Notice
2.3. McEliece Assumptions
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Descriptif
In this session, we willtalk about McEliece assumptions. The security of theMcEliece scheme is based on two assumptions as we havealready seen: the hardness of decoding a random linearcode and the problem of distinguishing a code with aprescribed structure from a random one. In this sequence, we will studyin detail these two assumptions. The first assumption claims thatdecoding a random linear code is difficult. First, notice that thegeneral decoding problem is basically a re-writing ofthe Syndrome Decoding problem. And both are equivalentto the problem of finding codewords of minimal weight.The Syndrome Decoding in the binary case is state as follows. Given a binary matrix, asyndrome S and a non-negative integer W, the weight. The decisionproblem faces the following question. There exists an errorpattern of weight at most w with syndrome S? while the computationalproblem is to find such a vector. The decoding problem wasproved to be NP-complete in 1978 by Berlekamp, McEliece andvan Tilborg in this article. For the q-ary case,see the article of Barg. Take notice that this prooftook place at the same time that the McEliececryptosystem was introduced. Thus, the worst case ofthe computational problem is known to be difficult in general. Of course, depending on theinput, some instances can be solved in polynomial time as wehave already seen in the first week. Actually, the instance ofSyndrome Decoding involved in breaking code-based systemsare in particularly a subclass of Syndrome Decoding where the weight w isbounded by half the minimum distance. This problem is not NP,however it is conjectured to be NP-Hard. But even more in theMcEliece cryptosystem, the chosen code is not completely random. Even if the matrix is notdistinguishable from a random binary matrix of the samesize, the decoding problem uses parameters of those of a Goppa code. This means that the code haslength 2^m and the dimension is n mt, where t isthe correction capacity.
Intervention
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