Systematically Quantifying Cryptanalytic Nonlinearities in Strong PUFs

Physically Unclonable Functions (PUFs) with large challenge space (also called Strong PUFs) are promoted for usage in authentications and various other cryptographic and security applications. In order to qualify for these cryptographic applications, the Boolean functions realized by PUFs need to possess a high nonlinearity (NL). However, with a large challenge space (usually
$\geq 64$
bits), measuring NL by classical techniques like the Walsh transformation is computationally infeasible. In this paper, we propose the usage of a heuristic-based measure called the non-homomorphicity test which estimates the cryptographic NL of Boolean functions with high accuracy in spite of not needing access to the entire challenge-response set. We also combine our analysis with a technique used in linear cryptanalysis, called Piling-up lemma, to measure the NL of popular PUF compositions. As a demonstration to justify the soundness of the metric, we perform extensive experimentation by first estimating the NL of constituent Arbiter/Bistable Ring PUFs using the non-homomorphicity test, and then applying them to quantify the same for their XOR compositions namely XOR Arbiter PUFs and XOR Bistable Ring PUF. Our findings show that the metric explains the impact of various parameter choices of these PUF compositions on the NL obtained and thus promises to be used as an important objective criterion for future efforts to evaluate PUF designs. While the framework is not representative of the machine learning robustness of PUFs, it can be a useful complementary tool to analyze the cryptanalytic strengths of PUF primitives.