2,014 research outputs found

    Applications of finite geometry in coding theory and cryptography

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    We present in this article the basic properties of projective geometry, coding theory, and cryptography, and show how finite geometry can contribute to coding theory and cryptography. In this way, we show links between three research areas, and in particular, show that finite geometry is not only interesting from a pure mathematical point of view, but also of interest for applications. We concentrate on introducing the basic concepts of these three research areas and give standard references for all these three research areas. We also mention particular results involving ideas from finite geometry, and particular results in cryptography involving ideas from coding theory

    Information-theoretic Physical Layer Security for Satellite Channels

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    Shannon introduced the classic model of a cryptosystem in 1949, where Eve has access to an identical copy of the cyphertext that Alice sends to Bob. Shannon defined perfect secrecy to be the case when the mutual information between the plaintext and the cyphertext is zero. Perfect secrecy is motivated by error-free transmission and requires that Bob and Alice share a secret key. Wyner in 1975 and later I.~Csisz\'ar and J.~K\"orner in 1978 modified the Shannon model assuming that the channels are noisy and proved that secrecy can be achieved without sharing a secret key. This model is called wiretap channel model and secrecy capacity is known when Eve's channel is noisier than Bob's channel. In this paper we review the concept of wiretap coding from the satellite channel viewpoint. We also review subsequently introduced stronger secrecy levels which can be numerically quantified and are keyless unconditionally secure under certain assumptions. We introduce the general construction of wiretap coding and analyse its applicability for a typical satellite channel. From our analysis we discuss the potential of keyless information theoretic physical layer security for satellite channels based on wiretap coding. We also identify system design implications for enabling simultaneous operation with additional information theoretic security protocols

    Nearly optimal robust secret sharing

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    Abstract: We prove that a known approach to improve Shamir's celebrated secret sharing scheme; i.e., adding an information-theoretic authentication tag to the secret, can make it robust for n parties against any collusion of size δn, for any constant δ ∈ (0; 1/2). This result holds in the so-called “nonrushing” model in which the n shares are submitted simultaneously for reconstruction. We thus finally obtain a simple, fully explicit, and robust secret sharing scheme in this model that is essentially optimal in all parameters including the share size which is k(1+o(1))+O(κ), where k is the secret length and κ is the security parameter. Like Shamir's scheme, in this modified scheme any set of more than δn honest parties can efficiently recover the secret. Using algebraic geometry codes instead of Reed-Solomon codes, the share length can be decreased to a constant (only depending on δ) while the number of shares n can grow independently. In this case, when n is large enough, the scheme satisfies the “threshold” requirement in an approximate sense; i.e., any set of δn(1 + ρ) honest parties, for arbitrarily small ρ > 0, can efficiently reconstruct the secret

    Computational complexity of reconstruction and isomorphism testing for designs and line graphs

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    Graphs with high symmetry or regularity are the main source for experimentally hard instances of the notoriously difficult graph isomorphism problem. In this paper, we study the computational complexity of isomorphism testing for line graphs of tt-(v,k,λ)(v,k,\lambda) designs. For this class of highly regular graphs, we obtain a worst-case running time of O(vlogv+O(1))O(v^{\log v + O(1)}) for bounded parameters t,k,λt,k,\lambda. In a first step, our approach makes use of the Babai--Luks algorithm to compute canonical forms of tt-designs. In a second step, we show that tt-designs can be reconstructed from their line graphs in polynomial-time. The first is algebraic in nature, the second purely combinatorial. For both, profound structural knowledge in design theory is required. Our results extend earlier complexity results about isomorphism testing of graphs generated from Steiner triple systems and block designs.Comment: 12 pages; to appear in: "Journal of Combinatorial Theory, Series A

    Problems on q-Analogs in Coding Theory

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    The interest in qq-analogs of codes and designs has been increased in the last few years as a consequence of their new application in error-correction for random network coding. There are many interesting theoretical, algebraic, and combinatorial coding problems concerning these q-analogs which remained unsolved. The first goal of this paper is to make a short summary of the large amount of research which was done in the area mainly in the last few years and to provide most of the relevant references. The second goal of this paper is to present one hundred open questions and problems for future research, whose solution will advance the knowledge in this area. The third goal of this paper is to present and start some directions in solving some of these problems.Comment: arXiv admin note: text overlap with arXiv:0805.3528 by other author
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