Efficient Decoding of Gabidulin Codes over Galois Rings

This paper presents the first decoding algorithm for Gabidulin codes over Galois rings with provable quadratic complexity. The new method consists of two steps: (1) solving a syndrome-based key equation to obtain the annihilator polynomial of the error and therefore the column space of the error, (2) solving a key equation based on the received word in order to reconstruct the error vector. This two-step approach became necessary since standard solutions as the Euclidean algorithm do not properly work over rings

A Subfield Lattice Attack on Overstretched NTRU Assumptions:Cryptanalysis of Some FHE and Graded Encoding Schemes

International audienc

Quaternary Affine-Invariant Codes

This thesis concerned with extended cyclic codes. The objective of this thesis is to give a full description of binary and quaternary affine-invariant codes of small dimensions. Extended cyclic codes are studied using group ring methods. Affine-invariant codes are described by their defining sets. Results are presented by enumeration of defining sets. Full description of affine-invariant codes is given for small dimension

Constructions in public-key cryptography over matrix groups

ISBN : 978-0-8218-4037-5International audienceThe purpose of the paper is to give new key agreement protocols (a multi-party extension of the protocol due to Anshel-Anshel-Goldfeld and a generalization of the Diffie-Hellman protocol from abelian to solvable groups) and a new homomorphic public-key cryptosystem. They rely on difficulty of the conjugacy and membership problems for subgroups of a given group. To support these and other known cryptographic schemes we present a general technique to produce a family of instances being matrix groups (over finite commutative rings) which play a role for these schemes similar to the groups $Z_n^*$ in the existing cryptographic constructions like RSA or discrete logarithm

Single-Shot Decoding of Linear Rate LDPC Quantum Codes With High Performance

We construct and analyze a family of low-density parity check (LDPC) quantum codes with a linear encoding rate, distance scaling as nϵ for ϵ>0 and efficient decoding schemes. The code family is based on tessellations of closed, four-dimensional, hyperbolic manifolds, as first suggested by Guth and Lubotzky. The main contribution of this work is the construction of suitable manifolds via finite presentations of Coxeter groups, their linear representations over Galois fields and topological coverings. We establish a lower bound on the encoding rate k/n of 13/72=0.180… and we show that the bound is tight for the examples that we construct. Numerical simulations give evidence that parallelizable decoding schemes of low computational complexity suffice to obtain high performance. These decoding schemes can deal with syndrome noise, so that parity check measurements do not have to be repeated to decode. Our data is consistent with a threshold of around 4% in the phenomenological noise model with syndrome noise in the single-shot regime