Existence of a fast algorithm to generate on-ellipse outputs from samples

Determine whether there exists an efficient algorithm that, given only a collection of sample log-probability vectors produced by a language model whose final normalization and linear projection constrain outputs to lie on a d-dimensional ellipsoid in R^v, can generate additional log-probability vectors that lie on the same model-specific ellipsoid without access to the model parameters or performing full O(d^6) ellipsoid fitting.

Background

The paper demonstrates that forging ellipse signatures—producing log-probabilities on a LLM’s ellipsoid without access to its parameters—is practically infeasible for production-scale models because recovering the ellipse requires super-cubic API query complexity (O(d^3 log d)) and ellipsoid fitting typically takes sextic time (O(d^6)).

Given these barriers, the authors explicitly pose whether there is a more direct route to forgery: a fast method that, using only samples, could synthesize new outputs that lie on the unknown model ellipse without extracting full ellipse parameters. They report unsuccessful attempts with GPU parallelization and approximation due to memory constraints and accuracy degradation, underscoring the unresolved status of this question.