Quantum Computers Can Find Quadratic Nonresidues in Deterministic Polynomial Time

TL;DR

This paper introduces a quantum algorithm that deterministically finds quadratic nonresidues in polynomial time, overcoming the limitations of classical methods without relying on unproven hypotheses.

Contribution

It presents the first known quantum algorithm capable of deterministically identifying quadratic nonresidues in polynomial time.

Findings

Quantum algorithm runs in deterministic polynomial time

Successfully generates quadratic nonresidues without probabilistic methods

Advances understanding of quantum algorithms for number theory problems

Abstract

An integer $a$ is a quadratic nonresidue for a prime $p$ if $x^2 \equiv a \bmod p$ has no solution. Quadratic nonresidues may be found by probabilistic methods in polynomial time. However, without assuming the Generalized Riemann Hypothesis, no deterministic polynomial-time algorithm is known. We present a quantum algorithm which generates a random quadratic nonresidue in deterministic polynomial time.