Peikert, C., Vaikuntanathan, V. (2008). Noninteractive Statistical Zero-Knowledge Proofs for Lattice Problems. In: Wagner, D. (eds) Advances in Cryptology – CRYPTO 2008. CRYPTO 2008. Lecture Notes in Computer Science, vol 5157. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85174-5_30
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.
Keywords: Proof System, Lattice Problem, Random String, Auxiliary Input, Promise Problem