## Cryptology ePrint Archive: Report 2008/061

Abelian varieties with prescribed embedding degree

David Freeman and Peter Stevenhagen and Marco Streng

Abstract: We present an algorithm that, on input of a CM-field $K$, an integer $k \ge 1$, and a prime $r \equiv 1 \bmod k$, constructs a $q$-Weil number $\pi \in \O_K$ corresponding to an ordinary, simple abelian variety $A$ over the field $\F$ of $q$ elements that has an $\F$-rational point of order $r$ and embedding degree $k$ with respect to $r$. We then discuss how CM-methods over $K$ can be used to explicitly construct $A$.

Category / Keywords: public-key cryptography / pairing-friendly curves, embedding degree, abelian varieties, hyperelliptic curves, CM method, complex multiplication