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Variational Probabilistic Quantization for Secret Key Generation

Xinyang Li, Vlad C. Andrei, Peter J. Gu, Yiqi Chen, Ullrich J. Mönich, Holger Boche

arXiv:2609.03205v1cs.ITmath.IT

TL;DR

Classical physical-layer key pipelines use hand-designed scalar quantization and defer secrecy, while learning-based approaches keep discretization outside training. VPQ trains discrete encoders against agreement, uniformity, and Eve-leakage objectives, then reconciles their outputs with a code-offset sketch; it provides induced-source key-rate guarantees and limits correlated-Eve leakage on the tested Gaussian fading channels.

  • Problem

    Classical physical-layer key pipelines use hand-designed scalar quantization and defer secrecy to privacy amplification, while learning-based methods separate discretization from training without a key-rate objective.

    Source: §I · p. 1

  • Method

    VPQ trains probabilistic encoders to map correlated observations directly to a discrete alphabet using entropy, mismatch, and adversarial leakage terms, then reconciles outputs with a code-offset secure sketch.

    Source: §I · p. 1, B. Code-Offset Reconciliation · p. 3

  • Results

    Against correlated Eve, VPQ keeps I(W; Z) ≤0.3 bits per symbol across all q and tested SNRs, versus 1.5 to 2.2 bits per symbol for the three baselines at 20 dB and above.

    Source: §V · p. 6

  • Takeaways & Limitations

    The VPQ losses lower-bound induced-source one-way key capacity, and q-ary linear sketches attain the optimal worst-case reconciliation rate over the stated source class.

    Source: B. Universal Reconciliation Rate · p. 4, §VI · p. 6

Abstract

from arXiv · show

Secret key generation from correlated observations at Alice and Bob, in the presence of an eavesdropper Eve, underpins physical-layer security. Classical pipelines quantize by hand, amplify privacy afterwards, and optimize no objective tied to a key rate. We propose Variational Probabilistic Quantization (VPQ): neural encoders that map the correlated sources directly into a discrete key alphabet, trained by a variational adversarial objective whose entropy, mismatch, and leakage terms match the three terms of the one-way secret key rate. A linear code-offset secure sketch then reconciles the encoder outputs into an identical key without a separate privacy amplification step. We prove that the VPQ losses lower-bound the one-way secret key capacity of the induced source, and derive in closed form the optimal worst-case key rate over the source class of a given alphabet size and mismatch probability, attained by finite-field linear sketches. On Gaussian fading channels, VPQ leaks less to a correlated eavesdropper than one classical and two recent learning-based baselines, and Reed--Solomon reconciliation operates within the predicted finite-blocklength gap.

I. INTRODUCTION

Conventional physical-layer key pipelines use hand-designed scalar quantization and defer secrecy to privacy amplification, while learning-based approaches still separate discretization from training. VPQ instead learns discrete keys with an objective for agreement, uniformity, and secrecy, and pairs the encoders with a secure sketch.

I. INTRODUCTION

  • Classical pipelines quantize a single scalar observable using hand-designed rules, ignoring Alice and Bob’s joint observations and treating secrecy as post-hoc decorrelation.The passage notes that secrecy degrades sharply when Eve is correlated with the legitimate channel.Source: §I · p. 1
  • Learning-based methods replace hand-designed quantizers with autoencoders, but keep discretization outside training and lack an objective tied to key rate.They use continuous reciprocal latent features followed by separate quantization; secrecy is deferred to privacy amplification or addressed with a domain-adversarial branch.Source: §I · p. 1
  • VPQ maps correlated sources directly into a discrete key alphabet, jointly training for uniformity, agreement, and secrecy rather than recovering these properties post hoc.A code-offset secure sketch reconciles the outputs, and the adversarial objective removes the need for separate privacy amplification, including with correlated Eve.Source: §I · p. 1
  • The VPQ losses lower-bound induced-source one-way key capacity, while q-ary linear sketches attain the optimal worst-case reconciliation rate for the stated source class.On the tested Gaussian fading sources, VPQ produces usable keys without privacy amplification against correlated Eve, while the three baselines leak up to 2.2 bits per symbol.Source: §I · p. 1

II. SYSTEM MODEL

The model considers Alice, Bob, and Eve observing i.i.d. correlated sequences, with Eve optionally absent. In one-way public discussion, Alice sends a message and the parties derive key variables; the paper studies this through the encoder-induced discrete source.

II. SYSTEM MODEL

  • Alice, Bob, and Eve observe i.i.d. triples drawn from a joint distribution, with Eve absent represented by Z = ∅.The one-way setting has Alice send a public message M = Φ(X^n), after which the parties extract key variables.Source: §II · p. 2
  • A key consists of variables K and L on a common alphabet, generated by mappings that may be deterministic or stochastic.The one-way secret-key capacity is expressed as a maximization of I(T; Y | U) − I(T; Z | U) over auxiliaries satisfying U−T−X−(Y, Z).Source: §II · p. 2
  • The proposed approach learns encoders that map observations to discrete variables (W, V), then uses linear code-offset sketches to produce an identical key satisfying the stated key conditions.This addresses practical extraction when the joint distribution is unknown and high-dimensional.Source: §II · p. 2

A. Variational Probabilistic Quantization

VPQ uses probabilistic encoders to map observations into a common q-ary alphabet and combines agreement, uniformity, and leakage objectives. For correlated Eve, a learnable predictor supplies a variational leakage surrogate; the training procedure alternates predictor and encoder updates after warm-up.

A. Variational Probabilistic Quantization

  • Two probabilistic encoders map observations elementwise to discrete variables on a common q-ary alphabet, with parameter sharing possible for statistically symmetric sources.Their final layers are a linear projection followed by a q-dimensional softmax; the three losses target agreement, near-uniform marginals, and low mutual information with Eve.Source: A. Variational Probabilistic… · p. 2
  • The mismatch loss is fully differentiable and drives paired inputs toward a common one-hot output by maximizing agreement between W and V.It uses the conditional probability of agreement, with conditional independence of W and V given (X, Y) enabling factorization.Source: A. Variational Probabilistic… · p. 2
  • Because agreement alone permits a constant-symbol solution, the uniformity loss pushes H(W) and H(V) toward log q.When the alphabet exceeds batch size, exponential moving averages track the marginal probabilities across training steps.Source: A. Variational Probabilistic… · p. 2
  • For correlated Eve, VPQ bounds leakage variationally with a learnable predictor: the VLB underestimates I(W; Z), while the VUB can overestimate it after a successful adversarial step.When Eve is absent or uncorrelated, the data processing inequality makes the leakage term zero, leaving the entropy and mismatch losses.Source: A. Variational Probabilistic… · p. 2
  • After a warm-up that updates only the encoders, predictor and encoder updates alternate one-to-one when Eve is correlated.The warm-up prevents predictor overfitting to a random encoder and an uninformative leakage signal.Source: A. Variational Probabilistic… · p. 3

B. Code-Offset Reconciliation

Because the encoder outputs can mismatch, the reconciliation stage uses a q-ary linear code-offset secure sketch. Alice sends one public offset message, and Bob recovers the same key whenever the block mismatch is within the code’s correction radius.

B. Code-Offset Reconciliation

  • With nonzero mismatch probability, the probability that W^n = V^n decays exponentially in n, making a public-discussion step necessary.Source: B. Code-Offset Reconciliation · p. 3
  • Alice samples K uniformly from a q-ary linear code and sends S = K − W^n; Bob decodes using V^n + S and outputs L = K when d_H(W^n, V^n) ≤ t.The code has parameters [n, m, 2t+1]_q, and each block yields (m/n) log q key bits.Source: B. Code-Offset Reconciliation · p. 3
  • The code-offset sketch reconciles the encoder outputs through a single one-way public message, producing identical keys when the mismatch lies within the correction radius.Source: B. Code-Offset Reconciliation · p. 3

IV. INFORMATION-THEORETIC GUARANTEES

The section connects the VPQ losses to achievable secret-key rates through an induced-source capacity bound and a universal reconciliation-rate characterization.

IV. INFORMATION-THEORETIC GUARANTEES

  • The analysis links the discrete source induced by the encoders to the one-way secret-key capacity.Source: §IV · p. 3
  • It also characterizes the optimal universal key rate achievable by code-offset reconciliation for an encoder with a given alphabet size and mismatch probability.Source: §IV · p. 3
  • Together, the two steps connect VPQ training losses to achievable secret-key rates.Source: §IV · p. 3

A. Achievable Rate of the Induced Source

The induced source admits a one-way secret-key capacity lower bound that accounts for Alice–Bob mismatch and Eve’s information. The VPQ losses correspond to terms in that bound, and data processing carries the induced-source guarantee to the original source.

A. Achievable Rate of the Induced Source

B. Universal Reconciliation Rate

For a q-ary source with mismatch probability p_e, the paper derives the optimal worst-case one-way reconciliation rate and shows q-ary linear code-offset sketches attain it. The rate is a uniform guarantee over the specified i.i.d. source class, not necessarily the best rate for each particular source law.

  • For prime-power q and p_e < 1−1/q, Theorem 2 states an optimal worst-case one-way reconciliation rate over D_pe, where W and V are uniform and I(W;Z)=0.Source: B. Universal Reconciliation Rate · p. 4
  • q-ary linear code-offset sketches achieve this rate uniformly over D_pe with reliability and zero leakage for every blocklength.Source: B. Universal Reconciliation Rate · p. 4
  • No scheme reliable and secure uniformly over D_pe can exceed R*univ, although a particular source law in the class may permit a higher rate.The converse uses a q-ary symmetric channel with constant Eve observation.Source: B. Universal Reconciliation Rate · p. 4
  • Random parity-check syndromes and Hamming-ball decoding achieve the rate because the collision bound depends only on p_e, giving a uniform construction.Source: B. Universal Reconciliation Rate · p. 4
  • At I(W; Z)=0, the universal reconciliation rate matches the encoder-induced rate, so reconciliation is not the bottleneck and no separate privacy amplification is needed.Source: B. Universal Reconciliation Rate · p. 4
  • When I(W; Z)≤ε, the construction bounds nI(K_n; S_n,Z_n) by ε while retaining rate at least (1−hq(p_e)) log q−δ.Source: B. Universal Reconciliation Rate · p. 4

V. EXPERIMENTS ON FADING CHANNELS

Experiments on Gaussian fading channels evaluate VPQ across Eve scenarios and alphabet sizes, examining training dynamics, reconciliation mismatch, and leakage.

V. EXPERIMENTS ON FADING CHANNELS

  • Mixed-SNR training draws legitimate noise variance across a range that covers 0–30 dB, allowing one encoder to serve that range without retraining or SNR feedback.The encoder is tested at specific SNRs after training.Source: §V · p. 5
  • H(W) approaches log q within tens of epochs; agreement stabilizes after adversarial fine-tuning begins, while the leakage gap settles near log q.The paper identifies a gap near log q as the signature of the zero-leakage regime.Source: §V · p. 5, Figure 2 · p. 5
  • VPQ keeps Eve leakage at most 0.3 bits per symbol across q and tested SNRs, versus baseline leakage of 1.5–2.2 bits per symbol at 20 dB and above.Fig. 4 reports the correlated-Eve comparison.Source: Figure 4 · p. 6, §V · p. 6
  • RS cliffs stay left of the universal limit; larger q shifts both limit and cliff right at moderate mismatch, but correlated Eve makes the smallest alphabets best at low rates.The RS gap is the expected finite-blocklength gap for a length-(q −1) code.Source: Figure 3 · p. 5, §V · p. 6

VI. CONCLUSION

VPQ combines variationally trained discrete encoders with a linear code-offset sketch for reconciliation. The conclusion reports theoretical guarantees and low leakage in Gaussian fading experiments.

VI. CONCLUSION

  • VPQ losses correspond term by term to a lower bound on the induced source’s one-way secret key capacity.The guarantee applies in particular under a correlated Eve.Source: §VI · p. 6
  • q-ary linear sketches attain the worst-case universal rate over the corresponding i.i.d. source class.The rate is characterized by Theorem 2.Source: §VI · p. 6
  • On Gaussian fading channels, VPQ leakage is at most 0.3 bits per symbol versus up to 2.2 for baselines, and Reed–Solomon reconciliation stays within the predicted finite-blocklength gap.The quantization stage is independent of public-discussion direction, so two-way reconciliation replaces only the sketch.Source: §VI · p. 6
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