This article describes the proposed differentially private (DP) algorithms that the US Census Bureau will use to release the Detailed Demographic and Housing Characteristics (DHC) Race & Ethnicity tabulations as part of the 2020 Decennial Census. The tabulations contain statistics (counts) of demographic and housing characteristics of the entire population of the US crossed with detailed races and tribes at varying levels of geography. We describe two differentially private algorithmic strategies, one based on adding noise drawn from a two-sided Geometric distribution that satisfies "pure"-DP, and another based on adding noise from a Discrete Gaussian distribution that satisfied a well studied variant of differential privacy, called Zero Concentrated Differential Privacy (zCDP). We analytically estimate the privacy loss parameters ensured by the two algorithms for comparable levels of error introduced in the statistics.