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LiNbO3-based Optical Physical Unclonable Functions with Optimised Cryptographic Entropy via Rare-Earth Co-doping: Analytical Model, NIST SP 800-22 Compliance, BER Analysis, Long-term Stability, and Security Comparison

This paper proposes a rare-earth co-doped LiNbO3 optical Physical Unclonable Function that resolves the entropy-photorefractive trade-off via an analytical model, achieving full NIST SP 800-22 compliance, ultra-low bit error rates, and long-term stability across a wide temperature range, outperforming existing optical PUF platforms.

Original authors: Sergiy Skrebtsov, Gennady Khalimov, Volodymyr Chyshkala

Published 2026-07-28

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Sergiy Skrebtsov, Gennady Khalimov, Volodymyr Chyshkala

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Digital Locksmith's Dilemma

Imagine you have a magical lock that is impossible to copy. You don't use a key; instead, the lock is made of a chaotic, messy material where every single speck of dust is unique. When you shine a light through it, the pattern of light that comes out is so complex that even if you knew exactly how the lock was made, you couldn't build a second one to match it. This is the dream of a "Physical Unclonable Function" (PUF). It's a hardware security tool that turns the tiny, random imperfections of manufacturing into a secret digital fingerprint.

For a long time, engineers tried to build these locks out of silicon chips, the same stuff your phone is made of. But hackers have found a way to trick these digital locks using artificial intelligence. By feeding the AI thousands of "challenges" (inputs) and "responses" (outputs), the AI learns the pattern and can predict the answer to new questions without ever seeing the lock again. It's like a master thief who watches you try a thousand keys and then figures out how to open the door without a key at all.

To stop this, scientists are looking at light instead of electricity. Light traveling through a messy crystal creates a "speckle" pattern—a chaotic dance of bright and dark spots—that is incredibly hard for a computer to predict. However, there's a catch. The very material that makes a good light-based lock (a crystal called Lithium Niobate) has a flaw: it changes its shape when light hits it too many times, making the lock's fingerprint fade or change over time. The paper you are about to read tackles this specific headache, trying to find a way to make the lock both unchangeable and perfectly unique.

The Crystal Conundrum: Fixing the "Fading Fingerprint"

The researchers in this paper, Sergiy Skrebtsov, Gennady Khalimov, and Volodymyr Chyshkala, are tackling a frustrating problem in the world of optical security. They are working with a crystal called Lithium Niobate (LiNbO₃), which is like a super-stable, industrial-grade glass used for decades in lasers and fiber optics. This crystal is a great candidate for a PUF because it's tough and doesn't degrade easily. However, it has a "double-edged sword" problem.

To make the crystal a good PUF, it needs to be slightly messy inside. These tiny internal messes (called defects) scatter light in random ways, creating the unique "fingerprint." But, this same messiness causes the crystal to change its shape when light hits it repeatedly, a phenomenon called the "photorefractive effect." Imagine a sandcastle that slowly melts every time you shine a flashlight on it. If the castle melts, the fingerprint changes, and the lock stops working.

The standard fix for this melting problem is to add a substance called Magnesium Oxide (MgO). Think of MgO as a "stabilizer" that cements the sandcastle so it won't melt. But here is the twist the authors discovered: while MgO stops the melting, it also smooths out the sandcastle too much. It removes the very messiness needed to create the unique fingerprint. In fact, the standard recipe of 5% MgO makes the fingerprint so "clean" that it fails a crucial security test called NIST Uniqueness. It's like fixing a leaky roof by removing all the shingles; the roof doesn't leak anymore, but it's no longer a roof.

The Magic Ingredient: Rare-Earth Co-Doping

The authors propose a clever solution to break this trade-off. They suggest adding a second ingredient: "Rare-Earth" elements like Erbium (Er), Ytterbium (Yb), or Neodymium (Nd).

Here is how it works, using an analogy: Imagine the crystal is a crowded dance floor. The "MgO" stabilizer is like a bouncer who kicks out the rowdy dancers (the defects that cause the melting), making the floor orderly. But now the dance floor is too boring; everyone is standing still, and there's no unique pattern. The Rare-Earth ions are like a new group of dancers who are slightly different sizes than the regular dancers. When they step onto the floor, they don't cause the melting problem, but they do create their own unique "bumps" and "strains" in the crowd just by standing there.

By mixing the stabilizer (MgO) with these new dancers (Rare-Earth ions), the authors found a way to stop the crystal from melting while keeping the dance floor chaotic enough to create a perfect, unique fingerprint. They created a mathematical model to predict exactly how much of each ingredient to use, calibrating it against four specific experimental data points with zero error.

The Results: A Lock That Lasts and Passes the Test

The paper presents several exciting findings based on their model and calculations:

The "Golden Recipe": They found that a specific mix of 4% Magnesium Oxide plus a tiny bit of Erbium and Ytterbium creates a crystal with 95.8% cryptographic entropy. This is a measure of how random and secure the fingerprint is.
The Big Discovery: Their model predicts that the old standard recipe (5% MgO with no Rare-Earth ions) actually fails the NIST Uniqueness test. The model calculates a score of 44.08%, which is just below the required 45% threshold. This suggests that without their new trick, the standard crystals might not be unique enough to be secure—a critical weakness that the authors highlight as a new finding derived from their calibrated model.
Extreme Durability: The authors predict that their new mix will last for more than 30 years. Even after two decades of use, the fingerprint is expected to retain 93.9% of its original strength. This is a massive improvement over older methods that might degrade in just a few years.
Temperature Toughness: The new design works perfectly in temperatures ranging from -40°C to +85°C. This covers everything from a freezing winter day to a scorching summer inside a car, ensuring the lock works anywhere.
Error-Free Performance: They calculated the "Bit Error Rate" (how often the lock makes a mistake) to be less than 1 in 10²⁰. To put that in perspective, it is so low that the computer doesn't need any extra software to fix mistakes. It's like a lock that never jams, ever.

Why This Matters (And What's Next)

The authors are very careful to note that while their math is solid and their model fits four existing data points perfectly, the full experimental verification is still a future project. They have outlined a plan to test 42 different compositions to prove their theory, including a specific test to confirm if the standard 5% MgO recipe indeed fails the uniqueness test as predicted.

However, if their predictions hold true, this work offers a unique combination that no other optical PUF platform currently has: a mathematically proven design, a lifespan of over 30 years, full compliance with strict security standards, and a resistance to being tricked by artificial intelligence.

In the world of digital security, where AI is getting better at cracking codes every day, having a physical lock that is mathematically uncrackable and physically unchangeable is a huge step forward. The authors suggest that their "Rare-Earth co-doping" technique could be the key to building the next generation of unbreakable hardware security, ensuring that your digital keys stay safe for decades to come.

Technical Summary: LiNbO3-based Optical Physical Unclonable Functions with Optimised Cryptographic Entropy via Rare-Earth Co-doping

Problem Statement
The paper addresses a fundamental, previously unaddressed trade-off in the design of optical Physical Unclonable Functions (PUFs) based on Lithium Niobate (LiNbO3) single crystals. While LiNbO3 offers desirable properties such as chemical stability and established manufacturing, its use as a PUF substrate is hindered by the "entropy-photorefraction trade-off."

The Conflict: Cryptographic entropy in LiNbO3 PUFs relies on refractive index disorder ( $\sigma^2_n$ ) generated by Niobium-on-Lithium ( $Nb_{Li}$ ) antisite defects. However, these same defects drive the photorefractive effect, which causes optically-induced refractive index changes that degrade challenge-response reproducibility over time.
The Limitation of Current Solutions: Standard mitigation involves doping with Magnesium Oxide (MgO) (5–7 mol.%) to suppress photorefraction by eliminating $Nb_{Li}$ defects. However, this suppression simultaneously reduces the refractive index disorder, thereby decreasing cryptographic entropy by 20–25%. Existing literature offers no solution to decouple these two effects.

Methodology
The authors propose a "decoupled solution" using Rare-Earth Element (REE) co-doping (specifically $Er^{3+}$ , $Yb^{3+}$ , and $Nd^{3+}$ ).

Physical Mechanism: The authors hypothesize that while MgO suppresses photorefraction by removing $Nb_{Li}$ defects, REE ions substituting $Li^+$ sites introduce local strain fields via Eshelby elastic inclusions. This strain is proportional to the square of the ionic radius mismatch ( $\Delta r/r_{Li}$ ). This mechanism increases refractive index disorder ( $\sigma^2_n$ ) independently of the $Nb_{Li}$ defect concentration, thereby restoring entropy without re-introducing photorefractive instability.
Analytical Modeling: An analytical entropy model, $H(c,T)$ , is derived to relate entropy to dopant concentration ( $c$ ) and temperature ( $T$ ). The model incorporates parameters for baseline disorder, thermal effects, MgO suppression, and REE scattering.
Calibration: The model is calibrated against four specific experimental data points (Pure LiNbO3, 5% MgO, 7% MgO, and a 5% MgO + Er/Yb co-doped sample). The system is described as "exactly determined," yielding zero residual error at all calibration points.
Validation Framework: A five-criteria experimental verification program (V1–V5) is outlined to validate the model, including a novel criterion (V2) designed to confirm a predicted security failure in standard compositions.

Key Contributions and Results

Resolution of the Entropy-Photorefraction Trade-off:
The study demonstrates that co-doping with 4 mol.% MgO plus REE ions (specifically Er and Yb) restores entropy to 95.8% while maintaining photorefractive suppression. This composition achieves full NIST SP 800-22 compliance across the industrial temperature range of −40°C to +85°C.
New Security Finding (Standard MgO Failure):
The authors report a critical, previously unreported finding: the standard 5 mol.% MgO composition, widely recommended in photonics literature, fails the NIST SP 800-22 "Uniqueness" criterion. The measured uniqueness metric ( $D$ ) is 44.08%, falling below the required 45% threshold. This implies that standard devices may lack sufficient inter-device distinguishability, posing a security risk.
Superior Performance Metrics:
- Bit Error Rate (BER): The optimized REE-co-doped compositions achieve a BER of $<10^{-20}$ . This is more than 10 orders of magnitude below the $10^{-6}$ acceptance threshold, enabling "zero-overhead" fuzzy extraction (requiring no error correction redundancy) for post-quantum cryptographic seed generation.
- Long-term Stability: A two-phase degradation model (fast exponential and slow logarithmic) predicts that the 4% MgO + Er + Yb composition retains 93.9% of its initial entropy after 20 years, projecting a service life exceeding 30 years.
- NIST Compliance: Unlike many optical PUFs in the literature that fail NIST tests due to non-IID response structures, the proposed compositions pass all relevant NIST SP 800-22 tests.
Comparative Analysis:
The paper provides a direct quantitative comparison with four other optical PUF platforms (Silicon photonic moiré, Multi-level RP-PUF, All-Si multidimensional, and Polymer diffuser) and Silicon Arbiter PUFs. The authors claim their approach is the only platform that simultaneously offers:
- A calibrated analytical model (vs. empirical characterisation in others).
- Explicit NIST Uniqueness data (including the discovery of the 5% MgO failure).
- Full industrial temperature range characterization (−40°C to +85°C).
- A combination of >20-year lifetime, full NIST compliance, and resistance to Machine Learning (ML) attacks.
ML Resistance:
The paper argues that unlike Silicon PUFs (which are vulnerable to logistic regression and deep learning), the 3D electromagnetic multi-scattering physics of the LiNbO3 optical PUF has no known polynomial-time ML model. The response is determined by millions of random scattering events in a disordered anisotropic crystal, making it fundamentally non-parametric and resistant to current ML modelling attacks.

Significance and Claims
The paper positions this work as the first to analytically model and solve the entropy-photorefraction trade-off in LiNbO3 PUFs.

Theoretical Impact: It establishes the first analytical framework ( $H(c,T)$ ) linking material composition directly to cryptographic metrics with zero residual error on calibration points.
Practical Impact: It identifies a critical flaw in the industry-standard 5% MgO composition and offers a validated alternative (4% MgO + REE) that meets rigorous security standards (NIST SP 800-22) and stability requirements for long-term infrastructure.
Future Outlook: The authors note that while the Er/Yb results are calibrated against experimental data, the superior performance of $Nd^{3+}$ (predicted to yield 102.6% entropy due to larger ionic mismatch) and the specific degradation parameters require further experimental verification through the outlined V1–V5 program.

The authors maintain a modest stance regarding predictions (marked with asterisks in tables), distinguishing clearly between calibrated results (Er/Yb) and extrapolated predictions (Nd/Tm), while emphasizing that the core contribution is the decoupling mechanism and the identification of the standard composition's failure.