Correlation Integral-Based Intrinsic Dimension: A Deep-Learning-Assisted Empirical Metric to Estimate the Robustness of Physically Unclonable Functions to Modeling Attacks

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Physically unclonable functions (PUFs) which are robust to modeling attacks, usually have a complex, high-dimensional, nonlinear relationship between challenges and responses. Often, it is difficult to derive closed-form analytical expressions for these relationships. Consequently, it becomes difficult to compare PUF variants with regard to their robustness to modeling attacks. In this article, we apply a data-driven empirical metric termed the intrinsic dimension (ID), to estimate the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">inherent complexity</i> of the relationship between the challenges and responses of a given PUF variant. This metric is computed from the linear projection layer of a deep neural network (DNN) aimed at modeling the PUF and has a unique advantage that it is independent of the architectural details and chosen hyperparameters of the DNN. It also does not require the knowledge of the structural and functional details of the PUF. The proposed approach is evaluated using two well-known ID estimation methods based on the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">nearest neighbor method</i> and the full correlation integral (FCI). Through detailed experimental results, we demonstrate that the numerical values of the FCI-based ID metric for different types of PUFs have consistently high positive correlation with the perceived difficulty of modeling several common PUF variants. We also show that the ID metric provides deep insight about various subtleties that affect the robustness of PUFs to modeling attack and provides a convenient mechanism to perform a systematic comparison between different PUF compositions.