Computational Complexity and Information Theory

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Proposal Number: 9912342PI: Madhu SudanInstitution: MITOne of the broad goals of the theory of computer science is to identify functions that seem hard to compute, and if possible to prove that they are indeed hard. A further goal is to quantify how hard a function is to compute, for some appropriate measure of hardness, and to find functions that are very hard under this measure. For example, in the most common application of computational hardness, namely cryptography, one needs to know that a given hard function, such as the discrete logarithm or RSA decryption, is hard on almost all inputs (rather than on some adversarially chosen inputs). In order to formalize such statements, one needs to find good measures of hardness and develop tools to analyze them.In the recent pasta number of research articles have proposed different notions of hardness and analyzed them. Many of these results can be thought of as abstracting quantitative notions of information based on computational complexity. The results show that given a hard function, one can construct a much harder one, in the sense that computing even a small amount of information about the harder function allows for efficient perfect computation of the given function. Further these results share a common theme of relying on state-of-the-art results on the efficient listdecodability of error-correcting codes.This research project will perform a systematic study of the influence of decoding algorithms on complexity theory. It will examine a series of topics where a connection may prove to be fruitful. The research project will also examine new questions in coding theory influenced by the search for new tools in complexity theory. The most ambitious element of the project is the exploration of a coding theoretic approach to average case hardness of problems in NP. The search for average-case hard problems within NP is one of the fundamental quests of complexity theory. Existence of problems that are hard on the average is a necessary condition for cryptography. It also explains seeming contrast between worst-case hardness and empirically observed easiness of some optimization problems. Thus progress in this direction would be of great impact to computer science.