Proving hard-core predicates using list decoding

We introduce a unifying framework for proving that predicate P is hard-core for a one-way function f, and apply it to a broad family of functions and predicates, reproving old results in an entirely different way as well as showing new hard-core predicates for well known one-way function candidates. Our framework extends the list-coding method of Goldreich and Levin for showing hard-core predicates. Namely, a predicate will correspond to some error correcting code, predicting a predicate will correspond to access to a corrupted codeword, and the task of inverting one-way functions will correspond to the task of list decoding a corrupted codeword. A characteristic of the error correcting codes which emerge and are addressed by our framework is that codewords can be approximated by a small number of heavy coefficients in their Fourier representation. Moreover, as long as corrupted words are close enough to legal codewords, they will share a heavy Fourier coefficient. We list decodes, by devising a learning algorithm applied to corrupted codewords for learning heavy Fourier coefficients. For codes defined over {0, 1}ⁿ domain, a learning algorithm by Kushilevitz and Mansour already exists. For codes defined over Z_N, which are the codes which emerge for predicates based on number theoretic one-way functions such as the RSA and Exponentiation modulo primes, we develop a new learning algorithm. This latter algorithm may be of independent interest outside the realm of hard-core predicates.