David Freeman (UC Berkeley)

Constructing abelian varieties for pairing-based cryptography

Many recent cryptographic protocols are based on the Weil and Tate pairings
on abelian varieties over finite fields.  For these protocols to be
efficient and secure, the abelian varieties in question must have large
prime-order subgroups and the pairings must take values in a field that is
a low-degree extension of the abelian variety's field of definition.  Such
"pairing-friendly" varieties are rare and require special constructions.

In this talk we will present two recent constructions of pairing-friendly
abelian varieties of arbitrary dimension.  Both generalize existing
constructions of pairing-friendly elliptic curves: the first generalizes
the method of Cocks and Pinch, while the second generalizes the method of
Brezing and Weng and leads to varieties defined over smaller fields than
the first.

The two constructions both use the theory of complex multiplication to
produce $q$-Weil numbers $\pi$ that correspond, in the sense of Honda-Tate
theory, to isogeny classes of pairing-friendly abelian varieties over
$\mathbb{F}_q$.  Explicit equations for varieties in these isogeny classes
can then be determined using complex multiplication methods.

The first construction is joint work with Peter Stevenhagen and Marco
Streng (Universiteit Leiden).