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In this paper, we introduce multiscale persistent functions for biomolecular structure characterization. The essential idea is to combine our multiscale rigidity functions with persistent homology analysis, so as to construct a series of multiscale persistent functions, particularly multiscale persistent entropies, for structure characterization. To clarify the fundamental idea of our method, the multiscale persistent entropy model is discussed in great detail. Mathematically, unlike the previous persistent entropy or topological entropy, a special resolution parameter is incorporated into our model. Various scales can be achieved by tuning its value. Physically, our multiscale persistent entropy can be used in conformation entropy evaluation. More specifically, it is found that our method incorporates in it a natural classification scheme. This is achieved through a density filtration of a multiscale rigidity function built from bond and/or dihedral angle distributions. To further validate our model, a systematical comparison with the traditional entropy evaluation model is done. It is found that our model is able to preserve the intrinsic topological features of biomolecular data much better than traditional approaches, particularly for resolutions in the mediate range. Moreover, our method can be successfully used in protein classification. For a test database with around nine hundred proteins, a clear separation between all-alpha and all-beta proteins can be achieved, using only the dihedral and pseudo-bond angle information. Finally, a special protein structure index (PSI) is proposed, for the first time, to describe the "regularity" of protein structures. Essentially, PSI can be used to describe the "regularity" information in any systems.

In this paper, we present a practical and provably secure two-pass authenticated key exchange protocol over ideal lattices, which is conceptually simple and has similarities to the Diffie-Hellman based protocols such as HMQV (CRYPTO 2005) and OAKE (CCS 2013). Our method does not involve other cryptographic primitives—in particular, it does not use signatures—which simplifies the protocol and enables us to base the security directly on the hardness of the ring learning with errors problem. The security is proven in the Bellare-Rogaway model with weak perfect forward secrecy in the random oracle model. We also give a one-pass variant of our two-pass protocol, which might be appealing in specific applications. Several concrete choices of parameters are provided, and a proof-of-concept implementation shows that our protocols are indeed practical.

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.

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