In recent years, how to learn a dictionary from input images for sparse modelling has been one very active topic in image processing and recognition. Most existing dictionary learning methods consider an over-complete dictionary, e.g. the K-SVD method. Often they require solving some minimization problem that is very challenging in terms of computational feasibility and efficiency. However, if the correlations among dictionary atoms are not well constrained, the redundancy of the dictionary does not necessarily improve the performance of sparse coding. This paper proposed a fast orthogonal dictionary learning method for sparse image representation. With comparable performance on several image restoration tasks, the proposed method is much more computationally efficient than the over-complete dictionary based learning methods.

Multivariate public key cryptosystems are being focused on as candidates for post-quantum cryptography. Rainbow is one of the most efficient signature schemes in multivariate public key cryptosystems. The main drawback of Rainbow is that their key size is much larger than that of RSA and ECC. In this paper, we propose an efficient variant of Rainbow that has a shorter secret key (and thus generates signatures faster) than the corresponding original Rainbow. In our scheme, we divide each layer of Rainbow into smaller blocks by using diagonal matrix representations. The size of the smaller blocks can be flexibly selected, and this enables us to carefully choose secure parameters so that our proposed scheme is secure against known attacks such as rank attacks, direct attacks, and UOV attack. We estimate that the secret key size of our proposed scheme with 100-bit security is smaller by about 40% than that of the original Rainbow. In addition, an implementation of our scheme in the C language is seen to generate signature faster by 40%.

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To a digraph with a choice of certain integral basis, we construct a CW complex, whose integral singular cohomology is canonically isomorphic to the path cohomology of the digraph as introduced in\cite {GLMY}. The homotopy type of the CW complex turns out to be independent of the choice of basis. After a very brief discussion of functoriality, this construction immediately implies some of the expected but perhaps combinatorially subtle properties of the digraph cohomology and homotopy proved very recently\cite {GLMY2}. Furthermore, one gets a very simple expected formula for the cup product of forms on the digraph. On the other hand, we present an approach of using sheaf theory to reformulate (di) graph cohomologies. The investigation of the path cohomology from this framework, leads to a subtle version of Poincare lemma for digraphs, which follows from the construction of the CW complex.