Error Correction for Index Coding With Coded Side Information

Index coding is a source coding problem in which a broadcaster seeks to meet the different demands of several users, each of whom is assumed to have some prior information on the data held by the sender. If the sender knows its clients' requests and their side-information sets, then the number of packet transmissions required to satisfy all users' demands can be greatly reduced if the data is encoded before sending. The collection of side-information indices as well as the indices of the requested data is described as an instance of the index coding with side-information (ICSI) problem. The encoding function is called the index code of the instance, and the number of transmissions employed by the code is referred to as its length. The main ICSI problem is to determine the optimal length of an index code for and instance. As this number is hard to compute, bounds approximating it are sought, as are algorithms to compute efficient index codes. Two interesting generalizations of the problem that have appeared in the literature are the subject of this work. The first of these is the case of index coding with coded side information, in which linear combinations of the source data are both requested by and held as users' side-information. The second is the introduction of error-correction in the problem, in which the broadcast channel is subject to noise. In this paper we characterize the optimal length of a scalar or vector linear index code with coded side information (ICCSI) over a finite field in terms of a generalized min-rank and give bounds on this number based on constructions of random codes for an arbitrary instance. We furthermore consider the length of an optimal error correcting code for an instance of the ICCSI problem and obtain bounds on this number, both for the Hamming metric and for rank-metric errors. We describe decoding algorithms for both categories of errors

Error-Correction Performance of Regular Ring-Linear LDPC Codes over Lee Channels

Most low-density parity-check (LDPC) code constructions are considered over finite fields. In this work, we focus on regular LDPC codes over integer residue rings and analyze their performance with respect to the Lee metric. Their error-correction performance is studied over two channel models, in the Lee metric. The first channel model is a discrete memoryless channel, whereas in the second channel model an error vector is drawn uniformly at random from all vectors of a fixed Lee weight. It is known that the two channel laws coincide in the asymptotic regime, meaning that their marginal distributions match. For both channel models, we derive upper bounds on the block error probability in terms of a random coding union bound as well as sphere packing bounds that make use of the marginal distribution of the considered channels. We estimate the decoding error probability of regular LDPC code ensembles over the channels using the marginal distribution and determining the expected Lee weight distribution of a random LDPC code over a finite integer ring. By means of density evolution and finite-length simulations, we estimate the error-correction performance of selected LDPC code ensembles under belief propagation decoding and a low-complexity symbol message passing decoding algorithm and compare the performances

Fundamental Properties of Sum-Rank Metric Codes

This paper investigates the theory of sum-rank metric codes for which the individual matrix blocks may have different sizes. Various bounds on the cardinality of a code are derived, along with their asymptotic extensions. The duality theory of sum-rank metric codes is also explored, showing that MSRD codes (the sum-rank analogue of MDS codes) dualize to MSRD codes only if all matrix blocks have the same number of columns. In the latter case, duality considerations lead to an upper bound on the number of blocks for MSRD codes. The paper also contains various constructions of sum-rank metric codes for variable block sizes, illustrating the possible behaviours of these objects with respect to bounds, existence, and duality properties

Analysis and Decoding of Linear Lee-Metric Codes with Application to Code-Based Cryptography

Lee-metric codes are defined over integer residue rings endowed with the Lee metric. Even though the metric is one of the oldest metric considered in coding-theroy and has interesting applications in, for instance, DNA storage and code-based cryptography, it received relatively few attentions compared to other distances like the Hamming metric or the rank metric. Hence, codes in the Lee metric are still less studied than codes in other metrics. Recently, the interest in the Lee metric increased due to its similarities with the Euclidean norm used in lattice-based cryptosystem. Additionally, it is a promising metric to reduce the key sizes or signature sizes in code-based cryptosystem. However, basic coding-theoretic concepts, such as a tight Singleton-like bound or the construction of optimal codes, are still open problems. Thus, in this thesis we focus on some open problems in the Lee metric and Lee-metric codes. Firstly, we introduce generalized weights for the Lee metric in different settings by adapting the existing theory for the Hamming metric over finite rings. We discuss their utility and derive new Singleton-like bounds in the Lee metric. Eventually, we abandon the classical idea of generalized weights and introduce generalized distances based on the algebraic structure of integer residue rings. This allows us to provide a novel and improved Singleton-like bound in the Lee metric over integer residue rings. For all the bounds we discuss the density of their optimal codes. Originally, the Lee metric has been introduced over a $q$-ary alphabet to cope with phase shift modulation. We consider two channel models in the Lee metric. The first is a memoryless channel matching to the Lee metric under the decoding rule ``decode to the nearest codeword''. The second model is a block-wise channel introducing an error of fixed Lee weight, motivated by code-based cryptography where errors of fixed weight are added intentionally. We show that both channels coincide in the limit of large block length, meaning that their marginal distributions match. This distribution enables to provide bounds on the asymptotic growth rate of the surface and volume spectrum of spheres and balls in the Lee metric, and to derive bounds on the block error probability of the two channel models in terms of random coding union bounds. As vectors of fixed Lee weight are also of interest to cryptographic applications, we discuss the problem of scalar multiplication in the Lee metric in the asymptotic regime and in a finite-length setting. The Lee weight of a vector may be increased or decreased by the product with a nontrivial scalar. From a cryptographic view point this problem is interesting, since an attacker may be able to reduce the weight of the error and hence reduce the complexity of the underlying problem. The construction of a vector with constant Lee weight using integer partitions is analyzed and an efficient method for drawing vectors of constant Lee weight uniformly at random from the set of all such vectors is given. We then focus on regular LDPC code families defined over integer residue rings and analyze their performance with respect to the Lee metric. We determine the expected Lee weight enumerator for a random code in fixed regular LDPC code ensemble and analyze its asymptotic growth rate. This allows us to estimate the expected decoding error probability. Eventually, we estimate the error-correction performance of selected LDPC code families under belief propagation decoding and symbol message passing decoding and compare the performances. The thesis is concluded with an application of the results derived to code-based cryptography. Namely, we apply the marginal distribution to improve the yet known fastest Lee-information set decoding algorithm

On Boolean functions, symmetric cryptography and algebraic coding theory

In the first part of this thesis we report results about some “linear” trapdoors that can be embedded in a block cipher. In particular we are interested in any block cipher which has invertible S-boxes and that acts as a permutation on the message space, once the key is chosen. The message space is a vector space and we can endow it with alternative operations (hidden sums) for which the structure of vector space is preserved. Each of this operation is related to a different copy of the affine group. So, our block cipher could be affine with respect to one of these hidden sums. We show conditions on the S-box able to prevent a type of trapdoors based on hidden sums, in particular we introduce the notion of Anti-Crooked function. Moreover we shows some properties of the translation groups related to these hidden sums, characterizing those that are generated by affine permutations. In that case we prove that hidden sum trapdoors are practical and we can perform a global reconstruction attack. We also analyze the role of the mixing layer obtaining results suggesting the possibility to have undetectable hidden sum trapdoors using MDS mixing layers. In the second part we take into account the index coding with side information (ICSI) problem. Firstly we investigate the optimal length of a linear index code, that is equal to the min-rank of the hypergraph related to the instance of the ICSI problem. In particular we extend the the so-called Sandwich Property from graphs to hypergraphs and also we give an upper bound on the min-rank of an hypergraph taking advantage of incidence structures such as 2-designs and projective planes. Then we consider the more general case when the side information are coded, the index coding with coded side information (ICCSI) problem. We extend some results on the error correction index codes to the ICCSI problem case and a syndrome decoding algorithm is also given