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Suppose I have an array of addresses which I append to somewhere in the contract. Do the miners (or anyone else) have a way of figuring out what the variables holds in terms of data? In other words, who can see the addresses in the array?

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In other words, who can see the addresses in the array?

Anyone. The data in the ethereum blockchain is public.

I supose there is only way to store data that must be private is encryption.

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Yes, anyone on the network can read the storage contents allocated to any given account so you must not assume the data is private, even if the variable you use is marked as private in Solidity.

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