Learnability of Optical Physical Unclonable Functions Through the Lens of Learning With Errors

Abstract

We show that a class of optical physical unclonable functions (PUFs) can be efficiently PAC-learned to arbitrary precision with arbitrarily high probability, even in the presence of intentionally injected noise, given access to polynomially many challenge-response pairs, under mild and practical assumptions about the distributions of the noise and challenge vectors. We motivate our analysis by identifying similarities between the integrated version of Pappu’s original optical PUF design and the post-quantum Learning with Errors (LWE) cryptosystem. We derive polynomial bounds for the required number of samples and the computational complexity of a linear regression algorithm, based on size parameters of the PUF, the distributions of the challenge and noise vectors, and the desired accuracy and probability of success of the regression algorithm. We use a similar analysis to that done by Bootle et al. [“LWE without modular reduction and improved side-channel attacks against BLISS,” in Advances in Cryptology – ASIACRYPT 2018], who demonstrated a learning attack on poorly implemented versions of LWE cryptosystems. This extends the results of Rührmair et al. [“Optical PUFs reloaded,” Cryptology ePrint Archive, 2013], who presented a theoretical framework showing that a subset of this class of PUFs is learnable in polynomial time in the absence of injected noise, under the assumption that the optics of the PUF were either linear or had negligible nonlinear effects. (Rührmair et al. also included an experimental validation of this technique, which of course included measurement uncertainty, demonstrating robustness to the presence of natural noise.) We recommend that the design of strong PUFs should be treated as a cryptographic engineering problem in physics, as PUF designs would benefit greatly from basing their physics and security on standard cryptographic assumptions. Finally, we identify future research directions, including suggestions for how to modify an LWE-based optical PUF design to better defend against cryptanalytic attacks.