Sigma parameter in ecm

storm5510 · 2019-10-09, 13:49

I have done a lot of web searching, including MersenneWiki, and I have not found anything which explains the purpose of Sigma in ECM factoring.

Anyone care to elaborate?

R.D. Silverman · 2019-10-09, 13:50

Quote:

Originally Posted by storm5510

I have done a lot of web searching, including MersenneWiki, and I have not found anything which explains the purpose of Sigma in ECM factoring.

Anyone care to elaborate?

It is a random number used to generate the coefficients of the elliptic curve.

storm5510 · 2019-10-09, 16:22

Quote:

Originally Posted by R.D. Silverman

It is a random number used to generate the coefficients of the elliptic curve.

Understood. Thank you very much!

PhilF · 2019-11-30, 20:36

Here's my question about sigmas. It will expose my ignorance, but at the same time the answer might help me visualize how ECM works.

Let's say you are running curves on a composite number that has only 2 rather large prime factors, and a lucky curve found one of those factors. Is the sigma that found that factor the only sigma that could have found it, or are there a number of different sigmas that could have found that same factor?

VBCurtis · 2019-11-30, 21:32

Lots of sigmas, usually. It's possible for a large-enough factor and 32-bit limit on sigma through GMP-ECM that just one of the ~4 billion sigmas would yield the factor for a particular B1/B2 choice, but that's a rather contrived case.

2019-10-09, 13:49	#1
storm5510 Random Account Aug 2009 Not U. + S.A. 101000110000₂ Posts	Sigma parameter in ecm I have done a lot of web searching, including MersenneWiki, and I have not found anything which explains the purpose of Sigma in ECM factoring. Anyone care to elaborate?

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2019-11-30, 20:36	#4
PhilF "6800 descendent" Feb 2005 Colorado 2²·5·37 Posts	Here's my question about sigmas. It will expose my ignorance, but at the same time the answer might help me visualize how ECM works. Let's say you are running curves on a composite number that has only 2 rather large prime factors, and a lucky curve found one of those factors. Is the sigma that found that factor the only sigma that could have found it, or are there a number of different sigmas that could have found that same factor?

2019-11-30, 21:32	#5
VBCurtis "Curtis" Feb 2005 Riverside, CA 1656₁₆ Posts	Lots of sigmas, usually. It's possible for a large-enough factor and 32-bit limit on sigma through GMP-ECM that just one of the ~4 billion sigmas would yield the factor for a particular B1/B2 choice, but that's a rather contrived case.