On Taking Square Roots without Quadratic Nonresidues over Finite Fields

Details On Taking Square Roots without Quadratic Nonresidues over Finite Fields On Taking Square Roots without Quadratic Nonresidues over Finite Fields

2008-12-14

arxiv.org/abs/0812.2591v3

Math. Comp., 80(275):1797-1811, July 2011

Mathematics

Number Theory

14 pages

Scientific paper

10.1090/S0025-5718-2011-02419-1

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 $N$.

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