curves inside a prover makes it possible to derive a checker for proving the primality of natural number...