Non-commutative digit expansions for arithmetic on supersingular elliptic curves

Abstract

We study non-commutative digit expansions in quaternion rings that arise in the context of supersingular elliptic curves. These digit expansions can be used in a \({\tau}\)-and-add method to speed up arithmetic (scalar multiplication and pairing) on certain families of supersingular elliptic curves in characteristic \({p \geqq 5}\). The basis \({\tau}\) is a quadratic algebraic integer that represents the Frobenius endomorphism of the curve, which is a very fast operation to evaluate. We prove the existence of a finite expansion for every element of the quaternion ring, as well as the equivalence between right and left digit expansions (i.e. the basis \({\tau}\) is placed right, resp. left, to the digit); this expansion turns out to be a non-adjacent form (NAF) for integers, i.e. in every two consecutive digits there is at least one 0.