471 research outputs found
Primal-dual distance bounds of linear codes with application to cryptography
Let denote the minimum length of a linear code with
and , where is the minimum Hamming distance of and
is the minimum Hamming distance of . In this paper, we
show a lower bound and an upper bound on . Further, for small
values of and , we determine 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
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
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
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
One of the main problems of subspace coding asks for the maximum possible
cardinality of a subspace code with minimum distance at least over
, where the dimensions of the codewords, which are vector
spaces, are contained in . In the special case of
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
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
In this article, the effective lengths of all -divisible linear codes
over with a non-negative integer are determined. For that
purpose, the -adic expansion of an integer is introduced. It is
shown that there exists a -divisible -linear code of
effective length if and only if the leading coefficient of the
-adic expansion of is non-negative. Furthermore, the maximum weight
of a -divisible code of effective length is at most ,
where denotes the cross-sum of the -adic expansion of .
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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