Steve Thakur
We study the isogenies of certain abelian varieties over finite fields with non-commutative endomorphism algebras with a view to potential use in isogeny-based cryptography. In particular, we show that any two such abelian varieties with endomorphism rings maximal orders in the endomorphism algebra are linked by a cyclic isogeny of prime degree.
Steve Thakur
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.