With the increasing amount of data in society, privacy concerns in data sharing have become widely recognized. Particularly, protecting personal attribute information is essential for a wide range of aims from crowdsourcing to realizing personalized medicine. Although various differentially private methods based on randomized response have been proposed for single attribute information or specific analysis purposes such as frequency estimation, there is a lack of studies on the mechanism for sharing individuals' multiple categorical information itself. The existing randomized response for sharing multi-attribute data uses the Kronecker product to perturb each attribute information in turn according to the respective privacy level but achieves only a weak privacy level for the entire dataset. Therefore, in this study, we propose a privacy-optimized randomized response that guarantees the strongest privacy in sharing multi-attribute data. Furthermore, we present an efficient heuristic algorithm for constructing a near-optimal mechanism. The time complexity of our algorithm is O(k^2), where k is the number of attributes, and it can be performed in about 1 second even for large datasets with k = 1,000. The experimental results demonstrate that both of our methods provide significantly stronger privacy guarantees for the entire dataset than the existing method. In addition, we show an analysis example using genome statistics to confirm that our methods can achieve less than half the output error compared with that of the existing method. Overall, this study is an important step toward trustworthy sharing and analysis of multi-attribute data. The Python implementation of our experiments and supplemental results are available at https://github.com/ay0408/Optimized-RR.