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 $L$ of degree $\geq 4$, prescribed integers $m$, $n$ and any prime $l\equiv 1 \pmod{mn}$, there exists an ordinary abelian variety over a finite field with endomorphism algebra $L$, embedding degree $n$ with respect to $l$ and the field extension generated by the $l$-torsion points of degree $mn$ over the field of definition. We also provide algorithms for the construction of such abelian varieties.