Trilinear maps for cryptography

We construct cryptographic trilinear maps that involve simple, non-ordinary abelian varieties over finite fields. In addition to the discrete logarithm problems on the abelian varieties, the cryptographic strength of the trilinear maps is based on a discrete logarithm problem on the quotient of certain modules defined through the N\'{e}ron-Severi groups. The discrete logarithm problem is reducible to constructing an explicit description of the algebra generated by two non-commuting endomorphisms, where the explicit description consists of a linear basis with the two endomorphisms expressed in the basis, and the multiplication table on the basis. It is also reducible to constructing an effective $\mathbb{Z}$-basis for the endomorphism ring of a simple non-ordinary abelian variety. Both problems appear to be challenging in general and require further investigation.