Boosting uniformity in quasirandom groups: fast and simple

Iterated multiplication of elements in a group is a fundamental problem that has a long history and wide-ranging applications, and is linked to long-standing open problems. Already in [LZ77] it has been pivotal to provide space-efficient algorithms for Dyck languages. Depending on the underlying group, iterated multiplication is complete for various complexity classes [KMR66, MC87, Mix89, BC92, IL95, Mil14]. For example, Barrington’s famous result [Mix89] shows that it is complete for $\text{NC}^{1}$ if and only if the underlying group is non-solvable. This in particular disproved a conjecture about the complexity of majority [BDFP83]. This type of results has then been taken further in the study of catalytic computation [BCK⁺14], leading to other surprising discoveries [BCK⁺14, CM20].

The focus of this paper is on number-on-forehead communication complexity [CFL83]. For a survey on the communication complexity of group products, see [Vio19], and see [KN97, RY19] for general background on communication complexity. Concretely, the input is a matrix of $k\times t$ elements $a_{ij}$ from a group $H$, and the goal is computing $\prod_{j=1}^{t}a_{1j}\cdots a_{kj}$. There are $k$ collaborating parties, with Party $i$ knowing all the input except row $i$.

This problem is also linked to central open problems in communication complexity. Specifically, [GV19] conjectured that over certain groups this problem remains hard even for $k$ larger than $\log n$. Establishing such bounds is arguably the most significant open problem in the area. A number of candidates have been put forward over the years, but many have been ruled out via ingenious protocols, e.g. in [PRS97, BGKL03, BC08, ACFN15]. Interestingly, for the iterated-product candidate proposed in [GV19], no non-trivial protocol is known.

Iterated group products are also candidate for providing strong separations between randomized and deterministic number-on-forehead communication. The current bounds (see [Vio19]) give a separation matching one in [BDPW07]. Stronger bounds could simplify and strengthen the recent exciting separation [KLM23].

Returning to the problem, we note that its complexity heavily depends on the underlying group. If it is abelian, then the problem can be solved with constant communication, using the public-coin protocol for equality. Over certain other groups a communication lower bound of $t/c^{k}$ follows via [Mix89] from the landmark lower bound in [BNS92] for generalized inner product; cf. [MV13]. However, this bound does not improve with the size of the group. In particular it is far from the (trivial) upper bound of $t\log H$, and it gives nothing when $t$ is constant. Motivated by a cryptographic application, [MV13] asked whether a lower bound that grows with the size of the group, ideally $ct\log H$, can be established over some group $H$.

Gowers and Viola [GV15, GV19] proved a bound of $(t\log H)/c^{c^{k}}$ for the group $\text{SL}(2,q)$ of $2\times 2$ invertible matrices over $\mathbb{F}_{q}$, which enables the motivating application from [MV13]. Subsequent work [DV23] simplified the proof and generalized it to any quasi-random group [Gow08], see also [GV19, Sha16]. While such bounds do grow with the size of the group, thus answering the question in [MV13] and enabling the motivating application in cryptography, the dependency on the number $k$ of parties is weak: One can only afford $k$ doubly-logarithmic in the input length.

In this work we give an exponential improvement and obtain bounds of the form $(t\log H)/c^{k}$, thus matching the state-of-the-art in number-on-forehead communication [BNS92]. As in [GV19], we prove stronger results that even bound the advantage such protocols have when the input is promised to multiply to one of two fixed elements.

The high-level proof technique is the same as in [GV19]. They reduced the problem to boosting uniformity over $m$ copies of $H$.

[GV19] showed that if $s$ is a $2$-uniform distribution over $H^{m}$ then the convolution (a.k.a. component-wise product) of some $\ell$ independent copies of $s$ is $H^{-m}$-uniform over the whole space $H^{m}$. Note that such a result is false for abelian groups – the convolution can remain only $2$-uniform. Quantitatively, they show that $\ell=c^{m}$ copies suffice. In the application to 1 one has $m=2^{k}$, which gives the doubly logarithmic dependence on $k$.

In this work we give a corresponding exponential improvement on the number of copies required to boost uniformity: we show that in fact $\ell=m^{c}$ copies suffice. Our proof is remarkably simple, especially if we start with $3$-uniform distributions, which we note suffices for 1. (We discuss below extensions to $2$-uniform and other groups.) We state this result next.

Our approach allows us to double the uniformity, i.e., go from $k$-uniform to $2k$-uniform using only a constant number of convolutions, independently of $k$, whereas [GV19] would use $\geq k$ convolutions. This points to a key difference in the techniques. In [GV19], boosting uniformity is achieved by reduction to interleaved products, and appears tailored to going from $2$-uniform to $3$-uniform. Our approach is different, and simpler, even taking into account the simple proof of interleaved mixing from [DV23]. It can be seen as a $k$-uniform version of the flattening lemmas discovered in [Gow08, BNP08, GV19]. In a nutshell, the $k$-uniformity assumption allows us to remove all “low-degree” Fourier coefficients, leaving only those of degree $>k$. Then the quasi-randomness assumption, combined with the tensor-product structure of the Fourier coefficients allows us to “flatten” distributions at a rate proportional to $H^{-ck}$, instead of $H^{-c}$ as in previous work. We note that while using $k$-uniformity to remove low-degree coefficients is a common proof technique, we are not aware of previous work where this is done in the non-abelian setting. This might indicate that our techniques might find other applications, and in general we advocate a systematic study of non-abelian analogues of the Fourier toolkit. Another step in this direction is discussed next.

In this section we fix some notation, especially about representation theory.

For a set $X$, we also write $X$ for its size $|X|$. We write $[i]$ for the set $\{0,1,\ldots,i-1\}$. Every occurrence of “$c$” denotes a possibly different universal constant. Replacing “$c$” with $O(1)$ everywhere is consistent with a common interpretation of the latter. For a function $f\colon G\to\mathbb{C}$ we denote by $|f|_{2}^{2}$ the un-normalized quantity $\sum_{x\in G}|f(x)|^{2}$.

Next we present the standard framework of representation theory. The books by Serre [Ser77], Diaconis [Dia88], and Terras [Ter99] are good references for representation theory and non-abelian Fourier analysis. The Barbados notes [Wig10] or Section 13 of [Gow17] or [GV22] provide briefer introductions. The exposition in these sources is not always consistent, and often has different aims from ours. So let us give a quick account of the theory that is most relevant for this work.

In this section we prove 3. The proof follows by repeated applications of the following theorem.

Note for small $k$ we may have $m=k+1$. But if $k\geq c$ then $m$ is a constant factor larger than $k$.

The choice of the error parameter is not too important because it can be boosted with convolutions:

In the rest of this section, we prove 7. The proof involves an excursion to 2-norms. The main step is the following new flattening lemma which can be seen as a $k$-wise variant of the flattening lemmas discovered in [Gow08, BNP08, GV19].

We need the following couple of claims to go back-and-forth between $\mathrm{\epsilon}$-uniform and 2-norms.

We can now present the proof of 7.

Let $m:=2^{k}$. As noted in [GV19], an application of the box norm (Corollary 3.11 in [VW08]) shows that the LHS in the conclusion is $\leq cH2^{d}$ times the statistical distance between the uniform distribution over $H^{m}$ and the convolution of $t$ independent copies of the following distribution $s$ over $H^{m}$.

We note that $s$ is not $4$-uniform, again just considering two coordinates.

To conclude the proof of 1, note that the convolution of $m^{c}$ copies of $s$ is $(H^{-m},m)$-uniform by 3. By 8 the convolution of $t$ copies is then $(H^{-m\cdot t/m^{c}},m)$-uniform, and the result follows.

In this section we prove the following generalization of [RX13].

First we note the following converse to 11.