## Abstract

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.

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## Abstract

Let G1 and G2 be simple graphs and let n_{1} = |V (G_{1})|, m_{1} = |E(G_{1})|, n_{2} = |V (G_{2})| and m_{2} = |E(G2)|. In this paper we derive sharp upper and lower bounds for the number of spanning trees τ in the Cartesian product G_{1} □G_{2} of G_{1} and G_{2}. We show that:

_{n1}□K

_{n2}which turns out to be

## Abstract

We say that a graph *F strongly arrows* a pair of graphs (*G,H*) and write *F*
*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*
*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 *G* is not connected providing also a sharp lower bound which is linear in both parameters.

## Abstract

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.

## Abstract

In this paper we study the Hausdorff approximation of the shifted Heaviside step function h_{t0}(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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