On the impossibility of entropy reversal, and its application to zero-knowledge proofs

Zero knowledge proof systems have been widely studied in cryptography. In the statistical setting, two classes of proof systems studied are Statistical Zero Knowledge (SZK) and Non-Interactive Statistical Zero Knowledge (NISZK), where the difference is that in NISZK only very limited communication is allowed between the verifier and the prover. It is an open problem whether these two classes are in fact equal. In this paper, we rule out efficient black box reductions between SZK and NISZK. We achieve this by studying algorithms which can reverse the entropy of a function. The problem of estimating the entropy of a circuit is complete for NISZK. Hence, reversing the entropy of a function is equivalent to a black box reduction of NISZK to its complement, which is known to be equivalent to a black box reduction of SZK to NISZK [Goldreich et al, CRYPTO 1999]. We show that any such black box algorithm incurs an exponential loss of parameters, and hence cannot be implemented efficiently

Discrete Fourier Analysis and Its Applications

The topic of discrete Fourier analysis has been extensively studied in recent decades. It plays an important role in theoretical computer science and discrete mathematics. One hand it is interesting to study the structure of boolean functions via discrete Fourier analysis. On the other hand, these structural results also provide a huge number of applications in theoretical computer science, including computational complexity, pseudorandomness, cryptography, learning theory. In this dissertation, we extend some more connections between discrete Fourier analysis and theoretical computer science. In particular, we study the following questions.\begin{itemize}\item Robust sensitivity of boolean function. In this part, we study the connection between the Fourier tail bound and the sensitivity tail bound of boolean functions, which is an analogue of the sensitivity conjecture, which was proposed by Nisan \cite{nisan1991crew}.\item DNF sparsification. The disjunctive normal form (or DNF) is a widely used representation of boolean functions. It is very interesting to study the structure of DNFs. There are two natural ways to measure the complexity of DNFs, the width and the size. In this thesis, we study a connection between these two measures. We propose a new approach by combing the swithing lemma (a combinatoric tool) and the hypercontrativity inequality (an analytic inequality). This framework does also suggest a new approach to the famous sunflower conjecture.\item Applications in learning theory. In 1989, the first Fourier-based learning algorithms was introduced by a seminar paper of Linial, Mansour and Nisan \cite{linial1989constant}. Followed by a series of subsequent works, people found that discrete Fourier analysis is powerful to design learning algorithms. One hand sparse Fourier functions are strong enough to approximate a lot of functions, on the other hand sparse Fourier functions are relatively easy to learn. Build on this framework, we give a more efficient algorithm to solve the \emph{population recovery} problem. That is how to recover a unknown distribution from noisy samples.\end{itemize