# Abelian varieties with prescribed embedding and full embedding degrees

## TL;DR

This paper demonstrates the existence of abelian varieties with specific embedding and full embedding degrees over finite fields, crucial for cryptography, by constructing varieties with prescribed CM fields and endomorphism properties.

## Contribution

It provides a method to construct abelian varieties with predetermined embedding degrees and endomorphism algebras, advancing cryptographic applications.

## Key findings

- Existence of abelian varieties with prescribed embedding degrees
- Construction over finite fields with specific CM fields
- Analysis of simple higher dimensional abelian varieties

## 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.