Jailbreak Scaling Laws for Large Language Models: Polynomial-Exponential Crossover

Adversarial attacks can reliably steer safety-aligned large language models toward unsafe behavior. Empirically, we find that adversarial prompt-injection attacks can amplify attack success rate from the slow polynomial growth observed without injection to exponential growth with the number of inference-time samples. We first identify a minimal statistical mechanism for these two regimes by giving a small set of assumptions on the distribution of safe generation across contexts under which both scaling laws follow. To explain this phenomenon further, we propose a theoretical generative model of proxy language in terms of a spin-glass system operating in a replica-symmetry-breaking regime, where generations are drawn from the associated Gibbs measure and a subset of low-energy, size-biased clusters is designated unsafe. We analytically show how this model naturally realizes the minimal assumptions. Short injected prompts correspond to a weak magnetic field aligned towards unsafe cluster centers and yield a power-law scaling of attack success rate with the number of inference-time samples, while long injected prompts, i.e., strong magnetic field, yield exponential scaling. We observe qualitatively consistent behavior across a broad range of large language models, spanning parameter scales from 3B to 70B. In particular, the main trends remain stable across multiple attack methods, such as GCG and AutoDAN, as well as across benchmark datasets such as AdvBench and HarmBench.