Description
Non-binary low-density parity-check (NB LDPC) codes significantly outperform their binary counterparts. Moreover, NB LDPC codes are especially good for the channels with burst errors and high-order modulations. This talk is devoted to the analysis of error-correcting capabilities of NB LDPC codes.<br/> We start with distance properties of NB LDPC codes. We consider two approaches to obtain upper bounds for such codes: shortening method and syndrome counting method. Both methods use a sparse structure of the parity-check matrix of NB LDPC codes. We compare obtained upper bounds to the Gallager’s lower bound and to upper bounds for the best non-binary codes. The new upper bounds are shown to lie under the Gilbert-Varshamov bound for some parameters.<br/> The usual way to decode NB LDPC codes is to apply Belief Propagation (BP) decoding. Unfortunately, BP decoding complexity is still large, that is why iterative hard and soft-reliability based decoding majority algorithms are of considerable interest for high-throughput practical applications. In the second part of the talk we investigate the error-correcting capabilities of NB LDPC codes decoded with a hard-decision low-complexity majority algorithm, which is a generalization of the bit-flipping algorithm for binary LDPC codes. We perform the worst-case analysis and estimate the decoding radius realized by this algorithm. By the decoding radius, we mean the number of errors that is guaranteed to be corrected. Our contribution is as follows. We suggest the majority-decoding algorithm with multiple thresholds and give a lower estimate on the decoding radius realized by the new algorithm. The estimate is shown to be at least 1.2 times better than the estimate for a single threshold majority decoder.<br/> At last, we consider BP decoding of NB LDPC codes. We consider an AWGN channel with high-order modulations. In order to reduce the decoding complexity, we propose to use multilevel coding schemes based on NB LDPC codes (NB-LDPC-MLC schemes) over smaller fields. The use of such schemes gives us a reasonable gain in complexity. At the same time the performance of NB-LDPC-MLC schemes is practically the same as the performance of LDPC codes over the field matching the modulation order. In particular, by means of simulations we showed that the performance of NB-LDPC-MLC schemes over GF(16) is the same as the performance of LDPC codes over GF(64) and GF(256) in AWGN channel with QAM 64 and QAM 256 accordingly.
Prochains exposés
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Efficient zero-knowledge proofs and arguments in the CL framework
Orateur : Agathe Beaugrand - Institut de Mathématiques de Bordeaux
The CL encryption scheme, proposed in 2015 by Castagnos and Laguillaumie, is a linearly homomorphic encryption scheme, based on class groups of imaginary quadratic fields. The specificity of these groups is that their order is hard to compute, which means it can be considered unknown. This particularity, while being key in the security of the scheme, brings technical challenges in working with CL,[…] -
Constant-time lattice reduction for SQIsign
Orateur : Sina Schaeffler - IBM Research
SQIsign is an isogeny-based signature scheme which has recently advanced to round 2 of NIST's call for additional post-quantum signatures. A central operation in SQIsign is lattice reduction of special full-rank lattices in dimension 4. As these input lattices are secret, this computation must be protected against side-channel attacks. However, known lattice reduction algorithms like the famous[…] -
Circuit optimisation problems in the context of homomorphic encryption
Orateur : Sergiu Carpov - Arcium
Fully homomorphic encryption (FHE) is an encryption scheme that enables the direct execution of arbitrary computations on encrypted data. The first generation of FHE schemes began with Gentry's groundbreaking work in 2019. It relies on a technique called bootstrapping, which reduces noise in FHE ciphertexts. This construction theoretically enables the execution of any arithmetic circuit, but[…] -
TBD
Orateur : Maria Corte-Real Santos - ENS Lyon
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Cryptography
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