We study systems of linear equations modulo two in $n$ variables
with three variables in each equation. We assume that the system has
a solution with $pn$ variables taking the value 1 for some value
$00$ it is hard to find a solution
of the same weight that satisfies at ... more >>>

We introduce the SHEDAG (Somewhere Honest Entropic sources over Directed Acyclic Graphs) source model, a general model for multi-block randomness sources with causal correlations.
A SHEDAG source is defined over a directed acyclic graph (DAG) $G$ whose nodes output $n$-bit blocks. Blocks output by honest nodes are independent (by ... more >>>

This paper explores the previously studied measure called block number of Boolean functions, that counts the maximum possible number of minimal sensitive blocks for any input. We present close to tight upper bounds on the block number in terms of the function’s sensitivity and the allowed block size, improving previous ... more >>>

In this short expository note, we provide an introduction to a distribution testing (and, more generally, indistinguishability) lower bound method based on moment-matching via polynomials. This method, which underlies several of the tight lower bounds on estimating symmetric properties, had for many years appeared mysterious and near-magical to the ... more >>>

Proving super-linear lower bounds on the size of circuits computing explicit linear functions $A:{\mathbb {F}}^n \to {\mathbb {F}}^n$ is a fundamental long-standing open problem in circuit complexity. We focus on the case where ${\mathbb {F}}$ is a finite field. The circuit can be either a Boolean circuit or an arithmetic ... more >>>

Res($\oplus$) is the simplest fragment of $\text{AC}^0[2]\text{-Frege}$ for which no super-polynomial lower bounds on the size of proofs are known. Bhattacharya and Chattopadhyay [BC25] recently proved lower bounds of the form $\exp(\tilde\Omega(N^{\varepsilon}))$ on the size of Res($\oplus$) proofs whose depth is upper bounded by $O(N^{2 - \varepsilon})$, where $N$ is ... more >>>

We prove that relative to a random oracle answering $O(\log n)$-bit queries, there exists a function computable in $O(n)$ time by a random-access machine (RAM) but requiring $n^2/polylog(n)$ time by any multitape Turing machine. This provides strong evidence that simulating RAMs on multitape Turing machines inherently incurs a nearly quadratic ... more >>>

We prove that $\mathrm{deg}(f) \leq 2 \, \mathrm{rdeg}(f)^4$ for every Boolean function $f$, where $\mathrm{deg}(f)$ is the degree of $f$ and $\mathrm{rdeg}(f)$ is the rational degree of $f$. This resolves the second of the three open problems stated by Nisan and Szegedy, and attributed to Fortnow, in 1994.

We present a new, simplified proof that the complexity class BPP is contained in the Polynomial Hierarchy (PH), using $k$-wise independent hashing as the main tool. We further extend this approach to recover several other previously known inclusions between complexity classes. Our techniques are inspired by the work of Bellare, ... more >>>

Let $g(X)$ be a polynomial over a finite field ${\mathbb F}_q$ with degree $o(q^{1/2})$, and let $\chi$ be the quadratic residue character. We give a polynomial time algorithm to recover $g(X)$ (up to perfect square factors) given the values of $\chi \circ g$ on ${\mathbb F}_q$, with up to a ... more >>>

In this work, we establish separation theorems for several subsystems of the Ideal Proof System (IPS), an algebraic proof system introduced by Grochow and Pitassi (J. ACM, 2018). Separation theorems are well-studied in the context of classical complexity theory, Boolean circuit complexity, and algebraic complexity.

In an important work ... more >>>

It is a long-standing open problem in algebraic complexity to prove lower bounds against multilinear algebraic branching programs (mABPs). The best lower bounds in this setting are still quadratic (Alon, Kumar and Volk (Combinatorica 2020)). At the same time, it remains a possibility that the “min-partition rank” method introduced by ... more >>>