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The total variation (TV) model is attractive in that it is able to preserve sharp attributes in images. However, the restored images from TV-based methods do not usually stay in a given dynamic range, and hence projection is required to bring them back into the dynamic range for visual presentation or for storage in digital media. This will affect the accuracy of the restoration as the projected image will no longer be the minimizer of the given TV model. In this paper, we show that one can get much more accurate solutions by imposing box constraints on the TV models and solving the resulting constrained models. Our numerical results show that for some images where there are many pixels with values lying on the boundary of the dynamic range, the gain can be as great as 10.28 decibel in the peak signal-to-noise ratio. One traditional hindrance using the constrained model is that it is difficult to solve. However, in this paper, we propose using the alternating direction method of multipliers (ADMM) to solve the constrained models. This leads to a fast and convergent algorithm that is applicable for both Gaussian and impulse noise. Numerical results show that our ADMM algorithm is better than some state-of-the-art algorithms for unconstrained models in terms of both accuracy and robustness with respect to the regularization parameter.

Short integer solution (SIS) and learning with errors (LWE) are two hard lattice problems. These two problems are believed having huge potential in application of cryptography. In 2012, Ding et al. [5] introduced the first provably secure key exchange based on LWE problem. On the other hand, we believe that it is very difficult to do key exchange on SIS problem only. In 2014, Wang et al. [6] did an attempt, but it was not successful. Mao et al. [7] broke the protocol by an attack based on CBi-SIS problem in 2016. However, their attack is not efficient. In this paper, we present a extremely straightforward and simple attack to Wang’s key exchange and then we will construct a key exchange based on SIS and LWE problems.

In this paper, we develop a discontinuous Galerkin (DG) method to solve the ideal special relativistic hydrodynamics (RHD) and design a bound-preserving (BP) limiter for this scheme by extending the idea in (X. Zhang and C.-W. Shu, Journal of Computational Physics, 229 (2010), 8918-8934). For RHD, the density and pressure are positive and the velocity is bounded by the speed of light. One difficulty in numerically solving the RHD in its conservative form is that the failure of preserving these physical bounds will result in ill-posedness of the problem and blowup of the code, especially in extreme relativistic cases. The standard way in dealing with this difficulty is to add extra numerical dissipation, while in doing so there is no guarantee in maintaining the high order of accuracy. Our BP limiter has the following features. It can theoretically guarantee to preserve the physical bounds for the numerical solution and maintain its designed high order accuracy. The limiter is local to the cell and hence is very easy to implement. Moreover, it renders $L^1$-stability to the numerical scheme. Numerical experiments are performed to demonstrate the good performance of this bound-preserving DG scheme. Even though we only discuss the BP limiter for DG schemes, it can be applied to high order finite volume schemes, such as weighted essentially non-oscillatory (WENO) finite volume schemes as well.

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