Sage Code for GGH Cryptanalysis by Hu and Jia

Recently, Yupu Hu and Huiwen Jia put a paper on the Cryptology ePrint Archive which describes a successful attack of the GGH (and GGHLite) candidate multilinear map. The attack does not try to recover the secret g or any other secret parameter of the map. Instead, it solves the Extraction \kappa-graded CDH (Ext-GCDH) problem directly.

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Google Summer of Code 2015

Both Sage and the Lmonade project were selected for Google’s Summer of Code 2015. If you are an eligible student, you should consider applying. If you need ideas what to work on, there are many fine projects/project ideas on either the Lmonade or the Sage GSoC pages. In particular, here are the fplll project ideas, for which I could be one of the two mentors.

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Sage Development with Emacs

A while back I described my (then current) setup to develop C code with Emacs. The other programming language I tend to spend a lot of time with is Python, specifically Sage’s Python. Here’s my Emacs setup for writing Sage code. For starters, it makes sense to highlight indentation in Python.

(use-package highlight-indentation
   :ensure t)

I use anaconda-mode for auto-completion and stuff, it runs jedi for me. In particular it offers:

  • M-. Goto definition for thing at point.
  • M-, Switch to buffer of most recent marker.
  • M-? Show documentation for context at point.
  • M-r Show usage for thing at point.
(use-package anaconda-mode
  :ensure t
  :diminish anaconda-mode
  :config (bind-key "M-," #'anaconda-nav-pop-marker anaconda-mode-map))

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Looking Back and Forward for Open-Source Mathematics Software (2014)

When a year ends people make lists. I can only guess that several people are currently busy with writing “The 5 most revised papers on eprint ” and “The 8 best IACR flagship conference rump session presentations of 2014”. Since all the good lists are taken, my list has to be a little bit more personal. Alas, here is my list of stuff that happened in open-source computational mathematics in 2014 around me. That is, below I list what developments happened in 2014 and try to provide an outlook for 2015 (so that I can come back in a year to notice that nothing played out as planned).

If you are interested in any of the projects below feel invited to get involved. Also, if you are student and you are interested in working on one of the (bigger) projects listed below over the summer, get in touch: we could try to turn it into a Google Summer of Code 2015 project.

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LMonade GSoC 2014 Accepted Projects

The list of accepted projects of this year’s Google Summer of Code is out. For the list of accepted projects for Sage see here, for the LMonade project see below, for all other accepted projects see Google’s site. I am going to mentor William’s M1RI project together with Clément Pernet. It’s going to be a blast.

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Sage & LMonade GSOC 2014

This year Sage and lmonade are mentoring organizations for the Google Summer of Code again. We have many exciting projects for students to work (from home) over the summer, get mentored by experts in the field and get paid by Google.

Note that projects are not limited to the ideas presented on these pages. That is, if you have a nice project you’d like to propose, by all means do so! The application deadline for students is March 21st. These days they should be talking to potential mentors and forming an applications. If you know any student who might be interested or have access to any forum such students might read, please forward this announcement.

Cryptanalysis of the FHE based on GACD?

Jintai Ding and Chengdong Tao published a new preprint on the IACR’s ePrint titled A New Algorithm for Solving the Approximate Common Divisor Problem and Cryptanalysis of the FHE based on GACD.

*Abstract. *In this paper, we propose a new algorithm for solving the approximate common divisors problems, which is based on LLL reduction algorithm of certain special lattice and linear equation solving algorithm over integers. Through both theoretical argument and experimental data, we show that our new algorithm is a polynomial time algorithm under reasonable assumptions on the parameters. We use our algorithm to solve concrete problems that no other algorithm could solve before. Further more, we show that our algorithm can break the fully homomorphic encryption schemes, which are based on the approximate common divisors problem, in polynomial time in terms of the system parameter λ.

It is worth emphasising that the Approximate GCD problem not only underpinsone of the few fully homomorphic encryption schemes we have but it is also somewhat related to one of two candidates for multilinear maps. So if it could be shown to be easy then this would be somewhat sad as the choice of problems for building fancy crypto schemes would have gotten a lot smaller. So what is the Approxmiate GCD problem?

Approximate Greatest Common Divisions Problem: Given polynomially many samples x_{i} = q_{i}· p + r_{i} where x_{i} = O(2^{γ}), r_{i} = O(2^{ρ}) and p = O(2^{η}), recover p.

The algorithm proceeds by using the LLL algorithm to find relations .

Note that if enough such relations can be found then this gives a linear system of equations in the r_{j} which we’d only need to solve. So how does the algorithm use LLL to recover the a_{ij}? It sets up a lattice basis of dimension (t+1) × (t+1) as follows:

Here, N is simply a random integer O(2^{γ}). Now, the authors claim that running LLL on the lattice spanned by B returns about t-2 of the desired relations. They are unable to rigorously argue why this should happen but offer the following intuition. Any  vector  in the lattice spanned by B has the form . Considering the last component = = they speculate that that LLL would focus on the left hand side of this expression as the right hand side would be rather small anyway. Making implies which in turn implies , if I understood correctly.

An implementation of the first step of this algorithm for Sage is given here (in a Sage cell). Indeed, if you plug in the parameters from the authors’ table on page 9, we do get our desired relations out.

Finally, let’s look at the application to parameters as they are used in cryptography. The authors consider the fully homomorphic encryption scheme due to Marten van Dijkm Craig Gentry, Shai Halevi, Vinod Vaikuntanathan which sets γ = λ^{5}, η = λ^{2} and ρ = λ for a security level of λ, i.e. 2^{λ} operations should be needed to break it. Here is what the authors write:

We apply our algorithm to the parameters in [ 6 ] and we could break all the cases where their parameter γ < 2^{20}.

It is not really clear to me if the authors actually ran their attack or not. On the one hand, we have that a choice of parameters where γ < 2^{20} would correspond to  λ=16 as (2^{20})^{(1/5)} = 2^{4}. Hence, attacking such dimensions would not mean much. On the other hand, the estimates by the authors about LLL would mean their attack would cost 2^{135} operations.

However, as far as I can tell, it does not work for these kind of parameters. That is, LLL fails to find the desired relations once we choose parameters as they are suggested in the cryptographic literature (cf. the example in the Sage cell above).

There are two reasons why the algorithm might fail:

  1. The target vectors might not be among the shortest vectors in the lattice. For the parameters on page 9 it seems this condition holds. It is not clear that the condition holds in general. While on page 7 the authors argue for the existence of target vectors within the approximation radius of LLL, the algorithm on page 8 expects vectors which are smaller than suggested by the Gaussian heuristic, which seems to be what is needed to me.
  2. The target vectors are the shortest vectors in the lattice. However, LLL cannot find them. In such a case it seems the situation is somewhat similar to the situation discussed in this comment from [[http://eprint.iacr.org/2009/616.pdf%5D%5B%5B6]]]:

On the other hand, when t is large, ~v likely is the shortest vector in L, but known lattice reductions algorithms will not be able to find it efficiently. Specifically, as a rule of thumb, they require time roughly 2^{(t/k)} to output a 2^{k} approximation of the shortest vector. Since clearly there are exponentially (in t) many vectors in L of length at most |x_{0}|√(t + 1) < 2^{γ}√(t + 1), which is about 2^{(η−ρ)} times longer than ~v, we need better than a 2^{(η−ρ)} approximation. For t ≥ γ/η, the time needed to guarantee a 2^{η} approximation (which is not even good enough to recover ~v) is roughly 2γ/η^{2}.  Thus setting γ/η^{2} = ω(log λ) foils this attack.

So if I understand this correctly, they should have a condition on t which implies that the target vectors are smaller than what the Gaussian heuristic suggests by the appropriate LLL Unique SVP factor. In particular, they ask if there are target vectors with

|μ_i| < 1/√(t +1) 2^{(γ/(t+1) + t/4)}

but it should be more like

|μ_i| < τ/√(t +1) 2^{(γ/(t+1) – t/4)}

i.e. within the LLL approximation radius there shouldn’t be any other vectors (where τ is the Unique-SVP factor ~0.5).

Update: Since the authors show that if a short vector exists it must satisfy their conditions, this argument is invalid. Still, the authors did not show that they are able to find short enough vectors for parameters as they are found in the literature.

Of course, I might have missed something.

Three sweet but short postdocs in France

The HPAC project has three one-year postdoc positions available:

Three research positions (postdoc or research engineer), offered by the French ANR project HPAC  (High Performance Algebraic Computation), are open.

Title: High Performance Algebraic Computing

Keywords: parallel computing, computer algebra, linear algebra, C/C++ programming

Locations:

  • Grenoble, France (LIG-MOAIS, LJK-CASYS),
  • Lyon, France (LIP-AriC),
  • Paris, France (LIP6-PolSys),

Starting date: between June 2014 and January 2015

Type of position: 3 postdoc or research engineer positions of 1 year each

Detailed descriptions:

General Context:

The ambition of the project HPAC is to provide international reference high-performance libraries for exact linear algebra and algebraic systems on multi-processor architectures and to influence parallel programming approaches for algebraic computing. It focuses on the design of new parallel algorithms and building blocks dedicated to exact linear algebra routines. These blocks will then be used for the parallelization of the sequential code of the LinBox and FGb libraries, state of the art for exact linear algebra and polynomial systems solving, and used in many computer algebra systems. The project combines several areas of expertise: parallel runtime and language, exact,
symbolic and symbolic/numeric algorithmic, and software engineering.

Profile of the positions:

We are seeking for candidates with solid expertise in software library design and developments (e.g. C, C++, OpenMP, Autotools, versioning,…) with preferably good background on mathematical software and computer algebra algorithmic. The main outcome of the work will depend on the type of the position (postdoc or engineer) and include code development in open-source C/C++ libraries such as LinBox, FGb, Kaapi and research publications in international journals or conferences.

Each location is seeking for candidates matching with the following keywords:

  • Lyon: (contact: Gilles….@ens-lyon.fr) High performance/parallel computer algebra, symbolic and mixed symbolic-numeric linear algebra,  validated computation, high performance Euclidean lattice computation, lattice basis reduction.
  • Grenoble: (contact: Jean-Guill…@imag.fr) Library design and development, LinBox, Sage, XKaapi, parallel exact linear algebra, work-stealing and data-flow tasks.
  • Paris: (contact: Jean-Charl…@groebner.org) Polynomial system solving, Gröbner basis computations, parallel exact linear algebra, algebraic cryptanalysis, distributed computing.

Feel free to exchange with the contact person of each site for further information.

Lazy Modulus Switching for the BKW Algorithm on LWE

our paper (with Jean-Charles FaugèreRobert Fitzpatrick and Ludovic Perret) on solving small secret LWE faster just hit ePrint (and was accepted for presentation at PKC 2014)

Abstract. Some recent constructions based on LWE do not sample the secret uniformly at random but rather from some distribution which produces small entries. The most prominent of these is the binary-LWE problem where the secret vector is sampled from {0, 1}* or {-1, 0, 1}*. We present a variant of the BKW algorithm for binary-LWE and other small secret variants and show that this variant reduces the complexity for solving binary-LWE. We also give estimates for the cost of solving binary-LWE instances in this setting and demonstrate the advantage of this BKW variant over standard BKW and lattice reduction techniques applied to the SIS problem. Our variant can be seen as a combination of the BKW algorithm with a lazy variant of modulus switching which might be of independent interest.

The code used to produce experimental data is available on bitbucket, source code to compute our complexity estimations is also available. Slides for a presentation discussing this work are also available on bitbucket.