2 research outputs found
Weil descent and cryptographic trilinear maps
It has recently been shown that cryptographic trilinear maps are sufficient
for achieving indistinguishability obfuscation. In this paper we develop a
method for constructing such maps on the Weil descent (restriction) of abelian
varieties over finite fields, including the Jacobian varieties of hyperelliptic
curves and elliptic curves. The security of these candidate cryptographic
trilinear maps raises several interesting questions, including the
computational complexity of a trapdoor discrete logarithm problem
Algebraic blinding and cryptographic trilinear maps
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 and involves variables,
where is the security parameter. Finding points on such algebraic sets in
general takes time exponential in 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