We cryptanalyse here two variants of the McEliece cryptosystem based on quasi-cyclic codes. Both aim at reducing the key size by restricting the public and secret generator matrices to be in quasi-cyclic form. The first variant considers subcodes of a primitive BCH code. The aforementioned constraint on the public and secret keys implies to choose very structured permutations. We prove that thi...

The main features of (finite) multidimensional convolutional codes are introduced as properties of the codewords supports, and connected with the polynomial matrices (encoders) adopted for their description. Observability and local detectability are shown to be equivalent to the kernel representation of a code via some parity check matrix H . The input/output representation of convolutional cod...

In this paper, we generalize the sampling theorem for bandlimited functions over the Boolean domain to a sampling theorem for bandlimited functions over a GF (q) domain. We also present a theorem for the relation between the parity check matrix of a linear code and any distinct error vectors. Lastly, we clarify the relation between the sampling theorem for functions over a GF (q) domain and lin...

Guo et al. recently presented a reaction attack against the QC-MDPC McEliece cryptosystem. Their attack is based on the observation that when a bit-flipping decoding algorithm is used in the QC-MDPC McEliece, then there exists a dependence between the secret matrix H and the failure probability of the bit-flipping algorithm. This dependence can be exploited to reveal the matrix H which constitu...

Quantum error correction is an indispensable ingredient for scalable quantum computing. In this Perspective we discuss a particular class of codes called low-density parity-check (LDPC) codes. The are alternatives to the surface code, which currently leading candidate implement fault-tolerance. We introduce zoo LDPC and their potential making computers robust against noise. particular, explain ...

Polarization-adjusted convolutional (PAC) codes have better error-correcting performance than polar mostly because of the improved weight distribution brought by pre-transformation. In this paper, we propose parity check PAC (PC-PAC) to further improve codes. The design principle is establish functions between bits with distinct row weights, such that information lower reliability are re-protec...

Low-density lattice codes (LDLC) designed for inter-symbol interference (ISI) channels are proposed in this paper, including the node-by-node belief propagation (BP) decoding algorithm and the construction of LDLC codes for ISI channels. A theorem is presented to illustrate that the nominal coding gain of LDLC codes can be increased by introducing ISI into the primary codes. Shaping operation m...

A mapping f from F2m to itself is almost perfect nonlinear (APN) if its directional derivatives in nonzero directions are all 2-to-1. Let Cf be the binary linear code of length 2 − 1, whose parity check matrix has its j-th column [ π f(π) ] , where π is a primitive element in F2m and j = 0, 1, · · · , 2 − 2. For m ≥ 3 and any quadratic APN function f(x) = ∑m−1 i,j=0 ai,jx 2+2 , ai,j ∈ F2m , it ...

In this paper, we introduce a new approach to side-channel key recovery, that combines the low time/memory complexity and noise tolerance of standard (divide and conquer) differential power analysis with the optimal data complexity of algebraic side-channel attacks. Our fundamental contribution for this purpose is to change the way of expressing the problem, from the system of equations used in...