Logic, Arithmetic, and Special Functions

Abstract

In this chapter we will give an outline of applications of logic functions and equations that are not directly related to the design of hardware. The applications in propositional logic are surely the oldest and reach back to the developments of science in ancient Greece. The names Boolean Algebra and Boolean Ring have been selected in honor of George Boole who wanted to formalize the ideas expressed by texts in natural language and to calculate the truth of complex constructions (in order to be precise and more accurate). Naturally we already had to use logic in this sense throughout the book, however, we will go back to an introductory or elementary point of view and show that the knowledge that has been prepared up to now is fully appropriate to solve these problems. There was an enormous development of logic (in a broad mathematical sense) during the last 150 years that resulted in a highly sophisticated and specialized level, however, the propositional calculus is still the foundation of all these developments, and it is still necessary and useful to have a good understanding of these concepts. Binary arithmetic is widely used in computer hardware, control systems, or other electronic devices. In this chapter we will give an outline of binary Mathematics before hardware-related problems will be discussed in more detail. Coding is a unique mapping of arbitrary objects to certain code words. Binary codes can be seen as bridges between many real world areas and logic functions. The code words can be a subset of a given finite set so that changes of one or more bits can be detected or even corrected. The introduced Specific Normal Form uniquely expresses a logic function, differs significantly from other known normal forms, and has several remarkable properties. This new normal form is used to find the most complex logic functions and to classify bent functions. Bent functions are logic functions having the largest distance to all linear function which is a useful property for cryptographic applications.