Unconditionally Secure Oblivious Polynomial Evaluation:A Survey and New Results

Oblivious polynomial evaluation(OPE)is a two-party protocol that allows a receiver,R to learn an evaluation f(α),of a sender,S's polynomial(f(x)),whilst keeping both a and f(x)private.This protocol has attracted a lot of attention recently,as it has wide ranging applications in the field of cryptography.In this article we review some of these applications and,additionally,take an in-depth look at the special case of information theoretic OPE.Specifically,we provide a current and critical review of the existing information theoretic OPE protocols in the literature.We divide these protocols into two distinct cases(three-party and distributed OPE)allowing for the easy distinction and classification of future information theoretic OPE protocols.In addition to this work,we also develop several modifications and extensions to existing schemes,resulting in increased security,flexibility and efficiency.Lastly,we also identify a security flaw in a previously published OPE scheme.

UNCONDITIONALLY OPTIMAL ERROR ESTIMATES OF THE BILINEAR-CONSTANT SCHEME FOR TIME-DEPENDENT NAVIER-STOKES EQUATIONS

In this paper,the unconditional error estimates are presented for the time-dependent Navier-Stokes equations by the bilinear-constant scheme.The corresponding optimal error estimates for the velocity and the pressure are derived unconditionally,while the previous works require certain time-step restrictions.The analysis is based on an iterated time-discrete system,with which the error function is split into a temporal error and a spatial error.The τ-independent(τ is the time stepsize)error estimate between the numerical solution and the solution of the time-discrete system is proven by a rigorous analysis,which implies that the numerical solution in L^(∞)-norm is bounded.Thus optimal error estimates can be obtained in a traditional way.Numerical results are provided to confirm the theoretical analysis.

A Kernel Based Unconditionally Stable Scheme for Nonlinear Parabolic Partial Differential Equations

In this paper,a class of high order numerical schemes is proposed to solve the nonlinear parabolic equations with variable coefficients.This method is based on our previous work[11]for convection-diffusion equations,which relies on a special kernel-based formulation of the solutions and successive convolution.However,disadvantages appear when we extend the previous method to our equations,such as inefficient choice of parameters and unprovable stability for high-dimensional problems.To overcome these difficulties,a new kernel-based formulation is designed to approach the spatial derivatives.It maintains the good properties of the original one,including the high order accuracy and unconditionally stable for one-dimensional problems,hence allowing much larger time step evolution compared with other explicit schemes.In additional,without extra computational cost,the proposed scheme can enlarge the available interval of the special parameter in the formulation,leading to less errors and higher efficiency.Moreover,theoretical investigations indicate that it is unconditionally stable for multi-dimensional problems as well.We present numerical tests for one-and two-dimensional scalar and system,demonstrating the designed high order accuracy and unconditionally stable property of the scheme.

New hybrid FDTD algorithm for electromagnetic problem analysis

Since the time step of the traditional finite-difference time-domain(FDTD) method is limited by the small grid size, it is inefficient when dealing with the electromagnetic problems of multi-scale structures.Therefore, the explicit and unconditionally stable FDTD(US-FDTD) approach has been developed to break through the limitation of Courant–Friedrich–Levy(CFL) condition.However, the eigenvalues and eigenvectors of the system matrix must be calculated before the time iteration in the explicit US-FDTD.Moreover, the eigenvalue decomposition is also time consuming, especially for complex electromagnetic problems in practical application.In addition, compared with the traditional FDTD method, the explicit US-FDTD method is more difficult to introduce the absorbing boundary and plane wave.To solve the drawbacks of the traditional FDTD and the explicit US-FDTD, a new hybrid FDTD algorithm is proposed in this paper.This combines the explicit US-FDTD with the traditional FDTD, which not only overcomes the limitation of CFL condition but also reduces the system matrix dimension, and introduces the plane wave and the perfectly matched layer(PML) absorption boundary conveniently.With the hybrid algorithm, the calculation of the eigenvalues is only required in the fine mesh region and adjacent coarse mesh region.Therefore, the calculation efficiency is greatly enhanced.Furthermore, the plane wave and the absorption boundary introduction of the traditional FDTD method can be directly utilized.Numerical results demonstrate the effectiveness, accuracy, stability, and convenience of this hybrid algorithm.

Reduced-order finite element method based on POD for fractional Tricomi-type equation

The reduced-order finite element method （FEM） based on a proper orthogo- nal decomposition （POD） theory is applied to the time fractional Tricomi-type equation. The present method is an improvement on the general FEM. It can significantly save mem- ory space and effectively relieve the computing load due to its reconstruction of POD basis functions. Furthermore, the reduced-order finite element （FE） scheme is shown to be un- conditionally stable, and error estimation is derived in detail. Two numerical examples are presented to show the feasibility and effectiveness of the method for time fractional differential equations （FDEs）.

Alternating segment explicit-implicit scheme for nonlinear third-order KdV equation

A group of asymmetric difference schemes to approach the Korteweg-de Vries （KdV） equation is given here. According to such schemes, the full explicit difference scheme and the full implicit one, an alternating segment explicit-implicit difference scheme for solving the KdV equation is constructed. The scheme is linear unconditionally stable by the analysis of linearization procedure, and is used directly on the parallel computer. The numerical experiments show that the method has high accuracy.

On precise time integration method for non-classically damped MDOF systems

In the complex mode superposition method, the equations of motion for non-classically damped multiple-degree-of-freedom （MDOF） discrete systems can be transferred into a combination of some generalized SDOF complex oscillators. Based on the state space theory, a precise recurrence relationship for these complex oscillators is set up; then a delicate general solution of non-classically damped MDOF systems, completely in real value form, is presented in this paper. In the proposed method, no calculation of the matrix exponential function is needed and the algorithm is unconditionally stable. A numerical example is given to demonstrate the validity and efficiency of the proposed method.

Space of Operators and Property (MB)

In this paper, a new class of Banach spaces, termed as Banach spaces with property (MB), will be introduced. It is stated that a space X has property (MB) if every V -subset of X* is an L-subset of X* . We describe those spaces which have property (MB) . Also, we show that if a Banach space X has property (MB) and Banach space Y does not contain , then every operator is completely continuous.

UNCONDITIONAL ERROR ANALYSIS OF VEMS FOR A GENERALIZED NONLINEAR SCHRODINGER EQUATION

In this work,we focus on the conforming and nonconforming leap-frog virtual element methods for the generalized nonlinear Schrodinger equation,and establish their unconditional stability and optimal error estimates.By constructing a time-discrete system,the error between the solutions of the continuous model and the numerical scheme is separated into the temporal error and the spatial error,which makes the spatial error τ-independent.The inverse inequalities in the existing conforming and new constructed nonconforming virtual element spaces are utilized to derive the L^(∞)-norm uniform boundedness of numerical solutions without any restrictions on time-space step ratio,and then unconditionally optimal error estimates of the numerical schemes are obtained naturally.What needs to be emphasized is that if we use the pre-existing nonconforming virtual elements,there is no way to derive the L^(∞)-norm uniform boundedness of the functions in the nonconforming virtual element spaces so as to be hard to get the corresponding inverse inequalities.Finally,several numerical examples are reported to confirm our theoretical results.

UNCONDITIONAL SUPERCONVERGENT ANALYSIS OF QUASI-WILSON ELEMENT FOR BENJAMIN-BONA-MAHONEY EQUATION

This article aims to study the unconditional superconvergent behavior of nonconforming quadrilateral quasi-Wilson element for nonlinear Benjamin Bona Mahoney(BBM)equation.For the generalized rectangular meshes including rectangular mesh,deformed rectangular mesh and piecewise deformed rectangular mesh,by use of the special character of this element,that is,the conforming part(bilinear element)has high accuracy estimates on the generalized rectangular meshes and the consistency error can reach order O(h^(2)),one order higher than its interpolation error,the superconvergent estimates with respect to mesh size h are obtained in the broken H^(1)-norm for the semi-/fully-discrete schemes.A striking ingredient is that the restrictions between mesh size h and time stepτrequired in the previous works are removed.Finally,some numerical results are provided to confirm the theoretical analysis.

A High Order Adaptive Time-Stepping Strategy and Local Discontinuous Galerkin Method for the Modified Phase Field Crystal Equation

In this paper,we will develop a first order and a second order convex splitting,and a first order linear energy stable fully discrete local discontinuous Galerkin(LDG)methods for the modified phase field crystal(MPFC)equation.In which,the first order linear scheme is based on the invariant energy quadratization approach.The MPFC equation is a damped wave equation,and to preserve an energy stability,it is necessary to introduce a pseudo energy,which all increase the difficulty of constructing numerical methods comparing with the phase field crystal(PFC)equation.Due to the severe time step restriction of explicit timemarchingmethods,we introduce the first order and second order semi-implicit schemes,which are proved to be unconditionally energy stable.In order to improve the temporal accuracy,the semi-implicit spectral deferred correction(SDC)method combining with the first order convex splitting scheme is employed.Numerical simulations of the MPFC equation always need long time to reach steady state,and then adaptive time-stepping method is necessary and of paramount importance.The schemes at the implicit time level are linear or nonlinear and we solve them by multigrid solver.Numerical experiments of the accuracy and long time simulations are presented demonstrating the capability and efficiency of the proposed methods,and the effectiveness of the adaptive time-stepping strategy.

An Adaptive Time-Stepping Strategy for the Cahn-Hilliard Equation

This paper studies the numerical simulations for the Cahn-Hilliard equation which describes a phase separation phenomenon.The numerical simulation of the Cahn-Hilliardmodel needs very long time to reach the steady state,and therefore large time-stepping methods become useful.The main objective of this work is to construct the unconditionally energy stable finite difference scheme so that the large time steps can be used in the numerical simulations.The equation is discretized by the central difference scheme in space and fully implicit second-order scheme in time.The proposed scheme is proved to be unconditionally energy stable and mass-conservative.An error estimate for the numerical solution is also obtained with second order in both space and time.By using this energy stable scheme,an adaptive time-stepping strategy is proposed,which selects time steps adaptively based on the variation of the free energy against time.The numerical experiments are presented to demonstrate the effectiveness of the adaptive time-stepping approach.

UNCONDITIONAL SUPERCONVERGENCE ANALYSIS OF AN Ri-GALERKIN MIXED FINITE ELEMENT METHOD FOR TWO-DIMENSIONAL GINZBURG-LANDAU EQUATION

An H^1-Galerkin mixed finite element method (MFEM) is discussed for the two-dimensional Ginzburg-Landau equation with the bilinear element and zero order Raviart-Thomas elemen t (Q11+Q10×Qo01). A linearized Crank-Nicolson fully-discrete scheme is developed and a time-discrete system is introduced to split the error into two parts which are called the temporal error and the spatial error, respectively. On one hand, the regularity of the time-discrete system is deduced through the temporal error estimation. On the other hand, the superconvergent estimates of u in H^1-norm and →q in H(div;Ω)-norm with order 0(h^2+τ^2) are obtained unconditionally based on the achievement of the spatial result. At last, a numerical experiment is included to illustrate the feasibility of the proposed method. Here, h is the subdivision parame ter and τ is the time step.

Summing Boolean Algebras

In this paper we will study some families and subalgebras■of■(N)that let us character- ize the unconditional convergence of series through the weak convergence of subseries ∑_(i∈A)x_i,A∈(?). As a consequence,we obtain a new version of the Orlicz Pettis theorem,for Banach spaces.We also study some relationships between algebraic properties of Boolean algebras and topological properties of the corresponding Stone spaces.

ENERGY STABLE NUMERICAL METHOD FOR THE TDGL EQUATION WITH THE RETICULAR FREE ENERGY IN HYDROGEL

Here we focus on the numerical simulation of the phase separation about macromolecule microsphere composite （MMC） hydrogel. The model is based on time-dependent Ginzburg- Landau （TDGL） equation with the reticular free energy. An unconditionally energy stable difference scheme is proposed based on the convex splitting of the corresponding energy functional. In the numerical experiments, we observe that simulating the whole process of the phase separation requires a considerably long time. We also notice that the total free energy changes significantly in initial stage and varies slightly in the following time. Based on these properties, we apply the adaptive time stepping strategy to improve the computational efficiency. It is found that the application of time step adaptivity can not only resolve the dynamical changes of the solution accurately but also significantly save CPU time for the long time simulation.