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.

Let S be a projective simply connected complex surface and  be a line bundle on S. We study the moduli space of stable compactly supported 2-dimensional sheaves on the total spaces of . The moduli space admits a ℂ∗-action induced by scaling the fibers of . We identify certain components of the fixed locus of the moduli space with the moduli space of torsion free sheaves and the nested Hilbert schemes on S. We define the localized Donaldson-Thomas invariants of  by virtual localization in the case that  twisted by the anti-canonical bundle of S admits a nonzero global section. When pg(S)>0, in combination with Mochizuki's formulas, we are able to express the localized DT invariants in terms of the invariants of the nested Hilbert schemes defined by the authors in [GSY17a], the Seiberg-Witten invariants of S, and the integrals over the products of Hilbert schemes of points on S. When  is the canonical bundle of S, the Vafa-Witten invariants defined recently by Tanaka-Thomas, can be extracted from these localized DT invariants. VW invariants are expected to have modular properties as predicted by S-duality.

Finding surface mappings with least distortion arises from many applications in various fields. Extremal Teichmüller maps are surface mappings with least conformality distortion. The existence and uniqueness of the extremal Teichmüller map between Riemann surfaces of finite type are theoretically guaranteed (see Fletcher and Markovic, Quasiconformal maps and Teichmüller theory, Oxford Graduate Texts in Math., vol. 11, Oxford University Press, Oxford, 2007). Recently, a simple iterative algorithm for computing the Teichmüller maps between connected Riemann surfaces with given boundary value was proposed by Lui, Lam, Yau, and Gu in Teichmüller extremal mapping and its applications to landmark matching registration, arXiv:1211.2569. Numerical results were reported in the paper to show the effectiveness of the algorithm. The method was successfully applied to landmarkmatching registration. The purpose of this paper is to prove the iterative algorithm proposed in loc. cit., indeed converges.

No abstract uploaded!

No abstract uploaded!