No abstract uploaded!

In this paper, we study how the algebraic attack on the HFE multivariate public key cryptosystem works if we build an HFE cryptosystem on a finite field whose characteristic is not two. Using some very basic algebraic geometry we argue that when the characteristic is not two the algebraic attack should not be polynomial in the range of the parameters which are used in practical applications. We further support our claims with extensive experiments using the Magma implementation of F_4, which is currently the best publicly available implementation of the Gröbner basis algorithm. We present a new variant of the HFE cryptosystems, where we project the public key of HFE to a space of one dimension lower. This protects the system from the Kipnis-Shamir attack and makes the decryption process avoid multiple candidates for the plaintext. We propose an example for a practical application on GF(11) and suggest a test challenge on GF(7).

The paper discusses quantum graphs with a vertex coupling which interpolates between the common one of the type and a coupling introduced recently by two of the authors which exhibits a preferred orientation. Describing the interpolation family in terms of circulant matrices, we analyze the spectral and scattering property of such vertices, and investigate the band spectrum of the corresponding square lattice graph.

We show that the hyperbolic structure on a closed, orientable, hyperbolic 3-manifold can be constructed from a solution to the hyperbolic gluing equations using any triangulation with essential edges. The key ingredients in the proof are Thurston's spinning construction and a volume rigidity result attributed by Dunfield to Thurston, Gromov and Goldman. As an application, we show that this gives a new algorithm to detect hyperbolic structures and small Seifert fibred structures on closed 3-manifolds.

No abstract uploaded!