A basic question about NP is whether or not search reduces in polynomial time to decision. We indicate that the answer is negative: under a complexity assumption (that deterministic and non-deterministic double-exponential time are unequal) we construct a language in NP for which search does not reduce to decision. These ideas extend in a natural way to interactive proofs and program checking. Under similar assumptions we present languages in NP for which it is harder to prove membership interactively than it is to decide this membership, and languages in NP which are not checkable. Keywords: NP-completeness, self-reducibility, interactive proofs, program checking, sparse sets,

We survey definitions, known results, and open questions in the area of locally random reductions and explore the ramifications of these reductions in complexity theory. This paper is based on a survey talk given at the DIMACS Workshop on Structural Complexity and Cryptography, Rutgers University, New Brunswick NJ, December 1990. 1 Introduction We consider the question of whether a probabilistic polynomial-time machine A can compute a function f in the following manner. A interacts with one or more machines B 1 , . . ., B k that are not restricted to probabilistic polynomial time. At the end of the interaction, A can use the information obtained from the B i 's to compute f(x). However, the information that A sends to the B i 's is locally random. Informally, this means that no individual B i can use it to figure out what A's private input x is. This study can be motivated by the practical problem of using shared resources for private computations. For example, f may be a financial ...

A basic question about NP is whether or not search reduces in polynomial time to decision. We indicate that the answer is negative: under a complexity assumption (that deterministic and nondeterministic doubleexponential time are unequal) we construct a language in NP for which search does not reduce to decision. These ideas extend in a natural way to interactive proofs and program checking. Under similar assumptions we present languages in NP for which it is harder to prove membership interactively than it is to decide this membership. Similarly we present languages where checking is harder than computing membership. Each of the following properties --- checkability, random-self-reducibility, reduction from search to decision, and interactive proofs in which the prover's power is limited to deciding membership in the language itself --- implies coherence, one of the weakest forms of self-reducibility. Under assumptions about triple-exponential time, we construct incoherent sets in NP....

Informally speaking, an instance-hiding proof system for the function f is a protocol in which a polynomial-time verifier is convinced of the value of f(x) but does not reveal the input x to the provers. We show here that a boolean function f has an instance-hiding proof system if and only if it is the characteristic function of a language in NEXP " coNEXP. We formalize the notion of zero-knowledge for instance-hiding proof systems with several provers and show that all such systems can be made perfect zero-knowledge.

We define the notion of an instance-hiding proof system (ihps) for a function f ; informally, an ihps is a protocol in which a polynomial-time verifier interacts with one or more all-powerful provers and is convinced of the value of f(x) but does not reveal the input x to the provers. We show here that a function f has a multiprover ihps if and only if it is computable in FNEXP. We formalize the notion of zero-knowledge for ihps's and show that any function that has a multiprover ihps in fact has one that is perfect zeroknowledge. Under the assumption that one-way permutations exist, we show that f has a one-prover, zero-knowledge ihps if and only if it is in FPSPACE and has a one-oracle instance-hiding scheme (ihs).

The interplay of coding theory and computational complexity theory is a rich source of results and problems. This article surveys three of the major themes in this area: ffl the use of codes to improve algorithmic efficiency ffl the theory of program testing and correcting, which is a complexity theoretic analogue of error detection and correction ffl the use of codes to obtain characterizations of traditional complexity classes such as NP and PSPACE; these new characterizations are in turn used to show that certain combinatorial optimization problems are as hard to approximate closely as they are to solve exactly. 1 Introduction Complexity theory is the study of efficient computation. Faced with a computational problem that can be modelled formally, a complexity theorist seeks first to find a solution that is provably efficient and, if such a solution is not found, to prove that none exists. Coding theory, which provides techniques for "robust representation" of information, ...