On a Family of Circulant Matrices for Quasi-Cyclic Low-Density Generator Matrix Codes

Abstract

We present a new class of sparse and easily invertible circulant matrices that can have a sparse inverse though not being permutation matrices. Their study is useful in the design of quasi-cyclic low-density generator matrix codes, that are able to join the inner structure of quasi-cyclic codes with sparse generator matrices, so limiting the number of elementary operations needed for encoding. Circulant matrices of the proposed class permit to hit both targets without resorting to identity or permutation matrices that may penalize the code minimum distance and often cause significant error floors.

Security gap analysis of some LDPC coded transmission schemes over the flat and fast fading Gaussian wire-tap channels

“…We consider QC-LDPC codes having a parity-check matrix in the form (5), with constant column weight equal to d v . In this case, it can easily be shown that the code minimum distance is ≤ 2d v and the corresponding multiplicity is ≈ n 0 2 [22].…”

“…Several ways of designing QC-LDPC codes are known in the literature. Two main classes are those using cyclic permutation matrices, like array LDPC codes [21], and those using general sparse circulant matrices [22].…”

An Improved Zero-knowledge Identification Scheme based on Quasi-Dyadic Codes

“…Unfortunately, Niederreiter's proposal was proved to be insecure subsequently. In the following years, a number of cryptosystems [5][6][7][8][9][10] were proposed by using different codes such as MDPC, QD-Goppa codes, etc. And the public key size can be reduced much more by exploiting the Toeplitz structure [11].…”

Squaring attacks on McEliece public-key cryptosystems using quasi-cyclic codes of even dimension

“…Despite its ability to resist attacks, several other underlying codes have been proposed. Among these, we find ReedSolomon codes, LDPC codes [1][2][3][4][5][6]14], rank-metric codes and more recently convolutional codes [16] and MDPC codes [20]. However, many of the proposals have turned out to be insecure.…”