Sets in &lt;mml:math altimg="si1.gif" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsevier.com/xml/ja/dtd" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmln

Haanpää, Harri; Huima, Antti; Östergård, Patric R. J.

doi:10.1016/s0166-218x(03)00273-7

Cited by 10 publications

(1 citation statement)

References 7 publications

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“…A subset X of a finite Abelian group G is said to be a sum cover of G if {x + x : x, x ∈ X} = G, a strict sum cover of G if {x + x : x, x ∈ X ∧ x = x } = G, and a difference cover of G {x − x : x, x ∈ X} = G. Swanson [19] gives some constructions and computational results for maximum difference packings of cyclic groups. Haanpää, Huima, andÖstergård compute maximum sum and strict sum packings of cyclic groups [11]. Fitch and Jamison [8] give minimum sum and strict sum covers of small cyclic groups, and Wiedemann [20] determines minimum difference covers for cyclic groups of order at most 133.…”

Section: Introductionmentioning

confidence: 99%

On Finding Small 2-Generating Sets

Fagnot

Fertin

2009

Lecture Notes in Computer Science

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Abstract. Given a set of positive integers S, we consider the problem of finding a minimum cardinality set of positive integers X (called a minimum 2-generating set of S) such that every element of S is an element of X or is the the sum of two (non-necessarily distinct) elements of X. We give some elementary properties of 2-generating sets and prove that finding a minimum cardinality 2-generating set is hard to approximate within ratio 1 + ε for any ε > 0. We then prove our main result, which consists in a standard representation lemma for minimum cardinality 2-generating sets.

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Section: Introductionmentioning

confidence: 99%

On Finding Small 2-Generating Sets

Fagnot

Fertin

2009

Lecture Notes in Computer Science

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Hard Instances of the Constrained Discrete Logarithm Problem

Mironov

Mityagin

Nissim

2006

Lecture Notes in Computer Science

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The discrete logarithm problem (DLP) generalizes to the constrained DLP, where the secret exponent x belongs to a set known to the attacker. The complexity of generic algorithms for solving the constrained DLP depends on the choice of the set. Motivated by cryptographic applications, we study sets with succinct representation for which the constrained DLP is hard. We draw on earlier results due to Erdös et al. and Schnorr, develop geometric tools such as generalized Menelaus' theorem for proving lower bounds on the complexity of the constrained DLP, and construct sets with succinct representation with provable non-trivial lower bounds.

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On constructions and parameters of symmetric configurations $$v_{k}$$ v k

Davydov

Faina

Giulietti

et al. 2015

Des. Codes Cryptogr.

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Tables of the currently known parameters of symmetric configurations are given. Formulas for parameters of the known infinite families of symmetric configurations are presented as well. The results of the recent paper [18] are used. This work can be viewed as an appendix to [18], in the sense that the tables given here cover a much larger set of parameters.(ii) If v = b and, hence, r = k, the configuration is symmetric, and it is referred to as a configuration v k .(A symmetric configuration v k is cyclic if there exists a permutation of the set of its points mappings blocks to blocks, and acting regularly on both points and blocks. Equivalently, v k is cyclic if one of its incidence matrix is circulant.Steiner systems are configurations with d = 0 [36]. The deficiency of a symmetric configuration v k is d = v − (k 2 − k + 1). The deficiency of v k is zero if and only if v k is a finite projective plane of order k − 1.A configuration (v r , b k ) can be viewed as a k-uniform r -regular linear hypergraph with v vertices and b hyperedges [34,36]. Connections of configurations (v r , b k ) with numerical semigroups are noted in [14,60]. Some analogies between configurations (v r , b k ), regular graphs, and molecule models of chemical elements are remarked in [32]. As an example of a practical application of configurations (both symmetric and non-symmetric), we mention also the problem of user privacy for using database; see [22,59] and the references therein.Denote by M(v, k) an incidence matrix of a symmetric configuration v k . Any matrix M(v, k) is a v × v 01-matrix with k ones in every row and column; moreover, the 2 × 2 matrix consisting of all ones is not a submatrix of M(v, k). Two incidence matrices of the same configuration may differ by a permutation on the rows and the columns.A matrix M(v, k) can be considered as a biadjacency matrix of the Levi graph of the configuration v k which is a k-regular bipartite graph without multiple edges [36, Sec. 7.2]. Clearly, the graph has girth at least six, i.e. it does not contain 4-cycles. Such graphs are useful for the construction of bipartite-graph codes that can be treated as low-density parity-check (LDPC) codes. If M(v, k) is circulant, then the corresponding LDPC code is quasi-cyclic; it can be encoded with the help of shift-registers with relatively small complexity; see [5,6,19,21,28,40,41,45] and the references therein.Matrices M(v, k) consisting of square circulant submatrices have a number of useful properties, e.g. they are more suitable for LDPC codes implementation. We say that a 01matrix A is block double-circulant (BDC for short) if A consists of square circulant blocks whose weights give rise to a circulant matrix (see Definition 3.1). A configuration v k with a BDC incidence matrix M(v, k) is called a BDC symmetric configuration. Symmetric and non-symmetric configurations with incidence matrices consisting of square circulant blocks are considered, for example, in [3, 4, 19-21, 40, 41, 52]. In [3,4], BDC symmetric configuration are considered in c...

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Cited by 10 publications

References 7 publications

On Finding Small 2-Generating Sets

On Finding Small 2-Generating Sets

Hard Instances of the Constrained Discrete Logarithm Problem

On constructions and parameters of symmetric configurations $$v_{k}$$ v k

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