Consider a pair of ordinary elliptic curves E and E ′ defined over the same finite field Fq. Suppose they have the same number of Fq-rational points, i.e. |E(Fq)| = |E ′ (Fq)|. In this paper we characterise for which finite field extensions F q k , k ≥ 1 (if any) the corresponding groups of F q k -rational points are isomorphic, i.e. E(F q k ) ∼ = E ′ (F q k ).

We describe a method to perform scalar multiplication on two classes of ordinary elliptic curves, namely E:y2=x3+Ax in prime characteristic p≡1normalmod0.2em4, and E:y2=x3+B in prime characteristic p≡1normalmod0.2em3. On these curves, the 4-th and 6-th roots of unity act as (computationally efficient) endomorphisms. In order to optimise the scalar multiplication, we consider a width-w-NAF (Non-Adjacent Form) digit expansion of positive integers to the complex base of τ, where τ is a zero of the characteristic polynomial x2−tx+p of the Frobenius endomorphism associated to the curve. We provide a precomputationless algorithm by means of a convenient factorisation of the unit group of residue classes modulo τ in the endomorphism ring, whereby we construct a digit set consisting of powers of subgroup generators, which are chosen as efficient endomorphisms of the curve.

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