An identification system based on the explicit isomorphism problem

We propose a new identification system based on algorithmic problems related to computing isomorphisms between central simple algebras. We design a statistical zero knowledge protocol which relies on the hardness of computing isomorphisms between orders in division algebras which generalizes a protocol by Hartung and Schnorr, which relies on the hardness of integral equivalence of quadratic forms.

Generalized Analysis of Division Algebras of Dimension 2

The mathematical properties of division algebras of dimension 2 are investigated on the basis of the analysis of possible values of the parameters introduced into the laws of composition of basic elements. Generalized expressions for calculating the inverse and neutral elements of the indicated algebras are given. The relations of the parameters defining the normalized division algebras are determined. Possibilities of application of linear orthogonal transformations for the analysis of isomorphism of such algebras are considered. The concept of an exponential function is introduced to represent the elements of the considered non-commutative division algebra in exponential form.

Normed Division Algebras Application to the Monopole Physics

We review some normed division algebras (R, C, H, O) applications to the monopole physics and MICZ-Kepler problems. More specifically, we will briefly review some results in applying the normed division algebras to interpret the existence of Dirac, Yang, and SO(8) monopoles. These monopoles also appear during the examination of the duality between isotropic harmonic oscillators and the MICZ-Kepler problems. We also revisit some of our newest results in the nine-dimensional MICZ-Kepler problem using the generalized Hurwitz transformation.

Reduced anisotropic unitary Whitehead groups of henselian division algebras with special residue fields of their centers

The reduced anisotropic unitary Whitehead groups of henselian division algebras with unitary involutions are computed in the cases where the centers of residue algebras are of special types.