  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 $L$ of degree $\geq 4$, prescribed integers
$m$, $n$ and any prime $\ell\equiv 1 \mod{mn}$ that splits completely in $L$,
there exists an ordinary abelian variety over a prime finite field with
endomorphism algebra $L$, embedding degree $n$ with respect to $\ell$ and the
field extension generated by the $\ell$-torsion points of degree $mn$ 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.

Number Theory;Cryptography and Security