On one Leontiev-Levin theorem

Lectures 11–12 - One Way Permutations, Goldreich Levin Theorem, Commitments

Proof: Just pick g at random. For every particular 2 √ n-time algorithm A, the expected number of inputs on which A(x) = g(x) is one, and the probability that A computes g successfully on an at least 2−n/10 fraction of the total 2n inputs can be shown to be less than 2−2 −n/2 . But a 2 √ n algorithm can be described by about 2 √ n 2n/2 bits and so the total number of such algorithms is much sma...

Cs787: Advanced Algorithms 9.1 Cook-levin Theorem

As we’ve already seen in the preceding lecture, two important classes of problems we can consider are P and NP . One very prominent open question is whether or not these two classes of problems are in fact equal. One tool that proves useful when considering this question is the concept of a problem being complete for a class. In the first two portions of this lecture, we show two problems to be...

Lecture 8 : Goldreich - Levin Theorem ( continued )

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 m...

A Mechanical Proof of the Cook-Levin Theorem

As is the case with many theorems in complexity theory, typical proofs of the celebrated Cook-Levin theorem showing the NPcompleteness of satisfiability are based on a clever construction. The Cook-Levin theorem is proved by carefully translating a possible computation of a Turing machine into a boolean expression. As the boolean expression is built, it is “obvious” that it can be satisfied if ...