Noninteractive Statistical Zero-Knowledge Proofs for Lattice Problems

We construct

noninteractive statistical zero-knowledge

(

NISZK

) proof systems for a variety of standard approximation problems on lattices, such as the shortest independent vectors problem and the complement of the shortest vector problem. Prior proof systems for lattice problems were either interactive or leaked knowledge (or both).

Our systems are the first known

NISZK

proofs for any cryptographically useful problems that are not related to integer factorization. In addition, they are proofs of knowledge, have reasonable complexity, and generally admit efficient prover algorithms (given appropriate auxiliary input). In some cases, they even imply the first known

interactive

statistical zero-knowledge proofs for certain cryptographically important lattice problems.

We also construct an

NISZK

proof for a special kind of disjunction (i.e., OR gate) related to the shortest vector problem. This may serve as a useful tool in potential constructions of noninteractive (computational) zero knowledge proofs for

NP

based on lattice assumptions.