The Top Twenty: Elliptic Curve Primality Proof

The Prime Pages keeps a list of the 5000 largest known primes, plus a few each of certain selected archivable forms and classes. These forms are defined in this collection's home page. This page is about one of those forms. Comments and suggestions requested.

This page lists record primes that were first proven prime by the elliptic curve primality proving algorithm. It is shown here as a convenience for those watching the heated contest between the chief ECPP programmers. Originally these were François Morain (who first set a titanic prime record for proving primality via ECPP) and Marcel Martin (who wrote a version called Primo for Windows machines). In 2003, J. Franke, T. Kleinjung and T. Wirth greatly increased the size of numbers that could be handled with a new program of their own. Morain has worked with this trio and they have both improved their programs [FKMW2003]. Martin's Primo is by far the easiest of these programs to set up and use. There seems to be some question which is fastest on a single CPU.

ECPP has replaced the groups of order n-1 and n+1 used in the classical test with a far larger range of group sizes (see our page on elliptic curve primality proving). The idea is that we can keep switching elliptic curves until we find one we can "factor". This improvement comes at the cost of having to do a great deal of work to find the actual size of these groups--but works for all numbers, not just those with very special forms.

About 1986 S. Goldwasser & J. Kilian [GK86] and A. O. L. Atkin [Atkin86] introduced elliptic curve primality proving methods. Atkin's method, ECPP, was implemented by a number of mathematicians, including Atkin & Morain [AM93]. Heuristically, ECPP is O((log n)^5+eps) (with fast multiplication techniques) for some eps > 0 [LL90]. It has been proven to be polynomial time for almost all choices of inputs. A version attributed to J. O. Shallit is O((log n)^4+eps). Franke, Kleinjung and Wirth combined with Morain to improve their respective programs (both now use Shallit's changes), creating what they "fastECPP" [FKMW2003].

The editors expect this page should remain our only Top Twenty Page dedicated to a proof method rather than a form of prime. Note that "fastECPP" is simply a name--their use of the adjective 'fast' should not be construed as a comparison to programs by other authors (which may also follow Shallit's approach).

rank

prime

digits

who

when

comment

p(1289844341)

40000

c84

Feb 2020

Partitions, ECPP

2¹¹⁶²²⁴ - 15905