On Calculating Square Roots in GF(p)
David S. Knight
TL;DR
This paper introduces a novel method for computing square roots in GF(p) using exponentiation in GF(p^3), compares it to existing algorithms, and discusses related conjectures.
Contribution
The paper presents a new GF(p^3)-based square root algorithm and explores its properties alongside existing methods and conjectures.
Findings
The new algorithm is similar to Cipolla-Lehmer but operates in GF(p^3).
Several conjectures about the algorithm's output are proposed.
Alternative quadratic sum-based square root methods are also discussed.
Abstract
This article presents a new method for calculating square roots in GF(p) by exponentiating in GF(p^3) or equivalently modulo irreducible cubic polynomials. This algorithm is in some ways similar to the Cipolla-Lehmer algorithm which is based on exponentiating in GF(p^2). Another less well known square root algorithm based on quadratic sums is also given. In addition to this, several conjectures about the output of this GF(p^3) square root algorithm are mentioned.
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Taxonomy
TopicsAnalytic Number Theory Research · Coding theory and cryptography · Algebraic Geometry and Number Theory