Gain-Dependent Entropy Characterization of a Zener Avalanche Noise TRNG via Dual NIST Standards: Toward Low-Cost Hardware Entropy for ICS Deployments

Introduction

True random number generators based on physical noise sources occupy a central role in hardware security. The random number generators provide the entropy for cryptographic key generation, nonce production, and challenge–response authentication. Unlike pseudorandom number generators, TRNGs derive unpredictability from irreducible physical phenomena such as thermal fluctuations, shot noise, and quantum tunneling, and are therefore resistant to algorithmic reconstruction by an adversary who knows the seed state. The practical challenge in TRNG engineering is not the existence of physical noise but rather the rigorous, standards-compliant characterization of how much entropy that noise actually delivers after digitization and conditioning.

Zener avalanche noise has attracted sustained interest as an entropy source precisely because of its simplicity [1, 2]. A reverse-biased $p$–$n$ junction operating in the avalanche-breakdown regime generates broadband shot-noise currents that, when amplified and thresholded by a comparator, produce a binary switching waveform. The resulting circuit requires only a Zener diode, an operational amplifier, and a handful of passive components. While oscillator-based and integrated-circuit TRNG topologies have received extensive characterization [7, 8, 9], the Zener avalanche-noise family has not: the literature contains no systematic characterization of how amplifier gain affects entropy quality. Published designs typically select a gain empirically and validate the output with a subset of NIST SP 800-22 tests [3], without examining the gain–entropy relationship or applying the more demanding SP 800-90B entropy-source assessment [4], which is the current NIST standard for qualifying entropy sources in cryptographic applications.

This paper closes that gap. We characterize an MCP6002-based Zener TRNG across five gain settings spanning nearly one decade ($G = 11$ to $G = 101$) and a secondary coupling-capacitor study ($C_g = 0.1\,\mu\mathrm{F}$ vs. $0.01\,\mu\mathrm{F}$), subjecting each configuration to both NIST SP 800-22 and SP 800-90B in parallel. The experimental dataset uses $1.44 \times10^{9}$ raw bits per configuration at 400 kHz and is the largest published for this circuit family, enabling statistical conclusions with confidence levels not achievable in shorter captures.

The principal contributions of this work are:

• Three-phase ACF structure. A three-phase positive–anti-correlated–independent autocorrelation structure appears in every gain configuration, attributable to MCP6002 comparator saturation-recovery dynamics.

• Gain–entropy relationship. SP 800-90B min-entropy rises monotonically with gain across the $C_g = 0.1$ µF sweep ($h_{\min} = 0.441$ at $G = 11$ to $0.832$ at $G = 101$), with the compression estimator the binding constraint throughout.

• Entropy source attribution. A DC substitution experiment replacing the Zener diode with a stable 23.5 V DC source at the same Node A bias point cuts mean switching frequency by 97% (11,841 Hz → 407 Hz), establishing Zener avalanche noise as the necessary entropy source for this circuit architecture.

• Comparator-overdrive mechanism. Node C amplitude and DC-floor offset govern the near-threshold oscillation rate, the physical mechanism linking gain to entropy quality and bias degradation.

• $C_g$ dual role. $C_g = 0.01$ µF achieves committed NodeREF crossings and a Pareto improvement in entropy, throughput, and bias over $C_g = 0.1$ µF.

• Minimum viable gain boundary. Convergent SP 800-22 and SP 800-90B evidence identifies $G = 48$ as optimal: 15/15 SP 800-22, $h_{\min} = 0.801$ b/b, IID pass, 11,568 b/s.

The remainder of this paper is organized as follows. Section II describes the circuit and apparatus. Section III details the data-processing methodology. Section IV presents the correlation-structure analysis. Section V analyzes the comparator-overdrive mechanism. Section VI characterizes the $C_g$ effect. Sections VII and VIII present the SP 800-22 and SP 800-90B results respectively. Section IX establishes the minimum viable gain boundary. Section X evaluates ICS applicability, and Section XI concludes.

Circuit Design and Experimental Apparatus

This section outlines the circuit design, experimental configurations, and measurement instrumentation used throughout the study.

Circuit Architecture

The TRNG circuit comprises three functional stages: a regulated high-voltage supply, a Zener avalanche-noise source with coupling network, and a two-stage MCP6002 signal chain. The following description traces the signal path from the physical noise source to the digital output at Node E, shown schematically in Figure 1.

Stage 1 - High-voltage supply: An MT3608 boost converter steps up the 5 V system supply to 27.9 V, establishing the reverse-bias voltage for the Zener diode. The boost converter output is stable to within ±0.3 V across all captures, ensuring a consistent Zener operating point throughout the gain sweep. Node A-the voltage at the Zener anode, outputs 23.6 V under all operating conditions, confirming stable avalanche-regime operation.

Stage 2 - Noise source and coupling: We reverse bias a 1N4749A Zener diode (24 V nominal) at Node A. In the avalanche-breakdown regime, the diode generates broadband shot noise arising from the stochastic avalanche-multiplication process [6]. The noise current flows through the Zener impedance, developing a small-signal voltage that constitutes the raw entropy input to the amplifier stage. A coupling capacitor $C_g$ in series with a gate resistor $R_g = 1$ kΩ forms an RC high-pass filter with cutoff frequency $f_c = 1/(2\pi R_g C_g)$. In the primary study, $C_g = 0.1$ µF gives $f_c = 1592$ Hz; a secondary study with $C_g = 0.01$ µF raises $f_c$ to 15,915 Hz. We analyze the role of the filter in shaping both the noise spectrum and the comparator dynamics in Section VI.

Stage 3 - Signal chain (MCP6002 dual op-amp): We configure the first op-amp section as a non-inverting voltage amplifier with gain $G = 1 + R_f/R_g$. The feedback resistance $R_f$ is the primary experimental variable, taking values in $\{10, 22, 30, 47, 100\}$ kΩ to produce gain settings $G \in\{11, 23, 31, 48, 101\}$. The amplified noise appears at Node C, where its peak-to-peak voltage is characterized independently by an Analog Discovery 3 (AD3) oscilloscope. We configure the second MCP6002 section as an open-loop voltage comparator, thresholding Node C against a reference voltage NodeREF. The comparator output at Node E is the binary TRNG output captured by the DSLogic U3Pro16 logic analyzer. We measure the NodeREF at 2.54 V under operating conditions, representing a +40 mV offset above the designed 2.50 V midpoint. Section V analyzes the implications of the offset.

The MCP6002 has a 1 MHz gain–bandwidth product (GBW). At $G = 101$, the closed-loop bandwidth of the amplifier stage is GBW/$G \approx9.9$ kHz-above the mean switching frequency of 11,841 Hz, ensuring that the amplifier does not become the bandwidth bottleneck at the highest gain setting. At $G = 11$, the closed-loop bandwidth extends to approximately 91 kHz, well beyond any relevant signal frequency. VDD stays stable at 5.04–5.06 V across all captures. The complete circuit sits on a standard breadboard with no specialized PCB or shielding.

Experimental Configurations

Table 1 summarizes the six capture configurations. Runs 1–5 vary $R_f$ while holding $C_g = 0.1$ µF constant. Run 6 retains $R_f = 100$ kΩ ($G = 101$) and reduces $C_g$ to 0.01 µF for the coupling-capacitor study. We compute the mean switching frequency $f_{\mathrm{avg}}$ from the DSLogic capture as half the transition count divided by the capture duration. Raw bias (ones fraction) is derived from the $1.44 \times10^9$ raw bits before any processing. Sections IV–VII discuss observations from these six configurations.

Run	$R_f$ (kΩ)	$G$	$C_g$ (µF)	VDD (V)	$f_{\mathrm{avg}}$ (Hz)	Bias (%)	Node A (V)
1	100	101	0.10	5.04	11,841	52.04	23.6
2	47	48	0.10	5.05	14,449	52.50	23.6
3	30	31	0.10	5.06	14,490	53.62	23.6
4	22	23	0.10	5.05	14,137	55.43	23.6
5	10	11	0.10	5.03	17,328	63.70	23.6
6	100	101	0.01	5.05	15,580	51.48	23.6

Measurement Instrumentation

Digital output capture: We use a DSLogic U3Pro16 logic analyzer to sample Node E at 400 kHz for 1 hour per configuration, yielding $1.44 \times10^9$ raw bits. The U3Pro16 captures all comparator transitions including rapid chatter events that analog instruments may miss.

Analog characterization: The Analog Discovery 3 (AD3) measures Node C peak-to-peak voltage, maximum, minimum, and mean level under continuous noise-driven operation. The AD3 also measures Node E frequency via trigger counting; however, a systematic discrepancy of 15–61% was observed between AD3 and DSLogic frequency estimates because the analog trigger misses rapid chatter transitions. All frequency values cited throughout this paper use DSLogic measurements as the authoritative reference.

Data Processing Methodology

This section outlines the data processing methodology and discusses the configurations for the NIST SP 800-22, and 800-90B assessment protocol.

Decimation and Independence Criterion

Raw 400 kHz captures exhibit inter-sample correlation: at a mean switching frequency of 11–17 kHz, the binary output changes state only once per 23–36 samples on average. Adjacent raw samples are therefore identical with probability exceeding 0.97. Applying statistical tests directly to the raw bitstream would characterize the sampling clock rather than the entropy source.

Decimation by factor $D = 120$ maps the 400 kHz capture to an effective rate of 3333 Hz, corresponding to a physical inter-sample spacing of $\Delta t = D/f_s = 300\ \mu\mathrm{s}$. The choice

of $D = 120$ is justified empirically by the autocorrelation analysis in Section IV: across all five $C_g = 0.1\ \mu\mathrm{F}$ configurations, the Autocorrelation Function (ACF) magnitude falls below $|r| < 0.005$ by 290 $\mu$s. Retained samples at 300 $\mu$s spacing thus reside in the statistically independent regime for every configuration. After decimation, $N_{\mathrm{dec}} = 12 \times10^6$ bits are available per configuration, packing to $1.50 \times10^6$ bytes and satisfying the SP 800-90B minimum of $10^6$ bytes.

Double Von Neumann Debiasing

Raw decimated bitstreams exhibit positive bias ranging from 52.0% to 63.7% ones (Table I), arising from the NodeREF offset and DC-floor dynamics detailed in § 5. The von Neumann (VN) extractor [5] processes non-overlapping consecutive bit pairs: discordant pairs (01 → 0, 10 → 1) are output; concordant pairs (00, 11) are discarded. This procedure removes first-order bias with output efficiency $\eta= 2p(1-p)$, where $p$ is the ones fraction.

A single VN pass reduces bias to 50.01–50.03% but leaves residual serial correlation $|r| \approx0.003$–0.005 in the output bitstream. At $n = 12 \times10^6$ bits, this correlation level is sufficient to cause statistically significant failures in SP 800-22 T03 (Runs), T11 (Serial), and T12 (Approximate Entropy). A second VN pass (Double VN, DVN) reduces correlation to $|r| < 0.003$ at the cost of approximately 75% of the single-pass output, yielding $N_{\mathrm{dvn}} \in[694\mathrm{k}, 751\mathrm{k}]$ bits per configuration, sufficient for the full fifteen-test SP 800-22 suite. We apply DVN uniformly across all six configurations to ensure methodological consistency.

NIST SP 800-22 Assessment Protocol

We implement fifteen SP 800-22 tests in Python, and validate each against the reference $p$-values in SP 800-22 Appendix A. The T10 (Linear Complexity) test employed a

Berlekamp–Massey algorithm over GF(2) capped at $N = 500$ blocks of $M = 500$ bits to avoid memory exhaustion, validated on analytically verified test vectors (all-zeros, all-ones, and alternating sequences with analytically known LFSR lengths). All fifteen tests are run on the DVN output for every configuration. The Python code evaluates the T14 (Random Excursions) and T15 (Excursions Variant) on applicability criteria ($J \ge500$ zero crossings) prior to execution.

NIST SP 800-90B Assessment Protocol

We apply the SP 800-90B bitstring estimator path to the full $N_{\mathrm{dec}} = 12 \times10^{6}$ decimated bits (prior to VN debiasing), packed into $1.50 \times10^{6}$ bytes. Using the validated sp800-90b Python package, the pipeline executes the ten bitstring estimators: Most Common Value (MCV), Collision, Markov, Compression, T-Tuple, Longest Repeated Substring (LRS), Multi-MCV, Multi-Markov, and the binding minimum $h_{\min}$. IID assessment used the SP 800-90B permutation-test procedure (Section 4.3 of the standard). All estimates are reported in bits per bit (b/b), obtained by dividing the package per-byte output by 8.

Correlation Structure Analysis

This section outlines the autocorrelation structure of Zener Avalanche TRNG across gain setting and determines the best sampling lag.

Config ($G =$)	$f_{\mathrm{avg}}$ (Hz)	Bias (%)	Peak $r$	Peak Lag (µs)	Stable Lag (µs)	Decorr. Lag (µs)
101	11,841	52.08	−0.101	77.5	662.5	37.5
48	14,449	52.46	−0.085	57.5	667.5	27.5
31	14,490	53.74	−0.077	67.5	547.5	32.5
23	14,137	55.58	−0.076	82.5	602.5	42.5
11	17,328	63.68	−0.063	77.5	952.5	32.5
101, 0.01 µF	15,580	51.46	−0.337	37.5	462.5	62.5

Three-Phase ACF Structure

Computation of the sample autocorrelation function on the raw 400 kHz bitstream reveals a three-phase structure that is consistent across all five $C_g = 0.1$ µF gain configurations.

Phase I - Positive correlation (0–40 µs): Within each switching half-cycle, the binary output maintains its current state, producing positive ACF for lags shorter than the mean half-period. All five configurations exhibit positive ACF out to approximately 40 µs, consistent with the 11–17 kHz mean switching frequencies observed across gain settings.

Phase II - Anti-correlation (40–290 µs): A negative ACF lobe follows the positive phase. We attribute this anti-correlation to MCP6002 comparator saturation recovery. When the amplified noise signal drives the comparator output to saturation, the recovery trajectory produces a transient overshoot in the opposite direction. This overshoot is reflected in the probability that the next sample takes the opposite value, manifesting as negative ACF. The depth of this phase scales with gain (Table II), consistent with deeper saturation at higher amplification levels.

Phase III - Independence (> 290 µs): The ACF magnitude decays below $|r| = 0.005$ for all configurations by 290 µs, entering the statistically independent regime (Figure 2, Top panel). Accordingly, we select a decimation spacing of 300 µs, which provides independent values for all gain settings.

Gain Dependence of Anti-Correlation Depth

Anti-correlation depth (peak negative $r$) decreases monotonically as gain decreases: from $r = -0.101$ at $G = 101$ to $r = -0.063$ at $G = 11$. This ordering reflects the saturation-recovery mechanism. Higher gain drives the MCP6002 output deeper into rail saturation, producing a longer and more energetic recovery overshoot. Notably, deeper anti-correlation does not imply lower entropy—the compression estimator results in Section VIII show that configurations with shallower anti-correlation (lower gain) have lower $h_{\min}$, not higher, because bit bursts arising from near-threshold oscillation are the dominant entropy-limiting mechanism at low gain.

Cg Effect on ACF

Replacing $C_g = 0.1$ µF with $C_g = 0.01$ µF at $G = 101$ increases the anti-correlation depth by 3.3× (peak $r = -0.337 vs. −0.101) and reduces the stable independence lag from 663 µs to 463 µs. The deeper anti-correlation reflects the larger Node C amplitude (158.6 mV vs. 51.3 mV) driving stronger comparator saturation and recovery ringing. The shorter independence lag is consistent with the higher switching frequency (15,580 Hz vs. 11,841 Hz) enabled by the reduced capacitive loading on the amplifier input. Both configurations satisfy the 300 µs independence criterion, and the decimation factor $D = 120$ remains valid for all six configurations.

Comparator Overdrive Mechanism

This section analyzes the comparator threshold geometry and overdrive mechanism, confirming that the observed noise originates from Zener avalanche dynamics rather than MCP6002 self-excitation.

DC Floor and Threshold Geometry

The measured NodeREF of 2.54 V (+40 mV above design) and a measured DC floor of 58 mV establish the threshold geometry for the comparator. Node C must swing to at least (NodeREF − 58 mV) = 2.482 V to produce a committed low-to-high transition at Node E. Table III shows that for all five $C_g = 0.1$ µF configurations, Node C minimum remains at 47.6–60.8 mV above NodeREF. The signal never crosses the threshold. We therefore characterize all observed binary switching at $C_g = 0.1$ µF as near-threshold oscillation (NTO)—Zener noise perturbing Node C close enough to NodeREF to repeatedly destabilize the comparator without producing a committed crossing. § 5-D verifies experimentally that this oscillation requires the Zener noise and does not arise from MCP6002 self-excitation.

Config	Vpp (mV)	Max (mV)	Min (mV)	OD (%)	NTO (%)	$f_{\mathrm{avg}}$ (Hz)
$G=101$, 0.1µF	51.3	+98.9	+47.6	0.44	13.3	11,841
$G=48$, 0.1µF	22.7	+80.6	+58.0	0.20	15.9	14,449
$G=31$, 0.1µF	15.5	+73.3	+57.7	0.13	19.2	14,490
$G=23$, 0.1µF	11.0	+69.2	+58.2	0.09	22.9	14,137
$G=11$, 0.1µF	6.96	+67.7	+60.8	0.06	29.2	17,328
$G=101$, 0.01µF	158.6	+143.8	−14.8	1.37	7.9	15,580

With $C_g = 0.01$ µF, Node C minimum reaches −14.8 mV relative to NodeREF, achieving committed threshold crossing by a margin of 72.8 mV. This single component change qualitatively alters the entropy-generation mechanism from near-threshold Zener-perturbed oscillation to committed crossing-driven switching with consequences for both entropy quality and waveform fidelity discussed in Section VI.

Near-Threshold Oscillation Rate and Bias Degradation

Although Node C does not achieve committed threshold crossings in the $C_g = 0.1$ µF configurations, Zener avalanche noise perturbs Node C sufficiently near NodeREF to destabilize the comparator repeatedly. The near-threshold oscillation (NTO) rate increases monotonically from 13.3% at $G = 101$ to 29.2% at $G = 11$ (Table III, Figure 3c), reflecting the decreasing Node C amplitude with decreasing gain (Figure 3a). At the lowest gain setting $G = 11$, the 6.96 mV swing is barely above the measurement noise floor.

The raw bitstream bias degrades in direct proportion to the NTO rate. At $G = 11$, the 63.7% ones fraction arises from an asymmetry in the MCP6002 settling behavior during the indeterminate near-threshold period. The output spends more time at the positive rail than the negative rail. This bias is deterministic and removable, Double VN debiasing restores all configurations to 49.95–50.11% within the VN output, demonstrating that the bias reflects a systematic circuit asymmetry rather than a fundamental entropy deficit.

Gain to Entropy Quality Relationship

The physical chain linking gain to entropy quality operates as follows. Higher $R_f$ increases Node C amplitude. Larger amplitude provides more decisive comparator perturbation, reducing the NTO fraction and producing a more symmetric bit distribution. Simultaneously, larger amplitude drives the MCP6002 deeper into saturation on each switching event, lengthening the recovery trajectory and producing deeper Phase II anti-correlation in the ACF. Lower gain inverts both effects. The small Node C swing perturbs the comparator weakly, generating dense near-threshold bit bursts that create correlated sequences and, at $G = 11$, ultimately cause IID failure under SP 800-90B permutation testing. The $C_g$ parameter modulates this mechanism by controlling both the noise bandwidth reaching the amplifier input and, critically, whether the resulting Node C amplitude is sufficient to achieve committed crossings, as analyzed in Section VI.

Entropy Source Verification: DC Substitution Test

To verify experimentally whether the near-threshold oscillation described above requires Zener avalanche noise, or could instead arise from MCP6002 self-excitation alone, we conducted a DC substitution experiment. We removed the 1N4749A Zener diode and retuned the existing MT3608 boost converter to 23.5 V, connecting its output directly to Node A through a 100 µF electrolytic capacitor and a 100 nF ceramic capacitor (suppressing the converter’s 1.2 MHz switching ripple) and a 10 kΩ dummy load (maintaining stable boost-converter regulation in the absence of the Zener’s current draw). We verified Node A ripple below 5 mV peak-to-peak with the AD3 at 1 µs/div before each capture. All other circuit parameters such as gain, $C_g$, NodeREF, and DSLogic capture settings remained identical to the original captures.

With $C_g = 0.1$ µF and $G = 101$, DC substitution reduced the switching frequency from 11,841 Hz (Table I) to approximately 407 Hz, a reduction of approximately 97%. Node E remained predominantly high, with rare, irregularly spaced dips consistent with environmental noise threshold crossings rather than self-sustaining comparator oscillation.

This result yields three conclusions. First, MCP6002 internal noise alone is insufficient to sustain comparator switching at this topology’s operating point. The DC-floor geometry places NodeREF too close to Node C’s resting potential for thermal noise to bridge the gap reliably. Second, Zener avalanche noise is the necessary entropy source for all six configurations, including those at $C_g=0.1\,\mu\mathrm{F}$ where Node C does not achieve a committed crossing. Third, the distinction between the two $C_g$ regimes is one of overdrive quality rather than entropy-source identity. In both regimes the Zener noise is essential, but $C_g=0.01\,\mu\mathrm{F}$ converts this noise into committed threshold crossings, whereas $C_g=0.1\,\mu\mathrm{F}$ produces near-threshold oscillation that is statistically weaker yet fundamentally Zener-driven.

Coupling Capacitor Effect Study

This section characterizes the dual filtering and stabilization roles of the coupling capacitor $C_g$ and quantifies its impact on entropy quality and throughput at $G=101$.

Dual Role of Cg

The coupling capacitor $C_g$ in series with $R_g$ forms an RC high-pass filter with cutoff frequency $f_c=1/(2\pi R_g C_g)$. With $R_g=1\,\mathrm{k}\Omega$ and $C_g=0.1\,\mu\mathrm{F}$, $f_c=1592$ Hz stays below the mean switching frequency $f_{\mathrm{avg}}=11{,}841$ Hz. The capacitor therefore attenuates the noise signal before it reaches the amplifier input, suppressing spectral content above 1592 Hz and limiting the Node C amplitude to 51.3 mV despite the high gain of 101. With $C_g=0.01\,\mu\mathrm{F}$, $f_c=15{,}915$ Hz slightly exceeds $f_{\mathrm{avg}}=15{,}580$ Hz, allowing the full noise bandwidth to reach the amplifier. This filter role together with the Zener-driven entropy source verified in § 5-D explains the 3.1× increase in $V_{pp}$ and the 31.6% increase in switching frequency when $C_g$ is reduced.

$C_g$ also plays a second, less obvious role as a comparator-stabilization element. By limiting the rate of change of the signal at the amplifier input, the larger $C_g=0.1\,\mu\mathrm{F}$ slows the transitions at Node C, reducing the comparator’s tendency toward near-threshold oscillation at the transition edges. With $C_g=0.01\,\mu\mathrm{F}$, the faster signal transitions and larger amplitude produce genuine threshold crossings at the cost of deeper ACF anti-correlation (peak $r=-0.337$ vs. $-0.101$, Figure 4), reflecting the more energetic saturation and recovery dynamics associated with clean crossing events.

Impact on Entropy Quality

Table IV summarizes the $C_g$ comparison at $G = 101$. Despite its deeper anti-correlation, $C_g = 0.01\,\mathrm{\mu F}$ achieves higher $h_{\min}$ (0.869 b/b vs. 0.832 b/b) under SP 800-90B. The compression estimator’s sensitivity rewards the faster alternation rate and reduced near-threshold oscillation of genuine crossing events, while the anti-correlation occurs on a timescale (37–463 $\mathrm{\mu s}$) below the 300 $\mathrm{\mu s}$ decimation threshold. Entropy throughput $\Phi= f_{\mathrm{avg}} \times h_{\min}$ reaches 13,534 b/s with $C_g = 0.01\,\mathrm{\mu F}$, a 37% improvement over the 9,856 b/s achieved with $C_g = 0.1\,\mathrm{\mu F}$ at the same gain setting.

Metric	$C_g = 0.1\,\mathrm{\mu F}$	$C_g = 0.01\,\mathrm{\mu F}$	Change
$f_c$ (Hz)	1,592	15,915	$\times10.0$
$V_{pp}$ at Node C (mV)	51.3	158.6	$\times3.09$
Crosses NodeREF	No	Yes	—
NTO (%)	13.3	7.9	$-40\%$
$f_{\mathrm{avg}}$ (Hz)	11,841	15,580	$+31.6\%$
Raw bias (%)	52.04	51.48	$-0.56\%$
ACF peak $r$	$-0.101$	$-0.337$	3.3× deeper
$h_{\min}$ (b/b)	0.832	0.869	$+0.037$
Throughput (b/s)	9,856	13,534	$+37\%$

These results establish that for this circuit topology, reducing $C_g$ from 0.1 $\mathrm{\mu F}$ to 0.01 $\mathrm{\mu F}$ represents a Pareto improvement; entropy quality, throughput, and bias all improve simultaneously. The trade-off is a deeper ACF anti-correlation that does not survive decimation and therefore does not affect the assessed entropy. For implementations where the single passive-component substitution is feasible, $C_g = 0.01\,\mathrm{\mu F}$ is unambiguously preferred.

NIST SP 800-22 Statistical Test Suite Results

This section presents the NIST SP 800-22 fifteen-test results across all six configurations and analyzes the marginal $p$-values observed at lower gain settings.

Summary of Results

Table V presents the SP 800-22 results on the Double VN output for each configuration. We execute fifteen tests for each of the six configurations. With fifteen simultaneous tests per configuration at $\alpha= 0.01$, a genuinely random source has probability $P = 1 - (0.99)^{15} \approx13.99\%$ of producing at least one marginal result by chance alone, so the four marginal $p$-values observed across the dataset are consistent with expected multiple-testing variation rather than a systematic entropy defect. The T14 results at $G = 101$ ($p = 0.008$) and $G = 23$ ($p = 0.005$) are due to multiple-testing statistical fluctuation.

Gain Config	T03	T06	T11	T13	T14	T15	Pass	Overall
101, 0.1$\mathrm{\mu F}$	✓	✓	✓	✓	0.008†	✓	14/15	MARG.
48, 0.1$\mathrm{\mu F}$	✓	✓	✓	✓	0.048	✓	15/15	PASS⋆
31, 0.1$\mathrm{\mu F}$	0.010†	✓	✓	✓	—‡	—‡	12/13	MARG.
23, 0.1$\mathrm{\mu F}$	0.008†	✓	✓	✓	0.005†	✓	13/15	MARG.
11, 0.1$\mathrm{\mu F}$	✓	✓	✓	✓	—‡	—‡	13/13	PASS
101, 0.01$\mathrm{\mu F}$	✓	✓	✓	✓	—‡	—‡	13/13	PASS

The T13 (Cumulative Sums) test, which is sensitive to the same walk drift that causes T14 failures, passes at all configurations ($p$ ranging from 0.21 to 0.84), ruling out systematic walk asymmetry as the cause. The T03 marginal failures at $G = 31$ ($p = 0.010$) and $G = 23$ ($p = 0.008$) are attributable to residual serial correlation $\lvert r \rvert\approx0.003$ in the Double VN output at these higher-bias configurations. At $n \approx745\mathrm{k}$ bits, T03 has statistical power to detect correlations of this magnitude. Both affected configurations achieve $h_{\min} > 0.67$ b/b under SP 800-90B, confirming that the T03 sensitivity at this sample size reflects a processing artifact rather than a fundamental entropy deficit.

T14 and T15 inapplicability ($J < 500$ zero crossings) at $G = 31$, $G = 11$, and the $C_g = 0.01\,\mathrm{\mu F}$ configuration arises from the combination of DVN output size ($\sim750\mathrm{k}$ bits) and residual bias below 0.05%. The random walk from these configurations does not return to zero frequently enough to satisfy the T14 applicability criterion. All three affected configurations pass all applicable tests.

$G = 48$ as the Optimal Configuration

$G = 48$ ($R_f = 47\,\mathrm{k\Omega}$, $C_g = 0.1\,\mathrm{\mu F}$) is the sole configuration achieving a perfect 15/15 score. T14 ($p = 0.048$) and T15 ($p = 0.031$) both pass with substantial margin, confirming symmetric random-walk behavior after debiasing. This result, combined with the SP 800-90B findings in Section VIII, provides a convergent identification of $G = 48$ as the optimal operating point by two independent assessment methodologies.

NIST SP 800-90B Entropy Source Assessment

This section presents the NIST SP 800-90B entropy source assessment, identifies the binding min-entropy estimator, and analyzes IID compliance and entropy throughput across configurations.

Min-Entropy Results and Compression Bottleneck

Table VI presents the SP 800-90B results for all six configurations, with key trends summarized in Fig. 5. The compression estimator is the binding constraint across the entire gain sweep: $h_{\min}^{(\mathrm{all})} = h_{\min}^{(\mathrm{compression})}$ for every configuration (Fig. 5b). The compression estimator detects exploitable patterns in the sequence of packed bytes, so its consistent dominance points to residual serial correlation in the decimated bitstream as the common cause.

The gap between $h_{\mathrm{compression}}$ and the next estimator ($h_{\mathrm{MCV}} \approx0.85$–0.96 b/b across configurations) widens monotonically as gain decreases: 0.109 b/b at $G = 101$, rising to 0.209 b/b at $G = 11$. This progressive widening reflects the increasing NTO rate at lower gain. The near-threshold oscillation produces correlated bit bursts that the compression estimator identifies as exploitable structure, while the distribution-based MCV estimator is insensitive to the temporal ordering of symbols. The gap therefore quantifies the entropic cost of near-threshold oscillation in a directly measurable form.

Config	IID	$h_{\mathrm{comp}}^{\dagger}$ (b/b)	$h_{\mathrm{MCV}}$ (b/b)	$h_{\mathrm{markov}}$ (b/b)	$h_{\mathrm{lrs}}$ (b/b)	$h_{\min}$ (b/b)	Throughput (b/s)	SP 800-90B Verdict
$G=101$, 0.1 $\mu$F	✓	0.832	0.941	0.949	0.900	0.832	9,856	Marginal
$G=48$, 0.1 $\mu$F	✓	0.801	0.929	0.935	0.995	0.801	11,568	Marginal$^{\star}$
$G=31$, 0.1 $\mu$F	✓	0.752	0.898	0.905	0.982	0.752	10,896	Marginal
$G=23$, 0.1 $\mu$F	✓	0.675	0.850	0.857	0.982	0.675	9,537	Insufficient
$G=11$, 0.1 $\mu$F	×	0.441	0.650	0.654	0.895	0.441	7,640	Insufficient
$G=101$, 0.01 $\mu$F	✓	0.869	0.957	0.958	0.981	0.869	13,534	Marginal

IID Assessment

IID failure is isolated exclusively to $G=11$. All five other configurations pass the SP 800-90B permutation test, establishing a binary IID boundary between $G=11$ and $G=23$. At $G=11$, the NTO rate of 29.2% creates correlated bit bursts sufficiently long and frequent to violate the IID assumption at the permutation test’s significance level. At $G=23$, the NTO rate of 22.9% remains below this threshold, suggesting that the IID boundary lies in the NTO-rate range 22.9–29.2%.

Throughput Analysis

Entropy throughput $\Phi=f_{\mathrm{avg}}\times h_{\min}$ is non-monotonic in gain, peaking at $G=48$ (11,568 b/s) within the $C_g=0.1\,\mu\mathrm{F}$ sweep (Figure 5c). $G=101$ delivers higher $h_{\min}$ (0.832 b/b) but lower $f_{\mathrm{avg}}$ (11,841 Hz), yielding lower throughput (9,856 b/s). This non-monotonicity arises because the switching frequency which sets the raw bit-generation rate before decimation does not scale monotonically with gain. $f_{\mathrm{avg}}$ peaks near $G=31$–48 (14,449–14,490 Hz) before declining at higher gain due to the MCP6002 GBW limitation and the longer RC time constant at larger $R_f$. The product $f_{\mathrm{avg}}\times h_{\min}$ therefore peaks at $G=48$, where neither factor is at its individual extreme but their product is maximized. The circuit achieves overall maximum throughput of 13,534 b/s at $G=101$ with $C_g=0.01\,\mu\mathrm{F}$, as analyzed in Section VI.

Minimum Viable Gain Boundary

This section establishes the minimum viable gain boundary from the convergent SP 800-22 and SP 800-90B results and provides gain-selection guidelines for deployment.

Cross-Standard Convergence

A distinguishing feature of this dataset is the independent convergence of two NIST standards on consistent gain-quality boundaries through entirely different mathematical mechanisms. SP 800-90B’s compression estimator detects temporal structure in byte sequences. SP 800-22’s T03 (Runs) test detects run-length imbalance in the bit sequence, and the SP 800-90B permutation test detects sequential structure absent in IID samples. All three respond to the same physical phenomenon; near-threshold oscillation producing correlated bit bursts. The agreement of their findings is therefore not circular and provides stronger evidence for the gain-boundary conclusions than either standard alone could furnish.

Gain Selection Guidelines

Based on the dual-standard assessment, the following guidelines apply to deployments using this circuit topology:

Minimum viable ($G = 23$, $R_f = 22\ \mathrm{k}\Omega$): $h_{\min} = 0.675$ b/b, IID pass, SP 800-90B Insufficient. Acceptable for applications requiring entropy diversity without formal SP 800-90B certification, where post-processing conditioning will be applied.

Recommended minimum ($G = 31$, $R_f = 30\ \mathrm{k}\Omega$): $h_{\min} = 0.752$ b/b, IID pass, SP 800-90B Marginal. The lowest gain meeting the Marginal threshold; suitable for conditioned entropy delivery.

Optimal ($G = 48$, $R_f = 47\ \mathrm{k}\Omega$): $h_{\min} = 0.801$ b/b, IID pass, 15/15 SP 800-22, peak throughput 11,568 b/s. Recommended operating point balancing entropy quality, throughput, and standards compliance.

High entropy ($G = 101$, $R_f = 100\ \mathrm{k}\Omega$, $C_g = 0.1\ \mu\mathrm{F}$): $h_{\min} = 0.832$ b/b, IID pass, 14/15 SP 800-22. Preferred when maximum $h_{\min}$ is prioritized over throughput.

Maximum throughput ($G = 101$, $C_g = 0.01\ \mu\mathrm{F}$): $h_{\min} = 0.869$ b/b, IID pass, 13/13 SP 800-22, 13,534 b/s. Recommended when throughput is the primary constraint and a single passive-component substitution is feasible.

The boundary $G = 11$ is unambiguously inadequate by both standards: IID failure, $h_{\min} = 0.441$ b/b (SP 800-90B Insufficient), and the highest NTO rate (29.2%) of any tested configuration. $G = 11$ should be excluded from any deployment requiring statistical independence of the entropy-source output.

Applicability to Industrial Control System Deployments

This section evaluates the applicability of the characterized TRNG to industrial control system deployments, addressing entropy budgets, conditioning, and integration cost.

ICS Entropy Requirements

Industrial control systems are increasingly subject to mandatory cybersecurity frameworks that presuppose access to hardware entropy sources at the field-device level. IEC 62443-3-3 [11] Security Level 2 requires cryptographic authentication for all inter-zone communications, which in practice demands fresh nonces and session-specific key material at each authentication event. NERC CIP-015-1 [13], the emerging internetwork security standard for bulk electric systems, mandates random session identifiers for all authenticated remote access sessions. IEC 62351-8 [12], covering role-based access control for power systems, requires entropy for privilege-token generation. IEEE 1815 (DNP3) [14], the dominant SCADA communication protocol in North American electric utilities, mandates random challenge values for Secure Authentication v5. Each of these standards implicitly requires a hardware entropy source at the PLC or IED level.

The field-device entropy budget is constrained by scan-cycle timing. ICS scan cycles range from 1 ms (substation protection IEDs) to 500 ms (water-treatment SCADA RTUs). At $G = 48$, the TRNG delivers 11,568 b/s, providing 11.6 bits per 1 ms cycle and 5,784 bits per 500 ms cycle. A 128-bit session nonce accumulates in 11.1 ms; a 256-bit key seed in 22.1 ms. These accumulation times fit comfortably within the timing budget of all ICS scan classes except the most latency-critical protection-relaying applications (sub-1 ms), which are typically handled by dedicated hardware security modules in any case.

Conditioning for Cryptographic Use

The SP 800-90B Marginal verdict at $G = 48$ reflects that the raw source output requires conditioning before cryptographic-strength key material can be derived. SP 800-90B Section 10 specifies approved conditioning constructions; the simplest appropriate for this source is the Hash_df construction using SHA-256 [10]. With $h_{\min} = 0.801$ b/b and 512 bits of TRNG input, Hash_df delivers 256 bits of output with full 256-bit security strength. At 11,568 b/s raw throughput, the conditioner can be refreshed approximately 22 times per second—more than adequate for all ICS authentication use cases. For non-cryptographic applications (nonce diversity, challenge randomization, seed diversification) the unconditioned TRNG output at $G = 48$ is directly usable.

Integration Pathway and Cost

The complete TRNG circuit (1N4749A, MT3608 module, MCP6002, passives) has a bill-of-materials cost below $5 at single-unit pricing, representing a cost reduction of two to four orders of magnitude relative to commercial HSMs. The circuit requires no specialized PCB, shielding, or calibration; the gain-selection guideline ($R_f = 47\ \mathrm{k}\Omega$ for $G = 48$) is a single resistor specification. The digital output at Node E is a 3.3–5 V logic-level square wave compatible with GPIO inputs on all major PLC families (Siemens S7-1200, Allen-Bradley MicroLogix, Schneider Modicon, Click C0-series, OpenPLC) without level shifting.

Future work can integrate the entropy source with multi-vendor PLC testbeds. For each vendor, potential work can characterize the entropy-accumulation latency within IEC 61131-3 scan-cycle constraints and Modbus/TCP authenticated session-key generation under operational traffic load. The present paper establishes the entropy-source characterization that such integration work requires as its foundation. Without knowing which gain setting produces IID-compliant output at sufficient entropy rate, integration experiments cannot be designed with statistical rigor. The recommendation of $G = 48$ as the optimal operating point provides that foundation.

Conclusion

This paper presents a dual-standard entropy characterization of a Zener avalanche-noise TRNG across a gain sweep ($G = 11$ to $G = 101$) and a coupling-capacitor study, using $1.44 \times10^{9}$ raw bits per configuration—the largest dataset reported for this circuit family. A foundational question in any such characterization is whether the observed switching arises from the intended noise source or from amplifier self-excitation. A DC substitution experiment, replacing the Zener diode with a stable 23.5 V source at the same Node A bias point, resolved this directly: mean switching frequency fell by approximately 97% (11,841 Hz $\rightarrow$ 407 Hz), establishing Zener avalanche noise as the necessary entropy source across all configurations. The coupling capacitor $C_g$ does not alter this attribution; it determines instead whether the Zener noise produces near-threshold oscillation ($C_g = 0.1$ µF) or committed threshold crossings ($C_g = 0.01$ µF). Both regimes share a three-phase autocorrelation structure (positive, anti-correlated, independent) attributed to MCP6002 saturation-recovery dynamics, which motivates the $D = 120$ decimation factor and places all assessed samples in the statistically independent regime. Under SP 800-90B, the compression estimator is the universally binding min-entropy constraint, and its gap from $h_{\mathrm{MCV}}$—widening from 0.109 b/b at $G = 101$ to 0.209 b/b at $G = 11$—directly quantifies the entropic cost of near-threshold oscillation as gain decreases. Reducing $C_g$ to 0.01 µF at $G = 101$ exploits committed-crossing dynamics to achieve a Pareto improvement from a single passive-component change: $h_{\min}$ rises from 0.832 to 0.869 b/b, throughput from 9,856 to 13,534 b/s, and the near-threshold oscillation rate falls from 13.3% to 7.9%. Within the $C_g = 0.1$ µF sweep, $G = 48$ emerges as the optimal operating point by the convergent evidence of both standards: the sole 15/15 SP 800-22 score, $h_{\min} = 0.801$ b/b, IID pass, and peak throughput of 11,568 b/s at a BOM cost below $5. At the other extreme, $G = 11$ fails IID outright, placing the minimum viable gain boundary at $G = 23$.

Future work will characterize temperature dependence across the industrial operating range ($-40\,^{\circ}\mathrm{C}$ to $+85\,^{\circ}\mathrm{C}$), evaluate NodeREF trimmer integration for bias correction, extend the assessment to the full SP 800-90B ten-estimator suite and temporal stationarity under extended operation, and integrate the TRNG with multi-vendor PLC testbeds—Siemens S7-1200, Allen-Bradley MicroLogix 1400, Schneider Modicon M221, and OpenPLC on Raspberry Pi—under IEC 61131-3 scan-cycle constraints.