Adleman-Manders-Miller Root Extraction Method Revisited

TL;DR

This paper revisits the Adleman-Manders-Miller method for extracting general rth roots over finite fields, clarifying its process and analyzing its computational complexity, especially addressing the challenge of discrete logarithms.

Contribution

It provides a detailed clarification and complexity analysis of the previously briefly described rth root extraction method, emphasizing the role of discrete logarithms.

Findings

Clarified the Adleman-Manders-Miller rth root extraction method

Analyzed the computational complexity involving discrete logarithms

Provided heuristic insights into the method's implementation

Abstract

In 1977, Adleman, Manders and Miller had briefly described how to extend their square root extraction method to the general $r$th root extraction over finite fields, but not shown enough details. Actually, there is a dramatic difference between the square root extraction and the general $r$th root extraction because one has to solve discrete logarithms for $r$th root extraction. In this paper, we clarify their method and analyze its complexity. Our heuristic presentation is helpful to grasp the method entirely and deeply.