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Besides their security, the efficiency of searchable encryption schemes is a major criteria when it comes to their adoption: in order to replace an unencrypted database by a more secure construction, it must scale to the systems which rely on it. Unfortunately, the relationship between the efficiency and the security of searchable encryption has not been widely studied, and the minimum cost of some crucial security properties is still unclear.

In this paper, we present new lower bounds on the trade-offs between the size of the client state, the efficiency and the security for searchable encryption schemes. These lower bounds target two kinds of schemes: schemes hiding the repetition of search queries, and forward-private dynamic schemes, for which updates are oblivious.

We also show that these lower bounds are tight, by either constructing schemes matching them, or by showing that even a small increase in the amount of leaked information allows for constructing schemes breaking the lower bounds.

R eferences [1] Cs. Bujtás, M.A. Henning and Zs. Tuza, Transversals and domination in uniform hypergraphs , European J. Combin. 33 (2012) 62–71. doi:10.1016/j.ejc.2011.08.002 [2] E. DeLaViña, R. Pepper and W. Waller, A note on dominating sets and average distance , Discrete Math. 309 (2009) 2615–2619. doi:10.1016/j.disc.2008.03.018 [3] E. DeLaViña, R. Pepper and W. Waller, Lower bounds for the domination number , Discuss. Math. Graph Theory 30 (2010) 475–487. doi:10.7151/dmgt.1508 [4] R. Gera, T.W. Haynes, S.T. Hedetniemi and M.A. Henning, An annotated


Let G1 and G2 be simple graphs and let n1 = |V (G1)|, m1 = |E(G1)|, n2 = |V (G2)| and m2 = |E(G2)|. In this paper we derive sharp upper and lower bounds for the number of spanning trees τ in the Cartesian product G1 □G2 of G1 and G2. We show that:

. We also characterize the graphs for which equality holds. As a by-product we derive a formula for the number of spanning trees in Kn1 □Kn2 which turns out to be .


We say that a graph F strongly arrows a pair of graphs (G,H) and write F ind(G,H) if any 2-coloring of its edges with red and blue leads to either a red G or a blue H appearing as induced subgraphs of F. The induced Ramsey number, IR(G,H) is defined as min{|V (F)| : F ind (G,H)}. We will consider two aspects of induced Ramsey numbers. Firstly we will show that the lower bound of the induced Ramsey number for a connected graph G with independence number α and a graph H with clique number ω is roughly ω2α2. This bound is sharp. Moreover we will also consider the case when G is not connected providing also a sharp lower bound which is linear in both parameters.


The goal of this overview article is to give a tangible presentation of the breakthrough works in discrepancy theory [3, 5] by M. B. Levin. These works provide proofs for the exact lower discrepancy bounds of Halton’s sequence and a certain class of (t, s)-sequences. Our survey aims at highlighting the major ideas of the proofs and we discuss further implications of the employed methods. Moreover, we derive extensions of Levin’s results.


In this paper we study the Hausdorff approximation of the shifted Heaviside step function ht0(t) by sigmoidal functions based on the Chen’s and Pham’s cumulative distribution functions and find an expression for the error of the best approximation. We give real examples with data provided by IBM entry software package and Apache HTTP Server using Chen’s software reliability model and Pham’s deterministic software reliability model. Some analyses are made.

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paraboloid and Diophantine approximation in R3 and R4. Math. Pannonica 10 (1999) 111-122. [7] W.G. Nowak: Diophantine approximation in Rs: On a method of Mordell and Armitage. In: Algebraic number theory and Diophantine analysis. Proceedings of the conference held in Graz, Austria, August 30 to September 5, 1998, W. de Gruyter, Berlin. (2000) 339-349. [8] W.G. Nowak: Lower bounds for simultaneous Diophantine approximation constants. Comm. Math. 22 (1) (2014) 71-76. [9] W.G. Nowak: Simultaneous Diophantine approximation: Searching for analogues of Hurwitz's theorem. In: T

. Ishaq, M. I. Qureshi, Upper and lower bounds for the Stanley depth of certain classes of monomial ideals and their residue class rings , Comm. Algebra, 41 (3), (2013) 1107-1116. [15] M. T. Keller, Y. Shen, N. Streib, S. J. Young, On the Stanley Depth of Squarefree Veronese Ideals , J. Algebraic Combin., 33 (2), (2011) 313-324. [16] M. T. Keller, S. J. Young, Combinatorial reductions for the Stanley depth of I and S/I , Electron. J. Combin. 24 (3), (2017) #P3.48. [17] M. C. Lin, D. Rautenbach, F. J. Soulignac, J. L. Szwarcfiter, Powers of cycles, powers of paths

.D. Fellner, On minimal graphs, Theoret. Comput. Sci. 17 (1982) 103-110. doi: 10.1016/0304-3975(82)90135-9 [9] R.L. Graham and N.J.A. Sloane, Lower bounds for constant weight codes, IEEE Trans. Inform. Theory 26 (1980) 37-43. doi: 10.1109/TIT.1980.1056141 [10] P. Hell and J. Nešetřil, On the complexity of H-coloring, J. Combin. Theory Ser. B 48 (1990) 92-110. doi: 10.1016/0095-8956(90)90132-J [11] M. Hosseini Dolama, E. Sopena and X. Zhu, Incidence coloring of k-degenerated graphs, Discrete Math. 283 (2004) 121-128. doi: 10.1016/j.disc.2004.01.015 [12] W. Klostermeyer and G