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arXiv · 1812.11479

Abelian varieties with prescribed embedding and full embedding degrees

Steve Thakur
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Abstract

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.

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Steve Thakur. 2018-12-30. Abelian varieties with prescribed embedding and full embedding degrees. https://arxiv.org/abs/1812.11479

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