Decoding of Repeated-Root Cyclic Codes up to New Bounds on Their Minimum Distance

The well-known approach of Bose, Ray-Chaudhuri and Hocquenghem and its generalization by Hartmann and Tzeng are lower bounds on the minimum distance of simple-root cyclic codes. We generalize these two bounds to the case of repeated-root cyclic codes and present a syndrome-based burst error decoding algorithm with guaranteed decoding radius based on an associated folded cyclic code. Furthermore, we present a third technique for bounding the minimum Hamming distance based on the embedding of a given repeated-root cyclic code into a repeated-root cyclic product code. A second quadratic-time probabilistic burst error decoding procedure based on the third bound is outlined. Index Terms Bound on the minimum distance, burst error, efficient decoding, folded code, repeated-root cyclic code, repeated-root cyclic product cod

Minimal Polynomial Algorithms for Finite Sequences

We show that a straightforward rewrite of a known minimal polynomial algorithm yields a simpler version of a recent algorithm of A. Salagean.Comment: Section 2 added, remarks and references expanded. To appear in IEEE Transactions on Information Theory

Decoding Reed-Solomon codes up to the Sudan radius with the Euclidean algorithm

International audienceWe modify the Euclidean algorithm of Feng and Tzeng to decode Reed-Solomon (RS) codes up to the Sudan radius. The basic steps are the virtual extension to an Interleaved RS code and the reformulation of the multi-sequence shift-register problem of varying length to a multi-sequence problem of equal length. We prove the reformulation and analyze the complexity of our new decoding approach. Furthermore, the extended key equation, that describes the multi-sequence problem, is derived in an alternative polynomial way

ANALYSIS OF SECURITY MEASURES FOR SEQUENCES

Stream ciphers are private key cryptosystems used for security in communication and data transmission systems. Because they are used to encrypt streams of data, it is necessary for stream ciphers to use primitives that are easy to implement and fast to operate. LFSRs and the recently invented FCSRs are two such primitives, which give rise to certain security measures for the cryptographic strength of sequences, which we refer to as complexity measures henceforth following the convention. The linear (resp. N-adic) complexity of a sequence is the length of the shortest LFSR (resp. FCSR) that can generate the sequence. Due to the availability of shift register synthesis algorithms, sequences used for cryptographic purposes should have high values for these complexity measures. It is also essential that the complexity of these sequences does not decrease when a few symbols are changed. The k-error complexity of a sequence is the smallest value of the complexity of a sequence obtained by altering k or fewer symbols in the given sequence. For a sequence to be considered cryptographically ‘strong’ it should have both high complexity and high error complexity values. An important problem regarding sequence complexity measures is to determine good bounds on a specific complexity measure for a given sequence. In this thesis we derive new nontrivial lower bounds on the k-operation complexity of periodic sequences in both the linear and N-adic cases. Here the operations considered are combinations of insertions, deletions, and substitutions. We show that our bounds are tight and also derive several auxiliary results based on them. A second problem on sequence complexity measures useful in the design and analysis of stream ciphers is to determine the number of sequences with a given fixed (error) complexity value. In this thesis we address this problem for the k-error linear complexity of 2n-periodic binary sequences. More specifically: 1. We characterize 2n-periodic binary sequences with fixed 2- or 3-error linear complexity and obtain the counting function for the number of such sequences with fixed k-error linear complexity for k = 2 or 3. 2. We obtain partial results on the number of 2n-periodic binary sequences with fixed k-error linear complexity when k is the minimum number of changes required to lower the linear complexity

Linear feedback shift registers and the minimal realization problem

[EN] The Berlekamp-Massey algorithm solves the problem of finding the shortest linear feedback shift register which generates a given finite sequence of scalars. This problem is reinterpreted from the point of view of the realization theory and several extensions to sequences of matrices are analyzed. We give a generalization of the result on which the Berlekamp-Massey algorithm is based in terms of the partial Brunovsky indices of a sequence of matrices and propose an algorithm to obtain them for sequences of vectors. The results we obtain hold for arbitrary fields.The first author is partially supported by grants MINECO MTM2017-83624-P, MTM2017-90682-REDT, and UPV/EHU GIU16/42. The second author is partially supported by grants MINECO MTM2017-83624-P and MTM2017-90682-REDT.Baragana, I.; Roca Martinez, A. (2019). Linear feedback shift registers and the minimal realization problem. Linear Algebra and its Applications. 576:200-227. https://doi.org/10.1016/j.laa.2018.06.009S20022757

Iterative List-Decoding of Gabidulin Codes via Gr\"obner Based Interpolation

We show how Gabidulin codes can be list decoded by using an iterative parametrization approach. For a given received word, our decoding algorithm processes its entries one by one, constructing four polynomials at each step. This then yields a parametrization of interpolating solutions for the data so far. From the final result a list of all codewords that are closest to the received word with respect to the rank metric is obtained.Comment: Submitted to IEEE Information Theory Workshop 2014 in Hobart, Australi

Fast Multi-Sequence Shift-Register Synthesis with the Euclidean Algorithm

International audienceFeng and Tzeng's generalization of the Extended Euclidean Algorithm synthesizes the shortest--length linear feedback shift--register for \$s \geq 1\$ sequences, where each sequence has the the same length \$n\$. In this contribution, it is shown that Feng and Tzeng's algorithm which solves this multi--sequence shift--register problem has time complexity \$\ONsn^2\$. An acceleration based on the Divide and Conquer strategy is proposed and it is proven that subquadratic time complexity is achieved

List-Decoding Gabidulin Codes via Interpolation and the Euclidean Algorithm

We show how Gabidulin codes can be list decoded by using a parametrization approach. For this we consider a certain module in the ring of linearized polynomials and find a minimal basis for this module using the Euclidean algorithm with respect to composition of polynomials. For a given received word, our decoding algorithm computes a list of all codewords that are closest to the received word with respect to the rank metric.Comment: Submitted to ISITA 2014, IEICE copyright upon acceptanc