# American Institute of Mathematical Sciences

ISSN:
1930-5346

eISSN:
1930-5338

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## Advances in Mathematics of Communications

August 2010 , Volume 4 , Issue 3

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2010, 4(3): 307-321 doi: 10.3934/amc.2010.4.307 +[Abstract](1676) +[PDF](209.9KB)
Abstract:
We extend the notion of an invalid-curve attack from elliptic curves to genus 2 hyperelliptic curves. We also show that invalid singular (hyper)elliptic curves can be used in mounting invalid-curve attacks on (hyper)elliptic curve cryptosystems, and make quantitative estimates of the practicality of these attacks. We thereby show that proper key validation is necessary even in cryptosystems based on hyperelliptic curves. As a byproduct, we enumerate the isomorphism classes of genus g hyperelliptic curves over a finite field by a new counting argument that is simpler than the previous methods.
2010, 4(3): 323-344 doi: 10.3934/amc.2010.4.323 +[Abstract](1337) +[PDF](351.1KB)
Abstract:
The linear programming method is developed in the space of unitary matrices in order to obtain bounds for unitary codes relative to the so-called diversity sum and diversity product. Theoretical and numerical results improving previously known bounds are derived.
2010, 4(3): 345-361 doi: 10.3934/amc.2010.4.345 +[Abstract](1279) +[PDF](232.5KB)
Abstract:
Let $n$ be an even positive integer and $\mathbb F$ be the field GF$(2)$. A word in $\mathbb F$n is called balanced if its Hamming weight is $n$/$2$. A subset $\mathcal C\subseteq\mathbb F$n is called a balancing set if for every word $\mathbf y\in\mathbb F$n there is a word $\mathbf x\in \mathcal C$ such that $\mathbf y + \mathbf x$ is balanced. It is shown that most linear subspaces of $\mathbb F$n of dimension slightly larger than $\frac{3}{2} \log$2$n$ are balancing sets. A generalization of this result to linear subspaces that are "almost balancing" is also presented. On the other hand, it is shown that the problem of deciding whether a given set of vectors in $\mathbb F$n spans a balancing set, is NP-hard. An application of linear balancing sets is presented for designing efficient error-correcting coding schemes in which the codewords are balanced.
2010, 4(3): 363-367 doi: 10.3934/amc.2010.4.363 +[Abstract](1466) +[PDF](109.6KB)
Abstract:
A new construction of codes from old ones is considered, it is an extension of the matrix-product construction. Several linear codes that improve the parameters of the known ones are presented.
2010, 4(3): 369-379 doi: 10.3934/amc.2010.4.369 +[Abstract](1519) +[PDF](168.4KB)
Abstract:
Recently, multisequences have gained increasing interest for applications in cryptography and quasi-Monte Carlo methods. We study the (generalized) joint linear complexity of a class of nonlinear pseudorandom multisequences introduced by the first two authors as well as the linear complexity of its coordinate sequences. We prove lower bounds which are much stronger than in the case of single sequences since the multidimensional case brings in new and favourable effects.
2010, 4(3): 381-398 doi: 10.3934/amc.2010.4.381 +[Abstract](1539) +[PDF](235.8KB)
Abstract:
Motivated by the problem of establishing a session key among parties based on the possession of certain credentials only, we discuss a notion of attribute-based key establishment. A number of new issues arise in this setting that are not present in the usual settings of group key establishment where unique user identities are assumed to be publicly available.
After detailing the security model, we give a two-round solution in the random oracle model. As main technical tool we introduce a notion of attribute-based signcryption, which may be of independent interest. We show that the type of signcryption needed can be realized through the encrypt-then-sign paradigm. Further, we discuss additional guarantees of the proposed protocol, that can be interpreted in terms of deniability and privacy.
2010, 4(3): 399-404 doi: 10.3934/amc.2010.4.399 +[Abstract](1359) +[PDF](109.5KB)
Abstract:
In this work the least covering radii of all binary linear codes of dimension 6 are determined. Codes of dimension up to 6 and lengths up to 15 having the least covering radius are classified and constructions of codes with $R=t$2$[n,k]$ of every length and dimension up to 6 are given. Examples of using this classification for the construction of codes with the least covering radius and dimensions greater than 6 are presented.
2010, 4(3): 405-417 doi: 10.3934/amc.2010.4.405 +[Abstract](1352) +[PDF](199.7KB)
Abstract:
We look at low density parity check codes over a finite field $\mathbb K$ associated with finite geometries $T$2*$(\mathcal K)$, where $\mathcal K$ is any subset of PG$(2,q)$, with $q=p$h, $p$≠char$\mathbb K$. This includes the geometry $LU(3,q)$D, the generalized quadrangle $T$2*$(\mathcal K)$ with $\mathcal K$ a hyperoval, the affine space AG$(3,q)$ and several partial and semi-partial geometries. In some cases the dimension and/or the code words of minimum weight are known. We prove an expression for the dimension and the minimum weight of the code. We classify the code words of minimum weight. We show that the code is generated completely by its words of minimum weight. We end with some practical considerations on the choice of $\mathcal K$.
2010, 4(3): 419-431 doi: 10.3934/amc.2010.4.419 +[Abstract](1463) +[PDF](229.5KB)
Abstract:
Combinatorial batch codes were defined by Paterson, Stinson, and Wei as purely combinatorial versions of the batch codes introduced by Ishai, Kushilevitz, Ostrovsky, and Sahai. There are $n$ items and $m$ servers each of which stores a subset of the items. It is required that, for prescribed integers $k$ and $t$, any $k$ items can be retrieved by reading at most $t$ items from each server. Only the case $t=1$ is considered here. An optimal combinatorial batch code is one in which the total storage required is a minimum. We establish an important connection between combinatorial batch codes and transversal matroids, and exploit this connection to characterize optimal combinatorial batch codes if $n=m+1$ and $n=m+2$.
2010, 4(3): 433-439 doi: 10.3934/amc.2010.4.433 +[Abstract](1350) +[PDF](131.7KB)
Abstract:
It is known that there are extremal formally self-dual even codes which are not self-dual only for lengths 6, 10, 12, 14, 18, 20, 22, 28 and 30. We complete the classification of extremal formally self-dual even codes by examining the case for length 30.

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