Matrix Multiplication over GF(p^e)

After my talk at Sage Days 35 in Warwick (that was in winter 2011) David Harvey had an idea on how to speed up matrix multiplication over \mathbb{F}_{p^n}. We spend some time on this in Warwick and developed this idea further (adding fun stuff like Mixed Integer Programming in the process) but did not get around to do much on this project in the mean time (I have explained the idea at the end of my talk in Mykonos, though).

Just now, in a conversation with Richard Parker I was reminded of this dormant project, i.e., the question of how many multiplications i \mathbb{F}_p it takes to do a multiplication in \mathbb{F}_{p^n}. In particular, I recalled to have written some code for Sage which gives some upper bound to this answer which is better than Karatsuba.

Well, here’s an interactive demo … gosh, I love the Sage cell server.

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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!

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.

Constraint Integer Programming in Sage

Now that version 4.6.2 of Sage is out which contains the new Mixed Integer Programming interface that Nathann (mainly) and I (a little bit) wrote, I took the time to get my SCIP interface enough into shape to open a ticket on Sage’s bugtracker for it. Continue reading “Constraint Integer Programming in Sage”

Cold Boot Key Recovery by Solving Polynomial Systems with Noise

Carlos and I finally managed to put our paper on polynomial system solving with noise and its application to the cold boot problem out.

Abstract: A method for extracting cryptographic key material from DRAM used in modern computers has been recently proposed in [9]; the technique was called Cold Boot attacks. When considering block ciphers, such as the AES and DES, simple algorithms were also proposed in [9] to recover the cryptographic key from the observed set of round subkeys in memory (computed via the cipher’s key schedule operation), which were however subject to errors due to memory bits decay. In this work we extend this analysis to consider key recovery for other ciphers used in Full Disk Encryption (FDE) products. Our algorithms are also based on closest code word decoding methods, however apply a novel method for solving a set of non-linear algebraic equations with noise based on Integer Programming. This method should have further applications in cryptology, and is likely to be of independent interest. We demonstrate the viability of the Integer Programming method by applying it against the Serpent block cipher, which has a much more complex key schedule than AES. Furthermore, we also consider the Twofish key schedule, to which we apply a dedicated method of recovery.

Btw. an older version of our code for Sage for solving polynomial systems with errors is available on bitbucket.org (… yes, I should update it to the most recent version). Here’s an example from my talk at the Tools for cryptanalysis workshop 2010:

sage: p = PRESENT(Nr=1,sbox_representation='lex')
sage: F = present_dc(,r=1,return_system=True,characteristic=True)
sage: H = F.gens()[:-64]
sage: S = F.gens()[-64:]
sage: S[:9]
(Y00100 + Y10100, Y00101 + Y10101, Y00102 + Y10102,  Y00103 + Y10103, Y00104 + Y10104, Y00105 + Y10105,  Y00106 + Y10106, Y00107 + Y10107 + 1, Y00108 + Y10108)

sage: F_prob = ProbabilisticMPolynomialSystem(F.ring(),H,S)
sage: s,t = F_prob.solve_mip(solver='SCIP')
Writing problem data to '/home/malb/.sage//temp/road/16007//tmp_1.mps'6605 records were writtenCPU Time: 0.20  Wall time: 25.95, Obj:  3.00

Not that this was a good way of attacking a blockcipher, but you get the idea.

Final Version of my PhD thesis

I passed my PhD defence on Friday. I’ve uploaded the final version of my thesis titled Algorithmic Algebraic Techniques and their Application to Block Cipher Cryptanalysis. Not much changed since the last draft, which I also posted on this blog, except a few typos and errors.

Continue reading “Final Version of my PhD thesis”