Invited Talk:

Recent Development in the Theory and Applications of
the Double-Base Number System

Vassil Dimitrov
Lab. of Signal Processing and Computer Technology
Helsinki University of Technology
Finland

vdimitro@wooster.hut.fi

Abstract:
The double-base number system (DBNS) is a special way of representing integers as a sum of mixed powers of two and three. This number representation scheme is unusually sparse, which is a good measure for potential applications. There are many purely number theoretic problems that remain to be solved in order to clarify the basic properties of DBNS. They are mainly associated with the theory of linear forms of logarithms and theory of exponential Diophantine equations. From an application viewpoint, we have concentrated our attention on three major issues. First of all, it is possible to use this number system for very fast addition and multiplication of positive integers. This number representation has a very simple geometric interpretation. To get a better understanding of the arithmetic properties of DBNS, especially in performing addition, we have done simulations with cellular neural networks (CNNs). The main goal is to obtain an appropriate implementation framework, which would be helpful for a possible VLSI design of a fast DBNS adders and multipliers. The second issue analysed is the possible applications of DBNS in the field of digital filtering. To achieve that, we have modified the representation scheme in order to accommodate negative numbers and represent, with suitable precision, real numbers. In this case our goal is to implement the main computational operation in digital filtering - multiply-and-accumulate - in a very efficient manner both in terms of time and space. The third application, perhaps, the most promising, is in the field of cryptography. We have demonstrated how the use of DBNS could speed up the performance of variety of cryptosystems - Diffie-Hellman key exchange algorithm, Elliptic Curve Cryptosystems (ECC), etc. Finally we discuss many open problems and possible generalisations.