Large language models (LLMs) are increasingly utilized in domains such as finance, healthcare, and interpersonal relationships to provide advice tailored to user traits and contexts. However, this personalization often relies on sensitive data, raising critical privacy concerns and necessitating data minimization. To address these challenges, we propose a framework that integrates zero-knowledge proof (ZKP) technology, specifically zkVM, with LLM-based chatbots. This integration enables privacy-preserving data sharing by verifying user traits without disclosing sensitive information. Our research introduces both an architecture and a prompting strategy for this approach. Through empirical evaluation, we clarify the current constraints and performance limitations of both zkVM and the proposed prompting strategy, thereby demonstrating their practical feasibility in real-world scenarios.