471 research outputs found

    Primal-dual distance bounds of linear codes with application to cryptography

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    Let N(d,d⊥)N(d,d^\perp) denote the minimum length nn of a linear code CC with dd and d⊥d^{\bot}, where dd is the minimum Hamming distance of CC and d⊥d^{\bot} is the minimum Hamming distance of C⊥C^{\bot}. In this paper, we show a lower bound and an upper bound on N(d,d⊥)N(d,d^\perp). Further, for small values of dd and d⊥d^\perp, we determine N(d,d⊥)N(d,d^\perp) and give a generator matrix of the optimum linear code. This problem is directly related to the design method of cryptographic Boolean functions suggested by Kurosawa et al.Comment: 6 pages, using IEEEtran.cls. To appear in IEEE Trans. Inform. Theory, Sept. 2006. Two authors were added in the revised versio

    Results on Binary Linear Codes With Minimum Distance 8 and 10

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    All codes with minimum distance 8 and codimension up to 14 and all codes with minimum distance 10 and codimension up to 18 are classified. Nonexistence of codes with parameters [33,18,8] and [33,14,10] is proved. This leads to 8 new exact bounds for binary linear codes. Primarily two algorithms considering the dual codes are used, namely extension of dual codes with a proper coordinate, and a fast algorithm for finding a maximum clique in a graph, which is modified to find a maximum set of vectors with the right dependency structure.Comment: Submitted to the IEEE Transactions on Information Theory, May 2010 To be presented at the ACCT 201

    A simple combinatorial treatment of constructions and threshold gaps of ramp schemes

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    We give easy proofs of some recent results concerning threshold gaps in ramp schemes. We then generalise a construction method for ramp schemes employing error-correcting codes so that it can be applied using nonlinear (as well as linear) codes. Finally, as an immediate consequence of these results, we provide a new explicit bound on the minimum length of a code having a specified distance and dual distance

    Hiding Symbols and Functions: New Metrics and Constructions for Information-Theoretic Security

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    We present information-theoretic definitions and results for analyzing symmetric-key encryption schemes beyond the perfect secrecy regime, i.e. when perfect secrecy is not attained. We adopt two lines of analysis, one based on lossless source coding, and another akin to rate-distortion theory. We start by presenting a new information-theoretic metric for security, called symbol secrecy, and derive associated fundamental bounds. We then introduce list-source codes (LSCs), which are a general framework for mapping a key length (entropy) to a list size that an eavesdropper has to resolve in order to recover a secret message. We provide explicit constructions of LSCs, and demonstrate that, when the source is uniformly distributed, the highest level of symbol secrecy for a fixed key length can be achieved through a construction based on minimum-distance separable (MDS) codes. Using an analysis related to rate-distortion theory, we then show how symbol secrecy can be used to determine the probability that an eavesdropper correctly reconstructs functions of the original plaintext. We illustrate how these bounds can be applied to characterize security properties of symmetric-key encryption schemes, and, in particular, extend security claims based on symbol secrecy to a functional setting.Comment: Submitted to IEEE Transactions on Information Theor

    Tables of subspace codes

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    One of the main problems of subspace coding asks for the maximum possible cardinality of a subspace code with minimum distance at least dd over Fqn\mathbb{F}_q^n, where the dimensions of the codewords, which are vector spaces, are contained in K⊆{0,1,…,n}K\subseteq\{0,1,\dots,n\}. In the special case of K={k}K=\{k\} one speaks of constant dimension codes. Since this (still) emerging field is very prosperous on the one hand side and there are a lot of connections to classical objects from Galois geometry it is a bit difficult to keep or to obtain an overview about the current state of knowledge. To this end we have implemented an on-line database of the (at least to us) known results at \url{subspacecodes.uni-bayreuth.de}. The aim of this recurrently updated technical report is to provide a user guide how this technical tool can be used in research projects and to describe the so far implemented theoretic and algorithmic knowledge.Comment: 44 pages, 6 tables, 7 screenshot

    Tensor-based trapdoors for CVP and their application to public key cryptography

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    We propose two trapdoors for the Closest-Vector-Problem in lattices (CVP) related to the lattice tensor product. Using these trapdoors we set up a lattice-based cryptosystem which resembles to the McEliece scheme

    On the lengths of divisible codes

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    In this article, the effective lengths of all qrq^r-divisible linear codes over Fq\mathbb{F}_q with a non-negative integer rr are determined. For that purpose, the Sq(r)S_q(r)-adic expansion of an integer nn is introduced. It is shown that there exists a qrq^r-divisible Fq\mathbb{F}_q-linear code of effective length nn if and only if the leading coefficient of the Sq(r)S_q(r)-adic expansion of nn is non-negative. Furthermore, the maximum weight of a qrq^r-divisible code of effective length nn is at most σqr\sigma q^r, where σ\sigma denotes the cross-sum of the Sq(r)S_q(r)-adic expansion of nn. This result has applications in Galois geometries. A recent theorem of N{\u{a}}stase and Sissokho on the maximum size of a partial spread follows as a corollary. Furthermore, we get an improvement of the Johnson bound for constant dimension subspace codes.Comment: 17 pages, typos corrected; the paper was originally named "An improvement of the Johnson bound for subspace codes
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