3.3. Information Set Decoding: the Power of Linear Algebra
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In this third session, wewill present the most important concept of the week:Information Set Decoding. The problem of decoding is notonly a combinatorial problem. Because we are dealing with linearcode, we may also use Linear Algebra. In particular, we are ableto transform the Computational Syndrome Decodingproblem by multiplying the matrix by apermutation P on the right and a nonsingular matrix U on the left. This will transform theproblem of syndrome decoding into an equivalent one. It is very easy to prove thatthe following equivalence holds. In fact, this implies thatthe two computational syndrome decoding that arepresented here are equivalent.If I solve one ofthem, I solve the other. In particular, I canchoose H'in the following form which is calledsystematic form or standard form. If the first n-k columns of matrix H*P areindependent, then I am able to compute such a form. If thisis the case, the last k column of the matrixforms an information set. If I am lucky, thepermutation P will send the w error position inthe left part of the position. If this happens,then the modified syndrome s'will have a weight equalto the weight of the error. And I can check that very easily. I can transformthat into an algorithm. So, I want to solve thecomputational syndrome decoding presented here. I will repeatedly picka permutation P, compute the following systematicform using Gaussian elimination, and check whether thetransformed syndrome s*(U transpose) is a weight w. When this hapens, I canreturn the following vector which is validsolution to the original syndrome decoding problem. Here, each iteration of thisalgorithm will cost n*(n-k) column operations that is thecost of the Gaussian elimination. And I have to repeat thata certain number of times, until one of thesolutions to my problem has all its non-zero coordinates "all left". In the next session, I will present Complexity Analysis of thatalgorithm in particular as long as tools to make thecomplexity analysis of the variance of informationset decoding I will present throughout this course.
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