A Convex Analysis Approach to Computational Entropy

The entropy accumulation theorem, and its subsequent generalized version, is a powerful tool in the security analysis of many device-dependent and device-independent cryptography protocols. However, it has the drawback that the finite-size bounds it yields are not necessarily optimal, and furthermore it relies on the construction of an affine min-tradeoff function, which can often be challenging to construct optimally in practice. In this work, we address both of these challe...

The information complexity of a function $f$ is the minimum amount of information Alice and Bob need to exchange to compute the function $f$. In this paper we provide an algorithm for approximating the information complexity of an arbitrary function $f$ to within any additive error $\alpha > 0$, thus resolving an open question as to whether information complexity is computable. In the process, we give the first explicit upper bound on the rate of convergence of the informat...

These lecture notes are an informal introduction to the theory of computational complexity and its links to quantum computing and statistical mechanics.

This research applies concepts from algorithmic probability to Boolean and quantum combinatorial logic circuits. A tutorial-style introduction to states and various notions of the complexity of states are presented. Thereafter, the probability of states in the circuit model of computation is defined. Classical and quantum gate sets are compared to select some characteristic sets. The reachability and expressibility in a space-time-bounded setting for these gate sets are enume...

Researchers have proposed formal definitions of quantitative information flow based on information theoretic notions such as the Shannon entropy, the min entropy, the guessing entropy, belief, and channel capacity. This paper investigates the hardness of precisely checking the quantitative information flow of a program according to such definitions. More precisely, we study the "bounding problem" of quantitative information flow, defined as follows: Given a program M and a po...

We study the space of functions computed by random-layered machines, including deep neural networks and Boolean circuits. Investigating the distribution of Boolean functions computed on the recurrent and layer-dependent architectures, we find that it is the same in both models. Depending on the initial conditions and computing elements used, we characterize the space of functions computed at the large depth limit and show that the macroscopic entropy of Boolean functions is e...

The decision problem version of estimating the Shannon entropy is the Entropy Difference problem (ED): given descriptions of two circuits, determine which circuit produces more entropy in its output when acting on a uniformly random input. The analogous problem with quantum circuits (QED) is to determine which circuit produces the state with greater von Neumann entropy, when acting on a fixed input state and after tracing out part of the output. Based on plausible complexity-...

More than a speculative technology, quantum computing seems to challenge our most basic intuitions about how the physical world should behave. In this thesis I show that, while some intuitions from classical computer science must be jettisoned in the light of modern physics, many others emerge nearly unscathed; and I use powerful tools from computational complexity theory to help determine which are which.

This manuscript includes some classical results we select apart from the new results we've found on the Analysis of Boolean Functions and Fourier-Entropy-Influence conjecture. We try to ensure the self-completeness of this work so that readers could probably read it independently. Among the new results, what is the most remarkable is that we prove that the entropy of a boolean function $f$ could be upper bounded by $O(I(f))+O(\sum\limits_{k}I_k(f)\log (1/I_k(f)))$.

This paper summarizes recent contributions of the authors and their co-workers in the area of information-theoretic security.