Quadratic Goldreich--Levin Theorems

Lecture 8 : Goldreich - Levin Theorem ( continued )

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...

A Quantum Goldreich-Levin Theorem with Cryptographic Applications

We investigate the Goldreich-Levin Theorem in the context of quantum information. This result is a reduction from the computational problem of inverting a one-way function to the problem of predicting a particular bit associated with that function. We show that the quantum version of the reduction—between quantum one-way functions and quantum hard-predicates—is quantitatively more efficient tha...

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...

Quantum lower bounds for the Goldreich-Levin problem

At the heart of the Goldreich–Levin theorem is the problem of determining an n-bit string a by making queries to two oracles, referred to as IP (inner product) and EQ (equivalence). The IP oracle, on input x, returns a bit that is biased towards a · x (the modulo two inner product of a with x) in the following sense. For a random x, the probability that IP(x)= a ·x is at least 2 (1+ ε). The EQ ...