Abelian varieties with prescribed embedding and full embedding degrees
This work addresses cryptographic applications by enabling tailored abelian varieties for pairing-based systems, though it appears incremental as it builds on known CM field methods.
The authors tackled the problem of constructing abelian varieties over finite fields with specific embedding degrees, which is crucial for pairing-based cryptography, by proving the existence of such varieties with prescribed CM fields and parameters, including prime conditions and torsion point degrees.
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.