56 research outputs found
List and Probabilistic Unique Decoding of Folded Subspace Codes
A new class of folded subspace codes for noncoherent network coding is
presented. The codes can correct insertions and deletions beyond the unique
decoding radius for any code rate . An efficient interpolation-based
decoding algorithm for this code construction is given which allows to correct
insertions and deletions up to the normalized radius ,
where is the folding parameter and is a decoding parameter. The
algorithm serves as a list decoder or as a probabilistic unique decoder that
outputs a unique solution with high probability. An upper bound on the average
list size of (folded) subspace codes and on the decoding failure probability is
derived. A major benefit of the decoding scheme is that it enables
probabilistic unique decoding up to the list decoding radius.Comment: 6 pages, 1 figure, accepted for ISIT 201
On Decoding Schemes for the MDPC-McEliece Cryptosystem
Recently, it has been shown how McEliece public-key cryptosystems based on
moderate-density parity-check (MDPC) codes allow for very compact keys compared
to variants based on other code families. In this paper, classical (iterative)
decoding schemes for MPDC codes are considered. The algorithms are analyzed
with respect to their error-correction capability as well as their resilience
against a recently proposed reaction-based key-recovery attack on a variant of
the MDPC-McEliece cryptosystem by Guo, Johansson and Stankovski (GJS). New
message-passing decoding algorithms are presented and analyzed. Two proposed
decoding algorithms have an improved error-correction performance compared to
existing hard-decision decoding schemes and are resilient against the GJS
reaction-based attack for an appropriate choice of the algorithm's parameters.
Finally, a modified belief propagation decoding algorithm that is resilient
against the GJS reaction-based attack is presented
Protograph-based Quasi-Cyclic MDPC Codes for McEliece Cryptosystems
In this paper, ensembles of quasi-cyclic moderate-density parity-check (MDPC)
codes based on protographs are introduced and analyzed in the context of a
McEliece-like cryptosystem. The proposed ensembles significantly improve the
error correction capability of the regular MDPC code ensembles that are
currently considered for post-quantum cryptosystems without increasing the
public key size. The proposed ensembles are analyzed in the asymptotic setting
via density evolution, both under the sum-product algorithm and a
low-complexity (error-and-erasure) message passing algorithm. The asymptotic
analysis is complemented at finite block lengths by Monte Carlo simulations.
The enhanced error correction capability remarkably improves the scheme
robustness with respect to (known) decoding attacks.Comment: 5 page
Interpolation-Based Decoding of Folded Variants of Linearized and Skew Reed-Solomon Codes
The sum-rank metric is a hybrid between the Hamming metric and the rank
metric and suitable for error correction in multishot network coding and
distributed storage as well as for the design of quantum-resistant
cryptosystems. In this work, we consider the construction and decoding of
folded linearized Reed-Solomon (FLRS) codes, which are shown to be maximum
sum-rank distance (MSRD) for appropriate parameter choices. We derive an
efficient interpolation-based decoding algorithm for FLRS codes that can be
used as a list decoder or as a probabilistic unique decoder. The proposed
decoding scheme can correct sum-rank errors beyond the unique decoding radius
with a computational complexity that is quadratic in the length of the unfolded
code. We show how the error-correction capability can be optimized for
high-rate codes by an alternative choice of interpolation points. We derive a
heuristic upper bound on the decoding failure probability of the probabilistic
unique decoder and verify its tightness by Monte Carlo simulations. Further, we
study the construction and decoding of folded skew Reed-Solomon codes in the
skew metric. Up to our knowledge, FLRS codes are the first MSRD codes with
different block sizes that come along with an efficient decoding algorithm.Comment: 32 pages, 3 figures, accepted at Designs, Codes and Cryptograph
Fast Decoding of Interleaved Linearized Reed-Solomon Codes and Variants
We construct s-interleaved linearized Reed-Solomon (ILRS) codes and variants
and propose efficient decoding schemes that can correct errors beyond the
unique decoding radius in the sum-rank, sum-subspace and skew metric. The
proposed interpolation-based scheme for ILRS codes can be used as a list
decoder or as a probabilistic unique decoder that corrects errors of sum-rank
up to , where s is the interleaving order, n the
length and k the dimension of the code. Upper bounds on the list size and the
decoding failure probability are given where the latter is based on a novel
Loidreau-Overbeck-like decoder for ILRS codes. The results are extended to
decoding of lifted interleaved linearized Reed-Solomon (LILRS) codes in the
sum-subspace metric and interleaved skew Reed-Solomon (ISRS) codes in the skew
metric. We generalize fast minimal approximant basis interpolation techniques
to obtain efficient decoding schemes for ILRS codes (and variants) with
subquadratic complexity in the code length. Up to our knowledge, the presented
decoding schemes are the first being able to correct errors beyond the unique
decoding region in the sum-rank, sum-subspace and skew metric. The results for
the proposed decoding schemes are validated via Monte Carlo simulations.Comment: submitted to IEEE Transactions on Information Theory, 57 pages, 10
figure
Error-Erasure Decoding of Linearized Reed-Solomon Codes in the Sum-Rank Metric
Codes in the sum-rank metric have various applications in error control for
multishot network coding, distributed storage and code-based cryptography.
Linearized Reed-Solomon (LRS) codes contain Reed-Solomon and Gabidulin codes as
subclasses and fulfill the Singleton-like bound in the sum-rank metric with
equality. We propose the first known error-erasure decoder for LRS codes to
unleash their full potential for multishot network coding. The presented
syndrome-based Berlekamp-Massey-like error-erasure decoder can correct
full errors, row erasures and column erasures up to in the sum-rank metric requiring at most
operations in , where is the code's length and its
dimension. We show how the proposed decoder can be used to correct errors in
the sum-subspace metric that occur in (noncoherent) multishot network coding.Comment: 6 pages, presented at ISIT 202
Fast Decoding of Codes in the Rank, Subspace, and Sum-Rank Metric
We speed up existing decoding algorithms for three code classes in different
metrics: interleaved Gabidulin codes in the rank metric, lifted interleaved
Gabidulin codes in the subspace metric, and linearized Reed-Solomon codes in
the sum-rank metric. The speed-ups are achieved by reducing the core of the
underlying computational problems of the decoders to one common tool: computing
left and right approximant bases of matrices over skew polynomial rings. To
accomplish this, we describe a skew-analogue of the existing PM-Basis algorithm
for matrices over usual polynomials. This captures the bulk of the work in
multiplication of skew polynomials, and the complexity benefit comes from
existing algorithms performing this faster than in classical quadratic
complexity. The new faster algorithms for the various decoding-related
computational problems are interesting in their own and have further
applications, in particular parts of decoders of several other codes and
foundational problems related to the remainder-evaluation of skew polynomials
Efficient Decoding of Folded Linearized Reed-Solomon Codes in the Sum-Rank Metric
Recently, codes in the sum-rank metric attracted attention due to several applications in e.g. multishot network coding, distributed storage and quantum-resistant cryptography. The sum-rank analogs of Reed-Solomon and Gabidulin codes are linearized Reed-Solomon codes. We show how to construct h-folded linearized Reed-Solomon (FLRS) codes and derive an interpolation-based decoding scheme that is capable of correcting sum-rank errors beyond the unique decoding radius. The presented decoder can be used for either list or probabilistic unique decoding and requires at most O(sn^2) operations in F_{q^m}, where s<=h is an interpolation parameter and n denotes the length of the unfolded code. We derive a heuristic upper bound on the failure probability of the probabilistic unique decoder and verify the results via Monte Carlo simulations
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