We introduce a pre-computation technique that drastically reduces the online computational complexity of mix-nets based on homomorphic cryptosystems. More precisely, we show that there is a permutation commitment scheme that allows a mix-server to: (1) commit to a permutation and efficiently prove knowledge of doing so correctly in the offline phase, and (2) shuffle its input and give an extremely efficient commitment-consistent proof of a shuffle in the online phase. We prove our result for a general class of shuffle maps that generalize all known types of shuffles, and even allows shuffling ciphertexts of different cryptosystems in parallel.
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