## Elliptic Curve Primality Proof |

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 1 p(1289844341)40000 c84 Feb 2020 Partitions, ECPP 2 2^{116224}- 1590534987 c87 Nov 2017 ECPP 3 (14665 · 10^{34110}- 56641)/999934111 c89 Mar 2018 ECPP, Palindrome 4 "10000000000000000000...(34053 other digits)...00000000000000532669"34093 c84 Nov 2016 ECPP 5 V(148091)30950 c81 Sep 2015 Lucas number, ECPP 6 U(148091)30949 x49 Sep 2021 Fibonacci number, ECPP 7 " - τ(331^{2128})"29492 c80 Sep 2015 ECPP 8 V(140057)29271 c76 Dec 2014 Lucas number, ECPP 9 (2^{95369}+ 1)/328709 x49 Aug 2021 Generalized Lucas number, Wagstaff, ECPP 10 - 30 · Bern(10264)/(1040513 · 252354668864651)28506 c94 May 2021 Irregular, ECPP 11 (10^{27669}+ 7)/831349383281865592944806559876345853111127630 c96 Feb 2021 ECPP 12 U(130021)27173 x48 May 2021 Fibonacci number, ECPP 13 546351925018076058 · Bern(9702)/12925504897610680478690425888051894126709 c77 Jan 2021 Irregular, ECPP 14 "τ(157^{2206})"26643 FE1 Apr 2011 ECPP 15 10^{25333}- 2 · 10^{5182}- 325333 c95 Aug 2020 ECPP 16 Phi(12345, 7176)/3153176024531352686503392125331 c54 Jun 2017 ECPP 17 (2^{84211}- 1)/1347377 / 31358793176711980763958121 / 331464167604234782416959156125291 c95 Sep 2020 Mersenne cofactor, ECPP 18 (2^{83339}+ 1)/325088 c54 Sep 2014 ECPP, generalized Lucas number, Wagstaff 19 6753^{5122}+ 5122^{6753}25050 FE1 Oct 2010 ECPP 20 (2^{82939}- 1)/88332390301254027803357181907324938 c84 Feb 2021 Mersenne cofactor, ECPP

- ECPP From the Prime Glossary
- Short introduction to ECPP

- AM93
A. O. L. AtkinandF. 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
Franke, J.,Kleinjung, T.,Morain, F.andWirth, T.,Proving the primality of very large numbers with fastECPP. In "Algorithmic number theory," Lecture Notes in Comput. Sci. Vol, 3076, Springer, Berlin, 2004. pp. 194--207, (http://dx.doi.org/10.1007/978-3-540-24847-7_14)MR 2137354- GK1999
S. GoldwasserandJ. Kilian, "Primality testing using elliptic curves,"J. ACM,46:4 (1999) 450--472.MR 2002e:11182- GK86
S. GoldwasserandJ. 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, May 1986. New York, NY, pp. 316--329,- LL90
Lenstra, Jr., A. K.andLenstra, 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,MR 2000i:11190

Chris K. Caldwell
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