Generating chaotic sequences is not difficult in principle. Both continuos-time and discrete-time systems can be simulated using a digital computer. If floating-point arithmetic is inadequate, user defined arithmetic can be used to keep track of more significant bits and arbitrarily extend the precision of math operations. Eventually the stress on computer resources (memory and computing time) will force a limit on how precise can one do arithmetic. For example, every iteration of the logistic map doubles the number of bits necessary to represent the result and also the number of operations. Obviously, this exponential growth in memory and computing time is not sustainable and round-off has to occur in arithmetic operations. In any case, the usefulness of chaotic sequences stems from the length of their period and there is nothing, in principle, that curtails a computer from generating long sequences.
Another option is to use a physical dynamical system to generate chaotic sequences. Circuits like Chua's oscillator can be built out of common components [8] [15] [16], can be synchronized [17] and can be controlled [18] [19] if the application requires it. Therefore it is safe to say that generating useful chaotic sequences is a practical reality.
How can we employ such sequences? Practical applications for them are abundant and varied, including music [20] [21], spread spectrum communications [22] [23], cryptography [24] [25] and estimation [26] [27]. In the rest of this paper we will concentrate our attention on two practical applications: cryptography and estimation.