Shizuo Kaji

Anti-commutative Dual Complex Numbers and 2D Rigid Transformation

We introduce a new presentation of the two dimensional rigid transformation which is more concise and efficient than the standard matrix presentation. By modifying the ordinary dual number construction for the complex numbers, we define the ring of anti-commutative dual complex numbers, which parametrizes two dimensional rotation and translation all together. With this presentation, one can easily interpolate or blend two or more rigid transformations at a low computational cost. We developed a library for C++ with the MIT-licensed source code [13].

Mathematical analysis on affine maps for 2D shape interpolation

This paper gives a simple mathematical framework for 2D shape interpolation methods that preserve rigidity. An interpolation technique in this framework works for given the source and target 2D shapes, which are compatibly triangulated. Focusing on the local affine maps between the corresponding triangles, we describe a global transformation as a piecewise affine map. Several existing rigid shape interpolation techniques are discussed and mathematically analyzed through this framework. This gives us not only a useful comprehensive understanding of existing approaches, but also new algorithms and a few improvements of previous approaches.

Equivariant Schubert Calculus of Coxeter Groups

We consider an equivariant extension for Hiller’s Schubert calculus on the coinvariant ring of a finite Coxeter group.

A mathematical problem for security analysis of hash functions and pseudorandom generators

The aim of this paper is to emphasize the significance of a certain mathematical problem in research on information security. We point out that the mathematical problem, which we refer to as Function Density Problem, has connections to the following two major cryptographic topics; security analysis of hash functions in the real world (like SHA-1), and construction of pseudorandom generators with some enhanced security property. We also provide a first example to show how a study of Function Density Problem can contribute to the progress of the above-mentioned two topics. Other potential applications of Function Density Problem to information security are also discussed.