Description
This paper examines the proposed "novel idea to compute square roots over finite fields, without being given any quadratic nonresidue, and without assuming any unproven hypothesis" as presented recently in Tsz-Wo Sze's paper "On Taking Square Roots Without Quadratic Nonresidues Over Finite Fields." Sze's algorithm claims a deterministic approach where "in some cases the algorithm runs in O((log q)_) bit operations over finite fields with q elements." A careful analysis carried out in the present work reveals that several conditions on the factors of q -- 1 must be satisfied in order for the algorithm to actually run in O((log q)_) bit operations. In addition, in this work a modification of Sze's approach is presented as an alternative probabilistic method that runs in polynomial time and without the need for quadratic nonresidues. Moreover, the approach being proposed is slightly faster than most common known probabilistic methods of its kind. An application of Sze's algorithm (and its probabilistic version) is a primality testing/proving algorithm that runs in polynomial time. We shall first introduce the reader to the background concerning the problem of finding square roots in finite fields along with explaining when the problem can be solved "easily." A thorough analysis of the Tonelli-Shanks algorithm, including a complexity analysis, will follow this for comparison's sake to our new proposed algorithm. This will be followed by a complete breakdown of Sze's algorithm, its complexity analysis, and its practical implementation. The new probabilistic approach will then be presented accompanied by its own complexity review and its application to a probabilistic primality testing algorithm
