Algebraic blinding and cryptographic trilinear maps

Abstract

It has been shown recently that cryptographic trilinear maps are sufficient for achieving indistinguishability obfuscation. In this paper we develop algebraic blinding techniques for constructing such maps. An earlier approach involving Weil restriction can be regarded as a special case of blinding in our framework. However, the techniques developed in this paper are more general, more robust, and easier to analyze. The trilinear maps constructed in this paper are efficiently computable. The relationship between the published entities and the hidden entities under the blinding scheme is described by algebraic conditions. Finding points on an algebraic set defined by such conditions for the purpose of unblinding is difficult as these algebraic sets have dimension at least linear in $n$ and involves $Ω(n^2)$ variables, where $n$ is the security parameter. Finding points on such algebraic sets in general takes time exponential in $n^2\log n$ with the best known methods. Additionally these algebraic sets are characterized as being {\em triply confusing} and most likely {\em uniformly confusing} as well. These properties provide additional evidence that efficient algorithms to find points on such algebraic sets seems unlikely to exist. In addition to algebraic blinding, the security of the trilinear maps also depends on the computational complexity of a trapdoor discrete logarithm problem which is defined in terms of an associative non-commutative polynomial algebra acting on torsion points of a blinded product of elliptic curves.