The Top Twenty--a Prime Page Collection

Elliptic Curve Primality Proof

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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.

(up) Definitions and Notes

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).  Both programs available).  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).

(up) Record Primes of this Type

rankprime digitswhowhencomment
1"τ(1572206)" 26643 FE1 Apr 2011 ECPP
267535122 + 51226753 25050 FE1 Oct 2010 ECPP
3" - τ(6911522)" 23770 c65 Mar 2014 ECPP
4273845 + 14717 22230 c61 Dec 2013 ECPP
5((((((25210088873 + 80)3 + 12)3 + 450)3 + 894)3 + 3636)3 + 70756)3 + 97220 20562 FE1 Jun 2006 ECPP, Mills' prime
6(263703 - 1)/42808417 19169 c59 Jan 2014 Mersenne cofactor, ECPP
7V(89849) 18778 c70 Jan 2014 Lucas number, ECPP
8primV(145353) 18689 c69 Dec 2013 ECPP, Lucas primitive part
9Phi(14943, - 100) 18688 c47 Mar 2014 Unique, ECPP
10Phi(741, - 638479)/44250132909040111 18666 c54 May 2013 ECPP
11587 · 43103#/2310 + 657402 18662 c35 Oct 2013 ECPP
12587 · 43103#/2310 - 455704 18662 c35 Jul 2013 ECPP
13261792 + 21661 18602 c61 Dec 2012 ECPP
14V(81671) 17069 c66 Sep 2013 Lucas number, ECPP
15256366 + 39079 16968 c61 Oct 2012 ECPP
1639161#/2310 - 510478 16901 c35 Apr 2013 ECPP
176521953289619 · 255555 - 5 16737 c58 Apr 2013 Triplet (1), ECPP
18"τ(5711090)" 16526 c65 Jan 2014 ECPP
19" - τ(791570)" 16386 c65 Feb 2014 ECPP
20"τ(6731018)" 15834 c65 Jul 2013 ECPP

(up) Related Pages

(up) References

AM93
A. O. L. Atkin and F. Morain, "Elliptic curves and primality proving," Math. Comp., 61:203 (July 1993) 29--68.  MR 93m:11136
Atkin86
A. O. L. Atkin, "Lecture notes of a conference," Boulder Colorado, (August 1986) Manuscript. [See also [AM93].]
FKMW2003
J. Franke, T. Kleinjung, F. Morain and T. Wirth, "Proving the primality of very large numbers with fastecpp," (2003) Available from ftp://lix,polytechnique.fr/pub/submissions/morain/Preprints/large.ps.gz.
GK1999
S. Goldwasser and J. Kilian, "Primality testing using elliptic curves," J. ACM, 46:4 (1999) 450--472.  MR 2002e:11182
GK86
S. Goldwasser and J. Kilian, Almost all primes can be quickly certified.  In "STOC'86, Proceedings of the 18th Annual ACM Symposium on the Theory of Computing (Berkeley, CA, 1986)," ACM, New York, NY, May 1986.  pp. 316--329,
LL90
Lenstra, Jr., A. K. and Lenstra, Jr., H. W., Algorithms in number theory.  In "Handbook of Theoretical Computer Science, Vol A: Algorithms and Complexity," The MIT Press, Amsterdam and New York, 1990.  pp. 673-715, MR 1 127 178
Morain98
F. Morain, Primality proving using elliptic curves: an update.  In "Algorithmic Number Theory, Third International Symposium, ANTS-III," J. P. Buhler editor, Lecture Notes in Comput. Sci. Vol, 1423, Springer-Verlag, June 1998.  pp. 111--127, Available from http://www.lix.polytechnique.fr/~morain/Articles/articles.english.htmlMR 2000i:11190
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