"Brian J. Beesley" <[EMAIL PROTECTED]> wrote
- From [EMAIL PROTECTED] Sun Nov 10 20:22:33 2002
- On Saturday 09 November 2002 04:45, George Woltman wrote:
- > A harder problem is finding some smooth ECM curves to test. I do not
- > have tools to compute group orders. If someone can help by finding a
- > couple of dozen smooth ECM test cases for exponents between 1000
- > and 500000, I would be most grateful.
-
- Here's one example:
-
- With sigma=1459848859275459, Prime95 v22.12 finds the factor
- 777288435261989969 of M1123:
-
- in stage 1 with B1 >= 535489
- in stage 2 with B1 >= 38917 & B2 >= 534241
-
- I'm not entirely sure why the B2 required to find the factor at the end of
- stage 2 is smaller than the B1 required to find it in stage 1. One of the
- improvements I guess.
-
- Regards
- Brian Beesley
Probably the 534241 entry is compared against several others,
including 1248 (= 535489 - 534241).
How does one map sigma to a curve (and initial point)?
What is the range of sigma (it seems to go beyond 2^32)?
The ECM tests should include a case where two primes are found at the
same time during step 2, because the largest primes dividing the
two group orders are equal. [That is, the GCD will be composite.]
This test may be hard to construct, however.
Either the ECM tests or the p-1 tests should include a case where
the group order (or p-1) is divisible by a power of a moderate prime,
such as 61^3 or 757^2 .
Peter Montgomery [EMAIL PROTECTED]
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