13 Search Results for "Gowers, W. T."

Document
DOI: 10.4230/LIPIcs.ITCS.2026.103
A General Framework for Low Soundness Homomorphism Testing

Authors: Tushant Mittal and Sourya Roy

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract
We introduce a general framework to design and analyze algorithms for the problem of testing homomorphisms between finite groups in the low-soundness regime. In this regime, we give the first constant-query tests for various families of groups. These include tests for: (i) homomorphisms between arbitrary cyclic groups, (ii) homomorphisms between any finite group and ℤ_p, (iii) automorphisms of dihedral and symmetric groups, (iv) inner automorphisms of non-abelian finite simple groups and extraspecial groups, and (v) testing linear characters of GL_n(F_q), and finite-dimensional Lie algebras over F_q. We also recover the result of Kiwi [TCS'03] for testing homomorphisms between F_qⁿ and F_q. Prior to this work, such tests were only known for abelian groups with a constant maximal order (such as F_qⁿ). No tests were known for non-abelian groups. As an additional corollary, our framework gives combinatorial list decoding bounds for cyclic groups with list size dependence of O(ε^{-2}) (for agreement parameter ε). This improves upon the currently best-known bound of O(ε^{-105}) due to Dinur, Grigorescu, Kopparty, and Sudan [STOC'08], and Guo and Sudan [RANDOM'14].
Cite as

Tushant Mittal and Sourya Roy. A General Framework for Low Soundness Homomorphism Testing. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 103:1-103:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{mittal_et_al:LIPIcs.ITCS.2026.103,
  author =	{Mittal, Tushant and Roy, Sourya},
  title =	{{A General Framework for Low Soundness Homomorphism Testing}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{103:1--103:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2026.103},
  URN =		{urn:nbn:de:0030-drops-253901},
  doi =		{10.4230/LIPIcs.ITCS.2026.103},
  annote =	{Keywords: Property Testing, Coding Theory}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2026.90
Slice Rank and Partition Rank of the Determinant

Authors: Amichai Lampert and Guy Moshkovitz

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract
The Laplace expansion expresses the n × n determinant det_n as a sum of n products. Do shorter expansions exist? In this paper we: - Fully determine the slice rank decompositions of det_n (where each product must contain a linear factor): In this case, we show that n summands are necessary, and moreover, the only such expansions with n summands are equivalent (in a precise sense) to the Laplace expansion. - Prove a logarithmic lower bound for the partition rank of det_n (where each product is of multilinear forms): In this case, we show that at least log₂(n)+1 summands are needed and we explain why existing techniques fail to yield any nontrivial lower bound. - Separate partition rank from slice rank for det_n: we find a quadratic expansion for det₄, over any field, with fewer summands than the Laplace expansion. This construction is related to a well-known example of Green-Tao and Lovett-Meshulam-Samorodnitsky disproving the naive version of the Gowers Inverse conjecture over small fields. An important motivation for these questions comes from the challenge of separating structure and randomness for tensors. On the one hand, we show that the random construction fails to separate: for a random tensor of partition rank r, the analytic rank is r-o(1) with high probability. On the other hand, our results imply that the determinant yields the first asymptotic separation between partition rank and analytic rank of d-tensors, with their ratio tending to infinity with d.
Cite as

Amichai Lampert and Guy Moshkovitz. Slice Rank and Partition Rank of the Determinant. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 90:1-90:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{lampert_et_al:LIPIcs.ITCS.2026.90,
  author =	{Lampert, Amichai and Moshkovitz, Guy},
  title =	{{Slice Rank and Partition Rank of the Determinant}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{90:1--90:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2026.90},
  URN =		{urn:nbn:de:0030-drops-253779},
  doi =		{10.4230/LIPIcs.ITCS.2026.90},
  annote =	{Keywords: Slice rank, partition rank, determinant}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2025.49
Pseudorandomness of Expander Walks via Fourier Analysis on Groups

Authors: Fernando Granha Jeronimo, Tushant Mittal, and Sourya Roy

Published in: LIPIcs, Volume 353, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)

Abstract
A long line of work has studied the pseudorandomness properties of walks on expander graphs. A central goal is to measure how closely the distribution over n-length walks on an expander approximates the uniform distribution of n-independent elements. One approach to do so is to label the vertices of an expander with elements from an alphabet Σ, and study closeness of the mean of functions over Σⁿ, under these two distributions. We say expander walks ε-fool a function if the expander walk mean is ε-close to the true mean. There has been a sequence of works studying this question for various functions, such as the XOR function, the AND function, etc. We show that: - The class of symmetric functions is O(|Σ|λ)-fooled by expander walks over any generic λ-expander, and any alphabet Σ . This generalizes the result of Cohen, Peri, Ta-Shma [STOC'21] which analyzes it for |Σ| = 2, and exponentially improves the previous bound of O(|Σ|^O(|Σ|) λ), by Golowich and Vadhan [CCC'22]. Moreover, if the expander is a Cayley graph over ℤ_|Σ|, we get a further improved bound of O(√{|Σ|} λ). Morever, when Σ is a finite group G, we show the following for functions over Gⁿ: - The class of symmetric class functions is O({√|G|}/D λ}-fooled by expander walks over "structured" λ-expanders, if G is D-quasirandom. - We show a lower bound of Ω(λ) for symmetric functions for any finite group G (even for "structured" λ-expanders). - We study the Fourier spectrum of a class of non-symmetric functions arising from word maps, and show that they are exponentially fooled by expander walks. Our proof employs Fourier analysis over general groups, which contrasts with earlier works that have studied either the case of ℤ₂ or ℤ. This enables us to get quantitatively better bounds even for unstructured sets.
Cite as

Fernando Granha Jeronimo, Tushant Mittal, and Sourya Roy. Pseudorandomness of Expander Walks via Fourier Analysis on Groups. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 49:1-49:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{jeronimo_et_al:LIPIcs.APPROX/RANDOM.2025.49,
  author =	{Jeronimo, Fernando Granha and Mittal, Tushant and Roy, Sourya},
  title =	{{Pseudorandomness of Expander Walks via Fourier Analysis on Groups}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
  pages =	{49:1--49:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-397-3},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{353},
  editor =	{Ene, Alina and Chattopadhyay, Eshan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.49},
  URN =		{urn:nbn:de:0030-drops-244157},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2025.49},
  annote =	{Keywords: Expander graphs, pseudorandomness}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2025.61
A Fast Coloring Oracle for Average Case Hypergraphs

Authors: Cassandra Marcussen, Edward Pyne, Ronitt Rubinfeld, Asaf Shapira, and Shlomo Tauber

Published in: LIPIcs, Volume 353, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)

Abstract
Hypergraph 2-colorability is one of the classical NP-hard problems. Person and Schacht [SODA'09] designed a deterministic algorithm whose expected running time is polynomial over a uniformly chosen 2-colorable 3-uniform hypergraph. Lee, Molla, and Nagle recently extended this to k-uniform hypergraphs for all k ≥ 3. Both papers relied heavily on the regularity lemma, hence their analysis was involved and their running time hid tower-type constants. Our first result in this paper is a new simple and elementary deterministic 2-coloring algorithm that reproves the theorems of Person-Schacht and Lee-Molla-Nagle while avoiding the use of the regularity lemma. We also show how to turn our new algorithm into a randomized one with average expected running time of only O(n). Our second and main result gives what we consider to be the ultimate evidence of just how easy it is to find a 2-coloring of an average 2-colorable hypergraph. We define a coloring oracle to be an algorithm which, given vertex v, assigns color red/blue to v while inspecting as few edges as possible, so that the answers to any sequence of queries to the oracle are consistent with a single legal 2-coloring of the input. Surprisingly, we show that there is a coloring oracle that, on average, can answer every vertex query in time O(1).
Cite as

Cassandra Marcussen, Edward Pyne, Ronitt Rubinfeld, Asaf Shapira, and Shlomo Tauber. A Fast Coloring Oracle for Average Case Hypergraphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 61:1-61:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{marcussen_et_al:LIPIcs.APPROX/RANDOM.2025.61,
  author =	{Marcussen, Cassandra and Pyne, Edward and Rubinfeld, Ronitt and Shapira, Asaf and Tauber, Shlomo},
  title =	{{A Fast Coloring Oracle for Average Case Hypergraphs}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
  pages =	{61:1--61:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-397-3},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{353},
  editor =	{Ene, Alina and Chattopadhyay, Eshan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.61},
  URN =		{urn:nbn:de:0030-drops-244272},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2025.61},
  annote =	{Keywords: average-case algorithms, local computation algorithms, graph coloring}
}
Document
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2025.62
Avoiding Range via Turan-Type Bounds

Authors: Neha Kuntewar and Jayalal Sarma

Published in: LIPIcs, Volume 353, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)

Abstract
Given a circuit C : {0,1}^n → {0,1}^m from a circuit class 𝒞, with m > n, finding a y ∈ {0,1}^m such that ∀ x ∈ {0,1}ⁿ, C(x) ≠ y, is the range avoidance problem (denoted by C-Avoid). Deterministic polynomial time algorithms (even with access to NP oracles) solving this problem are known to imply explicit constructions of various pseudorandom objects like hard Boolean functions, linear codes, PRGs etc. Deterministic polynomial time algorithms are known for NC⁰₂-Avoid when m > n, and for NC⁰₃-Avoid when m ≥ n²/log n, where NC⁰_k is the class of circuits with bounded fan-in which have constant depth and the output depends on at most k of the input bits. On the other hand, it is also known that NC⁰₃-Avoid when m = n+O(n^{2/3}) is at least as hard as explicit construction of rigid matrices. In fact, algorithms for solving range avoidance for even NC⁰₄ circuits imply new circuit lower bounds. In this paper, we propose a new approach to solving the range avoidance problem via hypergraphs. We formulate the problem in terms of Turan-type problems in hypergraphs of the following kind: for a fixed k-uniform hypergraph H, what is the maximum number of edges that can exist in H_C, which does not have a sub-hypergraph isomorphic to H? We show the following: - We first demonstrate the applicability of this approach by showing alternate proofs of some of the known results for the range avoidance problem using this framework. - We then use our approach to show (using several different hypergraph structures for which Turan-type bounds are known in the literature) that there is a constant c such that Monotone-NC⁰₃-Avoid can be solved in deterministic polynomial time when m > cn². - To improve the stretch constraint to linear, more precisely, to m > n, we show a new Turan-type theorem for a hypergraph structure (which we call the loose X_{2ℓ}-cycles). More specifically, we prove that any connected 3-uniform linear hypergraph with m > n edges must contain a loose X_{2ℓ} cycle. This may be of independent interest. - Using this, we show that Monotone-NC⁰₃-Avoid can be solved in deterministic polynomial time when m > n, thus improving the known bounds of NC⁰₃-Avoid for the case of monotone circuits. In contrast, we note that efficient algorithms for solving Monotone-NC⁰₆-Avoid, already imply explicit constructions for rigid matrices. - We also generalise our argument to solve the special case of range avoidance for NC⁰_k where each output function computed by the circuit is the majority function on its inputs, where m > n².
Cite as

Neha Kuntewar and Jayalal Sarma. Avoiding Range via Turan-Type Bounds. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 62:1-62:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{kuntewar_et_al:LIPIcs.APPROX/RANDOM.2025.62,
  author =	{Kuntewar, Neha and Sarma, Jayalal},
  title =	{{Avoiding Range via Turan-Type Bounds}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
  pages =	{62:1--62:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-397-3},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{353},
  editor =	{Ene, Alina and Chattopadhyay, Eshan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.62},
  URN =		{urn:nbn:de:0030-drops-244281},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2025.62},
  annote =	{Keywords: circuit lower bounds, explicit constructions, range avoidance, linear hypergraphs, Tur\'{a}n number of hypergraphs}
}
Document
DOI: 10.4230/LIPIcs.TQC.2025.9
Efficient Quantum Pseudorandomness from Hamiltonian Phase States

Authors: John Bostanci, Jonas Haferkamp, Dominik Hangleiter, and Alexander Poremba

Published in: LIPIcs, Volume 350, 20th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2025)

Abstract
Quantum pseudorandomness has found applications in many areas of quantum information, ranging from entanglement theory, to models of scrambling phenomena in chaotic quantum systems, and, more recently, in the foundations of quantum cryptography. Kretschmer (TQC '21) showed that both pseudorandom states and pseudorandom unitaries exist even in a world without classical one-way functions. To this day, however, all known constructions require classical cryptographic building blocks which are themselves synonymous with the existence of one-way functions, and which are also challenging to implement on realistic quantum hardware. In this work, we seek to make progress on both of these fronts simultaneously - by decoupling quantum pseudorandomness from classical cryptography altogether. We introduce a quantum hardness assumption called the Hamiltonian Phase State (HPS) problem, which is the task of decoding output states of a random instantaneous quantum polynomial-time (IQP) circuit. Hamiltonian phase states can be generated very efficiently using only Hadamard gates, single-qubit Z rotations and CNOT circuits. We show that the hardness of our problem reduces to a worst-case version of the problem, and we provide evidence that our assumption is plausibly fully quantum; meaning, it cannot be used to construct one-way functions. We also show information-theoretic hardness when only few copies of HPS are available by proving an approximate t-design property of our ensemble. Finally, we show that our HPS assumption and its variants allow us to efficiently construct many pseudorandom quantum primitives, ranging from pseudorandom states, to quantum pseudoentanglement, to pseudorandom unitaries, and even primitives such as public-key encryption with quantum keys.
Cite as

John Bostanci, Jonas Haferkamp, Dominik Hangleiter, and Alexander Poremba. Efficient Quantum Pseudorandomness from Hamiltonian Phase States. In 20th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 350, pp. 9:1-9:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bostanci_et_al:LIPIcs.TQC.2025.9,
  author =	{Bostanci, John and Haferkamp, Jonas and Hangleiter, Dominik and Poremba, Alexander},
  title =	{{Efficient Quantum Pseudorandomness from Hamiltonian Phase States}},
  booktitle =	{20th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2025)},
  pages =	{9:1--9:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-392-8},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{350},
  editor =	{Fefferman, Bill},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2025.9},
  URN =		{urn:nbn:de:0030-drops-240586},
  doi =		{10.4230/LIPIcs.TQC.2025.9},
  annote =	{Keywords: Quantum pseudorandomness, quantum phase states, quantum cryptography}
}
Document
DOI: 10.4230/LIPIcs.CCC.2025.1
List Decoding Quotient Reed-Muller Codes

Authors: Omri Gotlib, Tali Kaufman, and Shachar Lovett

Published in: LIPIcs, Volume 339, 40th Computational Complexity Conference (CCC 2025)

Abstract
Reed-Muller codes consist of evaluations of n-variate polynomials over a finite field 𝔽 with degree at most d. Much like every linear code, Reed-Muller codes can be characterized by constraints, where a codeword is valid if and only if it satisfies all degree-d constraints. For a subset X̃ ⊆ 𝔽ⁿ, we introduce the notion of X̃-quotient Reed-Muller code. A function F:X̃ → 𝔽 is a valid codeword in the quotient code if it satisfies all the constraints of degree-d polynomials lying in X̃. This gives rise to a novel phenomenon: a quotient codeword may have many extensions to original codewords. This weakens the connection between original codewords and quotient codewords which introduces a richer range of behaviors along with substantial new challenges. Our goal is to answer the following question: what properties of X̃ will imply that the quotient code inherits its distance and list-decoding radius from the original code? We address this question using techniques developed by Bhowmick and Lovett [Abhishek Bhowmick and Shachar Lovett, 2014], identifying key properties of 𝔽ⁿ used in their proof and extending them to general subsets X̃ ⊆ 𝔽ⁿ. By introducing a new tool, we overcome the novel challenge in analyzing the quotient code that arises from the weak connection between original and quotient codewords. This enables us to apply known results from additive combinatorics and algebraic geometry [David Kazhdan and Tamar Ziegler, 2018; David Kazhdan and Tamar Ziegler, 2019; Amichai Lampert and Tamar Ziegler, 2021] to show that when X̃ is a high rank variety, X̃-quotient Reed-Muller codes inherit the distance and list-decoding parameters from the original Reed-Muller codes.
Cite as

Omri Gotlib, Tali Kaufman, and Shachar Lovett. List Decoding Quotient Reed-Muller Codes. In 40th Computational Complexity Conference (CCC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 339, pp. 1:1-1:44, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{gotlib_et_al:LIPIcs.CCC.2025.1,
  author =	{Gotlib, Omri and Kaufman, Tali and Lovett, Shachar},
  title =	{{List Decoding Quotient Reed-Muller Codes}},
  booktitle =	{40th Computational Complexity Conference (CCC 2025)},
  pages =	{1:1--1:44},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-379-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{339},
  editor =	{Srinivasan, Srikanth},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2025.1},
  URN =		{urn:nbn:de:0030-drops-236957},
  doi =		{10.4230/LIPIcs.CCC.2025.1},
  annote =	{Keywords: Reed-Muller Codes, Quotient Code, Quotient Reed-Muller Code, List Decoding, High Rank Variety, High-Order Fourier Analysis, Error-Correcting Codes}
}
Document
DOI: 10.4230/LIPIcs.CCC.2025.9
Pseudorandom Bits for Non-Commutative Programs

Authors: Chin Ho Lee and Emanuele Viola

Published in: LIPIcs, Volume 339, 40th Computational Complexity Conference (CCC 2025)

Abstract
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.
Cite as

Chin Ho Lee and Emanuele Viola. Pseudorandom Bits for Non-Commutative Programs. In 40th Computational Complexity Conference (CCC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 339, pp. 9:1-9:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{lee_et_al:LIPIcs.CCC.2025.9,
  author =	{Lee, Chin Ho and Viola, Emanuele},
  title =	{{Pseudorandom Bits for Non-Commutative Programs}},
  booktitle =	{40th Computational Complexity Conference (CCC 2025)},
  pages =	{9:1--9:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-379-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{339},
  editor =	{Srinivasan, Srikanth},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2025.9},
  URN =		{urn:nbn:de:0030-drops-237039},
  doi =		{10.4230/LIPIcs.CCC.2025.9},
  annote =	{Keywords: Group programs, Space-bounded derandomization, Representation theory}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2025.2
Improved Lower Bounds for 3-Query Matching Vector Codes

Authors: Divesh Aggarwal, Pranjal Dutta, Zeyong Li, Maciej Obremski, and Sidhant Saraogi

Published in: LIPIcs, Volume 325, 16th Innovations in Theoretical Computer Science Conference (ITCS 2025)

Abstract
A Matching Vector (MV) family modulo a positive integer m ≥ 2 is a pair of ordered lists U = (u_1, ⋯, u_K) and V = (v_1, ⋯, v_K) where u_i, v_j ∈ ℤ_m^n with the following property: for any i ∈ [K], the inner product ⟨u_i, v_i⟩ = 0 mod m, and for any i ≠ j, ⟨u_i, v_j⟩ ≠ 0 mod m. An MV family is called r-restricted if inner products ⟨u_i, v_j⟩, for all i,j, take at most r different values. The r-restricted MV families are extremely important since the only known construction of constant-query subexponential locally decodable codes (LDCs) are based on them. Such LDCs constructed via matching vector families are called matching vector codes. Let MV(m,n) (respectively MV(m, n, r)) denote the largest K such that there exists an MV family (respectively r-restricted MV family) of size K in ℤ_m^n. Such a MV family can be transformed in a black-box manner to a good r-query locally decodable code taking messages of length K to codewords of length N = m^n. For small prime m, an almost tight bound MV(m,n) ≤ O(m^{n/2}) was first shown by Dvir, Gopalan, Yekhanin (FOCS'10, SICOMP'11), while for general m, the same paper established an upper bound of O(m^{n-1+o_m(1)}), with o_m(1) denoting a function that goes to zero when m grows. For any arbitrary constant r ≥ 3 and composite m, the best upper bound till date on MV(m,n,r) is O(m^{n/2}), is due to Bhowmick, Dvir and Lovett (STOC'13, SICOMP'14).In a breakthrough work, Alrabiah, Guruswami, Kothari and Manohar (STOC'23) implicitly improve this bound for 3-restricted families to MV(m, n, 3) ≤ O(m^{n/3}). In this work, we present an upper bound for r = 3 where MV(m,n,3) ≤ m^{n/6 +O(log n)}, and as a result, any 3-query matching vector code must have codeword length of N ≥ K^{6-o(1)}.
Cite as

Divesh Aggarwal, Pranjal Dutta, Zeyong Li, Maciej Obremski, and Sidhant Saraogi. Improved Lower Bounds for 3-Query Matching Vector Codes. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 2:1-2:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{aggarwal_et_al:LIPIcs.ITCS.2025.2,
  author =	{Aggarwal, Divesh and Dutta, Pranjal and Li, Zeyong and Obremski, Maciej and Saraogi, Sidhant},
  title =	{{Improved Lower Bounds for 3-Query Matching Vector Codes}},
  booktitle =	{16th Innovations in Theoretical Computer Science Conference (ITCS 2025)},
  pages =	{2:1--2:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-361-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{325},
  editor =	{Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2025.2},
  URN =		{urn:nbn:de:0030-drops-226308},
  doi =		{10.4230/LIPIcs.ITCS.2025.2},
  annote =	{Keywords: Locally Decodable Codes, Matching Vector Families}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2024.67
Small Sunflowers and the Structure of Slice Rank Decompositions

Authors: Thomas Karam

Published in: LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

Abstract
Let d ≥ 3 be an integer. We show that whenever an order-d tensor admits d+1 decompositions according to Tao’s slice rank, if the linear subspaces spanned by their one-variable functions constitute a sunflower for each choice of special coordinate, then the tensor admits a decomposition where these linear subspaces are contained in the centers of these respective sunflowers. As an application, we deduce that for every nonnegative integer k and every finite field 𝔽 there exists an integer C(d,k,|𝔽|) such that every order-d tensor with slice rank k over 𝔽 admits at most C(d,k,|𝔽|) decompositions with length k, up to a class of transformations that can be easily described.
Cite as

Thomas Karam. Small Sunflowers and the Structure of Slice Rank Decompositions. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 67:1-67:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{karam:LIPIcs.ITCS.2024.67,
  author =	{Karam, Thomas},
  title =	{{Small Sunflowers and the Structure of Slice Rank Decompositions}},
  booktitle =	{15th Innovations in Theoretical Computer Science Conference (ITCS 2024)},
  pages =	{67:1--67:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-309-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{287},
  editor =	{Guruswami, Venkatesan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.67},
  URN =		{urn:nbn:de:0030-drops-195953},
  doi =		{10.4230/LIPIcs.ITCS.2024.67},
  annote =	{Keywords: Slice rank, tensors, sunflowers, decompositions}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2022.80
Mixing in Non-Quasirandom Groups

Authors: W. T. Gowers and Emanuele Viola

Published in: LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

Abstract
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 2x2 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 ℤ₂ and ℤ_{n}, and F is multiplication by a certain subgroup, 3) G is Hⁿ 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.
Cite as

W. T. Gowers and Emanuele Viola. Mixing in Non-Quasirandom Groups. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 80:1-80:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{gowers_et_al:LIPIcs.ITCS.2022.80,
  author =	{Gowers, W. T. and Viola, Emanuele},
  title =	{{Mixing in Non-Quasirandom Groups}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{80:1--80:9},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-217-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{215},
  editor =	{Braverman, Mark},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.80},
  URN =		{urn:nbn:de:0030-drops-156761},
  doi =		{10.4230/LIPIcs.ITCS.2022.80},
  annote =	{Keywords: Groups, representation theory, mixing, communication complexity, quasi-random}
}
Document
DOI: 10.4230/LIPIcs.ITP.2021.4
A Graphical User Interface Framework for Formal Verification

Authors: Edward W. Ayers, Mateja Jamnik, and W. T. Gowers

Published in: LIPIcs, Volume 193, 12th International Conference on Interactive Theorem Proving (ITP 2021)

Abstract
We present the "ProofWidgets" framework for implementing general user interfaces (UIs) within an interactive theorem prover. The framework uses web technology and functional reactive programming, as well as metaprogramming features of advanced interactive theorem proving (ITP) systems to allow users to create arbitrary interactive UIs for representing the goal state. Users of the framework can create GUIs declaratively within the ITP’s metaprogramming language, without having to develop in multiple languages and without coordinated changes across multiple projects, which improves development time for new designs of UI. The ProofWidgets framework also allows UIs to make use of the full context of the theorem prover and the specialised libraries that ITPs offer, such as methods for dealing with expressions and tactics. The framework includes an extensible structured pretty-printing engine that enables advanced interaction with expressions such as interactive term rewriting. We exemplify the framework with an implementation for the https://leanprover-community.github.io. The framework is already in use by hundreds of contributors to the Lean mathematical library.
Cite as

Edward W. Ayers, Mateja Jamnik, and W. T. Gowers. A Graphical User Interface Framework for Formal Verification. In 12th International Conference on Interactive Theorem Proving (ITP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 193, pp. 4:1-4:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{ayers_et_al:LIPIcs.ITP.2021.4,
  author =	{Ayers, Edward W. and Jamnik, Mateja and Gowers, W. T.},
  title =	{{A Graphical User Interface Framework for Formal Verification}},
  booktitle =	{12th International Conference on Interactive Theorem Proving (ITP 2021)},
  pages =	{4:1--4:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-188-7},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{193},
  editor =	{Cohen, Liron and Kaliszyk, Cezary},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITP.2021.4},
  URN =		{urn:nbn:de:0030-drops-138996},
  doi =		{10.4230/LIPIcs.ITP.2021.4},
  annote =	{Keywords: User Interfaces, ITP}
}
Document
DOI: 10.4230/DagSemProc.09391.5
Weighted L_2 B Discrepancy and Approximation of Integrals over Reproducing Kernel Hilbert Spaces

Authors: Michael Gnewuch

Published in: Dagstuhl Seminar Proceedings, Volume 9391, Algorithms and Complexity for Continuous Problems (2009)

Abstract
We extend the notion of $L_2$ $B$ discrepancy provided in [E. Novak, H. Wo'zniakowski, $L_2$ discrepancy and multivariate integration, in: Analytic number theory. Essays in honour of Klaus Roth. W. W. L. Chen, W. T. Gowers, H. Halberstam, W. M. Schmidt, and R. C. Vaughan (Eds.), Cambridge University Press, Cambridge, 2009, 359 – 388] to the weighted $L_2$ $mathcal{B}$ discrepancy. This newly defined notion allows to consider weights, but also volume measures different from the Lebesgue measure and classes of test sets different from measurable subsets of some Euclidean space. We relate the weighted $L_2$ $mathcal{B}$ discrepancy to numerical integration defined over weighted reproducing kernel Hilbert spaces and settle in this way an open problem posed by Novak and Wo'zniakowski.
Cite as

Michael Gnewuch. Weighted L_2 B Discrepancy and Approximation of Integrals over Reproducing Kernel Hilbert Spaces. In Algorithms and Complexity for Continuous Problems. Dagstuhl Seminar Proceedings, Volume 9391, pp. 1-9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{gnewuch:DagSemProc.09391.5,
  author =	{Gnewuch, Michael},
  title =	{{Weighted L\underline2 B Discrepancy and Approximation of Integrals over Reproducing Kernel Hilbert Spaces}},
  booktitle =	{Algorithms and Complexity for Continuous Problems},
  pages =	{1--9},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2009},
  volume =	{9391},
  editor =	{Thomas M\"{u}ller-Gronbach and Leszek Plaskota and Joseph. F. Traub},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.09391.5},
  URN =		{urn:nbn:de:0030-drops-22966},
  doi =		{10.4230/DagSemProc.09391.5},
  annote =	{Keywords: Discrepancy, Numerical Integration, Quasi-Monte Carlo, Reproducing Kernel Hilbert Space}
}
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