Some mathematical aspects of security in the automated business environment

In the automated business world, personal contact and paper-based transactions are being replaced by digital electronic communications. This transition raises many new security issues, and has seen the emphasis of cryptography broaden from secrecy to include message integrity, user verification, digital signature, and access control. In this paper, we consider some of the ways in which mathematics is playing a crucial role in securing the automated business environment. We begin with a brief introduction to cryptography and the principles upon which the design of cipher systems is based. We then consider stream ciphers and block ciphers in turn, discussing their respective uses, design requirements, and cryptanalysis. Here statistical methods are of prime importance, as well as algebraic methods for generating and combining bit strings and blocks. The section on public-key systems includes a discussion of several mathematical techniques that provide one-way functions. These are functions that are easy to perform but difficult to invert, often based on some difficult mathematical problem, and are at the heart of public-key systems. Two public-key systems (RSA and El Gamal) are described, and the status of algorithms for solving the difficult problems upon which they are based – factoring and discrete logarithms – is reviewed. Finally we discuss applications of digital-signature and user-verification techniques. These are becoming of increasing importance in today's automated business world.