Ramp secret sharing with cheater identification in presence of rushing cheaters

Secret sharing allows one to share a piece of information among n participants in a way that only qualified subsets of participants can recover the secret whereas others cannot. Some of these participants involved may, however, want to forge their shares of the secret(s) in order to cheat other participants. Various cheater identifiable techniques have been devised in order to identify such cheaters in secret sharing schemes. On the other hand, Ramp secret sharing schemes are a practically efficient variant of usual secret sharing schemes with reduced share size and some loss in security. Ramp secret sharing schemes have many applications in secure information storage, information-theoretic private information retrieval and secret image sharing due to producing relatively smaller shares. However, to the best of our knowledge, there does not exist any cheater identifiable ramp secret sharing scheme. In this paper we define the security model for cheater identifiable ramp secret sharing schemes and provide two constructions for cheater identifiable ramp secret sharing schemes. In addition, the second construction is secure against rushing cheaters who are allowed to submit their shares during secret reconstruction after observing other participants’ responses in one round. Also, we do not make any computational assumptions for the cheaters, i.e., cheaters may be equipped with unlimited time and resources, yet, the cheating probability would be bounded above by a very small positive number.

A Coordination Mechanism for a Scheduling Game with Uniform-Batching Machines

In this paper, we consider a scheduling problem with [Formula: see text] uniform parallel-batching machines [Formula: see text] under game situation. There are [Formula: see text] jobs, each of which is associated with a load. Each machine [Formula: see text] has a speed [Formula: see text] and can handle up to [Formula: see text] jobs simultaneously as a batch. The load of a batch is the load of the longest job in the batch. All the jobs in a batch start and complete at the same time. Each job is owned by an agent and its individual cost is the completion time of the job. The social cost is the largest completion time over all jobs, i.e., the makespan. We design a coordination mechanism for the scheduling game problem. We discuss the existence of Nash Equilibrium and offer an upper bound on the price of anarchy (POA) of the coordination mechanism. We present a greedy algorithm and show that: (i) under the coordination mechanism, any instance of the scheduling game problem has a unique Nash Equilibrium and it is precisely the schedule returned by the greedy algorithm; (ii) the mechanism has a POA no more than [Formula: see text], where [Formula: see text], [Formula: see text], and [Formula: see text] is a small positive number that tends to 0.

Existence and Concentration of Positive Solutions For Coupled Nonlinear Schrödinger Systems in ℝN

The aim of this paper is to study the existence and concentration of positive solutions for the coupled nonlinear Schrödinger systemwhere ε is a small positive number, N ≥ 3, p, q > 1 satisfyand W(x), Q(x), K(x) are continuous and bounded positive functions defined in ℝ

A PSEUDO "FOLK" THEOREM IN THE STRATEGIC PROVISION OF STOCK EXTERNALITIES

The paper discusses the relationship between the efficient provision and the Nash equilibrium of stock externalities in a dynamic setting. The following proposition has been proved: under certain conditions, the maximal gains of an agent in the economy by deviating from the Pareto optimal provision of stock externalities is less ∊, an arbitrary small positive number, when the time discount rate of the agents are sufficiently close to 0. Namely, under the same conditions, a Pareto efficient path is an ∊-Nash equilibrium where ∊ could be smaller than any predetermined level. The propositions are different from the folk theorems in repeated games because supporting of the ∊-Nash equilibrium does not require the threat of retaliations from other agents. The policy implications of the above results are also discussed here.

Introduction

This monograph is devoted to studying worst case complexity results and optimal or nearly optimal methods for the approximation of solutions of nonlinear equations, approximation of fixed points, and computation of the topological degree. The methods are “global” in nature. They guarantee that the computed solution is within a specified error from the exact solution for every function in a given class. A common approach in numerical analysis is to study the rate of convergence and/or locally convergent methods that require special assumptions on the location of initial points of iterations to be “sufficiently” close to the actual solutions. This approach is briefly reviewed in the annotations to chapter 2, as well as in section 2.1.6, dealing with the asymptotic analysis of the bisection method. Extensive literature exists describing the iterative approach, with several monographs published over the last 30 years. We do not attempt a complete review of this work. The reader interested in this classical approach should consult the monographs listed in the annotations to chapter 2. We motivate our analysis and introduce basic notions in a simple example of zero finding for continuous function with different signs at the endpoints of an interval. Example 3.1 We want to approximate a zero of a function f from the class F = {f : [0,1] → R : f(0) ,< 0 and f(1) > 0, continuous}.By an approximate solution of this problem we understand any point x = x (f) such that the distance between x and some zero ∝ = ∝(f) of the function f , f (∝ ) = 0, is at most equal to a given small positive number ∈,|x — ∝ ≤ ∈. To compute x we first gather some information on the function f by sampling f at n sequentially chosen points ti in the interval [0,1]. Then, based on this information we select x. To minimize the time complexity we must select the minimal number of sampling points, that guarantee computing x(f) for any function f in the class F. This minimal number of samples (in the worst case) is called the information complexity of the problem.

APPROXIMATION OF BLOWING UP SOLUTIONS TO SEMILINEAR PARABOLIC EQUATIONS USING "MASS CONTROLLED" PARABOLIC SYSTEMS

This paper considers asymptotic approximations to the solutions of the semilinear parabolic equation: [Formula: see text] where the function f(u) is such that the solution to (0.1) blows up in a finite time Tb. In order to control the explosive behavior of this problem, we consider a "perturbation" to (0.1) defined by: [Formula: see text] where ε is a small positive number. The boundary and initial conditions on uε are those of u. For vε, the initial and boundary conditions are chosen to be 1. Note that system (0.2) belongs to a class of coupled semilinear parabolic equations, with positive solutions and "mass control" property, (see Ref. 10). The solution {uε, vε} of such systems is known to be global. As such, (0.2) appears to be a regular perturbation to a singular problem (0.1). In this work, our basic theorem is a convergence proof for uε and [Formula: see text] to u and ut, respectively, in the L∞ norm. These results constitute a framework for designing in subsequent work, numerical algorithms for the computation of blow-up times (see Ref. 6).

Lattice packing of nearly-euclidean balls in spaces of even dimension

We consider nearly-Euclidean balls of the shapewhere ε is a small positive number, and n is even. If ε is small enough, then the maximum lattice-packing density of this body is essentially greater than the Minkowski-Hlawka bound for large n.

Effective mean value estimates for complex multiplicative functions

Quantitative estimates for finite mean valuesof multiplicative functions are highly applicable tools in analytic and probabilistic number theory. Extending a result of Hall [4], Halberstam and Richert[3] proved a useful inequality valid for real, non-negative g satisfying for instance a Wirsing type condition, viz for all primes p, with constants λ1 ≥ 0, 0 ≤ λ2 < 2. Their upper bound is sharp to within a factor (l + o(l)), but even a weaker and easier to prove estimate, such as(where the implied constants depend on λ1 and λ2), may become a surprisingly strong device. For instance, setting g(p) = l ± ε, where ε is an arbitrarily small positive number, provides immediately a proof of the famous Hardy–Ramanujan theorem on the normal order of the number of prime factors of an integer. This example, and many others, are discussed in detail in our book [5] where we make extensive use of (2) for various problems connected with the structure of the set of divisors of a normal number.

On the distribution (mod 1) of polynomials of a prime variable

Throughout, ε is any small positive number, θ any real number, n, nj, k, N some positive integers and p, pj any primes. By ‖θ‖ we mean the distance from θ to the nearest integer. Write C(ε), C(ε, k) for positive constants which may depend on the quantities indicated inside the parentheses.

On random genetic drift in a subdivided population

Summary Generations are assumed to be non-overlapping. We consider a haploid population divided into K parts, each of which contain N adults in any generation. These are obtained by a random sampling of the offspring of the previous generation. We assume that the probability of an adult offspring of an individual in one subpopulation being in some other subpopulation is the same small positive number, no matter what two subpopulations are considered. If the population initially has individuals of two types, A and a, it is of interest to study approximations, if n is large, to (1) the rate at which A or a is lost between generations n-1 and n, (2) the probability that A and a are still present in generation n, (3) the joint distribution of frequencies of A in the subpopulations. A solution is given for the first problem. It is found that if the mean number of migrants per generation from one subpopulation to another is at least as large as 1, the population behaves almost as if it were not subdivided. But if this number is considerably less that 1, then the rate at which one or the other gene is lost is slower than in an undivided population. The other two problems are discussed for K = 2.