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When can the types of data be distinguished during the addition of encrypted data to a deterministic encrypted data set with equal proportions?

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This question is about adding new data with different proportions to a dataset with equal proportions for each type of data, and determining at what point the added data can be distinguished.

For example, in a hospital with doctors, nurses, cleaners, and security personnel, with proportions of [0.5, 0.2, 0.2, 0.1], there is a table of 1000 encrypted doctors, 1000 encrypted nurses, 1000 encrypted cleaners, and 1000 encrypted security personnel, but we do not know which is which. Now, new encrypted data is continuously inserted into the table with the same proportions as the original data. At what point can the four types of data be distinguished according to frequency?

If there are relevant algorithms and papers, please post them. Thank you

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