Quantum Computers Can Find Quadratic Nonresidues in Deterministic Polynomial Time

Abstract

An integer $a$ is a quadratic nonresidue for a prime $p$ if $x^2\equiv a\mathrm{mod}p$ 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.

7 pages, 6 figures