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CRYSTALS-Dilithium has been selected by the NIST as the new standard for post-quantum digital signatures. In this work, we revisit the side-channel countermeasures of Dilithium in three directions. First, we improve its sensitivity analysis by classifying intermediate computations according to their physical security requirements. Second, we provide improved gadgets dedicated to Dilithium, taking advantage of recent advances in masking conversion algorithms. Third, we combine these contributions and report performance for side-channel protected Dilithium implementations. Our benchmarking results additionally put forward that the randomized version of Dilithium can lead to significantly more efficient implementations (than its deterministic version) when side-channel attacks are a concern.