Sunflower Theorems in Monotone Circuit Complexity

Alexander Razborov (1985) developed the approximation method to obtain lower bounds on the size of monotone circuits deciding if a graph contains a clique. Given a "small" circuit, this technique consists in finding a monotone Boolean function which approximates the circuit in a distribution of interest, but makes computation errors in that same distribution. To prove that such a function is indeed a good approximation, Razborov used the sunflower lemma of Erd\H{o}s and Rado (1960). This technique was improved by Alon and Boppana (1987) to show lower bounds for a larger class of monotone computational problems. In that same work, the authors also improved the result of Razborov for the clique problem, using a relaxed variant of sunflowers. More recently, Rossman (2010) developed another variant of sunflowers, now called "robust sunflowers", to obtain lower bounds for the clique problem in random graphs. In the following years, the concept of robust sunflowers found applications in many areas of computational complexity, such as DNF sparsification, randomness extractors and lifting theorems. Even more recent was the breakthrough result of Alweiss, Lovett, Wu and Zhang (2020), which improved Rossman's bound on the size of hypergraphs without robust sunflowers. This result was employed to obtain a significant progress on the sunflower conjecture. In this work, we will show how the recent progress in sunflower theorems can be applied to improve monotone circuit lower bounds. In particular, we will show the best monotone circuit lower bound obtained up to now, breaking a 20-year old record of Harnik and Raz (2000). We will also improve the lower bound of Alon and Boppana for the clique function in a slightly more restricted range of clique sizes. Our exposition is self-contained. These results were obtained in a collaboration with Benjamin Rossman and Mrinal Kumar.

Decentralized Privacy Protection Framework Based on Attribute-Based Signcryption

In the last decade, the digital world has widely changed how we handle our daily actions on the internet. India has the second-largest internet users in the globe with about 370 million users. The Indian business market has understood the significance of the threats in managing digital data. This has led to the increased allocation of budgets for the implementation of security to protect data from breaches. The data security has been constantly improvised and the existing systems include various schemes focusing on central authority which must be fully trusted and is responsible for both attribute management and key generation. This system introduces multi-authority using cipher-text instead of key-policy ABSC scheme without destroying the attribute security. Since, the mechanism to verify does not require plaintext. It accepts any Monotone Boolean Function. Finally,a partial decryption algorithm is used to outsource operations to the cloud to save computational resources.

Monotone Boolean Functions with s Zeros Farthest from Threshold Functions

International audience Let $T_t$ denote the $t$-threshold function on the $n$-cube: $T_t(x) = 1$ if $|\{i : x_i=1\}| \geq t$, and $0$ otherwise. Define the distance between Boolean functions $g$ and $h$, $d(g,h)$, to be the number of points on which $g$ and $h$ disagree. We consider the following extremal problem: Over a monotone Boolean function $g$ on the $n$-cube with $s$ zeros, what is the maximum of $d(g,T_t)$? We show that the following monotone function $p_s$ maximizes the distance: For $x \in \{0,1\}^n$, $p_s(x)=0$ if and only if $N(x) < s$, where $N(x)$ is the integer whose $n$-bit binary representation is $x$. Our result generalizes the previous work for the case $t=\lceil n/2 \rceil$ and $s=2^{n-1}$ by Blum, Burch, and Langford [BBL98-FOCS98], who considered the problem to analyze the behavior of a learning algorithm for monotone Boolean functions, and the previous work for the same $t$ and $s$ by Amano and Maruoka [AM02-ALT02].