Trilinear maps for cryptography

TL;DR

This paper introduces cryptographic trilinear maps based on simple non-ordinary abelian varieties over finite fields, linking their security to complex discrete logarithm problems involving endomorphism rings.

Contribution

It constructs new cryptographic trilinear maps using abelian varieties and analyzes the associated hard discrete logarithm problems, expanding cryptographic tools.

Findings

Cryptographic trilinear maps are constructed from abelian varieties.

Security relies on discrete logarithm problems on endomorphism modules.

The problems involve constructing explicit algebraic descriptions of endomorphism rings.

Abstract

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.