Mathematical and algorithmic support for selecting parameters of multiplicative sequences

Abstract

The problem of selecting parameters for multiplicative sequences with specified period characteristics and a prescribed cyclic structure of nonzero states is considered. The arithmetic conditions for such selection are systematized, and the limits of transferring properties between structurally related bases are determined. The results of a computational experiment implemented in C# for two thousand sixty-seven bases using a sample of five hundred thousand consecutive prime moduli are analyzed. Comparison of groups by square-free part and power representation made it possible to distinguish between the proximity of empirical frequencies, shared necessary constraints, and exact equivalence with respect to primitivity. It is shown that a common square-free part implies certain shared arithmetic constraints but does not guarantee equality of multiplicative orders or of the sets of prime moduli for which the bases are primitive roots. At the same time, conditions are established for certain power families under which the property of primitivity can be correctly transferred between related bases. The investigated sample also shows that close empirical frequencies alone are not sufficient grounds for transferring properties between candidates. Based on known relationships for multiplicative order and index, an algorithmic procedure for parameter selection is formulated, including admissibility checking, rejection of incompatible candidates, exact order computation, and storage of suitable «base–modulus» pairs. An example of multiplier selection for forming two cycles of nonzero states modulo seventeen is presented. The proposed verification sequence combines the preliminary application of exact structural constraints with the final computation of the multiplicative order. The practical result is mathematically justified algorithmic support for selecting parameters of multiplicative sequences with a prescribed cyclic structure. Search acceleration and the statistical quality of the generators were not evaluated in this study.