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, http://scc2012.unican.es/

CALL FOR PAPERS

IMPORTANT DATES

  • 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, http://wmc2012.unican.es/) , 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:

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!

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.

Chen & Nguyen’s algorithm and Arora & Ge’s algorithm

In Faster Algorithms for Approximate Common Divisors: Breaking Fully-Homomorphic-Encryption Challenges over the Integers Yuanmi Chen and Phong Q. Nguyen (preprint here) propose a new algorithm for solving the approximate GCD problem. It drops the complexity from 2^{2\rho} to 2^{3/2\rho} in the general case and from 2^{\rho} to 2^{\rho/2} in the partial case (one multiple of p is given noise-free) which is a pretty big deal.

The algorithm is based on two key ideas (explained using the partial approximate GCD problem):

1. Noisy solving reduced to noise-free solving

Similar to Arora & Ge’s algorithm for solving LWE Chen and Nguyen reduce the approximate solving problem to a noise-free solving problem. In fact, the strategy is exactly the same (cf. also this post). Given noisy ideal elements f_i = \sum h_i g_i + r_i where g_i are generators of the ideal, h_i are ring elements and r_i are small noise terms, then

F_i = f_i \cdot \prod (f_i + j)(f_i - j)

will be elements of the ideal I spanned by g_i if j is big enough (depending on the exact setup we may drop the -j part). In the approximate GCD case g_0 is simply a small odd integer (often denoted p). Additionally, if we are given some sufficient “description” of some sufficiently big ideal \langle G_1,\dots,G_s \rangle = J \supset I (i.e., all elements of I are in J but not vice versa and J is considerably bigger than I) then we can compute

F_i = f_i \cdot \prod (f_i + j)(f_i - j) \mod J

which keeps the size of F_i small-ish. This is the role of x_0, the noise free multiple of p in the partial approximate GCD problem. Now, one simply solves the noise free system F_1,\dots,F_m. In the PAGCD case this means to compute a single GCD, in the multivariate polynomial case (including LWE) this means to compute a Gröbner basis (or linearise, which is the same thing for the cases we are concerned with). Hence, so far Arora&Ge and Chen&Nguyen are really the same thing (it should be mentioned that this ideal due to Nguyen was already mentioned in this paper) applied to different rings.

However, this is not really why the Chen & Nguyen algorithm is efficient (although this already provides a speed-up by a factor of 5).

2. Efficient multiplication

The key idea to drop the exponent from 2^{\rho} to 2^{\rho/2} is as follows. Instead of computing with integers we compute univariate polynomials mod x_0, i.e. one defines

f_j(x) = \prod_{j=0}^{j-1} (x_1 - (x + i)) \in \mathbb{F}_{x_0}[x]

and notices that for \rho' = \lfloor \rho/2 \rfloor:

\prod_{i=0}^{2^\rho-1} (x_1 - i) = \prod_{k=0}^{2^{\rho - \rho'} -1} f_{2^{\rho'}}(2^{\rho'}k)

i.e., we can reduce 2^\rho -1 multiplications to 2^{\rho - \rho'} - 1 multiplications and 2^{\rho - \rho'} - 1  polynomial evaluations. It turns out, this can be done in \mathcal{O}(2^{\rho'}). For the details read the paper.

But to get back to my previous point: It turns out the Arora&Ge perspective on noisy system solving is also useful for approximate GCDs. Which provides further evidence that it is useful to generalise LWE and AGCD to ideal theoretic problems in multivariate polynomial rings.

Slides: Introduction to Algebraic Techniques in Block Cipher Cryptanalysis

This morning I delivered my talk titled “Algebraic Techniques in Cryptanlysis (of block ciphers with a bias towards Gröbner bases)” at the ECrypt PhD Summerschool here in Albena, Bulgaria. I covered:

  1. Why bother
  2. Setting up equation systems
  3. Solving (GBs, SAT solvers, MIP, Cube Testers)
  4. “Advanced” Techniques

Well, here are the slides, which perhaps spend too much time explaining F4.

PS: This is as good as any opportunity to point to the paper “Algebraic Techniques in Differential Cryptanalysis Revisited” by Meiqin Wang, Yue Sun, Nicky Mouha and Bart Preneel accepted at ACISP 2011. I don’t agree with every statement in the paper – which revisits techniques Carlos and I proposed in 2009 – but our FSE 2009 paper does deserve a good whipping, i.e., we were way too optimistic about our attack.

Postdoc Position at LIP6

My team is hiring:

One-year post-doctoral position announcement

A 12-month postdoctoral position is available at the INRIA/LIP6/UPMC SALSA team on Campus Jussieu (located in the “Quartier Latin” of Paris, France). We are seeking candidates to apply for this position. The postdoc will work in the joint ANR/NSFC EXACTA project on polynomial system solving and its applications in cryptography, computational geometry, and biology.

Continue reading “Postdoc Position at LIP6”