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Nombre de programmes trouvés : 17838
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le (5m17s)

3.2. Combinatorial Solutions: Exhaustive Search and Birthday Decoding

In this session, I will detail two combinatorial solutions to the decoding problem. The first one is the Exhaustive Search. To find our w columns, we will simply enumerate all the tuples j1 to jw and check whether the corresponding column plus the syndrome is equal to zero modulo 2. In detail here is how we will do. We have w loops enumerating the indices from j1 to jw, and in the innermost loop, we add the w columns plus the syndrome and either we test the value of the syndrome or ...
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le (3m12s)

3.3. Information Set Decoding: the Power of Linear Algebra

In this third session, we will present the most important concept of the week: Information Set Decoding. The problem of decoding is not only a combinatorial problem. Because we are dealing with linear code, we may also use Linear Algebra. In particular, we are able to transform the Computational Syndrome Decoding problem by multiplying the matrix by a permutation P on the right and a nonsingular matrix U on the left. This will transform the problem of syndrome decoding into an equivalent one. It is very easy to prove that the ...
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le (5m30s)

3.4. Complexity Analysis

In this session, I will present the main technique to make the analysis of the various algorithms presented in this course. So, Information Set Decoding refers to a family of algorithms which is similar to the Prange algorithm that we have just seen. All variants of Information Set Decoding repeat a large number of independent iterations which all have a constant cost K and a success probability P. This means that this iteration has to be repeated an expected number of times N where N = 1/P. And the total workfactor ...
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le (3m8s)

3.5. Lee and Brickell Algorithm

In this fifth session, we will study a variant of information set decoding proposed by Lee and Brickell. So, the main idea consists in relaxing the Prange algorithm to amortize the cost of the Gaussian elimination. So, instead of error patterns with all positions on the left, we will allow error patterns of the form given in the slide. So, in the left part we have w-p coordinate to 1 and on the right hand side we allow a small number p of positions to have a value 1. So, at each ...
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le (6m37s)

3.6. Stern/Dumer Algorithm

In this session, we will present the Stern algorithm for decoding. In fact, the idea is to combine two algorithms that we have seen before, the Lee and Brickell algorithm and the Birthday Decoding.  So, instead of a full Gaussian elimination, we will simply have a partial Gaussian elimination as presented here. And if we look at the lower part, what is called step 1, in red here in this slide, it is, in fact, a smaller CSD problem with a smaller matrix H', with a smaller target syndrome s' and with ...
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le (7m28s)

3.7. May, Meurer, and Thomae Algorithm

So, with the session 7 we are entering the most advanced part of that course. The idea of what I called the  Improved Birthday Decoding is to use the so-called "representation technique" introduced by Howgrave-Graham and Joux in 2010 in which we will relax the way we construct the two lists in Birthday Decoding. So, if you remember, we could relax the size of the matrices H1 and H2 slightly to gain a polynomial factor on Birthday Decoding. But, we may push the idea further and increase the size of H1 and ...
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le (8m33s)

3.8. Becker, Joux, May, and Meurer Algorithm

Now in session 8, we will present yet another evolution of information set decoding. Before presenting this improvement, we will first improve the Birthday Decoding algorithm what I call a Further Improvement of Birthday Decoding. I will consider the two following lists. The difference between those two lists and those we had before is the + ɛ that you can find in the weight of the errors e1 and e2. Those lists depend on another parameter ɛ. What is the meaning of that parameter? Well, the idea is the following: if ...
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le (8m27s)

3.9. Generalized Birthday Algorithm for Decoding

The session nine is devoted to the application of the Generalized Birthday Algorithm to decoding. The Generalized Birthday Algorithm was presented by David Wagner in 2002, in a more general context. In fact, at order a, the Generalized Birthday Algorithm solves the following problem: we are given 2^a lists of vectors of size L and we want to find xi, one in every list Li, such that the sum of all the xi is 0. If the lists Li are large enough, then the algorithm runs in time 2^(l/(a+1)). Note that the ...
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le (8m37s)

3.10. Decoding One Out of Many

The final session of this week is devoted to Decoding One Out of Many. Decoding One Out of Many is interested in solving the following variant of Syndrome Decoding. In this variant, the only difference with the usual Syndrome Decoding is that we are interested in a set of syndromes rather than a single syndrome. So, the instance will be S, a set of syndromes of size N. H, a parity-check matrix and w an integer, the weight we are looking for. And we are interested in an error e, such that ...
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le (4m47s)

4.1. Introduction

Welcome to the fourth week of the MOOC Code-based Cryptography. Recall that we have mainly two ways of cryptanalyzing in the McEliece cryptosystem. We have Message Attacks, which address the problem of decoding a random linear code; these attacks has already been studied in the third week, by Nicolas Sendrier. Notice that efficient generic attack just makes the use of larger code in the McEliece scheme necessary. And we also have Key Attacks. These attacks try to retrieve the code structure, rather than attempting to use an specific decoding algorithm. These attacks ...
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