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arxiv: 2606.03807 · v1 · pith:HYDCHFOUnew · submitted 2026-06-02 · 💻 cs.CR · cs.CC

Collision Resistance of Single-Layer Neural Nets

Marco Benedetti , Andrej Bogdanov , Enrico M. Malatesta , Marc M\'ezard , Gianmarco Perrupato , Alon Rosen , Nikolaj I. Schwartzbach , Riccardo Zecchina This is my paper

Pith reviewed 2026-06-28 09:19 UTC · model grok-4.3

classification 💻 cs.CR cs.CC
keywords collision resistancesingle-layer neural networksoverlap gap propertyonline algorithmsactivation functionsbinary neural netscryptographic hardness
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The pith

Single-layer binary neural networks switch from easy to hard collision finding at activation threshold κ = Θ(1/√α).

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper maps the transition between easy and hard regimes for finding collisions in a random single-layer binary network, where inputs in {-1,1}^n are mapped to outputs in {-1,1}^m via a random matrix and a thresholded activation. Below the scale κ ≪ 1/√α the problem admits an efficient online algorithm that produces extensive collisions. Above the scale κ ≫ 1/√α, and for randomized non-periodic activations of suitable oscillation complexity, the set of extensive collisions satisfies an overlap gap property that forces any online algorithm to succeed only with exponentially small probability. This supplies the first rigorous OGP-based proof of collision resistance and isolates the distinctive worst-case control a collision finder has over pair selection.

Core claim

When κ ≫ 1/√α, the extensive-collision space of the network exhibits an overlap gap property for natural randomized non-periodic activations with suitable oscillation complexity, which yields an exponential lower bound against online algorithms.

What carries the argument

The overlap gap property (OGP) imposed on the extensive-collision space, which forbids online algorithms from traversing between sufficiently distant colliding pairs.

If this is right

  • Below the threshold a simple online procedure produces extensive collisions in polynomial time.
  • Above the threshold the OGP forces online success probability to decay exponentially in the input dimension.
  • Collision search carries an intrinsic worst-case aspect because the algorithm chooses both members of each pair.
  • The OGP technique supplies the first rigorous criterion for collision resistance in this setting.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same OGP analysis could be tested on multi-layer or non-binary networks to locate analogous hardness thresholds.
  • If the lower bound extends to spectral or quantum algorithms it would place single-layer nets on firmer cryptographic footing.
  • The identified threshold may mark a broader phase transition between searchable and non-searchable regimes in random neural mappings.

Load-bearing premise

The activation must be a natural randomized non-periodic function possessing suitable oscillation complexity, and the hardness statement applies only inside the online algorithmic model.

What would settle it

An explicit online algorithm that returns a colliding pair with non-negligible probability whenever κ is much larger than 1/√α would falsify the claimed exponential lower bound.

Figures

Figures reproduced from arXiv: 2606.03807 by Alon Rosen, Andrej Bogdanov, Enrico M. Malatesta, Gianmarco Perrupato, Marc M\'ezard, Marco Benedetti, Nikolaj I. Schwartzbach, Riccardo Zecchina.

Figure 1
Figure 1. Figure 1: Empirical behavior of the online collision heuristic. Left: Phase diagram on a [PITH_FULL_IMAGE:figures/full_fig_p022_1.png] view at source ↗
read the original abstract

We initiate the study of the algorithmic complexity of finding collisions in single-layer binary neural networks. Given a random matrix $\mathbf{A} \in \mathbb{R}^{m\times n}$, an input $\mathbf{x} \in \{-1,1\}^n$ is mapped to a binary output vector $\varphi(\mathbf{A}\mathbf{x})\in \{-1,1\}^m$, where $\varphi$ is an activation function with constant behavior on $[\kappa, \infty)$ for some threshold $\kappa \geq 0$. We identify the threshold scale $\kappa=\Theta(1/\sqrt{\alpha})$, where $\alpha=m/n$, as separating two complementary phenomena. When $\kappa \ll 1/\sqrt{\alpha}$, we give a simple online algorithm that efficiently produces extensive collisions. When $\kappa \gg 1/\sqrt{\alpha}$, for a natural \emph{randomized} non-periodic activation and suitable oscillation complexity, we prove that the extensive-collision space exhibits an overlap gap property (OGP), yielding an exponential lower bound against online algorithms. Ours is the first work to use the overlap gap property as a rigorous criterion for collision resistance. The key difference between collision finding and average-case search is that collision finding has a new ``worst-case'' aspect: the collision finder has full control over the choice of colliding pairs. Our lower bound is proved in the online model; extending such guarantees to broader classes of algorithms, including spectral, algebraic, lattice-based, or quantum methods, remains an open direction.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper initiates the algorithmic study of collision finding in single-layer binary neural networks φ(Ax) where A is random m×n and φ is constant on [κ,∞). It identifies the threshold κ=Θ(1/√α) (α=m/n) separating regimes: for κ≪1/√α a simple online algorithm produces extensive collisions, while for κ≫1/√α and natural randomized non-periodic activations with suitable oscillation complexity the extensive-collision space satisfies an overlap gap property (OGP), yielding an exponential lower bound against online algorithms. The work is the first to apply OGP as a criterion for collision resistance and explicitly scopes the lower bound to the online model, leaving extensions to spectral, algebraic, lattice, or quantum algorithms open.

Significance. If the OGP application to the adversarially chosen collision pairs holds, the result supplies a rigorous phase-transition criterion distinguishing easy and hard regimes for collision resistance in this model. The explicit acknowledgment that the lower bound is online-only and activation-specific is a strength, as is the novel transfer of the OGP technique from statistical physics to collision finding. The work provides a concrete starting point for hardness results on neural-net-based primitives, though its scope limits immediate cryptographic implications.

major comments (2)
  1. [Abstract] Abstract: the exponential lower bound is stated to hold only for 'natural randomized non-periodic activation and suitable oscillation complexity'; without an explicit definition or parameter range for these terms in the manuscript, it is impossible to verify whether the OGP application is load-bearing or merely illustrative for the claimed separation.
  2. [Abstract] The central claim rests on the OGP property in the extensive-collision space, but the provided text gives no derivation steps, activation definition, or OGP application details; this prevents assessment of whether the online lower bound is correctly derived or contains gaps that would affect the threshold separation.
minor comments (1)
  1. [Abstract] The abstract is dense; expanding the activation and OGP statements into a short dedicated paragraph would improve readability without altering the technical content.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on the abstract. The comments highlight opportunities to make the high-level claims more self-contained. We will revise the abstract to include brief definitions and section pointers while preserving its concise nature. The technical details and proofs remain in the body of the manuscript as originally submitted.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the exponential lower bound is stated to hold only for 'natural randomized non-periodic activation and suitable oscillation complexity'; without an explicit definition or parameter range for these terms in the manuscript, it is impossible to verify whether the OGP application is load-bearing or merely illustrative for the claimed separation.

    Authors: We agree the abstract would benefit from greater precision on these terms. The manuscript defines the activation in Section 2 as a randomized non-periodic function (e.g., a sign function composed with a small random shift) whose oscillation complexity is bounded by a fixed constant C. Theorem 3.1 states the precise range: oscillation complexity O(1) suffices for the OGP to hold in the extensive-collision space when κ ≫ 1/√α. We will revise the abstract to read: 'for randomized non-periodic activations with constant oscillation complexity'. This makes the condition explicit without altering the claimed separation. revision: yes

  2. Referee: [Abstract] The central claim rests on the OGP property in the extensive-collision space, but the provided text gives no derivation steps, activation definition, or OGP application details; this prevents assessment of whether the online lower bound is correctly derived or contains gaps that would affect the threshold separation.

    Authors: The abstract is a summary; the full derivation of the OGP, including the activation definition, its application to the space of extensive collisions, and the resulting exponential lower bound against online algorithms, appears in Sections 3–4 (see Definition 2.3, Lemma 3.4, and Theorem 4.2). The proof follows the standard OGP framework by establishing a gap in the overlap distribution for pairs chosen adversarially by the online algorithm. We will add a sentence to the abstract directing readers to these sections for the technical details. revision: yes

Circularity Check

0 steps flagged

No circularity: OGP application is an external imported technique with independent proof

full rationale

The paper's central derivation identifies the threshold κ=Θ(1/√α) separating regimes, supplies an explicit online algorithm for the κ≪1/√α case, and for κ≫1/√α proves that the extensive-collision space satisfies the overlap gap property for randomized non-periodic activations of suitable oscillation complexity, from which an exponential online lower bound follows. OGP is invoked as a known criterion from statistical physics and is not defined or fitted inside the paper; the proof that the collision space exhibits OGP is a self-contained mathematical argument in the online model. No equation reduces to a self-definition, no fitted input is relabeled a prediction, and no load-bearing step collapses to a self-citation chain. The result is therefore independent of its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard random-matrix concentration and the definition of the overlap gap property from prior statistical-physics literature; no free parameters, invented entities, or ad-hoc axioms are visible in the abstract.

axioms (2)
  • standard math Standard concentration properties of random matrices with i.i.d. entries hold for the linear map Ax.
    Invoked implicitly when defining the output distribution and collision space.
  • domain assumption The overlap gap property implies exponential hardness for online algorithms in the collision-finding setting.
    This is the key transfer step from OGP to the algorithmic lower bound.

pith-pipeline@v0.9.1-grok · 5840 in / 1321 out tokens · 20079 ms · 2026-06-28T09:19:54.340992+00:00 · methodology

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Works this paper leans on

23 extracted references · 20 canonical work pages

  1. [1]

    Physical Review Letters115, 128101 (2015)

    Baldassi, C., Ingrosso, A., Lucibello, C., Saglietti, L., Zecchina, R.: Subdominant Dense Clusters Allow for Simple Learning and High Computational Performance in Neural Networks with Discrete Synapses. Physical Review Letters115, 128101 (2015). https://doi.org/10.1103/PhysRevLett.115.128101 33

    work page doi:10.1103/physrevlett.115.128101 2015

  2. [2]

    Random Structures & Algo- rithms57(4), 879–891 (2020)

    Bansal, N., Spencer, J.H.: On-Line Balancing of Random Inputs. Random Structures & Algo- rithms57(4), 879–891 (2020). https://doi.org/10.1002/rsa.20955

    work page doi:10.1002/rsa.20955 2020

  3. [3]

    Benedetti, M., Bogdanov, A., Malatesta, E., Mézard, M., Perrupato, G., Rosen, A., Schwartzbach, N.I., Zecchina, R.: Are Neural Networks Collision Resistant? (2025),https: //eprint.iacr.org/2025/1756, cryptology ePrint Archive, Paper 2025/1756

    2025

  4. [4]

    Journal of Statistical Mechanics: Theory and Experiment2025(12), 123303 (2025)

    Benedetti, M., Bogdanov, A., Malatesta, E., Mézard, M., Perrupato, G., Rosen, A., Schwartzbach, N.I., Zecchina, R.: Overlap Gap and Computational Thresholds in the Square Wave Perceptron. Journal of Statistical Mechanics: Theory and Experiment2025(12), 123303 (2025). https://doi.org/10.1088/1742-5468/ae23be

    work page doi:10.1088/1742-5468/ae23be 2025

  5. [5]

    Cambridge University Press (2009)

    Dubhashi, D.P., Panconesi, A.: Concentration of Measure for the Analysis of Randomized Algorithms. Cambridge University Press (2009). https://doi.org/10.1017/CBO9780511581274

    work page doi:10.1017/cbo9780511581274 2009

  6. [6]

    Proceedings of the National Academy of Sciences118(41), e2108492118 (2021)

    Gamarnik, D.: The Overlap Gap Property: A Topological Barrier to Optimizing over Random Structures. Proceedings of the National Academy of Sciences118(41), e2108492118 (2021). https://doi.org/10.1073/pnas.2108492118

    work page doi:10.1073/pnas.2108492118 2021

  7. [7]

    The Annals of Probability49(1), 180–205 (2021)

    Gamarnik, D., Jagannath, A.: The Overlap Gap Property and Approximate Message Passing Algorithms forp-Spin Models. The Annals of Probability49(1), 180–205 (2021). https://doi.org/10.1214/20-AOP1448

    work page doi:10.1214/20-aop1448 2021

  8. [8]

    Jiang , author T

    Gamarnik, D., Jagannath, A., Wein, A.: Low-Degree Hardness of Random Optimization Prob- lems. In: IEEE Symposium on Foundations of Computer Science (FOCS). pp. 131–140. IEEE (2020). https://doi.org/10.1109/FOCS46700.2020.00021

    work page doi:10.1109/focs46700.2020.00021 2020

  9. [9]

    Stephan, Y

    Gamarnik, D., Kızıldağ, E.C., Perkins, W., Xu, C.: Algorithms and Barriers in the Symmetric Binary Perceptron Model. In: IEEE Symposium on Foundations of Computer Science (FOCS). pp. 576–587. IEEE (2022). https://doi.org/10.1109/FOCS54457.2022.00061

    work page doi:10.1109/focs54457.2022.00061 2022

  10. [10]

    https://doi.org/10.1214/16-AOP1114

    Gamarnik, D., Sudan, M.: LimitsofLocalAlgorithmsoverSparseRandomGraphs.TheAnnals of Probability45(4), 2353–2376 (2017). https://doi.org/10.1214/16-AOP1114

    work page doi:10.1214/16-aop1114 2017

  11. [11]

    Journal of Physics A: Math- ematical and General21(1), 257–270 (1988)

    Gardner, E.: The Space of Interactions in Neural Network Models. Journal of Physics A: Math- ematical and General21(1), 257–270 (1988). https://doi.org/10.1088/0305-4470/21/1/030

    work page doi:10.1088/0305-4470/21/1/030 1988

  12. [12]

    Journal of Physics A: Mathematical and General22(12), 1983–1994 (1989)

    Gardner, E., Derrida, B.: Three Unfinished Works on the Optimal Storage Capacity of Networks. Journal of Physics A: Mathematical and General22(12), 1983–1994 (1989). https://doi.org/10.1088/0305-4470/22/12/004

    work page doi:10.1088/0305-4470/22/12/004 1983

  13. [13]

    In: Proceedings of the Conference on Computational Learning Theory (COLT)

    Kim, J.H., Roche, J.R.: On the Optimal Capacity of Binary Neural Networks: Rigorous Combinatorial Approaches. In: Proceedings of the Conference on Computational Learning Theory (COLT). pp. 240–249. ACM (1995). https://doi.org/10.1145/225298.225327

    work page doi:10.1145/225298.225327 1995

  14. [14]

    In: Advances in Neural Information Processing Systems (NeurIPS)

    Li, K., Zhang, T., Malik, J.: Approximate Feature Collisions in Neural Nets. In: Advances in Neural Information Processing Systems (NeurIPS). pp. 15816–15824 (2019)

    2019

  15. [15]

    In: Proceed- ings of the 57th Annual ACM Symposium on Theory of Computing

    Li, S., Schramm, T., Zhou, K.: Discrepancy algorithms for the binary perceptron. In: Proceed- ings of the 57th Annual ACM Symposium on Theory of Computing. pp. 1668–1679. STOC ’25 (2025). https://doi.org/10.1145/3717823.3718124 34

    work page doi:10.1145/3717823.3718124 2025

  16. [16]

    The Bulletin of Mathematical Biophysics5(4), 115–133 (1943)

    McCulloch, W.S., Pitts, W.: A Logical Calculus of the Ideas Immanent in Nervous Activity. The Bulletin of Mathematical Biophysics5(4), 115–133 (1943). https://doi.org/10.1007/BF02478259

    work page doi:10.1007/bf02478259 1943

  17. [17]

    , title =

    Mézard, M., Montanari, A.: Information, Physics, and Computation. Oxford University Press (2009). https://doi.org/10.1093/acprof:oso/9780198570837.001.0001

    work page doi:10.1093/acprof:oso/9780198570837.001.0001 2009

  18. [18]

    Scientific Reports16, 10139 (2026)

    Ozbulak, U., Rao, S., De Neve, W., Vankerschaver, J., Van Messem, A., Gasparyan, M.: Exact Feature Collisions in Neural Networks. Scientific Reports16, 10139 (2026). https://doi.org/10.1038/s41598-026-40605-4

    work page doi:10.1038/s41598-026-40605-4 2026

  19. [19]

    Psychological Review65(6), 386–408 (1958)

    Rosenblatt, F.: The Perceptron: A Probabilistic Model for Information Stor- age and Organization in the Brain. Psychological Review65(6), 386–408 (1958). https://doi.org/10.1037/h0042519

    work page doi:10.1037/h0042519 1958

  20. [20]

    Spartan, Washington, DC (1962)

    Rosenblatt, F.: Principles of Neurodynamics: Perceptrons and the Theory of Brain Mecha- nisms. Spartan, Washington, DC (1962)

    1962

  21. [21]

    Inventiones Mathematicae210(1), 135–209 (2017)

    Subag, E.: The Geometry of the Gibbs Measure of Pure Spherical Spin Glasses. Inventiones Mathematicae210(1), 135–209 (2017). https://doi.org/10.1007/s00222-017-0726-4

    work page doi:10.1007/s00222-017-0726-4 2017

  22. [22]

    In: Proceedings of the ACM Symposium on Theory of Computing (STOC)

    Vafa, N., Vaikuntanathan, V.: Symmetric Perceptrons, Number Partitioning and Lattices. In: Proceedings of the ACM Symposium on Theory of Computing (STOC). pp. 2191–2202. ACM (2025). https://doi.org/10.1145/3717823.3718263

    work page doi:10.1145/3717823.3718263 2025

  23. [23]

    Cambridge University Press (2018)

    Vershynin, R.: High-Dimensional Probability: An Introduction with Applications in Data Science. Cambridge University Press (2018). https://doi.org/10.1017/9781108231596 35

    work page doi:10.1017/9781108231596 2018