No comments only because I haven't had a chance to really think about what would be required for a kilobit number. I just downloaded the M1039 paper.

For sieving, they used 16e with FB limits of 300M on both sides, which used about 1GB of memory. This shouldn't be a problem. They used 38-bit LPs, which seems a bit excessive. I would test with 33-bit or 34-bit LPs first. This would also help significantly with the filtering. It would reduce the number of relations for a "good" matrix to 1-2 billion.

I don't suspect filtering would be too much of a problem. I have a computer available with plenty of RAM for that.

Linear algebra is what I fear. They ended up with a 66.7M matrix. This would take more than a year using the BL in the current version of msieve. I have tried using the CADO BW, but I have not gotten good scaling with it either beyond 16 cores. I'm not sure there are publicly available tools that can solve this matrix in less than a year on the computers I have available. I can write a Teragrid research proposal for the computer time if necessary, but that doesn't help without a good implementation.

And, of course, the square roots shouldn't be a problem.

Ideas for solving the LA issue?

Linear algebra is what I fear.

.....

Ideas for solving the LA issue?

Does the latest msieve solve any of these problems?

Linear algebra is what I fear.

.....

Ideas for solving the LA issue?

Does the latest msieve solve any of these problems?

I'm not sure that we know the limits of parallel
Block Lanczos; but all of the recent records have
been set with LA done by block Wiedemann (RSA200,
SNFS1024, RSA768). Already the target 5, 409- C282
may be in "big iron" range, with a request for
substantial time on first-rate hardware (infiniband ...).
The most recent records have relied on distributing
the matrix, with each piece run in parallel, which
is a Wiedermann feature. -Bruce

Does the latest msieve solve any of these problems?

Up to at least a kilobit SNFS, I believe so. But I want to find out for sure, hence the increased importance of and credit boost for 16e sieving. Recruit all your friends whose computers have sufficient memory, and even your enemies!