Another look at square roots (and other less common operations) in fields of even characteristic

Abstract

We discuss a family of irreducible polynomials that can be used to speed up square root extraction in fields of characteristic two. They generalize trinomials discussed by Fong et al. [20]. We call such polynomials square root friendly. The main application is to point halving methods for elliptic curves (and to a lesser extent also divisor halving methods for hyperelliptic curves and pairing computations). We note the existence of square root friendly trinomials of a given degree when we already know that an irreducible trinomial of the same degree exists, and formulate a conjecture on the degrees of the terms of square root friendly polynomials. Following similar results by Bluher. we also give a partial result that goes in the direction of the conjecture. We also discuss how to improve the speed of solving quadratic equations. The increase in the time required to perform modular reduction is marginal and does not affect performance adversely. Estimates confirm that the new polynomials mantain their promises. Point halving gets a speed-up of 20% and scalar multiplication is improved by at least 11%. © Springer-Verlag Berlin Heidelberg 2007.