A (t, n)-locally random reduction maps a problem instance x into a set
of problem instances y(1), ..., y(n) in such a way that it is easy to
construct the answer to x from the answers to y(1), ..., y(n), and ye
t the distribution on t-element subsets of y(1), ..., y(n) depends onl
y on \x\. In this paper we formalize such reductions and give improved
methods for achieving them. Then we give a cryptographic application,
showing a new way to prove in perfect zero knowledge that committed b
its x(1), ..., x(m) satisfy some predicate Q. Unlike previous techniqu
es for such perfect zero-knowledge proofs, ours uses an amount of comm
unication that is bounded by a fixed polynomial in m, regardless of th
e computational complexity of Q.