We introduce Branch History Injection (BHI or Spectre-BHB), a new primitive to build cross-privilege BTI attacks on systems deploying isolation-based hardware defenses. BHI builds on the observation that, while the branch target state is now isolated across privilege domains, such isolation is not extended to other branch predictor elements tracking the branch history state—ultimately re-enabling cross-privilege attacks. We further analyze the guarantees of a hypothetical isolation-based mi...