Goldreich-Levin Theorem, Hardcore Predicates and Probabilistic Public-Key Encryption

Error Correcting Codes and Hardcore Predicates Error correcting codes (ECC) play an important role in both complexity theory and cryptography. For our purposes let an ECC be a mapping C : {0, 1} → {0, 1} (more generally the source and target alphabets can be arbitrary finite sets), such that if a string y which is close to a valid encoding C(x) is given, then it is possible to reconstruct the message x from this “corrupted encoding” y. To achieve this, it is necessary and sufficient that any two encodings C(x1) and C(x2) differ in many coordinates. The main motivation for using the ECC is reliable sending of information over a noisy channel. One of the earliest applications of ECC in complexity theory – in order to prove an average-case complexity result, is contained in the paper by Levin [Lev87]. In this paper pseudorandom generators are constructed from certain one-way functions and a first step towards this is to build hardcore predicates for such functions. Later Goldreich and Levin [GL89] introduced an efficient and general way of constructing hardcore predicates. The Goldreich-Levin approach can be seen as a list-decoding algorithm for an ECC. The coding theory leads to the construction of different and probably more efficient hardcore predicates using various codes and decoding algorithms [GRS00].