We obtain new explicit pseudorandom generators for several computational models involving groups. Our main results are as follows: 1) We consider read-once group-products over a finite group G, i.e., tests of the form ∏_{i=1}^n (g_i)^{x_i} where g_i ∈ G, a special case of read-once permutation branching programs. We give generators with optimal seed length c_G log(n/ε) over any p-group. The proof uses the small-bias plus noise paradigm, but derandomizes the noise to avoid the recursion in previous work. Our generator works when the bits are read in any order. Previously for any non-commutative group the best seed length was ≥ log n log(1/ε), even for a fixed order. 2) We give a reduction that "lifts" suitable generators for group products over G to a generator that fools width-w block products, i.e., tests of the form ∏ (g_i)^{f_i} where the f_i are arbitrary functions on disjoint blocks of w bits. Block products generalize several previously studied classes. The reduction applies to groups that are mixing in a representation-theoretic sense that we identify. 3) Combining (2) with (1) and other works we obtain new generators for block products over the quaternions or over any commutative group, with nearly optimal seed length. In particular, we obtain generators for read-once polynomials modulo any fixed m with nearly optimal seed length. Previously this was known only for m = 2. 4) We give a new generator for products over "mixing groups." The construction departs from previous work and uses representation theory. For constant error, we obtain optimal seed length, improving on previous work (which applied to any group). This paper identifies a challenge in the area that is reminiscent of a roadblock in circuit complexity - handling composite moduli - and points to several classes of groups to be attacked next.

In the trace reconstruction problem , an unknown source string x ∈ {0,1} n is sent through a probabilistic deletion channel that independently deletes each bit with probability δ and concatenates the surviving bits, yielding a trace of x . The problem is to reconstruct x given independent traces. This problem has received much attention in recent years both in the worst-case setting where x may be an arbitrary string in {0,1} n [ 7 , 8 , 10 , 11 , 12 , 23 ] and in the average-case setting where x is drawn uniformly at random from {0,1} n [ 7 , 8 , 12 , 13 , 25 ]. This article studies trace reconstruction in the smoothed analysis setting, in which a “worst-case” string x worst is chosen arbitrarily from {0,1} n , and then a perturbed version x of x worst is formed by independently replacing each coordinate by a uniform random bit with probability σ. The problem is to reconstruct x given independent traces from it. Our main result is an algorithm that, for any constant perturbation rate 0< σ < 1 and any constant deletion rate 0 < δ < 1, uses poly( n ) running time and traces and succeeds with high probability in reconstructing the string x . This stands in contrast with the worst-case version of the problem, for which \(\text{exp}(\tilde{O}(n^{1/5}))\) is the best known time and sample complexity [ 8 ]. Our approach is based on reconstructing x from the multiset of its short subwords and is quite different from previous algorithms for either the worst-case or average-case versions of the problem. The heart of our work is a new poly( n )-time procedure for reconstructing the multiset of all O (log n )-length subwords of any source string x ∈ {0,1} n given access to traces of x .

The goal of trace reconstruction is to reconstruct an unknown n-bit string x given only independent random traces of x, where a random trace of x is obtained by passing x through a deletion channel. A Statistical Query (SQ) algorithm for trace reconstruction is an algorithm which can only access statistical information about the distribution of random traces of x rather than individual traces themselves. Such an algorithm is said to be 𝓁-local if each of its statistical queries corresponds to an 𝓁-junta function over some block of 𝓁 consecutive bits in the trace. Since several - but not all - known algorithms for trace reconstruction fall under the local statistical query paradigm, it is interesting to understand the abilities and limitations of local SQ algorithms for trace reconstruction. In this paper we establish nearly-matching upper and lower bounds on local Statistical Query algorithms for both worst-case and average-case trace reconstruction. For the worst-case problem, we show that there is an Õ(n^{1/5})-local SQ algorithm that makes all its queries with tolerance τ ≥ 2^{-Õ(n^{1/5})}, and also that any Õ(n^{1/5})-local SQ algorithm must make some query with tolerance τ ≤ 2^{-Ω̃(n^{1/5})}. For the average-case problem, we show that there is an O(log n)-local SQ algorithm that makes all its queries with tolerance τ ≥ 1/poly(n), and also that any O(log n)-local SQ algorithm must make some query with tolerance τ ≤ 1/poly(n).

We prove several new results about bounded uniform and small-bias distributions. A main message is that, small-bias, even perturbed with noise, does not fool several classes of tests better than bounded uniformity. We prove this for threshold tests, small-space algorithms, and small-depth circuits. In particular, we obtain small-bias distributions that - achieve an optimal lower bound on their statistical distance to any bounded-uniform distribution. This closes a line of research initiated by Alon, Goldreich, and Mansour in 2003, and improves on a result by O'Donnell and Zhao. - have heavier tail mass than the uniform distribution. This answers a question posed by several researchers including Bun and Steinke. - rule out a popular paradigm for constructing pseudorandom generators, originating in a 1989 work by Ajtai and Wigderson. This again answers a question raised by several researchers. For branching programs, our result matches a bound by Forbes and Kelley. Our small-bias distributions above are symmetric. We show that the xor of any two symmetric small-bias distributions fools any bounded function. Hence our examples cannot be extended to the xor of two small-bias distributions, another popular paradigm whose power remains unknown. We also generalize and simplify the proof of a result of Bazzi.

We study the communication complexity of multiplying$k\times t$elements from the group$H$= SL$(2, q)$in the number-on-forehead model with$k$parties. We prove a lower bound of$(t\log H)/c^{k}$. This is an exponential improvement over previous work, and matches the state-of-the-art in the area. Relatedly, we show that the convolution of$k^{c}$independent copies of a 3-uniform distribution over$H^{m}$is close to a$k$- uniform distribution. This is again an exponential improvement over previous work which needed$c^{k}$copies. The proofs are remarkably simple; the results extend to other quasirandom groups. We also show that for any group$LI$, any distribution over$H^{m}$whose weight-k Fourier coefficients are small is close to a k-uniform distribution. This generalizes previous work in the abelian setting, and the proof is simpler.

For Boolean functions computed by read-once, depth-$D$ circuits with unbounded fan-in over the de Morgan basis, we present an explicit pseudorandom generator with seed length $\tilde{O}(\log^{D+1} n)$. The previous best seed length known for this model was $\tilde{O}(\log^{D+4} n)$, obtained by Trevisan and Xue (CCC `13) for all of $AC^0$ (not just read-once). Our work makes use of Fourier analytic techniques for pseudorandomness introduced by Reingold, Steinke, and Vadhan (RANDOM `13) to show that the generator of Gopalan et al. (FOCS `12) fools read-once $AC^0$. To this end, we prove a new Fourier growth bound for read-once circuits, namely that for every $F: \{0,1\}^n\to\{0,1\}$ computed by a read-once, depth-$D$ circuit, \begin{equation*}\sum_{s\subseteq[n], |s|=k}|\hat{F}[s]|\le O(\log^{D-1}n)^k,\end{equation*} where $\hat{F}$ denotes the Fourier transform of $F$ over $\mathbb{Z}^n_2$.

The well-known trace reconstruction problem is the problem of inferring an unknown source string x ∈ {0,1}n from independent “traces”, i.e. copies of x that have been corrupted by a δ-deletion channel which independently deletes each bit of x with probability δ and concatenates the surviving bits. The current paper considers the extreme data-limited regime in which only a single trace is provided to the reconstruction algorithm. In this setting exact reconstruction is of course impossible, and the question is to what accuracy the source string x can be approximately reconstructed.

We study query-to-communication lifting. The major open problem in this area is to prove a lifting theorem for gadgets of constant size. The recent paper [Paul Beame and Sajin Koroth, 2023] introduces semi-structured communication complexity, in which one of the players can only send parities of their input bits. They have shown that for any m ≥ 4 deterministic decision tree complexity of a function f can be lifted to the so called semi-structured communication complexity of f∘Ind_m, where Ind_m is the Indexing gadget. As our main contribution we extend these results to randomized setting. Our results also apply to a substantially larger set of gadgets. More specifically, we introduce a new complexity measure of gadgets, linear diversity. For all gadgets g with non-trivial linear diversity we show that randomized decision tree complexity of f lifts to randomized semi-structured communication complexity of f∘g. In particular, this gives tight lifting results for Indexing gadget Ind_m, Inner Product gadget IP_m for all m ≥ 2, and for Majority gadget MAJ_m for all m ≥ 4. We prove the same results for deterministic case. From our result it immediately follows that deterministic/randomized decision tree complexity lifts to deterministic/randomized parity decision tree complexity. For randomized case this is the first result of this type. For deterministic case, our result improves the bound in [Arkadev Chattopadhyay et al., 2023] for Inner Product gadget. To obtain our results we introduce a new secret sets approach to simulation of semi-structured communication protocols by decision trees. It allows us to simulate (restricted classes of) communication protocols on truly uniform distribution of inputs.

In the standard trace reconstruction problem, the goal is to exactly reconstruct an unknown source string x ∊ {0, 1}n from independent “traces”, which are copies of x that have been corrupted by a δ-deletion channel which independently deletes each bit of x with probability δ and concatenates the surviving bits. We study the approximate trace reconstruction problem, in which the goal is only to obtain a high-accuracy approximation of x rather than an exact reconstruction. We give an efficient algorithm, and a near-matching lower bound, for approximate reconstruction of a random source string x ∊ {0, 1}n from few traces. Our main algorithmic result is a polynomial-time algorithm with the following property: for any deletion rate 0 < δ < 1 (which may depend on n), for almost every source string x ∊ {0,1}n, given any number M ≤ Θ(1/δ) of traces from Delδ(x), the algorithm constructs a hypothesis string that has edit distance at most n · (δM)Ω(M) from x. We also prove a near-matching information-theoretic lower bound showing that given M ≤ Θ(1/δ) traces from Delδ(x) for a random n-bit string x, the smallest possible expected edit distance that any algorithm can achieve, regardless of its running time, is n · (δM)O(M).

We prove new results on the polarizing random walk framework introduced in recent works of Chattopadhyay et al. [Chattopadhyay et al., 2019; Eshan Chattopadhyay et al., 2019] that exploit L₁ Fourier tail bounds for classes of Boolean functions to construct pseudorandom generators (PRGs). We show that given a bound on the k-th level of the Fourier spectrum, one can construct a PRG with a seed length whose quality scales with k. This interpolates previous works, which either require Fourier bounds on all levels [Chattopadhyay et al., 2019], or have polynomial dependence on the error parameter in the seed length [Eshan Chattopadhyay et al., 2019], and thus answers an open question in [Eshan Chattopadhyay et al., 2019]. As an example, we show that for polynomial error, Fourier bounds on the first O(log n) levels is sufficient to recover the seed length in [Chattopadhyay et al., 2019], which requires bounds on the entire tail. We obtain our results by an alternate analysis of fractional PRGs using Taylor’s theorem and bounding the degree-k Lagrange remainder term using multilinearity and random restrictions. Interestingly, our analysis relies only on the level-k unsigned Fourier sum, which is potentially a much smaller quantity than the L₁ notion in previous works. By generalizing a connection established in [Chattopadhyay et al., 2020], we give a new reduction from constructing PRGs to proving correlation bounds. Finally, using these improvements we show how to obtain a PRG for 𝔽₂ polynomials with seed length close to the state-of-the-art construction due to Viola [Emanuele Viola, 2009].

In the \emph{trace reconstruction problem}, an unknown source string $x \in \{0,1\}^n$ is transmitted through a probabilistic \emph{deletion channel} which independently deletes each bit with some fixed probability $δ$ and concatenates the surviving bits, resulting in a \emph{trace} of $x$. The problem is to reconstruct $x$ given access to independent traces. Trace reconstruction of arbitrary (worst-case) strings is a challenging problem, with the current state of the art for poly$(n)$-time algorithms being the 2004 algorithm of Batu et al. \cite{BKKM04}. This algorithm can reconstruct an arbitrary source string $x \in \{0,1\}^n$ in poly$(n)$ time provided that the deletion rate $δ$ satisfies $δ\leq n^{-(1/2 + \varepsilon)}$ for some $\varepsilon > 0$. In this work we improve on the result of \cite{BKKM04} by giving a poly$(n)$-time algorithm for trace reconstruction for any deletion rate $δ\leq n^{-(1/3 + \varepsilon)}$. Our algorithm works by alternating an alignment-based procedure, which we show effectively reconstructs portions of the source string that are not "highly repetitive", with a novel procedure that efficiently determines the length of highly repetitive subwords of the source string.

In the trace reconstruction problem, an unknown source string x ∊ {0, 1}n is sent through a probabilistic deletion channel which independently deletes each bit with probability δ and concatenates the surviving bits, yielding a trace of x. The problem is to reconstruct x given independent traces. This problem has received much attention in recent years both in the worst-case setting where x may be an arbitrary string in {0, 1}n [6, 19, 7, 8, 4] and in the average-case setting where x is drawn uniformly at random from {0, 1}n [21, 9, 8, 4]. This paper studies trace reconstruction in the smoothed analysis setting, in which a “worst-case” string xworst is chosen arbitrarily from {0, 1}n, and then a perturbed version x of xworst is formed by independently replacing each coordinate by a uniform random bit with probability σ. The problem is to reconstruct x given independent traces from it. Our main result is an algorithm which, for any constant perturbation rate 0 < σ < 1 and any constant deletion rate 0 < δ < 1, uses poly(n) running time and traces and succeeds with high probability in reconstructing the string x. This stands in contrast with the worst-case version of the problem, for which the best known sample complexity is exp(Õ(n1/5)) [5], a recent improvement on exp(O(n1/3)) [6, 19]. Our approach is based on reconstructing x from the multiset of its short subwords and is quite different from previous algorithms for either the worst-case or average-case versions of the problem. The heart of our work is a new poly(n)-time procedure for reconstructing the multiset of all O(log n)-length subwords of any source string x ∊ {0, 1}n given access to traces of x.

We study the Fourier spectrum of functions f : {0,1}^{mk} -> {-1,0,1} which can be written as a product of k Boolean functions f_i on disjoint m-bit inputs. We prove that for every positive integer d, sum_{S subseteq [mk]: |S|=d} |hat{f_S}| = O(min{m, sqrt{m log(2k)}})^d . Our upper bounds are tight up to a constant factor in the O(*). Our proof uses Schur-convexity, and builds on a new "level-d inequality" that bounds above sum_{|S|=d} hat{f_S}^2 for any [0,1]-valued function f in terms of its expectation, which may be of independent interest. As a result, we construct pseudorandom generators for such functions with seed length O~(m + log(k/epsilon)), which is optimal up to polynomial factors in log m, log log k and log log(1/epsilon). Our generator in particular works for the well-studied class of combinatorial rectangles, where in addition we allow the bits to be read in any order. Even for this special case, previous generators have an extra O~(log(1/epsilon)) factor in their seed lengths. We also extend our results to functions f_i whose range is [-1,1].

Let $D$ be a $b$-wise independent distribution over $\{0,1\}^m$. Let $E$ be the “noise” distribution over $\{0,1\}^m$ where the bits are independent and each bit is 1 with probability $\eta/2$. We study which tests $f \colon \{0,1\}^m \to [-1,1]$ are $\varepsilon$-fooled by $D+E$, i.e., $|{\rm E}[f(D+E)] - {\rm E}[f(U)]| \le \varepsilon$, where $U$ is the uniform distribution. We show that $D+E$ $\varepsilon$-fools product tests $f : (\{0,1\}^n)^k \to [-1,1]$ given by the product of $k$ bounded functions on disjoint $n$-bit inputs with error $\varepsilon = k(1-\eta)^{\Omega(b^2/m)}$, where $m = nk$ and $b \ge n$. This bound is tight when $b = \Omega(m)$ and $\eta \ge (\log k)/m$. For $b \ge m^{2/3} \log m$ and any constant $\eta$ the distribution $D+E$ also $0.1$-fools log-space algorithms. We develop two applications of this type of results. First, we prove communication lower bounds for decoding noisy codewords of length $m$ split among $k$ parties. For Reed--Solomon codes of dimension $m/k$ where $k = O(1)$, communication $\Omega(\eta m) - O(\log m)$ is required to decode one message symbol from a codeword with $\eta m$ errors, and communication $O(\eta m \log m)$ suffices. Second, we obtain pseudorandom generators. We can $\varepsilon$-fool product tests $f\colon (\{0,1\}^n)^k \to [-1,1]$ under any permutation of the bits with seed lengths $2n + \tilde O(k^2 \log (1/\varepsilon))$ and $O(n) + \tilde O(\sqrt{nk \log 1/\varepsilon})$. Previous generators have seed lengths $\ge nk/2$ or $\ge n \sqrt{n k}$. For the special case where the $k$ bounded functions have range $\{0,1\}$ the previous generators have seed length $\ge (n+\log k)\log (1/\varepsilon)$.

Let k = k(n) be the largest integer such that there exists a k-wise uniform distribution over {0,1}^n that is supported on the set S_m := {x in {0,1}^n: sum_i x_i equiv 0 mod m}, where m is any integer. We show that Omega(n/m^2 log m) <= k <= 2n/m + 2. For k = O(n/m) we also show that any k-wise uniform distribution puts probability mass at most 1/m + 1/100 over S_m. For any fixed odd m there is k \ge (1 - Omega(1))n such that any k-wise uniform distribution lands in S_m with probability exponentially close to |S_m|/2^n; and this result is false for any even m.