Average-Case Complexity

We survey the average-case complexity of problems in NP. We discuss various notions of good-on-average algorithms, and present completeness results due to Impagliazzo and Levin. Such completeness results establish the fact that if a certain specific (but somewhat artificial) NP problem is easy-on-average with respect to the uniform distribution, then all problems in NP are easy-on-average with respect to all samplable distributions. Applying the theory to natural distributional problems remain an outstanding open question. We review some natural distributional problems whose average-case complexity is of particular interest and that do not yet fit into this theory. A major open question whether the existence of hard-on-average problems in NP can be based on the P$\neq$NP assumption or on related worst-case assumptions. We review negative results showing that certain proof techniques cannot prove such a result. While the relation between worst-case and average-case complexity for general NP problems remains open, there has been progress in understanding the relation between different ``degrees'' of average-case complexity. We discuss some of these ``hardness amplification'' results

PCD

Thesis (M. Eng.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2010.Page 96 blank. Cataloged from PDF version of thesis.Includes bibliographical references (p. 87-95).The security of systems can often be expressed as ensuring that some property is maintained at every step of a distributed computation conducted by untrusted parties. Special cases include integrity of programs running on untrusted platforms, various forms of confidentiality and side-channel resilience, and domain-specific invariants. We propose a new approach, proof-carrying data (PCD), which sidesteps the threat of faults and leakage by reasoning about properties of a computation's output data, regardless of the process that produced it. In PCD, the system designer prescribes the desired properties of a computation's outputs. Corresponding proofs are attached to every message flowing through the system, and are mutually verified by the system's components. Each such proof attests that the message's data and all of its history comply with the prescribed properties. We construct a general protocol compiler that generates, propagates, and verifies such proofs of compliance, while preserving the dynamics and efficiency of the original computation. Our main technical tool is the cryptographic construction of short non-interactive arguments (computationally-sound proofs) for statements whose truth depends on "hearsay evidence": previous arguments about other statements. To this end, we attain a particularly strong proof-of-knowledge property. We realize the above, under standard cryptographic assumptions, in a model where the prover has blackbox access to some simple functionality - essentially, a signature card.by Alessandro Chiesa.M.Eng

Efficient Batch Verification for UP

Consider a setting in which a prover wants to convince a verifier of the correctness of k NP statements. For example, the prover wants to convince the verifier that k given integers N_1,...,N_k are all RSA moduli (i.e., products of equal length primes). Clearly this problem can be solved by simply having the prover send the k NP witnesses, but this involves a lot of communication. Can interaction help? In particular, is it possible to construct interactive proofs for this task whose communication grows sub-linearly with k? Our main result is such an interactive proof for verifying the correctness of any k UP statements (i.e., NP statements that have a unique witness). The proof-system uses only a constant number of rounds and the communication complexity is k^delta * poly(m), where delta>0 is an arbitrarily small constant, m is the length of a single witness, and the poly term refers to a fixed polynomial that only depends on the language and not on delta. The (honest) prover strategy can be implemented in polynomial-time given access to the k (unique) witnesses. Our proof leverages "interactive witness verification" (IWV), a new type of proof-system that may be of independent interest. An IWV is a proof-system in which the verifier needs to verify the correctness of an NP statement using: (i) a sublinear number of queries to an alleged NP witness, and (ii) a short interaction with a powerful but untrusted prover. In contrast to the setting of PCPs and Interactive PCPs, here the verifier only has access to the raw NP witness, rather than some encoding thereof

Probabilistic Proof Systems

Various types of probabilistic proof systems have played a central role in the development of computer science in the last decade. In this exposition, we concentrate on three such proof systems -- interactive proofs, zero-knowledge proofs, and probabilistic checkable proofs -- stressing the essential role of randomness in each of them. This exposition is an expanded version of a survey written for the proceedings of the International Congress of Mathematicians (ICM94) held in Zurich in 1994. It is hope that this exposition may be accessible to a broad audience of computer scientists and mathematians

Some Applications of Coding Theory in Computational Complexity

Error-correcting codes and related combinatorial constructs play an important role in several recent (and old) results in computational complexity theory. In this paper we survey results on locally-testable and locally-decodable error-correcting codes, and their applications to complexity theory and to cryptography. Locally decodable codes are error-correcting codes with sub-linear time error-correcting algorithms. They are related to private information retrieval (a type of cryptographic protocol), and they are used in average-case complexity and to construct ``hard-core predicates'' for one-way permutations. Locally testable codes are error-correcting codes with sub-linear time error-detection algorithms, and they are the combinatorial core of probabilistically checkable proofs

Why Philosophers Should Care About Computational Complexity

One might think that, once we know something is computable, how efficiently it can be computed is a practical question with little further philosophical importance. In this essay, I offer a detailed case that one would be wrong. In particular, I argue that computational complexity theory---the field that studies the resources (such as time, space, and randomness) needed to solve computational problems---leads to new perspectives on the nature of mathematical knowledge, the strong AI debate, computationalism, the problem of logical omniscience, Hume's problem of induction, Goodman's grue riddle, the foundations of quantum mechanics, economic rationality, closed timelike curves, and several other topics of philosophical interest. I end by discussing aspects of complexity theory itself that could benefit from philosophical analysis.Comment: 58 pages, to appear in "Computability: G\"odel, Turing, Church, and beyond," MIT Press, 2012. Some minor clarifications and corrections; new references adde

Classical Cryptographic Protocols in a Quantum World

Cryptographic protocols, such as protocols for secure function evaluation (SFE), have played a crucial role in the development of modern cryptography. The extensive theory of these protocols, however, deals almost exclusively with classical attackers. If we accept that quantum information processing is the most realistic model of physically feasible computation, then we must ask: what classical protocols remain secure against quantum attackers? Our main contribution is showing the existence of classical two-party protocols for the secure evaluation of any polynomial-time function under reasonable computational assumptions (for example, it suffices that the learning with errors problem be hard for quantum polynomial time). Our result shows that the basic two-party feasibility picture from classical cryptography remains unchanged in a quantum world.Comment: Full version of an old paper in Crypto'11. Invited to IJQI. This is authors' copy with different formattin

Inapproximability of Combinatorial Optimization Problems

We survey results on the hardness of approximating combinatorial optimization problems

Complexity Theory

Computational Complexity Theory is the mathematical study of the intrinsic power and limitations of computational resources like time, space, or randomness. The current workshop focused on recent developments in various sub-areas including arithmetic complexity, Boolean complexity, communication complexity, cryptography, probabilistic proof systems, pseudorandomness, and quantum computation. Many of the developements are related to diverse mathematical fields such as algebraic geometry, combinatorial number theory, probability theory, quantum mechanics, representation theory, and the theory of error-correcting codes