Growing concerns over the theft and misuse of Large Language Models (LLMs) underscore the need for effective fingerprinting to link a model to its original version and detect misuse. We define five essential properties for a successful fingerprint: Transparency, Efficiency, Persistence, Robustness, and Unforgeability. We present a novel fingerprinting framework that provides verifiable proof of ownership while preserving fingerprint integrity. Our approach makes two main contributions. First, a "chain and hash" technique that cryptographically binds fingerprint prompts to their responses, preventing collisions and enabling irrefutable ownership claims. Second, we address a realistic threat model in which instruction-tuned models' output distribution can be significantly altered through meta-prompts. By incorporating random padding and varied meta-prompt configurations during training, our method maintains robustness even under significant output style changes. Experiments show that our framework securely proves ownership, resists both benign transformations (e.g., fine-tuning) and adversarial fingerprint removal, and extends to fingerprinting LoRA adapters. We release our code at: https://github.com/microsoft/Chain-Hash.