It is shown that there exists f:{0,1}^n/2×{0,1}^n/2→{0,1} in E ^NP such that for every 2^n/2×2^n/2 matrix M of rank ≤ρ we have ℙ_x,y[f(x,y)M_x,y]≥1/2-2^-Ω(k) , whenever logρ≤δn/k(logn+k) for a sufficiently small δ>0 , and n is large enough. This generalizes recent results which bound below the probability by 1/2-Ω(1) or apply to constant-depth circuits.

I give a very simple, apparently new proof of a tight communication lower bound for pointer chasing.

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 Pi (n) (i=1) g (xi) (i) where g(i) is an element of G, a special case of read-once permutation branching programs. We give generators with optimal seed length c(G) log( n/epsilon) 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/epsilon), 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 Pi g(fi) (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.

We study the communication complexity of multiplying kxt elements from the group H = SL(2, q) in the number-on-forehead model with k parties. We prove a lower bound of (t logH)/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 H, 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.

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 1) 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. 2) have heavier tail mass than the uniform distribution. This answers a question posed by several researchers including Bun and Steinke. 3) 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.

An n-bit boolean function is resilient to coalitions of size q if any fixed set of q bits is unlikely to influence the function when the other n-q bits are chosen uniformly. We give explicit constructions of depth-3 circuits that are resilient to coalitions of size cn/log^2n with bias n^-c. Previous explicit constructions with the same resilience had constant bias. Our construction is simpler and we generalize it to biased product distributions. Our proof builds on previous work; the main differences are the use of a tail bound for expander walks in combination with a refined analysis based on Janson's inequality.

Let $G$ be a group such that any non-trivial representation has dimension at least $d$. Let $X=(X_{1},X_{2},\ldots,X_{t})$ and $Y=(Y_{1},Y_{2},\ldots,Y_{t})$ be distributions over $G^{t}$. Suppose that $X$ is independent from $Y$. We show that for any $g\in G$ we have $|\mathbb{P}[X_{1}Y_{1}X_{2}Y_{2}\cdots X_{t}Y_{t}=g]-1/|G||\le\frac{|G|^{2t-1}}{d^{t-1}}\sqrt{\mathbb{E}_{h\in G^{t}}X(h)^{2}}\sqrt{\mathbb{E}_{h\in G^{t}}Y(h)^{2}}.$ Our results generalize, improve, and simplify previous works.

An n-bit boolean function is resilient to coalitions of size q if no fixed set of q bits is likely to influence the value of the function when the other n - q bits are chosen uniformly at random, even though the function is nearly balanced. We construct explicit functions resilient to coalitions of size q = n/(log n)(O(log log n)) = n(1-o)(1) computable by linear-size circuits and linear-time algorithms. We also obtain a tight size-depth tradeoff for computing such resilient functions. Constructions such as ours were not available even non-explicitly. It was known that functions resilient to coalitions of size q = n(0.63)... can be computed by linear-size circuits [BL85], and functions resilient to coalitions of size q = Theta(n/log(2) n) can be computed by quadratic-size circuits [AL93]. One component of our proofs is a new composition theorem for resilient functions.

We study the fundamental challenge of exhibiting explicit functions that have small correlation with low-degree polynomials over $\mathbb{F}_{2}$. Our main contributions include: 1. In STOC 2020, CHHLZ introduced a new technique to prove correlation bounds. Using their technique they established new correlation bounds for low-degree polynomials. They conjectured that their technique generalizes to higher degree polynomials as well. We give a counterexample to their conjecture, in fact ruling out weaker parameters and showing what they prove is essentially the best possible. 2. We propose a new approach for proving correlation bounds with the central "mod functions", consisting of two steps: (I) the polynomials that maximize correlation are symmetric and (II) symmetric polynomials have small correlation. Contrary to related results in the literature, we conjecture that (I) is true. We argue this approach is not affected by existing "barrier results". 3. We prove our conjecture for quadratic polynomials. Specifically, we determine the maximum possible correlation between quadratic polynomials modulo 2 and the functions $(x_{1},\dots,x_{n})\to z^{\sum x_{i}}$ for any $z$ on the complex unit circle; and show that it is achieved by symmetric polynomials. To obtain our results we develop a new proof technique: we express correlation in terms of directional derivatives and analyze it by slowly restricting the direction. 4. We make partial progress on the conjecture for cubic polynomials, in particular proving tight correlation bounds for cubic polynomials whose degree-3 part is symmetric.

Quasirandom groups enjoy interleaved mixing, Discrete Analysis 2023:14, 4 pp. In 1985 Babai and Sós asked whether there is a constant $c>0$ such that every group of order $n>1$ has a product-free subset of size at least $cn$, where this means a set $A$ such that it is not possible to find $a,b,c\in A$ with $ab=c$. A fairly straightforward argument shows that if the group is Abelian, then the answer is yes with $c=3/7$. (The constant comes from the fact that if the group is cyclic and $n$ is even then the odd elements form a sum-free set of size $n/2$, while for odd $n$ one can check that a suitable "middle third" of elements gives an example of size $\lfloor(n+1)/3\rfloor$. These examples are best possible, and the smallest ratio happens to occur when $n=7$.) However, for general groups there is no obvious construction, and in a 2008 paper Gowers showed that the answer is in general negative. More precisely, he defined a $d$-_quasirandom group_ to be a group with no non-trivial representation of dimension less than $d$, and proved that the largest product-free subset of a $d$-quasirandom group has cardinality $O(nd^{-1/3})$. This, combined with the fact that $d$-quasirandom groups exist with arbitrarily large $d$ (in fact, the group $\text{PSL}_2(q)$, which is of order roughly $q^3$, is $q$-quasirandom, so one can take $d$ to be of order of magnitude $n^{1/3}$), gives the negative answer in a strong form. In a later paper, motivated by an application to cryptography, Gowers and Viola (the second author of this paper) obtained an extension of the above result for the group $G=\text{SL}_2(q)$, and with less good bounds for all non-Abelian simple groups, which are quasirandom. (The bounds in the latter case were improved by Shalev.) In qualitative terms, their result stated that if $t$ is a fixed positive integer and $A$ and $B$ are large subsets of $G^t$, and if we pick $(a_1,\dots,a_t)$ and $(b_1,\dots,b_t)$ uniformly and independently at random from $A^t$ and $B^t$, respectively, then the "interleaved product" $a_1b_1a_2b_2\dots a_tb_t$ is approximately uniformly distributed in the strong sense that the probability that for every $x\in G$ the probability that it takes the value $x$ is approximately $1/|G|$. The proof was somewhat complicated, as it required a detailed analysis of the conjugacy classes of $\text{SL}_2(q)$ and a use of the Lang-Weil theorem. In this paper, the authors give a much simpler proof, with the added benefit that it applies not just to $\text{SL}_2(q)$ and to non-Abelian simple groups, but to all quasirandom groups, with a good dependence on the quasirandomness parameter in all cases. There have been various other cases of results first established for specific quasirandom groups, and only later proved for all quasirandom groups, usually with simpler arguments. It is a pleasant surprise each time this happens. The argument in this paper is a particularly good example of the phenomenon: indeed, it seems appropriate to say that it is the "book proof" of the result.

Suppose that a target distribution can be approximately sampled by a low-depth decision tree, or more generally by an efficient cell-probe algorithm. It is shown to be possible to restrict the input to the sampler so that its output distribution is still not too far from the target distribution

We exhibit a pseudorandom generator with nearly quadratic stretch for randomized Turing machines, which have a one-way random tape and a two-way work tape. This is the first generator for this model. Its stretch is essentially the best possible given current lower bounds. We use the generator to prove a time lower bound in the above Turing machine model extended with a two-way read-only input tape. The lower bound is of the form n1+s1(1) and is for a function computable in linear time with two quantifier alternations. Previously lower bounds were not known even for functions computable in simply exponential time.

We prove that the OR function on {-1,1\} n can be pointwise approximated with error ε by a polynomial of degree O ( k ) and weight 2 O ( n log (1/ε)/k) , for any k ≥ √ n log (1/ε). This result is tight for any k ≤ (1-Ω (1)) n . Previous results were either not tight or had ε = Ω (1). In general, we obtain a tight approximate degree-weight result for any symmetric function. Building on this, we also obtain an approximate degree-weight result for bounded-width CNF. For these two classes no such result was known. We prove that the \( \mathsf {OR} \) function on \( \lbrace -1,1\rbrace ^n \) can be pointwise approximated with error \( \epsilon \) by a polynomial of degree \( O(k) \) and weight \( 2^{O(n \log (1/\epsilon) /k)} \) , for any \( k \ge \sqrt {n \log (1/\epsilon)} \) . This result is tight for any \( k \le (1-\Omega (1))n \) . Previous results were either not tight or had \( \epsilon = \Omega (1) \) . In general, we obtain a tight approximate degree-weight result for any symmetric function. Building on this, we also obtain an approximate degree-weight result for bounded-width \( \mathsf {CNF} \) . For these two classes no such result was known. One motivation for such results comes from the study of indistinguishability. Two distributions \( P \) , \( Q \) over \( n \) -bit strings are \( (k,\delta) \) -indistinguishable if their projections on any \( k \) bits have statistical distance at most \( \delta \) . The above approximations give values of \( (k,\delta) \) that suffice to fool \( \mathsf {OR} \) , symmetric functions, and bounded-width \( \mathsf {CNF} \) , and the first result is tight for all \( k \) while the second result is tight for \( k \le (1-\Omega (1))n \) . We also show that any two \( (k, \delta) \) -indistinguishable distributions are \( O(n^{k/2}\delta) \) -close to two distributions that are \( (k,0) \) -indistinguishable, improving the previous bound of \( O(n)^k \delta \) . Finally, we present proofs of some known approximate degree lower bounds in the language of indistinguishability, which we find more intuitive.

We revisit the problem of constructing explicit pseudorandom generators that fool with error ϵ degree-d polynomials in n variables over the field F q , in the case of large q. Previous constructions either have seed length $\geq 2^{d}\log q$, and thus are only non-trivial when $d\lt \log n$, or else rely on a seminal reduction by Bogdanov (STOC 2005). This reduction yields seed length not less than $d^{4}\log n+\log q$ and requires fields of size $q\geq d^{6}/\epsilon^{2}$; and explicit generators meeting such bounds are known.Departing from Bogdanov’s reduction, we develop an algebraic analogue of the Bogdanov-Viola paradigm (FOCS 2007, SICOMP 2010) of summing generators for degree-one polynomials. Whereas previous analyses of the paradigm are restricted to degree $d\lt \log n$, we give a new analysis which handles large degrees. A main new idea is to show that the construction preserves indecomposability of polynomials. Apparently for the first time in the area, the proof uses invariant theory.Our approach in particular yields several new pseudorandom generators. In particular, for large enough fields we obtain seed length $O(d\log n+\log q)$ which is optimal up to constant factors. We also construct generators for fields of size as small as $O(d^{4})$. Further reducing the field size requires a significant change in techniques: Most or all generators for large-degree polynomials rely on Weil bounds; but such bounds are only applicable when $q\gt d^{4}$

The hardness vs.~randomness paradigm aims to explicitly construct pseudorandom generators $G:\{0,1\}^r \rightarrow \{0,1\}^m$ that fool circuits of size $m$, assuming the existence of explicit hard functions. A ``high-end PRG'' with seed length $r=O(\log m)$ (implying BPP=P) was achieved in a seminal work of Impagliazzo and Wigderson (STOC 1997), assuming the high-end hardness assumption: there exist constants $0<\beta < 1< B$, and functions computable in time $2^{B \cdot n}$ that cannot be computed by circuits of size $2^{\beta \cdot n}$. Recently, motivated by fast derandomization of randomized algorithms, Doron et al.~(FOCS 2020) and Chen and Tell (STOC 2021), construct ``extreme high-end PRGs'' with seed length $r=(1+o(1))\cdot \log m$, under qualitatively stronger assumptions. We study whether extreme high-end PRGs can be constructed from the following scaled version of the assumption which we call ``the extreme high-end hardness assumption'', and in which $\beta=1-o(1)$ and $B=1+o(1)$. We give a partial negative answer, showing that certain approaches cannot yield a black-box proof. (A longer abstract with more details appears in the PDF file)

We analyze the Fourier growth, i.e. the $L_1$ Fourier weight at level $k$ (denoted $L_{1,k}$), of various well-studied classes of"structured"$\mathbb{F}_2$-polynomials. This study is motivated by applications in pseudorandomness, in particular recent results and conjectures due to [CHHL19,CHLT19,CGLSS20] which show that upper bounds on Fourier growth (even at level $k=2$) give unconditional pseudorandom generators. Our main structural results on Fourier growth are as follows: - We show that any symmetric degree-$d$ $\mathbb{F}_2$-polynomial $p$ has $L_{1,k}(p) \le \Pr[p=1] \cdot O(d)^k$, and this is tight for any constant $k$. This quadratically strengthens an earlier bound that was implicit in [RSV13]. - We show that any read-$\Delta$ degree-$d$ $\mathbb{F}_2$-polynomial $p$ has $L_{1,k}(p) \le \Pr[p=1] \cdot (k \Delta d)^{O(k)}$. - We establish a composition theorem which gives $L_{1,k}$ bounds on disjoint compositions of functions that are closed under restrictions and admit $L_{1,k}$ bounds. Finally, we apply the above structural results to obtain new unconditional pseudorandom generators and new correlation bounds for various classes of $\mathbb{F}_2$-polynomials.

We initiate a systematic study of mixing in non-quasirandom groups. Let A and B be two independent, high-entropy distributions over a group G . We show that the product distribution AB is statistically close to the distribution F ( AB ) for several choices of G and F , including: (1) G is the affine group of 2 × 2 matrices, and F sets the top-right matrix entry to a uniform value, (2) G is the lamplighter group, that is the wreath product of Z 2 and Z n , and F is multiplication by a certain subgroup, (3) G is H n where H is non-abelian, and F selects a uniform coordinate and takes a uniform conjugate of it. The obtained bounds for (1) and (2) are tight. This work is motivated by and applied to problems in communication complexity. We consider the 3-party communication problem of deciding if the product of three group elements multiplies to the identity. We prove lower bounds for the groups above, which are tight for the affine and the lamplighter groups.

Suppose that a distribution S can be approximately sampled by an efficient cell-probe algorithm. It is shown to be possible to restrict the input to the algorithm so that its output distribution is still not too far from S, and at the same time many output coordinates are almost pairwise independent. Building on this several results are obtained, including: A lower bound for sampling prefix sums. A lower bound for sampling a variant of the predecessor problem. A separation between AC0 and cell-probe sampling. A separation between sampling with O(q) and q probes. A new proof of the Patrascu-Viola data-structure lower bound for prefix sums, demonstrating the feasibility of obtaining data-structure lower bounds via sampling. A separation between data structures making O(q) and q probes. The only previous cell-probe lower bounds for sampling followed from the AC0 lower bounds and applied to pseudorandom objects like error-correcting and extractors, making them inadequate for the above applications. ∗Supported by NSF CCF award 1813930. ISSN 1433-8092 Electronic Colloquium on Computational Complexity, Report No. 73 (2021)

Recently several conjectures were made regarding the Fourier spectrum of low-degree polynomials. We show that these conjectures imply new correlation bounds for functions related to Majority. Then we prove several new results on correlation bounds which aim to, but don’t, resolve the conjectures. In particular, we prove several new results on Majority which are of independent interest and complement Smolensky’s classic result.