BasedAI is a distributed network of machines which introduces decentralized infrastructure capable of integrating Fully Homomorphic Encryption (FHE) with any large language model (LLM) connected to its network. The proposed framework embeds a default mechanism, called "Cerberus Squeezing", into the mining process which enables the transformation of a standard LLMs into encrypted zero-knowledge LLMs, or "ZK-LLMs", leveraging insights from generative adversarial networks for data privacy. This novel quantization mechanism empowers BasedAI miners to process and respond to prompts derived from User interaction with LLMs without the need for decrypting either the queries or their corresponding responses. The introduction of Cerberus Squeezing significantly improves performance degradation caused by quantized functions in current FHE-compliant computing environments by proactively optimizing calls between users, miners, and validators.