Spectacular Exponents: A semi modular Approach to Fast Exponentiation

Robert Valenza

doi:10.24297/jam.v16i0.8301

Abstract

This paper introduces a computational scheme for calculating the exponential bw where b and w are positive integers. This two-step method is based on elementary number theory that is used routinely in this and similar contexts, especially the Chinese remainder theorem (CRT), Lagrange’s theorem, and a variation on Garner’s algorithm for inverting the CRT isomorphism. We compare the performance of the new method to the standard fast algorithm and show that for a certain class of exponents it is significantly more efficient as measured by the number of required extended multiplications.

Highlights

Introduction and Preliminary Estimates of MultiplicativeComplexityThroughout this analysis we are concerned with the multiplicative complexity of the exponential calculation, and we introduce two associated functions
We want to be clear at the outset that we shall be directing our attention exclusively to multiplicative complexity accruing from multiplication of extended integers and neglecting the computational cost of native arithmetic
Given a family of relatively prime positive integers m1,..., ms, the Chinese remainder theorem asserts that the following map is an isomorphism of rings with unity [4]: https://rajpub.com/index.php/jam s

Summary

A Semi modular Approach

We let Z/mZ denote the ring of integers mod m (whence the natural abstract algebraic identification of Z/0Z with Z is consistent with our previous convention). Given a family of relatively prime positive integers m1,..., ms , the Chinese remainder theorem asserts that the following map is an isomorphism of rings with unity [4]: https://rajpub.com/index.php/jam s. Note that the elements of the codomain look like coordinate vectors for which the j-th coordinate of the image of a mod mj is the projection of a into Z/mj Z. The most important particular elements of this assertion are the following: 1. The operations of addition and multiplication on the codomain are defined componentwise (similar to the operations of addition and scalar multiplication in linear algebra):.

The map is both additive and multiplicative:

Findings

A Few Trial Runs