11 Search Results for "Joux, Antoine"

Document
DOI: 10.4230/LIPIcs.SoCG.2026.88
Integer Points in Dilates of Polytopes

Authors: Shubhangi Saraf and Narmada Varadarajan

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract
In this paper we study how the number of integer points in a polytope grows as we dilate the polytope. We prove new and essentially tight bounds on this quantity by specifically studying dilates of the Hadamard polytope. The motivation for studying this question comes from the question of understanding the maximal number of monomials in a factor of a multivariate polynomial of s monomials. A recent result by Bhargava, Saraf and Volkovich [Bhargava et al., 2020] showed that if f is an n-variate polynomial, where each variable has degree d, and f has s monomials, then any factor of f has at most s^{O(d² log n)} monomials. The key technical ingredient of their proof was to show that any polytope with s vertices, where each vertex lies in {0,..,d}ⁿ can have at most s^{O(d² log n)} integer points. The precise dependence on d of the number of integer points was left open. We show that this bound, and in particular the dependence on d is essentially tight by studying dilates of the Hadamard polytope and proving new lower bounds on its number of integer points.
Cite as

Shubhangi Saraf and Narmada Varadarajan. Integer Points in Dilates of Polytopes. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 88:1-88:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{saraf_et_al:LIPIcs.SoCG.2026.88,
  author =	{Saraf, Shubhangi and Varadarajan, Narmada},
  title =	{{Integer Points in Dilates of Polytopes}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{88:1--88:9},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.88},
  URN =		{urn:nbn:de:0030-drops-258948},
  doi =		{10.4230/LIPIcs.SoCG.2026.88},
  annote =	{Keywords: Computational geometry, Complexity theory, Integer polytopes}
}
Document
DOI: 10.4230/LIPIcs.STACS.2026.57
Improving Lagarias-Odlyzko Algorithm for Average-Case Subset Sum: Modular Arithmetic Approach

Authors: Antoine Joux and Karol Węgrzycki

Published in: LIPIcs, Volume 364, 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)

Abstract
Lagarias and Odlyzko (J.ACM 1985) proposed a polynomial-time algorithm for solving "almost all" instances of the Subset Sum problem with n integers of size Ω(Γ_LO), where log₂(Γ_LO) > n² log₂(γ) and γ is a parameter of the lattice basis reduction (γ > √{4/3} for LLL). The algorithm of Lagarias and Odlyzko is a cornerstone of cryptography. However, the theoretical guarantee on the density of feasible instances has remained unimproved for almost 40 years. In this paper, we propose an algorithm that solves "almost all" instances of Subset Sum with integers of size Ω(√{Γ_LO}) after a single call to lattice reduction. Additionally, our approach allows solving the Subset Sum problem for multiple targets, whereas the previous method could handle only one target per call to lattice basis reduction. We introduce a modular arithmetic approach to the Subset Sum problem, leveraging lattice reduction to solve a linear system modulo a suitably large prime. By analyzing the lengths of the LLL-reduced basis vectors of both the primal and dual lattices simultaneously, we show that density guarantees can be improved.
Cite as

Antoine Joux and Karol Węgrzycki. Improving Lagarias-Odlyzko Algorithm for Average-Case Subset Sum: Modular Arithmetic Approach. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 57:1-57:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{joux_et_al:LIPIcs.STACS.2026.57,
  author =	{Joux, Antoine and W\k{e}grzycki, Karol},
  title =	{{Improving Lagarias-Odlyzko Algorithm for Average-Case Subset Sum: Modular Arithmetic Approach}},
  booktitle =	{43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
  pages =	{57:1--57:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-412-3},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{364},
  editor =	{Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.57},
  URN =		{urn:nbn:de:0030-drops-255462},
  doi =		{10.4230/LIPIcs.STACS.2026.57},
  annote =	{Keywords: Average-Case Analysis, Subset Sum, Lattice Reduction, LLL}
}
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.2026.23
Decoding Balanced Linear Codes with Preprocessing

Authors: Andrej Bogdanov, Rohit Chatterjee, Yunqi Li, and Prashant Nalini Vasudevan

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

Abstract
Prange’s information set algorithm is a well-known decoding algorithm for linear codes. It decodes corrupted codewords of most 𝔽₂-linear codes C of message length n up to relative error rate O(log n / n) in poly(n) time. We show that the error rate can be improved to O((log n)² / n), provided: (1) the decoder has access to a polynomial-length advice string that depends on C only, and (2) C is n^{-Ω(1)}-balanced. As a consequence we improve the error tolerance in decoding random linear codes if inefficient preprocessing of the code is allowed. This reveals potential vulnerabilities in cryptographic applications of Learning Noisy Parities with low noise rate. Our main technical result is that the Hamming weight of Hw, where the rows of H are a random sample of short dual codewords, measures the proximity of a received word w to the code in the regime of interest. Given such H as advice, our algorithm corrects errors by locally minimizing this measure. We show that for most codes, the error rate tolerated by our decoder is asymptotically optimal among all algorithms whose decision is based on thresholding Hw for an arbitrary polynomial-size advice matrix H.
Cite as

Andrej Bogdanov, Rohit Chatterjee, Yunqi Li, and Prashant Nalini Vasudevan. Decoding Balanced Linear Codes with Preprocessing. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 23:1-23:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{bogdanov_et_al:LIPIcs.ITCS.2026.23,
  author =	{Bogdanov, Andrej and Chatterjee, Rohit and Li, Yunqi and Vasudevan, Prashant Nalini},
  title =	{{Decoding Balanced Linear Codes with Preprocessing}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{23:1--23:23},
  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.2026.23},
  URN =		{urn:nbn:de:0030-drops-253107},
  doi =		{10.4230/LIPIcs.ITCS.2026.23},
  annote =	{Keywords: Linear codes, nearest codeword problem, learning parity with noise}
}
Document
DOI: 10.4230/LIPIcs.ESA.2025.95
The Planted Orthogonal Vectors Problem

Authors: David Kühnemann, Adam Polak, and Alon Rosen

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

Abstract
In the k-Orthogonal Vectors (k-OV) problem we are given k sets, each containing n binary vectors of dimension d = n^o(1), and our goal is to pick one vector from each set so that at each coordinate at least one vector has a zero. It is a central problem in fine-grained complexity, conjectured to require n^{k-o(1)} time in the worst case. We propose a way to plant a solution among vectors with i.i.d. p-biased entries, for appropriately chosen p, so that the planted solution is the unique one. Our conjecture is that the resulting k-OV instances still require time n^{k-o(1)} to solve, on average. Our planted distribution has the property that any subset of strictly less than k vectors has the same marginal distribution as in the model distribution, consisting of i.i.d. p-biased random vectors. We use this property to give average-case search-to-decision reductions for k-OV.
Cite as

David Kühnemann, Adam Polak, and Alon Rosen. The Planted Orthogonal Vectors Problem. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 95:1-95:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{kuhnemann_et_al:LIPIcs.ESA.2025.95,
  author =	{K\"{u}hnemann, David and Polak, Adam and Rosen, Alon},
  title =	{{The Planted Orthogonal Vectors Problem}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{95:1--95:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2025.95},
  URN =		{urn:nbn:de:0030-drops-245640},
  doi =		{10.4230/LIPIcs.ESA.2025.95},
  annote =	{Keywords: Average-case complexity, fine-grained complexity, orthogonal vectors}
}
Document
DOI: 10.4230/LIPIcs.ESA.2025.86
New Algorithms for Pigeonhole Equal Subset Sum

Authors: Ce Jin, Ryan Williams, and Stan Zhang

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

Abstract
We study the Pigeonhole Equal Subset Sum problem, which is a total-search variant of the Subset Sum problem introduced by Papadimitriou (1994): we are given a set of n positive integers {w₁,…,w_n} with the additional restriction that ∑_{i=1}^n w_i < 2ⁿ - 1, and want to find two different subsets A,B ⊆ [n] such that ∑_{i∈A} w_i = ∑_{i∈B} w_i. Very recently, Jin and Wu (ICALP 2024) gave a randomized algorithm solving Pigeonhole Equal Subset Sum in O^*(2^{0.4n}) time, beating the classical meet-in-the-middle algorithm with O^*(2^{n/2}) runtime. In this paper, we refine Jin and Wu’s techniques to improve the runtime even further to O^*(2^{n/3}).
Cite as

Ce Jin, Ryan Williams, and Stan Zhang. New Algorithms for Pigeonhole Equal Subset Sum. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 86:1-86:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{jin_et_al:LIPIcs.ESA.2025.86,
  author =	{Jin, Ce and Williams, Ryan and Zhang, Stan},
  title =	{{New Algorithms for Pigeonhole Equal Subset Sum}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{86:1--86:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2025.86},
  URN =		{urn:nbn:de:0030-drops-245541},
  doi =		{10.4230/LIPIcs.ESA.2025.86},
  annote =	{Keywords: pigeonhole principle, subset sums}
}
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.2025.14
The More the Merrier! On Total Coding and Lattice Problems and the Complexity of Finding Multicollisions

Authors: Huck Bennett, Surendra Ghentiyala, and Noah Stephens-Davidowitz

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

Abstract
We show a number of connections between two types of search problems: (1) the problem of finding an L-wise multicollision in the output of a function; and (2) the problem of finding two codewords in a code (or two vectors in a lattice) that are within distance d of each other. Specifically, we study these problems in the total regime, in which L and d are chosen so that such a solution is guaranteed to exist, though it might be hard to find. In more detail, we study the total search problem in which the input is a function 𝒞 : [A] → [B] (represented as a circuit) and the goal is to find L ≤ ⌈A/B⌉ distinct elements x_1,…, x_L ∈ A such that 𝒞(x_1) = ⋯ = 𝒞(x_L). The associated complexity classes Polynomial Multi-Pigeonhole Principle ((A,B)-PMPP^L) consist of all problems that reduce to this problem. We show close connections between (A,B)-PMPP^L and many celebrated upper bounds on the minimum distance of a code or lattice (and on the list-decoding radius). In particular, we show that the associated computational problems (i.e., the problem of finding two distinct codewords or lattice points that are close to each other) are in (A,B)-PMPP^L, with a more-or-less smooth tradeoff between the distance d and the parameters A, B, and L. These connections are particularly rich in the case of codes, in which case we show that multiple incomparable bounds on the minimum distance lie in seemingly incomparable complexity classes. Surprisingly, we also show that the computational problems associated with some bounds on the minimum distance of codes are actually hard for these classes (for codes represented by arbitrary circuits). In fact, we show that finding two vectors within a certain distance d is actually hard for the important (and well-studied) class PWPP = (B²,B)-PMPP² in essentially all parameter regimes for which an efficient algorithm is not known, so that our hardness results are essentially tight. In fact, for some d (depending on the block length, message length, and alphabet size), we obtain both hardness and containment. We therefore completely settle the complexity of this problem for such parameters and add coding problems to the short list of problems known to be complete for PWPP. We also study (A,B)-PMPP^L as an interesting family of complexity classes in its own right, and we uncover a rich structure. Specifically, we use recent techniques from the cryptographic literature on multicollision-resistant hash functions to (1) show inclusions of the form (A,B)-PMPP^L ⊆ (A',B')-PMPP^L' for certain non-trivial parameters; (2) black-box separations between such classes in different parameter regimes; and (3) a non-black-box proof that (A,B)-PMPP^L ∈ FP if (A',B')-PMPP^L' ∈ FP for yet another parameter regime. We also show that (A,B)-PMPP^L lies in the recently introduced complexity class Polynomial Long Choice for some parameters.
Cite as

Huck Bennett, Surendra Ghentiyala, and Noah Stephens-Davidowitz. The More the Merrier! On Total Coding and Lattice Problems and the Complexity of Finding Multicollisions. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 14:1-14:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bennett_et_al:LIPIcs.ITCS.2025.14,
  author =	{Bennett, Huck and Ghentiyala, Surendra and Stephens-Davidowitz, Noah},
  title =	{{The More the Merrier! On Total Coding and Lattice Problems and the Complexity of Finding Multicollisions}},
  booktitle =	{16th Innovations in Theoretical Computer Science Conference (ITCS 2025)},
  pages =	{14:1--14:22},
  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.2025.14},
  URN =		{urn:nbn:de:0030-drops-226424},
  doi =		{10.4230/LIPIcs.ITCS.2025.14},
  annote =	{Keywords: Multicollisions, Error-correcting codes, Lattices}
}
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.ESA.2022.6
Classical and Quantum Algorithms for Variants of Subset-Sum via Dynamic Programming

Authors: Jonathan Allcock, Yassine Hamoudi, Antoine Joux, Felix Klingelhöfer, and Miklos Santha

Published in: LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022)

Abstract
Subset-Sum is an NP-complete problem where one must decide if a multiset of n integers contains a subset whose elements sum to a target value m. The best known classical and quantum algorithms run in time Õ(2^{n/2}) and Õ(2^{n/3}), respectively, based on the well-known meet-in-the-middle technique. Here we introduce a novel classical dynamic-programming-based data structure with applications to Subset-Sum and a number of variants, including Equal-Sums (where one seeks two disjoint subsets with the same sum), 2-Subset-Sum (a relaxed version of Subset-Sum where each item in the input set can be used twice in the summation), and Shifted-Sums, a generalization of both of these variants, where one seeks two disjoint subsets whose sums differ by some specified value. Given any modulus p, our data structure can be constructed in time O(np), after which queries can be made in time O(n) to the lists of subsets summing to any value modulo p. We use this data structure in combination with variable-time amplitude amplification and a new quantum pair finding algorithm, extending the quantum claw finding algorithm to the multiple solutions case, to give an O(2^{0.504n}) quantum algorithm for Shifted-Sums. This provides a notable improvement on the best known O(2^{0.773n}) classical running time established by Mucha et al. [Mucha et al., 2019]. We also study Pigeonhole Equal-Sums, a variant of Equal-Sums where the existence of a solution is guaranteed by the pigeonhole principle. For this problem we give faster classical and quantum algorithms with running time Õ(2^{n/2}) and Õ(2^{2n/5}), respectively.
Cite as

Jonathan Allcock, Yassine Hamoudi, Antoine Joux, Felix Klingelhöfer, and Miklos Santha. Classical and Quantum Algorithms for Variants of Subset-Sum via Dynamic Programming. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 6:1-6:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

Copy BibTex To Clipboard

@InProceedings{allcock_et_al:LIPIcs.ESA.2022.6,
  author =	{Allcock, Jonathan and Hamoudi, Yassine and Joux, Antoine and Klingelh\"{o}fer, Felix and Santha, Miklos},
  title =	{{Classical and Quantum Algorithms for Variants of Subset-Sum via Dynamic Programming}},
  booktitle =	{30th Annual European Symposium on Algorithms (ESA 2022)},
  pages =	{6:1--6:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-247-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{244},
  editor =	{Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.6},
  URN =		{urn:nbn:de:0030-drops-169444},
  doi =		{10.4230/LIPIcs.ESA.2022.6},
  annote =	{Keywords: Quantum algorithm, classical algorithm, dynamic programming, representation technique, subset-sum, equal-sum, shifted-sum}
}
Document
Invited Talk
DOI: 10.4230/LIPIcs.STACS.2017.3
Discrete Logarithms in Small Characteristic Finite Fields: a Survey of Recent Advances (Invited Talk)

Authors: Antoine Joux

Published in: LIPIcs, Volume 66, 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)

Abstract
The discrete logarithm problem is one of the few hard problems on which public-key cryptography can be based. It was introduced in the field by the famous Diffie-Hellman key exchange protocol. Initially, the cryptographic use of the problem was considered in prime fields, but was readily generalized to arbitrary finite fields and, later, to elliptic or higher genus curves. In this talk, we survey the key technical ideas that can be used to compute discrete logarithms, especially in the case of small characteristic finite fields. These ideas stem from about 40 years of research on the topic. They appeared along the long road that leads from the initial belief that this problem was hard enough for cryptographic purpose to the current state of the art where it can no longer be considered for cryptographic use. Indeed, after the recent developments started in 2012, we now have some very efficient practical algorithms to solve this problem. Unfortunately, these algorithms remain heuristic and one important direction for future research is to lift the remaining heuristic assumptions.
Cite as

Antoine Joux. Discrete Logarithms in Small Characteristic Finite Fields: a Survey of Recent Advances (Invited Talk). In 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 66, p. 3:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{joux:LIPIcs.STACS.2017.3,
  author =	{Joux, Antoine},
  title =	{{Discrete Logarithms in Small Characteristic Finite Fields: a Survey of Recent Advances}},
  booktitle =	{34th Symposium on Theoretical Aspects of Computer Science (STACS 2017)},
  pages =	{3:1--3:1},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-028-6},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{66},
  editor =	{Vollmer, Heribert and Vall\'{e}e, Brigitte},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2017.3},
  URN =		{urn:nbn:de:0030-drops-70313},
  doi =		{10.4230/LIPIcs.STACS.2017.3},
  annote =	{Keywords: Cryptography, Discrete logarithms, Finite fields}
}
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