Recent studies have revealed the vulnerability of large language models to adversarial attacks, where adversaries craft specific inputs to induce incorrect or harmful outputs. Although various empirical defenses have been proposed, their worst-case robustness remains poorly understood, raising concerns about how they would behave against stronger future attacks. Here, we study the worst-case robustness of LLMs from both empirical and theoretical perspectives. First, we provide tighter empirical upper bounds for the tested deterministic defenses using stronger white-box attacks, under which most tested configurations achieve nearly 0\% robustness. Second, we formulate randomized-smoothing certification for bounded and binary base functions as fractional and 0-1 knapsack problems, respectively, and derive grouped solvers for two token kernels. For example, on AdvBench, we certify a GPT-4o safety detector smoothed by a uniform kernel to an average certified $\ell_0$ radius of 2.02. In our stated suffix-attack setting, the same kernel certifies a Llama-3-8B safety detector to an average certified radius of 6.41.

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