Jonathan M. Blackledge

Deterministic Chaos in Digital Cryptography

This paper studies the relationship between deterministic chaos and cryptographic systems. The theoretical background upon which this relationship is based, includes discussions on chaos, ergodicity, complexity, randomness, unpredictability, incompressibility. Exactly solvable and one-step unpredictable chaotic systems have a fundamental application in cryptography. Two approaches to the finite-state implementation of chaotic systems are considered: (i) floating-point approximation of continuous-state chaos; (ii) binary pseudo-chaos. Pseudo-Chaotic Number Generators (PCNG) are studied using a finite-state chaotic system to produce a sequence of bits. The essence of a PCNG is a nonlinear iterated function. PCNG are an important component of a cryptographic system and play the same role as Pseudo-Random Number Generators (PRNG) in conventional cryptographic systems. An overview is given of existing chaos-based encryption algorithms along with their strengths and weaknesses.

Finite Differences and Parabolic Equations

The first part of the chapter is concerned with finding finite difference approximations to partial derivatives with a view to replacing the full partial differential equation by a difference representation which then allows a numerical approach to be pursued. The result is that the partial derivatives are replaced by relationships between the function values at nodal points of some grid system. Hence the partial differential equations are approximated by a set of algebraic equations, for the function values at the nod al points. In order to perform this discretisation, expressions for such terms as at typical grid points are required in terms of the ϕ values at neighbouring grid points.

Solutions to Exercises

In this appendix all the exercises will be solved. In cases where the solutions are close replicas of the worked examples in the text then only the new numerical values at the major steps, or the specific difference from the earlier work will be included. Standard methods will not be exhibited in detail, hence for example a solution which requires Gaussian elimination will simply quote the result, as the process itself will be considered a standard tool. The same will apply to iterative methods in chapters where these are again used as tools. In fact readers will find that access to simple computing facilities will make the problems in this volume very much easier, as software can be built up as the work proceeds.

Hyperbolic Equations and Characteristics

In Section 1.6, general second order equations were classified using characteristics, and this subject is revisited here. In the first chapter, the characteristics were used to classify the equations and to form a transformation to allow reduction to canonical form. In the process an ordinary differential equation was also obtained which held along a characteristic (1.6.10) and in the case of real characteristics this equation can form the basis of a method of solution. The method is based on using finite differences along the characteristic curves which form a natural grid. This will be covered in Section 3.3 and a slightly different derivation to that in Chapter 1 will be given, as this approach is informative in its own right.

Finite Element Method for Ordinary Differential Equations

Finite element methods depend on a much more enlarged background of mathematics than the finite difference methods of the previous chapters. Most of these concepts can be applied to the solution of ordinary differential equations, and it is expedient to introduce these ideas through this medium. By this means the reader is less likely to become disorientated in the discussion on partial differential equations in the next chapter, as the underlying concepts will be dear.

Finite Elements for Partial Differential Equations

Finite element methods are essentially methods for finding approximate solutions to a problem, commonly a partial differential equation, in a finite region or domain. Generally the unknown in the problem varies continuously over the domain but the solution found by the finite element method will not possess the same degree of continuity. The basis of the method is to divide the region or domain of the problem into a number of finite elements, and then find a solution which minimises some measure of the error, and is continuous inside the elements where it is expressed in terms of simple functions. The solution may not be continuous where the elements fit together. The ultimate accuracy of the solution is dependent upon the number and size of the elements, and the types of approximate function used within the elements.

First-order Equations and Hyperbolic Second-order Equations

The concept of a characteristic curve for a second-order equation was introduced in Chapter 1, and led to a classification of these equations. When the characteristics are real as in the hyberbolic case, they can be used to solve partial differential equations directly. Along the characteristic curve, an ordinary differential equation holds, and in both an analytic and numerical context, can be used to solve the original partial differential equation. This chapter will begin with a consideration of first-order partial differential equations from an analytic point of view, where characteristics play an important role. This will be followed by the specific consideration of the wave equation and a characteristic based method known as d’Alembert’s method.

Separation of the Variables

In the previous chapter, some of the mathematical preliminaries were discussed. The principal physical equations which we will consider are those for heat flow, the wave equation and the potential equation of Laplace and are derived in the companion volume, Numerical methods for partial differential equations. These equations are very typical of second-order linear equations. It was shown in the previous chapter, §1.2 that there are just three canonical forms for the general second-order quasilinear equation of which the three physical problems are linear examples. Hence, in order to introduce methods of solution, these three equations will be considered in their own light and various methods will be studied.

Green’s Functions

Green’s functions are named after the mathematician and physicist George Green who was born in Nottingham in 1793 and “invented” the Green’s function in 1828. This invention was developed in an essay written by Green entitled “Mathematical Analysis to the Theories of Electricity and Magnetism” originally published in Nottingham in 1828 and reprinted by the George Green Memorial Committee to mark the bicentenary of the birth of George Green in 1993. In this essay, Green’s function solutions to the Laplace and Poisson equation are formulated (but not in the manner considered in this chapter, in which the Green’s function is defined using the delta function).