Another Look at Square Roots and Traces (and Quadratic Equations) in Fields of Even Characteristic

We discuss irreducible polynomials that can be used to speed up square root extraction in fields of characteristic two. The obvious applications are to point halving methods for elliptic curves and divisor halving methods for hyperelliptic curves. Irreducible polynomials P (X) such that the square root ζ of a zero x of P (X) is a sparse polynomial are considered and those for which ζ has minimal degree are characterized. We reveal a surprising connection between the minimality of this degree and the extremality of the the number of trace one elements in the polynomial base associated to P (X). We also show how to improve the speed of solving quadratic equations and that the increase in the time required to perform modular reduction is marginal and does not affect performance adversely. Experimental results confirm that the new polynomials mantain their promises; These results generalize work by Fong et al. to polynomials other than trinomials. Point halving gets a speed-up of 20% and the performance of scalar multiplication based on point halving is improved by at least 11%.