Sage 4.8 is out

If you care about the stuff I care about (and why else would you read this blog?) you might get excited about a few changes in Sage.

Rank-profile revealing Gaussian elimination and the CUP matrix decomposition

by Claude-Pierre Jeannerod, Clément Pernet, Arne Storjohann is now available on the archive. I like this paper a lot and we also referenced it in both the M4RI elimination paper and the M4RIE paper so three cheers that it’s now available.

Abstract: Transforming a matrix over a field to echelon form, or decomposing the matrix as a product of structured matrices that reveal the rank profile, is a fundamental building block of computational exact linear algebra. This paper surveys the well known variations of such decompositions and transformations that have been proposed in the literature. We present an algorithm to compute the CUP decomposition of a matrix, adapted from the LSP algorithm of Ibarra, Moran and Hui (1982), and show reductions from the other most common Gaussian elimination based matrix transformations and decompositions to the CUP decomposition. We discuss the advantages of the CUP algorithm over other existing algorithms by studying time and space complexities: the asymptotic time complexity is rank sensitive, and comparing the constants of the leading terms, the algorithms for computing matrix invariants based on the CUP decomposition are always at least as good except in one case. We also show that the CUP algorithm, as well as the computation of other invariants such as transformation to reduced column echelon form using the CUP algorithm, all work in place, allowing for example to compute the inverse of a matrix on the same storage as the input matrix.

http://arxiv.org/abs/1112.5717

Sage/FLINT Days aka Sage Days 35

I am writing this while waiting for my taxi to leave Sage Days 35. Although, I didn’t get much actual coding done, it was great fun and very useful. I met a lot of old friend, new faces and managed to put faces to e-mail addresses.

In terms of coding projects, first, I tried to speed up linear algebra mod p where p is a 32 or 64 bit prime. But it turns out that any trick I could think of could not improve on Frederik’s code. So that didn’t lead anywhere but I allowed me to read some code of FLINT2 (very readable) and admire how carefully it is written.

My other two projects both involved evaluate–pointwise-multiply–interpolate algorithms for fast matrix-matrix products over finite extension fields or for matrices with polynomial coefficients (over prime fields).  After my talk on M4RI(E) David Harvey worked out how to improve multiplication over from 17 multiplications over to 15, which then lead to a general approach for with composite . Much of it remains to be implemented (efficiently), but the example indeed shows a 10% speed-up as expected. The code is not clean yet, uses way too much memory and doesn’t deal with the more advanced finite field stuff appropriately. It should end up in M4RIE eventually though.

I also contributed a bit to #12177 which is about a “prime slice” implementation of matrices over . The idea is essentially to represent  these matrices as polynomials with matrix coefficients and to use fast polynomial multiplication algorithms for these polynomials. It turns out, this works very well even for small finite fields. Burcin Eröcal did all the coding, I only helped with some discussions. We need to polish the code a lot to be usable, so if you like matrices over head over to #12177 and help out.

Coldboot Code Available

After receiving two inquiries about the coldboot attack paper which were best answered by looking at the code or by comparing with our code, I figured it was about time I put it online. So here it is:

https://bitbucket.org/malb/algebraic_attacks/src/1af75effcc7d/coldboot

For this code to run you’ll need to apply this patch to Sage:

http://trac.sagemath.org/sage_trac/ticket/10879

which adds an interface to SCIP. Unfortunately, this patch crashes on OSX and I didn’t figure out yet why. Anybody willing to help, please step forward

Also, I assume the code on bitbucket needs some patching to work with the most recent version of Sage. Patches very welcome!

The M4RIE library for dense linear algebra over small fields with even characteristic

I finally uploaded a pre-print of the M4RIE paper to the arXiv:

Abstract: In this work, we present the M4RIE library which implements efficient algorithms for linear algebra with dense matrices over for . As the name of the library indicates, it makes heavy use of the M4RI library both directly (i.e., by calling it) and indirectly (i.e., by using its concepts). We provide an open-source GPLv2+ C library for efficient linear algebra over for e small. In this library we implemented an idea due to Bradshaw and Boothby which reduces matrix multiplication over to a series of matrix multiplications over . Furthermore, we propose a caching technique – Newton-John tables – to avoid finite field multiplications which is inspired by Kronrod’s method (“M4RM”) for matrix multiplication over . Using these two techniques we provide asymptotically fast triangular solving with matrices (TRSM) and PLE-based Gaussian elimination. As a result, we are able to significantly improve upon the state of the art in dense linear algebra over with .

Efficient dense Gaussian elimination over the field with two elements

Finally, we finished our paper about Gaussian elimination in the M4RI library.

Abstract: In this work we describe an efficient implementation of a hierarchy of algorithms for Gaussian elimination upon dense matrices over the field with two elements (). We discuss both well-known and new algorithms as well as our implementations in the M4RI library, which has been adopted into Sage. The focus of our discussion is a block iterative algorithm for PLE decomposition which is inspired by the M4RI algorithm. The implementation presented in this work provides considerable performance gains in practice when compared to the previously fastest implementation. We provide performance figures on x86_64 CPUs to demonstrate the alacrity of our approach.

The sources of this document are available on bitbucket. But I also compiled a PDF.

Update: arXiv link.

Ring-LWE and the GB(N) Problem

Over at the Bristol Cryptography Blog Martijn Stam writes about our “Polly Cracker, Revisted” paper:

We did not discuss the paper  in great detail, but Jake did mention one interesting avenue for continued research. Given that this new approach allows one to cast both LWE and approximate GCD in the same framework, can one also capture ring-LWE. If so, this might enable a better comparison of the various fully homomorphic encryption (FHE) schemes out there. The hope expressed by Jake was that this might allow a reduction to standard LWE (for the current batch of ring-LWE schemes), which would boost our confidence in those schemes.

This motivated me to express the Ring-LWE problem in a language of Gröbner bases, here’s what I could come up with so far.

Sage/FLINT Days in Warwick 17 – 22nd December 2011

“A Sage Days workshop around the theme of Algorithms in Number Theory and FLINT.”

 

 

See http://wiki.sagemath.org/SageFlintDays for more information and registration.

PS: I’ll be talking about M4RI(E) … big surprise.

Challenge matrices

Now, that we have a decent PNG reader/writer in M4RI, it’s much easier to get some challenge matrices out of the library. Below, I list and link a few such matrices as they appear during Gröbner basis computations.

file	matrix dimensions	density	PLE	M4RI	GB
HFE 25 matrix 5 (5.1M)	12307 x 13508	0.07600	1.03	0.59	0.81
HFE 30 matrix 5 (16M)	19907 x 29323	0.06731	4.79	2.70	4.76
HFE 35 matrix 5 (37M)	29969 x 55800	0.05949	19.33	9.28	19.51
Mutant matrix (39M)	26075 x 26407	0.18497	5.71	3.98	2.10
random n=24, m=26 matrix 3 (30M)	37587 x 38483	0.03832	20.69	21.08	19.36
random n=24_ m=26 matrix 4 (24M)	37576 x 32288	0.04073	18.65	28.44	17.05
SR(2,2,2,4) compressed, matrix 2 (328K)	5640 x 14297	0.00333	0.40	0.29	0.18
SR(2,2,2,4) compressed, matrix 4 (2.4M)	13665 x 17394	0.01376	2.18	3.04	2.04
SR(2,2,2,4) compressed, matrix 5 (2.8M)	11606 x 16282	0.03532	1.94	4.46	1.59
SR(2,2,2,4) matrix 6 (1.4M)	13067 x 17511	0.00892	1.90	2.09	1.38
SR(2,2,2,4) matrix 7 (1.7M)	12058 x 16662	0.01536	1.53	1.93	1.66
SR(2,2,2,4) matrix 9 (36M)	115834 x 118589	0.00376	528.21	578.54	522.98

The first three rows are from GB computations for the hidden field equations cryptosystem (those matrices were provided by Michael Brickenstein). The “mutant” row is a matrix as it appears during a run of the MXL2 algorithm on a random system (I believe). It was contributed by Wael Said. The rows “random n=25,m=26″ are matrices as they appear during a GB computation with PolyBoRi for a random system of equations in 24 variables and 26 equations. The remaining rows are matrices from PolyBoRi computations on small scale AES instances. Those rows which have “compressed” in their description correspond to systems where “linear variables” were eliminate before running the Gröbner basis algorithm.

The last three columns give running times (quite rough ones!) for computing an echelon form (not reduced) using (a) the M4RI algorithm, (b) PLE decomposition and (c) a first implementation of the TRSM for trivial pivots trick. As you can see, currently it’s not straight-forward to pick which strategy to use to eliminate matrices appearing during Gröbner basis computations: the best algorithm to pick varies between different problems and the differences can be dramatic.

M4RIE Paper

I’ve been writing up the ideas that went into the M4RIE library for dense linear algebra over small extensions of . I think it is now in a state to be readable enough to up a PDF of  the current draft online. Hence, here it is. While the paper does explain what we mean by “Travolta tables” it doesn’t explain why we call them that way … but the image below does: