In this paper we derive lower bounds for the degree of polynomials that approximate the square root of the discrete logarithm for Elliptic Curves with orders of various specific types. These bounds can serve as evidence for the difficulty in the computation of the square root of discrete logarithms for such elliptic curves, with properly chosen parameters that result in the curve having order of any of types studied in this paper. The techniques are potentially applicable to elliptic curves of order of any specific, allowable (by Hasse's bounds), order type that is of interest for the application in hand.
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