Retrieval-Augmented Generation (RAG) empowers LLMs with external knowledge, making cross-institutional domain-specific knowledge base integration a highly promising deployment paradigm. Despite this potential, strict privacy regulations create severe "data silos" that obstruct such collaboration. Building federated RAG systems requires distributed inference, but the Transformer's self-attention mechanism fundamentally conflicts with this by mandating cross-node access to distributed Key-Value caches. To address this challenge, we present FedRAG, a high-throughput, privacy-preserving federated RAG framework. At its core is a novel Scrambled Distributed Attention protocol that utilizes numerically stable feature scrambling and token permutation. By dynamically delegating scrambled computations to collaborating nodes, our system successfully decouples attention execution from data localization without exposing plaintext. Crucially, our approach requires no specialized hardware or model retraining, circumventing the prohibitive latency and communication overheads of cryptographic solutions while robustly defending against intermediate state inversion attacks. Extensive evaluations demonstrate our framework preserves negligible (<0.1\%) model utility degradation and achieves up to a 62$\times$ latency reduction over existing secure baselines, sustaining practical, human-reading throughput for cross-institutional knowledge synergy.

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