Notice
3.6. Stern/Dumer Algorithm
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Descriptif
In this session, we willpresent the Stern algorithm for decoding. In fact, the idea is to combine two algorithmsthat we have seen before, the Lee and Brickell algorithmand the Birthday Decoding. So, instead of a fullGaussian elimination, we will simply have a partial Gaussianelimination as presented here. And if we look at the lowerpart, what is called step 1, in red here in thisslide, it is, in fact, a smaller CSD problem witha smaller matrix H', with a smaller target syndrome s' and with the weight p. So, now we are going tosolve this smaller problem in the first step. In fact, weare going to find all or most solutions to that problem e'. And for every e'foundin step 1, we will check in step 2 whetherthis particular solution e'can be extended to asolution of the full CSD problem. If we solve the firststep using enumeration, as I presented as the veryfirst algorithm, then we obtain an algorithm which is very similarto the Lee and Brickell algorithm. But, if we use for step 1a Birthday Decoding, then we obtain a significantspeedup for decoding and it is in fact the Dumer Algorithm. Now, I will describe this algorithm. So, we are trying to solve CSD of H, s, w and this time wehave two additional integer parameters p and l. And we repeat the following:we pick a permutation matrix P, we make a partial Gaussianelimination as you can see on the right of the slide,obtaining U, H', H'', s', s''. We solve the small CSD problem with H', s' and p - we solve this onewith Birthday Decoding - and for every solution of theproblem we will construct the second part of the error e''. And if we are lucky,this second part of the error pattern e''has asufficiently small weight, and then, provides thesolution to our problem. So, what I presentedis the Dumer algorithm. In fact, this algorithmwas published in 1991. But Sternalgorithm, which is slightly more expensive, waspresented two years before.
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