We present a cryptanalysis of a signature scheme HIMQ-3 due to Kyung-Ah Shim et al [10], which is a submission to National Institute of Standards and Technology (NIST) standardization process of post-quantum cryptosystems in 2017. We will show that inherent to the signing process is a leakage of information of the private key. Using this information one can forge a signature.

Suppose A and B are normed division algebras, i.e. R,C,H or O, we introduce and study Grassmannians of linear subspaces in (A ⊗ B)n which are complex/Lagrangian/maximal isotropic with respect to natural two tensors on (A ⊗ B)n. We show that every irreducible compact symmetric space must be one of these Grassmannian spaces, possibly up to a finite cover. This gives a simple and uniform description of all compact symmetric spaces. This generalizes the Tits magic square description for simple Lie algebras to compact symmetric spaces.

In this short note, we give a new proof of a theorem of Arezzo-Tian on the existence of smooth geodesic rays tamed by a special degeneration with smooth central fiber.

We present a general existence proof for a wide class of non-linear elliptic equations which can be applied to problems with barrier conditions without specifying any assumptions guaranteeing the uniqueness or local uniqueness of particular solutions. As an application we prove the existence of closed hypersurfaces with curvature prescribed in the tangent bundle of an ambient Riemannian manifold $N$ without supposing any sign condition on the sectional curvatures $K_N$. A curvature flow wouldn't work in this situation, neither the method of successive approximation.

Inspired by the work of Z. Lu and G. Tian (Duke Math. J. 125:351–387,2004) in the compact setting, in this paper we address the problem of studying the Szegö kernel of the disk bundle over a noncompact Kähler manifold. In particular we compute the Szegö kernel of the disk bundle over a Cartan–Hartogs domain based on a bounded symmetric domain. The main ingredients in our analysis are the fact that every Cartan–Hartogs domain can be viewed as an “iterated” disk bundle over its base and the ideas given in (Arezzo, Loi and Zuddas in Math. Z. 275:1207–1216,2013) for the computation of the Szegö kernel of the disk bundle over an Hermitian symmetric space of compact type.