2,436 research outputs found
Abelian Varieties with Prescribed Embedding Degree
We present an algorithm that, on input of a CM-field , an integer ,
and a prime , constructs a -Weil number \pi \in \O_K
corresponding to an ordinary, simple abelian variety over the field \F of
elements that has an \F-rational point of order and embedding degree
with respect to . We then discuss how CM-methods over can be used to
explicitly construct .Comment: to appear in ANTS-VII
Abelian varieties in pairing-based cryptography
We study the problem of the embedding degree of an abelian variety over a
finite field which is vital in pairing-based cryptography. In particular, we
show that for a prescribed CM field of degree , prescribed integers
, and any prime , there exists an ordinary abelian
variety over a finite field with endomorphism algebra , embedding degree
with respect to and the field extension generated by the -torsion points
of degree over the field of definition. We also study a class of
absolutely simple higher dimensional abelian varieties whose endomorphism
algebras are central over imaginary quadratic fields.Comment: Typos fixe
A CM construction for curves of genus 2 with p-rank 1
We construct Weil numbers corresponding to genus-2 curves with -rank 1
over the finite field \F_{p^2} of elements. The corresponding curves
can be constructed using explicit CM constructions. In one of our algorithms,
the group of \F_{p^2}-valued points of the Jacobian has prime order, while
another allows for a prescribed embedding degree with respect to a subgroup of
prescribed order. The curves are defined over \F_{p^2} out of necessity: we
show that curves of -rank 1 over \F_p for large cannot be efficiently
constructed using explicit CM constructions.Comment: 19 page
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