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Key Reconciliation with RC-LDPC/Error Estimation for Satellite-based FSO/QKD Systems

Cuong T. Nguyen, Hoang D. Le, Anshul Jaiswal, Swaminathan R., Anh T. Pham

arXiv:2609.25646v1cs.NIeess.SY

TL;DR

Satellite FSO/QKD links make key reconciliation challenging because QBER fluctuates and reconciliation can bottleneck throughput. The paper combines protograph RC-LDPC blind reconciliation with syndrome-based error estimation and develops an SKT framework. Its numerical findings report higher SKT than conventional blind reconciliation, with a complexity trade-off.

  • Problem

    Fluctuating QBER complicates key reconciliation, a computationally intensive post-processing step and throughput bottleneck in QKD.

    Source: §I · p. 1

  • Method

    The paper combines blind reconciliation using protograph RC-LDPC codes with syndrome-based error estimation and develops an end-to-end SKT framework.

    Source: §I · p. 1

  • Results

    The proposed design outperforms conventional blind reconciliation in SKT, especially when public-channel data rate and the number of code members are high.

    Source: §VII · p. 14

  • Takeaways & Limitations

    The paper offers design guidelines for system settings and pass-time optimization, and reports a case study using Starlink LEO satellite and ground-vehicle data.

    Source: §I · p. 2, §VII · p. 14

    The proposed design has increased complexity due to its additional error-estimation step.

    Source: §VII · p. 14

Abstract

from arXiv · show

Satellite-based free-space optics (FSO) quantum key distribution (QKD) systems have recently attracted significant research interest due to their potential to enable globally secured applications. However, the inherent uncertainty of FSO channels, caused by weather conditions and satellite mobility, induces severe fluctuations in quantum bit-error rate (QBER) between legitimate users. This makes designing an efficient key reconciliation, an essential step in the QKD post-processing, particularly challenging. In this work, we propose a key reconciliation scheme that combines protograph rate-compatible (RC) low-density parity-check (LDPC) codes with a syndrome-based error estimation method. The proposed error estimation method reduces the number of communication rounds without requiring additional information disclosure. Furthermore, to our best knowledge, an analytical framework is first developed to evaluate end-to-end secret-key throughput (SKT), accounting for the impact of imperfect error estimation and dynamic FSO channel conditions. Numerical results demonstrate that the proposed scheme consistently outperforms conventional blind reconciliation under diverse FSO channel conditions and provide practical guidelines for system parameter selection. Finally, we validate the proposed framework through a case study that incorporates data from a Starlink low-Earth orbit (LEO) satellite and moving ground vehicles.

I. INTRODUCTION

The paper targets efficient key reconciliation for satellite FSO/QKD under fluctuating channel conditions. It presents a joint reconciliation design, an end-to-end SKT framework, and a LEO-to-vehicle system model.

I. INTRODUCTION

  • The paper jointly designs blind reconciliation with protograph RC-LDPC codes and syndrome-based error estimation for satellite-specific challenges.Source: §I · p. 1
  • The proposed theoretical framework evaluates end-to-end SKT for conventional and proposed reconciliation while accounting for FSO impairments and imperfect error estimation.It also derives a fitted block-wise QBER probability density function for system-level analysis.Source: §I · p. 1
  • The introduction reports a Starlink LEO satellite and ground-vehicle case study as evidence of the proposed design’s effectiveness.Source: §I · p. 2

II. RELATED WORK

  • Prior adaptive-rate schemes rely on accurate QBER estimates, while blind reconciliation can lower code rates incrementally but may require multiple rounds over high-latency satellite links.Source: §II · pp. 2–3
  • Random sampling consumes sifted bits and an exchange round, while estimates from previous blocks may be inaccurate under strongly fluctuating satellite-link QBER.The paper motivates syndrome-based estimation as an alternative and identifies end-to-end SKT analysis under imperfect estimation as an open need.Source: §II · p. 3

III. SYSTEM DESCRIPTION

  • The system models a LEO satellite distributing keys to a ground vehicle, with quantum transmission and public-channel post-processing continuing within each superframe.Source: §III · p. 3, A. System Model · p. 3

A. System Model

  • Each superframe ends after its last sifted block completes reconciliation; the quantum phase uses non-coherent CV-QKD, followed by sifting, reconciliation, and privacy amplification.The model assumes an authenticated, error-free public channel and negligible probability of undetected reconciliation failure.Source: A. System Model · pp. 3–4

B. FSO Channel Model

The LEO-to-vehicle channel combines cloud attenuation, atmospheric turbulence, and pointing-error fluctuations. The model describes each impairment and accounts for distinct pointing conditions at Bob and Eve.

B. FSO Channel Model

  • The composite channel coefficient combines cloud attenuation, atmospheric turbulence, and pointing errors for Bob or Eve.The channel is written as hX = hchthp,X, with X distinguishing Bob and Eve.Source: B. FSO Channel Model · p. 4

1) Cloud Attenuation:

  • Cloud attenuation models optical-power reduction from scattering by liquid water particles, with attenuation dependent on cloud extent, zenith angle, wavelength, and visibility.Visibility is related to cloud liquid water content and droplet number concentration.Source: 1) Cloud Attenuation · p. 4

2) Atmospheric Turbulence:

  • Atmospheric turbulence is modeled with a Gamma-Gamma distribution to represent scintillation-induced fluctuations in received power.Its scale parameters represent effective large- and small-scale scattering-cell counts.Source: 2) Atmospheric Turbulence · p. 4

3) Generalized Pointing Error Model:

  • Pointing errors reduce collected power through beam-detector misalignment, with Bob’s displacement modeled using independent Gaussian x- and y-axis jitter.The radial displacement is approximated by a modified Rayleigh distribution; Eve’s model assumes a detector on Bob’s receiver plane, dE meters away, affected by the same misalignment conditions.Source: 3) Generalized Pointing Error Model · pp. 4–5

4) Composite Channel Model:

The composite channel model derives numerical approximations for Bob’s and Eve’s channel distributions.

4) Composite Channel Model:

  • Bob’s composite-channel probability density function is derived by approximating the Gamma-Gamma distribution with a mixture-Gamma distribution.The numerical approximation uses Gauss-Laguerre weights and abscissae.Source: 4) Composite Channel Model · p. 5
  • Eve’s composite-channel model is numerically approximated using Gauss-Hermite weights and abscissae.The approximation order is denoted by m.Source: 4) Composite Channel Model · p. 5

C. CV-QKD with Dual-threshold/Direct Detection (DT/DD)

This section describes a non-coherent CV-QKD protocol using SIM/BPSK and dual-threshold detection. It also specifies received-noise modeling for the system.

C. CV-QKD with Dual-threshold/Direct Detection (DT/DD)

  • The quantum phase uses non-coherent CV-QKD with SIM/BPSK and a dual-threshold detector, a protocol inspired by BB84.The passage describes it as low-complexity, low-cost, and compatible with conventional communication devices.Source: C. CV-QKD with Dual-threshold/Direct… · p. 5
  • Alice modulates random bits into SIM/BPSK signals, while Bob detects the received electrical signal using dual thresholds.A signal below d0 maps to 0, one above d1 maps to 1, and an intermediate value is recorded as X.Source: C. CV-QKD with Dual-threshold/Direct… · p. 6
  • The received noise is modeled as zero-mean additive white Gaussian noise, with shot, background, and thermal noise components.The passages specify parameters for these components, including data rate, background radiation, receiver temperature, noise figure, and load resistance.Source: C. CV-QKD with Dual-threshold/Direct… · p. 6

IV. PROPOSED METHOD

The proposed blind reconciliation combines protograph-based RC-LDPC codes with syndrome-based QBER estimation. The receiver uses the initial syndrome to help select a code rate before further reconciliation exchanges.

1) Review of Blind Reconciliation:

  • Conventional blind reconciliation starts at the highest code rate and sends additional syndrome bits after a failed decoding attempt, lowering the rate progressively.The process stops after successful decoding or after all rates have been tried, when the sifted blocks are discarded.Source: 1) Review of Blind Reconciliation · p. 6

A. Blind Reconciliation Design

  • When high average QBER makes initial decoding failure common, the proposed design uses the initial syndrome to estimate QBER and select a code rate for the next round.The estimated QBER and rate index are shared with the sender; the passage identifies high satellite propagation delay as an added concern.Source: 1) Review of Blind Reconciliation · p. 6, 3) Operation Steps · p. 7

2) Protograph-based Rate-compatible (RC)-LDPC Code Family:

  • The RC-LDPC family is constructed from protographs, extending a high-rate protomatrix with rows selected for the lowest decoding threshold.The resulting parity-check matrices are nested, allowing lower-rate syndromes to add bits to the higher-rate syndrome.Source: 2) Protograph-based Rate-compatible… · pp. 6–7

3) Operation Steps:

  • Alice first sends a syndrome, and Bob attempts decoding; after failure, he estimates QBER and sends the selected rate index so Alice can provide an incremental syndrome.Bob concatenates the new syndrome bits with the initial syndrome and retries, continuing through rates until decoding succeeds or all rates are exhausted.Source: 3) Operation Steps · p. 7

B. Syndrome-based Error Estimation

The section estimates sifted-block bit-flip probability from syndrome differences and uses that estimate to select a code rate. Its protograph RC-LDPC design is intended to simplify implementation while covering a wide QBER range.

B. Syndrome-based Error Estimation

  • For the illustrated RC-LDPC family, code-rate thresholds {0.005, 0.0135, 0.027, 0.054, 0.0835} correspond to FER 10^-3.The family has rates {0.9, 0.8, 0.7, 0.6, 0.5} and a block length of 10,000 bits.Source: B. Syndrome-based Error Estimation · p. 7, Figure 6 · p. 7
  • Bob compares syndrome bits from minimum-row-weight checks, estimates the syndrome bit-flipping probability, and uses its relation to the sifted-key bit-flipping probability to estimate QBER.The syndrome vectors are formed from rows of equal weight; the theoretical syndrome bit-flipping probability is modeled as a function of the bit-flipping probability.Source: B. Syndrome-based Error Estimation · p. 8
  • Protograph RC-LDPC codes provide structured parity-check matrices for efficient hardware decoding and a code family designed to cover a wide QBER range.The proposed estimation method uses fixed row weights to reduce computational complexity relative to irregular-LDPC approaches requiring exhaustive searches.Source: B. Syndrome-based Error Estimation · p. 8

V. PERFORMANCE ANALYSIS

The analysis formulates end-to-end secret-key throughput for conventional blind reconciliation and outlines how the framework accounts for QBER variation across sifted blocks. It models that QBER distribution with an exponentiated Weibull fit to simulation data.

V. PERFORMANCE ANALYSIS

  • The performance framework covers conventional blind reconciliation and proposed reconciliation with imperfect error estimation, alongside statistical modeling of sifted-block QBER.The analysis treats end-to-end SKT as average secret bits produced per second.Source: §V · p. 8

A. Conventional Blind Reconciliation

  • Conventional blind-reconciliation SKT combines expected secret bits per superframe with the average superframe duration, accounting for reconciliation success and secret-key information terms.The formulation includes the reconciliation efficiency, code rate, Alice–Bob mutual information, and Eve's information measure.Source: A. Conventional Blind Reconciliation · p. 8
  • Because fading makes finite sifted blocks span different coherence times and error patterns, the analysis fits an exponentiated Weibull distribution to simulated block-QBER data.The fitted distribution supplies the QBER CDF used in the system analysis.Source: A. Conventional Blind Reconciliation · p. 9

B. The Proposed Design with Imperfect Error Estimation

The proposed-design analysis incorporates finite-syndrome estimation errors into SKT and delay calculations. It distinguishes the costs of underestimating and overestimating QBER and uses numerical approximations for otherwise intractable integrations.

B. The Proposed Design with Imperfect Error Estimation

  • Finite syndrome length makes the estimated QBER discrete, so a wrong state can require extra communication rounds when underestimated or leak more information when overestimated.The analysis distinguishes the actual state of ρ from the estimated state.Source: B. The Proposed Design with Imperfect… · p. 9
  • The imperfect-estimation formulation accounts for secret bits and post-processing delay across actual and estimated QBER states.It uses joint-state probabilities and state-dependent delays in calculating these quantities.Source: B. The Proposed Design with Imperfect… · pp. 9–10
  • Gauss-Legendre quadrature approximates the analytically intractable integration, and the reported numerical approximations converge quickly after ℓ=10.The convergence statement applies to approximations (28) and (30).Source: B. The Proposed Design with Imperfect… · p. 10

C. Statistical Distributions of QBER of Sifted Blocks

Across the considered scenarios, the exponentiated Weibull distribution best fits simulated sifted-block QBER data, with R2 above 0.99 in most scenarios. The fitted distribution is then used to model QBER for the rest of the paper.

C. Statistical Distributions of QBER of Sifted Blocks

  • The fitting procedure compares candidate PDFs with simulated QBER histograms using the R2 goodness-of-fit measure, for which values closer to 1 indicate a better fit.The study generates 10^4 samples per scenario and uses 50-bin histograms.Source: C. Statistical Distributions of QBER… · p. 10
  • R2 > 0.99 in most scenarios, and the exponentiated Weibull distribution provides the best fit among the four considered QBER distributions.The comparison covers normal, log-normal, Weibull, and exponentiated Weibull distributions for different DT scale coefficients ζ.Source: C. Statistical Distributions of QBER… · p. 10
  • The authors use the exponentiated Weibull distribution to model sifted-block QBER in the remainder of the paper.They attribute its fit to the greater freedom provided by its three parameters.Source: C. Statistical Distributions of QBER… · p. 10

VI. NUMERICAL RESULTS & DISCUSSIONS

The numerical evaluation compares the proposed design with conventional blind reconciliation and examines code and public-channel settings. The proposal achieves higher SKT across the considered CLWC range, and simulations match the analytical results.

A. Parameter Settings

  • The evaluation uses protograph RC-LDPC families with block lengths of 10,000 and 40,000 and five rates from 0.9 to 0.5.The rate-specific decoding thresholds are empirically determined for each block length.Source: A. Parameter Settings · p. 11

B. Performance Comparison

  • The proposed design outperforms conventional blind reconciliation across the considered CLWC range, and Monte Carlo simulations match the analytical results.The comparison includes sifted block lengths of 10,000 and 40,000; increasing block length also improves SKT.Source: B. Performance Comparison · pp. 11–12, Figure 9 · p. 12
  • Higher public-channel data rates improve SKT by reducing transmission and post-processing delay, while propagation delay remains dominant.The comparison considers 250 Mbps and 2 Gbps.Source: B. Performance Comparison · p. 12
  • With seven code rates, the proposed design performs better than with three, while conventional blind reconciliation degrades as communication rounds increase.More rates maintain high reconciliation efficiency over a wider QBER range.Source: B. Performance Comparison · p. 12

C. Impact and Selection of Proposed Design Parameters

The design outperforms conventional blind reconciliation in SKT, particularly with high public-channel data rates and more code members, at the cost of an extra error-estimation step. The results also give parameter-selection guidance for throughput across channel conditions and a satellite pass.

C. Impact and Selection of Proposed Design Parameters

  • Public-channel throughput saturates once transmission delay becomes insignificant; at CLWC 3 mg/m3, 0.5 Gbps retains throughput of 1.25 Mbps.The study recommends selecting the minimum public-channel rate that maintains the desired throughput.Source: C. Impact and Selection of Proposed… · p. 12, Figure 10 · p. 12
  • Above 500 Mbps, imperfect error estimation creates a more pronounced throughput gap from perfect estimation; using the maximum possible syndrome bits is recommended.The passage attributes this to propagation delay dominating transmission delay and recommends more syndrome bits for more accurate estimates.Source: C. Impact and Selection of Proposed… · pp. 12–13, Figure 11 · p. 12

D. Performance Investigation Over A Satellite Pass

VII. CONCLUSION

  • The proposed design outperforms conventional blind reconciliation in SKT, especially with high public-channel data rates and more code members, but adds complexity through error estimation.The analytical framework accounts for major FSO impairments and imperfect error estimation; Monte Carlo simulations verify it.Source: §VII · pp. 13–14
  • Practical feasibility remains future work, including validation with experimental data from a LEO-to-ground optical link.Source: §VII · p. 14

APPENDIX A RC-LDPC CODE FAMILY DESIGN EXAMPLE

The appendix describes constructing an RC-LDPC code family by extending protomatrices, then gives proof steps for the sift-probability and syndrome-estimation results. The proofs use joint detection probabilities and a binomial model for syndrome weight.

APPENDIX A RC-LDPC CODE FAMILY DESIGN EXAMPLE

  • The code-family example starts with a rate-0.9 protomatrix and extends the base matrix by adding rows while searching for low decoding thresholds.The construction uses a rate-1/2 R4JA code as its starting point for lengthening.Source: Appendix A · p. 14

APPENDIX B PROOF OF LEMMA 1

  • The Lemma 1 proof derives Bob’s average sift probability from joint Alice–Bob detection probabilities and derives Eve’s error probability using her optimal threshold.It obtains numerical expressions using a Gaussian-Laguerre polynomial approximation.Source: Appendix B · p. 14

APPENDIX C PROOF OF LEMMA 2

  • The Lemma 2 proof models syndrome weight as binomial and uses its CDF, expressed with the regularized incomplete beta function, to obtain the estimate bound.Source: Appendix C · p. 15
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