trying to implement block lanczos on GF2...

paul0 · 2020-10-04, 17:33

Hi, I'm currently trying to implement Block Lanczos, but I'm having some trouble understanding the notation on calculating S_i and W_inv. I'm reading Montgomery's paper and Yang et. al.'s pseudocode (attached).

On line 13 of the attached file, the algorithm seems to create a set S of row & col where a 1 is found. Eventually I need to compute S_i*S_i^T based on S. I think to form S_i*S_i^T I just need to set M[x,x] = 1 for each x ∈ S. M is a square matrix with side size of machine word (as defined by the paper) rows and cols. Is this correct?

Also, S_i-1 is an input to this function, but I don't see it was used there. What am I missing?

paul0 · 2020-10-04, 19:12

I tried implementing M[x,x] = 1 as I mentioned above. It worked! V_i^T*A*V_i becomes zero. However, it seems that not all X - Y are nullspaces of B. I got 10 out of 32 valid nullspace, then 6 out of 32 on a matrix with more dependencies. The paper says I need to combine these vectors. How do I do that? gaussian elimination?

I'd like to mention that I'm avoiding optimizations for now, like the ANDing of S_i*S_i^T.

paul0 · 2020-10-05, 02:07

It seems that I missed an entire paragraph about the last step with gaussian elmination. will try to implement that first. also, RIP Peter Montgomery.

jasonp · 2020-10-05, 15:06

You figured it out, but yes the block Lanczos algorithm finds the nullspace of A^T*A and not of A, since the algorithm only works for symmetric matrices.To get the answers you need, Gauss elimination of the nullspace vectors you have is necessary.

I remember how great it felt when I managed to get the code running on a real problem, BL is hard to figure out from papers, or even from other people's code.

henryzz · 2020-10-06, 05:20

It looks like Yang's paper is an improvement to clarity. I have tried and failed with other references in the past. Time for another go at some point. Congratulations on getting this working.

paul0 · 2020-10-06, 05:52

Quote:

Originally Posted by jasonp

You figured it out, but yes the block Lanczos algorithm finds the nullspace of A^T*A and not of A, since the algorithm only works for symmetric matrices.To get the answers you need, Gauss elimination of the nullspace vectors you have is necessary.

I remember how great it felt when I managed to get the code running on a real problem, BL is hard to figure out from papers, or even from other people's code.

Quote:

Originally Posted by henryzz

It looks like Yang's paper is an improvement to clarity. I have tried and failed with other references in the past. Time for another go at some point. Congratulations on getting this working.

I agree with you both. I kept trying sporadically since 2015. Aside from Yang's paper, this recent C++ implementation also helped me (https://github.com/SebWouters/blanczos). It is well commented and seemed to written as teaching tool, though there are a few optimizations which may confuse beginners.

Here are the rest of the pseudocode from Yang's paper (full paper behind paywall): https://www.sparrho.com/item/an-impr...zation/9c93ad/

I got the last step working and got it to work with my toy NFS implementation. The terms Montgomery used for the last step (page 114) were unfamiliar to me, so I'm a bit suspicious that by running gaussian elimination, I may have bypassed the entire Block Lanczos process. Also, I don't know yet how to not compute U explicitly.

I have 2 questions:
1. Is it normal for the first dependency to be always trivial?
2. The very last output is "a basis for ZU". This just means the nullspaces of B are the columns of ZU?

EDIT: it seems that i'm in the wrong subforum, feel free to move this thread mods.

paul0 · 2020-10-06, 07:10

I'd like to add a third question:
3. In Montgomery's paper p. 115, he writes: "Afterwards I check whether all nonzero columns of V_i+1 were chosen in S_i and/or S_i-1". This assertion is implemented in the c++ implementation I linked above. What would happen if I do not implement this assertion? Will my implementation silently fail for some inputs?

Chris Card · 2020-10-06, 08:12

For what it's worth, here's my C++ implementation of block lanczos that I wrote a few years ago, based on Montgomery's paper:
https://github.com/ChrisCGH/factor-b...ockLanczos.cpp

paul0 · 2020-10-08, 04:09

Quote:

Originally Posted by Chris Card

For what it's worth, here's my C++ implementation of block lanczos that I wrote a few years ago, based on Montgomery's paper:
https://github.com/ChrisCGH/factor-b...ockLanczos.cpp

appreciate it, thanks

2020-10-04, 17:33	#1
paul0 Sep 2011 3·19 Posts	trying to implement block lanczos on GF2... Hi, I'm currently trying to implement Block Lanczos, but I'm having some trouble understanding the notation on calculating S_i and W_inv. I'm reading Montgomery's paper and Yang et. al.'s pseudocode (attached). On line 13 of the attached file, the algorithm seems to create a set S of row & col where a 1 is found. Eventually I need to compute S_iS_i^T based on S. I think to form S_iS_i^T I just need to set M[x,x] = 1 for each x ∈ S. M is a square matrix with side size of machine word (as defined by the paper) rows and cols. Is this correct? Also, S_i-1 is an input to this function, but I don't see it was used there. What am I missing? Attached Thumbnails Last fiddled with by paul0 on 2020-10-04 at 17:56

2020-10-04, 19:12	#2
paul0 Sep 2011 3·19 Posts	I tried implementing M[x,x] = 1 as I mentioned above. It worked! V_i^TAV_i becomes zero. However, it seems that not all X - Y are nullspaces of B. I got 10 out of 32 valid nullspace, then 6 out of 32 on a matrix with more dependencies. The paper says I need to combine these vectors. How do I do that? gaussian elimination? I'd like to mention that I'm avoiding optimizations for now, like the ANDing of S_iS_i^T. Last fiddled with by paul0 on 2020-10-04 at 19:53*

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2020-10-05, 02:07	#3
paul0 Sep 2011 3×19 Posts	It seems that I missed an entire paragraph about the last step with gaussian elmination. will try to implement that first. also, RIP Peter Montgomery.

2020-10-05, 15:06	#4
jasonp Tribal Bullet Oct 2004 DCE₁₆ Posts	You figured it out, but yes the block Lanczos algorithm finds the nullspace of A^TA and not of A, since the algorithm only works for symmetric matrices.To get the answers you need, Gauss elimination of the nullspace vectors you have is necessary. I remember how great it felt when I managed to get the code running on a real problem, BL is hard* to figure out from papers, or even from other people's code.

2020-10-06, 05:20	#5
henryzz Just call me Henry "David" Sep 2007 Cambridge (GMT/BST) 16BE₁₆ Posts	It looks like Yang's paper is an improvement to clarity. I have tried and failed with other references in the past. Time for another go at some point. Congratulations on getting this working.

2020-10-06, 07:10	#7
paul0 Sep 2011 3×19 Posts	I'd like to add a third question: 3. In Montgomery's paper p. 115, he writes: "Afterwards I check whether all nonzero columns of V_i+1 were chosen in S_i and/or S_i-1". This assertion is implemented in the c++ implementation I linked above. What would happen if I do not implement this assertion? Will my implementation silently fail for some inputs?

2020-10-06, 08:12	#8
Chris Card Aug 2004 2×5×13 Posts	For what it's worth, here's my C++ implementation of block lanczos that I wrote a few years ago, based on Montgomery's paper: https://github.com/ChrisCGH/factor-b...ockLanczos.cpp