Mathematical Foundations for Security

This chapter is devoted to the theoretical foundations of generalized Gray codes, encompassing both classical “left-sided” and proposed alternative “right-sided” Gray codes. Left-sided codes develop from left to right, whereas right-sided codes develop from right to left. Left- and right-sided Gray codes, both in direct and inverse forms, when supplemented with reverse permutation operators, constitute a subset of so-called composite Gray codes. Such codes have been constructively applied to solving a number of problems of synthesis and analysis of discrete systems of Vilenkin–Chrestenson functions (VCF), whose special cases include systems of discrete exponential functions (DEF) and Walsh functions. So-called indicator matrices (IM) have been proposed for the first time, which are bijectively related to VCF systems. Indicator matrices form the basis for the synthesis of VCF systems in the non-discrete space of their parameters (the basis of the number systems and the code length). The orders of IM are logarithmically related to the orders of VCF systems, which makes it possible to represent VCF matrices compactly. Unique Walsh–Cully function systems have been developed; these are the only Walsh function systems that ensure linear connectivity of the frequency scales of DFT processors.