arXiv subjects
Publications and source records attributed to Swastik Kopparty.
The recent breakthrough work of Chatterjee, Ghosh, Gurjar, Raj and Thierauf [CGGRT26] gives the first deterministic NC algorithm for the bipartite matching problem. They show how to detect as well as find perfect matchings in bipartite graphs in NC. In this note we present an arguably simpler-to-state variation of the NC detection criterion of [CGGRT26], with improved parameters.
Expander (Tanner) codes combine sparse graphs with local constraints, enabling linear-time decoding and asymptotically good distance--rate tradeoffs. A standard constraint-counting argument yields the global-rate lower bound $R\ge 2r-1$ for a Tanner code with local rate $r$, which gives no positive-rate guarantee in the low-rate regime $r\le 1/2$. This regime is nonetheless important in applications that require algebraic local constraints (e.g., Reed--Solomon locality and the Schur-product/multiplication property). We introduce \emph{Algebraic Expander Codes}, an explicit algebraic family of Tanner-type codes whose local constraints are Reed--Solomon and whose global rate remains bounded away from $0$ for every fixed $r\in(0,1)$ (in particular, for $r\le 1/2$), while achieving constant relative distance. Our codes are defined by evaluating a structured subspace of polynomials on an orbit of a non-commutative subgroup of $\mathrm{AGL}(1,\mathbb{F})$ generated by translations and scalings. The resulting sparse coset geometry forms a strong spectral expander, proved via additive character-sum estimates, while the rate analysis uses a new notion of polynomial degree and a polytope-volume/dimension-counting argument.
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 constant fraction of the values having errors. This was previously unknown even for the case of no errors. We give a similar algorithm for additive characters of polynomials over fields of characteristic $2$. This gives the first polynomial time algorithm for decoding dual-BCH codes of polynomial dimension from a constant fraction of errors. Our algorithms use ideas from Stepanov's polynomial method proof of the classical Weil bounds on character sums, as well as from the Berlekamp-Welch decoding algorithm for Reed-Solomon codes. A crucial role is played by what we call *pseudopolynomials*: high degree polynomials, all of whose derivatives behave like low degree polynomials on ${\mathbb F}_q$. Both these results can be viewed as algorithmic versions of the Weil bounds for this setting.
In this paper we study a basic and natural question about Fourier analysis of Boolean functions, which has applications to the study of Matching Vector based Private Information Retrieval (PIR) schemes. For integers m and r, define a delta function on {0,1}^r to be a function f: Z_m^r -> C with f(0) = 1 and f(x) = 0 for all nonzero Boolean x. The basic question we study is how small the Fourier sparsity of a delta function can be; namely how sparse such an f can be in the Fourier basis? In addition to being intrinsically interesting and natural, such questions arise naturally when studying "S-decoding polynomials" for the known matching vector families. Finding S-decoding polynomials of reduced sparsity, which corresponds to finding delta functions with low Fourier sparsity, would improve the current best PIR schemes. We show nontrivial upper and lower bounds on the Fourier sparsity of delta functions. Our proofs are elementary and clean. These results imply limitations on improving Matching Vector PIR schemes simply by finding better S-decoding polynomials. In particular, there are no S-decoding polynomials that can make Matching Vector PIRs based on the known matching vector families achieve polylogarithmic communication with a constant number of servers. Many interesting questions remain open.
This paper is motivated by basic complexity and probability questions about permanents of random matrices over finite fields, and in particular, about properties separating the permanent and the determinant. Fix $q = p^m$ some power of an odd prime, and let $k \leq n$ both be growing. For a uniformly random $n \times k$ matrix $A$ over $\mathbb{F}_q$, we study the probability that all $k \times k$ submatrices of $A$ have zero permanent; namely that $A$ does not have full "permanental rank". When $k = n$, this is simply the probability that a random square matrix over $\mathbb{F}_q$ has zero permanent, which we do not understand. We believe that the probability in this case is $\frac{1}{q} + o(1)$, which would be in contrast to the case of the determinant, where the answer is $\frac{1}{q} + \Omega_q(1)$. Our main result is that when $k$ is $O(\sqrt{n})$, the probability that a random $n \times k$ matrix does not have full permanental rank is essentially the same as the probability that the matrix has a $0$ column, namely $(1 +o(1)) \frac{k}{q^n}$. In contrast, for determinantal (standard) rank the analogous probability is $\Theta(\frac{q^k}{q^n})$. At the core of our result are some basic linear algebraic properties of the permanent that distinguish it from the determinant.
In this paper, we construct new t-server Private Information Retrieval (PIR) schemes with communication complexity subpolynomial in the previously best known, for all but finitely many t. Our results are based on combining derivatives (in the spirit of Woodruff-Yekhanin) with the Matching Vector based PIRs of Yekhanin and Efremenko. Previously such a combination was achieved in an ingenious way by Dvir and Gopi, using polynomials and derivatives over certain exotic rings, en route to their fundamental result giving the first 2-server PIR with subpolynomial communication. Our improved PIRs are based on two ingredients: - We develop a new and direct approach to combine derivatives with Matching Vector based PIRs. This approach is much simpler than that of Dvir-Gopi: it works over the same field as the original PIRs, and only uses elementary properties of polynomials and derivatives. - A key subproblem that arises in the above approach is a higher-order polynomial interpolation problem. We show how "sparse S-decoding polynomials", a powerful tool from the original constructions of Matching Vector PIRs, can be used to solve this higher-order polynomial interpolation problem using surprisingly few higer-order evaluations. Using the known sparse S-decoding polynomials, in combination with our ideas leads to our improved PIRs. Notably, we get a 3-server PIR scheme with communication $2^{O^{\sim}( (\log n)^{1/3}) }$, improving upon the previously best known communication of $2^{O^{\sim}( \sqrt{\log n})}$ due to Efremenko.
We study the problem of partitioning the unit cube $[0,1]^n$ into $c$ parts so that each $d$-dimensional axis-parallel projection has small volume. This natural combinatorial/geometric question was first studied by Kopparty and Nagargoje [KN23] as a reformulation of the problem of determining the achievable parameters for seedless multimergers -- which extract randomness from `$d$-where' random sources (generalizing somewhere random sources). This question is closely related to influences of variables and is about a partition analogue of Shearer's lemma. Our main result answers a question of [KN23]: for $d = n-1$, we show that for $c$ even as large as $2^{o(n)}$, it is possible to partition $[0,1]^n$ into $c$ parts so that every $n-1$-dimensional axis-parallel projection has volume at most $(1/c) ( 1 + o(1) )$. Previously, this was shown by [KN23] for $c$ up to $O(\sqrt{n})$. The construction of our partition is related to influences of functions, and we present a clean geometric/combinatorial conjecture about this partitioning problem that would imply the KKL theorem on influences of Boolean functions.
The classical Reed-Muller codes over a finite field $\mathbb{F}_q$ are based on evaluations of $m$-variate polynomials of degree at most $d$ over a product set $U^m$, for some $d$ less than $|U|$. Because of their good distance properties, as well as the ubiquity and expressive power of polynomials, these codes have played an influential role in coding theory and complexity theory. This is especially so in the setting of $U$ being ${\mathbb{F}}_q$ where they possess deep locality properties. However, these Reed-Muller codes have a significant limitation in terms of the rate achievable -- the rate cannot be more than $\frac{1}{m{!}} = \exp(-m \log m)$. In this work, we give the first constructions of multivariate polynomial evaluation codes which overcome the rate limitation -- concretely, we give explicit evaluation domains $S \subseteq \mathbb{F}_q^m$ on which evaluating $m$-variate polynomials of degree at most $d$ gives a good code. For $m= O(1)$, these new codes have relative distance $\Omega(1)$ and rate $1 - \epsilon$ for any $\epsilon > 0$. In fact, we give two quite different constructions, and for both we develop efficient decoding algorithms for these codes that can decode from half the minimum distance. The first of these codes is based on evaluating multivariate polynomials on simplex-like sets whereas the second construction is more algebraic, and surprisingly (to us), has some strong locality properties, specifically, we show that they are locally testable.
In this paper, we construct Error-Correcting Graph Codes. An error-correcting graph code of distance $\delta$ is a family $C$ of graphs on a common vertex set of size $n$, such that if we start with any graph in $C$, we would have to modify the neighborhoods of at least $\delta n$ vertices in order to obtain some other graph in $C$. This is a natural graph generalization of the standard Hamming distance error-correcting codes for binary strings. Yohananov and Yaakobi were the first to construct codes in this metric, constructing good codes for $\delta < 1/2$, and optimal codes for a large-alphabet analogue. We extend their work by showing 1. Combinatorial results determining the optimal rate vs. distance trade-off nonconstructively. 2. Graph code analogues of Reed-Solomon codes and code concatenation, leading to positive distance codes for all rates and positive rate codes for all distances. 3. Graph code analogues of dual-BCH codes, yielding large codes with distance $\delta = 1-o(1)$. This gives an explicit ''graph code of Ramsey graphs''. Several recent works, starting with the paper of Alon, Gujgiczer, K\"orner, Milojevi\'c, and Simonyi, have studied more general graph codes; where the symmetric difference between any two graphs in the code is required to have some desired property. Error-correcting graph codes are a particularly interesting instantiation of this concept.
In this note, we give very simple constructions of unique neighbor expander graphs starting from spectral or combinatorial expander graphs of mild expansion. These constructions and their analysis are simple variants of the constructions of LDPC error-correcting codes from expanders, given by Sipser-Spielman [SS96] (and Tanner [Tan81]), and their analysis. We also show how to obtain expanders with many unique neighbors using similar ideas. There were many exciting results on this topic recently, starting with Asherov-Dinur [AD23] and Hsieh-McKenzie-Mohanty-Paredes [HMMP23], who gave a similar construction of unique neighbor expander graphs, but using more sophisticated ingredients (such as almost-Ramanujan graphs) and a more involved analysis. Subsequent beautiful works of Cohen-Roth-TaShma [CRT23] and Golowich [Gol23] gave even stronger objects (lossless expanders), but also using sophisticated ingredients. The main contribution of this work is that we get much more elementary constructions of unique neighbor expanders and with a simpler analysis.
For an odd prime $p$, we say $f(X) \in {\mathbb F}_p[X]$ computes square roots in $\mathbb F_p$ if, for all nonzero perfect squares $a \in \mathbb F_p$, we have $f(a)^2 = a$. When $p \equiv 3 \mod 4$, it is well known that $f(X) = X^{(p+1)/4}$ computes square roots. This degree is surprisingly low (and in fact lowest possible), since we have specified $(p-1)/2$ evaluations (up to sign) of the polynomial $f(X)$. On the other hand, for $p \equiv 1 \mod 4$ there was previously no nontrivial bound known on the lowest degree of a polynomial computing square roots in $\mathbb F_p$; it could have been anywhere between $\frac{p}{4}$ and $\frac{p}{2}$. We show that for all $p \equiv 1 \mod 4$, the degree of a polynomial computing square roots has degree at least $p/3$. Our main new ingredient is a general lemma which may be of independent interest: powers of a low degree polynomial cannot have too many consecutive zero coefficients. The proof method also yields a robust version: any polynomial that computes square roots for 99\% of the squares also has degree almost $p/3$. In the other direction, a result of Agou, Deliglése, and Nicolas (Designs, Codes, and Cryptography, 2003) shows that for infinitely many $p \equiv 1 \mod 4$, the degree of a polynomial computing square roots can be as small as $3p/8$.
We study the problem of extracting randomness from somewhere-random sources, and related combinatorial phenomena: partition analogues of Shearer's lemma on projections. A somewhere-random source is a tuple $(X_1, \ldots, X_t)$ of (possibly correlated) $\{0,1\}^n$-valued random variables $X_i$ where for some unknown $i \in [t]$, $X_i$ is guaranteed to be uniformly distributed. An $extracting$ $merger$ is a seeded device that takes a somewhere-random source as input and outputs nearly uniform random bits. We study the seed-length needed for extracting mergers with constant $t$ and constant error. We show: $\cdot$ Just like in the case of standard extractors, seedless extracting mergers with even just one output bit do not exist. $\cdot$ Unlike the case of standard extractors, it $is$ possible to have extracting mergers that output a constant number of bits using only constant seed. Furthermore, a random choice of merger does not work for this purpose! $\cdot$ Nevertheless, just like in the case of standard extractors, an extracting merger which gets most of the entropy out (namely, having $Ω$ $(n)$ output bits) must have $Ω$ $(\log n)$ seed. This is the main technical result of our work, and is proved by a second-moment strengthening of the graph-theoretic approach of Radhakrishnan and Ta-Shma to extractors. In contrast, seed-length/output-length tradeoffs for condensing mergers (where the output is only required to have high min-entropy), can be fully explained by using standard condensers. Inspired by such considerations, we also formulate a new and basic class of problems in combinatorics: partition analogues of Shearer's lemma. We show basic results in this direction; in particular, we prove that in any partition of the $3$-dimensional cube $[0,1]^3$ into two parts, one of the parts has an axis parallel $2$-dimensional projection of area at least $3/4$.
Motivated by problems in algebraic complexity theory (e.g., matrix multiplication) and extremal combinatorics (e.g., the cap set problem and the sunflower problem), we introduce the geometric rank as a new tool in the study of tensors and hypergraphs. We prove that the geometric rank is an upper bound on the subrank of tensors and the independence number of hypergraphs. We prove that the geometric rank is smaller than the slice rank of Tao, and relate geometric rank to the analytic rank of Gowers and Wolf in an asymptotic fashion. As a first application, we use geometric rank to prove a tight upper bound on the (border) subrank of the matrix multiplication tensors, matching Strassen's well-known lower bound from 1987.
For smooth finite fields $F_q$ (i.e., when $q-1$ factors into small primes) the Fast Fourier Transform (FFT) leads to the fastest known algebraic algorithms for many basic polynomial operations, such as multiplication, division, interpolation and multi-point evaluation. However, the same operations over fields with no smooth order root of unity suffer from an asymptotic slowdown. The classical algorithm of Schonhage and Strassen incurred a multiplicative slowdown factor of $\log \log n$ on top of the smooth case. Recent remarkable results of Harvey, van der Hoeven and Lecerf dramatically reduced this multiplicative overhead to $\exp(\log^* (n))$. We introduce a new approach to fast algorithms for polynomial operations over all large finite fields. The key idea is to replace the group of roots of unity with a set of points $L \subset F$ suitably related to a well-chosen elliptic curve group (the set $L$ itself is not a group). The key advantage of this approach is that elliptic curve groups can be of any size in the Hasse-Weil interval $[q+1 \pm 2\sqrt{q}]$ and thus can have subgroups of large, smooth order, which an FFT-like divide and conquer algorithm can exploit. Compare this with multiplicative subgroups over whose order must divide $q-1$. For polynomials represented by their evaluation over subsets of $L$, we show that multiplication, division, degree-computation, interpolation, evaluation and Reed-Solomon encoding (also known as low-degree extension) with fixed evaluation points can all be computed with arithmetic circuits of size similar to what is achievable with the classical FFTs when the field size is special. For several problems, this yields the asymptotically smallest known arithmetic circuits even in the standard monomial representation of polynomials.
Motivated by the quest for scalable and succinct zero knowledge arguments, we revisit worst-case-to-average-case reductions for linear spaces, raised by [Rothblum, Vadhan, Wigderson, STOC 2013]. We first show a sharp quantitative form of a theorem which says that if an affine space $U$ is $δ$-far in relative Hamming distance from a linear code $V$ - this is the worst-case assumption - then most elements of $U$ are almost $δ$-far from $V$ - this is the average case. This leads to an optimal analysis of the soundness of the FRI protocol of [Ben-Sasson, et.al., eprint 2018] for proving proximity to Reed-Solomon codes. To further improve soundness, we sample outside the box. We suggest a new protocol which asks a prover for values of a polynomial at points outside the domain of evaluation of the Reed-Solomon code. We call this technique Domain Extending for Eliminating Pretenders (DEEP). We use the DEEP technique to devise two new protocols: (1) An Interactive Oracle Proof of Proximity (IOPP) for RS codes, called DEEP-FRI. This soundness of the protocol improves upon that of the FRI protocol while retaining linear arithmetic proving complexity and logarithmic verifier arithmetic complexity. (2) An Interactive Oracle Proof (IOP) for the Algebraic Linking IOP (ALI) protocol used to construct zero knowledge scalable transparent arguments of knowledge (ZK-STARKs) in [Ben-Sasson et al., eprint 2018]. The new protocol, called DEEP-ALI, improves soundness of this crucial step from a small constant $< 1/8$ to a constant arbitrarily close to $1$.
In this work, we show new and improved error-correcting properties of folded Reed-Solomon codes and multiplicity codes. Both of these families of codes are based on polynomials over finite fields, and both have been the sources of recent advances in coding theory. Folded Reed-Solomon codes were the first explicit constructions of codes known to achieve list-decoding capacity; multivariate multiplicity codes were the first constructions of high-rate locally correctable codes; and univariate multiplicity codes are also known to achieve list-decoding capacity. However, previous analyses of the error-correction properties of these codes did not yield optimal results. In particular, in the list-decoding setting, the guarantees on the list-sizes were polynomial in the block length, rather than constant; and for multivariate multiplicity codes, local list-decoding algorithms could not go beyond the Johnson bound. In this paper, we show that Folded Reed-Solomon codes and multiplicity codes are in fact better than previously known in the context of list-decoding and local list-decoding. More precisely, we first show that Folded RS codes achieve list-decoding capacity with constant list sizes, independent of the block length; and that high-rate univariate multiplicity codes can also be list-recovered with constant list sizes. Using our result on univariate multiplicity codes, we show that multivariate multiplicity codes are high-rate, locally list-recoverable codes. Finally, we show how to combine the above results with standard tools to obtain capacity achieving locally list decodable codes with query complexity significantly lower than was known before.
In this paper, we prove new relations between the bias of multilinear forms, the correlation between multilinear forms and lower degree polynomials, and the rank of tensors over $GF(2)= \{0,1\}$. We show the following results for multilinear forms and tensors. 1. Correlation bounds : We show that a random $d$-linear form has exponentially low correlation with low-degree polynomials. More precisely, for $d \ll 2^{o(k)}$, we show that a random $d$-linear form $f(X_1,X_2, \dots, X_d) : \left(GF(2)^{k}\right)^d \rightarrow GF(2)$ has correlation $2^{-k(1-o(1))}$ with any polynomial of degree at most $d/10$. This result is proved by giving near-optimal bounds on the bias of random $d$-linear form, which is in turn proved by giving near-optimal bounds on the probability that a random rank-$t$ $d$-linear form is identically zero. 2. Tensor-rank vs Bias : We show that if a $d$-dimensional tensor has small rank, then the bias of the associated $d$-linear form is large. More precisely, given any $d$-dimensional tensor $$T :\underbrace{[k]\times \ldots [k]}_{\text{$d$ times}}\to GF(2)$$ of rank at most $t$, the bias of the associated $d$-linear form $$f_T(X_1,\ldots,X_d) := \sum_{(i_1,\dots,i_d) \in [k]^d} T(i_1,i_2,\ldots, i_d) X_{1,i_1}\cdot X_{1,i_2}\cdots X_{d,i_d}$$ is at least $\left(1-\frac1{2^{d-1}}\right)^t$. The above bias vs tensor-rank connection suggests a natural approach to proving nontrivial tensor-rank lower bounds for $d=3$. In particular, we use this approach to prove that the finite field multiplication tensor has tensor rank at least $3.52 k$ matching the best known lower bound for any explicit tensor in three dimensions over $GF(2)$.
In the simultaneous Max-Cut problem, we are given $k$ weighted graphs on the same set of $n$ vertices, and the goal is to find a cut of the vertex set so that the minimum, over the $k$ graphs, of the cut value is as large as possible. Previous work [BKS15] gave a polynomial time algorithm which achieved an approximation factor of $1/2 - o(1)$ for this problem (and an approximation factor of $1/2 + ε_k$ in the unweighted case, where $ε_k \rightarrow 0$ as $k \rightarrow \infty$). In this work, we give a polynomial time approximation algorithm for simultaneous Max-Cut with an approximation factor of $0.8780$ (for all constant $k$). The natural SDP formulation for simultaneous Max-Cut was shown to have an integrality gap of $1/2+ε_k$ in [BKS15]. In achieving the better approximation guarantee, we use a stronger Sum-of-Squares hierarchy SDP relaxation and a rounding algorithm based on Raghavendra-Tan [RT12], in addition to techniques from [BKS15].