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What you need for this is something called an $n$-wise independent hash function (like "pairwise independent" but $n$ instead). Such a hash function has the property that when applied to at most $n$ different inputs, its outputs are completely random. These can be constructed efficiently; e.g., a random polynomial of the appropriate degree works. What you ...


Perfect secrecy requires that your key be as long as the plaintext you are encrypting. If you want to use the same key to encrypt multiple messages, you can use each part of the key once as a one-time pad. Your idea (assuming you fix the issue raised by poncho) seems to be just a more complicated way to achieve the same thing. If you remove the ...

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