Abstract
Elliptic curves over a finite field with j-invariant 0 or 1728, both supersingular and ordinary, whose embedding degree k is low are studied. In the ordinary case we give conditions characterizing such elliptic curves with fixed embedding degree with respect to a subgroup of prime order ℓ. For , these conditions give parameterizations of q in terms of ℓ and two integers m, n. We show several examples of families with infinitely many curves. Similar parameterizations for need a fixed kth root of the unity in the underlying field. Moreover, when the elliptic curve admits distortion maps, an example is provided.
Highlights
Let E be an elliptic curve defined over a finite field Fq, with q = pr, p prime, p ≥ 5, given by its Weiertrass model y2 = x3 + Ax + B; A, B ∈ Fq
Distortion maps always exist on supersingular elliptic curves but never for ordinary elliptic curves with embedding degree greater than 1
Most of them are based on the following idea: Given an embedding degree k, look for a suitable equation t2 − 4q = Dh2 with a small D and determine an elliptic curve with discriminant D and cardinality q + 1 − t using the complex multiplication method
Summary
Distortion maps always exist on supersingular elliptic curves but never for ordinary elliptic curves with embedding degree greater than 1 (see [22]). We will study the embedding degree and distortion maps for curves with invariant j = 1728 (i.e. with Weiertrass equation y2 = x3 + Ax) and j = 0 (curves with equation y2 = x3 + B). These curves are well studied in the classical theory of elliptic curves [21]. We can deduce that the number of elliptic curves of the families with embedding degree 1 or 2 closely approaches the expected value given by the Bateman-Horn’s conjecture [3]
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