  We continue to study the construction of cryptographic trilinear maps
involving abelian varieties over finite fields. We introduce Weil descent as a
tool to strengthen the security of a trilinear map. We form the trilinear map
on the descent variety of an abelian variety of small dimension defined over a
finite field of a large extension degree over a ground field. The descent
bases, with respect to which the descents are performed, are trapdoor secrets
for efficient construction of the trilinear map which pairs three trapdoor
DDH-groups. The trilinear map also provides efficient public identity testing
for the third group. We present a concrete construction involving the jacobian
varieties of hyperelliptic curves.
