We construct several new statistical zero-knowledge proofs with efficient
provers, i.e. ones where the prover strategy runs in probabilistic polynomial
time given an NP witness for the input string. Our first proof systems are
for approximate versions of the Shorttest Vector Problem (SVP) and Closest
Vector Problem (CVP), where the witness is simply a short vector in the lattice
or a lattice vector close to the target, respectively. Our proof systems are in
fact proofs of knowledge, and as a result, we immediately obtain efficient
lattice-based identification schemes which can be implemented with arbitrary
families of lattices in which the approximate SVP or CVP are hard.
We then turn to the general question of whether all problems in SZK intersect NP admit
statistical zero-knowledge proofs with efficient provers. Towards this end, we
give a statistical zero-knowledge proof system with an efficient prover for a
natural restriction of Statistical Difference, a complete problem for SZK. We
also suggest a plausible approach to resolving the general question in the
positive.