Detecting squarefree numbers

We present an algorithm, based on the explicit formula for $L$-functions and conditional on GRH, for proving that a given integer is squarefree with little or no knowledge of its factorization. We analyze the algorithm both theoretically and practically, and use it to prove that several RSA challenge numbers are not squarefull.Comment: 31 pages, 3 figures, latest versio

Positive trigonometric polynomials for strong stability of difference equations

We follow a polynomial approach to analyse strong stability of linear difference equations with rationally independent delays. Upon application of the Hermite stability criterion on the discrete-time homogeneous characteristic polynomial, assessing strong stability amounts to deciding positive definiteness of a multivariate trigonometric polynomial matrix. This latter problem is addressed with a converging hierarchy of linear matrix inequalities (LMIs). Numerical experiments indicate that certificates of strong stability can be obtained at a reasonable computational cost for state dimension and number of delays not exceeding 4 or 5

Receivers for Diffusion-Based Molecular Communication: Exploiting Memory and Sampling Rate

In this paper, a diffusion-based molecular communication channel between two nano-machines is considered. The effect of the amount of memory on performance is characterized, and a simple memory-limited decoder is proposed and its performance is shown to be close to that of the best possible imaginable decoder (without any restriction on the computational complexity or its functional form), using Genie-aided upper bounds. This effect is specialized for the case of Molecular Concentration Shift Keying; it is shown that a four-bits memory achieved nearly the same performance as infinite memory. Then a general class of threshold decoders is considered and shown not to be optimal for Poisson channel with memory, unless SNR is higher than a value specified in the paper. Another contribution is to show that receiver sampling at a rate higher than the transmission rate, i.e., a multi-read system, can significantly improve the performance. The associated decision rule for this system is shown to be a weighted sum of the samples during each symbol interval. The performance of the system is analyzed using the saddle point approximation. The best performance gains are achieved for an oversampling factor of three.Comment: Submitted to JSA

Real-Time Heuristic Algorithms for the Static Weapon-Target Assignment Problem

The problem of targeting and engaging individual missiles (targets) with an arsenal of interceptors (weapons) is known as the weapon target assignment problem. As many solution techniques are based upon a transformation of the objective function, their final solutions rarely produce optimal solutions. We propose a nonlinear branch and bound algorithm to provide the first optimization approach to the untransformed problem found in the literature. Further, we propose a new heuristic based upon the branch and bound algorithm which dominates other heuristics explored in optimality gap. We also propose a heuristic based upon the optimal solution to the quiz problem which finds solutions within 6% of optimal for small problems and provides statistically similar results as one of the best heuristics found in the literature for larger problems while solving these problems in ten thousandths of the time

User-Base Station Association in HetSNets: Complexity and Efficient Algorithms

This work considers the problem of user association to small-cell base stations (SBSs) in a heterogeneous and small-cell network (HetSNet). Two optimization problems are investigated, which are maximizing the set of associated users to the SBSs (the unweighted problem) and maximizing the set of weighted associated users to the SBSs (the weighted problem), under signal-to-interference-plus-noise ratio (SINR) constraints. Both problems are formulated as linear integer programs. The weighted problem is known to be NP-hard and, in this paper, the unweighted problem is proved to be NP-hard as well. Therefore, this paper develops two heuristic polynomial-time algorithms to solve both problems. The computational complexity of the proposed algorithms is evaluated and is shown to be far more efficient than the complexity of the optimal brute-force (BF) algorithm. Moreover, the paper benchmarks the performance of the proposed algorithms against the BF algorithm, the branch-and-bound (B\&B) algorithm and standard algorithms, through numerical simulations. The results demonstrate the close-to-optimal performance of the proposed algorithms. They also show that the weighted problem can be solved to provide solutions that are fair between users or to balance the load among SBSs

Coarse Graining of Nonbonded Inter-particle Potentials Using Automatic Simplex Optimization to Fit Structural Properties

We implemented a coarse-graining procedure to construct mesoscopic models of complex molecules. The final aim is to obtain better results on properties depending on slow modes of the molecules. Therefore the number of particles considered in molecular dynamics simulations is reduced while conserving as many properties of the original substance as possible. We address the problem of finding nonbonded interaction parameters which reproduce structural properties from experiment or atomistic simulations. The approach consists of optimizing automatically nonbonded parameters using the simplex algorithm to fit structural properties like the radial distribution function as target functions. Moreover, any mix of structural and thermodynamic properties can be included in the target function. Different spherically symmetric inter-particle potentials are discussed. Besides demonstrating the method for Lennard--Jones liquids, it is applied to several more complex molecular liquids such as diphenyl carbonate, tetrahydrofurane, and monomers of poly(isoprene).Comment: 24 pages, 3 tables, 14 figures submitted to the Journal of Chemical Physics (JCP

Approximating the Largest Root and Applications to Interlacing Families

We study the problem of approximating the largest root of a real-rooted polynomial of degree $n$ using its top $k$ coefficients and give nearly matching upper and lower bounds. We present algorithms with running time polynomial in $k$ that use the top $k$ coefficients to approximate the maximum root within a factor of $n^{1/k}$ and $1+O(\tfrac{\log n}{k})^2$ when $k\leq \log n$ and $k>\log n$ respectively. We also prove corresponding information-theoretic lower bounds of $n^{\Omega(1/k)}$ and $1+\Omega\left(\frac{\log \frac{2n}{k}}{k}\right)^2$, and show strong lower bounds for noisy version of the problem in which one is given access to approximate coefficients. This problem has applications in the context of the method of interlacing families of polynomials, which was used for proving the existence of Ramanujan graphs of all degrees, the solution of the Kadison-Singer problem, and bounding the integrality gap of the asymmetric traveling salesman problem. All of these involve computing the maximum root of certain real-rooted polynomials for which the top few coefficients are accessible in subexponential time. Our results yield an algorithm with the running time of $2^{\tilde O(\sqrt[3]n)}$ for all of them

Foundations, Properties, and Security Applications of Puzzles: A Survey

Cryptographic algorithms have been used not only to create robust ciphertexts but also to generate cryptograms that, contrary to the classic goal of cryptography, are meant to be broken. These cryptograms, generally called puzzles, require the use of a certain amount of resources to be solved, hence introducing a cost that is often regarded as a time delay---though it could involve other metrics as well, such as bandwidth. These powerful features have made puzzles the core of many security protocols, acquiring increasing importance in the IT security landscape. The concept of a puzzle has subsequently been extended to other types of schemes that do not use cryptographic functions, such as CAPTCHAs, which are used to discriminate humans from machines. Overall, puzzles have experienced a renewed interest with the advent of Bitcoin, which uses a CPU-intensive puzzle as proof of work. In this paper, we provide a comprehensive study of the most important puzzle construction schemes available in the literature, categorizing them according to several attributes, such as resource type, verification type, and applications. We have redefined the term puzzle by collecting and integrating the scattered notions used in different works, to cover all the existing applications. Moreover, we provide an overview of the possible applications, identifying key requirements and different design approaches. Finally, we highlight the features and limitations of each approach, providing a useful guide for the future development of new puzzle schemes.Comment: This article has been accepted for publication in ACM Computing Survey

Testing for Stochastic Dominance with Diversification Possibilities

We derive empirical tests for stochastic dominance that allow for diversification betweenchoice alternatives. The tests can be computed using straightforward linearprogramming. Bootstrapping techniques and asymptotic distribution theory canapproximate the sampling properties of the test results and allow for statistical inference.Our results could provide a stimulus to the further proliferation of stochastic dominancefor the problem of portfolio selection and evaluation (as well as other choice problemsunder uncertainty that involve diversification possibilities). An empirical application forUS stock market data illustrates our approach.stochastic dominance;portfolio selection;linear programming;portfolio diversification;portfolio evaluation