Consider a data structure problem with possible data coming from a set D, queries coming from a set Q, and in the dynamic case updates coming from a set U. Then, the current state of the art in data structure lower bounds is t = Ω(log|Q|) for static data structure problems, and max(tq,tu) = Ω((logn)2) where n = max(|Q|,|U|,log|D|) for dynamic. We port Razborov and Rudich’s natural-proofs framework to the setting of static and dynamic data structures in the cell probe model, in a way that strongly suggests this state of the art is unlikely to be improved anytime soon. A similar direction was recently taken also by Korten, Pitassi and Impagliazzo (FOCS 2025) who look at static data structure lower bounds in a different regime of parameters. Our contribution is: We define notions analogous to pseudo-random functions (PRF). We call these primitives local PRFs, in the context of static data structures, and local and locally updatable (LLU) PRFs, in the context of dynamic data structures. We then formulate cryptographic conjectures, namely, that secure local PRFs and secure LLU PRFs exist, precisely at the frontier where we are no longer able to prove static, respectively dynamic, data structure lower bounds. If these conjectures are true, it follows that the current state of the art in data structure lower bounds cannot be improved by a natural proof. We show that (almost) every single known data structure lower bound proof is a natural proof, by surveying all lower bounds in the literature known to us. (The only exception is proofs based on lifting theorems.) It follows that, if our cryptographic conjecture is true, then all known lower bound proof techniques (minus the one exception) are unable to improve upon the state of the art. (We also attempt to address the exception.) Further, we provide concrete candidate constructions for our two pseudo-random primitives. We conjecture that our constructions are secure for parameters just above the state-of-the-art lower bounds. We also show that, whether or not they are secure, our candidate PRFs at least satisfy the natural properties appearing in all (but one) known proofs. So if one is interested in improving upon the state of the art in static or dynamic data structure lower bounds, one must either find a non-natural method of proving such lower bounds (no such method currently exists), or one may as well begin by trying to break our PRF candidates.

Rossman [In $\textit{Proc. $34$th Comput. Complexity Conf.}$, 2019] introduced the notion of $\textit{criticality}$. The criticality of a Boolean function $f : \{0,1\}^n \to \{0,1\}$ is the minimum $λ\geq 1$ such that for all positive integers $t$, \[ \Pr_{ρ\sim \mathcal{R}_p}\left[\text{DT}_{\text{depth}}(f|_ρ) \geq t\right] \leq (pλ)^t. \] Hästad's celebrated switching lemma shows that the criticality of any $k$-DNF is at most $O(k)$. Subsequent improvements to correlation bounds of $\text{AC}^0$-circuits against parity showed that the criticality of any $\text{AC}^0$-$\textit{circuit}$ of size $S$ and depth $d+1$ is at most $O(\log S)^d$ and any $\textit{regular}$ $\text{AC}^0$-$\textit{formula}$ of size $S$ and depth $d+1$ is at most $O\left(\frac1d \cdot \log S\right)^d$. We strengthen these results by showing that the criticality of $\textit{any}$ $\text{AC}^0$-formula (not necessarily regular) of size $S$ and depth $d+1$ is at most $O\left(\frac1d\cdot {\log S}\right)^d$, resolving a conjecture due to Rossman. This result also implies Rossman's optimal lower bound on the size of any depth-$d$ $\text{AC}^0$-formula computing parity [$\textit{Comput. Complexity, 27(2):209--223, 2018.}$]. Our result implies tight correlation bounds against parity, tight Fourier concentration results and improved $\#$SAT algorithm for $\text{AC}^0$-formulae.

Chang’s lemma (Duke Mathematical Journal, 2002) is a classical result in mathematics, with applications spanning across additive combinatorics, combinatorial number theory, analysis of Boolean functions, communication complexity and algorithm design. For a Boolean function f that takes values in {-1, 1} let r(f) denote its Fourier rank (i.e., the dimension of the span of its Fourier support). For each positive threshold t, Chang’s lemma provides a lower bound on δ(f):= Pr[f(x) = -1] in terms of the dimension of the span of its characters with Fourier coefficients of magnitude at least 1/t. In this work we examine the tightness of Chang’s lemma with respect to the following three natural settings of the threshold: the Fourier sparsity of f, denoted k(f), the Fourier max-supp-entropy of f, denoted k′(f), defined to be the maximum value of the reciprocal of the absolute value of a non-zero Fourier coefficient, the Fourier max-rank-entropy of f, denoted k′′(f), defined to be the minimum t such that characters whose coefficients are at least 1/t in magnitude span a r(f)-dimensional space. In this work we prove new lower bounds on δ(f) in terms of the above measures. One of our lower bounds, δ(f) = Ω (r(f)2/(k(f) log2 k(f))), subsumes and refines the previously best known upper bound r(f) = O(pk(f) log k(f)) on r(f) in terms of k(f) by Sanyal (Theory of Computing, 2019). We improve upon this bound and show r(f) = O(pk(f)δ(f) log k(f)). Another lower bound, δ(f) = Ω (r(f)/(k′′(f) log k(f))), is based on our improvement of a bound by Chattopadhyay, Hatami, Lovett and Tal (ITCS, 2019) on the sum of absolute values of level-1 Fourier coefficients in terms of F2-degree. We further show that Chang’s lemma for the above-mentioned choices of the threshold is asymptotically outperformed by our bounds for most settings of the parameters involved. Next, we show that our bounds are tight for a wide range of the parameters involved, by constructing functions witnessing their tightness. All the functions we construct are modifications of the Addressing function, where we replace certain input variables by suitable functions. Our final contribution is to construct Boolean functions f for which our lower bounds asymptotically match δ(f), and for any choice of the threshold t, the lower bound obtained from Chang’s lemma is asymptotically smaller than δ(f). Our results imply more refined deterministic one-way communication complexity upper bounds for XOR functions. Given the wide-ranging application of Chang’s lemma to areas like additive combinatorics, learning theory and communication complexity, we strongly feel that our refinements of Chang’s lemma will find many more applications.