A method for Calculating the Positional Characteristics of a Modular Representation with Linear Complexity

A method has been developed for selecting base modules for generating modular number systems and modular arithmetic, in which the calculation of the positional characteristic of the modular representation of a numerical quantity, which is a nonlinear function of many variables, is performed with linear complexity from the number of bases of the modular number system when calculated in the range of a single base of the modular system. This significantly reduces the bit depth (hence the amount of hardware) of additional modular processor blocks. Modular algorithmics previously lacked methods for calculating positional characteristics for such parameters. All non-modular (not parallelizable in modular arithmetic) operations of a specialized processor with multiple processor elements (data streams) and a single instruction stream are based on the calculation of positional characteristics. For numerical quantities in modular data formats, the method allows them to be performed with minimal linear complexity. The article substantiates the formulation of the problem and the goal of fast calculation of positional features in modular data encoding. A new method is described and justified. The results of numerical modeling of the method and examples of modular number systems allowing its use are presented. The analysis is given the obtained of new results.