24 Search Results for "Qiao, Youming"

Document
DOI: 10.4230/LIPIcs.ITCS.2026.7
An Unholy Trinity: TFNP, Polynomial Systems, and the Quantum Satisfiability Problem

Authors: Marco Aldi, Sevag Gharibian, and Dorian Rudolph

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

Abstract
The theory of Total Function NP (TFNP) and its subclasses says that, even if one is promised an efficiently verifiable proof exists for a problem, finding this proof can be intractable. Despite the success of the theory at showing intractability of problems such as computing Brouwer fixed points and Nash equilibria, subclasses of TFNP remain arguably few and far between. In this work, we define two new subclasses of TFNP borne of the study of complex polynomial systems: Multi-homogeneous Systems (MHS) and Sparse Fundamental Theorem of Algebra (SFTA). The first of these is based on Bézout’s theorem from algebraic geometry, marking the first TFNP subclass based on an algebraic geometric principle. At the heart of our study is the computational problem known as Quantum SAT (QSAT) with a System of Distinct Representatives (SDR), first studied by [Laumann, Läuchli, Moessner, Scardicchio, and Sondhi 2010]. Among other results, we show that QSAT with SDR is MHS-complete, thus giving not only the first link between quantum complexity theory and TFNP, but also the first TFNP problem whose classical variant (SAT with SDR) is easy but whose quantum variant is hard. We also show how to embed the roots of a sparse, high-degree, univariate polynomial into QSAT with SDR, obtaining that SFTA is contained in a zero-error version of MHS. We conjecture this construction also works in the low-error setting, which would imply SFTA ⊆ MHS.
Cite as

Marco Aldi, Sevag Gharibian, and Dorian Rudolph. An Unholy Trinity: TFNP, Polynomial Systems, and the Quantum Satisfiability Problem. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 7:1-7:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{aldi_et_al:LIPIcs.ITCS.2026.7,
  author =	{Aldi, Marco and Gharibian, Sevag and Rudolph, Dorian},
  title =	{{An Unholy Trinity: TFNP, Polynomial Systems, and the Quantum Satisfiability Problem}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{7:1--7:24},
  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.2026.7},
  URN =		{urn:nbn:de:0030-drops-252946},
  doi =		{10.4230/LIPIcs.ITCS.2026.7},
  annote =	{Keywords: quantum complexity theory, Quantum Merlin Arthur (QMA), Quantum Satisfiability Problem (QSAT), total function NP (TFNP)}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2026.52
Diffie-Hellman Key Exchange from Commutativity to Group Laws

Authors: Dung Hoang Duong, Youming Qiao, and Chuanqi Zhang

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

Abstract
In Diffie-Hellman key exchange, the commutativity of power operations is instrumental in the agreement of keys. Viewing commutativity as a law in abelian groups, we propose Diffie-Hellman key exchange in the group action framework (Brassard-Yung, Crypto'90; Ji-Qiao-Song-Yun, TCC'19), for actions of non-abelian groups with laws. The security of this protocol is shown, following Fischlin, Günther, Schmidt, and Warinschi (IEEE S&P'16), based on a pseudorandom group action assumption. A concrete instantiation is proposed based on the monomial code equivalence problem.
Cite as

Dung Hoang Duong, Youming Qiao, and Chuanqi Zhang. Diffie-Hellman Key Exchange from Commutativity to Group Laws. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 52:1-52:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{duong_et_al:LIPIcs.ITCS.2026.52,
  author =	{Duong, Dung Hoang and Qiao, Youming and Zhang, Chuanqi},
  title =	{{Diffie-Hellman Key Exchange from Commutativity to Group Laws}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{52:1--52:20},
  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.2026.52},
  URN =		{urn:nbn:de:0030-drops-253396},
  doi =		{10.4230/LIPIcs.ITCS.2026.52},
  annote =	{Keywords: Diffie-Hellman, Key Exchange, Group Laws, Group Actions, Code Equivalence}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2026.32
Vanishing Signatures, Orbit Closure, and the Converse of the Holant Theorem

Authors: Jin-Yi Cai and Ben Young

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

Abstract
Valiant’s Holant theorem is a powerful tool for algorithms and reductions for counting problems. It states that if two sets ℱ and 𝒢 of tensors (a.k.a. constraint functions or signatures) are related by a holographic transformation, then ℱ and 𝒢 are Holant-indistinguishable, i.e., every tensor network using tensors from ℱ, respectively from 𝒢, contracts to the same value. Xia (ICALP 2010) conjectured the converse of the Holant theorem, but a counterexample was found based on vanishing signatures, those which are Holant-indistinguishable from 0. We prove two near-converses of the Holant theorem using techniques from invariant theory. (I) Holant-indistinguishable ℱ and 𝒢 always admit two sequences of holographic transformations mapping them arbitrarily close to each other, i.e., their GL_q-orbit closures intersect. (II) We show that vanishing signatures are the only true obstacle to a converse of the Holant theorem. As corollaries of the two theorems we obtain the first characterization of homomorphism-indistinguishability over graphs of bounded degree, a long standing open problem, and show that two graphs with invertible adjacency matrices are isomorphic if and only if they are homomorphism-indistinguishable over graphs with maximum degree at most three. We also show that Holant-indistinguishability is complete for a complexity class TOCI introduced by Lysikov and Walter [Vladimir Lysikov and Michael Walter, 2024], and hence hard for graph isomorphism.
Cite as

Jin-Yi Cai and Ben Young. Vanishing Signatures, Orbit Closure, and the Converse of the Holant Theorem. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 32:1-32:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{cai_et_al:LIPIcs.ITCS.2026.32,
  author =	{Cai, Jin-Yi and Young, Ben},
  title =	{{Vanishing Signatures, Orbit Closure, and the Converse of the Holant Theorem}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{32:1--32:20},
  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.2026.32},
  URN =		{urn:nbn:de:0030-drops-253198},
  doi =		{10.4230/LIPIcs.ITCS.2026.32},
  annote =	{Keywords: Holant, Orbit Closure Intersection, Homomorphism Indistinguishability, Tensor Network}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2025.78
Strong Keys for Tensor Isomorphism Cryptography

Authors: Anand Kumar Narayanan

Published in: LIPIcs, Volume 345, 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)

Abstract
Sampling a non degenerate (that is, invertible) square matrix over a finite field is easy, draw a random square matrix and discard if the determinant is zero. We address the problem in higher dimensions, and sample non degenerate boundary format tensors, which generalise square matrices. Testing degeneracy is conjectured to be hard in more than two dimensions [Hillar and Lim, 2013], precluding the "draw a random tensor and discard if degenerate" recipe. The difficulty is in computing hyperdeterminants, higher dimensional analogues of determinants. Instead, we start with a structured random non degenerate tensor and scramble it by infusing more randomness while still preserving non degeneracy. We propose two kinds of scrambling. The first is multiplication in each dimension by random invertible matrices, which preserves dimension and format. Assuming pseudo randomness of this action, which also underlies tensor isomorphism based cryptography, our samples are computationally indistinguishable from uniform non degenerate tensors. The second scrambling employs tensor convolution (that generalises multiplication by matrices) and can increase dimension. Inspired by hyperdeterminant multiplicativity, we devise a recursive sampler that uses tensor convolution to reduce the problem from arbitrary to three dimensions. Our sampling is a candidate solution for drawing public keys in tensor isomorphism based cryptography, since non degenerate tensors elude recent weak key attacks targeting public key tensors either containing geometric structures such as "triangles" [Lars Ran and Simona Samardjiska, 2024] or being deficient in tensor rank [Gilchrist et al., 2024]. To accommodate our sampling, tensor isomorphism based schemes need to be instantiated in boundary formats such as (2k+1) × (k+1) × (k+1), away from the more familiar k × k × k cubic formats. Our sampling (along with the recent tensor trapdoor one-way functions [Anand Kumar Narayanan, 2025]) makes an enticing case to transition tensor isomorphism cryptography to boundary formats tensors, which are true analogues of square matrices.
Cite as

Anand Kumar Narayanan. Strong Keys for Tensor Isomorphism Cryptography. In 50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 345, pp. 78:1-78:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{narayanan:LIPIcs.MFCS.2025.78,
  author =	{Narayanan, Anand Kumar},
  title =	{{Strong Keys for Tensor Isomorphism Cryptography}},
  booktitle =	{50th International Symposium on Mathematical Foundations of Computer Science (MFCS 2025)},
  pages =	{78:1--78:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-388-1},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{345},
  editor =	{Gawrychowski, Pawe{\l} and Mazowiecki, Filip and Skrzypczak, Micha{\l}},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2025.78},
  URN =		{urn:nbn:de:0030-drops-241857},
  doi =		{10.4230/LIPIcs.MFCS.2025.78},
  annote =	{Keywords: tensors, finite fields, post-quantum cryptography}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2025.99
Algorithmic Aspects of Semistability of Quiver Representations

Authors: Yuni Iwamasa, Taihei Oki, and Tasuku Soma

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract
We study the semistability of quiver representations from an algorithmic perspective. We present efficient algorithms for several fundamental computational problems on the semistability of quiver representations: deciding the semistability and σ-semistability, finding the maximizers of King’s criterion, and computing the Harder-Narasimhan filtration. We also investigate a class of polyhedral cones defined by the linear system in King’s criterion, which we refer to as King cones. For rank-one representations, we demonstrate that these King cones can be encoded by submodular flow polytopes, enabling us to decide the σ-semistability in strongly polynomial time. Our approach employs submodularity in quiver representations, which may be of independent interest.
Cite as

Yuni Iwamasa, Taihei Oki, and Tasuku Soma. Algorithmic Aspects of Semistability of Quiver Representations. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 99:1-99:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{iwamasa_et_al:LIPIcs.ICALP.2025.99,
  author =	{Iwamasa, Yuni and Oki, Taihei and Soma, Tasuku},
  title =	{{Algorithmic Aspects of Semistability of Quiver Representations}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{99:1--99:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.99},
  URN =		{urn:nbn:de:0030-drops-234762},
  doi =		{10.4230/LIPIcs.ICALP.2025.99},
  annote =	{Keywords: quivers, \sigma-semistability, King’s criterion, operator scaling, submodular flow}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2025.62
A High Dimensional Cramer’s Rule Connecting Homogeneous Multilinear Equations to Hyperdeterminants

Authors: Antoine Joux and Anand Kumar Narayanan

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

Abstract
We present a new algorithm for solving homogeneous multilinear equations, which are high dimensional generalisations of solving homogeneous linear equations. First, we present a linear time reduction from solving generic homogeneous multilinear equations to computing hyperdeterminants, via a high dimensional Cramer’s rule. Hyperdeterminants are generalisations of determinants, associated with tensors of formats generalising square matrices. Second, we devise arithmetic circuits to compute hyperdeterminants of boundary format tensors. Boundary format tensors are those that generalise square matrices in the strictest sense. Consequently, we obtain arithmetic circuits for solving generic homogeneous boundary format multilinear equations. The complexity as a function of the input dimension varies across boundary format families, ranging from quasi-polynomial to sub exponential. Curiously, the quasi-polynomial complexity arises for families of increasing dimension, including the family of multipartite quantum systems made of d qubits and one qudit. Finally, we identify potential directions to resolve the hardness the hyperdeterminants, notably modulo prime numbers through the cryptographically significant tensor isomorphism complexity class.
Cite as

Antoine Joux and Anand Kumar Narayanan. A High Dimensional Cramer’s Rule Connecting Homogeneous Multilinear Equations to Hyperdeterminants. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 62:1-62:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{joux_et_al:LIPIcs.ITCS.2025.62,
  author =	{Joux, Antoine and Narayanan, Anand Kumar},
  title =	{{A High Dimensional Cramer’s Rule Connecting Homogeneous Multilinear Equations to Hyperdeterminants}},
  booktitle =	{16th Innovations in Theoretical Computer Science Conference (ITCS 2025)},
  pages =	{62:1--62:16},
  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.2025.62},
  URN =		{urn:nbn:de:0030-drops-226904},
  doi =		{10.4230/LIPIcs.ITCS.2025.62},
  annote =	{Keywords: arithmetic circuits, tensors, determinants, hyperdeterminants}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2024.31
On the Complexity of Isomorphism Problems for Tensors, Groups, and Polynomials III: Actions by Classical Groups

Authors: Zhili Chen, Joshua A. Grochow, Youming Qiao, Gang Tang, and Chuanqi Zhang

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

Abstract
We study the complexity of isomorphism problems for d-way arrays, or tensors, under natural actions by classical groups such as orthogonal, unitary, and symplectic groups. These problems arise naturally in statistical data analysis and quantum information. We study two types of complexity-theoretic questions. First, for a fixed action type (isomorphism, conjugacy, etc.), we relate the complexity of the isomorphism problem over a classical group to that over the general linear group. Second, for a fixed group type (orthogonal, unitary, or symplectic), we compare the complexity of the isomorphism problems for different actions. Our main results are as follows. First, for orthogonal and symplectic groups acting on 3-way arrays, the isomorphism problems reduce to the corresponding problems over the general linear group. Second, for orthogonal and unitary groups, the isomorphism problems of five natural actions on 3-way arrays are polynomial-time equivalent, and the d-tensor isomorphism problem reduces to the 3-tensor isomorphism problem for any fixed d > 3. For unitary groups, the preceding result implies that LOCC classification of tripartite quantum states is at least as difficult as LOCC classification of d-partite quantum states for any d. Lastly, we also show that the graph isomorphism problem reduces to the tensor isomorphism problem over orthogonal and unitary groups.
Cite as

Zhili Chen, Joshua A. Grochow, Youming Qiao, Gang Tang, and Chuanqi Zhang. On the Complexity of Isomorphism Problems for Tensors, Groups, and Polynomials III: Actions by Classical Groups. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 31:1-31:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{chen_et_al:LIPIcs.ITCS.2024.31,
  author =	{Chen, Zhili and Grochow, Joshua A. and Qiao, Youming and Tang, Gang and Zhang, Chuanqi},
  title =	{{On the Complexity of Isomorphism Problems for Tensors, Groups, and Polynomials III: Actions by Classical Groups}},
  booktitle =	{15th Innovations in Theoretical Computer Science Conference (ITCS 2024)},
  pages =	{31:1--31:23},
  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.2024.31},
  URN =		{urn:nbn:de:0030-drops-195595},
  doi =		{10.4230/LIPIcs.ITCS.2024.31},
  annote =	{Keywords: complexity class, tensor isomorphism, polynomial isomorphism, group isomorphism, local operations and classical communication}
}
Document
DOI: 10.4230/LIPIcs.STACS.2022.26
The Isomorphism Problem for Plain Groups Is in Σ₃^{𝖯}

Authors: Heiko Dietrich, Murray Elder, Adam Piggott, Youming Qiao, and Armin Weiß

Published in: LIPIcs, Volume 219, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)

Abstract
Testing isomorphism of infinite groups is a classical topic, but from the complexity theory viewpoint, few results are known. Sénizergues and the fifth author (ICALP2018) proved that the isomorphism problem for virtually free groups is decidable in PSPACE when the input is given in terms of so-called virtually free presentations. Here we consider the isomorphism problem for the class of plain groups, that is, groups that are isomorphic to a free product of finitely many finite groups and finitely many copies of the infinite cyclic group. Every plain group is naturally and efficiently presented via an inverse-closed finite convergent length-reducing rewriting system. We prove that the isomorphism problem for plain groups given in this form lies in the polynomial time hierarchy, more precisely, in Σ₃^𝖯. This result is achieved by combining new geometric and algebraic characterisations of groups presented by inverse-closed finite convergent length-reducing rewriting systems developed in recent work of the second and third authors (2021) with classical finite group isomorphism results of Babai and Szemerédi (1984).
Cite as

Heiko Dietrich, Murray Elder, Adam Piggott, Youming Qiao, and Armin Weiß. The Isomorphism Problem for Plain Groups Is in Σ₃^{𝖯}. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 26:1-26:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{dietrich_et_al:LIPIcs.STACS.2022.26,
  author =	{Dietrich, Heiko and Elder, Murray and Piggott, Adam and Qiao, Youming and Wei{\ss}, Armin},
  title =	{{The Isomorphism Problem for Plain Groups Is in \Sigma₃^\{𝖯\}}},
  booktitle =	{39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
  pages =	{26:1--26:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-222-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{219},
  editor =	{Berenbrink, Petra and Monmege, Benjamin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.26},
  URN =		{urn:nbn:de:0030-drops-158368},
  doi =		{10.4230/LIPIcs.STACS.2022.26},
  annote =	{Keywords: plain group, isomorphism problem, polynomial hierarchy, \Sigma₃^\{𝖯\} complexity class, inverse-closed finite convergent length-reducing rewriting system}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2022.87
Symbolic Determinant Identity Testing and Non-Commutative Ranks of Matrix Lie Algebras

Authors: Gábor Ivanyos, Tushant Mittal, and Youming Qiao

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

Abstract
One approach to make progress on the symbolic determinant identity testing (SDIT) problem is to study the structure of singular matrix spaces. After settling the non-commutative rank problem (Garg-Gurvits-Oliveira-Wigderson, Found. Comput. Math. 2020; Ivanyos-Qiao-Subrahmanyam, Comput. Complex. 2018), a natural next step is to understand singular matrix spaces whose non-commutative rank is full. At present, examples of such matrix spaces are mostly sporadic, so it is desirable to discover them in a more systematic way. In this paper, we make a step towards this direction, by studying the family of matrix spaces that are closed under the commutator operation, that is, matrix Lie algebras. On the one hand, we demonstrate that matrix Lie algebras over the complex number field give rise to singular matrix spaces with full non-commutative ranks. On the other hand, we show that SDIT of such spaces can be decided in deterministic polynomial time. Moreover, we give a characterization for the matrix Lie algebras to yield a matrix space possessing singularity certificates as studied by Lovász (B. Braz. Math. Soc., 1989) and Raz and Wigderson (Building Bridges II, 2019).
Cite as

Gábor Ivanyos, Tushant Mittal, and Youming Qiao. Symbolic Determinant Identity Testing and Non-Commutative Ranks of Matrix Lie Algebras. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 87:1-87:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{ivanyos_et_al:LIPIcs.ITCS.2022.87,
  author =	{Ivanyos, G\'{a}bor and Mittal, Tushant and Qiao, Youming},
  title =	{{Symbolic Determinant Identity Testing and Non-Commutative Ranks of Matrix Lie Algebras}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{87:1--87:21},
  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.2022.87},
  URN =		{urn:nbn:de:0030-drops-156837},
  doi =		{10.4230/LIPIcs.ITCS.2022.87},
  annote =	{Keywords: derandomization, polynomial identity testing, symbolic determinant, non-commutative rank, Lie algebras}
}
Document
DOI: 10.4230/LIPIcs.CCC.2021.16
On p-Group Isomorphism: Search-To-Decision, Counting-To-Decision, and Nilpotency Class Reductions via Tensors

Authors: Joshua A. Grochow and Youming Qiao

Published in: LIPIcs, Volume 200, 36th Computational Complexity Conference (CCC 2021)

Abstract
In this paper we study some classical complexity-theoretic questions regarding Group Isomorphism (GpI). We focus on p-groups (groups of prime power order) with odd p, which are believed to be a bottleneck case for GpI, and work in the model of matrix groups over finite fields. Our main results are as follows. - Although search-to-decision and counting-to-decision reductions have been known for over four decades for Graph Isomorphism (GI), they had remained open for GpI, explicitly asked by Arvind & Torán (Bull. EATCS, 2005). Extending methods from Tensor Isomorphism (Grochow & Qiao, ITCS 2021), we show moderately exponential-time such reductions within p-groups of class 2 and exponent p. - Despite the widely held belief that p-groups of class 2 and exponent p are the hardest cases of GpI, there was no reduction to these groups from any larger class of groups. Again using methods from Tensor Isomorphism (ibid.), we show the first such reduction, namely from isomorphism testing of p-groups of "small" class and exponent p to those of class two and exponent p. For the first results, our main innovation is to develop linear-algebraic analogues of classical graph coloring gadgets, a key technique in studying the structural complexity of GI. Unlike the graph coloring gadgets, which support restricting to various subgroups of the symmetric group, the problems we study require restricting to various subgroups of the general linear group, which entails significantly different and more complicated gadgets. The analysis of one of our gadgets relies on a classical result from group theory regarding random generation of classical groups (Kantor & Lubotzky, Geom. Dedicata, 1990). For the nilpotency class reduction, we combine a runtime analysis of the Lazard Correspondence with Tensor Isomorphism-completeness results (Grochow & Qiao, ibid.).
Cite as

Joshua A. Grochow and Youming Qiao. On p-Group Isomorphism: Search-To-Decision, Counting-To-Decision, and Nilpotency Class Reductions via Tensors. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 16:1-16:38, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{grochow_et_al:LIPIcs.CCC.2021.16,
  author =	{Grochow, Joshua A. and Qiao, Youming},
  title =	{{On p-Group Isomorphism: Search-To-Decision, Counting-To-Decision, and Nilpotency Class Reductions via Tensors}},
  booktitle =	{36th Computational Complexity Conference (CCC 2021)},
  pages =	{16:1--16:38},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-193-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{200},
  editor =	{Kabanets, Valentine},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2021.16},
  URN =		{urn:nbn:de:0030-drops-142905},
  doi =		{10.4230/LIPIcs.CCC.2021.16},
  annote =	{Keywords: group isomorphism, search-to-decision reduction, counting-to-decision reduction, nilpotent group isomorphism, p-group isomorphism, tensor isomorphism}
}
Document
DOI: 10.4230/LIPIcs.STACS.2021.38
Average-Case Algorithms for Testing Isomorphism of Polynomials, Algebras, and Multilinear Forms

Authors: Joshua A. Grochow, Youming Qiao, and Gang Tang

Published in: LIPIcs, Volume 187, 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)

Abstract
We study the problems of testing isomorphism of polynomials, algebras, and multilinear forms. Our first main results are average-case algorithms for these problems. For example, we develop an algorithm that takes two cubic forms f, g ∈ 𝔽_q[x_1, … , x_n], and decides whether f and g are isomorphic in time q^O(n) for most f. This average-case setting has direct practical implications, having been studied in multivariate cryptography since the 1990s. Our second result concerns the complexity of testing equivalence of alternating trilinear forms. This problem is of interest in both mathematics and cryptography. We show that this problem is polynomial-time equivalent to testing equivalence of symmetric trilinear forms, by showing that they are both Tensor Isomorphism-complete (Grochow & Qiao, ITCS 2021), therefore is equivalent to testing isomorphism of cubic forms over most fields.
Cite as

Joshua A. Grochow, Youming Qiao, and Gang Tang. Average-Case Algorithms for Testing Isomorphism of Polynomials, Algebras, and Multilinear Forms. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 38:1-38:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{grochow_et_al:LIPIcs.STACS.2021.38,
  author =	{Grochow, Joshua A. and Qiao, Youming and Tang, Gang},
  title =	{{Average-Case Algorithms for Testing Isomorphism of Polynomials, Algebras, and Multilinear Forms}},
  booktitle =	{38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021)},
  pages =	{38:1--38:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-180-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{187},
  editor =	{Bl\"{a}ser, Markus and Monmege, Benjamin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2021.38},
  URN =		{urn:nbn:de:0030-drops-136836},
  doi =		{10.4230/LIPIcs.STACS.2021.38},
  annote =	{Keywords: polynomial isomorphism, trilinear form equivalence, algebra isomorphism, average-case algorithms, tensor isomorphism complete, symmetric and alternating bilinear maps}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2021.11
Sample Efficient Identity Testing and Independence Testing of Quantum States

Authors: Nengkun Yu

Published in: LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

Abstract
In this paper, we study the quantum identity testing problem, i.e., testing whether two given quantum states are identical, and quantum independence testing problem, i.e., testing whether a given multipartite quantum state is in tensor product form. For the quantum identity testing problem of 𝒟(ℂ^d) system, we provide a deterministic measurement scheme that uses 𝒪(d²/ε²) copies via independent measurements with d being the dimension of the state and ε being the additive error. For the independence testing problem 𝒟(ℂ^d₁⊗ℂ^{d₂}⊗⋯⊗ℂ^{d_m}) system, we show that the sample complexity is Θ̃((Π_{i = 1}^m d_i)/ε²) via collective measurements, and 𝒪((Π_{i = 1}^m d_i²)/ε²) via independent measurements. If randomized choice of independent measurements are allowed, the sample complexity is Θ(d^{3/2}/ε²) for the quantum identity testing problem, and Θ̃((Π_{i = 1}^m d_i^{3/2})/ε²) for the quantum independence testing problem.
Cite as

Nengkun Yu. Sample Efficient Identity Testing and Independence Testing of Quantum States. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 11:1-11:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{yu:LIPIcs.ITCS.2021.11,
  author =	{Yu, Nengkun},
  title =	{{Sample Efficient Identity Testing and Independence Testing of Quantum States}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{11:1--11:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-177-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{185},
  editor =	{Lee, James R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.11},
  URN =		{urn:nbn:de:0030-drops-135504},
  doi =		{10.4230/LIPIcs.ITCS.2021.11},
  annote =	{Keywords: Quantum property testing}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2021.31
On the Complexity of Isomorphism Problems for Tensors, Groups, and Polynomials I: Tensor Isomorphism-Completeness

Authors: Joshua A. Grochow and Youming Qiao

Published in: LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

Abstract
We study the complexity of isomorphism problems for tensors, groups, and polynomials. These problems have been studied in multivariate cryptography, machine learning, quantum information, and computational group theory. We show that these problems are all polynomial-time equivalent, creating bridges between problems traditionally studied in myriad research areas. This prompts us to define the complexity class TI, namely problems that reduce to the Tensor Isomorphism (TI) problem in polynomial time. Our main technical result is a polynomial-time reduction from d-tensor isomorphism to 3-tensor isomorphism. In the context of quantum information, this result gives multipartite-to-tripartite entanglement transformation procedure, that preserves equivalence under stochastic local operations and classical communication (SLOCC).
Cite as

Joshua A. Grochow and Youming Qiao. On the Complexity of Isomorphism Problems for Tensors, Groups, and Polynomials I: Tensor Isomorphism-Completeness. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 31:1-31:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{grochow_et_al:LIPIcs.ITCS.2021.31,
  author =	{Grochow, Joshua A. and Qiao, Youming},
  title =	{{On the Complexity of Isomorphism Problems for Tensors, Groups, and Polynomials I: Tensor Isomorphism-Completeness}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{31:1--31:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-177-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{185},
  editor =	{Lee, James R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.31},
  URN =		{urn:nbn:de:0030-drops-135702},
  doi =		{10.4230/LIPIcs.ITCS.2021.31},
  annote =	{Keywords: complexity class, tensor isomorphism, polynomial isomorphism, group isomorphism, stochastic local operations and classical communication}
}
Document
DOI: 10.4230/LIPIcs.ESA.2020.26
Improved Algorithms for Alternating Matrix Space Isometry: From Theory to Practice

Authors: Peter A. Brooksbank, Yinan Li, Youming Qiao, and James B. Wilson

Published in: LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)

Abstract
Motivated by testing isomorphism of p-groups, we study the alternating matrix space isometry problem (AltMatSpIso), which asks to decide whether two m-dimensional subspaces of n×n alternating (skew-symmetric if the field is not of characteristic 2) matrices are the same up to a change of basis. Over a finite field 𝔽_p with some prime p≠2, solving AltMatSpIso in time p^O(n+m) is equivalent to testing isomorphism of p-groups of class 2 and exponent p in time polynomial in the group order. The latter problem has long been considered a bottleneck case for the group isomorphism problem. Recently, Li and Qiao presented an average-case algorithm for AltMatSpIso in time p^O(n) when n and m are linearly related (FOCS '17). In this paper, we present an average-case algorithm for AltMatSpIso in time p^O(n+m). Besides removing the restriction on the relation between n and m, our algorithm is considerably simpler, and the average-case analysis is stronger. We then implement our algorithm, with suitable modifications, in Magma. Our experiments indicate that it improves significantly over default (brute-force) algorithms for this problem.
Cite as

Peter A. Brooksbank, Yinan Li, Youming Qiao, and James B. Wilson. Improved Algorithms for Alternating Matrix Space Isometry: From Theory to Practice. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 26:1-26:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{brooksbank_et_al:LIPIcs.ESA.2020.26,
  author =	{Brooksbank, Peter A. and Li, Yinan and Qiao, Youming and Wilson, James B.},
  title =	{{Improved Algorithms for Alternating Matrix Space Isometry: From Theory to Practice}},
  booktitle =	{28th Annual European Symposium on Algorithms (ESA 2020)},
  pages =	{26:1--26:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-162-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{173},
  editor =	{Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.26},
  URN =		{urn:nbn:de:0030-drops-128920},
  doi =		{10.4230/LIPIcs.ESA.2020.26},
  annote =	{Keywords: Alternating Matrix Spaces, Average-case Algorithm, p-groups of Class 2nd Exponent p, Magma}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2020.72
Randomized Polynomial-Time Equivalence Between Determinant and Trace-IMM Equivalence Tests

Authors: Janaky Murthy, Vineet Nair, and Chandan Saha

Published in: LIPIcs, Volume 170, 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)

Abstract
Equivalence testing for a polynomial family {g_m}_{m ∈ ℕ} over a field 𝔽 is the following problem: Given black-box access to an n-variate polynomial f({𝐱}), where n is the number of variables in g_m for some m ∈ ℕ, check if there exists an A ∈ GL(n,𝔽) such that f({𝐱}) = g_m(A{𝐱}). If yes, then output such an A. The complexity of equivalence testing has been studied for a number of important polynomial families, including the determinant (Det) and the family of iterated matrix multiplication polynomials. Two popular variants of the iterated matrix multiplication polynomial are: IMM_{w,d} (the (1,1) entry of the product of d many w× w symbolic matrices) and Tr-IMM_{w,d} (the trace of the product of d many w× w symbolic matrices). The families - Det, IMM and Tr-IMM - are VBP-complete under p-projections, and so, in this sense, they have the same complexity. But, do they have the same equivalence testing complexity? We show that the answer is "yes" for Det and Tr-IMM (modulo the use of randomness). The above result may appear a bit surprising as the complexity of equivalence testing for IMM and that for Det are quite different over ℚ: a randomized poly-time equivalence testing for IMM over ℚ is known [Neeraj Kayal et al., 2019], whereas [Ankit Garg et al., 2019] showed that equivalence testing for Det over ℚ is integer factoring hard (under randomized reductions and assuming GRH). To our knowledge, the complexity of equivalence testing for Tr-IMM was not known before this work. We show that, despite the syntactic similarity between IMM and Tr-IMM, equivalence testing for Tr-IMM and that for Det are randomized poly-time Turing reducible to each other over any field of characteristic zero or sufficiently large. The result is obtained by connecting the two problems via another well-studied problem in computer algebra, namely the full matrix algebra isomorphism problem (FMAI). In particular, we prove the following: 1) Testing equivalence of polynomials to Tr-IMM_{w,d}, for d ≥ 3 and w ≥ 2, is randomized polynomial-time Turing reducible to testing equivalence of polynomials to Det_w, the determinant of the w × w matrix of formal variables. (Here, d need not be a constant.) 2) FMAI is randomized polynomial-time Turing reducible to equivalence testing (in fact, to tensor isomorphism testing) for the family of matrix multiplication tensors {Tr-IMM_{w,3}}_{w ∈ ℕ}. These results, in conjunction with the randomized poly-time reduction (shown in [Ankit Garg et al., 2019]) from determinant equivalence testing to FMAI, imply that the four problems - FMAI, equivalence testing for Tr-IMM and for Det, and the 3-tensor isomorphism problem for the family of matrix multiplication tensors - are randomized poly-time equivalent under Turing reductions.
Cite as

Janaky Murthy, Vineet Nair, and Chandan Saha. Randomized Polynomial-Time Equivalence Between Determinant and Trace-IMM Equivalence Tests. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 72:1-72:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{murthy_et_al:LIPIcs.MFCS.2020.72,
  author =	{Murthy, Janaky and Nair, Vineet and Saha, Chandan},
  title =	{{Randomized Polynomial-Time Equivalence Between Determinant and Trace-IMM Equivalence Tests}},
  booktitle =	{45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
  pages =	{72:1--72:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-159-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{170},
  editor =	{Esparza, Javier and Kr\'{a}l', Daniel},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2020.72},
  URN =		{urn:nbn:de:0030-drops-127419},
  doi =		{10.4230/LIPIcs.MFCS.2020.72},
  annote =	{Keywords: equivalence testing, determinant, trace of the matrix product, full-matrix algebra isomorphism}
}
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