libFES : Fast Exhaustive Search for Polynomial Systems over F2

Charles Bouillaguet set up a nice shiny website for libFES the library for exhaustive search on polynomial systems over \mathbb{F}_2. The library has a Sage interface, so it’s easy to get started. It’s also integrated in Charles’ upcoming one-stop boolean system solving patch.

He calls his benchmarketing “bragging rights” … and boy has he earned those rights! Check it out, libFES is fast!

SAT Solvers for Sage

One of the most efficient techniques for solving polynomial systems over \mathbb{F}_2 is to convert the problem to a satisfiability problem and to use a standard SAT solver. In the past, I have used CryptoMiniSat and either my own ANF to CNF converter scripts based on Gregory Bard’s ideas or PolyBoRi’s script.

However, this setup leaves much to be desired:

  1. It’s all based on string parsing which has some overhead.
  2. Usually the instances produced using PolyBoRi’s conversion method are faster to solve. However, as the number of variables per equation increase this method becomes essentially exponentially more expensive. Hence, a compromise between the two techniques is needed.
  3. We don’t have access to learnt clauses and conflict clauses.
  4. It all feels a bit duct taped and fragile, partly because the code is not shipped with Sage.

At #418 I just finished a much nicer interface to various SAT solvers. Here are some features. Continue reading “SAT Solvers for Sage”

Call for Papers: 3nd International Conference on Symbolic Computation and Cryptography

CIEM – Castro Urdiales, Spain, 11-13 July 2012,



  • Deadline for submission: April 28, 2012
  • Notification of acceptance or rejection: May 18, 2012
  • Deadline for final version: May 30, 2012
  • Deadline for registration: June 12, 2012
  • Deadline for special issue JSC: September 30, 2012

SCC 2012 is the third edition of a new series of conferences where  research and development in symbolic computation and cryptography may be presented and discussed. It is organized in response to the growing  interest in applying and developing methods, techniques, and software  tools of symbolic computation for cryptography. The use of Lattice  Reduction algorithms in cryptology and the application of Groebner  bases in the context of algebraic attacks are typical examples of  explored applications. The SCC 2012 conference is co-located with  third Workshop on Mathematical Cryptology (WMC 2012, , an event also organized by research group Algorithmic Mathematics  And Cryptography (AMAC), which will be held on 9-11 July 2012.

Continue reading “Call for Papers: 3nd International Conference on Symbolic Computation and Cryptography”

Summer School on Tools :: Mykonos, Greece :: 28.5 – 1.6.

Slightly redacted announcement for the 2012 Summer School on Tools below.

Following the success of the ECRYPT Workshop on Tools for Cryptanalysis 2010,the ECRYPT II Symmetric Techniques Virtual Lab (SymLab) is pleased to announce the 2012 Summer School on Tools. Covering selected topics in both symmetric and asymmetric cryptography, this summer school will provide a thorough overview of some of the most important cryptographic tools that emerged in recent years. While the summer school is aimed primarily at postgraduate students, attendance is open to all. Continue reading “Summer School on Tools :: Mykonos, Greece :: 28.5 – 1.6.”

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:

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

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!

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.