Analysis of Digital Expansions of Minimal Weight

Digital expansions are one method for efficient implementations of scalar multiplication (or linear combinations) in Abelian groups, such as the point group of elliptic curves. One application is in public key cryptography, where scalar multiplication is a key ingredient. Let G be an Abelian group. A positive integer n ∈ Z can also be seen as an endomorphism of G by setting nQ = Q+ · · ·+Q (with n summands) for Q ∈ G. This immediately carries over to all integers in Z. We consider an algebraic integer τ which also acts as an endomorphism on G. This action can easily be extended to an action of the ring Z[τ ] on G. In order to compute nP for some n ∈ Z[τ ] and some P ∈ G, we consider a digital expansion