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Documents authored by Narayanan, Anand Kumar

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
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
RANDOM
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2021.54
Candidate Tree Codes via Pascal Determinant Cubes

Authors: Inbar Ben Yaacov, Gil Cohen, and Anand Kumar Narayanan

Published in: LIPIcs, Volume 207, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)

Abstract
Tree codes are combinatorial structures introduced by Schulman [Schulman, 1993] as key ingredients in interactive coding schemes. Asymptotically-good tree codes are long known to exist, yet their explicit construction remains a notoriously hard open problem. Even proposing a plausible construction, without the burden of proof, is difficult and the defining tree code property requires structure that remains elusive. To the best of our knowledge, only one candidate appears in the literature, due to Moore and Schulman [Moore and Schulman, 2014]. We put forth a new candidate for an explicit asymptotically-good tree code. Our construction is an extension of the vanishing rate tree code by Cohen-Haeupler-Schulman [Cohen et al., 2018], and its correctness relies on a conjecture that we introduce on certain Pascal determinants indexed by the points of the Boolean hypercube. Furthermore, using the vanishing distance tree code by Gelles et al. [Gelles et al., 2016] enables us to present a construction that relies on an even weaker assumption. We furnish evidence supporting our conjecture through numerical computation, combinatorial arguments from planar path graphs and based on well-studied heuristics from arithmetic geometry.
Cite as

Inbar Ben Yaacov, Gil Cohen, and Anand Kumar Narayanan. Candidate Tree Codes via Pascal Determinant Cubes. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 54:1-54:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{benyaacov_et_al:LIPIcs.APPROX/RANDOM.2021.54,
  author =	{Ben Yaacov, Inbar and Cohen, Gil and Narayanan, Anand Kumar},
  title =	{{Candidate Tree Codes via Pascal Determinant Cubes}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)},
  pages =	{54:1--54:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-207-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{207},
  editor =	{Wootters, Mary and Sanit\`{a}, Laura},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.54},
  URN =		{urn:nbn:de:0030-drops-147474},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2021.54},
  annote =	{Keywords: Tree codes, Sparse polynomials, Explicit constructions}
}
Document
DOI: 10.4230/LIPIcs.MFCS.2016.47
Algebraic Problems Equivalent to Beating Exponent 3/2 for Polynomial Factorization over Finite Fields

Authors: Zeyu Guo, Anand Kumar Narayanan, and Chris Umans

Published in: LIPIcs, Volume 58, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)

Abstract
The fastest known algorithm for factoring univariate polynomials over finite fields is the Kedlaya-Umans (fast modular composition) implementation of the Kaltofen-Shoup algorithm. It is randomized and takes ~O(n^{3/2}*log(q)+n*log^2(q)) time to factor polynomials of degree n over the finite field F_q with q elements. A significant open problem is if the 3/2 exponent can be improved. We study a collection of algebraic problems and establish a web of reductions between them. A consequence is that an algorithm for any one of these problems with exponent better than 3/2 would yield an algorithm for polynomial factorization with exponent better than 3/2.
Cite as

Zeyu Guo, Anand Kumar Narayanan, and Chris Umans. Algebraic Problems Equivalent to Beating Exponent 3/2 for Polynomial Factorization over Finite Fields. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 47:1-47:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{guo_et_al:LIPIcs.MFCS.2016.47,
  author =	{Guo, Zeyu and Narayanan, Anand Kumar and Umans, Chris},
  title =	{{Algebraic Problems Equivalent to Beating Exponent 3/2 for Polynomial Factorization over Finite Fields}},
  booktitle =	{41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)},
  pages =	{47:1--47:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-016-3},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{58},
  editor =	{Faliszewski, Piotr and Muscholl, Anca and Niedermeier, Rolf},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.47},
  URN =		{urn:nbn:de:0030-drops-64609},
  doi =		{10.4230/LIPIcs.MFCS.2016.47},
  annote =	{Keywords: algorithms, complexity, finite fields, polynomials, factorization}
}
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