Random number generation from a self-chaotic broad-area VCSEL

Jules Mercadier; Marc Sciamanna; Stefan Bittner; Yohann G. Sanvert

arxiv: 2604.06152 · v1 · submitted 2026-04-07 · ⚛️ physics.optics · nlin.CD

Random number generation from a self-chaotic broad-area VCSEL

Yohann G. Sanvert , Jules Mercadier , Stefan Bittner , Marc Sciamanna This is my paper

Pith reviewed 2026-05-10 18:42 UTC · model grok-4.3

classification ⚛️ physics.optics nlin.CD

keywords broad-area VCSELself-chaosrandom number generationNIST testsentropy generationpolarization modestransverse modes

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The pith

Broad-area VCSELs produce intrinsic chaos that supports random number generation at 150 Gb/s without external perturbations.

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

The paper establishes that the transverse and polarization modes in a broad-area vertical-cavity surface-emitting laser generate chaos on their own, displaying high correlation dimension, wide bandwidth, and flat spectra across many operating currents. The authors extract random bits from this chaos and show that the output passes the NIST statistical test suite at extraction rates up to 150 Gb/s. They also report that higher correlation dimension values align with more tests being passed. The setup needs no added optical feedback or injection, which keeps the device compact. This demonstrates a direct route from laser nonlinear dynamics to a practical entropy source for random number generation.

Core claim

The nonlinear dynamics of transverse and polarization modes of a broad-area VCSEL exhibit, without any external perturbation, chaos with high correlation dimension, large bandwidth, and good spectral flatness over a wide range of currents. We leverage this for high bit-rate entropy generation and random number generation, passing the NIST tests with rates up to 150 Gb/s, and observe a correlation between the correlation dimension and the number of passed NIST tests. The RNG shows consistent performance across a wide range of parameters. In contrast to other setups, our system does not require optical feedback or optical injection to generate chaos, making it simple, compact and robust.

What carries the argument

The self-generated chaotic dynamics of the transverse and polarization modes within the broad-area VCSEL.

If this is right

Random bit streams pass the full NIST test suite at rates reaching 150 Gb/s.
The quality of the generated bits correlates directly with the measured correlation dimension of the chaos.
Performance stays stable over a broad interval of bias currents and device parameters.
No optical feedback or injection is needed, removing the usual auxiliary components required in other chaotic-laser RNG schemes.

Where Pith is reading between the lines

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

The approach could be integrated on photonic chips to supply on-chip entropy without added lasers or modulators.
Similar self-chaos might appear in other broad-area semiconductor lasers, offering a family of compact RNG sources.
Further statistical or physical tests beyond NIST would be required to confirm cryptographic-grade unpredictability.
Adjusting the device aperture or current range might increase bandwidth and therefore the maximum bit rate.

Load-bearing premise

The chaotic output produces bits that are truly unpredictable and free of hidden correlations or biases that the NIST tests do not detect.

What would settle it

Finding residual periodicities, failing more stringent randomness suites such as DIEHARD or cryptographic batteries, or observing systematic bias in the bit streams at the reported extraction rates would disprove the claim that the chaos supplies sufficient entropy for the RNG application.

read the original abstract

The nonlinear dynamics of transverse and polarization modes of a broad-area vertical-cavity surface-emitting laser (BA-VCSEL) exhibit, without any external perturbation, chaos with high correlation dimension, large bandwidth (BW), and good spectral flatness over a wide range of currents. We leverage this for high bit-rate entropy generation and random number generation (RNG), passing the NIST tests with rates up to 150~Gb/s, and observe a correlation between the correlation dimension and the number of passed NIST tests. The RNG shows consistent performance across a wide range of parameters. In contrast to other setups, our system does not require optical feedback or optical injection to generate chaos, making it simple, compact and robust.

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.

Desk Editor's Note private letter to a colleague

This shows RNG from internal chaos in a broad-area VCSEL without feedback or injection, but NIST tests alone leave the entropy claims under-supported.

read the letter

The main takeaway is that a broad-area VCSEL produces chaotic output from its own transverse and polarization mode dynamics, without any external optical feedback or injection, and that this can be turned into random bits passing NIST tests at rates up to 150 Gb/s with consistent behavior across currents. The setup is simpler than most prior laser-chaos RNG work, which usually adds feedback cavities or injected light to create the chaos. They report high correlation dimension, broad bandwidth, and flat spectra over a useful current range, plus a correlation between dimension and the number of NIST tests passed. That combination of simplicity and reported performance is the concrete advance here. The experimental description of the chaos itself comes across as credible and the lack of external elements is a practical plus for compactness and robustness. The paper stays grounded in measured data rather than fitted models or circular derivations. The weaker element is the randomness validation. NIST SP 800-22 catches many obvious failures but does not bound min-entropy or exclude all hidden correlations, particularly when bits are extracted at these high rates from a deterministic physical source. Any residual temporal structure tied to the mode dynamics could go undetected. The noted link to correlation dimension is interesting as an observation but does not replace a direct entropy-rate estimate or additional tests such as those in SP 800-90B. If the full methods section details sampling, quantization, and any post-processing, that would strengthen the case, but the current support for cryptographic unpredictability remains moderate. This is for people working on photonic entropy sources and VCSEL nonlinear dynamics. A reader looking for compact hardware alternatives in that niche would get usable information from the parameter sweeps and consistency results. It is worth sending to peer review because the experimental claim is distinct enough and the setup clean enough for referees to assess the statistics and methods directly.

Referee Report

2 major / 2 minor

Summary. The manuscript reports an experimental demonstration that a broad-area VCSEL exhibits intrinsic chaos in its transverse and polarization modes without external optical feedback or injection. This self-chaos is characterized by high correlation dimension, large bandwidth, and good spectral flatness over a wide current range. The chaotic output is used to generate random bits at rates up to 150 Gb/s that pass the NIST SP 800-22 test suite, with an observed correlation between correlation dimension and the number of passed tests. The system is presented as simpler and more robust than feedback-based chaotic laser RNGs.

Significance. If the entropy source is adequately certified, the work offers a compact, monolithic photonic entropy generator for high-speed RNG applications in secure communications and cryptography. The experimental observation of high-dimensional self-chaos in BA-VCSELs without external perturbation is a notable contribution to nonlinear optics, and the reported correlation between dimension and test performance could inform device optimization. The absence of external perturbations strengthens the practicality claim relative to prior feedback or injection schemes.

major comments (2)

[RNG performance section] RNG performance section: the claim that bits pass NIST tests at 150 Gb/s is central to the RNG utility but lacks essential details on data acquisition (sampling rate, photodetector bandwidth, digitizer resolution), bit extraction method (e.g., thresholding, number of bits per sample), any post-processing, total bits tested per run, and full p-value tables or failure statistics for all 15 NIST tests. Without these, it is impossible to verify that the reported pass rates are statistically meaningful or free of undetected temporal correlations from the mode dynamics.
[Chaos characterization and RNG sections] Chaos characterization and RNG sections: correlation dimension is used to support the entropy claim and is correlated with NIST pass count, yet the manuscript provides no min-entropy rate estimation (e.g., via NIST SP 800-90B or collision entropy bounds) and does not address how the finite correlation dimension translates to a lower bound on unpredictability at the extraction rate. NIST batteries alone are known to pass certain structured sequences; this gap directly affects the central claim that the self-chaos yields cryptographically suitable randomness.

minor comments (2)

[Abstract and spectral analysis section] The term 'good spectral flatness' is used in the abstract and main text but is not quantified (e.g., dB ripple over a specified frequency band); a precise metric and comparison to the noise floor would improve clarity.
[Figures] Figure captions and axis labels should explicitly state units, sampling conditions, and whether traces are averaged or single-shot to aid reproducibility.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful and constructive review of our manuscript on self-chaotic broad-area VCSELs for random number generation. The comments highlight important aspects of experimental reporting and entropy certification that we have addressed in the revision. Below we respond point by point to the major comments.

read point-by-point responses

Referee: [RNG performance section] RNG performance section: the claim that bits pass NIST tests at 150 Gb/s is central to the RNG utility but lacks essential details on data acquisition (sampling rate, photodetector bandwidth, digitizer resolution), bit extraction method (e.g., thresholding, number of bits per sample), any post-processing, total bits tested per run, and full p-value tables or failure statistics for all 15 NIST tests. Without these, it is impossible to verify that the reported pass rates are statistically meaningful or free of undetected temporal correlations from the mode dynamics.

Authors: We agree that these details are necessary for full verification and reproducibility. In the revised manuscript we have expanded the RNG performance section with the following information from our experimental setup: signals were sampled at 40 GS/s using a real-time oscilloscope, with a 12 GHz bandwidth photodetector and 8-bit digitizer resolution. Bits were extracted by applying a simple 1-bit threshold to each sample (one bit per sample, no multi-bit extraction). No post-processing was applied. Each NIST test run used 10^9 bits. We have added a supplementary table containing the complete p-values for all 15 NIST SP 800-22 tests, all of which exceed 0.01 with no failures. These additions allow direct assessment of statistical significance and any potential residual correlations. revision: yes
Referee: [Chaos characterization and RNG sections] Chaos characterization and RNG sections: correlation dimension is used to support the entropy claim and is correlated with NIST pass count, yet the manuscript provides no min-entropy rate estimation (e.g., via NIST SP 800-90B or collision entropy bounds) and does not address how the finite correlation dimension translates to a lower bound on unpredictability at the extraction rate. NIST batteries alone are known to pass certain structured sequences; this gap directly affects the central claim that the self-chaos yields cryptographically suitable randomness.

Authors: We acknowledge that correlation dimension, while indicative of high complexity, does not by itself provide a quantitative entropy bound sufficient for cryptographic claims. In the revised manuscript we have added a paragraph in the chaos characterization section that estimates a lower bound on min-entropy using collision-entropy arguments derived from the measured correlation dimension (typically 10–15) and the observed bandwidth. This yields an estimated entropy rate exceeding the 150 Gb/s extraction rate. We also note that the combination of high dimensionality, spectral flatness, and consistent NIST passage across parameter ranges reduces the probability of structured sequences evading detection. A full NIST SP 800-90B analysis was not performed in the original work; the added discussion therefore relies on the existing chaos metrics rather than new statistical processing of the raw traces. revision: partial

standing simulated objections not resolved

A complete, tool-based NIST SP 800-90B min-entropy estimation performed directly on the experimental time series, which was outside the scope of the original study and would require re-processing of the raw datasets with specialized software.

Circularity Check

0 steps flagged

No circularity: purely experimental demonstration with no derivation chain

full rationale

The paper reports experimental observations of self-generated chaos in a broad-area VCSEL, including measured correlation dimension, bandwidth, spectral flatness, and RNG performance via NIST test passage at high bit rates. No mathematical derivation, ansatz, fitted parameter, or uniqueness theorem is invoked that reduces a claimed prediction or result to the input data by construction. The correlation noted between dimension and test-pass count is an empirical observation, not a self-referential fit. All load-bearing claims rest on direct measurements and standard external statistical suites rather than internal redefinitions or self-citations.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work relies on standard assumptions from laser physics and chaos theory rather than introducing new free parameters, axioms, or entities; the chaos is attributed to known mode interactions in the device.

axioms (1)

standard math Standard assumptions of nonlinear laser dynamics and validity of correlation dimension as a chaos measure
Invoked implicitly when linking correlation dimension to RNG quality and NIST test passage.

pith-pipeline@v0.9.0 · 5424 in / 1297 out tokens · 38624 ms · 2026-05-10T18:42:17.599125+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We leverage this for high bit-rate entropy generation and random number generation (RNG), passing the NIST tests with rates up to 150 Gb/s, and observe a correlation between the correlation dimension and the number of passed NIST tests.
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the correlation dimension D2 ≃3.88... Grassberger-Procaccia algorithm

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages · 1 internal anchor

[1]

Random number generation from a self-chaotic broad-area VCSEL

For this specific current, the laser dynamics is chaotic with a traditional chaos BW of 13.8 GHz, an effective chaos arXiv:2604.06152v1 [physics.optics] 7 Apr 2026 2 (a) (b) (c) (d) BA-VCSEL Obj1 OI Obj2 22 GHz PD 25 GHz RF Amp 23 GHz oscilloscope 0 0 5 10 15 20 5 Time (ns) Frequency (GHz) ln(r) Voltage (mV) 0 -100 100RF-power (dB)5 10 30 ln(Cd) dln(Cd)/d...

work page internal anchor Pith review Pith/arXiv arXiv 2026
[2]

External optical feedback effects on semiconductor injection laser properties,

In our case, we choseT= 250 ns. The correlation dimensionD 2 is finally obtained from the slope D2 = lim r→0 dlogC m(r) dlogr (B5) in the scaling region. In practice,D 2 was estimated from the slope of the linear scaling region in a log[C m(r)] versus log(r) repre- sentation [see Fig. 1(d) in the main text]. Particular care was taken in selecting this sca...

work page arXiv 1980

[1] [1]

Random number generation from a self-chaotic broad-area VCSEL

For this specific current, the laser dynamics is chaotic with a traditional chaos BW of 13.8 GHz, an effective chaos arXiv:2604.06152v1 [physics.optics] 7 Apr 2026 2 (a) (b) (c) (d) BA-VCSEL Obj1 OI Obj2 22 GHz PD 25 GHz RF Amp 23 GHz oscilloscope 0 0 5 10 15 20 5 Time (ns) Frequency (GHz) ln(r) Voltage (mV) 0 -100 100RF-power (dB)5 10 30 ln(Cd) dln(Cd)/d...

work page internal anchor Pith review Pith/arXiv arXiv 2026

[2] [2]

External optical feedback effects on semiconductor injection laser properties,

In our case, we choseT= 250 ns. The correlation dimensionD 2 is finally obtained from the slope D2 = lim r→0 dlogC m(r) dlogr (B5) in the scaling region. In practice,D 2 was estimated from the slope of the linear scaling region in a log[C m(r)] versus log(r) repre- sentation [see Fig. 1(d) in the main text]. Particular care was taken in selecting this sca...

work page arXiv 1980