Exact zCDP for k-Ary Randomized Response

Abstract

This note gives a self-contained exact zero-concentrated differential privacy (zCDP) characterization of k-ary randomized response for every integer k >= 2 and every privacy parameter epsilon >= 0. It proves that the necessary endpoint condition identified by Harrison and Manurangsi is sufficient, including equality. When that condition fails, the optimum is characterized by a unique scalar root with an explicit finite bracket. The proof applies established monotonicity principles, with elementary derivations included, and yields bisection enclosures and geometric error bounds in exact real arithmetic. The author used AI tools (GPT-6 Astra Max and Opus 5.5 Max) for code review, assistance with manuscript translation, and the development of counterexamples. The author takes full responsibility for the code, the original manuscript, the final translation, and all other related outputs.