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by Roger J. Camrass. / Bibliography: p.98. / Prepared under Grant NSF-ENG75-14103. Originally presented as the author's thesis, (M.S.) in the M.I.T. Dept. of Electrical Engineering and Computer Science, 1976.

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D.Ing. (Electrical And Electronic Engineering) / We address the issue of synchronization, using sync-words (or markers), for encoded data. We focus on data that is encoded using permutation codes or Reed-Solomon codes. For each type of code (permutation code and Reed-Solomon code) we give a synchronization procedure or algorithm such that synchronization is improved compared to when the procedure is not employed. The gure of merit for judging the performance is probability of synchronization (acquisition). The word acquisition is used to indicate that a sync-word is acquired or found in the right place in a frame. A new synchronization procedure for permutation codes is presented. This procedure is about nding sync-words that can be used speci cally with permutation codes, such that acceptable synchronization performance is possible even under channels with frequency selective fading/jamming, such as the power line communication channel. Our new procedure is tested with permutation codes known as distance-preserving mappings (DPMs). DPMs were chosen because they have de ned encoding and decoding procedures. Another new procedure for avoiding symbols in Reed-Solomon codes is presented. We call the procedure symbol avoidance. The symbol avoidance procedure is then used to improve the synchronization performance of Reed-Solomon codes, where known binary sync-words are used for synchronization. We give performance comparison results, in terms of probability of synchronization, where we compare Reed-Solomon with and without symbol avoidance applied.

D.Ing. / In this study we consider the application of Reed-Solomon codes on two distinct channels. The first channel is the packet erasure channel, where packets are either received errorfree,or packets are lost. The second channel is the power line channel where additive Gaussian noise,, impulsive noise and narrowband noise are encountered In this thesis we focus on M-FSK modulation for the power line channel. For the packet erasure channel,we develop two new erasures-only decoders .The distinction between the two decoders is that the one is optimized for burst erasure correction, while the second decoder is optimized for random erasure correction.It is found that for single burst erasures, or bursterasures contained within n - k elements (where n is the code length and k is the number of information elements to be encoded), the best performance is obtained with the burst erasure decoder. When used in conjunction with a modified interleaver found in the literature, the developed random erasure decoder yields the best all-round performance. The main contribution to the power line channel is the development of a combined demodulation and decoding strategy to detect narrowband interference When standard Reed-Solomon codes operate in a channel where narrowband interference is present, undetected errors result. With the proposed decoding strategy it is found that wideband noise (a generalization of impulsive noise) affects the performance more negatively than narrowband noise for M-FSK modulation. .

Le sujet de cette thèse est la cryptographie et son interconnexions avec la théorie des codes. En particulier, on utilise des techniques issues de la théorie des codes pour construire et analyser des protocoles cryptographiques avec des propriétés nouvelles ou plus avancées. On se concentre d'abord sur le partage de secret ou secret sharing, un sujet important avec de nombreuses applications pour la Cryptographie actuelle. Dans la variante à laquelle on s'intéresse, un schéma de partage de secret reçoit en entrée un élément secret, et renvoie en sortie n parts de telle façon que chaque ensemble de parts de taille suffisamment petite ne donne aucune information sur le secret (confidentialité), tandis que chaque ensemble de taille suffisamment grande permet de reconstituer le secret (reconstruction). Un schéma de partage de secret peut donc être vu comme une solution à un problème de communication où un émetteur Alice est connectée avec un destinataire Bob par n canaux distincts, dont certains sont contrôlés par un adversaire Ève. Alice peut utiliser un schéma de partage de secret pour communiquer un message secret a Bob de telle façon qu'Ève n'apprenne aucune information sur le secret en lisant les données transmises sur les canaux qu'elle contrôle, tandis que Bob peut recevoir le message même si Ève bloque ces dits canaux. Notre contributions au partage de secret concernent ses liens avec la théorie des codes ; comme les deux domaines partagent un même but (récupérer des données à partir d'informations partielles), ce n'est pas surprenant qu'ils aient connu une interaction longue et fertile. Plus précisément, Massey commença une analyse fructueuse à propos de la construction et de l'étude d'un schéma de partage de secret à partir d'un code correcteur. L'inconvénient de cette analyse est que la confidentialité d'un schéma de partage de secret est estimé grâce au dual du code sous-jacent ; cela peut être problématique vu qu'il pourrait ne pas être possible d'obtenir des codes avec des propriétés souhaitables qui aient aussi un bon code dual. On contourne ce problème en établissant une connexion nouvelle entre les deux domaines, telle que la confidentialité d'un schéma de partage de secrets n'est plus contrôlée par le dual du code sous-jacent. Cela nous permet d'exploiter complètement le potentiel de certaines constructions récentes de codes pour obtenir des meilleurs schémas; on illustre ceci avec deux applications. Premièrement, en utilisant des codes avec codage et décodage en temps linéaire on obtient une famille de schémas de partage de secret où le partage (calcul des parts issues du secret) tout comme la reconstruction peuvent s'effectuer en temps linéaire ; pour des seuils de confidentialité et de reconstruction croissants, ceci restait jusqu'à présent un problème ouvert. Deuxièmement, on utilise des codes avec décodage en liste pour construire des schémas de partage de secret robustes, c'est-à-dire des schémas qui peuvent reconstituer le secret même si certaines parts sont incorrectes, sauf avec une petite probabilité d'erreur. etc... / The topic of this dissertation is Cryptography, and its connections with Coding Theory. Concretely, we make use of techniques from Coding Theory to construct and analyze cryptographic protocols with new and/or enhanced properties. We first focus on Secret Sharing, an important topic with many applications to modern Cryptography, which also forms the common ground for most of the concepts discussed in this thesis. In the flavor we are interested in, a secret-sharing scheme takes as input a secret value, and produces as output n shares in such a way that small enough sets of shares yield no information at all on the secret (privacy), while large enough sets of shares allow to recover the secret (reconstruction). A secret-sharing scheme can thus be seen as a solution to a secure communication problem where a sender Alice is connected to a receiver Bob via $n$ distinct channels, some of which are controlled by an adversary Eve. Alice can use a secret-sharing scheme to communicate a secret message to Bob in such a way that Eve learns no information on the message by eavesdropping on the channels she controls, while Bob can receive the message even if Eve blocks the channels under her control. Our contributions to Secret Sharing concern its connection with Coding Theory; since the two fields share the goal of recovering data from incomplete information, it is not surprising that Secret Sharing and Coding Theory have known a long and fruitful interplay. In particular, Massey initiated a very successful analysis on how to construct and study secret-sharing schemes from error-correcting codes. The downside of this analysis is that the privacy of secret-sharing schemes is estimated in terms of the dual of the underlying code; this can be problematic as it might not be possible to obtain codes with desirable properties that have good duals as well. We circumvent this problem by establishing a new connection between the two fields, where the privacy of secret-sharing schemes is no longer controlled by the dual of the underlying code. This allows us to fully harness the potential of recent code constructions to obtain improved schemes; we exemplify this by means of two applications. First, by making use of linear-time encodable and decodable codes we obtain a family of secret-sharing schemes where both the sharing (computation of the shares from the secret) and the reconstruction can be performed in linear time; for growing privacy and reconstruction thresholds, this was an hitherto open problem. Second, we make use of list-decodable codes to construct robust secret-sharing schemes, i.e., schemes that can recover the secret even if some of the shares are incorrect, except with a small error probability. The family we present optimizes the trade-off between the extra data that needs to be appended to the share to achieve robustness and the error probability in the reconstruction, reaching the best possible value. etc...

D.Ing. / This thesis is dedicated to an analysis of fundamental topics and issues related to deterministic insertion/deletion correcting and detecting codes. The most important contributions in this respect are the construction of a multiple insertion/deletion correcting code for run-length limited sequences and the construction and applications of multiple deletion (insertion) detecting codes. It is shown how run-length constraints and higher order moments can be combined to create a code which is simultaneously multiple insertion/deletion error correcting and runlength constrained. A systematic form of this code is presented, whereby any arbitrary run-length constrained sequence can be made into a multiple insertion/deletion correcting codeword by adding a prefix. This prefix is appended to the run-length constrained sequence in such a way that the resulting codeword is itself run-length constrained. Furthermore, it is shown that, despite the run-length constraints, the resulting code is guaranteed to have a better asymptotic rate than the Helberg code, the only other known non-trivial deterministic multiple insertion/deletion correcting code. We consider insertion/deletion detecting codes and present a multiple deletion (insertion) detecting code. It is shown that this code, which is systematic, is optimal in the sense that there exists no other systematic multiple deletion (insertion) detecting code with a better rate. Furthermore, we present a number of applications of such codes. In addition, further related topics of interest are considered. Firstly, jitter as a fundamental cause of insertion/deletion errors is investigated and as a result a counterpart to the signal-to-noise ratio in the amplitude domain is proposed for the time domain. Secondly, motivated by the correspondence of Levenshtein and Varshamov-Tenengol’ts codes, we investigate the insertion/deletion correcting capability of the single asymmetric error correcting Constantin-Rao codes within a wider framework of asymmetric error correcting and insertion/deletion correcting code structure correspondences. Finally, we propose a generalisation of Tenengol’ts’ construction for multiple non-binary insertion/deletion correction.

M.Ing.

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