Constructing Cryptographic Multilinear Maps Using Affine Automorphisms

TL;DR

This paper introduces a novel method for constructing cryptographic multilinear maps using affine automorphisms, linking algebraic geometry with multivariate cryptography to enhance security assumptions.

Contribution

It presents a new construction of multilinear maps based on affine automorphisms, connecting algebraic geometry techniques with cryptographic applications.

Findings

Multiple versions of the discrete logarithm problem are analyzed.

Efficient solutions to certain problems imply algorithms for inverting multivariate polynomials.

The work relates the difficulty of these problems to the security of multivariate encryption.

Abstract

The point of this paper is to use affine automorphisms from algebraic geometry to build cryptographic multivariate mappings. We will construct groups G,H, both isomorphic to the cyclic group with a prime number of elements and multilinear pairings from the k-fold product of G to H. The construction is reminiscent of techniques in multivariate encryption. We display several different versions of the discrete logarithm problem for these groups. We show that the efficient solution of some of these problems result in efficient algorithms for inverting systems of multivariate polynomials corresponding to affine automorphisms, which implies that such problems are as computationally difficult as breaking multivariate encryption.