On The Security of Individual Data

We will consider the following problem in this paper: Assume that there are n numerical data {x(1),x(2),.., x(n)} (like salaries of n individuals) stored in a database and some subsums of these numbers are made public or just available for persons not eligible to learn the original data. Our motivating question is: At most how many of these subsums may be disclosed such that none of the numbers x1; x2;...; xn can be uniquely determined from these sums. These types of problems arise in the cases when certain tasks concerning a database are done by subcontractors who are not eligible to learn the elements of the database, but naturally should be given some data to fulfill there task. In database theory such examples are called statistical databases as they are used for statistical purposes and no individual data are supposed to be obtained using a restricted list of SUM queries. This problem was originally introduced by [ 1], originally solved by Miller et al. [ 7] and revisited by Griggs [ 4, 5]. It was shown in [ 7] that no more than ((n)(n/2)) subsums of a given set of secure data may be disclosed without disclosing at least one of the data, which upper bound is sharp as well. To calculate a subsum, it might need some operations whose number is limited. This is why it is natural to assume that the disclosed subsums of the original elements of the database will contain only a limited number of elements, say at most k. The goal of the present paper is to determine the maximum number of subsums of size at most k which can be disclosed without making possible to calculate any of the individual data x(i). The maximum is exactly determined for the case when the number of data is much larger than the size restriction k

Diszkrét matematika és alkalmazásai = Discrete mathematics and its applications

Folytattuk kutatásainkat a gráfelmélet, egyéb kombinatorikai ágazatok, az algebra és a kombinatorika határa illetve az elméleti számítástudomány területén. Eredményeinket (véletlenül!) éppen 100 cikkben publikáltuk. Ezeket mintegy 170 előadásban terjesztettük a világban, ezek körülbelül harmada meghívott vagy plenáris előadás volt. Gráfelmélet. Különféle színezési paraméterekre vonatkozó eredmények, egy gráf maximális élszámának meghatározása, ha bizonyos konfigurációk kizártak, Hamilton-tipusú tételek használata kombinatorikus konstrukciókra. Más kombinatorika. Egy halmazrendszerben (hipergráfban) lévő halmazok (hiperélek) maximális számának meghatározása, ha bizonyos konfigurációk tiltottak Analóg eredmények 0,1 mátrixokra, ahol bizonyos részmátrix-konfigurációk kizártak (két változatban: sorrend számít, vagy nem). Néhány algebrai eredmény, amit kombinatorikus gondolkodással sikerült elérni. Bizonyos algebrai eredményeknek viszont fontos gráfelméleti interpretációi vannak. Elméleti számítástudomány. A legjobb ujjlenyomat-kódok meghatározása. Az egyéni adatok biztonságára vonatkozó eredmények, ahol részösszegek adhatók ki. Új adatbázis-modellek. Bizonyos adatbázisbeli kulcsrendszerek legjobb reprezentálása. Új modellek és eredmények a kereséselmélet területén. | We continued our research in the areas of graph theory, other combinatorial theories, combinatorial geometry, border areas of algebra and combinatorics and theoretical computer science. The results are published in (incidentally!) 100 papers. Their contents were disseminated around the world in about 170 lectures, one third of them were invited or plenary talks. Graph theory. Results on different chromatic parameters, determination of the maximum number of edges in a graph if certain configurations are excluded, usage of Hamiltonian theorems in combinatorial constructions. Other combinatorics. Determination of the maximum number of subsets (hyperedges) in a family of subsets (hypergraph) if certain configurations are forbidden, analogous extremal results for 0,1 matrices where certain submatrix-configurations are excluded (in two different settings: order does, or does not matter). Some algebraic problems were solved using combinatorial way of thinking. Results in algebra have interpretations in graph theory. Theoretical computer science. Determination of the best fingerprint codes. Results on the security of individual data, when subsums can be given out. New models in database theory. Determination of the best representations of certain given key systems in databases. New models and results in search theory