  The field of Explainable Artificial Intelligence (XAI) focuses on techniques
for providing explanations to end-users about the decision-making processes
that underlie modern-day machine learning (ML) models. Within the vast universe
of XAI techniques, counterfactual (CF) explanations are often preferred by
end-users as they help explain the predictions of ML models by providing an
easy-to-understand & actionable recourse (or contrastive) case to individual
end-users who are adversely impacted by predicted outcomes. However, recent
studies have shown significant security concerns with using CF explanations in
real-world applications; in particular, malicious adversaries can exploit CF
explanations to perform query-efficient model extraction attacks on proprietary
ML models. In this paper, we propose a model-agnostic watermarking framework
(for adding watermarks to CF explanations) that can be leveraged to detect
unauthorized model extraction attacks (which rely on the watermarked CF
explanations). Our novel framework solves a bi-level optimization problem to
embed an indistinguishable watermark into the generated CF explanation such
that any future model extraction attacks that rely on these watermarked CF
explanations can be detected using a null hypothesis significance testing
(NHST) scheme, while ensuring that these embedded watermarks do not compromise
the quality of the generated CF explanations. We evaluate this framework's
performance across a diverse set of real-world datasets, CF explanation
methods, and model extraction techniques, and show that our watermarking
detection system can be used to accurately identify extracted ML models that
are trained using the watermarked CF explanations. Our work paves the way for
the secure adoption of CF explanations in real-world applications.
