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2.7. Reducing the Key Size - LDPC codes
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Descriptif
LDPC codes have aninteresting feature: they are free of algebraic structure. We will study in detailthis proposal for the McEliece cryptosystem in thissession. LDPC codes were originally introduced by Gallager, inhis doctoral thesis, in 1963. One of the characteristic ofLDPC codes is the existence of several iterativedecoding algorithms which achieve excellent performances.Tanner, later, in the 1981, introduced a graphicalrepresentation to these codes as bipartite graph. However, they remainedalmost forgotten by the coding theory community until1996 when MacKay and Neal re-discovered these codes,promoting them to the select group of good linear codes fortelecommunication applications. LDPC codes admit twodifferent representations: on one hand, we have thematrix representation. LDPC codes admit a sparseparity-check matrix, that is, it contains few non-zero entries incomparison to the amount of zeros. And we have also agraphical representation. LDPC codes could berepresented with a graph which is also known as the Tannergraph. First of all, recall the definition of a bipartitegraph which is a graph that we can partition the set ofvertices into two nonempty disjoint sets such that no twovertices within the same set are adjacent. Now, let H be a sparse matrix. We will denote theset of variable nodes
to the column of theparity-check matrix and the check nodes to the rows ofthe parity-check matrix. And we define an edge betweenthe j check nodes and the i variable nodes if theentry (i,j) at the matrix H is non-zero. Forexample, if we have a binary LDPC codes with this parity-checkmatrix then we can associate the following Tanner graph. The first parity-checkequation gives us this relation. The second parity-checkequation gives us this other relation and the third onegives us the complete graph. We explain here an iterativedecoding algorithm for LDPC codes which is called theBit-Flipping algorithm, which was already introduced by Gallager.
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