On Taking Square Roots without Quadratic Nonresidues over Finite Fields

TL;DR

This paper introduces a deterministic, elementary algorithm for computing square roots over finite fields without quadratic nonresidues, enabling efficient primality proofs without unproven assumptions.

Contribution

It presents a novel deterministic method for square root extraction over finite fields that does not rely on quadratic nonresidues or unproven hypotheses.

Findings

Algorithm runs in  O(log^2 q)  bit operations

Constructs a deterministic primality proof algorithm

Operates efficiently for large integers N

Abstract

We present a novel idea to compute square roots over finite fields, without being given any quadratic nonresidue, and without assuming any unproven hypothesis. The algorithm is deterministic and the proof is elementary. In some cases, the square root algorithm runs in $\tilde{O}(\log^2 q)$ bit operations over finite fields with $q$ elements. As an application, we construct a deterministic primality proving algorithm, which runs in $\tilde{O}(\log^3 N)$ for some integers $N$.