Large language models (LLMs) power modern AI applications, but processing sensitive data on untrusted servers raises privacy concerns. Homomorphic encryption (HE) enables computation on encrypted data for secure inference. However, neural text generation requires decoding methods like argmax and sampling, which are non-polynomial and thus computationally expensive under encryption, creating a significant performance bottleneck. We introduce cutmax, an HE-friendly argmax algorithm that reduces ciphertext operations compared to prior methods, enabling practical greedy decoding under encryption. We also propose the first HE-compatible nucleus (top-p) sampling method, leveraging cutmax for efficient stochastic decoding with provable privacy guarantees. Both techniques are polynomial, supporting efficient inference in privacy-preserving settings. Moreover, their differentiability facilitates gradient-based sequence-level optimization as a polynomial alternative to straight-through estimators. We further provide strong theoretical guarantees for cutmax, proving it converges globally to a unique two-level fixed point, independent of the input values beyond the identity of the maximizer, which explains its rapid convergence in just a few iterations. Evaluations on realistic LLM outputs show latency reductions of 24x-35x over baselines, advancing secure text generation.