Notice
2.8. Reducing the Key Size - MDPC codes
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Descriptif
This is the last session where wewill talk about reducing the key size. Here we willintroduce the MDPC codes.
In 2012, the MDPC codes wereproposed for the McEliece schemes. An MDPC code is acode that admits a binary
moderate density-parity check matrix. Typically, the Hammingweight of each row is of the order the square of the length. In this sequence, I willdescribe this scheme of quasi-cyclic MDPC McEliece fora binary code of rate one half. So, we use circulantmatrices of blocks of size p to define the codes. The lengthwill be 2p and the dimension p. Other parameters are theweight of the parity check equations and the numberof correctable errors. So, let us explain the McElieceschemes using quasi-cyclic MDPC code. First of all, we pickrandomly two vectors of weight p,
such that the concatenatedvector has a weight smaller than w. We will repeat until thecorresponding polynomial h0 is invertible. In particular,we ask the weight to be odd. Then, the secret key and the publickey will be the corresponding matrices. To encrypt a message, weapply the following function, that is, we encode themessage and we add random errors of weight smaller than t. But we will describethem in terms of polynomial. To decrypt, we use anMDPC-like iterative decoding algorithm as theGallager's Bit-Flipping algorithm, already explained inthe previous session. The quasi-cyclic MDPC proposalis secure under two assumptions. First of all, the problem ofdistinguishing a public key from a random quasi-cyclicmatrix or equivalently the problem of findingcodewords of weight w in the dual of an MDCP code; and thehardness of decoding random quasi-cyclic codes. Thesecurity reduction can be translated in terms ofpolynomials as follows.
Intervention
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