Trilinear maps for cryptography
This work addresses cryptographic security for applications requiring multilinear maps, but it is incremental as it builds on existing concepts with unresolved challenges.
The authors tackled the problem of constructing cryptographic trilinear maps using simple, non-ordinary abelian varieties over finite fields, with security based on discrete logarithm problems and challenges in describing endomorphism algebras and rings.
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 Néron-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.