Generalized numerical-theoretical transformation

TL;DR

This paper introduces a generalized number-theoretic transformation based on the exponential function theorem, unifying various known transformations like Mersen, Gauss, and Fourier within a single theoretical framework.

Contribution

It formulates a new generalized NPT using the exponential function theorem, establishing its properties and unifying multiple classical transformations.

Findings

Formulated the main theorems and properties of the generalized NPT.

Proved duality and weight function properties of the transformation.

Unified several classical transformations under a generalized framework.

Abstract

The generalized number-theoretic transformation (NPT) is formulated on the basis of the exponential function theorem, which allows us to replace operations modulo the expression as a whole by modulo operations on the exponent of this function, which makes this theorem fundamental for NPT, since it is such a function used in NPT as a weight conversion function. On the basis of this theorem, all the main theorems of the generalized NPT, their duality, as well as the properties of the weight functions of this transformation are formulated and proved. The choice of the basis of this function, as which any number can be chosen, including a complex one, determines not only one or another type of transformation, but also the module of the transformation itself. This allows us to generalize a number of well-known NPTs, such as Mersen, Gauss, and even Fourier, in the form of a unified theory of discrete transformations.